Distribution Theory (and its Applications) Lecturer ... · Distribution Theory (and its Applications) Lecturer: Anthony Ashton Scribe: Paul Minter ... Recommended book: Hörmander,

Distribution Theory (and its Applications)

Lecturer Anthony Ashton

Scribe Paul Minter

Michaelmas Term 2018

These notes are produced entirely from the course I took and my subsequent thoughtsThey are not necessarily an accurate representation of what was presented and may have

in places been substantially edited Please send any corrections to pdtwm2camacuk

Recommended book Houmlrmander Analysis of linear partial differential operators Volume 1

Distribution Theory (and its Applications) Paul Minter

Contents

0 Motivation 2

1 Distributions 5

11 Test Functions and Distributions 5

12 Limits in 994963 prime(X ) 8

13 Basic Operations 9

131 Differentiation and Multiplication by Smooth Functions 9

132 Reflection and Translation 11

133 Convolution between 994963 prime(995014n) and 994963(995014n) 13

14 Density of 994963(995014n) in 994963 prime(995014n) 15

2 Distributions with Compact Support 18


22 Convolution between 994964 prime(995014n) and 994963 prime(995014n) 21

3 Tempered Distributions and Fourier Analysis 23

31 Functions of Rapid Decay 23

32 Fourier Transform on S(995014n) 24

33 Fourier Transform of Sprime(995014n) 28

34 Sobolev Spaces 30

4 Applications of the Fourier Transform 32

41 Elliptic Regularity 32

42 Fundamental Solutions 37

43 The Structure Theorem for 994964 prime(X ) 44

44 The Paley-Wiener-Schwartz Theorem 46

5 Oscillatory Integrals 51

1


51 Singular Supports 61

2


0 MOTIVATION

We start with some examples of motivate the things we will be talking about in the course

Example 01 (Derivative of Dirac Delta)

We often define the ldquoDirac Deltardquo via requring983133 +infin

minusinfinδ(x minus x0) f (x) dx = f (x0) for all suitable f

No such function exists however in terms of LebesgueRiemann integrability But let us supposeit did Then what would δprime its derivative be Well we could differentiate the above under theintegral sign ie983133 infin

minusinfinδprime(x minus x0) f (x) dx ldquo= rdquo lim

hrarr0

983133 infin

minusinfin

δ(x minus x0 + h)minusδ(x minus x0)h

middot f (x) dx

= limhrarr0

1h

983133 infin

minusinfinδ(x minus (x0 minus h)) f (x)minusδ(x minus x0) f (x) dx

= limhrarr0

1h[ f (x0 minus h)minus f (x0)]

= minus f prime(x0)

or in other words 983133 infin

minusinfinδprime(x minus x0) f (x) dx = minus

983133 infin

minusinfinδ(x minus x0) f

prime(x) dx

This suggests the integration by parts formula holds

Example 02 (Fourier Trasnforms of Polynomials) The standard definition of the Fourier trans-form is

f (λ) =

983133 infin

minusinfineminusiλx f (x) dx for λ isin 995014

If f is absolutely integrable (ie f isin L1(995014)) then

| f (λ)|le983133 infin

minusinfin

994802994802eminusiλx f (x)994802994802 dx =

983133 infin

minusinfin| f (x)| dx = 983042 f 983042L1(995014)

But what if f ∕rarr 0 as |x |rarrinfin (eg if not in L1(995014)) You might have come across the lsquoequalityrsquo

δ(λ) =1

2π

983133 infin

minusinfineminusiλx dx

before although this integral does not make sense However if lsquotruersquo this would imply that

1(λ) = 2πδ(λ)

ie the Fourier transform of 1 is related to this mysterious lsquofunctionrsquo δ But then using this whatwould f (λ) be when f (x) = xn Well we have983133 infin

minusinfinxneminusiλx dx =

983133 infin

minusinfin

994858i

ddλ

994868neminusiλx dx =994858

id

dλ

994868n[2πδ(λ)] = 2π(i)nδ(n)(λ)

3


So if we could understand the derivatives of δ we could potentially make sense of Fourier transformsof polynomials

Alternatively we could invoke Parsevalrsquos theorem which says for all sufficiently nice f g983133 infin

minusinfinf (λ) g(λ) =

983133 infin

minusinfinf (λ)g(λ) dλ

We could use this to extend the Fourier transform to more general functions by requiring this tohold Eg if f (x) = x then we could define f by requiring that for all ldquonicerdquo g we have

983133 infin

minusinfinf g(λ) =

983133 infin

minusinfinλ983167983166983165983168

= f (λ)

g(λ) dλ

So to compute f we would need to find the function such that this holds

Example 03 (Discontinuous Solutions to PDE) We often want solutions to certain PDEs to havediscontinuities For example an explosion creates a shockwave The pressure will be discontinuousover the shock wave and we want the solution to reflect this

With one spatial dimension we would want the pressure p = p(x t) to satisfy the wave equation

(983183) p =1c2

part 2ppart t2minus part

2ppart x2

= 0

[Wlog set c = 1] So assume that p isin C2(9950142) Then we want to make sense of the PDE p = 0 fora larger family of functions and not just C2 ones By integration the above we get

0=

983133

9950142

f (x t)p dxdt forall f isin C2c (9950142)

If we integrate by parts we then get

0=

983133

9950142

p(x t) f dxdt

where all the boundary terms vanish due to the compact support of f But now this equations makessense if p is just integrable eg if p was just continuous This gives the notion of a weak solutionWe say p = p(x t) is a weak solution to (983183) if

0=

983133

9950142

p f dxdt forall f isin C2c (9950142)

Note In each of the above examples we have to introduce a space of auxiliary (ldquonicerdquo) functions sowe could extend the range of some classical definitions This is the essence of distribution theory

In distribution theory functions are replaced by distributions which are defined as linear maps fromsome auxiliary space of test functions to 994999 So start with some topological vector space V of testfunctions (so we have a notion of convergence from the topology) Then we say u isin V lowast (ie in thetopological dual) if u V rarr 994999 is linear and respects convergence in V (ie is continuous)

4


ie if the pairing between V lowast and V is denoted by langmiddot middotrang(i) then u isin V lowast implies langu fnrang rarr langu f rang iffnrarr f in V

Example 04 Let V = Cinfin(995014) with the topology of local uniform convergence In this topologywe have fnrarr f if and only if fn|K rarr f |K uniformly for all K sub 995014 compact

Then define δx0 V rarr 995014 by langδx0

f rang = f (x0) Then δx0is a distribution (ie δx0

isin V lowast) and isexactly the Dirac delta from before

(i)By the pairing we mean langmiddot middotrang V lowast times V rarr 994999 defined by langu f rang = u( f )

5


1 DISTRIBUTIONS

11 Test Functions and Distributions

Let X sub 995014n be open Then the first space of test functions we look at is

Definition 11 Define 994963(X) to be the topological vector space Cinfinc (X ) with the topology definedby

ϕmrarr 0 in 994963(X ) if exist compact K sub X with supp(ϕm) sub K forallm and

part αϕmrarr 0 uniformly in X for every multi-index α

Note The compact set K in the above definition must be the same for the whole sequence

This is a nice set of functions Compact support ensures that ϕ and its derivatives vanish at theboundary of X which means that integration by parts is easy ie if ϕψ isin 994963(X ) then

983133

Xϕ middot part αψ dx = (minus1)|α|

983133

Xψ middot part αϕ dx

ie we have no boundary terms

We also have Taylorrsquos theorem to arbitrary order ie

ϕ(x + h) =983131

|α|leN

hα

αpart αϕ(x) + RN (x h)

where RN (x h) = o994790|h|N+1994791

uniformly in x [See Example Sheet 1]

As a general philosophy for distribution theory lsquonicerrsquo test functions will have more assumptions onthem and will give us more distributions

Definition 12 A linear map u D(X )rarr 994999 is called a distribution written u isin 994963 prime(X ) if for eachcompact K sub X exist constants C = C(K) N = N(K) such that

(11) |languϕrang|le C983131

|α|leN

supX|part αϕ|

for all ϕ isin 994963(X ) with supp(ϕ) sub K

If N can be chosen independent of K then the smallest such N is called the order of u and is denotedord(u)

We refer to an inequality like (11) as a semi-norm estimate (semi-norm since it is only for a givencompact set K) Note that the supremum can be over K instead of X since all the ϕ have support inK

6


Example 11 Recall the Dirac delta δx0 for x0 isin X This was defined by

langδx0ϕrang = ϕ(x0)

and so thus|langδx0

ϕrang|= |ϕ(x0)|le sup |ϕ|and thus we have (11) with C equiv 1 and N equiv 0 independent of supp(ϕ) So hence δx0

isin 994963 prime(X )and ord(δx0

) = 0

Example 12 A more interesting example is to define the linear map T 994963(X )rarr 994999 by

langTϕrang =983131

|α|leM

983133

Xfα(x) middot part αϕ(x) dx

where fα isin C(X ) Note that this is well-defined since the ϕ have compact support and so theintegrals exist since then fα middot part αϕ is continuous with compact support

We can see that this is a distribution Indeed fix a compact set K sub X Then if ϕ isin D(X ) withsupp(ϕ) sub K we have

|langTϕrang|le983131

|α|leM

983133

X| fα(x)| middot |part αϕ(x)| dx

=983131

|α|leM

983133

K| fα(x)| middot |part αϕ(x)| dx

le983131

|α|leM

supK|part αϕ| middot983133

K| fα(x)| dx

le994808

max|α|leM

983133

K| fα(x)| dx

994809middot983131

|α|leM

supK|part αϕ|

which is exactly (11) with C =max|α|leM

983125K | fα(x)| dx and N = M

So hence this shows that T isin 994963 prime(X ) and ord(T )le M (as some of the fα could be 0)

Note that we still have T isin 994963 prime(X ) is the fα are just locally integrable (ie fα isin L1loc(X ) ie

fα isin L1(K) for all compact K sub X)

In general if g isin L1loc(X ) we can define Tg isin 994963 prime(X ) via

langTg ϕrang =983133

Xg middotϕ dx

Example 12 above is essentially a linear combination of such distributions We often abuse notationand write Tg equiv g For example on 994963 prime(995014) we might say ldquoconsider the distribution u(x) = xrdquo Clearlythis is a function not a distribution But what we mean is the distribution

languϕrang =983133

995014xϕ(x) dx

7


Example 13 (Distribution with lsquoinfinitersquo order) Consider on 994963(995014) the linear map

languϕrang =infin983131

m=0

ϕ(m)(m)

where ϕ(m) denotes m-th derivative Note that for each ϕ isin 994963(995014) this is a finite sum since ϕ hascompact support So in particular we get

|languϕrang|leinfin983131

m=0

|ϕ(m)(m)|leinfin983131

m=0

sup |ϕ(m)|

and so if supp(ϕ) sub [minusK K] we see that |languϕrang|le983123K

m=0 sup |ϕ(m)| So hence if we take K rarrinfin(which we can do since we can always find some function which has support contained in [minusK K]for each K) this shows that there is no universal N value as in (11) which works for all K sub 995014compact Hence the order of this u is not defined (it is lsquoinfinitersquo if you like)

So how does this definition of distribution relate to continuity The key result from analysis is thatfor a linear map (between normed vector spaces) continuity is equivalent to boundedness Thisgives the so-called sequential continuity definition of 994963 prime(X )

Lemma 11 (Sequential Continuity Definition of 994963 prime(X ))

Let u 994963(X )rarr 994999 be a linear map Then

(12) u isin 994963 prime(X ) hArr limnrarrinfinlanguϕmrang= 0 for all sequences (ϕm)m with ϕmrarr 0 in 994963(X )

Proof (rArr) Suppose u isin 994963 prime(X ) and ϕmrarr 0 in 994963(X ) In particular we know existK sub X compact withsupp(ϕm) sub K for all m Then from (11) we have

|languϕmrang|le C983131

|α|leN

sup |part αϕm|rarr 0

since sup |part αϕm|rarr 0 (from ϕmrarr 0 in 994963(X ))

(lArr) Suppose the implication is not true Then we could have the RHS of (12) holding but noestimate of the form of (11) holds Hence existK sub X compact such that (11) fails for every choiceof C N So take C = N = m isin 995010gt0 Then since (11) fails with these choices of C N we knowexistϕm isin 994963(X ) with suppm(ϕ) sub K and

|languϕmrang|gt m983131

|α|lem

sup |part αϕm|

In particular since the RHS here is ge 0 this tells us that |languϕmrang|gt 0 So define

ϕm = ϕmlanguϕmrangThen we have

1= |langu ϕmrang|gt m983131

|α|lem

sup |part αϕm|

8


ie 983131

|α|lem

sup |part αϕm|lt1m

=rArr sup |part αϕm|lt1mforall|α|le m

But then this shows that ϕmrarr 0 in 994963(X ) But then we can apply (12) to this sequence to see thatwe require

languϕmrang rarr 0 as mrarrinfin

But we know languϕmrang= 1 for all m and so we have a contradiction So the implication must be true

12 Limits in 994963 prime(X )

Often we have sequences of distributions (um)m sub 994963 prime(X )

Definition 13 We say um rarr 0 in 994963 prime(X ) if limmrarrinfinlangumϕrang = 0 for all ϕ isin 994963(X ) ie wlowast-convergence

Remark So far we have only defined convergence to zero in the spaces 994963(X ) and 994963 prime(X ) Howeversince these are topological vector spaces we get natural definitions of convergence to other limits

bull ϕmrarr ϕ in 994963(X ) means ϕm minusϕrarr 0 in 994963(X )bull umrarr u in 994963 prime(X ) means um minus urarr 0 in 994963 prime(X )

An important result is that 994963 prime(X ) is that pointwise limits of distributions are also distributions

Theorem 11 (Non-Examinable) Suppose (um)m sub 994963 prime(X ) and suppose that languϕrang =limmrarrinfinlangumϕrang exists for all ϕ isin 994963(X ) Then u isin 994963 prime(X )

Proof This is just an application of the uniform-boundedness principleBanach-Steinhaus See Houmlr-mander Vol 1 for more details

Limits in 994963 prime(X ) can look strange however

Example 14 Consider 994963 prime(995014) and um(x) = sin(mx) Then

langumϕrang=983133

995014sin(mx)ϕ(x) dx

=1m

983133

995014cos(mx)ϕprime(x) dx

9


via integrating by parts which thus shows

|langumϕrang|le 1m

983133

995014|ϕprime(x)| dx rarr 0 as mrarrinfin

for any ϕ isin 994963(995014) which shows that umrarr 0 in 994963 prime(995014) So hence

ldquo limmrarrinfin

sin(mx) = 0 in 994963 prime(995014)rdquoif you will (just remember we are viewing these as distributions not functions)

13 Basic Operations

131 Differentiation and Multiplication by Smooth Functions

To motivate the definition of derivatives of distributions we look at the duality with the smoothfunction case If u isin Cinfin(X ) then we can define part αu isin 994963 prime(X ) via (for ϕ isin 994963(X ))

langpart αuϕrang=983133

Xpart αu middotϕ dx

= (minus1)|α|983133

Xu middot part αϕ dx

= (minus1)|α|langu Dαϕrangwhere we have integrated by parts which we can do since everything is smooth Thus we definemore generally

Definition 14 For u isin 994963 prime(X ) and f isin Cinfin(X ) we define the derivative of the distribution f uby

langpart α( f u)ϕrang = (minus1)|α|langu f part αϕrangfor f isin 994963(X )

We call part αu the distributionalweak derivative of u

Note In the case of u isin Cinfin(X ) then these distributional derivatives are exactly the derivatives of uHowever more generally we can define them for distributions and we see that for any distributionall the distributional derivatives are defined

Remark Taking α = 0 in the definition this also tells us that we define multiplication by a smoothfunction of a distribution by

lang f uϕrang = langu f ϕrang

Example 15 For δx the Dirac Delta we have

langpart αδx ϕrang= (minus1)|α|langδx part αϕrang= (minus1)|α|part αϕ(x)

10


Compare this with our heuristic definition of part αδx from sect0

Example 16 Define the Heaviside function by

H(x) =

9948160 if x le 01 if x gt 0

Then since H isin L1loc(995014) it defines a distribution H isin 994963 prime(995014) Then we have

langH primeϕrang = minuslangHϕprimerang= minus983133

995014H(x)ϕprime(x) dx

= minus983133 infin

0

ϕprime(x)

= minus[ϕ(x)]x=infinx=0

= ϕ(0)

= langδ0ϕrangwhere we have used the fact the ϕ has compact support and so it is zero nearinfin Hence this showsthat H prime = δ0 in 994963 prime(995014)

Remark In general clearly if u v isin 994963 prime(X ) and we have languϕrang = langvϕrang for all ϕ isin 994963(X ) then wesay that u= v in 994963 prime(X )

With this knowledge we are now able to solve the simplest of distributional equations uprime = 0

Lemma 12 Suppose uprime = 0 in 994963 prime(995014) Then u= constant in 994963 prime(995014)

Key point Fix ϕ isin 994963(995014) We want to consider languϕrang Now if we could find ψ such that ψprime = ϕthen we would get

languϕrang= languψprimerang= minuslanguprimeψrang= lang0ψrangThe only issue with such an argument is that ψ would be defined via an integral of ϕ and thus doesnot necessarily have compact support But we need ψ isin 994963 prime(995014) for this to work We cannot justsubtract a constant to make it have compact support since it would mess up either end Thus weneed to subtract a constant off in a lsquoweightedrsquo way as the proof demonstrates

Proof Fix θ0 isin 994963(995014) with

1=

983133

995014θ0 dx = lang1θ0rang

Take ϕ isin 994963(995014) arbitrary Then write

ϕ = (ϕ minus lang1ϕrangθ0)983167 983166983165 983168=ϕA

+ lang1ϕrangθ0983167 983166983165 983168=ϕB

11


Clearly we have lang1ϕArang=983125995014ϕA dx = 0 Now set

ψA(x) =

983133 x

minusinfinϕA(y) dy

which is smooth since it is the integral of a smooth function

Claim ψA isin 994963(995014)

Proof of Claim Since θ0 ϕ have compact support so does ϕA Hence for x suf-ficiently negative (outside the support of ϕA) this integral is just the integral of 0and so we see ψA(x) = 0 for all x le minusC for some C gt 0

Also since ϕA = 0 for all x ge C prime we see that ψA is constant for x ge C prime But weknow that ψA(x)rarr 0 as x rarrinfin since

983125995014ϕA dx = 0 and so this constant must be

0 ie ψA(x) = 0 for all x ge C

Hence this shows ψA has compact support and so ψA isin 994963(995014)

So ψA isin 994963(995014) and ψprimeA = ϕA So we have

languϕrang= languϕArang+ languϕBrang= languψprimeArang+ lang1ϕrang middot languθ0rang983167 983166983165 983168

=c a constant independent of ϕ

= minuslang uprime983167983166983165983168=0 by assumption

ψArang+ langcϕrang

= langcϕrangie languϕrang= langcϕrang for all ϕ isin 994963(995014) and thus u= c in 994963 prime(995014)

Remark We can use similar arugments to solve other distributional equations We can also use whatwe know about the distributional derivatives of other distributions to help us

132 Reflection and Translation

For ϕ isin 994963(995014n) and h isin 995014n we can define the translate of ϕ by

(τhϕ)(x) = ϕ(x minus h)

and the reflection of ϕ byϕ(x) = ϕ(minusx)

τh is called the translation operator andor the reflection operator We can extend these definitionsto 994963 prime(995014n) by duality(ii) as we have been doing

(ii)By duality we just mean see the result first for distributions defined by smooth functions and then define the generalcase obey the same relation just like we did for eg derivatives

12


Definition 15 For u isin 994963 prime(995014n) and h isin 995014n we define the distributional reflection and reflec-tions by

langτhuϕrang = languτminushϕrang and languϕrang = langu ϕrang

Remark If u we smooth then one can check that these definitions agree with what we would expecteg

languϕrang=983133

995014n

u(minusx)ϕ(x) dx =

983133

995014n

u(x)ϕ(minusx) dx = langu ϕrangetc

So do the definitions of translation and distributional derivative coincide in the usual way It turnsout that they do and the following lemma shows this for directional derivatives

Lemma 13 (Directional Distributional Derivatives) Let u isin 994963 prime(995014n) and h isin 995014n0 Thendefine

vh =τminush(u)minus u|h|

Then if h|h|rarr m isin Snminus1 then we have vhrarr m middot part u in 994963 prime(995014n) as hrarr 0

Remark If u were smooth then we would have

vh(y) =u(y + h)minus u(y)

|h|

Proof By definition we have

langvhϕrang= 1|h|languτhϕ minusϕrang

Then by Taylorrsquos theorem we know

(τhϕ)(x)minusϕ(x) = ϕ(x minus h)minusϕ(x) = minus983131

i

hipart ϕ

part x i(x) + R1(x h)

where R1(x h) = o(|h|) in 994963(995014n) [See Example Sheet 1 Question 2] Then by the sequentiallycontinuity definition of 994963 prime(995014n) (this is to pull the R1(x h) term out) and the convergence h|h|rarr mwe have

langvhϕrang= minus994903

u983131

i

hi

|h| middotpart ϕ

part x i

994906+ o(1)

= minus983131

i

hi

|h|

994902upart ϕ

part x i

994905+ o(1)

minusrarr983131

i

mi

994902upart ϕ

part x i

994905=

994903983131

i

mipart upart x i

ϕ

994906= langm middot part uϕrang

ie vhrarr m middot part u in 994963 prime(995014n) as hrarr 0

13


133 Convolution between 994963 prime(995014n) and 994963(995014n)

By combining reflection and translation we see that for ϕ isin 994963(995014n) we have

(τx ϕ)(y) = ϕ(y minus x) = ϕ(x minus y)

So if u isin Cinfin(995014n) we have the classical notion of convolution via

(u lowastϕ)(x) =983133

995014n

u(x minus y)ϕ(y) dy

=

983133

995014n

u(y)ϕ(x minus y) dy

=

983133

995014n

u(y)(τx ϕ)(y) dy

= languτx ϕrang

and this last equality makes sense for any distribution on 995014n Hence we define (by duality if youwill)

Definition 16 If u isin 994963 prime(995014n) then the (distributional) convolution of u with ϕ isin 994963(995014n) isdefined by

(u lowastϕ)(x) = languτx ϕrangand thus is a function (u lowastϕ) 995014nrarr 995014

So the convolution of a distribution with a test function gives another function What can be saidabout this function In the case where the distribution is defined by a smooth function we knowthat convolutions are usually smooth as well Can the same be said here With an extra conditionit turns out that it can be and the usual derivative formula holds

Lemma 14 (Derivatives of (Distributional) Convolution) Let ϕ isin Cinfin(995014n times 995014n) and defineΦx(y) = ϕ(x y) Suppose that for each x isin 995014n exist a neighbourhood Nx of x on which for allx prime isin Nx we have supp (Φx prime) sub K for some K compact (independent of x prime isin Nx) Then

part αx languΦxrang= langupart αx Φxrang

Proof The assumption on Φx means that for each x isin 995014n we can find an open neighbourhood Nxof x and K compact such that supp

994790ϕ(middot middot)|Nxtimes995014n

994791sub Nx times K [Shown in Figure 1]

By Taylorrsquos theorem for fixed x isin 995014n we have

Φx+h(y)minusΦx(y) =983131

i

hipart ϕ

part x i(x y) + R1(x y h)

where R1 = o(|h|) and supp(R1(x middot h)) sub K for some compact K for |h| sufficiently small [this is byour assumption on ϕ as for |h| sufficiently small we will have x + h isin Nx and so supp(Φx+h) sub K ]

