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London Mathematical Society Lecture Note Series. 202

The Technique ofPseudodifferential OperatorsH.O. CordesEmeritus, University of California, Berkeley

AMBRIDGEUNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESSCambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo

Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.orgInformation on this title: www.cambridge.org/9780521378642

© Cambridge University Press 1995

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the writtenpermission of Cambridge University Press.

First published 1995

A catalogue record for this publication is available from the British Library

ISBN 978-0-521-37864-2 paperback

Transferred to digital printing 2008

To my 6 children, Stefan and Susan

Sabine and Art, Eva and Sam

TABLE OF CONTENTS

Chapter 0. Introductory discussions 1

0.0. Some special notations, used in the book 1

0.1. The Fourier transform; elementary facts 3

0.2. Fourier analysis for temperate distributions onIn

9

0.3. The Paley-Wiener theorem; Fourier transform for

general uE D' 14

0.4. The Fourier-Laplace method; examples 20

0.5. Abstract solutions and hypo-ellipticity 30

0.6. Exponentiating a first order linear differential

operator 31

0.7. Solving a nonlinear first order partial differen-

tial equation 36

0.8. Characteristics and bicharacteristics of a linear

PDE 40

0.9. Lie groups and Lie algebras for classical analysts 45

Chapter 1. Calculus of pseudodifferential operators 52

1.0. Introduction 52

I.I. Definition of pdo's 52

1.2. Elementary properties of ipdo's 56

1.3. Hoermander symbols; Weyl pdo's; distribution

kernels 60

1.4. The composition formulas of Beals 64

1.5. The Leibniz' formulas with integral remainder 69

1.6. Calculus of 1pdo's for symbols of Hoermander type 72

1.7. Strictly classical symbols; some lemmata for

application 78

Chapter 2. Elliptic operators and parametrices in Mn 81


2.1. Elliptic and md-elliptic Vdo's 82

2.2. Formally hypo-elliptic pdo's 84

2.3. Local md-ellipticity and local md-hypo-ellipticity 87

2.4. Formally hypo-elliptic differential expressions 91

2.5. The wave front set and its invariance under yxlo's 93

viii Contents

2.6. Systems of ,do's 97

Chapter 3. L2-Sobolev theory and applications 99


3.1. L2-boundedness of zero-order do's 99

3.2. L2-boundedness for the case of 6>0 103

3.3. Weighted Sobolev spaces; K-parametrix and Green

inverse 106

3.4. Existence of a Green inverse 113

3.5. Hs-compactness for ftpdo's of negative order 117

Chapter 4. Pseudodifferential operators on manifolds with

conical ends 118


4.1. Distributions and temperate distributions on

manifolds 119

4.2. Distributions on S-manifolds; manifolds with

conical ends 123

4.3. Coordinate invariance of pseudodifferential

operators 129

4.4. Pseudodifferential operators on S-manifolds 134

4.5. Order classes and Green inverses on S-manifolds 139

Chapter 5. Elliptic and parabolic problems 144


5.1. Elliptic problems in free space; a summary 147

5.2. The elliptic boundary problem 149

5.3. Conversion to an &n-problem of Riemann-Hilbert

type 154

5.4. Boundary hypo-ellipticity; asymptotic expansion

mod av 157

5.5. A system of fide's for the Vj of problem 3.4 162

5.6. Lopatinskij-Shapiro conditions; normal solvabi-

lity of (2.2). 169

5.7. Hypo-ellipticity, and the classical parabolic

problem 174

5.8. Spectral and semi-group theory for ado's 179

5.9. Self-adjointness for boundary problems 186

5.10. C*-algebras of tpdo's; comparison algebras 189

Chapter 6. Hyperbolic first order systems 196


6.1. First order symmetric hyperbolic systems of PDE 196

6.2. First order symmetric hyperbolic systems of

fide's on n. 200

6.3. The evolution operator and its properties 206

Contents ix

6.4. N-th order strictly hyperbolic systems and

symmetrizers. 210

6.5. The particle flow of a single hyperbolic pde 215

6.6. The action of the particle flow on symbols 219

6.7. Propagation of maximal ideals and propagation

of singularities 223

Chapter 7. Hyperbolic differential equations 226


7.1. Algebra of hyperbolic polynomials 227

7.2. Hyperbolic polynomials and characteristic surfaces 230

7.3. The hyperbolic Cauchy problem for variable

coefficients 235

7.4. The cone h for a strictly hyperbolic expression

of type e° 238

7.5. Regions of dependence and influence; finite

propagation speed 241

7.6. The local Cauchy problem; hyperbolic problems

on manifolds 244

Chapter 8. Pseudodifferential operators as smooth

operators of L(H) 247


8.1. ,do's as smooth operators of L(H0) 248

8.2. The 11DO-theorem 251

8.3. The other half of the gbbO-theorem 257

8.4. Smooth operators; the V -algebra property;

'Wdo-calculus 261

8.5. The operator classes 'S and 'IVL , and their

symbols 265

8.6 The Frechet algebras y"x0, and the Weinstein-

Zelditch class 271

8.7 Polynomials in x and ax with coefficients in 'TX 275

8.8 Characterization of qtX by the Lie algebra 279

Chapter 9. Particle flow and invariant algebra of a semi-

strictly hyperbolic system; coordinate invariance

of OpWxm. 282


9.1. Flow invariance of V10 283

9.2. Invariance of Vsm under particle flows 286

9.3. Conjugation of Optpx with eiKt , KE Opy)ce 289

9.4. Coordinate and gauge invariance; extension to

S-manifolds 293

9.5. Conjugation with eiKt for a matrix-valued K=k(x,D) 296

x Contents

9.6. A technical discussion of commutator equations 301

9.7. Completion of the proof of theorem 5.4 305

Chapter 10. The invariant algebra of the Dirac equation 310


10.1. A refinement of the concept of observable 314

10.2. The invariant algebra and the Foldy-Wouthuysen

transform 319

10.3. The geometrical optics approach for the Dirac

algebra P 324

10.4. Some identities for the Dirac matrices 329

10.5. The first correction z0 for standard observables 334

10.6. Proof of the Foldy-Wouthuysen theorem 343

10.7. Nonscalar symbols in diagonal coordinates of 350

10.8. The full symmetrized first correction symbol zS 356

10.9. Some final remarks 367

References 370

Index 380

P R E F A C E

It is generally well known that the Fourier-Laplace trans-

form converts a linear constant coefficient PDE P(D)u=f on Rn to

an equation P()u ()=f"(), for the transforms u , f- of u and f,

so that solving P(D)u=f just amounts to division by the polynomial

The practical application was suspect, and ill understood,

however, until theory of distributions provided a basis for a log-

ically consistent theory. Thereafter it became the Fourier-Laplace

method for solving initial-boundary problems for standard PDE. We

recall these facts in some detail in sec's 1-4 of ch.0.

The technique of pseudodifferential operator extends the

Fourier-Laplace method to cover PDE with variable coefficients,

and to apply to more general compact and noncompact domains or

manifolds with boundary. Concepts remain simple, but, as a rule,

integrals are divergent and infinite sums do not converge, forcing

lengthy, often endlessly repetitive, discussions of 'finite parts'

(a type of divergent oscillatory integral existing as distribution

integral) and asymptotic sums (modulo order -oo).

Of course, pseudodifferential operators (abbreviated ado's)

are (generate) abstract linear operators between Hilbert or Banach

spaces, and our results amount to 'well-posedness' (or normal sol-

vability) of certain such abstract linear operators. Accordingly

both, the Fourier-Laplace method and theory of ipdo's, must be seen

in the context of modern operator theory.

To this author it always was most fascinating that the same

type of results (as offered by elliptic theory of ipdo;'s) may be

obtained by studying certain examples of Banach algebras of linear

operators. The symbol of a ipdo has its abstract meaning as Gelfand

function of the coset modulo compact operators of the abstract ope-

rator in the algebra.

On the other hand, hyperbolic theory, generally dealing with

a group exp(Kt) (or an evolution operator U(t)) also has its mani-

festation with respect to such operator algebras: conjugation with

xii Preface

exp(Kt) amounts to an automorphism of the operator algebra, and

of the quotient algebra. It generates a flow in the symbol space

essentially the characteristic flow of singularities. In [Ci],

[C2] we were going into details discussing this abstract approach.

We believe to have demonstrated that ado's are not necessary

to understand these fact. But the technique of pdo's, in spite of

its endless formalisms (as a rule integrals are always 'distribut-

ion integrals', and infinite series are asymptotically convergent,

not convergent), still provides a strongly simplifying principle,

once the technique is mastered. Thus our present discussion of

this technique may be justified.

On the other hand, our hyperbolic discussions focus on in-

variance of ido-algebras under conjugation with evolution opera-

tors, and do not touch the type of oscillatory integral and fur-

ther discussions needed to reveal the structure of such evolution

operators as Fourier integral operators. In terms of Quantum mecha-

nics we prefer the Heisenberg representation, not the Schroedinger

representation.

In particular this leads us into a discussion of the Dirac

equation and its invariant algebra, in chapter X. We propose it as

algebra of observables.

The basis for this volume is (i) a set of notes of lectures

given at Berkeley in 1974-80 (chapters I-IV) published as preprint

at U. of Bonn, and (ii) a set of notes on a seminar given in 1984

also at Berkeley (chapters VI-IX). The first covers elliptic (and

parabolic) theory, the second hyperbolic theory. One might say

that we have tried an old fashiened PDE lecture in modern style.

In our experience a newcomer will have to reinvent the theo-

ry before he can feel at home with it. Accordingly, we did not try

to push generality to its limits. Rather, we tend to focus on the

simplest nontrivial case, leaving generalizations to the reader.

In that respect, perhaps we should mention the problems (partly of

research level) in chapters I-IV, pointing to manifolds with coni-

cal tips or cylindrical ends, where the 'Fredholm-significant sym-

bol' becomes operator-valued.

The material has been with the author for a long time, and

was subject of many discussions with students and collaborators.

Especially we are indebted to R. McOwen, A.Erkip, H. Sohrab, E.

Schrohe, in chronological order. We are grateful to Cambridge Uni-

versity Press for its patience, waiting for the manuscript.

Berkeley, November 1993 Heinz 0. Cordes

Chapter 0. INTRODUCTORY DISCUSSIONS.

In the present introductory chapter we give comprehensive

discussions of a variety of nonrelated topics. All of these bear

on the concept of pseudo-differential operator, at least in the

author's mind. Some are only there to make studying Ado's appear

a natural thing, reflecting the author's inhibitions to think

along these lines.

In sec.! we discuss the elementary facts of the Fourier

transform, in sec.'s 2 and 3 we develop Fourier-Laplace trans-

forms of temperate and nontemperace distributions. In sec.4 we

discuss the Fourier-Laplace method of solving initial-value pro-

blems and free space problems of constant coefficient partial

differential equations. Sec.5 discusses another problem in PDE,

showing how the solving of an abstract operator equation together

with results on hypo-ellipticity and "boundary-hypo-ellipticity"

can lead to existence proofs for classical solutions of initial-

boundary problems. Sec.6 is concerned with the operator eLt , for

a first order differential expression L . Sec.'s 7 and 8 deal with

the concept of characteristics of a linear differential expression

and learning how to solve a first order PDE. Sec.9 gives a mini-

introduction to Lie groups, focusing on the mutual relationship

between Lie groups and Lie algebras. (Note the relation to i,do's

discussed in ch.8).

We should expect the reader to glance over ch.0 and use it

to have certain prerequisites handy, or to get oriented in the

serious reading of later chapters.

0. Some special notations.

The following notations, abbreviations, and conventions will

be used throughout this book.

(a) xn (2n)-n/2 , ox = xndx1dx2...dxn = xndx .

(b) (x) = (l+Ix12)!/2 , (1+1 12)!,2 , etc.

!

2 0. Introductory discussions

(c) Derivatives are written in various ways, at convenience:

For u=u(x)=u(x1,...,xn) we write u(a)=aXU=axlax2...u =

a a=aatlax",...a n/ax nu. Or, ulx =ax u, ulx to denote the n-vector

j J

with components ulx , V"u for the k-dimensional array with compo-J

nents ulxi xi .. For a function of (x,i;)=(xl " '.'fin)z '

it is often convenient to write u(a)=aaXU.

(d) A multi-index is an n-tuple of integers a=(a1,...an)

We write laI=lall+...+lanl , al=all...an! , ((3)=(R )...( ),

a axa= xl 1...xn n , etc., IIn={all multi-indices} .

(e) Some standard spaces: &n = n-dimensional Euclidean space

Bn=directional compactification of Ien (one infinite point «x added

in every direction (of a unit vector x).

(f) Spaces of continuous or differentiable complex-valued

functions over a domain or differentiable manifold X (or sometimes

only X=1en): C(X) = continuous functions on X ; CB(X)= bounded con-

tinuous functions on X; CO(X)= continuous functions on X vanishing

at a ; CS(X) = continuous functions with directional limits; CO(X)

= continuous functions with compact support; Ck(X)= functions with

derivatives in C, to order k, (incl. k=oo). CB00(X)="all derivatives

exist and are bounded". The Laurent-Schwartz notations D(X)=CO(X),

E(X)=C°(X) are used. Also S= S(Jk)= "rapidly decreasing functions"

(All derivatives decay stronger as any power of x). Also, distri-

bution spaces D', E', SO.