14


FIGURE 1 An illustration of the assumption on ϕ in Lemma 14

So hence we see that for each such x we have R1(x middot h) = o(|h|) as an equality in 994963(995014n) So usingthis and sequential continuity (Lemma 11) we have (using the above)

languΦx+hrang minus languΦxrang=994903

u983131

i

hipartΦx

part x i

994906+ o(|h|)

=983131

i

hi

994902upartΦx

part x i

994905+ o(|h|)

So hence this must be the Taylor expansion of languΦxrang wrt x and thus we see (combinging o(h)terms)

part

part x ilanguΦxrang=994902

upartΦx

part x i

994905

Thus this proves the result for first order derivatives The general result then follows from repeatingthe above by induction

Corollary 11 (Important) Suppose u isin 994963 prime(995014n) and ϕ isin 994963(995014n) Then

u lowastϕ isin Cinfin(995014n) and part α(u lowastϕ) = u lowast part αϕ

Proof Fix such a uϕ Then define ϕ isin Cinfin(995014n times995014n) by

ϕ(x y) = τx ϕ(y) = ϕ(x minus y)

Then for x isin 995014n we can clearly find a neighbourhood Nx of x and K sub 995014n compact as in Lemma 14since ϕ has compact support (eg just take a ball about x and the relevant translate of the supportof ϕ)

Thus we can apply Lemma 14 to this ϕ which proves the result since languΦxrang= (u lowastϕ)(x) here

15


Remark Heuristically we can ldquowriterdquo δ(x) = limnrarrinfin δn(x) Then since we can write

u(x) = (u lowastδ)(x) = limnrarrinfin

(u lowastδn)(x)

which is a limit of smooth functions So heuristically this says that distributions can be ldquoviewedrdquo asa limit of smooth functions which makes our lives very simple if we can make sense if thisif it istrue because we know how to understand limits of smooth functions

14 Density of 994963(995014n) in 994963 prime(995014n)

We have now seen that for u isin 994963 prime(995014n) and ϕ isin 994963(995014n) that u lowastϕ isin Cinfin(995014n) no matter how ldquoweirdrdquou is We often call u lowastϕ a regularisation of u

Lemma 15 Suppose u isin 994963 prime(995014n) and ϕψ isin 994963(995014n) Then

(u lowastϕ) lowastψ = u lowast (ϕ lowastψ)ie lowast is associative (when we have two test functions 1 distribution)(iii)

Note On the LHS the second (lsquoouterrsquo) convolution is actually a classical convolution of smoothfunctions since u lowast ϕ is smooth whilst on the RHS the outer convolution is the one defined fordistributions

Proof For fixed x isin 995014n we have

[(u lowastϕ) lowastψ](x) =

983133(u lowastϕ)(x minus y)ψ(y) dy

=

983133languτxminusy ϕrangψ(y) dy

=

983133langu(z) (τxminusy ϕ)(z)rangψ(y) dy

=

983133langu(z)ϕ(x minus y minus z)ψ(y)rang dy

where we have brought in the factor ofψ(y) into the distribution by linearity since for each y ψ(y)is just a constant

Now to see that this is what we are after write the final integral as a Riemann sum and so we get

= limhrarr0

983131

misin995022n

langu(z)ϕ(x minus z minus hm)ψ(hm)hnrang

= limhrarr0

994903u(z)983131

misin995022n

ϕ(x minus z minus hm)ψ(hm)hn

994906(983183)

where we are able to bring in the lsquoinfinite sumrsquo into the brackets by linearity of distributions sincefor each hgt 0 the sum is actually a finite sum since ψ has compact support (so for |m| large enoughψ(hm) = 0)

(iii)Although this is the only way we can make sense of this since we canrsquot convolute a distribution with anotherdistribution

16


Now setFh(z) =983131

misin995022n

ϕ(x minus z minus hm)ψ(hm)hm

We can then show that supp(Fh) is contained inside a fixed compact K sub 995014n for |h|le 1 and that wehave

Fh(z)rarr (ϕ lowastψ)(x minus z) pointwiseand we also have part αFh rarr part α(ϕ lowastψ) pointwise But for convergence in 994963(995014n) we want uniformconvergence of these

But note that for each multi-index α we have |part αFh| le Mα and thus we have equicontinuity of allderivatives Hence by Arzelaacute-Ascoli by passing to a subsequence if necessary we get that Fh(z)rarrϕ lowastψ(x minus z) uniformly Hence983131

misin995022n

ϕ(x minus z minus hm)ψ(hm)hnrarr (ϕ lowastψ)(x minus z) in 994963(995014n) as hrarr 0

[We may need to take a diagonal subsequence when invoking Arzelaacute-Ascoli to get all derivativesconverge uniformly as well]

Thus by sequential continuity we can exchange the limit in (983183) with langmiddot middotrang and thus we get

(983183) =

994903u(z) lim

hrarr0

983131

misin995022n


994906

= langu(z) (ϕ lowastψ)(x minus z)rang=994800uτx

ˇ(ϕ lowastψ)994801

(iv)

= = [u lowast (ϕ lowastψ)](x)which completes the proof

Theorem 12 (Density Theorem) Let u isin 994963 prime(995014n) Then exist a sequence (ϕm)m sub 994963(995014n) such thatϕmrarr u in 994963 prime(995014n) ie

langϕmθ rang rarr languθ rang forallθ isin 994963(995014n)

Proof First fix ψ isin 994963(995014n) with983125ψ dx = 1 Then define

ψm(x) = mnψ(mx)(v)

Also fix χ isin 994963(995014n) with

χ(x) =

9948161 on |x |lt 10 on |x |gt 2

Then defineχm(x) = χ(xm)

This χ will function as a cut off function to ensure compact support of our ϕm

Finally defineϕm = χm(u lowastψ)

(iv)Apologies I canrsquot seem to get a wider lsquocheckrsquo symbol(v)This is a bit like an approximation to a δ-function - the function gets lsquosquishedlsquoand more lsquospikedrsquo as mrarrinfin

17


From the above we know that ϕm isin 994963(995014n) Then for θ isin 994963(995014n) we have

langϕmθ rang= langu lowastψmχmθ rang=994792(u lowastψm) lowast ˇ(χmθ )

994793(0)

= u lowast994792ψm lowast ˇ(χmθ )994793(0)

(983183)

where in the second line we have used the fact that in general languϕrang= (ulowast ϕ)(0) and then we haveused Lemma 15 (associativity of convolution)

Now994792ψ lowast ˇ(χθ )994793(x) =

983133ψm(x minus y)χm(minusy)θ (minusy) dy

=

983133mnψ(m(x minus y))χ(minusym)θ (minusy) dy

=

983133ψ(y)χ994806 y

m2minus x

m

994807θ994806 y

mminus x994807

dy [change variables y prime = m(x minus y) to get this]

= θ (minusx) +

983133ψ(y)994844χ994806 y

m2minus x

m

994807θ994806 y

mminus x994807minus θ (minusx)994845

dy

= θ (minusx) + Rm(minusx)

= θ (x) + Rm(x)

where we have used that983125ψ(y) dy = 1 and where

R(x) =

983133ψ(y)994844χ994806 y

m2+

xm

994807θ994806 y

m+ x994807minus θ (x)994845

dy

Now (dagger) giveslangϕmθ rang= (u lowast θ )(0) + (u lowast Rm)(0) = languθ rang+ langu Rmrang

Then it is straightforward to show that Rm rarr 0 in 994963(995014n) [Exercise to check] and thus by thesequential continuity of 994963 prime(995014n) we get

langϕmθ rang rarr languθ rangand so since θ isin 994963(995014n) was arbitrary we deduce that ϕmrarr u in 994963 prime(995014n)

Remark So we see that test functions (or more precisely their associated distributions) are densein 994963 prime(995014n) It is a viewpoint of some mathematicians (eg Terry Tao) that this is the best wayto do everything with distributions first prove the result for the test function case and then usedensitycontinuity to get the result for all distributions [Terry Tao wrote about this on his blog]

18


2 DISTRIBUTIONS WITH COMPACT SUPPORT

Definition 21 For u isin 994963 prime(X ) we say that u vanishes on Y sub X if languϕrang = 0 for all ϕ isin 994963(X )with supp(ϕ) sub Y

We then define the support of u supp(u) as the complement of the largest open set on which uvanishes ie

supp(u) = X983134Y sub X u vanishes on Y

In practice usually it is easy to see what the support of a distribution is

Example 21 Consider the Dirac Delta δx Then since langδx ϕrang = ϕ(x) consider Y = XxThen Y is open and any ϕ isin 994963(X ) with supp(ϕ) sub Y will have langδx ϕrang = ϕ(y) = 0 Thus uvanishes on Y and clearly u does not vanish on X Thus we have supp(δx) = x


We have seen that if we have more restrictions on the test functions we are considering (so thereare less of them) then we have more distributions and so there is a seesaw effect For example wehave so far only considered test functions which were smooth and had compact support and so theassociated distributions did not care about the boundary of our domain However if we drop thecompact support assumption then we expect the distributions to care about what happens at theboundary and so they change

So let us define a new space of test functions

Definition 22 Define a space of test functions by 994964 (X ) which is the space of smooth functionsϕ X rarr 994999 with the topology

ϕmrarr 0 in 994964 (X ) if part αϕmrarr 0 uniformly on each compact K sub X for each α

ie 994964 (X ) = Cinfin(X ) with the topology of local uniform convergence

Remark This makes 994964 (X ) into a Frecheacutet space

Note that clearly 994963(X ) sub 994964 (X ) So intuitively since the distributions are the dual spaces of these weexpect 994964 prime(X ) sub 994963 prime(X )

Remark How are the topologies on 994963(X ) 994964 (X ) related Are they compatible ie is the topologyon 994963(X ) just the induced subspace topology

19


Definition 23 We say that a linear map u 994964 (X )rarr 994999 defines an element of 994964 prime(X) the spaceof distributions on 994964(X) if exist a compact set K sub X and constants C N such that

|languϕrang|le C983131

|α|leN

supK|part αϕ| forallϕ isin 994964 (X )

Note These distributions necessarily have finite order due to the uniformality of N

As before we then have

Lemma 21 (Sequential Continuity Definition of 994964 prime(X )) Let u 994964 (X ) rarr 994999 be a linear mapThen

u isin 994964 prime(X ) hArr limnrarrinfinlanguϕmrang= 0 for each (ϕm)m sub 994964 (X ) with ϕmrarr 0 in 994964 (X )

Proof Almost identical as the 994963 prime(X ) version - Exercise to prove [need to use a compact exhaustionof X ]

Note If u isin 994964 prime(X ) then it is true that supp(u) sub K where K is as in the definition of 994964 prime(X ) (this isimmediate from the definitions) But we do not always have supp(u) = K

So how do 994964 prime(X ) and 994963 prime(X ) relate to one another It turns out that we have 994964 prime(X ) sub 994963 prime(X ) (inter-preted suitably) just as we expected Also elements of 994963 prime(X ) with compact support actually defineelements of 994964 prime(X ) and thus the two spaces the compatible in the nicest way This essential tells usthat elements of 994964 prime(X ) are just elements of 994963 prime(X ) with compact support

Lemma 22 Let u isin 994964 prime(X ) Then u|994963(X ) defines an element of 994963 prime(X ) that has compact supportand finite order

Conversely if u isin 994963 prime(X ) has compact support then exist u isin 994964 prime(X ) such that u|994963(X ) = u in 994963 prime(X )and supp(u) = supp(u)

Remark In the converse statement we do not need any assumption on the order of u (since weknow elements of 994964 prime(X ) have finite order we might expect this) This is because finite order isimplied by u having compact support Indeed suppose u isin 994963 prime(X ) has compact support Then setρ isin Cinfinc (X ) be 1 on supp(u) Then we have u = ρu and if K sub X is compact for all ϕ isin 994963(X ) withsupp(ϕ) sub K we have

|languϕrang|= |langρuϕrang|= |languρϕrang|

20


Now ρϕ is smooth of compact support in supp(u) Thus applying the definition of 994963 prime(X ) withK = supp(ϕ) we get

|languϕrang|= |languρϕrang|le C983131

|α|leN

supsupp(u)capK

|part α(ρϕ)|= C983131

|α|leN

supK|part α(ρϕ)|le C983131

|α|leN

supKpart αϕ

where in the last inequality we have used the Leibniz rule for derivatives and then pulled all supre-mumrsquos involving ρ out So finally note that this N used was the one for the compact set supp(u)(and so is independent of K) and the above holds for any ϕ isin 994963(X ) with supp(ϕ) sub K Hence thisshows that u has order le N ltinfin

Now onto proving the lemma

Proof Since 994963(X ) sub 994964 (X ) for u isin 994964 prime(X ) u|994963(X ) is well-defined in 994963 prime(X ) Moreover by definition of994964 prime(X ) exist a compact K sub X and constants C N such that


|α|leN

supK|part αϕ|

for all ϕ isin 994963(X ) (since 994963(X ) sub 994964 (X ))

Thus clearly u|994963(X ) isin 994963 prime(X ) and supp(u) sub K and ord(u)le N

For the converse suppose that u isin 994963 prime(X ) has compact support say K We will first show existenceand then uniqueness Then define u 994964 (X )rarr 994999 via

languϕrang = languρϕrangfor all ϕ isin 994964 (X ) where ρ isin 994963(X ) is chosen so that ρ equiv 1 on supp(u) (ie u = ρu)(vi) Then wehave


|α|leN

supK|part α(ρϕ)|

since supp(ρϕ) sub K is compact So by Leibniz we can expand part α(ρϕ) and as N ltinfin we can pullout the (finite) factor max|α|leN

994798supK |part αρ|994799

to get


|α|leN

supK|part αϕ|

and so hence this shows u isin 994964 prime(X ) So we have existence of such a u

For uniqueness suppose existv isin 994964 prime(X ) such that u|994963(X ) = v|994963(X ) and supp(u) = supp(u) = supp(v)Then for any ϕ isin 994964 (X ) write

ϕ = ρϕ983167983166983165983168=ϕ0

+(1minusρ)ϕ983167 983166983165 983168=ϕ1

where ρ is as before Then clearly ϕ0 isin 994963(X ) and ϕ1 equiv 0 on supp(u) = supp(v) So by the definitionof support we have

languϕrang= languϕ0rang+ languϕ1rang983167 983166983165 983168=0

= langvϕ0rang+ 0= langvϕ0rang+ langvϕ1rang983167 983166983165 983168=0

= langvϕrang

since u|994963(X ) = v|994963(X ) Hence u= v and we are done

(vi)The idea is that u only sees things in supp(u) = K So if we kill off things outside supp(u) our new u should work -we use ρ to do this killing off

21


Note This means that we can treat 994964 prime(X ) as the subspace of 994963 prime(X ) containing the distributions ofcompact support Thus as 994964 prime(X ) sub 994963 prime(X ) is a subspace all previous definitions of differentiationconvolutions etc for 994963 prime(X ) carry over to 994964 prime(X )

[So essentially we have shifted the compact support assumption from the test functions to the dis-tributions]

22 Convolution between 994964 prime(995014n) and 994963 prime(995014n)

Definition 24 For ϕ isin 994964 (995014n) and u isin 994964 prime(995014n) define (as usual) the convolution by

(u lowastϕ)(x) = languτx ϕrang

As before we have u lowastϕ isin Cinfin(995014n) and part α(u lowastϕ) = u lowast part αϕ

So we have defined convolutions for 994963(X ) with 994963 prime(X ) and 994964 (X ) with 994964 prime(X ) Since 994963(X ) sub 994964 (X ) wecan thus define the convolution of 994963(X ) with 994964 prime(X ) But both of these have compact support andso we might expect the convolution to have compact support

Indeed if we have ϕ isin 994963(995014n) and u isin 994964 prime(995014n) then

0= (u lowastϕ)(x) = languτx ϕrang ldquo= rdquo langu(y)ϕ(x minus y)rangand thus this is zero unless (x minus y) isin supp(ϕ) for some y isin supp(u) So we see that

supp(u lowastϕ) sub supp(u) + supp(ϕ)

and so supp(u lowast ϕ) is compact as both sets on the RHS are comapct So hence in this case we seethat u lowastϕ isin 994963(995014n) ie

lowast 994964 (995014n)times994964 prime(995014n)rarr994964 (995014n) and lowast 994963(995014n)times994964 prime(995014n)rarr994963(995014n)

and from before we knowlowast 994963(995014n)times994963 prime(995014n)rarr994964 (995014n)

Thus as the result gives another function in 994963(995014n) in this case we can convolute it with any elementof 994963 prime(X ) Thus we can define

Definition 25 (Convolution of Distributions) Suppose u1 u2 isin 994963 prime(Rn) with at least one ofu1 u2 having compact support (ie in 994964 prime(995014n)) Then we can define u1 lowast u2 by

(u1 lowast u2)(ϕ) = u1 lowast (u2 lowastϕ) forallϕ isin 994963(995014n)

Note As above this is well-defined Indeed if u1 isin 994964 prime(995014n) then u2 lowastϕ isin 994964 (995014n) and so u1 lowast (u2 lowastϕ)is well-defined via Definition 24 However if u2 isin 994964 prime(995014n) then u2 lowastϕ isin 994963(995014n) and so u1 lowast (u2 lowastϕ)makes sense in the way of Definition 16

22


Remark One can show that u1 lowast u2 isin 994963 prime(Rn) - see Example Sheet 2

Lemma 23 For u1 u2 isin 994963 prime(995014n) with at least one of u1 u2 in 994964 prime(995014n) we have

u1 lowast u2 = u2 lowast u1

ie lowast is commutative

Proof Let ϕψ isin 994963(995014n) Then by repeated application of Lemma 15 we have

(u1 lowast u2) lowast (ϕ lowastψ) = u1 lowast (u2 lowast (ϕ lowastψ))= u1 lowast ((u2 lowastϕ) lowastψ) by Lemma 15

= u1 lowast (ψ lowast (u2 lowastϕ)) since f lowast g = g lowast f for smooth functions

= (u1 lowastψ) lowast (u2 lowastϕ) by Lemma 15

Now by symmetry in the above we have

(u2 lowast u1) lowast (ϕ lowastψ) = (u2 lowast u1) lowast (ψ lowastϕ) = (u2 lowastϕ) lowast (u1 lowastψ) = (u1 lowastψ) lowast (u2 lowastϕ)where in the first and third equalities we have used the fact that f lowast g = g lowast f for smooth functionsHence we see

(u1 lowast u2) lowast (ϕ lowastψ) = (u2 lowast u1) lowast (ϕ lowastψ)ie setting E = u1 lowast u2 minus u2 lowast u1

E lowast (ϕ lowastψ) = 0 forallϕψ isin 994963(995014n)

But then using the trick that languϕrang= (u lowast ϕ)(0) we have

langE lowastϕψrang= ((E lowastϕ) lowast ψ)(0) = 0 forallϕψ =rArr E lowastϕ = 0 forallϕand so

langEϕrang= (E lowast ϕ)(0) = 0 forallϕ =rArr E = 0 in 994963 prime(995014n)and so we are done

Now if u isin 994963 prime(995014n) we can make sense of u lowastδ0 Indeed we have

(u lowastδ0) lowastψ = u lowast (δ0 lowastψ) = u lowastψfor all ψ isin 994963(995014n) where we have used that

(δ0 lowastψ)(x) = langδ0τx τrang= (τxψ)(0) =ψ(0) ie δ0 lowastψ =ψ as functions

So hence we see that u lowast δ0 = u as distributions for all u isin 994963 prime(995014n) [compare this with the heuristicformula ldquou(x) =

983125δ0(x minus y)u(y) dyrdquo]

23


3 TEMPERED DISTRIBUTIONS AND FOURIER ANALYSIS

31 Functions of Rapid Decay

We have another new set of test functions

Definition 31 A smooth function ϕ 995014nrarr 994999 is called a Schwartz function if

983042ϕ983042αβ = sup995014n

994802994802xαpart βϕ994802994802ltinfin

for each multi-index αβ (ie each semi-norm is finite)

We write S(995014n) for the space of all Schwartz functions on 995014n with the topology induced by thesemi-norms ie

ϕnrarr 0 in S(995014n) if 983042ϕm983042αβ rarr 0 forallαβ

Note This makes S(995014n) into a locally convex space

The intuition is that Schwartz functions are those smooth functions whose derivatives tends to 0faster than any polynomial The standard example of a Schwartz function is the Gaussian ϕ(x) =eminus|x |

2

Note Clearly we have Cinfinc (995014n) sub S(995014n) sub Cinfin(995014n)

Definition 32 A linear map u S(995014n)rarr 994999 is called a tempered distribution if exist constants C Nsuch that


|α||β |leN

983042ϕ983042αβ

ie they are distributions on S(995014n) We write Sprime(995014n) for the space of tempered distributions

Again we can give an equivalent definition of Sprime(995014n) in terms of sequential continuity [Exercise tocheck]

Remark We can show that994964 prime(995014n) 991590rarr Sprime(995014n) 991590rarr994963 prime(995014n)

and those embeddings are in fact continuous embeddings(vii) This is essentially a consequence of994963(995014n) 991590rarr S(995014n) 991590rarr994964 (995014n)

We can define differentiation on Sprime(995014n) as before as well as convolution between S(995014n) and Sprime(995014n)Note that we cannot however multiply a Schwartz function by a smooth function and hope to obtainanother Schwartz function consider

exp(minus|x |2) middot exp(|x |2) = 1

(vii)We say that X is continuously embedded in Y written X 991590rarr Y if X sub Y (or at least X is isomorphic to a subspaceof Y ) and whenever xn rarr 0 in X we have xn rarr 0 in Y Since the subspace topology on X is the smallest topology on Xfor which the inclusion is continuous this means the topology on X contains the subspace topology

24


which is the product of a Schwartz function with a smooth function which doesnrsquot give a Schwartzfunction This means that we canrsquot define multiplication by arbitrary smooth functions on Sprime(995014n)but only by those which preserve Schwartz functions

32 Fourier Transform on S(995014n)

Definition 33 We define the Fourier transform on L1(995014n) 994965 f 991596rarr f by

f (λ) =

983133

995014n

eminusiλmiddotx f (x) dx for λ isin 995014n

Note that we have S(995014n) sub L1(995014n) since for ϕ isin S(995014n)983133

995014n

|ϕ| dx =

983133

995014n

(1+ |x |)minusN middot (1+ |x |)n|ϕ| dx

le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin

for N sufficiently large

So hence we can take the Fourier transform of Schwartz functions

Lemma 31 We have 994965 L1(995014n)rarr C(995014n) ie if f isin L1(995014n) then f isin C(995014n)

Proof This is just dominated convergence since if λmrarr λ in 995014n we have

f (λ) =

983133

995014n

eminusiλmmiddotx f (x)rarr983133

995014n

eminusiλmiddotx f (x) dx = f (λ)

where the integrals convergence by dominated convergence since |eminusiλmmiddotx f (x)| le | f (x)| isin L1(995014n)Hence this shows that f isin C(995014n)

This lemma tells us that if f decays fast enough atinfin (eg so is in L1) then the Fourier transform off is more regular (eg continuous) So what happens if f were to decay even faster atinfin Would994965 ( f ) be more regular This turns out to be the key point of the Fourier transform it exchanges decay(which tends to be easy to check in practice) with regularity (which tends to be hard to check)

[So when solving a Fourier transformed PDE just by checking the decay rate of the Fourier trans-formed solution can tell us about the regularity of the original solution]

25


Lemma 32 If ϕ isin S(995014n) then

994923(Dαϕ)(λ) = λαϕ(λ) and 994923994790xβϕ994791(λ) = (minus1)|β |Dβ ϕ(λ)

where we write D equiv minusipart for part the usual derivative(viii)