(g) LP-spaces: For a measure space X with measure dµ we wri-

te Lp(X)=Lp(X,dµ)={measurable functions u(x) with luIp integrable}

for lsp<oo; L00(X)={essentially bounded functions}.

(h) Maps between general spaces: C(X,Y) denotes the conti-

nuous maps X->Y . Similar for the other symbols under (f), i.e.,

CB(X,Y) ..... .

(i) Classes of linear operators (X= Banach space) : L(X)

(K(%))= continuous (compact) operators; GL(X) (U(H)) = invertible

(unitary) operators of L(X) (of L(H), H=Hilbert space); Un U(Mn).

For operators X- Y, again, L(X,Y), etc.

(j) The convolution product: For u,v E LI(len) we write w(x)

=(u*v)(x)=xnfdyu(x-y)v(y) (Note the factor xn (2n)-n/2).

(k) Special notation: " X CC Y " means that X is contained

in a compact subset of Y .

0.1. The Fourier transform 3

(1) For technical reason we may write limE-Oa(E)= alE..O

(m) Abbreviations used: ODE (PDE) = ordinary (partial) diff-

erential equation (or "expression"). FOLPDE (or folpde)= first or-

der linear partial differential equation (or "expression"); 'pdo=

pseudodifferential operator.

(n) Integrals need not be existing (proper or improper) Rie-

mann or Lebesgue integrals, unless explicitly stated, but may be

distribution integrals By this term we mean that either (i) the in-

tegral may be interpreted as value of a distribution at a testing

function-the integrand may be a distribution, or (ii) the limit of

Riemann sums exists in the sense of weak convergence of a sequence

of (temperate) distributions, or (iii) the limit defining an impro-

per Riemann integral exists in the sense of weak convergence, as

above, or (iv) the integral may be a 'finite part' (cf. 1,4).

(o) Adjoints: For a linear operator A we use 'distribution

adjoint' A and 'Hilbert space adjoint' A*, corresponding to trans-

pose AT and adjointAT=A*,

in case of a matrix A=((a.k)), respecti-

vely. For a symbols a* (or a+) may denote the symbol of

the adjoint 'tpdo of a(x,D) , as specified in each section.

(p) supp u (sing supp u (or s.s.u)) denotes the (singular)

support of the distribution u.

1. The Fourier transform; elementary facts.

Let u E L1(&n) be a complex-valued integrable function.

Then we define the Fourier transform u^= Fu of u by the integral

(1.1) u^(x) = J x E &n ,

with x =x. = an existing Lebesgue integral. Clearly,

(1.2) u^ (x) I s IIu11L1 n9lxlu(x)l=

Note that u^ is uniformly continuous over In : We get

u^ (x)-u^ (y) I s 2f¢tI

(1.3)

s NIX-111 llullL1 + R1

where the right hand side is <E if N is chosen for <E/4,


and then Ix-yi< a/(2NIIuiI ) Moreover, we get uÊ CO(&n), i.e.,L

limlxI.00u^(x)=0, a fact, known as the Riemann-Lebesgue lemma.

To prove the latter, we reduce it to the case of u E CO(&n):00

is known to be dense in LI . By (1.1) we getThe space CO

(1.4) Iu^ (x)-v^ (x) I s IIu-vilL1 < e/2 , as v E CO , IIu-vuIL1<E/2.

Hence limI X I

OOv^ (x)=0 implies I u^ I s I u^ -v^ I+ I v^ I< e wheneverx is chosen according to IvÎ < e/2 .

But for v E CO the Fourier integral extends over a ball ICI

s N only, since v=0 outside. We may integrate by parts for

(1.5) IxI2u^(x) =-J0 =_f =-(Av)^(x)

with the Laplace differential operator e _=1a 2 Clearly we

have tv E CO C LI as well, whence (1.1) applies to AV , for

(1.6) Iv^ (x) I s 11 AV ll

L1/1X12 0 as IxI

completing the proof.The above partial integration describes a general method to

be applied frequently in the sequel. (1.6) may be derived under

the weaker assumptions that vE C2, and that all derivatives v(a),

Ials2 , are in LI (cf. pbm.5). On the other hand, there are simple

examples of u(4 LI such that u^ does not decay as rapidly as (1.6)

indicates. In particular, uE L' exists with u^(4 L' (cf.pbm.4).

This matter becomes important if we think of inverting the

linear operator F:LI - CO defined by (1.1), because formally an

inverse seems to be given by almost the same integral. Indeed,

define the (complex) conjugate Fourier transform F:LI -> CO by

Fu = (Fu) , or, u"= Fu , where

(1.7) u" (x) = f u E LI (len) .

Then, in essence, it will be seen that F is the inverse of

the operator F. More precisely we will have to restrict F to a

(dense) subspace of LI, for this result. Or else, the definition

of the operator F must be extended to certain non-integrable func-

tions, for which existence of the Lebesgue integral (1.7) cannot

be expected. Both things will be done, eventually.


It turns out that F induces a unitary operator of the Hil-

bert space L2(&n): We have Parseval's relation:

(1.8) JRn4IxIu^(x)12 = fRnoxlu(x)12 , for all u E_= L'(,n) ,2 (e)

Formula (1.8) is easier to prove as the Fourier inversion

formula, asserting u^'=u`^=u for certain u: We may write

. ( .-1. )(1.9)

Qglxu^ (x)v^ (x) = _1

Neix

7 j -"j qIx.N -N

assuming that u,vE LI(&n) , with the 'cube' QN {lx.IsN,j=1,..,N}

some integer N>0. Indeed, the interchange of integrals leading to

(1.9) is legal, since the integrand is L'(QNxinxlen)

Note that fNeistdt = 2sis sNs340 , = 2N , s=0, allowing

-N

evaluation of the inner integrals at right of (1.9). With f A A _f Af 911 , and it = - IN , A = N n,,

, (1.9) assumes the form

(1.10) fQN 41xu^ (x)v^ (x) = j

where cp(t) = (2/n) 1/2 sin t , t340, continuously extended into t=0.

For vE C(Rn), as N-- , the function will converge

to independent of . Thus one expects the inner integral at

right of (1.10) to converge to j = since

(1.11) P sin t dt/t = n/20

Legalization of this argument will confirm Parseval's rela-

tion, since the right hand converges to the right hand side of

(1.8), as N-. With uE L' and vE Co (setting cen(t)=ikp(t.)) write

(1.12) f Rng1xuv

To show that the inner integral at left goes to 0 as N->oo it is

more skilful to use the integration variable O=1;/N, dt=NndO. For

n=1 , fsin NO f101sb + fjolab = I0 + I. .

Here we get (with w(O) =

JI01 sOIIv'III N((w(O)cos(NO)le=&b+

f1OjZbcos(NO)wl0(O)dO).

The latter gives 100 s (11vIILW+ 11v' 11L00) , with a constant c, only

depending on the volume of supp v, i.e., it is fixed after fixing

v . The estimates imply the inner integral to go to 0, uniformly


as xE lgn. For uE L' the Lebesgue theorem then implies the left

hand side of (1.12) to tend to 0, as N-- , for each fixed vE CO

For general n the proof is a bit less transparent, but remains

the same: Split the inner integral into a sum of integrals over a

small neighbourhood of 0 and its complement. In the first term use

differentiability of v; in the second an integration by parts.

We now have a 'polarized' Parseval relation, in the form

(1.13) RngIxu^ V. ngxuv , for u E L1 , v E COO

For u E LIf1L2 pick a sequence ujECO with IIu-ujIILl - 0, 11u-uill L2 0,

as is possible. Then, since uj-ulE CO C L2 , (1.13) with u=v=uj-vj

implies Iluj^-u1ÎIL2=IIuj-u11IL2 0, j,l - . In other words, uj and

uj^ both converge in L2 . Clearly, uj u^. Indeed, initially we

showed uniform convergence over &n, while the V -limit z=lim u

satisfies (u^ ,(p)=fz pdx for all cpE Cc". This yields f (u^ -z)cpdx=0 for

all such cp, hence u '=z (almost everywhere), since CO is dense in

V. Substituting u=v=uj in (1.13), letting j-3, it follows that

(1.8) is valid for all u ELI(1L2 , confirming Parseval's relation.

Clearly (1.13) also holds for all u,vE L'(lL2. We use it to

prove the Fourier inversion Let n=1. For vE L'("IP, u=X[0,x0 ]'

some

xo>0 apply (1.13). Confirm by calculation of the integral that

(1.14) (2n)1/2u^(x) =(e-ixxo_

1)/(-ix) = hxo(x) , x # 0 ,

hence

xo(1.15) v(x)dx = gIxv^ (x)hxo (x)dx

0

The Fourier inversion formula is a matter of differentiating

(1.15) for xo under the integral sign, assuming that this is legal

Consider the difference quotient:

(1.16) (20)-Ixo+b

v(x)dx = JgIxv^(x)eixxo sin 6x

xo -b

Assuming only that v , v^ both are in LI , it follows indeed that

(1.17) limn-0(26)-nfQ v(x)dx =f9lxv^(x)eixxo= (v^ )I (xo ), xoE 2n.

xo ,s(Actually, our proof works for n=1 , xo > 0 only , but can easily

be extended to all xo , and general n . One must replace the deri-

vative d/dxo by a mixed derivative an/(axol...axo n). ) Indeed,


letting 6-0 in (1.17) we obtain (1.15), using that sin(6x) /(Sx)

- 1 uniformly on compact sets, and boundedly on & , as 6-0 .

If v is continuous at x0 then clearly the left hand side

of (1.17) equals v(xo) , giving the Fourier inversion formula, as

it is well known. For n=1, if v has a jump at xo then the left

hand side of (1.17) equals the mean value (v(xo+0)+v(xo-0))/2

Again for n=1 a limit of (1.16), as exists, if only

(1.18) limaloo v^ (x)pix ,-a

the principal value, exists (cf. pbm.6), without requiring VA E-= L'

We summarize our results thus far:

Proposition 1.1. The Fourier transform u^ of (1.1) and its com-

plex conjugate u" =(u^) are defined for all u E L1(&n) , and we

have u^ , u" E CO(1Qn) . For u E L1(In)1L2(In) we have Parseval'srelation (1.8) . If both u E L1(In) and u^ E L1(1en) hold, then we

have u^" (x) = u"^ (x) = u(x) for almost all x E In

It is known that the Banach space L1(&n) is a commutative

Banach algebra under the convolution product

(1.19) u*v = w , w(x) = fOyu(x-Y)w(Y) = f41Yv(x-Y)u(Y)

Indeed,

(1.20) IIwIIL1=JIw(x)Idx s xnfdxfdyIu(x-Y)IIv(Y)I = KnIIuIIL1IIvIIL1

Prop.1.2, below, clarifies the role of the Fourier transform F for

this Banach-algebra: F provides the Gelfand homomorphism.

Proposition 1.2. For u,v E L' let w = u*v . Then we have

(1.21) w^ ( ) = u^ E &n

Proof. We have

fgixe fpiye

The substitution x-y=z , dy=dz thus confirms (1.21), q.e.d.

The importance of the Fourier transform for PDE's hinges on

Proposition 1.3. If u(P) E L1 for all (3s a then

(1.22) u(a)^ h E &n .


Proof.Partial integration gives fdxe

(with vanishing boundary integrals), implying (1.21), q.e.d.

Given a linear partial differential equation

(1.23) P(D)u = f , P(D) = a Daja sN ax

where fE L1(&n), Dx =-iax , one might attempt to find solutions byj J

taking the Fourier transform. With proper assumptions (1.21) gives

(1.24)

Assuming that e = (P(X))' exists, (1.24) will assume the form

(1.25) u" e,

which by prop.1.2 (and Fourier inversion) is equivalent to

(1.26) u(x) = fgIye(x-y)f(y) .

Presently, (1.26) can only have a formal meaning, since nor-

mally (1/P)(4 L', or f (4 L' , or u(4 LL, in practical applications.

However, as to be discussed in the sections below, the Four-

ier transform may be extended to more general classes of functions

and to generalized functions. Then (1.26) yields a powerful tool

for solving problems in constant coefficient PDE's (cf. sec.4).

Problems. 1) For n=1 obtain the Fourier transforms of the

functions a) (a2+x2)-1, a>O; b) (sin2ax)/x2, a>O; c) 1/cosh x

22) For general n obtain the Fourier transform of

a_ ax, a>0

3) Obtain the Fourier transform of f(x) = (1+IxI2)-v, where v>n/2

(Hint: A knowledge of Bessel functions is required for this pro-

blem). 4) Construct a function f(x) E L1(&n) such that f^(4 L1

5)The Riemann-Lebesgue lemma states that fÊ CO whenever f E L1

Is it true that even f (x) = O((x)-E) for each f E L1 with some

8>O ? 6) Combining some facts, derived above, show that, for n=1,

every piecewise smooth function f(x) E L1(&) has a Fourier trans-

form satisfying f(x) = O(1/x) , as Jxj is large, and satisfying

(1.27) (f(x+0)+f(x-0))/2 =limfa91yelxyf^(y) , x E It

Here 'piecewise smooth' means, that & may be divided into finitely

many closed subintervals in each of which f is C1 , possibly after

changing its value at boundary points.

0.2. Fourier analysis for temperate distributions

2. Fourier analysis for temperate distributions on &n

9

We assume that the reader is familiar with the concept of

distribution, as a continuous linear functional on the space

D(In) = CO(&n) . A linear functional f:D - C is said to be conti-

nuous if (f,gj)-.0 whenever (pj-0 in D. The latter means that (i) fj

E D, j=1,2,..., (ii) supp cpjE KCC &n , K independent of j, (iii)

sup{I cp(a)(x)I: xE &n} - 0, as j->oo, for every a. The space of dist-

ributions on In is called D'=D'(&n). The space Llloc(&n) of local-

ly integrable functions is naturally imbedded in DI by defining

(2.1) (f,(p) = ff(x)g(x)dx , for fE L11oc

The derivatives f(a)=aaf of a distribution f E D' are defined by

( 2 . 2 )(f(a),(P)

= ( - 1 ) I a l ( f , ( , ( a ) ) , q E D

the product of a distribution f E D' and a C"O(&n) function g by

( 2 . 3 ) (gf,lp) = (f,g(q) , p E D .

Thus Lf is defined for any distribution fE D,(Rn) and linear dif-

ferential operator L=Yaaact with coefficients aa() E C,(&n)

While the value f(x) of a distribution at a point x is a

meaningless concept, one may talk about the restriction fl!a of fE

D.(&n) to an open subset S2 , and its properties: First of all, the

space DI(S2) of distributions over a consists of the continuous lin-

ear functionals on D(c)=CO(SZ), with continuity defined as for &n.

For fE D,(&n), the restriction fID(St) defines a distribution of

D'(a), denoted by f192. Thus, for example, it is meaningful to say

that fE DI(&n) is a function (a Ck(f)-function, etc.) in an open

set 0C &n - it means that fISz has this property. For a distribut-

ion fE DI (a) on an open set the derivatives and product with gE

COO(S2) is defined as in (2.2) , (2.3) . The support supp f (singu-

lar support sing supp f) of fE D' is defined as the smallest clo-

sed set E (intersection of all closed sets E) such that f=0 (such

that f isC00

) in the complement of E .

The concept of Fourier transform can be generalized to distri-

butions on &n, with multiple benefit: Some non-L'-functions will

get distributions as Fourier transforms. Certain distributions

will get functions as Fourier transforms. The Fourier inversion

formula and many assumptions (limit interchanges) will simplify.


We used the Fourier integral of (1.1) only for uE L'(&n).

It is practical to introduce a growth restriction for uE D'(&n) if

we want u^ to be a distribution again. Later on (sec.3) we also

define u^ for general uE D'(&n) , but it no longer will be a dist-

ribution in D'(&n). We follow [Schwl] here, but [GS] in sec.3.

The growth restriction is imposed by requesting that uE D'

allows an extension to a larger space of testing functions called

S. Here S - the space of rapidly decreasing functions- consists of

all cpE C"O(&n) such that for all multi-indices a and k=1,2,...,

(2.4) sP(a)(x) = 0((X)-k)

- the derivatives of y decay faster than every power(x)-k

Note that, equivalently, we could have prescribed that for eve-

ry a one (and the same) of the following conditions be satisfied:

(x)ku(a)(x) (for every k=1,2,..), or xOu(a)(x) (for every (3),

(2.5) or (AM) (a) (for every 0), is 0(1) , or is o(1) , or is

CB , or CO , or L2 , or LP (for some l sps )

Indeed, for a given a one of these conditions may be weaker

or stronger than the other. However for all a simultaneously all

conditions are equally strong. One must use Leibniz' formula to

handle interchanges of as and multiplications (cf. lemma 2.8).

The above at once gives the following:

Theorem 2.1. We have SC L1(&n), so that u^ of (1.1) (and u") are

defined on S. Moreover, for uE S, we have u^, u"E S, and

(2.6) (u^ )" (x) = (u" )^ (x) = u(x) , x E In .

The Fourier transform and its conjugate therefore define bijec-

tive linear maps S -S , inverting each other.

Proof. Using repeated partial integration and

get fdxe

=iIaI+ISIa hence

(2.7) (xRu(a))A

In fact, we get xsu(a) E L' , for every a,(3 , by the equivalence

0.2. Fourier analysis for temperate distributions 11

(2.5) , for u E S . Therefore the right hand side is in CO , for

every a,(3 , so that uÊ S , again by the equivalence (2.5). Thus

we get u^ E S for all u E S . Similarly for """ . Also, the Fou-

rier inversion formula holds for u E S , and the left hand side of

(1.17) equals v(x). This implies (2.5), also by taking complex

conjugates. The bijectivity then follows at once, q.e.d.

Following Schwartz we introduce distributions with controlled

growth at infinity - so called temperate distributions - over sz=&n

as continuous linear functionals over S. The space of all tempera-

te distributions is denoted by S'. Clearly, S DD , so that a func-

tional u over S induces a functional over D - its restriction uID.

Definition 2.2. A sequence of functions cpj ES is said to converge

to 0 (in S) if for every multi-index a and k = 0,1,2,... the

sequence (x)kcpj(a)(x) converges to zero uniformly for all x E In

Definition 2.3. A linear functional u over S is said to be conti-

nuous if cpjE S , (p,-.0 in S implies (u,gj) - 0Temperate distributions are distributions. More precisely

speaking: For uE S' the restriction uID determines u uniquely, and

uIDE D' (1n). To confirm this we must prove:

Lemma 2.4. a) If cpjE D, (Pj-> 0 in D, then we also have cpj- 0 in S.

b) For cpE S there exists a sequence cpjE D such that q)-pj->0 in S

From lemma 2.4 it follows that for WE S' the restriction v=

uID is continuous over D : If cpj-0 in D , then c.->0 in S (by (a)),

hence (v, p) =(u, (pj) -0. Hence vE D' . Furthermore, if u, wE S' haveuID=wID=vE D', then for gE S let (p. be a sequence of (b) above.

Get u-wE S' , (u-w, c-cpj) -0. Hence 0=(u-w, (pj) _(v-v, cpj) -+( u-w, (p) , im-plying that (u, cp) =(v, (p) for all y e S, or u=v, so that indeed uE S'is uniquely determined by its restriction v=uIDE D' .

Proof of lemma 2.4. (a): IfkgjE D, cpj-0 in D then supp Tea) C K=

In, while the functions (x) are bounded in K. Thus the uniform

convergence (x)kcp.(a)(x)-,0 in pn follows from the uniform conver-7

gence j(a)(x)-0 in I , and we have y.- 0 in S, proving (a).

To prove (b), let X(x)EE

CoO(&n)

satisfy x(x)=l near 0. For a

qE S define cpj(x)=T(x)x(x/j), j=1,2,... , so that qjE D. Setting

w.(x)=1-x(x/j), get p.=g-c.=Tw.=0 in Ixlsl for large j. Note,

x) is a linear combination of ORY ' j=(x) kcp(') wj (Y) , (3+y =a

where sup{IOPY'j(x)I:x E &n} s

sup at right goes to 0 as 1-- (i.e.,as j-co).

Also, sup{Iwj(Y)I}=j-IYIsup{w(x):xE 2n}s c. Thus Vj->0 in S, q.e.d.


Note that polynomials, and delta functions S(a)(x-a) are ex-

amples of temperate distribution. However, ex (4 S(le) (pbms.2,3).

To generalize F we still require the following.

Corollary 2.5. The transforms F and F both have the property that

cjE S , Tj- 0 in S implies Fcpj- 0 FT j- 0 in SIt is sufficient to prove this for F. Again we need an equi-

valence like (2.2), now for the property in S' :

Proposition 2.6. Let q E S , j=1..... . Then 'ypj- 0 in S ' is

equivalent to each of the following conditions:

(x)kgpj(a) 0 , or Apj(a) a 0 , or (xs(pj)(a) 0

(2.8) for all multi-indices a , or k=0,1,2,..., in one (and

the same) of the norms of CB(1n) or Lp(&n) , 1spso .

For the proof cf. lemma 2.8.

Using prop.2.6, lemma 2.5 is a matter of (1.2), and (2.7).

Indeed, if pj- 0 in S , we have IlxRcpj(a)IILI-> 0 , j hence

(xaq)j^ ) (P) IICB_ 0 , implying (pj^ - 0 , q.e.d.For a given u E S' , observe that u^ , defined by

(2.9) (u^ ,p) = q E S ,

defines a functional in S', since q.- 0 in S implies p^- 0 in S

(by cor.2.5) , hence (u,cp^) - 0 . If U E LI(2n) then it follows

that u E S' (cf. pbm.3). In that case we have

(2.10) (u,cP^) = ix= (p E S

by Fubini's theorem, since the integrand is L'(12n). Thus, for uE

L', (2.10) implies that the functional (2.9) coincides with that

of the Fourier transform u^ of (1.1). Accordingly, for a general u

E S' we define the Fourier transform u^ as the functional of (2.9)

and the conjugate Fourier transform uv by

(2.11) (u" ,(P) = (u,cp°) , (P E S .

It is clear at once that we have

Theorem 2.7. The (conjugate) Fourier transform coincides with the

0.2. Fourier analysis for temperate distributions 13

(conjugate) Fourier transform previously defined for L1-functions

(cf.(1.1), and (1.7)) . We have the Fourier inversion formula

(2.12) (u^ )` = (u` )^ = u , for all u E S'.

Also, for u E S' we have E S' , and (2.7) holds as well.

Prop.2.6 and (2.2) follow from the (evident) lemma, below.

Lemma 2.8. a) We have (using Leibniz' formula and its adjoint)

(2.13) (xau) (R) = IC a(3Yxa-Yu(R-Y) , xau(1) = EdaRY

(xa-Y u) (R-Y) ,

with finite sums and constantscaRY , daRY

b) We have

(2.14) Ixals(x)lal , and (x)ksc (x011 with a constant ck laIsk

k

c) We have

(2.15) (lullLp

s II(x)-kllLp

II(x)kull Loo, Isp<oo , k>n/p

d) We have

Ilull scIIU^ II lscll (1+Ixl )n+1u^ II c 7, Ilxau^ II(2.16) L L L lalsn+l L

= C E II (U(a)

)^ II "o s c I IN(a)

II

lalsn+l L lalsn+l L

Problems: 1) Show that the following functionals define distribu-

tions in D'(&n): a) (f,(p)=q(a)(x°), for given multiindex a and x°

E &n; b) (f,(p)= J (p(x)dS, dS=surface measure ; c) (p.v.X,cp)=Ixl=1

rp(x)dxX (for n=1). 2) Obtain the first partials of theIxarl

distributions of pbm.1. 3) Show that distributions f.E DI(I) are

defined by (f+,T)=lime-0,s>0 -J cp()xE. Relate f+ with p.v.X of

pbm.1. 4) The distribution derivative satisfies Leibniz' formula

and its adjoint (cf. [C,],I,(1.23)). 5) Show that a distribution f

E D'(R) with f(a)E C(st), lalsk is a function in Ck(S). 6) Let L1po1

be the class of all uE L'1oc(Rn) with (x) -kuE L' (i") for some k=

k(u). Show that Lpo1C S'. 7) Show that p(x)= aax' E Lpo1C S'.asm

Also that CB(&n)C Lpol, and LP(&n)C Lpol, lspso. 8) Show that eax


E D'(2), but eax S', as Re a O. 9) Let Tpol be the class of all

kaE C00(2n) with a(a)(x)=O((x) a), for some kaE Z, for every a. Show

that differentiation and multiplication by aE Tpol leaves S' inva-

riant. That is, for uE S' , aE Tpol, aE 2+ we have auE S', u(a)E

S'. 10) Obtain the Fourier transform of the following distribu-

tions (If necessary, show, they are in S'): a) xa, a E 2+; b)

S (P), E Zn ; c) eiax , aE 2n. 11) Obtain (p.v.1)^, for the dis-xo + 3F

1tribution of pbm.1. 12) Define a distribution x E S',

using the same kind of 'principal-value integral' as in pbm 1.

Calculate (p.v.sln-h X)^ 13) Obtain the Fourier transform of a

2n-periodic C00(2)- function a(x). Hint: Use that a(x) has a uni-

2nformly convergent Fourier series a(x)=y00 ameimx, am ae-iz°xdx

0

14) Let f(x)=sin xJ . Show that fE S' and evaluate f^.

3. The Paley-Wiener theorem; Fourier transform of a general uE D'.

The support of a distribution uE D' was defined as smallest

closed set Q with u=O in 1\Q. We now consider u with supp u0 a.

A simple but important remark is that a compactly supported

distribution uE D'(fz), as linear functional over D(fz), admits a

natural extension to the larger space E=C00(c). (The notation was

introduced by Schwartz again.) Indeed, for a given X(x)E CD(f)

with x(x)=1 near supp u, define the extension of (u,.) to E by

(3.1) (u,(p) = (u,x(p) , for all q) E E(a) = C,(fz) .