Proof On integrating by parts

994923(Dαxϕ)(λ) =983133

995014n

eminusiλmiddotx Dαxϕ dx

= (minus1)|α|983133

995014n

ϕ middot Dαx994792eminusiλmiddotx994793983167 983166983165 983168(minus1)|α|λαeminusiλmiddotx

dx

= λα983133

995014n

ϕeminusiλmiddotx dx

= λαϕ(λ)

And by differentiating under the integral

Dβλϕ(λ) = Dβ

λ

994808983133

995014n

eminusiλmiddotxϕ(x) dx

994809

=

983133

995014n

(minus1)|β |xβ eminusiλmiddotxϕ(x) dx

= (minus1)|β |994923994790xβϕ994791(λ)

where we have used dominated convergence to justify differentiating under the integral sign

Important The above lemma tells us exactly how the Fourier transform interchanges local regularitywith decay atinfin and vice versa For example Dαϕ is lsquoless regularrsquo than ϕ and so we would expect

that994923(Dαϕ) should lsquodecay slowerrsquo than ϕ which is what the above shows since it gets multiplied bya polynomial term λα

We may have seen the ldquoFourier inversion formulardquo before which gives

ϕ(x) =1

(2π)n

983133

995014n

eiλmiddotx ϕ(λ) dλ

We will prove this in the following result

Theorem 31 The Fourier transform maps S(995014n) to itself and is a continuous isomorphism hereie

994965 S(995014n)rarr S(995014n) is a continuous isomorphism

(viii)This is convention for Fourier transforms as it removes factors of i from the derivatives

26


[so in particular it is bounded (as 994965 is linear) and if ϕmrarr 0 in S(995014n) then ϕmrarr 0 in S(995014n)]

Proof First we need to show that 994965 maps S(995014n) to S(995014n) So if ϕ isin S(995014n) we have

|λαDβ ϕ|=994802994802994802994802983133

995014n

eminusiλmiddotx Dα(xβϕ) dx

994802994802994802994802lt983133

995014n

|Dα(xβϕ)| dx ltinfin

and thus we see that 983042ϕ983042αβ ltinfin for each αβ Also ϕ is smooth from Lemmas 31 and 32 Thusϕ isin S(995014n)

We also see that ϕmrarr 0 in S(995014n) if ϕmrarr 0 in S(995014n)

So we now just need infectivity and surjectivity which will come from Fourier inversion So considerthe quantity

1(2π)n

983133

995014n

eiλmiddotx ϕ(λ) dλ =1

(2π)n

983133

995014n

eiλmiddotx994810983133

995014n

eminusiλmiddotyϕ(y) dy

994811dx

Note that the function (λ y) 991596rarr eiλmiddot(xminusy)ϕ(y) is not necessarily absolutely integrable and so we can-not apply Fubini to interchange the order of integration(ix) So to get around this we use dominatedconvergence and insert a regulariser Indeed note by dominated convergence that

983133

995014n

eiλmiddotx ϕ(λ) dλ = lim983171darr0

983133

995014n

eiλmiddotxminus983171|λ|2ϕ(λ) dλ

where we can use dominated convergence since |eiλmiddotx eminus983171|λ|2ϕ|le |ϕ| isin L1(995014n)

Now for each 983171 gt 0 we can use Fubini and so983133

995014n

eiλmiddotxminus983171|λ|2ϕ(λ) dλ =

983133

995014n

ϕ(y)

994810983133

995014n

eiλmiddot(xminusy)minus983171|λ|2 dλ

994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133

995014eiλ j(x jminusy j)minus983171λ2

j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n

ϕ(y)eminus|xminusy |2

4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n

ϕ(x minus 2991598983171 y prime)eminus|y

prime|2 dy prime

= πn2 middot 2n middot983133

995014n

ϕ(x minus 2991598983171 y)eminus|y|

2dy

(ix)Recall that Fubinirsquos theorem says us that if983125 983125| f (x y)| dxdy ltinfin ie f is absolutely integrable then we can

interchange the order of integration

27


where in (dagger) we changed variables y prime = (x minus y)2991598983171 and we shall explain (983183) momentarily We can

then use dominated convergence in this last line since pointwise ϕ(x minus 2991598983171 y)rarr ϕ(x) as 983171rarr 0 to

get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)

since the last integral is a product of n-Gaussian integrals and so equals πn2 So combining all ofthis we see

ϕ(x) =1

(2π)n

983133

995014n


So hence 994965 is injective since this shows if ϕ then we must have ϕ equiv 0 Moreover this shows that994965 is surjective since

ϕ(x) =1

(2π)n

983133

995014n

eminusiλmiddotx ϕ(minusλ) dy =

983133

995014n

eminusiλmiddotx994878

1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)

and thus we are done

Aside Fokas and Gelfand provided a beautiful proof of this theorem via the Riemann-Hilbert prob-lem

Note At (983183) in the above proof we have used that983133 infin

minusinfineizαminus983171z2

dz =

983133 infin

minusinfineminus983171(zminus

iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133

Im(z)=α2983171eminus983171(zminus

iα2983171 )

2

dz by Cauchyrsquos theorem from Complex Analysis

= eminusα24983171

983133 infin

minusinfineminus983171z

2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171

We apply Cauchyrsquos theorem on the boundary of a box of the form

z isin 994999 984158(z) isin [minusN N]984143(z) isin [0α2983171]where the integrals on the side parts (of constant real part) vanish as we take N rarrinfin

We will now use all of this to define the Fourier transform of a tempered distribution

28


33 Fourier Transform of Sprime(995014n)

We need the following lemma to motivative the definition of 994965 on Sprime(995014n)

Lemma 33 (Parsevalrsquos Theorem) For ϕψ isin S(995014n) we have983133

995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx

or alternatively if (middot middot) is the L2-inner product this says (ϕ994965 (ψ)) = (994965 (ϕ)ψ) ie 994965 is self-adjoint here

Proof By Fubinirsquos theorem we have983133

995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n

eminusiλmiddotxψ(λ) dλ

994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ

where we have been able to apply Fubini since the integral is absolutely convergent ie since(x λ) 991596rarr eminusiλmiddotxϕ(x)ψ(λ) is in L1(995014n times995014n) Hence we are done

Treating u isin S(995014n) as its associated tempered distribution this reads

languϕrang= langu ϕrang

But the RHS os well-defined for any u isin Sprime(995014n) since994965 S(995014n)rarr S(995014n) Thus we define in general

Definition 34 For u isin Sprime(995014n) we define the Fourier transform of the tempered distribution uby

languϕrang = langu ϕrang forallϕ isin S(995014n)

So we can always take the Fourier transform of a tempered distribution

Example 31 Take u= δ0 isin Sprime(995014n) (noting that all distributions of compact support are tempereddistributions) Then

langδ0ϕrang = langδ0 ϕrang= ϕ(0) =983133

995014n

ϕ(x) dx = lang1ϕrang

and this is true for all ϕ isin S(995014n) and thus δ0 = 1

29


Also

lang1ϕrang = lang1 ϕrang=983133

995014n

ϕ(λ) dλ = (2π)nϕ(0) = lang(2π)nδ0ϕrang

where we have used Fourier inversion to evaluate the integral of ϕ Thus we see

1= (2π)nδ0

or in old heuristic language

ldquoδ0(x) =1

(2π)n

983133

995014n

eminusiλmiddotx dλrdquo

We can extend Lemma 32 to Sprime(995014n) so that

993486Dαu= λαu and 993486xβu= (minus1)|β |Dβ u

for all u isin Sprime(995014n) where the derivatives here are distributional derivatives [see Example Sheet 2 fora proof]

So now we shall see that the Fourier transform of a tempered distribution is another tempered dis-tribution This will enable us to try to solve PDEs in the space of tempered distributions (whichincludes a lot of functions when we identify functions by their associated functions) by taking theFourier transform of the PDE

Theorem 32 The Fourier transform extends to a continuous isomorphism

994965 Sprime(995014n)rarr Sprime(995014n)

Proof Suppose that ϕm rarr 0 in S(995014n) Then we know by Theorem 31 that ϕm rarr 0 in S(995014n) Sohence

languϕmrang = langu ϕmrang rarr 0

since u isin Sprime(995014n) So by the sequential continuity definition of Sprime(995014n) we deduce that u isin Sprime(995014n) (so994965 does map Sprime(995014n) into Sprime(995014n))

We also claim that

u= (2π)minusn994920(993484)u

Indeed

languϕrang= langu ϕrang (dagger)=994901u (2π)minusn994921(993485)ϕ994904=994901(2π)minusn994920(993484)u ϕ994904

for all ϕ isin S(995014n) where (dagger) follows from the Fourier inversion theorem since

ϕ(x) = ϕ(minusx) =1

(2π)n

983133

995014n

eminusiλmiddotx ϕ(λ) dλ = (2π)minusn994921(993485)ϕ (x)

and so we have a Fourier inversion result for tempered distributions which as before implies that994965is injective and surjective

30


Finally for continuity we have that if umrarr 0 in Sprime(995014n) then

langumϕrang rarr 0 forallϕ isin S(995014n)

hArr langum ϕrang rarr 0 forallϕ isin S(995014n)

hArr langumϕrang rarr 0 forallϕ isin S(995014n)

ie umrarr 0 in Sprime(995014n) So done

34 Sobolev Spaces

The Sobolev spaces are a very useful space of functions

Definition 35 For s isin 995014 let Hs(995014n) denote the set of u isin Sprime(995014n) for which u isin Sprime(995014n) can beidentified with a locally integrable function λ 991596rarr u(λ) for which

983042u9830422Hs =

983133

995014n

|u(λ)|2langλrang2s dλ ltinfin

where langλrang = (1+ |λ|2)12 is called the Japanese bracket (for some reason)

Note Since the Fourier transform exchanges differentiability and decay this is essentially asking fors-differentiability of u

We can define Sobolev spaces on X sub 995014n via localisation note that if u isin 994963 prime(X ) and ϕ isin 994963(X ) thenϕu isin 994964 prime(995014n) sub Sprime(995014n)

Definition 36 We say u isinHsloc(X) if u isin 994963 prime(X ) and ϕu isin Hs(995014n) for every ϕ isin 994963(X ) This is

the local Sobolev space

A key first result is the Sobolev embedding which tells us when we can infer regularity of u isin Hs(995014n)

Lemma 34 (Sobolev Lemma) Let u isin Hs(995014n) Then if s gt n2 then u isin C(995014n)

Proof By Cauchy-Schwarz983133

995014n

|u(λ)| dλ =983133

995014n

langλrangminuss middot langλrangs|u(λ)| dλ

le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n

|u(λ)|2langλrang2s dλ

99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0

(1+ r2)minussrnminus1 dr

99480912

31


where we have changed to polar coordinates (and so dλ = dσ middot rnminus1dr where dσ is the area elementon Snminus1)

Now note that if s = n2 + 983171 for 983171 gt 0 then

(1+ r2)minussrnminus1 sim rminus2s+nminus1 = rminus1minus2983171

for r ≫ 0 which is integrable Hence this shows that the integral on the RHS is ltinfin and so983133

995014n

|u(λ)| dλ ltinfin ie u isin L1(995014n)

Now for ϕ isin S(995014n)

langu ϕrang= languϕrang=983133

995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n

eiλmiddotx ϕ(x) dx

994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n

eiλmiddotx u(λ) dλ

994811

983167 983166983165 983168=F(x)

dx

= langF ϕrangwhere we have used Fubinirsquos theorem which we can do since u isin L1(995014n) and so (x λ) 991596rarr ϕ(x)u(λ)eiλmiddotx

is absolutely integrable (ie in L1(995014n times995014n))

So hence we can identify u isin Sprime(995014n) with the function F above (as the above says u= F) and so wehave

u(x) =1

(2π)n

983133

995014n


But we know the Fourier transform of an integrable function is continuous (by dominated conver-gence - this is Lemma 31) and thus this shows u isin C(995014n)

Note In the above we are trying to deduce a Fourier inversion formula for functions in L1 whichwe donrsquot know holds a priori as we have only seen it for S(995014n) Hence we need to check it directlyin the above

Corollary 31 Suppose u isin Hs(995014n) for every s gt n2 Then u isin Cinfin(995014n)

Proof Follows from Lemma 34 since Dαu will be in Hs(995014n) for all s gt n2 (as differentiating reducesthe exponent s but we have this for all s gt n2) So hence we can just apply Lemma 34 inductivelyto get all the Dαu can be identified with continuous functions and thus u can be identified with asmooth function [Exercise to check details more thoroughly]

32


4 APPLICATIONS OF THE FOURIER TRANSFORM

41 Elliptic Regularity

We are now interested in PDErsquos of the form P(D)u = f where u f isin 994963 prime(X ) and P is a polynomial(recall that D = minusipart ) Note that these are PDEs with constant coefficients For example if P(λ) =minus(λ2

1 + middot middot middot+λ2n) then

P(D) = minus994808994858minusipart

part x1

9948682+ middot middot middot+994858minusipart

part xn

9948682994809=∆

is the Laplacian

The main question we are interested in here is one of regularity if f isin Hsloc(X ) then what can be

said of the solution u (assuming it exists)

Definition 41 For a differential operator P(D) =983123|α|leN cαDα the order of P is N (assuming

that not all aα with |α|= N are 0)

The principal symbol of P is thenσP(λ) =983131

|α|=N

cαλα

We say that P(D) is elliptic if σP(λ) ∕= 0 for all λ isin 995014n0 ie if ker(σ) = 0

Lemma 41 If P(D) is an Nrsquoth order elliptic operator then exist constants C R such that

|P(λ)|ge ClangλrangN for all |λ|gt R

Remark Heuristically if we tried to solve P(D)u = f we could take the Fourier transform of theequation to get P(λ)u= f since the Fourier transform changes Drsquos into λrsquos So then we would wantto say u= f P(λ) and take the inverse Fourier transform to find the solution u but P could be zeroand this may not be integrable The above Lemma 41 tells us that this isnrsquot the case if we take λ farenough from 0

Proof Since P is elliptic we know σP(λ) ∕= 0 for all λ isin Snminus1 Thus by compactness of Snminus1 σPattains its (non-zero) minima on Snminus1 and thus existc0 gt 0 such that

minλisinSnminus1

|σP(λ)|= c0 gt 0

Then for any λ ∕= 0

|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33


But by the triangle inequality

|P(λ)|ge |σP(λ)|minus |P(λ)minusσP(λ)|

ge c0λN minus |λ|N994878

P(λ)minusσP(λ)|λ|N994881

Now since P(λ)minusσP(λ) is a polynomial of degree N minus1 we can choose |λ| sufficiently large so that994802994802994802994802P(λ)minusσP(λ)|λ|N

994802994802994802994802lec0

2

Hence combining we get

|P(λ)|ge 12

c0|λ|N ge ClangλrangN

for all |λ| sufficiently large and so we are done

Theorem 41 (Elliptic Regularity) If P(D) is an Nrsquoth order elliptic operator and P(D)u= f withf isin Hs

loc(X ) then u isin Hs+Nloc (X )

Remark The order of the Sobolev space is essentially saying how many times differentiable thefunction is (see Example sheet 2) Indeed if

983125995014n |u|langλrang2s dλ ltinfin then we have

983133

995014n

|Dαu|2 dλ = C

983133

995014n

|λα|2 middot |u| dλ forall |α|le s

Proof Later

Corollary 41 Suppose P(D) is elliptic and that P(D)u isin Cinfin(X ) Then in fact u isin Cinfin(X )

ie elliptic operators are hypoelliptic

Proof Apply Theorem 41 and Corollary 31

We first prove an easier version of the elliptic regularity theorem using a parametrix

Definition 42 We say that E isin 994963 prime(995014n) is a parametrix for P(D) if

P(D)E = δ0 + w

for some w isin 994964 (995014n) ie E inverts P(D) up to some smooth function

The intuition for this definition comes from using Greenrsquos functions to solve PDEs Note that due tothe presence of the δ0 we donrsquot expect a parametrix to be smooth at the origin This is exactly whatthe next lemma gives for elliptic PDEs

34


Lemma 42 Suppose P(D) is elliptic Then it admits a parametrix E which is smooth away fromthe origin

Proof If P is elliptic then by Lemma 41 we know that existRgt 0 such that

|P(λ)|≳ (x)langλrangN for |λ|gt R

Now fix χ isin 994963(995014n) with χ equiv 1 on |λ|lt R and χ equiv 0 on |λ|gt R+ 1 Then set

E(λ) =1minusχ(λ)

P(λ)

where the lsquohatrsquo notation is suggestive - we will take the inverse Fourier transform of this quantity toget the parametrix we want Note that E is smooth and |E(λ)|≲ langλrangminusN for |λ|gt R+ 1

So we have E isin Sprime(995014n) and so by the Fourier transform

P(λ)E = 1minusχ(λ) rArr P(D)E = δ0 minus w

where E = 994965minus1(E) and w = χ isin 994963(995014n) sub S(995014n) Hence w isin S(995014n) and so in particular it smoothThen note (using properties of the Fourier transform)

994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN

for |λ| gt R+ 1 (this follows just from checking it as P(λ) is a polynomial of degree N) So hencewe see

994925994790Dα(xβ E)994791isin L1(995014n) for |β | sufficiently large

So hence by Lemma 31 we have that Dα(xβ E) is continuous for each α for |β | sufficiently large Sohence E is smooth away from x = 0 (since xβ is always differentiable) and so done

We can now prove the easy version of Theorem 41 if u f isin 994964 prime(995014n) P(D)u= f

Theorem 42 (Easy Version of Elliptic Regularity) Suppose P(D) is an Nrsquoth order elliptic operatorand we have P(D)u= f with u f isin 994964 prime(995014n) and f isin Hs

loc(995014n) Then u isin Hs+N

loc (995014n)

(x)Recall that we write A≲ B to mean existC gt 0 such that Ale CB

35


Proof By Lemma 42 we know P admits a parametrix E which is smooth away from the origin andso P(D)E = δ0 + w for some w smooth Then we have

u= u lowastδ0 = u lowast (P(D)E minus w)

= [P(D)u] lowast E minus u lowastw

= f lowast E minus u lowastw

Let us look at each term on the RHS separately For u lowastw we have (from example sheet 2)

994923(u lowastw) = uw

and thus as w isin S(995014n) and u is smooth with |u| le Clangλrangm we have uw isin S(995014n) ie994920u lowastw isin S(995014n)and so (from what we know about the Fourier transform)rArr u lowastw isin S(995014n)

Hence u lowastw is smooth and so will not effect the regularity of u above

For f lowast E we have 994802994802994802994920f lowast E994802994802994802= |E(λ)| middot | f (λ)|≲ langλrangminusN | f (λ)|

Now since f isin Hs(995014n) ie 983133

995014n

| f (λ)|2langλrang2s dλ ltinfin

and so with the above we see 983133

995014n

994802994802994802994920f lowast E9948029948029948022langλrang2s+2N dλ ltinfin

and so f lowast E isin Hs+N (995014n) Hence

u= E lowast f983167983166983165983168isinHs+N (995014n)

minus u lowastw983167983166983165983168isinS(995014n)

isin Hs+N (995014n)

and so we are done

Remark The idea for the general result is as follows For u isin Hsloc(X ) we have ϕu isin Hs(995014n) for any

ϕ isin D(X ) Now looking at P(D)u= f we have ϕ middot P(D)u= ϕ f ie

P(D)[uϕ] + (ϕ middot P(D)uminus P(D)(uϕ)) = ϕ f

and upon expanding the derivatives

P(D)(uϕ) = ϕP(D)u+ (lower order terms)

So all the highest order derivatives of u as contained in the first term on the LHS of the above whichis a good term we can work with We will use this idea and a bootstrapping argument to proveTheorem 41

Proof of Theorem 41 We have P(D) an N rsquoth order elliptic operator (with constant coefficients) andP(D)u = f isin Hs

loc(X ) Recall the following elementary facts about Sobolev spaces (many of theseare exercises on Example Sheet 2)

(i) If u isin 994964 prime(995014n) then u isin H t(995014n) for some t isin 995014

(ii) If u isin H t(995014n) then Dαu isin H tminus|α|(995014n)

36


(iii) If s gt t then Hs(995014n) sub H t(995014n)

(iv) If ϕ isin S(995014n) and u isin H t(995014n) then ϕu isin H t(995014n)

To show u isin Hs+Nloc (X ) it suffices to show that ϕu isin Hs+N (995014n) for ϕ isin 994963(X ) arbitrary So fix

ϕ isin 994963(X )

Now introduce ψ0 ψM sub S(995014n) where

supp(ϕ) sub supp(ψM ) sub middot middot middot sub supp(ψ0)

and ψiminus1|supp(ψi) equiv 1 Note that ψ0u isin 994964 prime(995014n) Hence by property (i) above existt isin 995014 such thatψ0u isin H t(995014n) Now note that

P(D)(ψ1u) =ψ1P(D)u+ [P(D)ψ1uminusψ1P(D)u]

=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)

where [g h] = ghminushg is the commutator and in the last equality we have used the fact thatψ0 equiv 1on supp(ψ1) (and so this doesnrsquot change anything)

FIGURE 2 An illustration of the ψi maps in relation to one another and ϕ

Note that (by the Leibniz rule) [P(D)ψ1] is a differential operator of order N minus 1 with coeffi-cients in S(995014n) (since ψ1 isin S(995014n)) Hence as ψ0u isin H t(995014n) (by property (iv) above) we have[P(D)ψ1](ψ0u) isin H tminus(Nminus1)(995014n) from property (ii) and (iii) So

P(D)(ψ1u) = ψ1 f983167983166983165983168isinHs(995014n) as ψ1 is in S(995014n)

+

isinH tminus(Nminus1)(995014n)983165 983168983167 983166[P(D)ψ1](ψ0u) =rArr P(D)(ψ1u) isin H A1(995014n)

where A1 =mins t minus N + 1 This means that (by definition of Sobolev spaces)

983133

995014n

994802994802994802P(λ)994922(ψ1u)(λ)9948029948029948022langλrang2A1 dλ ltinfin

37


and hence (since P is of N rsquoth orderLemma 41)983133

995014n

994802994802994802994922(ψ1u)(λ)9948029948029948022langλrang2A1+2N dλ ≲

983133

995014n

994802994802994802994802P(λ)langλrangN middot994922(ψ1u)(λ)

9948029948029948029948022

middot langλrang2A1+2N dλ

=

983133

995014n

994802994802994802P(λ)994922(ψ1u)(λ)9948029948029948022langλrang2A1 dλ

ltinfinand thus ψ1u isin HA1(995014n) where A1 = A1 + N =mins+ N t + 1

Now repeating this we have

P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)

since ψ1 = 1 on supp(ψ2) Then continuing as before (just with t replaced by A1) we find thatψ2u isin HA2(995014n) where

A2 =mins+ N A1 + 1=mins+ N mins+ N + 1 t + 2=mins+ N s+ N + 1 t + 2=mins+ N t + 2

Thus continuing inductively we see that ψM u isin HAM (995014n) where

AM =mins+ N t +MSo choosing M gt s + N minus t so that AM = s + N then since ψM = 1 on supp(ϕ) we get thatϕu isin Hs+N (995014n)

But then since ϕ isin 994963(X ) was arbitrary this gives that u isin Hs+Nloc (X ) and so we are done

42 Fundamental Solutions

Again we study P(D)u= f with u f isin 994963 prime(995014n)

Definition 43 We say E isin 994963 prime(995014n) is a fundamental solution for P(D) if

P(D)E = δ0

Note At least formally if P(D)E = δ0 and we want to solve P(D)u = f then we have u = E lowast f since

P(D)u= (P(D)Elowast) f = δ0 lowast f = f

However we have only defined convolutions if one of E f has compact support and so this doesnrsquotquite work But it is the key idea

(xi)We can do things like Fourier transforms when we have a test function multiplied by u This is why we are so keento get ψrsquos in here