This defines an extension: if yE D(fz) , then (1-x)TE D(fz), andsupp (1-x)TC supp (1-x) is disjoint from supp u, hence (u,(1-x)(0=

0, or, (u,(p)=(u,x(p). The extension is independent of the choice

of X. If OE D(Q) has the property of x then r-x=0 near supp u,

(3.2) (u,Ocp) = (u,xq)) , for all q> E E(c) .

The class of all distributions uE D'(fz) with compact support

is commonly denoted by E'(fz). We have seen that E'(fz) is naturally

identified with a class of linear functionals on the space E(fz).

Proposition 3.1. The set El(fl) of all (above extensions of) com-

pactly supported uE D'(fz) coincides with the set of continuous lin-

0.3. The Paley-Weiner theorem 15

ear functionals over E(n) (i.e., the functionals u over E(at) such

that jE E, q .-'O in E implies (u, (pj) ->0) . Here ypj-0 in E means thatTj (a)(x) - 0 uniformly on compact sets of f, for all a .

Clearly the extension (3.1) to E of uE D' with supp u Cc n

is a continuous linear functional over E, in the above sense: If

ypjE E, (pj- 0 in E , then xroj* 0 in D, as a consequence of Leibniz'

formula. Vice versa, for a continuous linear functional u over E

the restriction v=uID is a distribution in D', since cpjE D, Tj-0

in D trivially implies Tj-. 0 in E. Prop.3.1 follows if we can

show that supp v CC st. Suppose not, then a sequence of balls Bj

may be constructed such that um0 in Bj, while every set KCC st is

disjoint from all but finitely many of the B. Construct T.E D ,

supp W C Bj with (u,(pj)=1 . Observe that cpj-> 0 in E while ?u,(Pj)=1 does not tend to zero, a contradiction. Q.E.D.

For a compactly supported distribution on &n we always have

a Fourier transform in the sense of sec.2, i.e.,we get E'(len)C S':

Theorem 3.2. All compactly supported distributions over mn are

temperate. Moreover, for uEE'C S', u^ is a Co function given by

(3.3) u^ (x) = fne (u,ex) , ex( ) = e ix

with a distribution integral, given by the third expression (3.3).

In fact, the function u^(x) is entire analytic, in the n

complex variables xj, in the sense that v(z)=(u,ez), ez(x)=a-izx,

is meaningful for all zE Cn, (not only &n), and defines an exten-

sion of u^ of (3.3) to Cn having continuous partial derivatives in

the complex sense with respect to each of the variables zl,...,zn.

Note that formula (3.3) is meaningful only by virtue of

our extension (3.1) of u E=-El to all of E .

Proof. For uE D'(En), supp u CC &n, the natural extension to E may

be restricted to S again to provide a continuous linear functional

on S, since "(pj- 0 in S " implies "q)j->0 in E". Hence uE S'. The

function v(z) indeed is meaningful for all zE Cn. Existence of

av/azj is a matter of the continuity of the functional u over E:

For a fixed z , h E Cn, form the difference quotient

(3.4) we = (v(z+eh)-v(z))/E = (u,(ez+Eh ez)/e) , E > 0

For the directional derivative Vhez of ez at z , we get

(3.5) V. = (ez+eh ez)/E - Ohez 0 in E

Indeed, this only means that aXVE, 0 uniformly on KCC 2n, as rea-


dily verified for Continuity of u then implies

(3.6) limE-0,Ex0wE _ (u,Vhez) ,

confirming that v(z) is analytic for all z. Formally we then get

(3.7) (u^ ,q)) = rP(U)) = f91 (u,e

with v(x) as defined, where the interchange of limit leading to

the second equality remains to be confirmed. Clearly (3.7) implies

u^=v, i.e., (3.3) and thm.3.2 follows. For the interchange of

limit show existence of the improper Riemann integral f g

in the sense of convergence in E: For KCC in we must show that

in E, as k-. Here Sk is any sequence of Riemann

sums, with maximum partition diameter tending to 0 as k-'o. Also,

that fR n \K

as K runs through a sequence Kj with LX.=&n,

again, with convergence in E. Again, convergence in E just means

local uniform convergence with all derivatives. One confirms easi-

ly the local uniform convergence in the parameter x , since the

function e,(x) = e-'x is continuous. Similarly for the x-deriva-

tives, again continuous in x and . This, and the fact that the

x-derivatives of the Riemann sums are Riemann sums again, indeed

allows to confirm the desired convergences. Q.E.D.

As examples for Fourier transforms of compactly supported

distributions we mention those of the delta-function and its

derivatives. As seen in 2,pbm.5 we get 60(a)^= i1alxnxa . In fact,

this is an immediate consequence of (3.3), above.

We observe that the entire analytic function u^(z) of (3.3),

as a function of complex arguments z , has a growth property which

characterizes the Fourier transforms of compactly supported distri-

butions. The result is called the Paley-Wiener theorem.

Theorem 3.3. An entire analytic function v(z) over Tn is the Fou-

rier transform of a compactly supported distribution uE D'(&n) if

and only if there exists an integer k > 0 and a real i>O such that

(3.8) v(z) = O((z)ke'11Im z1) for all z ETn , (z)=(I+Ilzj12)1/2

Moreover, the constant I may be chosen as the radius of the

smallest ball lxlsr containing supp u . Furthermore, uE D(&n) if

and only if (3.8) holds for all k with rl=max{lxi: xE supp u}


Proof. For u E E' we must have

(3.9) I(u,(P) I s c sup{ IyP (a) (x) I: x E K Ialsk }

for some c, k, and some compact K J supp u and all TEE E. Otherwise

for every c=k=j and Ixlsj there exists T=q E E with (u,T.)=1 , and

">" holds in (3.9). Or, I(pja)(x)Is for all lalsj Ix1sj , j=1,

2,... , implying uniform convergence Tja)(x)- 0 ,j-*oo , a contra-

diction, since 1=(u, c).) does not tend to 0.We get u^(z)=(u,Xzez) xz=X(IzI(IxI-i)) where xEC00(R)

x(t)=1 , t<? , =0, t>1 , x decreasing. It follows that supp X. C

{IxI511+ } so that (XZez)(a)(x)= O(e'1IImzI+1(z)k). Combining this

with (3.9) we get (3.8) with the proper constant i

Next assume uE C0(f). We trivially get (3.9) with k=0 and K=

supp u CC n, since uE L1. Similarly for u(a). Accordingly, for all

a we get 1zau^(z)I=I(u,e(a))I=O(e1IIm z1), hence (3.8) for all k.

Vice versa, (3.8) for all k implies xavllnE L'C S'. Then v"

is given by the conjugate Fourier integral. We have u=(vl&n)È CO,

and even u(a)E CO, i.e., uE Coo(&n). To show that supp uCC in write

(3.10) u(x) = J

If 0 E In is given arbitrary then we also may write (3.10) as

(3.11) u(x) =

Indeed, this is a matter of Cauchy's integral theorem, applied for

a rectangle in the complex i;j-plane with sides Re tj=±A, In =0

or Oj. In such a rectangle the integrand elx'v(i;) is holomorphic

as a function of ti for constant other variables, so that the com-

plex integral over the boundary vanishes. For A-- the integrals

over Re tj=±A, O<Im i;j<0il tend to zero, in view of (3.8) for k=-2

for example. The integration pathes have length 0i and the inte-

grand is O((A)-2e(1-x)IOI). The integral (3.10) may be written as

n-fold iterated integral over It. The above proceedure allows the

transfer of the integration from R to the line { j+i0 xjE I}

Next let us estimate (3.11):

(3.12) u(x) = O(e1101-xO)

setting k=n+1 (it holds for every k), and using that (1+i0) z M.

The 'Q(.)- constant' is independent of 0. Hence we can set 0=tx,

t>0, for u(x)=O(etlxl(11-Ixl)), The exponent is <0 as 1xI>1, and


u(x)=0 follows. Thus supp uC {IxIsi}, uE D, if (3.8) for all k.

Finally, if (3.8) holds for some k, let X(x)E=- D, supp x C

{jxIs1}, x(x)2:0, fx(x)¢x=1. For e>0 let XE(x)=nx(E). Note that

x ( )=x^ (ei;)-+x^ (0)=1, as Moreover, for any cpE S get qe^-w+0

in S. Since supp XEC {IxjsE} we have (3.8) for xE^(t) with rf rep-

laced by a for all k. Hence the product v(z)xE^(z) satisfies (3.8)

with ti replaced by T1+e again for all k. It follows that vxe^=ue^

with uEE D, supp uEC {jxjsri+E}. Also (uE,(P}=(uE^ ,Cpl}=(v,xe^V")(v, q)-) _(u, q) , for all qE S. The latter implies that supp uCxI2M+e}, all a>O. It follows that uE E, supp uC {IxIsr1}, q.e.d.

Let Z denote the space of all entire analytic functions v(z)

=v(z1,...,zn) in n complex variables such that for k=0,1,2,...,

and some rjz0 we have (3.8) satisfied. We shall say that a sequence

vj E Z tends to 0 in Z if (i) estimates (3.8) hold with constants

independent of j, and (ii) mj=Max{Ivj(x)I: x ENn}-0,

as

Corollary 3.4. The Fourier transform F: u -, u^ establishes a li-

near bijection D H Z which is continuous in either direction, in

the sense that u 0 in D holds if and only if uj^-> 0 in Z .

Proof. After thm.3.3 we focus on continuity only. If uj-> 0 in D

then supp uj C {jxIsa} for a independent of j . This yields (3.8)

with r)=a independent of j, by thm.3.3. Inspecting the first part

of thm.3.3's proof we also find the 0(.) constant independent of J.

Vice versa, if in Z, then (3.8) with r) independent of j

implies supp uj° C But (3.8), for real z=x, implies vj=

O((x)-k), uniformly in x and j, for every k. Thus conclude from

cdn.(ii) that jlxav1I, 1-> 0, as j-. -. For the inverse Fourier trans-

form uj=vj^ we get 1Iuj(a)ii 0, so that indeed in D. Q.E.D.00 _>L

Following [GS] we now define a Fourier transform of a gene-

ral distribution f E D'(111) regardless of growth at infinity, as

a continuous linear functional f^:Z-'Q. Here of course "f^ contin-

uous" means that "(f^ ,(pj) -'0, whenever qj->0 in Z". We define f^ by

(3.13) (f^ ,(p) = (f,(p^) , for all p E Z ,

taking into account that p^ E D for T E Z

This definition is compatible with the earlier ones. Indeed,

we have Z C S , in the sense that for uE Z the restriction ulin

determines u uniquely and is contained in S . Moreover, Z is dense

in S, since Z=D^ , and D is dense in S while F and F are continu-


ous maps S->S. Also T j-0 in Z implies cp.-> 0 in S. For uE S' the res-triction u1Z determines u and we have ulZE V. Hence get a naturalimbedding S' 4Z' . For uE S' C Z' we earlier defined (u^ , (P) _(u, V )for TE S. The restriction uÎZ gives our present functional,q.e.d.

Notice that u^ , for uE D' in general is not a distribution,

as defined in sec.2. It is a linear functional on Z , not on D .

Recall that for a function f E LIloc(R) with f=0 in x<O and

f=O(ecx) , some c, one commonly defines the Laplace transform by

(3.14) f-(r;)= f glxe lxtf(x) , Im t < -c0

where the integral exists and defines a holomorphic function in

the complex half-plane Im i; <-c (we have modified the standard

definition, by a factor i). The inverse transform then is given by

(3.15) f(x) = +«'+iy plteixif (i;)-oo+iy

with a complex curve integral along the parallel Im = y <-c

We now will identify f- with the Fourier transform fÊ Z' of

the distribution fE D'. For cpE D , supp T C {jxjsi} , we know that

q is entire analytic, satisfying (3.8). For y<-c we have

(3.16) f- (t) q), (t) dt = P91xf (x) f eixtg. (t) dt _ (f,g)Im =y Imt=y

The integral dxdIl;I exists absolutely, hence the interchange, by

Fubini's theorem. Also, we get f = fR , at right, by the pro-

perties of the (analytic) integrand. Then (3.16) follows from

Fourier inversion for functions in D. Or, fÊ Z' may be written as

(3.17) (f^ t)) = f f- (t)cp(z)dt , cpE ZIm?=y

where we must choose y < -c with c of (3.14) (for f)

Thus for a function fE LIloc(R) of exponential growth and =0

in x<0 the Fourier transform f^ is given as the complex integral

(3.17) involving the Laplace transform f- of f .

Problems. 1) Obtain the Laplace transforms of the following func-

tions (Each is extended zero for x<0). a) xk k=0,1,..., b) eax;

c) cos bx ; d) eaxsin bx ; e)sin

. In each case, discuss the

Fourier transform - i.e., the linear functional on Z. 2) Obtain

the inverse Laplace transform of a) ; b) log(1+Z ). (In each

case specify a branch of the (multi-valued) function well defined

in a half-plane Im z < y .) 3) For uE D'(ien) with supp uC {xix0}=


In and ecx'uE S', for some c, show that u^ may be defined by a

complex integral like (3.17), with u replaced by ut

with respect to (x2,...,xn). 4) The convolution product w=u*v, so

far defined for u,vE L'(&n), by (1.19), may be defined for general

distributions u,vE D'(&n) under a support restriction -for example

(i) if supp U= 1n, supp v general, or (ii) if supp vC {xlzo},

supp v C {jxjscx,} . One then defines (w,(p>=ffdxdyu(x)v(y)T(x+y),

with a distribution integral (for precise definition cf.[Schw1],

or, Show that (1.21) is valid for this convolution

product as well, assuming in case (ii) the cdns. of pbm.3 for u,v.

5) Let TA be the space of all entire functions x(z) satisfying

(3.8) for some k. Show that XPE Z, for pE Z , XE TA . Moreover,

show that fE Z' allows definition of a product XfE Z' , setting

(xf,(p)=(f,xcp), (pE Z . All polynomials p(x) belong to TA . 6) Show

that (1.22) is valid for general distributions u E D'(&n)

4. The Fourier-Laplace method; examples.

We now will discuss the 'Fourier-Laplace method' for 'free

space'-problems of the following constant coefficient operators:

(4.1) A ==lax 2 (the Laplace operator)

(4.2) A + k2 (the Helmholtz operator) ,

(4.3) H = axo - A = at - A (the heat operator) ,

(4.4) = axo2 - A = at A (the wave operator)

(4.5) +m2 (the Klein-Gordon operator) .

The last 3 operators act on the n+1 variables x0=t, (x1,...,xn)=x.

The first two act on x only, distinguishing xo from the others.

The discussion around (1.23)-(1.26) was a formal attempt to

solve constant coefficient PDE in free space (in all In). We found

e= (- )' , for a P(D), of special interest. Now we are prepared

to implicate this technique, called the Fourier-Laplace method.

Certain initial-boundary problems may be converted into free

space problems: (a) An initial-value problem for (4.3),(4.4), or

0.4. The Fourier-Laplace method 21

(4.5) seeks to find solutions u of P(D)u=f in some half-space,

say, t=xo>O, where f is given in ta0, together with initial data

of u at t=0. Such problem may be written as a free space problem

by extending u=0 and f=0 into t<O, letting v and g be the exten-

ded functions. We will not have P(D)v =g then, but, rather, P(D)v

=g+h , with a distribution h, supp hC {t=o}, since normally v will

jump at t=0. The initial conditions on u often are well posed if

they allow to determine h, making the initial-value problem equiva-

lent to the free space problem P(D)v=g+h , where g+h is given.

(b) Another example: If Au=f (A of (4.1)) is to be solved

in a half-space under Dirichlet bondary conditions - say, Au=f in

x1>0, u=0 as x1=0, then consider the odd extensions of u and f to

&n: v(xl,...,xn)=u(x), x1>0, =-u(-xl,x2,...,xn), x,<0, similarly

g extending f. It follows that Av=g in Rn, again converting the

half-space Dirichlet problem of A to a free space problem over I .

Similarly for the Neumann problem, using even extensions.

Technique (a) works as well for a more general initial sur-

face t=0(x) , x(-= &n. Both techniqes may be combined to reduce cer-

tain initial-boundary problems to free space problems.

The above will emphasize the power of the Fourier Laplace

method. (4.1)-(4.5) are a crossection of popular PDE's. We control

parabolic and hyperbolic initial value problems, elliptic boundary

problems and initial-boundary problems in half spaces, etc., with

Green's (Riemann) functions, using results on special functions.

From now on interprete the equation P(D)v=g , xE Rn, as a

PDE involving distributions v,gE D'(1n). The Fourier transform ex-

ists without restrictions: Using 3,pbm.6, we get P(x)v^=g^ , where

v^, gÊ Z'. If eE D'(&n) solves P(D)e=(2n)n/2S we get P(x)e^=1.

In the cases corresponding to (4.1)-(4.5) we get, respectively,

(4.6) P(x) =-Ix12 , =k2-Ix12 , =it+x12 , =Ixl2-t2 , =m2+Ix12-t2,

where t=x0 again. Generally, P(x) (4 Liloc' except for (4.3) and

(4.1), n2t3, due to zeros of P. Some pbm's of sec's 2,3 (and, more

generally, [C,],II) discuss distributions p.v.a associated to a

Llloc' P(x)z=1 may have many solutions zE D' (or E Z'). For (4.3)-

(4.5) we will be interested in z=e^, P(x)z=1, with supp eC {to},

because then supp a*gC {t2-0} whenever supp gC {tao}, so that u=

(e*(g+h))I{t2-0} will solve the initial-value problem for P(D)u=f

ta0, we started with in (a) above. Indeed, a proper z exists: In

pbm's I and 3 of sec.2 we defined p.v.1 , and f+ , all 3 distinct,

xf=1. Only f has its inverse Fourier transform =0 for x<0.


For (4.3)-(4.5) we will construct such eED,(Rn+1) solving

P(D)e--%r2_xn+I6 , supp e C {t2-0}, using the setup of sec.3, pbm.3:

Such e, if ecxeE S', will have a Fourier transform in x=(xi,..)

and a Laplace transform in xo=t. Accordingly we must seek an in-

verse Laplace transform of an inverse Fourier transform of a sui-

table solution z of Pz=1, or vice versa, in appropriate variables.

Our proofs will be sketchy, in part, due to overflow of details.

The lemma below is convenient, due to spherical symmetry of P.

Lemma 4.1. Given a spherically symmetric function f(x)=co(Ixi) ,

where w(r) E LI()e+,rn-1dr). Then the transforms f^ and f, are sphe-

rically symmetric as well: f^(x)=f"(x)=X(Ixl), where X(r) and co(r)

are related by the Hankel transform A , v=2 1. In detail we have

r(n-1)/2X(r) = Hn/2-1(r(n-1)/2(o(r))(4.7)

r(n-1)/2w(r) = Hn/2-1(r(n-1)/2X(r)).

where

(4.8) H (X(r))(p) = fV pr Jv(rp) ?(r) dr , Re v > -20

with the Bessel function J (z) . The second formula (4.7) is validv

if in addition X E L1(&+,rn-1dr).

TProof. For an orthogonal nxn-matrix 0 get paxe-ix

Jxe-i(OTX)

Tw(IOTxI) = folye Thus f^ has the same

symmetry: with some X(p). We may set

(4.9) f^ I l ; I , 0 , ... , 0)=f RIXe-ipx w(r)=Kn f rn-1w(r)f e-irpz, dS0

where the inner integral I is over the unit sphere Izl=1. Evaluate

this inner integral by converting it to an integral on the n-1-di-

mensional ball IXI2 sl, setting z=(z1,?,). We know that dS=

With a contribution from the upper and lower hemisphere where z,=

1-72 , writing dX=an-2dodl, o=IXI, etc., we get

1

I=2 on-2 1662dEcos(rp)=tan-1 f 6n-2cos(rp)0

A substitution a= sin 0 of integration variable yields

(4.10) 1 = 2an-1x /2

d9 sinn-20 cos(rpcos 0)0


2 (n-1)/2with an-1 the area of the n-l-dimensional='((n-1)/2),

unit sphere. Using Poisson's formula ([MOS],p.79) we get

(4.11)fn /2

cos(rpcosO) sine-28 d9 =2n/2-2V.(n21)Jn/2-1(rp)

0

Substituting into (4.10) and I into (4.9) confirm (4.7). No change

ifa-ix in (4.9) is replaced by elx . Thus lemma 4.1 follows.

Recall also (1.26), now under the aspect of 3,pbm.4. In det-

ails, the convolution product v*w of two distributions v,wE D'(&n)

may be defined by setting (with a distribution integral)

(4.12) (v*w,(P) = fv(x)w(y)g(x+y)dxdy = (v(ga,V) , V(x,y)=T(x+y)

for all q)E D(&n)=Cp(&n) , if v and w satisfy the support condition

(4.13) Ka (supp v x supp w)fl{jx+yIsa} is compact for every a>0 .

Here supp v and supp w are regarded as subsets of in of the varia-

bles x and y, respectively (cf. [Schwl] or [C1], 1,8).

The distribution v=e*g is defined for e as constructed above

whenever g E E' for (4.1)-(4.3) , and gE D'(1en+1) , g=o as t<0

for (4.4) and (4.5) , since condition (4.13) holds under these

assumptions. Moreover, P(D)v=g follows, leading to a solution of

the free space problem, and the related initial-boundary problems.

Now let us attempt a detailed construction of the proper e.

Ia) Consider the operator A of (4.1), i.e., the potential

equation Au=f. For n2:3 the function -IX is Lpol, hence a distri-

bution in S'. This is a homogeneous distribution of degree -2

Hence e is homogeneous of degree 2-n . It is also sphe-

rically symmetric. Conclusion:e(x)=cnIxI2-n

, with a constant cn.

Clearly e E Lpol . The constant cn may be evaluated by looking at

(2n)n/2g(0)=(e,Ag)= cnfAW(x)1x12-ndx =cnlims-0 fdSicpdrr2-n

r=e=cnan(2-n)T(0) . It follows that

(4.14) en(x) =(n2)w 1xI2-n n- n 2 nn

For n=1 we first define a distribution

(4.15) e^=-p.f.XZ = !I (p.v.1

involving the distribution derivative and p.v.X of pbm.l,sec.2.

We confirm that e^ solves -xz e^ =l . Using a=e^ one finds that

(4.16) e(x) _jxI .


For n=2 we can define (cf.[C1],II,(2.11), for 1=2)

(4.17) a"(x)=-p.f.IX :=-Lj=1( x Zxjloglxl)Ix.7=-2A((log Ix2

again with distribution derivatives. However, it is easier to con-

firm directly that

(4.18) e(x) = x1loglxl

is a spherically symmetric L11oc-function satisfying Ae=b (justevaluate the integral (Ae, q) _(e, A(p) , using partial integration).

Ib) Consider (4.2), i.e, the Helmholtz equation (D+X)u=f

also known as the time-independent wave equation if X=k2>0, and as

the resolvent equation of the Laplace operator A if XE C, X#k2 .

In the latter case get a^(x)=(X-1xI2)-1E C°n Lpol. An evaluation

of e(x) is possible, using lemma 4.1, as long as ns3 . We get

(4.19) e(x) =IxI1-n/2

J Pn/2X Jn/2-1(Plxl)

For larger n this integral ceases to exist. However, it still

will exist as improper integral in the sense of distributions -

Athat is, as limAO , where the limit exists in weak convergence

0

of D' (i.e., (lim.,(p) = lim(.,(p) ). For odd n the Bessel function

Jn/2-1 may be expressed by trigononmetric functions. For example,

in case n=3 we get J1/2(z) Or,

(4.20) e(x) = VnII J-- 24 sinpixl0

We may write X = x2 , picking the root x with In x>O . Then

(4.21)

sinpr = 2 KPp sinpr =41 eirpxdP = 2ixlxl Hence00-00

e(x) =-7 e'xlxl/lxl

(4.21) may be confirmed, noting that e=eiicr/r solves (A+X)e=2n6

For x=k real P(x)=k2-lx,2 vanishes at the set Ixl=k, and

is not Ltioc Then look at p.f.(P(1x)). Or else, observe that

(4.22) lim£.,0,s>0 ek+is(x) = e(x) , ex(x) as e(x) in (4.21)

in the sense of distributions. This implies that

(4.23) a+(x) _ -Ve±iklx1/IxI


both will solve (A+k2)e=(2n)3/26 . The proper sign may be chosen

by imposing a 'radiation condition' at - .

For general n2:2 we still may evaluate the integral (4.19).

Using a formula by Sonine and Gegenbauer (cf. [MOS], p.105) we get

(4.24) e(x)=-(1XI

)n/2-1Kn/2_1(KIxI) , x > 0

with the modified Hankel function Kv(z). Again get (4.24) more dir-

ectly, observing that e(x)=y(Ixl) solves (A+X)e=0, hence y(r) solv-

es the ODE y"+n-1y'-x2y=0. Substituting y=rvb, v n-1, we obtain

the modified Bessel equation S"+rS'-(1+j.)5=0, showing that the

only spherically symmetric solutions of (A+X)u=0 in S' are the

multiples of (4.24). A partial integration shows that fe(A+X)tpdx=

g(x) for all TEED , fixing the remaining multiplicative constant.

II) In the case (4.3) of the heat equation Hu=ut Au=f we may

use (4.24): Applying FX1 to P for the second and third polynomial

(4.6) gives the same result, if we set ?=k2=-x2=it. That is, x=

e-in14Jt,Re x >0, in (4.24) will define F-'(.) , and we then

must obtain the inverse Laplace transform.

It is more practical, however, to first obtain Ft

Note 8(t)=(it+a)-1 has inverse Laplace transform 04(t)- 2ne-at,

ta0, =0, t<0 , calculating f e-ate-ittdt. For e'=F- 1(P) get0

(4.25) e' (t) =(2n)1/2e-t1x1 2

, as ta0 , e' (t)=0 , as t<0 .