38


Example 41 Let z = x1 + i x2 isin 994999 sim= 9950142 Then define the Cauchy-Riemann operator bypart

part z=

12

994858part

part x1+ ipart

part y

994868

We claim that E = 1πz is the fundamental solution for part

part z

Proof Fix ϕ isin 994963(9950142) Note that on |z|ge 983171 (for any 983171 gt 0) we havepart

part zE = 0

Also E is locally integrable and so E isin 994963 prime(995014n) Now we have994902part

part zEϕ994905= minus983133

9950142

E middot part ϕpart z

dx1dx2

= minus lim983171darr0

983133

|z|gt983171E middot part ϕpart z

dx1dx2


983133

|z|gt983171

part

part z(ϕE) dx1dx2 since

part

part zE = 0 here

Now note that (by definition of part part z)part

part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881

and so by Greenrsquos theorem(xii)(setting Q = ϕE2 P = minusiϕE2) we get994902part

part zEϕ994905= lim983171darr0

12

983133

|z|=983171minusiϕEdx1 +ϕEdx2

= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i

dz since dz = dx1 + idx2

= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i

middot 1π983171eiθ

middot i983171eiθ dθ setting z = 983171eiθ (as E = 1πz)

= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)

= langδ0ϕrangHence we see that as distributions part E

part z = δ0 and so we are done

(xii)Recall that Greenrsquos theorem says that for suitable PQ A983116

part A

Pdx +Qdy =

983133 983133

A

994858partQpart xminus part Ppart y

994868dxdy

39


Example 42 (Fundamental Solution of the Heat Equation) If L = partpart t minus∆x is the heat operator

on 995014n times995014 then a fundamental solution is

E(x t) =

9948301

(4πt)n2 eminus|x |24t if t gt 0

0 if t le 0

Proof Note on t ge 983171 (for any 983171 gt 0) we have that994858part

part tminus∆x

994868E = 0

and that E is locally integrable So E isin 994963 prime(995014n times995014)

Then by definition for any ϕ isin 994963(995014n times995014) we have

langLEϕrang=983133 infin

0

dt

983133

995014n

dx E994858minuspart ϕpart tminus∆xϕ

994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n

dx minus partpart t(ϕE) + lim

983171darr0

983133 infin

983171

dt

983133

995014n

dx ϕ994858part Epart tminus∆x E994868

983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2

eminus|x |24983171 and setting y =

x2991598983171

= lim983171darr0

983133

995014n

dy ϕ(2991598983171 y983171) middot (4π983171)minusn2eminus|y|

2 middot 983171n22n

= lim983171darr0πminusn2

983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)

= langδ0ϕrang(where in the first equality we get a minus sign on minuspart ϕpart t as this comes from integrating byparts once whilst the other term we integrate by parts twice and so the sign does not change)Thus we see that as distributions LE = δ0 and so E is the fundamental solution

40


Remark So how do we find fundamental solutions in general Our intuitionfirst guess might beto consider

Eϕ 991596minusrarr langEϕrang = 1(2π)n

983133

995014n

ϕ(minusλ)P(λ)

dλ

Then we would (heuristically) have

langP(D)Eϕrang= 1(2π)n

983133

995014n

994925[P(minusD)ϕ](minusλ)P(λ)

dλ =1

(2π)n

983133

995014n

P(λ)ϕ(minusλ)P(λ)

dλ = ϕ(0) = langδ0ϕrang

where we have used the Fourier inversion formula So this would appear to work but this E may notbe well-defined as we could have P(λ) = 0 We will get around this by constructing Houmlrmanderrsquosstaircase The idea is to ldquowobblerdquo the surface of integration

Lemma 43 For x isin 995014n write x = (x prime xn) with x prime isin 995014nminus1 Then for ϕ isin 994963(995014n) the functionz 991596rarr ϕ(λprime z) is complex analytic in z isin 994999 for each λprime isin 995014nminus1 Also994802994802ϕ(λprime z)

994802994802≲m (1+ |z|)minusmeδ|Im(z)| for all m= 0 1 2

for some δ gt 0 (ie ϕ decays rapidly in each horizontal z-direction)

Remark Here ≲m means ≲ except the constant can depend on m

Proof By definition of the Fourier transform and Fubinirsquos theorem we have

ϕ(λprime z) =

983133

995014nminus1

eminusiλprimemiddotx prime994808983133

995014eminusizxnϕ(x prime xn) dxn

994809dx prime

Thus we see that z 991596rarr ϕ(λprime z) is analytic from checking the Cauchy-Riemann equations (we candifferentiate under the integral sign by dominated convergence)(xiii) Also on integrating by parts

|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn

994881ϕ(x prime xn) dxn

994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1


995014eminusizxn middot part

mϕ

part xmn(x prime xn) dxn

994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|

where we have used thatϕ equiv 0 for |xn|gt δ gt 0 for some δ gt 0 so we only need to consider |xn|le δin the integral so we can bound this by the integral of

994802994802994802 partmϕpart xm

n

994802994802994802 and the maximum the exponential term

is which is this

Note The above lemma tells us (along with Cauchyrsquos theorem from complex analysis) that983133

Im(z)=cϕ(λprime z) dz

(xiii)Alternatively we can use Morerarsquos theorem and Fubinirsquos theorem to check this

41


is independent of c isin 995014

FIGURE 3 Lemma 43 tells us the integrals along the vertical sides tend to 0 as theyget pushed out Thus the integral along the horizontal sides is independent of c byCauchyrsquos theorem

Theorem 43 (Malgrange-Ehrenpries) Every non-zero constant coefficient partial differentialoperator P(D) admits a fundamental solution

Proof By a rotation of our coordinate axes we can assume wlog that P(D) is of the form

P(λprimeλn) = λMn +

Mminus1983131

i=0

ai(λprime)λi

n

ie treat P as a polynomial in λn with coefficients depending on λprime Wlog by scaling the leadingcoefficient is 1

Then for fixed microprime isin 995014nminus1 we can write

P(microprimeλn) =M983132

i=1

(λn minusτi(microprime))

where τi(microprime)Mi=1 are the roots of the M rsquoth order polynomial P(microprime middot)

Now exist a horizontal line Im(λn) = c(microprime) in the complex λnminusplane with |Im(λn)| le (M + 1)(1+ 983171)for 983171 gt 0 small such that

|Im(λn)minusτi(microprime)|gt 1 for all i = 1 M

This is seen by considering 2M + 2 strips of width 1 + 983171 Then by the pigeonhole principle exist twoadjacent strips that donrsquot contain any roots τi(microprime) Then take ldquoIm(λn) = crdquo for c the line betweenthese two strips (then clearly c = c(microprime)) [See Figure 4]

So on Im(λn) = c(microprime) we have

|P(microprimeλn)|=M983132

i=1

|Im(λn)minusτi(microprime)|gt 1

since each term in the product is gt 1 Then by continuity (as the roots of a polynomial vary contin-uously with the polynomials coefficients which is seen from the formula expression the coefficients

42


FIGURE 4 An illustration of how the constants ci are constructed using the pigeon-hole principle There are M roots but 2M + 2 strips so two adjacent strips do notcontain any roots

in terms of the roots) the τi(microprime) depend continuously on microprime and thus the same estimate holds on aneighbourhood of microprime Nmicroprime say

Then since this argument holds for any microprime isin 995014nminus1 we can cover 995014nminus1 with such neighbourhoodsNmicroprimemicroprimeisin995014nminus1 Then by Heine-Borel we can extract a locally finite subcover Ni = Nmicroprimei so that|P(λprimeλn)|gt 1 for λprime isin Ni Im(λn) = c(microprimei) = ci

[ie get a finite subcover on compact balls of increasing radii - this gives a countable subcover whichis finite on every compact subset of 995014nminus1]

So we have that we can find a horizontal line Im(λn) = ci on each Ni to avoid the roots of P Howeverwe need to be careful on overlaps Thus therefore define the disjoint sets

∆i = Niiminus1983134

j=1

N j

for which we see |P(λprimeλn)|gt 1 for λprime isin∆i and Im(λn) = ci Moreover we have983126infin

i=1∆i = 995014nminus1

Now define(xiv)

langEϕrang = 1(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci

ϕ(minusλprimeminusλn)P(λprimeλn)

dλn

994882dλprime

(xiv)Heuristically we wanted to define


983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0

ϕ(minusλprimeminusλn)P(λ)

dλn

994882dλprime

since Im(λn) = 0 equiv 995014 Then here we know from Lemma 43 that we can change983125

Im(λn)=0to983125

Im(λn)=ci and we can

change the integral983125995014nminus1 dλprime into983123

i

983125∆i

dλprime as the sets ∆i are disjoint and cover (up to a set of measure zero) So we get

the formula given in the proof which is good because it avoids the roots of P

43


By construction we have

|P(λprimeλn)|gt 1 for (λprimeλn) isin∆i times λ Im(λn) = ciand it follows [from Example Sheet 3] that E isin 994963 prime(995014n) Also we have


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci

P(λ)ϕ(minusλprimeminusλn)

P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0

ϕ(minusλprimeminusλn) dλn

994882dλprime by Cauchyrsquos theorem and Lemma 43

=1

(2π)n

983133

995014nminus1

983133

995014ϕ(minusλprimeminusλn) dλndλprime

=1

(2π)n

983133

995014n

ϕ(minusλ) dλ

= ϕ(0) by Fourier inversion

= langδ0ϕrangand so as ϕ isin 994963(995014n) was arbitrary in the above we get P(D)E = δ0 and so we are done

Remark As said before this method of proof is called Houmlrmanderrsquos staircase since the integrationalong 995014nminus1 looks like a staircase instead of a plane

FIGURE 5 An illustration of Houmlrmanderrsquos staircase We essentially split up the planeIm(λn) = 0sim= 995014 into disjoint regions and translate them in such a way to avoid theP(λ) = 0 surface

This result establishes the existence of a fundamental solution in 994963 prime(995014n) Using more sophisticationone can show that exist a fundamental solution in Sprime(995014n) [Wagner gave an explicit construction witha nice formula as above using complex analysis]

44


43 The Structure Theorem for 994964 prime(X )

We know that if f isin C(X ) then we can always define part α f isin 994963 prime(X ) (ie as a distribution not afunction) via

langpart α f ϕrang = (minus1)|α|983133

Xf part αϕ dx

for ϕ isin 994963(X ) A natural question you might ask then is whether all distributions can be written asa linear combination of such distributions (a bit like a Taylor series) ie whether we can find somecollection ( fα)α sub 994999(X ) with

u=983131

α

part α fα in 994963 prime(X )

We will prove that this is true for u isin 994964 (X )

So if u isin 994964 prime(995014n) sub Sprime(995014n) then define the Fourier transform u isin Sprime(995014n) in the usual way

languϕrang = languϕrang forallϕ isin S(995014n)

Then by definition of the Fourier transform we have

languϕrang= langu(x) ϕ(x)rang=994818

u(x)

983133

995014n

eminusiλmiddotxϕ(λ) dλ

994819

Now fix(xv) χ isin 994963(995014n) with χ equiv 1 on |λ|lt 1 and χ equiv 0 on |λ|gt 2 Then define ϕm isin 994963(995014n) by

ϕm(λ) = χ994858λ

m

994868ϕ(λ)

Then we can show ϕm rarr ϕ in S(995014n) and hence ϕm rarr ϕ in S(995014n) by continuity of the Fouriertransform So

languϕrang= limmrarrinfinlangu ϕmrang= lim

mrarrinfin

994818u(x)

983133

995014m

eminusiλmiddotxϕm(λ) dλ

994819

Now apply the same Riemann sum argument (from Lemma 15) to interchange langmiddot middotrangwith983125995014nlangmiddot middotrang dλ

and so hence

= limmrarrinfin

983133

995014n

langu(x) eminusiλmiddotxrangϕm(λ) dλ =

983133

995014n

langu(x) eminusiλmiddotxrangϕ(λ) dλ

where this last equality comes from the dominated convergence theorem and a semi-norm estimate(see below)

So we can identify u isin Sprime(995014n) with the function

λ 991596minusrarr u(λ) = langu(x) eminusiλmiddotxrangfor λ isin 995014n We can then in fact show that u isin Cinfin(995014n) and by the semi-norm definition of 994964 prime(995014n)exist constants C N ge 0 and compact K sub 995014n such that

|u(λ)|=994802994802langu(x) eminusiλmiddotxrang994802994802

le C983131

|α|leN

supK

994802994802part αx (eminusiλmiddotx)994802994802

≲ langλrangN

and so hence we can apply the dominated convergence theorem in the above

(xv)The point is we want to do the same Riemann integralsum argument we did before in Lemma 15 However thesums arenrsquot finite here as we do not have compact support So we need to modify by such a function to get this

45


In summary we see that if u isin 994964 prime(995014n) sub S(995014n) then its Fourier transform u isin Sprime(995014n) can be identifiedwith the function u = langu fλrang where fλ(x) = eminusiλmiddotx Moreover we have u isin Cinfin(995014n) and |u(λ)| ≲langλrangN for some N ge 0(xvi)

This is just all preamble to the following theorem

Theorem 44 For u isin 994964 prime(X ) exist a finite collection ( fα)α sub C(X ) such that

u=983131

α


and supp( fα) sub X for all α

Remark We can use this to prove results if we can just prove a result for distributions of the formpart α f for f isin C(X ) then as the above sum is finite we can extend the result to all u isin 994964 (X )

Proof Fix ρ isin 994963(X ) have ρ equiv 1 on supp(u) Then for ϕ isin 994964 (X ) we have

languϕrang= languρϕrang+ langu (1minusρ)ϕrang983167 983166983165 983168=0 as 1minusρequiv0 on supp(u)

ielanguϕrang= languρϕrang

Then since ρϕ isin 994963(X ) we can treat both u and ρϕ as elements of 994964 prime(995014n) and 994963(995014n) respectivelyby extension by zero

Then since ρϕ isin 994963(995014n) sub S(995014n) we can write

ρϕ =993484993484ψ for some ψ isin S(995014n)

(ie double Fourier transform so ρϕ = (2π)nψ) Hence

languϕrang= langu994841994841ψrang= langu ψrangUsing the Laplacian ∆ =

983123ni=1(part part x i)2 we can write (as the Fourier transform changes minus∆ into

λ1 + middot middot middot+λ2n = |λ|2)

ψ = langλrangminus2M 994925[(1minus∆)Mψ]and so

languϕrang=994901langλrangminus2M u 994925[(1minus∆)Mψ]

994904

Now since |u| ≲ langλrangN for some N ge 0 (from the preamble to this theorem) we can choose M suchthat langλrangminus2M u isin L1(995014n) Then define

f (x) =1

(2π)n

983133

995014n

eminusiλmiddotxlangλrangminus2M u(λ) dλ

So f isin C(995014n) by the dominated convergence theorem (ie Lemma 31) Hence since

f =1

(2π)n994925994792langλrangminus2M ˆ994793

u

(xvi)This demonstrates a general rule taking the Fourier transform of a distribution of compact support gives a verynice function So to solve a problem we can take the Fourier transform solve the problem with this nice function andthen take the inverse Fourier transform to solve the original problem

46


we see

languϕrang=994902994925994792langλrangminus2M ˆ994793

u (1minus∆)Mψ994905

=994800(2π)n f (1minus∆)Mψ

994801

=994800

f (1minus∆)M ˇ(ρϕ)994801

=994800

f (1minus∆)M (ρϕ)994801

which is good since we know f is continuous By the Leibniz rule we have

(1minus∆)M (ρϕ) =983131

α

(minus1)|α|ραpartαϕ

where ρα isin 994963(X ) are defined accordingly (just by the Leibniz rule) Thus we see

languϕrang=994818

f 983131

α


994819

=

994818983131

α

(minus1)|α|part α(ρα f ) ϕ

994819

So setting fα = ρα f we see fα isin C(X ) and supp( fα) sub X and the above gives

u=983131

α

(minus1)|α|part α fα in 994964 prime(X )

and so we are done

Remark Note that this proof is constructive we can actually write down these fα from u

Aside For u isin 994963 prime(X ) a similar approach can be taken and we can show that

u=983131

α


but the sum can be infinite although the sum is locally finite ie

languϕrang=983131

α

langpart α fαϕrang for each ϕ isin 994963(X )

with only finitely many terms on the RHS being non-zero (so a finite sum for each ϕ although whichfα are included can change) To prove this we just need to invoke a partition of unity of X (ie for(ρβ )β a partition of unity on X u =

983123β ρβu is a sum of distributions of compact support on which

the above can be applied) although we shanrsquot go into the details here

[Look at Reed and Simonrsquos book in particular the chapter on tempered distributions and the quantumharmonic oscillator]

44 The Paley-Wiener-Schwartz Theorem

For u isin 994964 prime(995014n) we know we can identify the distribution u with the smooth function

λ 991596minusrarr u(λ) = langu(x) eminusiλmiddotxrang for λ isin 995014n

47


So what happens if we let λ isin 994999n Then we can take the complex extension (λrarr z isin 994999) ie define

u(z) = langu(x) eminusizmiddotxrangwhich is called the Fourier-Laplace transform of u (ie the complex extension of the functionidentifying the Fourier transform of u)

We can see that in fact the Fourier-Laplace transform is complex analytic since it satisfies the Cauchy-Riemann conditions part

part z u= 0 since

part

part z ilangu(x) eminusizmiddotxrang=

994902u(x)

part

part z ieminusizmiddotx994905= 0

since z 991596rarr eminusizmiddotx is analytic and so has partpart z i

eminusizmiddotx = 0 for each i = 1 n

We also have a nice growth bound on u(z)

|u(z)|le C983131

|α|leN

supK

994802994802part αx994790eminusizmiddotx994791994802994802

for some compact K sub 995014n and C N ge 0 This comes from the semi-norm definition of 994964 prime(995014n)

Lemma 44 Suppose u isin 994964 prime(995014n) has supp(u) sub Bδ(0) Then exist constants C N ge 0 such that

|u(z)|le C(1+ |z|)N eδ|Im(z)| for all z isin 994999n

Proof Fix ϕ isin Cinfin(995014) such that ϕ equiv 1 for τge minus12 and ϕ equiv 0 on τle minus1 Then define ϕ983171 for 983171 gt 0by

ϕ983171(x) = ϕ (983171(δminus |x |)) for x isin 995014n

Then we have ϕ983171 isin 994963(995014n) and

FIGURE 6 A sketch of ϕ

ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171

Thus in particular ϕ983171 equiv 1 on supp(u) So we have

u(z) = langu(x) eminusizmiddotxrang= langu(x)ϕ983171(x)eminusizmiddotxrang

48


So by the semi-norm estimate

|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e

minusizmiddotx994791994802994802

where K983171 = supp(ϕ983171) Then on K983171

|part βx ϕ983171|≲ 983171|β |

and 994802994802part αx (eminusizmiddotx)994802994802le |z||α|eIm(z)middotx le |z||α|e(δ+ 1

983171 )|Im(z)|

and so by Liebniz we have

|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|

Now choose 983171 = |z| and then combining we get

|u(z)|≲ (1+ |z|)N eδ|Im(z)|

for any z isin 994999n

So in particular we see from Lemma 44 and the proceeding discussion that if u isin 994964 prime(995014n) andsupp(u) sub Bδ then u(z) is entire (ie analytic for all z isin 994999n) and

|u(z)|≲ (1+ |z|)N eδ|Im(z)|

The Paley-Wiener-Schwartz theorem is about the converse to this

Theorem 45 (Paley-Wiener-Schwartz) We have the following

(A) If u isin 994964 prime(995014n) and supp(u) sub Bδ(0) then u(z) is entire and obeys the estimate

(dagger) |u(z)|≲ (1+ |z|)N eδ|Im(z)|

for all z isin 994999n for some N ge 0

Conversely if U = U(z) is an entire function and obeys (dagger) then U = u for someu isin 994964 prime(995014n) with supp(u) sub Bδ(0)

(B) If u isin 994963(995014n) and supp(u) sub Bδ(0) then u(z) is entire and obeys the estimate

(Dagger) |u(z)|≲m (1+ |z|)minusmeδ|Im(z)| for all m= 0 1 2 z isin 994999n

Conversely if U = U(z) is entire and satisfies (Dagger) then U = u where u isin 994963(995014n) andsupp(u) sub Bδ(0)

Proof (B) For u isin 994963(995014n) then we know (as before) that

u(z) =

983133

995014n

eminusi x middotzu(x) dx

49


is an entire function of z isin 994999n and

|zαu(z)|=994802994802994802994802983133

995014n

u(x)Dαx994790eminusizmiddotx994791 dx

994802994802994802994802

=

994802994802994802994802983133

995014n

eminusizmiddotx Dαu dx

994802994802994802994802 integrating by parts

le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|

where m= |α| This then implies that |u(z)|≲m (1+ |z|)minusmeδ|Im(z)| for all z isin 994999

For the converse if U is as stated then U = U(λ) (λ isin 995014n) decays rapidly as |λ|rarrinfin λ isin 995014n andso by Fourier inversion we have U = u where

u(x) =1

(2π)n

983133

995014n

eiλmiddotx U(λ) dλ

By differentiating under the integral and using dominated convergence we have u isin Cinfin(995014n)

Then by Cauchyrsquos theorem and (Dagger) we have (using Cauchyrsquos theorem to change the line we areintegrating over) 983133

995014n

eiλmiddotx U(λ) dλ =

983133

995014n

ei(λ+iη)middotx U(λ+ iη) dλ

So

|u(x)|≲994802994802994802994802983133

995014n

U(λ+ iη)ei x middot(λ+iη) dλ

994802994802994802994802

≲m

983133

995014n

(1+ |λ+ iη|)minusmeδ|η|eminusx middotη dλ

≲m eδ|η|minusx middotη

Now take η = t x|x | to get |u(z)| ≲m eδtminust|x | = et(δminus|x |) Now if |x | gt δ we can take t rarrinfin todeduce u(x) = 0 Hence supp(u) sub Bδ(0)

FIGURE 7 An illustration of how we use Cauchyrsquos theorem here As the integralsalong the vertical sides rarr 0 as we take the width to infin we get for f such that| f | rarr 0 sufficiently rapidly in the horizontal directions

983125Im(λi)=c f (λi) dλi =983125

Im(λi)=cprime f (λi) dλi

50


(A) We have already seen in Lemma 44 that u(z) is entire and (dagger) is satisfied if u is as stated

For the converse suppose that U is entire and satisfies (dagger) Then in particular

|U(λ)|le C(1+ |λ|)N for all λ isin 995014n

and so U isin Sprime(995014n) So by Theorem 32 (Fourier transform is an isomorphism on Sprime(995014n)) we haveexistu isin Sprime(995014n) with U = u

Now fix ϕ isin 994963(995014n) such that983125995014n ϕ dx = 1 and supp(ϕ) sub B1(0) Then for 983171 gt 0 set ϕ983171(x) =

983171minusnϕ(x983171) so that supp(ϕ983171) sub B983171(0) and ϕ983171 rarr δ0 in Sprime(995014n) Then set u983171 = u lowastϕ983171 [The idea hereis to approximate and use (B)]

Then (by a question on Example Sheet 2) u983171(z) = u(z)ϕ983171(z) = U(z)ϕ983171(z) for z isin 994999n Thus we seethat u983171 is entire and

|u983171(z)|≲m C(1+ |z|)N eδ|Im(z)| middot (1+ |z|)minusme983171|Im(z)|

for some N ge 0 and each m = 0 1 2 (from (B)) So by taking m = N +mprime mprime = 0 1 2 wesee that

|u983171(z)|≲mprime (1+ |z|)minusmprimee(δ+983171)|Im(z)| for all mprime = 0 1 2

So hence u983171 isin 994963(995014n) and supp(u983171) sub Bδ+983171(0) by (B) Then since u983171 rarr u in Sprime(995014n) as 983171 darr 0 wededuce from this by taking 983171 darr 0 that supp(u) sub Bδ(0) So done

We finish looking at the Paley-Wiener-Schwartz theorem with an example of its use

Example 43 Suppose we have ( fm)Nm=1 which are entire functions fm 994999nrarr 994999 Suppose that wehave 994802994802ei ymmiddotz fm(z)

994802994802≲m (1+ |z|)me|Im(z)|m+1

for m= 1 N and ym isin 995022n

Question Are the ( fm)m linearly independent

Answer The bound we have on the fm looks very similar to the bounds in the Paley-Wiener-Schwartz theorem except we have extra factors of ei ymmiddotz on the LHS However these just correspond

to Fourier transforms of the translated fm ie994925(τminusymfm)(z) = ei ymmiddotz fm(z)

So hence if fm = um then the Paley-Wiener-Schwartz theorem gives supp(um) sub B 1m+1(ym)

Hence if we had983123

mαm fm(z) = 0 then by taking the Fourier transform this is true if and only if983131

m

αmum = 0

But then by the above the um have disjoint supports and so this is true if and only if αm = 0 for allm So hence the ( fm)m are linearly independent

[Pictorially we have a lattice with the supports being disjoint balls centred at the lattice points]

51


5 OSCILLATORY INTEGRALS

At the start of the course we considered ldquostrangerdquo integrals like983133 infin

minusinfineiλx dx

This is an example of an oscillatory integral and it is not to be interpreted in terms of RiemannLebesgueintegrability We will look at objects of the form

983133

995014k

eiΦ(x θ )a(x θ ) dθ

where (x θ ) isin X times 995014k for X sub 995014 open Here Φ is called the phase function and a is called thesymbol

We will allow a(x θ ) to grow as |θ |rarrinfin eg we could take a(x θ ) = p(θ ) for some polynomialp

Key idea Just like alternating series donrsquot need as much decay to converge(xvii) we can the oscillationand integration by parts to control the integral and hopefully ldquocancel outrdquo the growth of a

For example for ϕ isin 994963(995014) we know that983133

995014eiλθϕ(θ ) dθ =

983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ

where L = 1iλ middot partpart θ is a differential operator which has L(eiλθ ) = eiλθ So on integrating by parts

983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133

995014(Llowast)(N) [ϕ] eiλθ dθ

= 994974 (λminusN )

as Llowast = minus 1iλpartpart θ So we see that we have rapid decay here as λrarrinfin since N isin 995010 was arbitrary

More generally considering983125995014 eiλΦ(θ )ϕ(θ ) dθ for Φ isin Cinfin(995014) we can play the same game we have983133

995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d

dθ(let us just assume for the moment that Φprime ∕= 0 on supp(ϕ) so that L is well-defined here) Againintegrating by parts N times using the fact that Llowast = minus d

dθ

9948771

iλΦprime(θ )

994880 we get

983133

995014eiλΦ(θ )ϕ(θ ) dθ = 994974 (λminusN )

So what we are seeing is that ldquooscillation implies decayrdquo and lots of it if Φprime ∕= 0

(xvii)Consider 1minus 12 +

13 minus 1

4 + middot middot middot vs 11 +

12 +

13 +

14 + middot middot middot

52


FIGURE 8 These two illustrations demonstrate the type types of behaviour we can see Inthe top image we see that Φ larger and larger (ie period smaller and smaller) the bumpslabelled 1 those labelled 2 etc cancel each other out more and more as the peaks are thenon smaller and smaller intervals and ϕ does not change so much on them This means thatboth the peaks will be around the same size and so cancel In the second image we see whathappens when Φprime = 0 somewhere - the shape of eiλΦ(θ )ϕ(θ ) (or at least the realimaginaryparts of) flattens out at these points and so we donrsquot have the cancellation as before In somesense there is a double zero for Φ here (note that d

dθ (eiλΦϕ) = ϕprimeeiλΦ+ iλΦprimeeiλΦϕ) since both

Φ and Φprime are zero at such a point So we need to check the integral about these points stilldecay as λrarrinfin This is what stationary phase says

Lemma 51 (Stationary Phase) Let Φ isin Cinfin(995014) have Φ(0) = Φprime(0) = 0 (wlog by translating)with Φprimeprime(0) ∕= 0 and Φprime(θ ) ∕= 0 for θ isin 9950140 Then for ϕ isin 994963(995014) we have983133

995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12

994867as λrarrinfin

Note Before when Φprime ∕= 0 we saw that we got 994974 (λminusN ) decay for arbitrary N So these points ofstationary phase are bad indeed but we can get some control

Proof Fix ρ isin 994963(995014) with ρ equiv 1 on |θ |lt 1 and ρ equiv 0 on |θ |gt 2 Then for δ gt 0 write983133

995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)

then since ρ(θδ) = 0 on |θ |gt 2δ we have I1(λ) = 994974 (δ)

53


Now note that since 1 minus ρ(θδ) = 0 on |θ | lt δ we can integrate by parts in I2(λ) (since we canwrite the integral as one over |θ | gt δ and we can integrate by parts everywhere where Φprime ∕= 0) Sosetting L = 1

iλΦprime(θ )d

dθ upon integrating by parts (since L[eiλΦ] = eiλΦ) we have (there are never anyboundary terms due to compact support)

I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ

994878(1minusρ(θδ))ϕ(θ )

Φprime(θ )

994881

=1iλmiddotΦprimeprime(θ )9947901minusρ994790θδ

994791994791ϕ(θ )

Φprime(θ )2+

1iλmiddot 1δmiddotρprime994790θδ

994791ϕ(θ )

Φprime(θ )minus 1

iλmiddot9947901minusρ994790θδ

994791994791ϕprime(θ )

Φprime(θ )

Note

Φprime(θ ) = Φprime(θ )minusΦprime(0)983167983166983165983168=0 by assumption

=

983133 θ

0

Φprimeprime(τ) dτ= θ

983133 1

0

Φprimeprime(τθ ) dτ

and soΦprime(θ )θ=

983133 1

0


So since Θprimeprime(θ ) ∕= 0 on a neighbourhood of 0 (as Φprimeprime(0) ∕= 0 and by continuity of Φprimeprime) we see that

|θ ||Φprime(θ )| ≲ 1

near 0 Now since supp(ϕ) is compact this gives that we have 1|Φprime(θ )| ≲

1|θ | on the entire range of

integration in I2 Hence

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868

since the third term does not matter

Then iterating this N times we see

(Llowast)(N) [middot middot middot ]≲ 1λNθN

9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN

since on I2 we have |θ |gt δ and so 1|θ | lt

1δ So

I2(λ)≲ λminusNδminusN

983133

|θ |gtδ|θ |minusN dθ ≲ λminusNδminus2N+1

since when we integrate xminusN we get C middot xminusN+1 So combining we see983133

995014eiλΦ(θ )ϕ(θ ) dθ ≲maxδλminusNδminus2N+1

To minimise the RHS we should choose δ such that δ = λminusNδminus2N+1 ie δ = λminus12 Hence983133

995014eiλΦ(θ )ϕ(θ ) dθ = 994974 (λminus12) as λrarrinfin

54


This lemma of stationary phase suggests that control of983125995014k eiΦ(x θ )a(x θ ) dθ will depend heavily on

|984162θ a(x θ )| where

984162θ a =994858part apart θ1

part apart θk

994868

Now let us look at defining oscillatory integrals properly via distributions

Definition 51 Let X sub 995014n be open Then a smooth function a X times995014k rarr 994999 is called a symbolof order N for for all K sub X compact and for all pairs of multi-indices (αβ) exist a constant C suchthat 994802994802994802Dαx Dβ

θa(x θ )994802994802994802le Clangθ rangNminus|β |

for (x θ ) isin K times995014k for K sub X compact We call the space of all such symbols Sym(X 995014k N)

Remark If a(x θ ) = P(θ ) for a polynomial P of degree N then a isin Sym(X 995014k N) Intuitivelyfrom the definition of a symbol we expect symbols to look like a polynomial in θ with coefficientsthat are smooth functions in x (although they are more general than this since a symbol can havenegative order)

We only care about the large |θ | behaviour of such symbols because we can always write983133

995014k

eiΦ(x θ )a(x θ ) dθ =

983133

995014k

eiΦ(x θ )a(x θ )ρ(θ ) dθ

983167 983166983165 983168classically well-defined

+

983133

995014k

eiΦ(x θ )a(x θ ) (1minusρ(θ )) dθ

983167 983166983165 983168oscillatory integral with symbol = 0 on |θ |ltR

where ρ isin 994963(995014k) has ρ equiv 1 on |θ | lt R So we only care about the symbol for |θ | gt R but R gt 0was arbitrary

Definition 52 A function Φ X times995014krarr 995014 is called a phase function if

(i) Φ is continuous and (positively) homogeneous of degree 1 in θ (ie Φ(x τθ ) = τΦ(x θ )for all τgt 0)

(ii) Φ is smooth on X times994790995014k0994791

(iii) dΦ =984162xΦ middot dx +984162θΦ middot dθ is non-vanishing on X times994790995014k0994791

Remark The reason behind these conditions for a phase function are as follows To start with wewant to try to make sense of

983125995014n eiλmiddotx ie here we have Φ(x θ ) = x middot θ So we try to prove results

for this Φ Then afterwards we look back at the proofs and ask ldquowhat properties of the Φ did weactually useneedrdquo Then we would see that the above conditions of a phase function are exactlythose [We will work with the general definition above and not a specific Φ but this is a reason forwhy the phase function definition is how it is]

The standard example of a phase function is Φ(x θ ) = x middot θ for (x θ ) isin 995014n times995014n To see that this isa phase function note that (i) (ii) are immediate and

dΦ = θ middot dx + x middot θθ

55


But dθ1 dθn dx1 dxn are linearly independent and so this is non-zero on 995014n times (995014n0)

Then the classical example of an oscillatory integral is983133

995014n

θα

(2π)nei x middotθ dθ for (x θ ) isin 995014n times995014n

A natural question is whether this equals Dαδ0(x) in the sense of distributions

Lemma 52 Let a isin Sym(X 995014k N) and ai isin Sym(X 995014k Ni) for i = 1 2 Then

(i) Dαx Dβθ

a isin Sym(X 995014k N minus |β |) ie only derivatives in θ change order of the symbol(ii) a1a2 isin Sym(X 995014k N1 + N2)

Proof (i) If a isin Sym(X 995014k N) then Dαx Dβθ

a is smooth and994802994802994802Dαprimex Dβ

prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime

θa994802994802994802≲αprimeβ primeK langθ rangNminus|β |minus|β

prime|

for K sub X compact So Dαx Dβθ

a isin Sym(X 995014k N minus |β |)

(ii) Now if ai isin Sym(X 995014k Ni) for i = 1 2 then a1a2 is smooth and

994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime

θa1 middot Dαminusα

primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ

994802994802994802Dαprimex Dβprime

θa1

994802994802994802 middot994802994802994802Dαminusαprimex Dβminusβ

prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131

β primeleβlangθ rangN1minus|β prime|langθ rangN2minus(|β |minus|β prime|)

≲αβ K langθ rangN1+N2minus|β |

and thus we see a1a2 isin Sym(X 995014k N1 + N2)

Now to define oscillatory integrals we could define983125995014k eiΦ(x θ )a(x θ ) dθ as a linear form on 994963(X )

via

ϕ 991596minusrarr langIΦ(a)ϕrang =983133

995014k

994808983133

XeiΦ(x θ )a(x θ )ϕ(x) dx

994809dθ

for ϕ isin 994963(X )

However this definition is cumbersome due to the lack of absolute convergence in the double integral(as there is no decay guaranteed in the θ -direction The x-integral is fine) So instead we define

langIΦ(a)ϕrang = lim983171darr0langIΦ983171(a)ϕrang

56


where

IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ ) dθ

where χ isin 994963(995014k) has χ equiv 1 on |θ |lt 1

Using this we can interchange the orders of integration etc before taking 983171 darr 0

Theorem 51 (Oscillatory integrals are distributions of finite order) For Φ a phase function anda isin Sym(X 995014k N) the oscillatory integral IΦ(a) = lim983171darr0 IΦ983171(a) defines an element of 994963 prime(X ) oforder le N + k+ 1

Proof Later

Remark We can actually improve this result to get that the order is le N + k but we wonrsquot discussthis

To establish Theorem 51 we need an integration by parts trick (similar to that in Lemma 51Sta-tionary Phase) to get more control of the integral as |θ |rarrinfin

Lemma 53 If a differential operator L has the form

(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )

where a j isin Sym(X 995014k 0) and b j c isin Sym(X 995014kminus1) then the formal adjoint Llowast hastakes thesame form

Note This is just like how in Lemma 51stationary phase we had L = 1iλΦprime

ddθ

Proof The formal adjoint of such an L is

Llowast = minusk983131

j=1

part

part θ j

994792a j(x middot) bull994793minus

n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c

where we can see that a j = minusa j isin Sym(X 995014k 0) b j = minusb j isin Sym(X 995014kminus1) and

c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j

part x jisin Sym(X 995014kminus1)

since c isin Sym(X 995014kminus1)part a j

part θ jisin Sym(X 995014kminus1) by Lemma 52(i) and

part b j

part x jisin Sym(X 995014kminus1) by

Lemma 52(i) Hence this takes on the same form and so we are done

57


Note By ldquoformal adjointrdquo we just mean the differential operator which obeys the integration byparts rule 983133

(L f ) middot g =983133

f middot (Llowastg)

It is not the adjoint with respect to the L2-inner product (where complex conjugates get involved)

Lemma 54 For a given phase function Φ existL of the form of (983183) in Lemma 53 such that

Llowast994792eiΦ994793= eiΦ

Aside What is the point of having L as the form of (983183) The point is to get absolute integrability inthe double integral983133

995014k

983133

XeiΦ(x θ )a(x θ )χ(983171θ )ϕ(x) dxdθ =

983133

995014k

983133

X

994790LlowasteiΦ994791

a(x θ )χ(983171θ )ϕ(x) dxdθ

=

983133

995014k

983133

XeiΦL (a(x θ )χ(983171θ )ϕ(x)) dxdθ

Now a(x θ )χ(983171θ )ϕ(x) is like a symbol of order N So what does L do to a symbol of order NWell acting by part

part θ jlowers the order by 1 whilst multiplying by a j does not change the order Hence

a jpartpart θ j

lowers the order by 1 Similarly acting by partpart x j

does not change the order whilst multiplying

by b j lowers the order by 1 Hence acting by b jpartpart x j

lowers the order by 1 Hence L lowers the orderby 1 and so we expect L(a(x θ )χ(983171θ )ϕ(x)) to act like a symbol of order N minus 1

But then we could repeat this to get983133

995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168

symbol of order N minusM

dxdθ

and so if we keep doing this until N minus M lt 0 we get absolute integrability and so can interchangethe order of integration etc

Proof of Lemma 54 Note thatpart

part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ

Then note994822minusi|θ |2

k983131

j=1

partΦ

part θ jmiddot partpart θ jminus i

n983131

j=1

partΦ

part x jmiddot partpart x j

994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1

π(x θ )eiΦ(x θ )

Now note that since Φ(x θ ) is positively homogeneous of degree 1 in θ we have for τ gt 0 (aconstant)

τpart

part x jΦ(x θ ) =

part

part x jΦ(x τθ ) =

partΦ

part x j(x τθ )

58


and so partΦpart x j

is positively homogeneous of degree 1 in θ Also

τpart

part θ jΦ(x θ ) =

part

part θ jΦ(x τθ ) = τ

partΦ

part θ j(x τθ )

and so partΦpart θ j

is positively homogeneous of degree 0 in θ So

(x θ ) 991596minusrarr |θ |2 middot |984162θΦ|2 + |984162xΦ|2

is positively homogeneous of degree 2 in θ since the first term on the RHS is a product of a termof homogeneous degree 2 and one of degree 0 whilst the one of the right is the square of a termof homogeneous degree 1 and so also has degree 2 Hence π(x θ ) is homogeneous of degree -2Then define

L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096

By the above we know that LeiΦ = eiΦ but π(x θ ) could be singular at θ = 0 (ie if |984162xΦ| = 0 atθ = 0 which isnrsquot ruled out by the assumption on Φ that dΦ ∕= 0 on X times

994790995014k0994791)

So to deal with this fix ρ isin 994963(995014k) with ρ = 1 on |θ |lt 1 Set

Llowast = (1minusρ(θ )) L +ρ(θ )Then we have

LlowasteiΦ = (1minusρ(θ ))eiΦ +ρ(θ )eiΦ = eiΦ

and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where

a j = minusi|θ |2(1minusρ(θ )) partΦpart θ j

|θ |2 middot |984162θΦ|2 + |984162xΦ|2is a smooth function (as we have removed the problem at θ = 0 with the 1minus ρ(θ ) factor) and ishomogeneous of degree 0 by construction Moreover we have

b j =minusi(1minusρ(θ )) partΦpart x j

|θ |2 middot |984162θΦ|2 + |984162xΦ|2which is smooth and homogeneous of degree minus1 since the numerator has degree 1 and the denom-inator has degree 2 Finally we have

c = ρ(θ )So since a j is homogeneous of degree 0 on |θ |gt 1 and smooth we have a j isin Sym(X 995014k 0) Similarlyb j c isin Sym(X 995014kminus1) So by Lemma 53 we have L = (Llowast)lowast has the same form and so we are done

Note By definition of L if a isin Sym(X 995014k N) then we have already argued that

L(a) isin Sym(X 995014k N minus 1)

ie L lowers the order by 1 for a symbol More generally if ϕ is smooth we have

L(M) [a(x θ )ϕ(x)] =983131

|α|leM

aα(x θ )part αϕ(x)

59


where aα isin Sym(X 995014k N minusM) This can be proved via induction on M

Proof of Theorem 51 By definition we have IΦ(a) = lim983171darr0 IΦ983171(a) where

IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )

and for ϕ isin 994963(X )langIΦ983171(a)ϕrang=983133

995014k

983133

XeiΦ(x θ )a(x θ )χ(983171θ )ϕ(x) dxdθ

By Lemma 54 we know existL of the form of (983183) in Lemma 53 such that LlowasteiΦ = eiΦ Using this (Mtimes) and integrating by parts we see that

langIΦ983171(a)ϕrang=983133

995014k

983133

XeiΦ(x θ )L(M) [a(x θ )χ(983171θ )ϕ(x)] dxdθ

Now observe that 994802994802994802994802994858part

part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )

≲ 983171|α|lang983171θ rangminus|α|

= Cα middot983171|α|

(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2

≲α langθ rangminus|α|

for 983171 isin (0 1] where the second line is true since χ is smooth of compact support and so |part αχ(θ )|≲αlangθ rangminus|α| Moreover the last line holds uniformly for 983171 isin (0 1] So hence this tells us that uniformlyin 983171 we can treat θ 991596rarr χ(983171θ ) as a symbol of order 0 Hence

a(x θ )χ(983171θ ) isin Sym(X 995014k N)

uniformly in 983171 isin (0 1] So

L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM

aα(x θ 983171)part αϕ(x)

uniformly in 983171 where aα isin Sym(X 995014k N minus M) Now if we choose M such that N minus M lt minusk thenthe aα are integrable Thus by dominated convergence

langIΦ(a)ϕrang= lim983171darr0

983133

995014k

983133

XeiΦ(x θ )983131

|α|leM

aα(x θ 983171)part αϕ(x) dxdθ

=983131

|α|leM

983133

995014k

983133

XeiΦ(x θ )aα(x θ )part αϕ(x) dxdθ

where aα(x θ )equiv aα(x θ 0) Then if supp(ϕ) sub K in X we have

|langIΦ(a)ϕrang|le983131

|α|leM

983133

995014k

983133

X|aα(x θ )| middot |part αϕ| dxdθ

≲K

983131

|α|leM

supK|part αϕ|

60


as the |aα| are integrable by construction Then since we just need M gt N +k Taking M = N +k+1this shows that ord(IΦ(a))le N + k+ 1 since the above is the semi-norm estimate for a distributionHence we are done

So now we know what983125995014k eiΦa(x θ ) dθ is But what are the (distributional) derivatives of this Do

they agree with what we might naiumlvely expect Well we have994902part IΦ(a)part x i

ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133

XeiΦa(x θ )χ(983171θ )

part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )

994793middotϕ dxdθ via integrating by parts

= lim983171darr0

983133

995014k

983133

XeiΦ994878part apart x i

+ iapartΦ

part x i

994881χ(983171θ )ϕ dxdθ (dagger)

= lim983171darr0

983133

995014k

983133

XeiΦa(x θ )χ(983171θ )ϕ dxdθ

= langIΦ(a)ϕrang

where a = part apart x i+ ia partΦpart x i

isin Sym(X 995014l N minus 1) and in (dagger) we are above to pull out χ(983171θ ) from the

derivative since it does not depend on x So hence we see part IΦ(a)part x i

= IΦ(a) as distributions whichsays

part

part x i

983133

995014k


983133

995014k

eiΦ994878part apart x i

+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ

ie we can just differentiate under the integral sign as we might have naiumlvely expected So Physicistswere fine all along

Example 51 (An Oscillatory Integral) Consider

IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ

for (x θ ) isin 995014n times995014n ie a equiv 1 Φ(x θ ) = x middot θ Then by our definition we have

langIΦ(a)ϕrang = lim983171rarr0

1(2π)n

983133

995014n

983133

995014n

ei x middotθϕ(x)χ(983171θ ) dxdθ

= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n

ei x middot(θ983171)ϕ(x)983171minusnχ(θ ) dθdx (setting θ prime = 983171θ)

= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n

ei x middotθϕ(983171x)χ(θ ) dθdx (setting x prime = x983171)

61


= lim983171rarr0

983133

995014n

ϕ(983171x)994792(2π)minusnχ(minusx)994793

dx by definition of the Fourier transform

= ϕ(0)χ(0) by the inverse Fourier transform

= ϕ(0)

since 1= χ(0) = 1(2π)n983125995014n χ(minusx) dx Hence

langIΦ(a)ϕrang= ϕ(0) = langδ0ϕrangie

δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ

51 Singular Supports

It is natural to ask when IΦ(a) coincides with a smooth function We use the idea of singular sup-ports

Definition 53 For a distribution u isin 994963 prime(X ) the singular support of u denoted sing supp(u) isthe complement of the largest open set on which u can be identified with a smooth function

Example 52 Consider δ0 isin 994963 prime(995014n) Then for ϕ isin 994963(995014n0) we have langδ0ϕrang= 0 and so as theconstant function 0 is a smooth function we see that δ0 equiv 0 on995014n0 Hence sing supp(δ0) sub 0

By the stationary phase lemma (Lemma 51) we expect IΦ(a) to be better at x0 isin X for which984162θΦ(x0θ ) ∕= 0 for any θ isin 995014k0 The reason we are not interested in θ = 0 is because we areonly interested in the large θ behaviour (as we saw before by splitting the integral into two partswith ρ(θ ) and 1minusρ(θ )) This is seen by the following lemma

Lemma 55 For ρ isin 994963(995014k) the function

x 991596minusrarr983133

995014k


is smooth

Proof This follows from the definitions and the dominated convergence theorem

62


Equipped with this we can always write

IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k

eiΦ(x θ ) a(x θ )(1minusρ(θ ))983167 983166983165 983168=a(x θ )

dθ

So if we choose ρ isin 994963(995014k) with ρ equiv 1 on |θ |lt 1 then a isin Sym(X 995014k N) if a isin Sym(X 995014k N) anda(x θ ) = 0 on |θ |lt 1

So if there is any part of the oscillatory integral which is not smooth (ie truly a distribution) thenit comes from the second part of the integral above which vanishes on a neighbourhood of 0 So wecan assume wlog a is 0 near 0