Recall that (e - x 2/2)"=e- xjz/2 (in n dimensions), by a complex

integration. Also for the function ga(x) = g(ax) we get

(4.26) g0" (x) = a ng" (x/a) , a E &+ ,

as shown by an integral substitution. Choosing a--V_2t we thus get

_I X12ta0 =0 t<0(4.27) e(t,x) _() 11

This is the well known fundamental solution of the heat opera

tor. It is not L1(Rn+1). Use it to solve the initial value problem

(4.28) Hu = ax u-Au = f , x0a0 , u(x0,x) = q(x) ,

0

where f,p are given Setting u and f zero in t<0 to


obtain functions v and g we get

(4.29) Hv = g + 6(t)(9p(x) = h

Thus v=e*h , or,

J9Tye(t,x-y)w(y).(4.30) u(t,x)=xn+lftstgltgTye(t-t,x-y)f(y) + xn+1r

III) Now we look at (4.4), or, the wave equation

(4.31) u = (at - O)u = f

We apply the Fourier-Laplace method as for (II): The function

(4.32) P = IxI -t' 21x7{t+Ixi - tIxI}

has inverse Laplace transform (in t) given by

itjxl_ itlxl sin( x t) ta0(4.33) 2i Ix (e- ) = V2n1xI

,

(and zero for t<0). Looking for F;1 of the function (4.33) we can-

not apply a Fourier integral, since the function (4.33) is not L1.

First set n=2. Writing FXlw=w° for a moment, we have

(4.34) (e,cp) = (cos2rt (Plt) = (f,mlt), roE D(Rn) , r=IxI

using that (cos Ixlt)lt = -Ixl sin IxIt . Now the inverse Fourier

integral of f= cos2rt may be calculated as improper Riemann

integral limAjl xIsA , although still that function is not LI

Using (4.7), (4.8) - where v=2 ,

j,(,)=V2z , for n=3 - we get

f°=w(IXI) w(r)=(V_2n)(N(_2

00

)

f '(r) sin r13 cos2ptdp . Or,

0

w(r) = r Psin rp cos ptp

=r fco(sin p(t-r) - sin p(t+r)) P

= r{ sgn(t-r) - 2} . Conclusion:

(4.35) e(t,x) =li atH(t-IxI) , as t>0 , =0 , as t<0 , n=3

with the distribution derivative at , and the Heaviside function

H(t)=1, tZ0 , H(t)=0 , t<O. We are tempted to write atH(t-IxI) as

6(t-IxI), but then must remember the proper interpretation.

Converting the Cauchy problem for the wave equation,

0.4. The Fourier-Laplace method

(4.36) ultt-Au=p , t>0 , u=p , ult=V at t=0

into a full space problem we get

(4.37) vltt Av = S'(t)Clp(x) + 6(t)W(x) + g = h

where v and g are u and f , extended zero for t<0 . Evaluating

e*h we get the solution of (4.36) in the form

u(t,x) =4iJlx-ylst

yf(t-lx-yl,y)(4.38)

27

+ 4n at(t,Ilzl=1T(x+tz)dSZ)+ 4n Jlz1=1V(x+tz)dSz

(4.38) is known as the Kirchhoff formula.

For general n2t2 let j=[n21] . As in (4.35) write (e,y)) _

V-(f j,((at)j(p)where fj=±sPn°+t or fj=tcPs,1t (with p=lxl) foreven (odd) J. The inverse Fourier integral of fj exist at least as

limAJ lxlsA' we apply lemma 4.1 as above. For odd n we get

(4.39) fj°(t,x)=wj(t,lxl) , wj(t,r)=*r v P7V(rp)ls'ncos pt}

for even n this will be of the form

(4.40) fj°(t,x)=wj(t,lxl) wj(t.r)=trv

J JV(rp){cos pt} dp0

with a convergent improper Riemann integral. More precisely, in

(4.39) and (4.40) we have "(-1)1cos pt " as j is odd, and

"(-1)lsin pt " as j is even, with 1=[Z] . Our distribution e then

is given by (4.41) below, with a distribution derivative at

(4.41) e(t,x) = (-1)JV-2natwj(t,lxl)

Now we must consult different formulas on Bessel function

integrals where as a general reference we quote [MOS] (or [MO]).

Case (a): n even, j odd. Then j=[n21]-2-1=v, 1=, n=0(mod 4).

We use a Weber-Schafheitlin integral ([MOS] p.99, last formula).

Since v=j is odd, we have for t>r the formula

(4.42) I = f Jv(rp)cos pt dp= (rz-t2)-1/2cos(vsin-1(r))=4Pj_1(F)'0

with a certain polynomial Pj-1(t) of degree j-1. Note that the

integral enters (4.41) under the derivative at , so that a=0 for


t>lxl For r<t we get I= r-j(t+ tx-rx)-j by the sametr-formula. Summarizing: For n=0(mod 4) we get

e(t,x) _-r 821+1(Z1(t,r)) , with(4.43)

Z1(t,r) = r 21Q214P, t>r (t+ -rL)21+1, t<r

whereQ21(2) =

=0(-i)1 m(22m1)T2m(1_i2)1-m

Case (b): n even, j even. We still have j=v=?-1, 1=2, n=2(mod 4)

Use formula 3 on p. 51 of [MO]. The result is as in (4.43) with

Z1 (t,r) = r n/2Q21-14P , t>r , =-p (t+ t - ) 21 , t<r ,

(4.43)bQ (t) 1(_1)l-m( 21 )t 2m+1(i_T2 )1-m-1 , 1 n!2>0,

21 -1 =_-0

2m+1 4

while for n=2 (i.e., 1=0) we set Q2i_1=0.

Case (c): n,j odd,

[nn-1] n2

=j, n=2j+1, 1='-', j=21+1, n=3(mod 4).2

Let I= f00

Jv(rp)cospt =v 2 f Jj-112(rp)J-112(tp)dp. Use formula 2,

p.50 of [MO] with (a,b)=(r,t), (v,n)=(1+?,1). Get v-n=2, I=O, t>r,

(4.44) I=N""1.3.5... (21-1)xF, t<r2r 2 4 6 2.1 2 2 r 2r

where the hypergeometric function (hence Q21(z))

x1 (11(4.45) 2Fi(1+-1;2;z)=1-1(21+1)z+li =](21+1)(21+3)2r + ...

is a polynomial in z of degree 1=n43. Combining (4.39) and (4.44),

(4.46) e(t,x)=(_i)1+1xr 21-1x21+1 t __n _

t IQ21(r)H(r ,)} 1--n-4 , r-lxl

Case (d): n odd, j even, j=n41, n=2j+1, 1= , n=i(mod 4) . Now we

need I= f Jv(rp)sinpt r = n2 f Jj_i,,2(rp)Ji/2(tp)dp . Again use0

the formula of case (c), now with Get

(4.47) I=V7(t)3.5.. 21+1 2F, (1+1,1-1;3;tx )_`/ (t, t<r,2r r 2.4 2t-2 2 2 r 21-1 rand 1=0 for t>r , with a polynomial Q21-1(z) of degree 21-1. Hence

(4.48) e(t,x) _ (-1)lnr 21at1{Q21-1(r)H(r-t)} n41 , r=IxI

IV) Finally, look at the Klein-Gordon equation (4.5), n=3 only


(4.49) (p+mz ) u = utt Au + mz u = f

Equation (4.32) now assumes the form

p = Xz ltz ={t+a. - tip} a(x)- m2+ x

FesintVx mx VTP m

For n=3, v=2, j=n21-1 use (4.7),(4.8) on x(IxI) of (4.51), for

(4.52) Vrw(r) = nVt f

00

J1/2(rp)J1/2(t )p3/2/Vg -+W dp0

This integral diverges, just as for m=0. Applying a formula of So-

nine (1880) and Gegenbauer (1884) (first on p.104 of [MOS]), for

(a,b)=(t,r) , (µ,v)=(2,2 E) , m=z , s>O , we get

foo

J1/2(rp)J1/2+E (t +p ) pJ mp"I ',(4.53) 0

IF

1-Elimn Js-1 (MV-t2

with existing weak limit in D'(13) . Or,

(4.54) e(t,x) = nm limE->O,E>O{H(t-IxI)JE-l(m t2 rz)/ t-E

Recall, for v=s-1 we have

Z/2) E-1 z 2(4.55) Jv(z) = r(s) - 0 r( +m+E m+

6>0

The second term, at right, goes to -J1(z), as E-0, uniformly on KM &n , and weakly in D'. For a cE D write (e, (p) =T, +Tz , by the de-composition (4.55). Then Tz - -nm(Jl (m tz -rz )/ t2-rz ,(p) , Ti

2n lime-Ofrd(t)

(tt x )1-E , where r(E)=E(1+c,E+ ...). HenceIxlst

T1=2n lime.,O{f dxdt t+rx (tEr)E =2n limfdtdr{(t-r)E}t

(4.56) IxIst

=-2n1imE-Of1xlstdxdtat{r+t x }(t-r)E =-2nfH(t-r){ }tdtdx

In other words, TI=n(-:-.a tH(t-r),g) . Result:

(4.57) e(t,x) = r tH(t-Ixl) - J1(m tz-xz)


5. Abstract solutions and hypo-ellipticity.

In sections 1,2,3,4 we have deployed the Fourier-Laplace

method of solving certain problems involving constant coefficient

PDE's under sufficiently simple initial boundary conditions. The

focus was on the Fourier transform. In a distribution setting it

reduces the problem of solving P(D)v=g to the division problem

P(x)v^=g^, providing an 'inverse' for P(D) in the form spea-

king roughly. In later chapters, this type of inverse construction

will be extended to variable coefficients - and more general boun-

daries, but the central role of the Fourier transform is maintai-

ned, and the inverses obtained have similar features.

On the other hand, the theory of constructing an inverse of

an abstract operator between two linear spaces is well studied, in

present times. It will be ever present in the background of our

discussions. The role of certain Hilbert spaces- L2-Sobolev spaces

- in theory of 1pdo's will have to be studied. Theory of (bounded

and unbounded) Fredholm operators in Hilbert spaces will be cru-

cial for elliptic i,do's, as well as compactness properties.

Often existence of a generalized solution to a PDE (a boun-

dary problem) can be derived by purely abstract arguments. But

then theory of pdo's may have to be used to derive properties of

such solution -even differentiability, and that we have a solution

in the classical sense. Let us discuss an example.

For an open domain czC &n, consider the Hilbert space H=L2(n)

with norm and inner product

(5.1) (u,v) = 1lull2 = (u,u) , u,v E H

The Laplace operator A= ax 2 defines a linear map CD(cZ)-C'(S)C H7

which may be interpreted as an unbounded operator H0 of H with

domain dom H0=Co(st), using that C0(a) is a dense subspace of H

In fact, H0 is hermitian and negative, i.e., we have

(5.2) (u,H0u) real and s0 , for every u E dom H0 .

This implies that H0 has self-adjoint extensions. A distin-

guished such extension H , called the Friedrichs extension, may be

constructed using a rather simple form-closing principle (cf.

[Ka1],[RN],[Wm1],[Yo1], or [C2],I, thm.2.7 ).To express the above in different terms: H is also a linear

0.6. Exponentiating a fast order operator 31

H

We have (5.2) for H instead of HO, but H is maximal with respect

to this property: No proper extension of H still satisfies (5.2).

This maximality has important consequences: Any operator H

satisfying (5.2) must be hermitian. A maximal such operator must

be self-adjoint- it possesses a spectral measure, as we will not

discuss here. Moreover, it follows that the unbounded operator K=

1-H is invertible in the following sense: There exists a continu-

ous operator R E L(H), called K 1 = (1-H)-1, such that R:H-> dom H

C H is a bi-jection inverting the linear map K:dom H -> H .

Accordingly, due to existence of the inverse R, it follows

that the linear equation

(5.3) (1-H)u = f , f E H ,

admits one and only one solution u E dom H C H , for every f E H

After this abstract discussion we now ask: Does u of (5.3),

a function of H whose existence is shown by a chain of abstract

arguments, solve the differential equation u-Au=f ? Also, does

u satisfy any kind of boundary condition?

The corresponding question is meaningful not only for A

but for any smooth differential expression a(x,D) defined in sa

such that the corresponding operator L0: C0 ->C0C H satisfies (5.2).

For general a(x,D), with the Friedrichs extension L of LO ,

the unique uE dom. LC H solving (1-L)u=f, for given fE HC L'loc(n)

C Du(st) is a distribution solution of u-a(x,D)u =f, since

(u,c)-a(x,D)"cp) = (u,g)-((Lu) q,ED(R), usingthat (Lu,T)=(u,a(x,D)p), gE D(st)=CO(st)

Question: (a) If, in addition, we have fE C"(c), will u be

smooth - and a classical solution of u-a(x,D)u=f? (b) If, in addi-

tion, st has a smooth boundary r, and fE COO(clLE') , will also u beC0O(szlf), and will it satisfy boundary conditions?

For a(x,D)=A both questions have a positive answer. In par-

ticular, u must satisfy the Dirichlet condition u=0 on r. The same

remains true if a(x,D) is elliptic and of second order (cf. ch.5).

Generally, if a(x,D) is hypoelliptic the answer to (a) will be

positive. An answer to (b) is given in V,4. We speak of a concept

called boundary hypoellipticity, in that respect.

6. Exponentiating a first order partial differential operator.

In this section we discuss the formal linear operator e,G


with a first order differential expression (a 'folpde')

(6.1) G = b3a+ p , aa/a=1 jj=

xJ

with real Co coefficients bj(x) , j=1,...,n, and complex-valued

p()E C, all defined in some domain S2 C &n. By definition eG will

be the solution operator of the initial-value problem

(6.2) atu=au/at =Gu , xES2 , to 0 , u=u0 , t = 0 ,

i.e., u(x,t)=eGtuo(x), uoEC00

(12), is the unique solution of (6.2).