So just like our intuition above about when oscillatory integrals are nice our main result is

Theorem 52 For an oscillatory integral IΦ(a) we have

sing supp(IΦ(a)) sub x 984162θΦ(x θ ) = 0 for some θ isin 995014k0

Remark Thus if 984162θΦ(x θ ) ∕= 0 for any θ isin 995014k0 then IΦ(a) is smooth about x

Proof We will prove the converse (ie the opposite inclusion for the complements) Wlog as abovewe can assume that a = 0 on |θ |lt 1

Suppose that x0 isin X has 984162θΦ(x0θ ) ∕= 0 for all θ isin 995014k0 Now since Φ(x θ ) is homogeneousof degree 1 in θ we see that |984162θΦ(x θ )| is homogeneous of degree 0 in θ So |984162θΦ(x θ )| isdetermined by its values on any sphere |θ | = R R gt 0 So by compactness |984162θΦ(x θ )| is boundedbelow on |θ |= R (for each x fixed) and so

|984162θΦ(x0θ )|≳ 1

for all θ isin 995014k0 Then by continuity the same estimate holds for x isin B983171(x0) for some 983171 gt 0 small

Now fix ψ isin 994963(X ) with supp(ψ) sub B983171(x0) Then define

Llowast = minusi983123k

j=1partΦpart θ jmiddot partpart θ j

|984162θΦ|2

which is similar to the Lrsquos we used before in Lemma 53 (except we have removed an xminusderivativesas we donrsquot want them to hit ψ as we shall see)

Then we know Llowast is well-defined on supp(a) where a(x θ ) = a(x θ )ψ(x) Now

ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63


So since984162θΦ is homogeneous of degree 0 in θ since a isin Sym(X 995014k N)we have L(a) isin Sym(X 995014k Nminus1) and so by definition

langψIΦ(a)ϕrang = lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133

XeiΦ(x θ )L(M) [a(x θ )χ(983171θ )ϕ(x)] dxdθ integrating by parts and Llowast[eiΦ] = eiΦ

= lim983171rarr0

983133

995014k

983133

XeiΦ(x θ )L(M) [a(x θ )χ(983171θ )]ϕ(x) dxdθ

where in the last line we have used that L has no xminusderivatives and so we can pull ϕ out of the L-term Now choose M gt 0 sufficiently large so that L(M) [a(x θ )] is a symbol of sufficiently negativedegree so that this integral is absolutely integrable So then we can exchange the order of integrationand so we get

langψIΦ(a)ϕrang=983133

X

994808983133

995014k

eiΦ(x θ )L(M) [a(x θ )] dθ

994809ϕ(x) dx

= lang(middot middot middot )ϕrang

and so we see by the arbitrariness of ϕ

ψIΦ(a) =

983133

995014k


Now since this was true for M arbitrary (sufficiently large) we can differentiate under the integraland conclude that IΦ(a) is smooth on a neighbourhood of x

Example 53 By previous calculations we know that if δ0(x) =1

(2π)n983125995014n ei x middotθ dθ has

Dαδ0(x) =1

(2π)n

983133

995014k

θαei x middotθ dθ

So applying Theorem 52 we see

sing supp(Dαδ0) sub x 984162θ (x middot θ )983167 983166983165 983168=x

= 0 for some θ isin 995014k0= x = 0

What else can we do with these oscillatory integrals To finish the course we shall show how theycan be used to solve PDEs

Example 54 Consider the PDE994816part upart t + c middot984162yu= 0 for (y t) isin 995014n times (0infin)u(x 0) = δ0(x)

64


for some constant c Then we guess via taking the Fourier transform and then solving the equationand taking the inverse Fourier transform that the solution is

u(y t) =1

(2π)n

983133

995014n

eiθ middot(yminusc t) dθ

Then in the usual notation here we have X = 995014n times [0infin) x = (y t) So

Φ(x θ )equiv Φ(y tθ ) = θ middot (y minus c t)

Then we can check thatpart upart t+ c middot984162yu=

1(2π)n

983133

995014n

eiθ middot(yminusc t) [minusc middot θ + c middot θ] dθ = 0

and so this solves the equation and formally

u(y 0) =1

(2π)n

983133

995014n

ei ymiddotθ dθ = δ0(y)

ie u(y t)rarr δ0(y) as t rarr 0 in 994963 prime(995014n) Then by Theorem 52 we have

sing supp(u) sub (y t) 984162θ (θ middot (y minus c t)) = 0 for some θ isin 995014k0= (y t) y = c tand so we see that the singularity propagates with time This is perhaps not unexpected as it is awave but it is nice to see this mathematically

End of Lecture Course

65


Contents

0 Motivation 2

1 Distributions 5


12 Limits in 994963 prime(X ) 8

13 Basic Operations 9

131 Differentiation and Multiplication by Smooth Functions 9

132 Reflection and Translation 11

133 Convolution between 994963 prime(995014n) and 994963(995014n) 13

14 Density of 994963(995014n) in 994963 prime(995014n) 15

2 Distributions with Compact Support 18


22 Convolution between 994964 prime(995014n) and 994963 prime(995014n) 21

3 Tempered Distributions and Fourier Analysis 23

31 Functions of Rapid Decay 23

32 Fourier Transform on S(995014n) 24

33 Fourier Transform of Sprime(995014n) 28

34 Sobolev Spaces 30

4 Applications of the Fourier Transform 32

41 Elliptic Regularity 32

42 Fundamental Solutions 37

43 The Structure Theorem for 994964 prime(X ) 44

44 The Paley-Wiener-Schwartz Theorem 46

5 Oscillatory Integrals 51

1



2


0 MOTIVATION







hrarr0

983133 infin

minusinfin


middot f (x) dx

= limhrarr0

1h

983133 infin


= limhrarr0


= minus f prime(x0)



983133 infin


prime(x) dx



f (λ) =

983133 infin




minusinfin

994802994802eminusiλx f (x)994802994802 dx =

983133 infin



δ(λ) =1

2π

983133 infin



1(λ) = 2πδ(λ)



983133 infin

minusinfin

994858i

ddλ


id

dλ

994868n[2πδ(λ)] = 2π(i)nδ(n)(λ)

3





983133 infin



983133 infin

minusinfinf g(λ) =

983133 infin

minusinfinλ983167983166983165983168

= f (λ)

g(λ) dλ




(983183) p =1c2


2ppart x2

= 0


0=

983133

9950142



0=

983133

9950142

p(x t) f dxdt


0=

983133

9950142




4








5


1 DISTRIBUTIONS








983133


983133




ϕ(x + h) =983131

|α|leN

hα


where RN (x h) = o994790|h|N+1994791





|α|leN

supX|part αϕ|




6







) = 0


langTϕrang =983131

|α|leM

983133





|α|leM

983133


=983131

|α|leM

983133


le983131

|α|leM


K| fα(x)| dx

le994808

max|α|leM

983133

K| fα(x)| dx

994809middot983131

|α|leM

supK|part αϕ|


983125K | fα(x)| dx and N = M






Xg middotϕ dx


languϕrang =983133

995014xϕ(x) dx

7




m=0

ϕ(m)(m)



m=0


m=0

sup |ϕ(m)|









|α|leN





|α|lem

sup |part αϕm|




|α|lem

sup |part αϕm|

8


ie 983131

|α|lem
















langumϕrang=983133


=1m

983133


9



|langumϕrang|le 1m

983133



ldquo limmrarrinfin


13 Basic Operations





= (minus1)|α|983133











10




H(x) =

9948160 if x le 01 if x gt 0




= minus983133 infin

0

ϕprime(x)


= ϕ(0)








1=

983133




+ lang1ϕrangθ0983167 983166983165 983168=ϕB

11



ψA(x) =

983133 x

minusinfinϕA(y) dy


Claim ψA isin 994963(995014)



















12





languϕrang=983133

995014n

u(minusx)ϕ(x) dx =

983133

995014n








|h|





i

hipart ϕ




u983131

i

hi

|h| middotpart ϕ

part x i

994906+ o(1)

= minus983131

i

hi

|h|

994902upart ϕ

part x i

994905+ o(1)

minusrarr983131

i

mi

994902upart ϕ

part x i

994905=

994903983131

i

mipart upart x i

ϕ



13







995014n


=

983133

995014n


=

983133

995014n

u(y)(τx ϕ)(y) dy

= languτx ϕrang












i

hipart ϕ



14





u983131

i

hipartΦx

part x i

994906+ o(|h|)

=983131

i

hi

994902upartΦx

part x i

994905+ o(|h|)


part


upartΦx

part x i

994905








15




(u lowastδn)(x)










=


=


=




= limhrarr0

983131

misin995022n


= limhrarr0

994903u(z)983131

misin995022n


994906(983183)



16



misin995022n





misin995022n




(983183) =

994903u(z) lim

hrarr0

983131

misin995022n


994906



(iv)







χ(x) =

9948161 on |x |lt 10 on |x |gt 2





17




994793(0)


(983183)




=


=

983133ψ(y)χ994806 y

m2minus x

m

994807θ994806 y

mminus x994807


= θ (minusx) +

983133ψ(y)994844χ994806 y

m2minus x

m

994807θ994806 y


dy


= θ (x) + Rm(x)


R(x) =

983133ψ(y)994844χ994806 y

m2+

xm

994807θ994806 y

m+ x994807minus θ (x)994845

dy





18

















19




|α|leN













20




|α|leN

supsupp(u)capK

|part α(ρϕ)|= C983131

|α|leN


|α|leN

supKpart αϕ





|α|leN

supK|part αϕ|






|α|leN

supK|part α(ρϕ)|


994798supK |part αρ|994799

to get


|α|leN

supK|part αϕ|



ϕ = ρϕ983167983166983165983168=ϕ0

+(1minusρ)ϕ983167 983166983165 983168=ϕ1




= langvϕrang



21



















22






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65



2


0 MOTIVATION







hrarr0

983133 infin

minusinfin


middot f (x) dx

= limhrarr0

1h

983133 infin


= limhrarr0


= minus f prime(x0)



983133 infin


prime(x) dx



f (λ) =

983133 infin




minusinfin

994802994802eminusiλx f (x)994802994802 dx =

983133 infin



δ(λ) =1

2π

983133 infin



1(λ) = 2πδ(λ)



983133 infin

minusinfin

994858i

ddλ


id

dλ

994868n[2πδ(λ)] = 2π(i)nδ(n)(λ)

3





983133 infin



983133 infin

minusinfinf g(λ) =

983133 infin

minusinfinλ983167983166983165983168

= f (λ)

g(λ) dλ




(983183) p =1c2


2ppart x2

= 0


0=

983133

9950142



0=

983133

9950142

p(x t) f dxdt


0=

983133

9950142




4








5


1 DISTRIBUTIONS








983133


983133




ϕ(x + h) =983131

|α|leN

hα


where RN (x h) = o994790|h|N+1994791





|α|leN

supX|part αϕ|




6







) = 0


langTϕrang =983131

|α|leM

983133





|α|leM

983133


=983131

|α|leM

983133


le983131

|α|leM


K| fα(x)| dx

le994808

max|α|leM

983133

K| fα(x)| dx

994809middot983131

|α|leM

supK|part αϕ|


983125K | fα(x)| dx and N = M






Xg middotϕ dx


languϕrang =983133

995014xϕ(x) dx

7




m=0

ϕ(m)(m)



m=0


m=0

sup |ϕ(m)|









|α|leN





|α|lem

sup |part αϕm|




|α|lem

sup |part αϕm|

8


ie 983131

|α|lem
















langumϕrang=983133


=1m

983133


9



|langumϕrang|le 1m

983133



ldquo limmrarrinfin


13 Basic Operations





= (minus1)|α|983133











10




H(x) =

9948160 if x le 01 if x gt 0




= minus983133 infin

0

ϕprime(x)


= ϕ(0)








1=

983133




+ lang1ϕrangθ0983167 983166983165 983168=ϕB

11



ψA(x) =

983133 x

minusinfinϕA(y) dy


Claim ψA isin 994963(995014)



















12





languϕrang=983133

995014n

u(minusx)ϕ(x) dx =

983133

995014n








|h|





i

hipart ϕ




u983131

i

hi

|h| middotpart ϕ

part x i

994906+ o(1)

= minus983131

i

hi

|h|

994902upart ϕ

part x i

994905+ o(1)

minusrarr983131

i

mi

994902upart ϕ

part x i

994905=

994903983131

i

mipart upart x i

ϕ



13







995014n


=

983133

995014n


=

983133

995014n

u(y)(τx ϕ)(y) dy

= languτx ϕrang












i

hipart ϕ



14





u983131

i

hipartΦx

part x i

994906+ o(|h|)

=983131

i

hi

994902upartΦx

part x i

994905+ o(|h|)


part


upartΦx

part x i

994905








15




(u lowastδn)(x)










=


=


=




= limhrarr0

983131

misin995022n


= limhrarr0

994903u(z)983131

misin995022n


994906(983183)



16



misin995022n





misin995022n




(983183) =

994903u(z) lim

hrarr0

983131

misin995022n


994906



(iv)







χ(x) =

9948161 on |x |lt 10 on |x |gt 2





17




994793(0)


(983183)




=


=

983133ψ(y)χ994806 y

m2minus x

m

994807θ994806 y

mminus x994807


= θ (minusx) +

983133ψ(y)994844χ994806 y

m2minus x

m

994807θ994806 y


dy


= θ (x) + Rm(x)


R(x) =

983133ψ(y)994844χ994806 y

m2+

xm

994807θ994806 y

m+ x994807minus θ (x)994845

dy





18

















19




|α|leN













20




|α|leN

supsupp(u)capK

|part α(ρϕ)|= C983131

|α|leN


|α|leN

supKpart αϕ





|α|leN

supK|part αϕ|






|α|leN

supK|part α(ρϕ)|


994798supK |part αρ|994799

to get


|α|leN

supK|part αϕ|



ϕ = ρϕ983167983166983165983168=ϕ0

+(1minusρ)ϕ983167 983166983165 983168=ϕ1




= langvϕrang



21



















22






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65


0 MOTIVATION







hrarr0

983133 infin

minusinfin


middot f (x) dx

= limhrarr0

1h

983133 infin


= limhrarr0


= minus f prime(x0)



983133 infin


prime(x) dx



f (λ) =

983133 infin




minusinfin

994802994802eminusiλx f (x)994802994802 dx =

983133 infin



δ(λ) =1

2π

983133 infin



1(λ) = 2πδ(λ)



983133 infin

minusinfin

994858i

ddλ


id

dλ

994868n[2πδ(λ)] = 2π(i)nδ(n)(λ)

3





983133 infin



983133 infin

minusinfinf g(λ) =

983133 infin

minusinfinλ983167983166983165983168

= f (λ)

g(λ) dλ




(983183) p =1c2


2ppart x2

= 0


0=

983133

9950142



0=

983133

9950142

p(x t) f dxdt


0=

983133

9950142




4








5


1 DISTRIBUTIONS








983133


983133




ϕ(x + h) =983131

|α|leN

hα


where RN (x h) = o994790|h|N+1994791





|α|leN

supX|part αϕ|




6







) = 0


langTϕrang =983131

|α|leM

983133





|α|leM

983133


=983131

|α|leM

983133


le983131

|α|leM


K| fα(x)| dx

le994808

max|α|leM

983133

K| fα(x)| dx

994809middot983131

|α|leM

supK|part αϕ|


983125K | fα(x)| dx and N = M






Xg middotϕ dx


languϕrang =983133

995014xϕ(x) dx

7




m=0

ϕ(m)(m)



m=0


m=0

sup |ϕ(m)|









|α|leN





|α|lem

sup |part αϕm|




|α|lem

sup |part αϕm|

8


ie 983131

|α|lem
















langumϕrang=983133


=1m

983133


9



|langumϕrang|le 1m

983133



ldquo limmrarrinfin


13 Basic Operations





= (minus1)|α|983133











10




H(x) =

9948160 if x le 01 if x gt 0




= minus983133 infin

0

ϕprime(x)


= ϕ(0)








1=

983133




+ lang1ϕrangθ0983167 983166983165 983168=ϕB

11



ψA(x) =

983133 x

minusinfinϕA(y) dy


Claim ψA isin 994963(995014)



















12





languϕrang=983133

995014n

u(minusx)ϕ(x) dx =

983133

995014n








|h|





i

hipart ϕ




u983131

i

hi

|h| middotpart ϕ

part x i

994906+ o(1)

= minus983131

i

hi

|h|

994902upart ϕ

part x i

994905+ o(1)

minusrarr983131

i

mi

994902upart ϕ

part x i

994905=

994903983131

i

mipart upart x i

ϕ



13







995014n


=

983133

995014n


=

983133

995014n

u(y)(τx ϕ)(y) dy

= languτx ϕrang












i

hipart ϕ



14





u983131

i

hipartΦx

part x i

994906+ o(|h|)

=983131

i

hi

994902upartΦx

part x i

994905+ o(|h|)


part


upartΦx

part x i

994905








15




(u lowastδn)(x)










=


=


=




= limhrarr0

983131

misin995022n


= limhrarr0

994903u(z)983131

misin995022n


994906(983183)



16



misin995022n





misin995022n




(983183) =

994903u(z) lim

hrarr0

983131

misin995022n


994906



(iv)







χ(x) =

9948161 on |x |lt 10 on |x |gt 2





17




994793(0)


(983183)




=


=

983133ψ(y)χ994806 y

m2minus x

m

994807θ994806 y

mminus x994807


= θ (minusx) +

983133ψ(y)994844χ994806 y

m2minus x

m

994807θ994806 y


dy


= θ (x) + Rm(x)


R(x) =

983133ψ(y)994844χ994806 y

m2+

xm

994807θ994806 y

m+ x994807minus θ (x)994845

dy





18

















19




|α|leN













20




|α|leN

supsupp(u)capK

|part α(ρϕ)|= C983131

|α|leN


|α|leN

supKpart αϕ





|α|leN

supK|part αϕ|






|α|leN

supK|part α(ρϕ)|


994798supK |part αρ|994799

to get


|α|leN

supK|part αϕ|



ϕ = ρϕ983167983166983165983168=ϕ0

+(1minusρ)ϕ983167 983166983165 983168=ϕ1




= langvϕrang



21



















22






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65





983133 infin



983133 infin

minusinfinf g(λ) =

983133 infin

minusinfinλ983167983166983165983168

= f (λ)

g(λ) dλ




(983183) p =1c2


2ppart x2

= 0


0=

983133

9950142



0=

983133

9950142

p(x t) f dxdt


0=

983133

9950142




4








5


1 DISTRIBUTIONS








983133


983133




ϕ(x + h) =983131

|α|leN

hα


where RN (x h) = o994790|h|N+1994791





|α|leN

supX|part αϕ|




6







) = 0


langTϕrang =983131

|α|leM

983133





|α|leM

983133


=983131

|α|leM

983133


le983131

|α|leM


K| fα(x)| dx

le994808

max|α|leM

983133

K| fα(x)| dx

994809middot983131

|α|leM

supK|part αϕ|


983125K | fα(x)| dx and N = M






Xg middotϕ dx


languϕrang =983133

995014xϕ(x) dx

7




m=0

ϕ(m)(m)



m=0


m=0

sup |ϕ(m)|









|α|leN





|α|lem

sup |part αϕm|




|α|lem

sup |part αϕm|

8


ie 983131

|α|lem
















langumϕrang=983133


=1m

983133


9



|langumϕrang|le 1m

983133



ldquo limmrarrinfin


13 Basic Operations





= (minus1)|α|983133











10




H(x) =

9948160 if x le 01 if x gt 0




= minus983133 infin

0

ϕprime(x)


= ϕ(0)








1=

983133




+ lang1ϕrangθ0983167 983166983165 983168=ϕB

11



ψA(x) =

983133 x

minusinfinϕA(y) dy


Claim ψA isin 994963(995014)



















12





languϕrang=983133

995014n

u(minusx)ϕ(x) dx =

983133

995014n








|h|





i

hipart ϕ




u983131

i

hi

|h| middotpart ϕ

part x i

994906+ o(1)

= minus983131

i

hi

|h|

994902upart ϕ

part x i

994905+ o(1)

minusrarr983131

i

mi

994902upart ϕ

part x i

994905=

994903983131

i

mipart upart x i

ϕ



13







995014n


=

983133

995014n


=

983133

995014n

u(y)(τx ϕ)(y) dy

= languτx ϕrang












i

hipart ϕ



14





u983131

i

hipartΦx

part x i

994906+ o(|h|)

=983131

i

hi

994902upartΦx

part x i

994905+ o(|h|)


part


upartΦx

part x i

994905








15




(u lowastδn)(x)










=


=


=




= limhrarr0

983131

misin995022n


= limhrarr0

994903u(z)983131

misin995022n


994906(983183)



16



misin995022n





misin995022n




(983183) =

994903u(z) lim

hrarr0

983131

misin995022n


994906



(iv)







χ(x) =

9948161 on |x |lt 10 on |x |gt 2





17




994793(0)


(983183)




=


=

983133ψ(y)χ994806 y

m2minus x

m

994807θ994806 y

mminus x994807


= θ (minusx) +

983133ψ(y)994844χ994806 y

m2minus x

m

994807θ994806 y


dy


= θ (x) + Rm(x)


R(x) =

983133ψ(y)994844χ994806 y

m2+

xm

994807θ994806 y

m+ x994807minus θ (x)994845

dy





18

















19




|α|leN













20




|α|leN

supsupp(u)capK

|part α(ρϕ)|= C983131

|α|leN


|α|leN

supKpart αϕ





|α|leN

supK|part αϕ|






|α|leN

supK|part α(ρϕ)|


994798supK |part αρ|994799

to get


|α|leN

supK|part αϕ|



ϕ = ρϕ983167983166983165983168=ϕ0

+(1minusρ)ϕ983167 983166983165 983168=ϕ1




= langvϕrang



21



















22






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65








5


1 DISTRIBUTIONS








983133


983133




ϕ(x + h) =983131

|α|leN

hα


where RN (x h) = o994790|h|N+1994791





|α|leN

supX|part αϕ|




6







) = 0


langTϕrang =983131

|α|leM

983133





|α|leM

983133


=983131

|α|leM

983133


le983131

|α|leM


K| fα(x)| dx

le994808

max|α|leM

983133

K| fα(x)| dx

994809middot983131

|α|leM

supK|part αϕ|


983125K | fα(x)| dx and N = M






Xg middotϕ dx


languϕrang =983133

995014xϕ(x) dx

7




m=0

ϕ(m)(m)



m=0


m=0

sup |ϕ(m)|









|α|leN





|α|lem

sup |part αϕm|




|α|lem

sup |part αϕm|

8


ie 983131

|α|lem
















langumϕrang=983133


=1m

983133


9



|langumϕrang|le 1m

983133



ldquo limmrarrinfin


13 Basic Operations





= (minus1)|α|983133











10




H(x) =

9948160 if x le 01 if x gt 0




= minus983133 infin

0

ϕprime(x)


= ϕ(0)








1=

983133




+ lang1ϕrangθ0983167 983166983165 983168=ϕB

11



ψA(x) =

983133 x

minusinfinϕA(y) dy


Claim ψA isin 994963(995014)



















12





languϕrang=983133

995014n

u(minusx)ϕ(x) dx =

983133

995014n








|h|





i

hipart ϕ




u983131

i

hi

|h| middotpart ϕ

part x i

994906+ o(1)

= minus983131

i

hi

|h|

994902upart ϕ

part x i

994905+ o(1)

minusrarr983131

i

mi

994902upart ϕ

part x i

994905=

994903983131

i

mipart upart x i

ϕ



13







995014n


=

983133

995014n


=

983133

995014n

u(y)(τx ϕ)(y) dy

= languτx ϕrang












i

hipart ϕ



14





u983131

i

hipartΦx

part x i

994906+ o(|h|)

=983131

i

hi

994902upartΦx

part x i

994905+ o(|h|)


part


upartΦx

part x i

994905








15




(u lowastδn)(x)










=


=


=




= limhrarr0

983131

misin995022n


= limhrarr0

994903u(z)983131

misin995022n


994906(983183)



16



misin995022n





misin995022n




(983183) =

994903u(z) lim

hrarr0

983131

misin995022n


994906



(iv)







χ(x) =

9948161 on |x |lt 10 on |x |gt 2





17




994793(0)


(983183)




=


=

983133ψ(y)χ994806 y

m2minus x

m

994807θ994806 y

mminus x994807


= θ (minusx) +

983133ψ(y)994844χ994806 y

m2minus x

m

994807θ994806 y


dy


= θ (x) + Rm(x)


R(x) =

983133ψ(y)994844χ994806 y

m2+

xm

994807θ994806 y

m+ x994807minus θ (x)994845

dy





18

















19




|α|leN













20




|α|leN

supsupp(u)capK

|part α(ρϕ)|= C983131

|α|leN


|α|leN

supKpart αϕ





|α|leN

supK|part αϕ|






|α|leN

supK|part α(ρϕ)|


994798supK |part αρ|994799

to get


|α|leN

supK|part αϕ|



ϕ = ρϕ983167983166983165983168=ϕ0

+(1minusρ)ϕ983167 983166983165 983168=ϕ1




= langvϕrang



21



















22






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65


1 DISTRIBUTIONS








983133


983133




ϕ(x + h) =983131

|α|leN

hα


where RN (x h) = o994790|h|N+1994791





|α|leN

supX|part αϕ|




6







) = 0


langTϕrang =983131

|α|leM

983133





|α|leM

983133


=983131

|α|leM

983133


le983131

|α|leM


K| fα(x)| dx

le994808

max|α|leM

983133

K| fα(x)| dx

994809middot983131

|α|leM

supK|part αϕ|


983125K | fα(x)| dx and N = M






Xg middotϕ dx


languϕrang =983133

995014xϕ(x) dx

7




m=0

ϕ(m)(m)



m=0


m=0

sup |ϕ(m)|









|α|leN





|α|lem

sup |part αϕm|




|α|lem

sup |part αϕm|

8


ie 983131

|α|lem
















langumϕrang=983133


=1m

983133


9



|langumϕrang|le 1m

983133



ldquo limmrarrinfin


13 Basic Operations





= (minus1)|α|983133











10




H(x) =

9948160 if x le 01 if x gt 0




= minus983133 infin

0

ϕprime(x)


= ϕ(0)








1=

983133




+ lang1ϕrangθ0983167 983166983165 983168=ϕB

11



ψA(x) =

983133 x

minusinfinϕA(y) dy


Claim ψA isin 994963(995014)



















12





languϕrang=983133

995014n

u(minusx)ϕ(x) dx =

983133

995014n








|h|





i

hipart ϕ




u983131

i

hi

|h| middotpart ϕ

part x i

994906+ o(1)

= minus983131

i

hi

|h|

994902upart ϕ

part x i

994905+ o(1)

minusrarr983131

i

mi

994902upart ϕ

part x i

994905=

994903983131

i

mipart upart x i

ϕ



13







995014n


=

983133

995014n


=

983133

995014n

u(y)(τx ϕ)(y) dy

= languτx ϕrang












i

hipart ϕ



14





u983131

i

hipartΦx

part x i

994906+ o(|h|)

=983131

i

hi

994902upartΦx

part x i

994905+ o(|h|)


part


upartΦx

part x i

994905








15




(u lowastδn)(x)










=


=


=




= limhrarr0

983131

misin995022n


= limhrarr0

994903u(z)983131

misin995022n


994906(983183)



16



misin995022n





misin995022n




(983183) =

994903u(z) lim

hrarr0

983131

misin995022n


994906



(iv)







χ(x) =

9948161 on |x |lt 10 on |x |gt 2





17




994793(0)


(983183)




=


=

983133ψ(y)χ994806 y

m2minus x

m

994807θ994806 y

mminus x994807


= θ (minusx) +

983133ψ(y)994844χ994806 y

m2minus x

m

994807θ994806 y


dy


= θ (x) + Rm(x)


R(x) =

983133ψ(y)994844χ994806 y

m2+

xm

994807θ994806 y

m+ x994807minus θ (x)994845

dy





18

















19




|α|leN













20




|α|leN

supsupp(u)capK

|part α(ρϕ)|= C983131

|α|leN


|α|leN

supKpart αϕ





|α|leN

supK|part αϕ|






|α|leN

supK|part α(ρϕ)|


994798supK |part αρ|994799

to get


|α|leN

supK|part αϕ|



ϕ = ρϕ983167983166983165983168=ϕ0

+(1minusρ)ϕ983167 983166983165 983168=ϕ1




= langvϕrang



21



















22






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65







) = 0


langTϕrang =983131

|α|leM

983133





|α|leM

983133


=983131

|α|leM

983133


le983131

|α|leM


K| fα(x)| dx

le994808

max|α|leM

983133

K| fα(x)| dx

994809middot983131

|α|leM

supK|part αϕ|


983125K | fα(x)| dx and N = M






Xg middotϕ dx


languϕrang =983133

995014xϕ(x) dx

7




m=0

ϕ(m)(m)



m=0


m=0

sup |ϕ(m)|









|α|leN





|α|lem

sup |part αϕm|




|α|lem

sup |part αϕm|

8


ie 983131

|α|lem
















langumϕrang=983133


=1m

983133


9



|langumϕrang|le 1m

983133



ldquo limmrarrinfin


13 Basic Operations





= (minus1)|α|983133











10




H(x) =

9948160 if x le 01 if x gt 0




= minus983133 infin

0

ϕprime(x)


= ϕ(0)








1=

983133




+ lang1ϕrangθ0983167 983166983165 983168=ϕB

11



ψA(x) =

983133 x

minusinfinϕA(y) dy


Claim ψA isin 994963(995014)



















12





languϕrang=983133

995014n

u(minusx)ϕ(x) dx =

983133

995014n








|h|





i

hipart ϕ




u983131

i

hi

|h| middotpart ϕ

part x i

994906+ o(1)

= minus983131

i

hi

|h|

994902upart ϕ

part x i

994905+ o(1)

minusrarr983131

i

mi

994902upart ϕ

part x i

994905=

994903983131

i

mipart upart x i

ϕ



13







995014n


=

983133

995014n


=

983133

995014n

u(y)(τx ϕ)(y) dy

= languτx ϕrang












i

hipart ϕ



14





u983131

i

hipartΦx

part x i

994906+ o(|h|)

=983131

i

hi

994902upartΦx

part x i

994905+ o(|h|)


part


upartΦx

part x i

994905








15




(u lowastδn)(x)










=


=


=




= limhrarr0

983131

misin995022n


= limhrarr0

994903u(z)983131

misin995022n


994906(983183)



16



misin995022n





misin995022n




(983183) =

994903u(z) lim

hrarr0

983131

misin995022n


994906



(iv)







χ(x) =

9948161 on |x |lt 10 on |x |gt 2





17




994793(0)


(983183)




=


=

983133ψ(y)χ994806 y

m2minus x

m

994807θ994806 y

mminus x994807


= θ (minusx) +

983133ψ(y)994844χ994806 y

m2minus x

m

994807θ994806 y


dy


= θ (x) + Rm(x)


R(x) =

983133ψ(y)994844χ994806 y

m2+

xm

994807θ994806 y

m+ x994807minus θ (x)994845

dy





18

















19




|α|leN













20




|α|leN

supsupp(u)capK

|part α(ρϕ)|= C983131

|α|leN


|α|leN

supKpart αϕ





|α|leN

supK|part αϕ|






|α|leN

supK|part α(ρϕ)|


994798supK |part αρ|994799

to get


|α|leN

supK|part αϕ|



ϕ = ρϕ983167983166983165983168=ϕ0

+(1minusρ)ϕ983167 983166983165 983168=ϕ1




= langvϕrang



21



















22






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65




m=0

ϕ(m)(m)



m=0


m=0

sup |ϕ(m)|









|α|leN





|α|lem

sup |part αϕm|




|α|lem

sup |part αϕm|

8


ie 983131

|α|lem
















langumϕrang=983133


=1m

983133


9



|langumϕrang|le 1m

983133



ldquo limmrarrinfin


13 Basic Operations





= (minus1)|α|983133











10




H(x) =

9948160 if x le 01 if x gt 0




= minus983133 infin

0

ϕprime(x)


= ϕ(0)








1=

983133




+ lang1ϕrangθ0983167 983166983165 983168=ϕB

11



ψA(x) =

983133 x

minusinfinϕA(y) dy


Claim ψA isin 994963(995014)



















12





languϕrang=983133

995014n

u(minusx)ϕ(x) dx =

983133

995014n








|h|





i

hipart ϕ




u983131

i

hi

|h| middotpart ϕ

part x i

994906+ o(1)

= minus983131

i

hi

|h|

994902upart ϕ

part x i

994905+ o(1)

minusrarr983131

i

mi

994902upart ϕ

part x i

994905=

994903983131

i

mipart upart x i

ϕ



13







995014n


=

983133

995014n


=

983133

995014n

u(y)(τx ϕ)(y) dy

= languτx ϕrang












i

hipart ϕ



14





u983131

i

hipartΦx

part x i

994906+ o(|h|)

=983131

i

hi

994902upartΦx

part x i

994905+ o(|h|)


part


upartΦx

part x i

994905








15




(u lowastδn)(x)










=


=


=




= limhrarr0

983131

misin995022n


= limhrarr0

994903u(z)983131

misin995022n


994906(983183)



16



misin995022n





misin995022n




(983183) =

994903u(z) lim

hrarr0

983131

misin995022n


994906



(iv)







χ(x) =

9948161 on |x |lt 10 on |x |gt 2





17




994793(0)


(983183)




=


=

983133ψ(y)χ994806 y

m2minus x

m

994807θ994806 y

mminus x994807


= θ (minusx) +

983133ψ(y)994844χ994806 y

m2minus x

m

994807θ994806 y


dy


= θ (x) + Rm(x)


R(x) =

983133ψ(y)994844χ994806 y

m2+

xm

994807θ994806 y

m+ x994807minus θ (x)994845

dy





18

















19




|α|leN













20




|α|leN

supsupp(u)capK

|part α(ρϕ)|= C983131

|α|leN


|α|leN

supKpart αϕ





|α|leN

supK|part αϕ|






|α|leN

supK|part α(ρϕ)|


994798supK |part αρ|994799

to get


|α|leN

supK|part αϕ|



ϕ = ρϕ983167983166983165983168=ϕ0

+(1minusρ)ϕ983167 983166983165 983168=ϕ1




= langvϕrang



21



















22






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65


ie 983131

|α|lem
















langumϕrang=983133


=1m

983133


9



|langumϕrang|le 1m

983133



ldquo limmrarrinfin


13 Basic Operations





= (minus1)|α|983133











10




H(x) =

9948160 if x le 01 if x gt 0




= minus983133 infin

0

ϕprime(x)


= ϕ(0)








1=

983133




+ lang1ϕrangθ0983167 983166983165 983168=ϕB

11



ψA(x) =

983133 x

minusinfinϕA(y) dy


Claim ψA isin 994963(995014)



















12





languϕrang=983133

995014n

u(minusx)ϕ(x) dx =

983133

995014n








|h|





i

hipart ϕ




u983131

i

hi

|h| middotpart ϕ

part x i

994906+ o(1)

= minus983131

i

hi

|h|

994902upart ϕ

part x i

994905+ o(1)

minusrarr983131

i

mi

994902upart ϕ

part x i

994905=

994903983131

i

mipart upart x i

ϕ



13







995014n


=

983133

995014n


=

983133

995014n

u(y)(τx ϕ)(y) dy

= languτx ϕrang












i

hipart ϕ



14





u983131

i

hipartΦx

part x i

994906+ o(|h|)

=983131

i

hi

994902upartΦx

part x i

994905+ o(|h|)


part


upartΦx

part x i

994905








15




(u lowastδn)(x)










=


=


=




= limhrarr0

983131

misin995022n


= limhrarr0

994903u(z)983131

misin995022n


994906(983183)



16



misin995022n





misin995022n




(983183) =

994903u(z) lim

hrarr0

983131

misin995022n


994906



(iv)







χ(x) =

9948161 on |x |lt 10 on |x |gt 2





17




994793(0)


(983183)




=


=

983133ψ(y)χ994806 y

m2minus x

m

994807θ994806 y

mminus x994807


= θ (minusx) +

983133ψ(y)994844χ994806 y

m2minus x

m

994807θ994806 y


dy


= θ (x) + Rm(x)


R(x) =

983133ψ(y)994844χ994806 y

m2+

xm

994807θ994806 y

m+ x994807minus θ (x)994845

dy





18

















19




|α|leN













20




|α|leN

supsupp(u)capK

|part α(ρϕ)|= C983131

|α|leN


|α|leN

supKpart αϕ





|α|leN

supK|part αϕ|






|α|leN

supK|part α(ρϕ)|


994798supK |part αρ|994799

to get


|α|leN

supK|part αϕ|



ϕ = ρϕ983167983166983165983168=ϕ0

+(1minusρ)ϕ983167 983166983165 983168=ϕ1




= langvϕrang



21



















22






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65



|langumϕrang|le 1m

983133



ldquo limmrarrinfin


13 Basic Operations





= (minus1)|α|983133











10




H(x) =

9948160 if x le 01 if x gt 0




= minus983133 infin

0

ϕprime(x)


= ϕ(0)








1=

983133




+ lang1ϕrangθ0983167 983166983165 983168=ϕB

11



ψA(x) =

983133 x

minusinfinϕA(y) dy


Claim ψA isin 994963(995014)



















12





languϕrang=983133

995014n

u(minusx)ϕ(x) dx =

983133

995014n








|h|





i

hipart ϕ




u983131

i

hi

|h| middotpart ϕ

part x i

994906+ o(1)

= minus983131

i

hi

|h|

994902upart ϕ

part x i

994905+ o(1)

minusrarr983131

i

mi

994902upart ϕ

part x i

994905=

994903983131

i

mipart upart x i

ϕ



13







995014n


=

983133

995014n


=

983133

995014n

u(y)(τx ϕ)(y) dy

= languτx ϕrang












i

hipart ϕ



14





u983131

i

hipartΦx

part x i

994906+ o(|h|)

=983131

i

hi

994902upartΦx

part x i

994905+ o(|h|)


part


upartΦx

part x i

994905








15




(u lowastδn)(x)










=


=


=




= limhrarr0

983131

misin995022n


= limhrarr0

994903u(z)983131

misin995022n


994906(983183)



16



misin995022n





misin995022n




(983183) =

994903u(z) lim

hrarr0

983131

misin995022n


994906



(iv)







χ(x) =

9948161 on |x |lt 10 on |x |gt 2





17




994793(0)


(983183)




=


=

983133ψ(y)χ994806 y

m2minus x

m

994807θ994806 y

mminus x994807


= θ (minusx) +

983133ψ(y)994844χ994806 y

m2minus x

m

994807θ994806 y


dy


= θ (x) + Rm(x)


R(x) =

983133ψ(y)994844χ994806 y

m2+

xm

994807θ994806 y

m+ x994807minus θ (x)994845

dy





18

















19




|α|leN













20




|α|leN

supsupp(u)capK

|part α(ρϕ)|= C983131

|α|leN


|α|leN

supKpart αϕ





|α|leN

supK|part αϕ|






|α|leN

supK|part α(ρϕ)|


994798supK |part αρ|994799

to get


|α|leN

supK|part αϕ|



ϕ = ρϕ983167983166983165983168=ϕ0

+(1minusρ)ϕ983167 983166983165 983168=ϕ1




= langvϕrang



21



















22






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65




H(x) =

9948160 if x le 01 if x gt 0




= minus983133 infin

0

ϕprime(x)


= ϕ(0)








1=

983133




+ lang1ϕrangθ0983167 983166983165 983168=ϕB

11



ψA(x) =

983133 x

minusinfinϕA(y) dy


Claim ψA isin 994963(995014)



















12





languϕrang=983133

995014n

u(minusx)ϕ(x) dx =

983133

995014n








|h|





i

hipart ϕ




u983131

i

hi

|h| middotpart ϕ

part x i

994906+ o(1)

= minus983131

i

hi

|h|

994902upart ϕ

part x i

994905+ o(1)

minusrarr983131

i

mi

994902upart ϕ

part x i

994905=

994903983131

i

mipart upart x i

ϕ



13







995014n


=

983133

995014n


=

983133

995014n

u(y)(τx ϕ)(y) dy

= languτx ϕrang












i

hipart ϕ



14





u983131

i

hipartΦx

part x i

994906+ o(|h|)

=983131

i

hi

994902upartΦx

part x i

994905+ o(|h|)


part


upartΦx

part x i

994905








15




(u lowastδn)(x)










=


=


=




= limhrarr0

983131

misin995022n


= limhrarr0

994903u(z)983131

misin995022n


994906(983183)



16



misin995022n





misin995022n




(983183) =

994903u(z) lim

hrarr0

983131

misin995022n


994906



(iv)







χ(x) =

9948161 on |x |lt 10 on |x |gt 2





17




994793(0)


(983183)




=


=

983133ψ(y)χ994806 y

m2minus x

m

994807θ994806 y

mminus x994807


= θ (minusx) +

983133ψ(y)994844χ994806 y

m2minus x

m

994807θ994806 y


dy


= θ (x) + Rm(x)


R(x) =

983133ψ(y)994844χ994806 y

m2+

xm

994807θ994806 y

m+ x994807minus θ (x)994845

dy





18

















19




|α|leN













20




|α|leN

supsupp(u)capK

|part α(ρϕ)|= C983131

|α|leN


|α|leN

supKpart αϕ





|α|leN

supK|part αϕ|






|α|leN

supK|part α(ρϕ)|


994798supK |part αρ|994799

to get


|α|leN

supK|part αϕ|



ϕ = ρϕ983167983166983165983168=ϕ0

+(1minusρ)ϕ983167 983166983165 983168=ϕ1




= langvϕrang



21



















22






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65



ψA(x) =

983133 x

minusinfinϕA(y) dy


Claim ψA isin 994963(995014)



















12





languϕrang=983133

995014n

u(minusx)ϕ(x) dx =

983133

995014n








|h|





i

hipart ϕ




u983131

i

hi

|h| middotpart ϕ

part x i

994906+ o(1)

= minus983131

i

hi

|h|

994902upart ϕ

part x i

994905+ o(1)

minusrarr983131

i

mi

994902upart ϕ

part x i

994905=

994903983131

i

mipart upart x i

ϕ



13







995014n


=

983133

995014n


=

983133

995014n

u(y)(τx ϕ)(y) dy

= languτx ϕrang












i

hipart ϕ



14





u983131

i

hipartΦx

part x i

994906+ o(|h|)

=983131

i

hi

994902upartΦx

part x i

994905+ o(|h|)


part


upartΦx

part x i

994905








15




(u lowastδn)(x)










=


=


=




= limhrarr0

983131

misin995022n


= limhrarr0

994903u(z)983131

misin995022n


994906(983183)



16



misin995022n





misin995022n




(983183) =

994903u(z) lim

hrarr0

983131

misin995022n


994906



(iv)







χ(x) =

9948161 on |x |lt 10 on |x |gt 2





17




994793(0)


(983183)




=


=

983133ψ(y)χ994806 y

m2minus x

m

994807θ994806 y

mminus x994807


= θ (minusx) +

983133ψ(y)994844χ994806 y

m2minus x

m

994807θ994806 y


dy


= θ (x) + Rm(x)


R(x) =

983133ψ(y)994844χ994806 y

m2+

xm

994807θ994806 y

m+ x994807minus θ (x)994845

dy





18

















19




|α|leN













20




|α|leN

supsupp(u)capK

|part α(ρϕ)|= C983131

|α|leN


|α|leN

supKpart αϕ





|α|leN

supK|part αϕ|






|α|leN

supK|part α(ρϕ)|


994798supK |part αρ|994799

to get


|α|leN

supK|part αϕ|



ϕ = ρϕ983167983166983165983168=ϕ0

+(1minusρ)ϕ983167 983166983165 983168=ϕ1




= langvϕrang



21



















22






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65





languϕrang=983133

995014n

u(minusx)ϕ(x) dx =

983133

995014n








|h|





i

hipart ϕ




u983131

i

hi

|h| middotpart ϕ

part x i

994906+ o(1)

= minus983131

i

hi

|h|

994902upart ϕ

part x i

994905+ o(1)

minusrarr983131

i

mi

994902upart ϕ

part x i

994905=

994903983131

i

mipart upart x i

ϕ



13







995014n


=

983133

995014n


=

983133

995014n

u(y)(τx ϕ)(y) dy

= languτx ϕrang












i

hipart ϕ



14





u983131

i

hipartΦx

part x i

994906+ o(|h|)

=983131

i

hi

994902upartΦx

part x i

994905+ o(|h|)


part


upartΦx

part x i

994905








15




(u lowastδn)(x)










=


=


=




= limhrarr0

983131

misin995022n


= limhrarr0

994903u(z)983131

misin995022n


994906(983183)



16



misin995022n





misin995022n




(983183) =

994903u(z) lim

hrarr0

983131

misin995022n


994906



(iv)







χ(x) =

9948161 on |x |lt 10 on |x |gt 2





17




994793(0)


(983183)




=


=

983133ψ(y)χ994806 y

m2minus x

m

994807θ994806 y

mminus x994807


= θ (minusx) +

983133ψ(y)994844χ994806 y

m2minus x

m

994807θ994806 y


dy


= θ (x) + Rm(x)


R(x) =

983133ψ(y)994844χ994806 y

m2+

xm

994807θ994806 y

m+ x994807minus θ (x)994845

dy





18

















19




|α|leN













20




|α|leN

supsupp(u)capK

|part α(ρϕ)|= C983131

|α|leN


|α|leN

supKpart αϕ





|α|leN

supK|part αϕ|






|α|leN

supK|part α(ρϕ)|


994798supK |part αρ|994799

to get


|α|leN

supK|part αϕ|



ϕ = ρϕ983167983166983165983168=ϕ0

+(1minusρ)ϕ983167 983166983165 983168=ϕ1




= langvϕrang



21



















22






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65







995014n


=

983133

995014n


=

983133

995014n

u(y)(τx ϕ)(y) dy

= languτx ϕrang












i

hipart ϕ



14





u983131

i

hipartΦx

part x i

994906+ o(|h|)

=983131

i

hi

994902upartΦx

part x i

994905+ o(|h|)


part


upartΦx

part x i

994905








15




(u lowastδn)(x)










=


=


=




= limhrarr0

983131

misin995022n


= limhrarr0

994903u(z)983131

misin995022n


994906(983183)



16



misin995022n





misin995022n




(983183) =

994903u(z) lim

hrarr0

983131

misin995022n


994906



(iv)







χ(x) =

9948161 on |x |lt 10 on |x |gt 2





17




994793(0)


(983183)




=


=

983133ψ(y)χ994806 y

m2minus x

m

994807θ994806 y

mminus x994807


= θ (minusx) +

983133ψ(y)994844χ994806 y

m2minus x

m

994807θ994806 y


dy


= θ (x) + Rm(x)


R(x) =

983133ψ(y)994844χ994806 y

m2+

xm

994807θ994806 y

m+ x994807minus θ (x)994845

dy





18

















19




|α|leN













20




|α|leN

supsupp(u)capK

|part α(ρϕ)|= C983131

|α|leN


|α|leN

supKpart αϕ





|α|leN

supK|part αϕ|






|α|leN

supK|part α(ρϕ)|


994798supK |part αρ|994799

to get


|α|leN

supK|part αϕ|



ϕ = ρϕ983167983166983165983168=ϕ0

+(1minusρ)ϕ983167 983166983165 983168=ϕ1




= langvϕrang



21



















22






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65





u983131

i

hipartΦx

part x i

994906+ o(|h|)