The single first order PDE atu=Gu becomes an ODE along cer-

tain curves, called characteristic curves, defined as solutions of

a system of ODE's, the system of characteristic equations,

(6.3) t' = 1 , xI = -bj(x) , j = 1,...,n , " " = d/ds

for (t(s),x(s))=(t(s),x1(s),...,xn(s)) with real-valued functions

t,xj of a parameter s. The first equation (6.3) gives t-s=c=const.

Thus set t=s, reducing (6.3) to the autonomous system (for xj(t))

(6.4) -bj(xl,...,xn) , j = 1,...,n , d/dt .

Since the functions bj(x) are real-valued and Coo it is clear that

the orbits (i.e. the curves in Sz, given by the parametric repre-

sentation x=x(t), for x(t) solving (6.4)), provide a family of non-

intersecting smooth curves covering the entire domain R . A solu-

tion x(t) with x(0)=x0 will exist in a maximal interval Ost<t0,

where either t0=oo or x (t) E St\K for tE [ tK, t0 ) for every compactset KC Sz with some tK<t0 . In particular t0=co whenever for every

t1>0 a compact set Kt C Sz can be found such that the 'apriori

estimate' x(t)E Kt , as 0stst,, can be verified from the fact that

x(t) solves (6.4) and x(0)=x0. An orbit may degenete to a point,

or, may be closed -i.e., x(t) may be periodic.

The relation between (6.2) and (6.3) is this: A function uE

C1, defined for (t,x) near (t° ,x° ), x°E S2, solves the PDE (6.2) ifand only if along every solution curve (t,x(t)) of (6.3) the com-

posite function u(t,x(t))=y(t) solves the ODE

(6.5) dt = Y(t)q , Y(t)=p(x(t))

Indeed, atj(t) _ -b.(x(t)) implies that

(6.6) EbJulx )(t,x(t))J j

Hence if u solves (6.2), we will get (6.5) along every curve x(t)

0.6. Exponentiating a first order operator 33

for which the composition u(t,x(t)) is defined. Vice versa, if

u(t,x) is a function defined and C1 near (t°,x°) such that p(t)

satisfies (6.5) for every curve x(t) solving (6.4) in some neigh-

bourhood of x° . Then u(t,x) solves (6.2) near (t°,x°).

Let u(O,x)=uo(x) be given for all xE St. Then a unique solu-

tion u(t,x) of (6.2) may be constructed as follows: At each x°E $1

construct the solution x(t) of (6.4) through x°, defined for a max-

imal t-interval (t-,t+), tt=t(x°). Define y(t)=p(x(t), and p(t)=

u0(x°)exp{ f y(r)dt}, tE (t-,t+). Then set u(t,x(t)) (t) along0

the curve x=x(t), tE (t-,t+), and do this for all such curves.

Assume that we have t+=tao for every x° E 12. Then the map x°x(t°,x°) defined by following the orbit through x° from t=O to t=

t° defines a diffeomorphism v,to :S2 - n. Indeed, this map S2-S2 is 1-1by construction, and it may be inverted by solving the reverse

initial-value problem (follow x(t) through x° for a t-interval of

length t° in negative t-direction). The map vt is Coo by standard

results of dependence of solutions of an ODE on initial values.

We thus obtain a 1-parameter family F = {vt : tE &} of dif-

feomorphisms vt : St - St , having the group property

(6.7) Vto nt = nt+t , t' t E 1e ,

since following the curves for t+t units has the same effect than

following first for t then for t units.- Such a group of diffeomor-

phisms is commonly called a flow. The flow defined by (6.4) is

called the characteristic flow of the PDE (6.2)

In this terminology we have proven:

Theorem 6.1. Assume that all solutions of the characteristic sys-

tem (6.4) extend for tE Then the problem (6.2) has a uni-

que solution u(t,x)E Coo(&xn), for uoE C00(SZ), where u is given by

(6.8) u(t,x) = ft(pov-t)(x)dt}0

Indeed our above description of u(t,x) translates to (6.8).

In particular the diffeomorphism vt is inverted by v-t, hence u

is defined for all x E 12 and all t , and (6.8) follows.

Note that (6.8) establishes an abelian group {etG : tE le} of

linear maps etG:C"O(S2)-C00(St) , defined by

(6.9) etGuO(x) = u(t,x) , u0 E C'(c) , t e & .

We also have

34

(6.10)

0. Introductory discussions

etG: CO(Sc) - CO(SZ) , t E &

The relation between equations (6.2) and (6.4) remains inva-

riant under a coordinate transform of 11, i.e., the characteristic

equations (6.4) go into those of the transformed equation (6.2).

Accordingly theorem 6.1 also holds for a folpde G defined on a

differentiable manifold a with local representation (6.2) and

characteristic equations locally given by (6.4) , assuming that

our condition remains satisfied - that all solutions of x'=-b(x),

regardless of the choice of initial value x°E n , extend into an

infinite time interval, passing of x(t) between charts of diffe-

rent coordinates being permitted.

Now we want to address the problem of finding a more concrete

condition for the assumption of thm.6.1 that all solutions of the

characteristic system (6.3) extend indefinitely. Assume that a

carries a Riemmannian metric ds2=EhJkdxJ,dxk, under which it is

complete. Let d(x,x0) denote the distance from x to x0 on n .

Proposition 6.2. Assume that the principal part tensor (b3) of

the expression G of (6.1) satisfies the estimate

(6.11) Ib(x)I = {(Ehjkb3bk)(x)}1/2 = 0(1+d(x,x0)) , x E 11 ,

with some fixed x0E n. Then all x(t) of (6.4) extend to -oc<t- .

Proof. By well known results on continuation of solutions of ODE'S

it suffices to get an apriori estimate. Let x(t) solving (6.4) be

defined for tE [t0,t1] , x(to )=x° . Let y (t)=1+d(x° ,x(t)). We knowy(t) is Lipschitz continuous in t and that

(6.12) Idy/dtl s {Ehjkxj'xk'}1/2 = Ib(x(t))I = O(y(t)) ,

noting that (6.11) trivially holds for each fixed x0 E n if it is

valid for only one x0. Now (6.12) implies log y(t) =O(t-t0), or

(6.13) x(t) E {d(x,x0) s ec(t-to) -1 } .

Since n is complete, the sphere (6.13) is compact for t=t1, and we

have shown that x(t) stays in it. This proves the proposition.

Now we look at G and eG in a Hilbert space H=L2(n,dµ), with

(6.14) (u,v) = fnuvdµ , IIu!I=

as inner product and norm, dµ denoting a positive Co measure on Q,

locally of the form dµ=Kdx, 0<x(=-C00. We still assume that the cha-

0.6. Exponentiating a fast order operator 35

racteristic flow F is defined: All solutions of (6.4) extend inde-

finitely. Clearly the restriction GO=GICO(st) may be interpreted as

unbounded operator G0 of H with domain dom G0 = CO(n).

Assume G skew-selfadjoint: with real-valued bj , p we have

(6.15) G = Ebjaj + 1/2 x- I(xbJ)Ij

+ iplx

then G0 with domain CO(n) clearly is skew-hermitian: We have00

(6.16) (u,GOu) + (G0u,u) = 0 , u E dom G0 = CO(ft)

If u(t,x) solves (6.2) , for some u0E CO(sz), (6.10) implies that

u(t,.)E D(st), for each fixed t. The y(t)=Ilu(t,.)III EC00(1),

and

(6.17) dy/dt = (u,ult) + (ult,u) = (u,GOu) + (GOu,u) = 0 , tE t

Accordingly,

(6.18) IletGuOII = IIu(t,.)II = I1uOII = const., for all u0E Co(st).

This shows that the operator etG: CO(ft)-+CO(ft) defines an isometry

in the norm of H . Clearly this is an invertible isometry, using

the group property of etG. Let us first assume that p of (6.15)

vanishes. Then v(x)=etGu0, for some fixed t, is of the form

(6.19) v(x) = (u0ox)(x)w(x) , x(x) = v-t(x) ,

with a real-valued positive Co°-function w(x) independent of u0, by

(6.8), with p=(2x)-I(xbj).

. From (6.18) and (6.19) we conclude1

rf

x

(6.20) 11v112 = I(u0w)(x(x))12dµ = .IIu0(x)12ds = IIu0II2,u0E Co.

With new integration variables y=x(x) in the first integral we get

(6.21)f11Iu0(x)I2(w2(x)(dµovt)/dµ

- 1) = 0 , for all u0E Co(st).

But (6.21) implies that

(6.22) w(x(x)) = {(d1iov_t)/dµ}I/2

For p general, etGuO(x) will carry an additional exponential

factor. All other functions remain the same. We have proven

Proposition 6.3. For a skew-selfadjoint G (6.15) assumes the form

Jt

(6.23) etGuO() = ((u0,v_t){a1Lov_t/a1'}I/2)(x)exp{i pov_z(x)dT},0

defining an invertible isometry COO-COO the sense of the norm of H


Then eGt:co->C admits a continuous extension to H, a unitary000

operator. We get a group {U(t)=(eGot)closure : tE &} of unitary

operators, strongly continuous in t. Its infinitesimal generator

(6.24) G1=dU/dt(0) , dom GI={uE H: lim£-,oU(£)E-u =G1u exists in H}

is a skew-selfadjoint realization of the folpde G , extending G0.

This realization is unique, since domeGot=CO

is dense in H. G1 is

the closure of G0, and iG0 is essentially self-adjoint.

Finally consider a dissipative expression G (i.e. GO+G*sO).

G must have the form (6.15) with real-valued bj but complex p sat-

isfying In p(x)a0, xE S. Hence (6.16) now assumes the form

(6.25) (u,GOu)+(GOu,u) _ -Jdµ(Im p)lul2 s 0 , u E C,(a)

Under this assumption the map etGO , still of the form (6.23), is

a contraction under the norm of H, for t2t0, i.e., IleGtllsl, t2!0.

This follows by repeating the above argument.

7. Solving a nonlinear first order partial differential equation.

In this section we are going to consider the Cauchy problem

for a single first order partial differential equation

(7.1) F(x,u,p) = 0 , p = ulx = (ax u, ... ax u) ,

1 n

where F is a given real-valued Co'-function of the 2n+1 real varia-

bles x=(xi,...,xn) , u , p=(pi.... ,pn). For simplicity assume F

defined in S2xlex1n, with some domain RC ten. Our main applicationwill be the characteristic equation (8.3), in sec.8, below. There

F=aN(x,p) is independent of u and a homogeneous polynomial in p.

The Cauchy problem seeks a solution of (7.1) satisfying an in-

itial condition at some n-i-dimensional submanifold rC tt. Let I' be

given by x=x(s), s=(si,...,sn-i)E EC &n-i, with ax/6s=((axj/asl))

of rank n-1. Require as initial condition that

(7.1') u(x(s)) = w(s) is given for s E E .

Then differentiation of (7.1') for s) determines all tangential

derivatives of u at r, while (7.1) gives an implicit relation for

normal derivative of u. It is natural to assume that the n equat-

ions resulting for pj=ulxj locally admit precisely one solution,

so that ulx is fully determined along the hypersurface E.

0.7. Solving a first order PDE 37

Thus it is convenient to assume not only u=w(s) but also p=

ulx given for sE E. Of course certain conditions on w(s),p(s) must

express the fact that p=ulx. In other words, we assume given

(7.2) x = x(s) , p = p(s) , u = w(s) , s E E ,

with ax/as of maximal rank n-1, while x(s), w(s), p(s) satisfy

(7.3) lp -aslx = aslw , s E E , 1 = 1,...,n-1

(called the strip condition) and

(7.4) F(x(s),w(s),p(s)) = 0 , s E I .

A (2n+1)-tuple of functions (7.2) satisfying (7.3) is called an

((n-1)-dimensional) strip: The plane in (x,u)-space through the

point (x(s),u(s)) with normal vector p(s) is tangential to the

n-l-surface (x(s),u(s)) by condition (7.3). If in addition (7.4)

holds, then we talk about an integral strip of (7.1).

Instead of asking for a function u(x) satisfying (7.1) and

(7.1') it is more convenient to ask for a solution of (7.1) exten-

ding a given n-i-dimensional integral strip (7.2). That is,

(7.5) u(x(s)) = w(s) , uIx(x(s)) = p(s) , s E I .

Thus the Cauchy problem for (7.1) seeks to find u(x) satis-

fying (7.1) and (7.5) for a given 'initial integral strip' (7.2).

We shall see that this reformulation already holds the key for a

solution of the problem, while in fact only w(s) may be freely

chosen in many cases, perhaps up to a finite choice of p(s).