=983131

i

hi

994902upartΦx

part x i

994905+ o(|h|)


part


upartΦx

part x i

994905








15




(u lowastδn)(x)










=


=


=




= limhrarr0

983131

misin995022n


= limhrarr0

994903u(z)983131

misin995022n


994906(983183)



16



misin995022n





misin995022n




(983183) =

994903u(z) lim

hrarr0

983131

misin995022n


994906



(iv)







χ(x) =

9948161 on |x |lt 10 on |x |gt 2





17




994793(0)


(983183)




=


=

983133ψ(y)χ994806 y

m2minus x

m

994807θ994806 y

mminus x994807


= θ (minusx) +

983133ψ(y)994844χ994806 y

m2minus x

m

994807θ994806 y


dy


= θ (x) + Rm(x)


R(x) =

983133ψ(y)994844χ994806 y

m2+

xm

994807θ994806 y

m+ x994807minus θ (x)994845

dy





18

















19




|α|leN













20




|α|leN

supsupp(u)capK

|part α(ρϕ)|= C983131

|α|leN


|α|leN

supKpart αϕ





|α|leN

supK|part αϕ|






|α|leN

supK|part α(ρϕ)|


994798supK |part αρ|994799

to get


|α|leN

supK|part αϕ|



ϕ = ρϕ983167983166983165983168=ϕ0

+(1minusρ)ϕ983167 983166983165 983168=ϕ1




= langvϕrang



21



















22






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65




(u lowastδn)(x)










=


=


=




= limhrarr0

983131

misin995022n


= limhrarr0

994903u(z)983131

misin995022n


994906(983183)



16



misin995022n





misin995022n




(983183) =

994903u(z) lim

hrarr0

983131

misin995022n


994906



(iv)







χ(x) =

9948161 on |x |lt 10 on |x |gt 2





17




994793(0)


(983183)




=


=

983133ψ(y)χ994806 y

m2minus x

m

994807θ994806 y

mminus x994807


= θ (minusx) +

983133ψ(y)994844χ994806 y

m2minus x

m

994807θ994806 y


dy


= θ (x) + Rm(x)


R(x) =

983133ψ(y)994844χ994806 y

m2+

xm

994807θ994806 y

m+ x994807minus θ (x)994845

dy





18

















19




|α|leN













20




|α|leN

supsupp(u)capK

|part α(ρϕ)|= C983131

|α|leN


|α|leN

supKpart αϕ





|α|leN

supK|part αϕ|






|α|leN

supK|part α(ρϕ)|


994798supK |part αρ|994799

to get


|α|leN

supK|part αϕ|



ϕ = ρϕ983167983166983165983168=ϕ0

+(1minusρ)ϕ983167 983166983165 983168=ϕ1




= langvϕrang



21



















22






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65



misin995022n





misin995022n




(983183) =

994903u(z) lim

hrarr0

983131

misin995022n


994906



(iv)







χ(x) =

9948161 on |x |lt 10 on |x |gt 2





17




994793(0)


(983183)




=


=

983133ψ(y)χ994806 y

m2minus x

m

994807θ994806 y

mminus x994807


= θ (minusx) +

983133ψ(y)994844χ994806 y

m2minus x

m

994807θ994806 y


dy


= θ (x) + Rm(x)


R(x) =

983133ψ(y)994844χ994806 y

m2+

xm

994807θ994806 y

m+ x994807minus θ (x)994845

dy





18

















19




|α|leN













20




|α|leN

supsupp(u)capK

|part α(ρϕ)|= C983131

|α|leN


|α|leN

supKpart αϕ





|α|leN

supK|part αϕ|






|α|leN

supK|part α(ρϕ)|


994798supK |part αρ|994799

to get


|α|leN

supK|part αϕ|



ϕ = ρϕ983167983166983165983168=ϕ0

+(1minusρ)ϕ983167 983166983165 983168=ϕ1




= langvϕrang



21



















22






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65




994793(0)


(983183)




=


=

983133ψ(y)χ994806 y

m2minus x

m

994807θ994806 y

mminus x994807


= θ (minusx) +

983133ψ(y)994844χ994806 y

m2minus x

m

994807θ994806 y


dy


= θ (x) + Rm(x)


R(x) =

983133ψ(y)994844χ994806 y

m2+

xm

994807θ994806 y

m+ x994807minus θ (x)994845

dy





18

















19




|α|leN













20




|α|leN

supsupp(u)capK

|part α(ρϕ)|= C983131

|α|leN


|α|leN

supKpart αϕ





|α|leN

supK|part αϕ|






|α|leN

supK|part α(ρϕ)|


994798supK |part αρ|994799

to get


|α|leN

supK|part αϕ|



ϕ = ρϕ983167983166983165983168=ϕ0

+(1minusρ)ϕ983167 983166983165 983168=ϕ1




= langvϕrang



21



















22






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65

















19




|α|leN













20




|α|leN

supsupp(u)capK

|part α(ρϕ)|= C983131

|α|leN


|α|leN

supKpart αϕ





|α|leN

supK|part αϕ|






|α|leN

supK|part α(ρϕ)|


994798supK |part αρ|994799

to get


|α|leN

supK|part αϕ|



ϕ = ρϕ983167983166983165983168=ϕ0

+(1minusρ)ϕ983167 983166983165 983168=ϕ1




= langvϕrang



21



















22






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65




|α|leN













20




|α|leN

supsupp(u)capK

|part α(ρϕ)|= C983131

|α|leN


|α|leN

supKpart αϕ





|α|leN

supK|part αϕ|






|α|leN

supK|part α(ρϕ)|


994798supK |part αρ|994799

to get


|α|leN

supK|part αϕ|



ϕ = ρϕ983167983166983165983168=ϕ0

+(1minusρ)ϕ983167 983166983165 983168=ϕ1




= langvϕrang



21



















22






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65




|α|leN

supsupp(u)capK

|part α(ρϕ)|= C983131

|α|leN


|α|leN

supKpart αϕ





|α|leN

supK|part αϕ|






|α|leN

supK|part α(ρϕ)|


994798supK |part αρ|994799

to get


|α|leN

supK|part αϕ|



ϕ = ρϕ983167983166983165983168=ϕ0

+(1minusρ)ϕ983167 983166983165 983168=ϕ1




= langvϕrang



21



















22






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65



















22






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65






















23






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65






983042ϕ983042αβ = sup995014n







2




|α||β |leN

983042ϕ983042αβ








24





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65





f (λ) =

983133

995014n



995014n

|ϕ| dx =

983133

995014n


le C983131

|α|leN

983042ϕ983042α0

983133

995014n

1(1+ |x |)N dx

ltinfin





f (λ) =

983133

995014n


995014n





25






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65






994923(Dαxϕ)(λ) =983133

995014n


= (minus1)|α|983133

995014n


dx

= λα983133

995014n


= λαϕ(λ)


Dβλϕ(λ) = Dβ

λ

994808983133

995014n


994809

=

983133

995014n


= (minus1)|β |994923994790xβϕ994791(λ)





ϕ(x) =1

(2π)n

983133

995014n






26




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65




|λαDβ ϕ|=994802994802994802994802983133

995014n


994802994802994802994802lt983133

995014n





1(2π)n

983133

995014n


(2π)n

983133

995014n


995014n


994811dx


983133

995014n


983133

995014n




995014n


983133

995014n

ϕ(y)

994810983133

995014n


994811dy

=

983133

995014n

ϕ(y)

983093983095

n983132

j=1

983133


j dλ j

983094983096 dy

(983183) =

983133

995014n

ϕ(y)

983093983095

n983132

j=1

994806π983171

99480712eminus

(x jminusy j )2

4983171

983094983096 dy

=994806π983171

994807n2983133

995014n


4983171 dy

(dagger) =994806π983171

994807n22n middot 983171n2

983133

995014n


prime|2 dy prime


995014n


2dy



27




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65




get that this

rarr πn22n

983133

995014n

ϕ(x)eminus|y|2

dy

= πn22nϕ(x)

983133

995014n

eminus|y|2

dy

= (2π)nϕ(x)


ϕ(x) =1

(2π)n

983133

995014n



ϕ(x) =1

(2π)n

983133

995014n


983133

995014n


1(2π)n

ϕ(minusλ)994881

dλ =994965 [994878

1(2π)n

ϕ(minusλ)994881

983167 983166983165 983168isinS(995014n)





dz =

983133 infin


iα2983171 )

2

middot eminus α2

4983171 dz

= eminusα24983171

983133 infin


iα2983171 )

2

dz

= eminusα24983171

983133


iα2983171 )

2


= eminusα24983171

983133 infin


2dz

983167 983166983165 983168=(π983171)12

=994806π983171

9948072eminusα

24983171




28





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65





995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)ψ(x) dx



995014n

ϕ(x)ψ(x) dx =

983133

995014n

ϕ(x)

994810983133

995014n


994811dx

=

983133

995014n

ψ(λ)

994810983133

995014n


994811dλ

=

983133

995014n

ψ(λ)ϕ(λ) dλ










995014n



29


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65


Also


995014n



1= (2π)nδ0


ldquoδ0(x) =1

(2π)n

983133

995014n











We also claim that

u= (2π)minusn994920(993484)u

Indeed




(2π)n

983133

995014n



30







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65







34 Sobolev Spaces



983042u9830422Hs =

983133

995014n










995014n

|u(λ)| dλ =983133

995014n


le994808983133

995014n

langλrangminus2s

99480912994808983133

995014n


99480912

= 983042u983042Hs middot994808983133

Snminus1

dσ

983133 infin

0


99480912

31






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65






995014n




995014n

u(λ)ϕ(λ) dλ

=

983133

995014n

u(λ)

9948101

(2π)n

983133

995014n


994811dλ

=

983133

995014n

ϕ(x)

9948101

(2π)n

983133

995014n


994811

983167 983166983165 983168=F(x)

dx




u(x) =1

(2π)n

983133

995014n






32







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65







part x1


part xn

9948682994809=∆

is the Laplacian






|α|=N

cαλα






minλisinSnminus1

|σP(λ)|= c0 gt 0


|σP(λ)|=994802994802994802994802994802983131

|α|=N

cαλα

994802994802994802994802994802= |λ|N middot994802994802994802994802994802983131

|α|=N

cα

994858λ

|λ|

994868α994802994802994802994802994802983167 983166983165 983168=σP (λ|λ|) ge c0

ge c0|λ|N

33







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65







994802994802994802994802lec0

2


|P(λ)|ge 12







983133

995014n

|Dαu|2 dλ = C

983133

995014n


Proof Later






P(D)E = δ0 + w



34






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65






E(λ) =1minusχ(λ)

P(λ)





994802994802994802994802 994925994790Dα(xβ E)994791(λ)

994802994802994802994802=994802994802λαDβ (E)994802994802

=

994802994802994802994802λαDβ994858

1P(λ)

994868994802994802994802994802≲ |λ||α|minus|β |minusN







loc (995014n)


35












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65












995014n



995014n





isin Hs+N (995014n)

and so we are done











36





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65





ϕ isin 994963(X )





=ψ1 f + [P(D)ψ1](u)

=ψ1 f + [P(D)ψ1](ψ0u)





+



983133

995014n


37



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65



995014n


983133

995014n


9948029948029948029948022


=

983133

995014n




P(D)(ψ2u) =ψ2P(D)u+ [P(D)ψ2](u)

=ψ2 f + [P(D)ψ2](ψ1u)(xi)









P(D)E = δ0





38



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65



part z=

12

994858part

part x1+ ipart

part y

994868


part z


part zE = 0



9950142


dx1dx2


983133


dx1dx2


983133

|z|gt983171

part


part

part zE = 0 here


part z(ϕE) =

12

994878part

part x1(ϕE) + i

part

part x2(ϕE)994881



12

983133


= lim983171darr0

983133

|z|=983171

ϕE2i[dx1 + idx2]

= lim983171darr0

983133

|z|=983171

ϕE2i


= lim983171darr0

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ ))2i



= lim983171darr0

12π

983133 2π

0

ϕ(983171 cos(θ )983171 sin(θ )) dθ

= ϕ(0)




part A

Pdx +Qdy =

983133 983133

A


994868dxdy

39




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65




E(x t) =

9948301


0 if t le 0


part tminus∆x

994868E = 0




0

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


994868

= lim983171darr0

983133 infin

983171

dt

983133

995014n


983171darr0

983133 infin

983171

dt

983133

995014n


983167 983166983165 983168=0

= lim983171darr0

983133

995014n

dx ϕ(x 983171)E(x 983171)

= lim983171darr0

983133

995014n

dx ϕ(x 983171) middot 1(4π983171)n2


x2991598983171

= lim983171darr0

983133

995014n


2 middot 983171n22n


983133

995014n

dy ϕ(2991598983171 y983171)eminus|y|

2

= πminusn2ϕ(0 0)

983133

995014n

eminus|y|2

dy

= ϕ(0 0)


40




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65




983133

995014n

ϕ(minusλ)P(λ)

dλ



983133

995014n


dλ =1

(2π)n

983133

995014n









ϕ(λprime z) =

983133

995014nminus1



994809dx prime


|zmϕ(λprime z)|=994802994802994802994802983133

995014nminus1


995014

994878994858part

part xn

994868meminusizxn


994809dx prime994802994802994802994802

=

994802994802994802994802983133

995014nminus1



mϕ


994809dx prime994802994802994802994802

le983133

995014n

eIm(z)xn

994802994802994802994802part mϕ

part xmn

994802994802994802994802 dx

≲m eδ|Im(z)|


994802994802994802 partmϕpart xm

n


is which is this




41







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65







Mminus1983131

i=0

ai(λprime)λi

n




i=1








i=1



42








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65








j=1

N j


i=1∆i = 995014nminus1

Now define(xiv)


infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


dλn

994882dλprime



983133

995014n

ϕ(minusλ)P(λ)

dλ =1

(2π)n

983133

995014nminus1

994879983133

Im(λn)=0


dλn

994882dλprime


Im(λn)=0to983125



i

983125∆i



43





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65





infin983131

i=1

983133

∆i

994879983133

Im(λn)=ci


P(λ)dλn

994882dλprime

=1

(2π)n

infin983131

i=1

983133

∆i

994879983133

Im(λn)=0



=1

(2π)n

983133

995014nminus1

983133


=1

(2π)n

983133

995014n

ϕ(minusλ) dλ






44





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65





Xf part αϕ dx


u=983131

α







u(x)

983133

995014n


994819


ϕm(λ) = χ994858λ

m

994868ϕ(λ)



mrarrinfin

994818u(x)

983133

995014m


994819


and so hence

= limmrarrinfin

983133

995014n


983133

995014n






le C983131

|α|leN

supK


≲ langλrangN



45





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65





u=983131

α
















994904


f (x) =1

(2π)n

983133

995014n



f =1


u


46


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65


we see



=994800(2π)n f (1minus∆)Mψ

994801

=994800


=994800

f (1minus∆)M (ρϕ)994801


(1minus∆)M (ρϕ) =983131

α



languϕrang=994818

f 983131

α


994819

=

994818983131

α


994819


u=983131

α


and so we are done



u=983131

α



languϕrang=983131

α









47





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65





part z u= 0 since

part


994902u(x)

part





|u(z)|le C983131

|α|leN

supK









ϕ983171 equiv994816

0 on |x |ge δ+ 1983171

1 on |x |le δ+ 12983171



48



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65



|u(z)|le C983131

|α|leN

supK983171

994802994802part αx994790ϕ983171(x)e



|part βx ϕ983171|≲ 983171|β |


983171 )|Im(z)|


|u(z)|le C983131

|α|+|β |leN

983171|β ||z||α|e(δ+ 1983171 )|Im(z)|


|u(z)|≲ (1+ |z|)N eδ|Im(z)|



|u(z)|≲ (1+ |z|)N eδ|Im(z)|











u(z) =

983133

995014n


49



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65



|zαu(z)|=994802994802994802994802983133

995014n


994802994802994802994802

=

994802994802994802994802983133

995014n



le eδ|Im(z)|983133

995014n

|Dαu| dx

≲m eδ|Im(z)|



u(x) =1

(2π)n

983133

995014n




995014n


983133

995014n


So

|u(x)|≲994802994802994802994802983133

995014n


994802994802994802994802

≲m

983133

995014n





983125Im(λi)=c f (λi) dλi =983125


50















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65















994802994802≲m (1+ |z|)me|Im(z)|m+1








m

αmum = 0



51




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65




minusinfineiλx dx


983133

995014k







983133

995014

1iλ

ddθ

994790eiλθ994791ϕ(θ ) dθ equiv983133

995014L994792eiλθ994793ϕ(θ ) dθ


983133


983133

995014L(N)994792eiλθ994793ϕ(θ ) dθ

983133


= 994974 (λminusN )



995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014L994792eiλΦ(θ )994793ϕ(θ ) dθ

where

L =1

iλΦprime(θ )d


dθ

9948771

iλΦprime(θ )

994880 we get

983133




13 minus 1


12 +

13 +


52






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65






995014eiλΦ(θ )ϕ(θ ) dθ = 994974

994857λminus

12




995014eiλΦ(θ )ϕ(θ ) dθ =

983133

995014eiλΦ(θ )ρ

994858θ

δ

994868ϕ(θ ) dθ

983167 983166983165 983168=I1(λ)

+

983133

995014

9948581minusρ994858θ

δ

994868994868eiλΦ(θ )ϕ(θ ) dθ

983167 983166983165 983168=I2(λ)


53



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65



iλΦprime(θ )d


I2(λ) =

983133

995014eiλΦ(θ )Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

dθ

But then

Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881

= minus 1iλ

ddθ


Φprime(θ )

994881


994791994791ϕ(θ )

Φprime(θ )2+


994791ϕ(θ )

Φprime(θ )minus 1


994791994791ϕprime(θ )

Φprime(θ )

Note


=

983133 θ

0


983133 1

0



983133 1

0







Llowast994878994858

1minusρ994858θ

δ

994868994868ϕ(θ )994881= 994974994858

1λ|θ |2994868+ 994974994858

1λmiddot 1δ|θ |

994868= 994974994858

1λ|θ |

9948781|θ | +

1δ

994881994868




9948581|θ | +

1δ

994868N≲ 1λN |θ |NδN


1δ So


983133






54



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65



|984162θ a(x θ )| where


part apart θk

994868







995014k


983133

995014k



+

983133

995014k













55




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65




995014n

θα




(i) Dαx Dβθ




prime

θ

994857Dαx Dβ

θa994867994802994802994802=994802994802994802Dα+αprimex Dβ+β

prime


prime|




994802994802994802Dαx Dβθ(a1a2)994802994802994802=994802994802994802994802994802983131

αprimeleα

983131

β primeleβ

994858α

αprime

994868994858β

β prime

994868Dαprime

x Dβprime


primex Dβminusβ

prime

θa2

994802994802994802994802994802

≲αβ

983131

αprimeleα

983131

β primeleβ


θa1


prime

θa2

994802994802994802

≲αβ K

983131

αprimeleα

983131





via


995014k

994808983133


994809dθ




56


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65


where

IΦ983171(a) =

983133

995014k





Proof Later




(983183) L =k983131

j=1

a j(x θ )part

part θ j+

n983131

j=1

b j(x θ )part

part x j+ c(x θ )



ddθ



j=1

part

part θ j


n983131

j=1

part

part x j

994792b j(x θ ) bull994793+ c(x θ )

=k983131

j=1

a jpart

part θ j+

n983131

j=1

b jpart

part x j+ c


c = c minusk983131

j=1

part a j

part θ jminus

n983131

j=1

part b j




part b j



57



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65



(L f ) middot g =983133

f middot (Llowastg)





995014k

983133


983133

995014k

983133

X



=

983133

995014k

983133




a jpartpart θ j






995014k

983133

XeiΦ L(M) [a(x θ )χ(983171θ )ϕ(x)]983167 983166983165 983168


dxdθ



part θ jeiΦ = i

partΦ

part θ jeiΦ and

part

part x jeiΦ = i

partΦ

part x jeiΦ


k983131

j=1

partΦ


n983131

j=1

partΦ


994823eiΦ =

994822|θ |2

k983131

j=1

994802994802994802994802partΦ

part θ j

9948029948029948029948022

+n983131

j=1

994802994802994802994802partΦ

part x j

9948029948029948029948022994823

eiΦ

=994790|θ |2 |984162θΦ|2 + |984162xΦ|2

994791eiΦ

=1



τpart

part x jΦ(x θ ) =

part


partΦ

part x j(x τθ )

58




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65




τpart


part


partΦ

part θ j(x τθ )





L = π(x θ )

983093983095minusi|θ |2

k983131

j=1

partΦ


n983131

j=1

partΦ


983094983096


994790995014k0994791)




and

Llowast =983131

j

a jpart

part θ j+983131

j

b jpart

part x j+ c

where









L(M) [a(x θ )ϕ(x)] =983131

|α|leM


59




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65




IΦ983171(a) =

983133

995014k

eiΦ(x θ )a(x θ )χ(983171θ )


995014k

983133




995014k

983133



part θ

994868αχ(983171θ )

994802994802994802994802= |983171||α|(part αχ)(983171θ )



(1+ |983171θ |2)α2

= Cα

99485819831712+ |θ |2994868minus|α|2





L(M) [a(x θ )χ(983171θ )ϕ(x)] =983131

|α|leM




983133

995014k

983133

XeiΦ(x θ )983131

|α|leM


=983131

|α|leM

983133

995014k

983133




|α|leM

983133

995014k

983133


≲K

983131

|α|leM

supK|part αϕ|

60





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65





ϕ994905

= minus994902

IΦ(a)part ϕ

part x i

994905


983133

995014k

983133


part ϕ

part x idxdθ

= lim983171darr0

983133

995014k

983133

X

part

part x i

994792eiΦa(x θ )χ(983171θ )


= lim983171darr0

983133

995014k

983133


+ iapartΦ

part x i


= lim983171darr0

983133

995014k

983133


= langIΦ(a)ϕrang





part

part x i

983133

995014k


983133

995014k


+ iapartΦ

part x i

994881dθ

=

983133

995014k

part

part x i

994792eiΦ(x θ )a(x θ )994793

dθ



IΦ(a) =1

(2π)n

983133

995014n

ei x middotθ dθ



1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


= lim983171rarr0

1(2π)n

983133

995014n

983133

995014n


61


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65


= lim983171rarr0

983133

995014n




= ϕ(0)



δ0 =1

(2π)n

983133

995014n

ei x middotθ dθ








995014k


is smooth


62



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65



IΦ(a) =

983133

995014k


983167 983166983165 983168smooth by Lemma 55

+

983133

995014k


dθ









|984162θΦ(x0θ )|≳ 1





|984162θΦ|2



ψIΦ(a) =

983133

995014k

eiΦ(x θ ) a(x θ )ψ(x)983167 983166983165 983168=a

dθ

63




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65




983133

995014k

983133


= lim983171rarr0

983133

995014k

983133


= lim983171rarr0

983133

995014k

983133




X

994808983133

995014k


994809ϕ(x) dx



ψIΦ(a) =

983133

995014k





Dαδ0(x) =1

(2π)n

983133

995014k







64



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65



u(y t) =1

(2π)n

983133

995014n





1(2π)n

983133

995014n



u(y 0) =1

(2π)n

983133

995014n





65

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Distribution Theory (and its Applications) Lecturer ... · Distribution Theory (and its Applications) Lecturer: Anthony Ashton Scribe: Paul Minter ... Recommended book: Hörmander,