In sec.6 we already solved this problem for a linear equation

(6.2). In the present more general setting the system of characte-

ristic equations (6.3) or(6.4) must be replaced by the following:

(7.6) x =Flp(x,u,p), u =PFIP(x,u,P), p =-Flx(x,u,p)-pFlu(x,u,p),

with 'aF'_''' . This is a system of 2n+1 first order ODE's in 2n+1

unknowns x(t), u(t), p(t). The coefficients of (7.6) are Cw in x,

u, p, independent of t, hence there exists a unique local solution

defined for ltI<S for sufficiently small S, satisfying

(7.7) x(0) = x0 , u(0) = u0 , p(0) = p0

for arbitrary x0E St, u0E &, p0E 2n. The solutions may be extended

as long as they stay inside szxIx&n. Their orbits fill SZxXxin as a

nowhere intersecting family of curves. Generally we assume that


(7.8) Flp(x,u,P) # 0 , (x,u,p) E 1 x & x In .

Then the orbits are nondegenerate C° curves. Moreover, x=x(t) de-

fines a nondegenerate Cc'--curve in x-space ien.

We claim that (x(t),u(t),p(t)) also may be interpreted as a

1-dimensional strip in (x,u)-space, along which F is constant.

That is, x=x(t) , u=u(t) defines a Co curve E in (x,u)-space; at

(x(t),u(t))E 'E, the plane through that point with normal p(t), i.e7

(7.9) u - u(t) = Lj=1p(t)(x-x(t)) = P(t)-(x-x(t))

defines a plane tangent to EE at (x(t),u(t)). Indeed the vector

(x (t),u clearly lies in the plane (7.9). We get

(7.10)d/dt(F(x(t),u(t),p(t)) = Flxx + Fluu + Flpp

FIxFIP + FIu(pFIP) - FIP(Flx + PFIu) = 0

confirming that F is constant along the strips discussed.

A characteristic integral strip is defined to be a 1-dimensio-

nal strip as described above,with the additional property that

(7.11) F(x(t),u(t),p(t)) = 0 along 8 .

For the construction of a solution of the Cauchy problem (7.1)

and (7.5) we start from a given n-l-dimensional integral strip

(7.2). For each (x(s),w(s),p(s)), sE E, we obtain the unique char-

acteristic integral strip through (x,w,p). That is we construct

(7.12) x(s,t) = x(s1,...,sn-1,t) , w(s,t) , p(s,t) , s E E

solving (7.6) and the initial conditions

(7.13) x(s,0) = x(s) , w(s,0) = w(s) , p(s,0) = p(s) , s E E

The functions (7.12) exist only for (s,t) E no , with

(7.14) no = {(s,t) : s E E , Itl < t1(s)}

with a suitable function ti(s) > 0 , defined over E .

Suppose the function x(s,t) can be inverted near t=O.Since we

assume ax/as of maximal rank this means that ax/at=x , at r, is

linearly independent of the ax/asj. In other words, the projection

of the characteristic strip through (x(s),w(s),p(s)) onto x-space

must never be tangent to r. Under this condition we call r an

0.7. Solving a fast order PDE 39

admissible initial integral strip. The implicit function theorem

then implies that x(s,t) is invertible whenever the initial strip

is admissible, and tl(s) is choosen sufficiently small. Let X(x)=

(s(x),t(x)) denote the inverse. We claim that a solution of the

Cauchy problem (7.1), (7.5) is defined by setting

(7.15) u = w,X , i.e., u(x) = w(s(x),t(x)) .

Indeed for x e l' we have t = 0 , so that u(x)=u(x(s))=w(s) . From

(7.15) we get u(x(s,t)) = w(s,t) . Differentiating this for sj and

t , using (7.6) and the strip conditions (7.3) , we get

(7.16) ulxjaslxj = asl =p151

ulxjatxj = atw =i

pjatxj , (s,t) E I

Since the Jacobian ax/a(s,t) is invertible, (7.16) implies that

(7.17) p(s,t) = ulx(x(s,t)) (s,t) E E .

In particular we again get t=0 , as xE r hence ulx(x(s)) =p(s).

Hence we have (7.5) Also we know that F(x(s,t),w(s,t),p(s,t))= 0.

Thus (7.15) and (7.17) imply that the PDE (7.1) is satisfied by u

of (7.15) Thus indeed we solved the Cauchy problem.

Next let us assume that u(x) satisfies (7.1) and (7.5) , for

an admissible strip (7.2) . We then may consider the system

(7.18) x = Flp(x,u(x),ulx(x)) , n'n = d/dt

of n first order ODE's in n unknowns x(t). Let y(s,t) solve (7.18)

and the initial cdn's y(s,O)=x(s). Let K(s,t)= u(y(s,t)), q(s,t)=

ulx(y(s,t)). We will show that y,K,q solves (7.6) with the same

initial conditions as our constructed x(s,t) , w(s,t) , p(s,t)

so that we must have x=y , K=w , p=q , for all sE I , ltl<tl(s)

Since K = uoy = u,x = w , we then find that u = w,X coincides with

our previously constructed solution (7.15) of (7.1) and (7.5) , so

that we have uniqueness of the solution of this Cauchy problem.

Indeed, from (7.18) get y =Flp(y,x,q), assuming u E C2. Also,

F(x,u(x),ulx(x))=0 , since u solves (7.1). Differentiating we get

(7.19) Flx(x,u,ulx) + Flu(x,u,ulx) + Flp(x,u,ulx)ulxx = 0 .

In (7.18) let x = y(s,t). Then q= ulxx(y)Flp(y,K,q) =-Flx(y,K,q)

- Flu(y,x,6)q so that indeed the system (7.6) for y,K,q follows

We have proven the result below.


Theorem 7.1. Let r = {x(s) : s E E} be an n-l-dimensional C"-sur-

face in In with local parametric representation, rank as/ax = n-1.

Then for every admissible integral strip (x(s),w(s),p(s)) over r

the Cauchy problem (7.1),(7.5) admits a solution uE C°°(SZo), with

some neighbourhood r& of r. The solution is unique within C' (SL ).

Remark 7.2. Suppose an n-l-dimensional surface r={xE c:ip(x)=V(x0)}

is given, where VIx#O , and F(x,p(x),Vlx(x))=O on r. Then, if x=

x(s) represents (some part of) r, we find that w(s)=p(x(s)), p(s)=

Vlx(x(s)) defines an n-l-dimensional integral strip of F. Suppose

this strip is admissible. Then thm.7.1 yields a solution cp(x), de-

fined near r, such that p(x)=p(x), and l (x)=yrlx(x) on r. It fol-

lows that r also is given in the form T = c , at least locally.

This construction will be useful in sec.8, where we ask whe-

ther a characteristic surface always may be given as a surface of

constancy of a solution cp of the characteristic equation (8.3).

However, a closer inspection shows that, for an equation (8.3) the

above integral strip is never admissible: We get y=const on r ,

hence the surface normal is a multiple of iplx(x(s))=p(s) for all s

On the other hand, the projection of the characteristic integral

strip at x(s) has direction x'=Flp(x(s),p(s)). Also, since F(x,p)

=aN(x,p) of (8.3) is homogeneous of degree N in p , we get p.x' _

p.Flp(x(s),p(s))= NF(x(s),p(s))= 0 along a characteristic integral

strip. Accordingly, x' must be in the tangent space of x(s), since

it is perpendicular to the surface normal.

8.Characteristics and bicharacteristics of a linear PDE.

The conventional approach to the concept of a characteristic

(hyper-)surface for a differential expression

(8.1) L = a(x,D) = a(x)DaI aTsK

a

is the Cauchy-Kowalewska theorem, discussing existence of a unique

analytic solution for a PDE with analytic coefficients satisfying

analytic initial data. The result directs the attention to certain

surfaces along which data cannot be prescribed freely.

The concepts also make sense for real-valued solutions of

a PDE with real coefficients. However, one has to work with two

different sets of assumptions: Either assume real Co coefficients

or complex analytic coefficients. The latter case is mostly analo-

gous to the real Co case, and will not be discussed in detail.

0.8. Characteristics 41

Accordingly we assume that L of (8.1) has complex C" coeffi-

cients, but that the coefficients of the principal part polynomial

(8.2) a

laaa 0

=Nare real-valued C°°(S), for a domain nC &n, up to a common factor

O#yE C00(n) (here assumed 1). The characteristic equation of L is

(8.3) aN(x,Cplx) = 0 ,

with the gradient Tlx of T. Clearly (8.3) is a single first order

PDE for the unknown (real-valued) function T(x)EE C00(tt).

In sec.7 we discussed solving such PDE. Since F(x,p,u)=aN(x,p)

is independent of u, the characteristic system reduces to

(8.4) x =

a system of 2n first order ODE for 2n unknowns x(s), F;(s), where

we wrote (s) instead of p(s). The equation for u' of (7.6) is p'=

NaN(x,(plx)=0, due to (8.3), using Eulers relation for the homogen-

eous function of degree N in plx). (7.6) splits into this equation

and (8.4) to be solved separately. The integral strips are given

as solutions of (8.4) together with g(s)=const. A sys-

tem of the form (8.4) is commonly called a Hamiltonian system.

The graph of a solution T of (8.3) is fibered strips x(s)

(s) solving (8.4), cp=const.: Through any point (x°,g(x°)) on the

graph there is a unique characteristic integral strip

(x(s),c)(x° with x(O)=x° , (0)=9Plx(x°where solves (8.4). In sec.7 we have seen how solut-

ions of (8.3) may be composed of such strips. Moreover, since

g is constant along such strips, it is clear that they stay on the

same level surface of cp . Thus also the level surfaces of solut-

ions are fibered by characteristic integral strips of (8.3).

A characteristic (hyper-)surface r of the differential ex-

pression (8.1) is defined as a surface of constancy of a solution

T of the characteristic equation (8.3), with PIx'O. To be precise

we require that for every x°E I' there exists a solution T = q) x of

(8.3), defined in B={Ix-x°l<a} such that Tf1B = {xE B: cp(x)=c}

for some constants c, a>0 .

Actually one may define somewhat more generally by dropping

the assumption that c(x) solves (8.3) near the surface t , and

requiring (8.3) on the surface t only.

It is clear from the above that a characteristic surface r

is fibered by characteristic integral strips: There is precisely


one strip through each point x E r . These strips

are called bicharacteristic strips of the expression L .

We speak of a simply characteristic surface r of the expres-

sion L of (8.1) if r is given by cp(x)=0 with g(x)=0 , cplx(x)#0 ,

aN(x,cplx(x))=0 on r , as above, and in addition aN1,(x,T,x(x))#0

on r . For a simple characteristic (surface) it is no loss of ecn-

erality to assume that p(x) solves (8.3) near r .

Indeed, we may assume thataNI

(x°,c)lx(x°))#0, and then sol-

ve aN(x,T;)=0 for ' . Get a function p, (x,n") with pi (x° cplx(x° )" )=gplx.(x°), where p =(p2,...,pn). The plane perpendicular to the in-

tegral strip through x° is noncharacteristic near x°. In that pla-

ne 10 set w(x)=cp(x), p" (x)=cplx(x)- , with our given T(x) solving(8.3) on r near x° . Also set pi =p, (x,p" ). We get an admissible

integral strip, and a solution w(x) near x°. We have w(x)=cp(x°) on

the strip through x°. Also, w(x)=p(x)=cp(x°) on rn E0. Hence also

w(x)=q(x°)=c on the intersection with r of a small ball around x°.

Thus w(x) indeed solves (3.3) and is constant on r near x°.

A characteristic surface r along which Fly= aNI (x,Tlx)=0

will be called a multiple characteristic. An example of an expres-

sion with multiple characteristics is the heat operator (example

b), below). For hyperbolic equations one tends to avoid multiple

characteristics, introducing the concept of strictly hyperbolic

expressions (cf. VII,2). An expression L having no multiple real

characteritics is said to be of principal type (VII,2).

As mentioned initially, characteristic surfaces are important

for the initial-value problem of (8.1): For a given (hyper-) sur-

face rC &n and given 'data' V. defined on r and f defined near r

we ask for existence of u (defined near r such that

(8.5) Lu = f near r, u(a)= aau = 1pa, x(= r , for all lal s N-1

The bicharacteristic strips, on the other hand, will be

recognized as carriers of singularities of solutions of Lu = f,

(cf. ch.6, ch.9). The key result is our version of Egorov's theo-

rem (V1,thm.5.1), accessible only after a study of i,do's.

Solving (8.5), it is clear that not all functions 1ya may be

prescribed arbitrarily. Assuming r and 1pa smooth, let v(x) be a

smooth transversal vector, defined (locally) near r, and never tan-

gential to r, let us start prescribing arbitrarily the functions

u=1p0 , Dvu = 1U1 , ... , DNu = 1IrN-1 , on r , where

(8.6)

0.8. Characteristics 43

This will determine all u(a) : lals N-1, on r, as tangential deri-

vatives of the Vj: For some local parametric representation x=x(k)

'xn-1) of r the derivatives u(a)(x), xE T, become linear

expressions in the derivatives a.yPj. Accordingly, Lu=f, along IT,

translates into a linear equation of the form

(8.7) pDNu + Fpaaa j = f

Now if q E C1 withTlx

yd 0 is constant on F we find that

(8.8)

ax7= (TIxj/DvT)Dv + Tj , j = 1 , ... n ,

with vector fields Ti satisfying T,T = 0 . Substituting (8.8) into

(8.1) one finds that p in (8.7) is given by

(8.9) p =(Dvq))-N

I

-N aa(x)cplxa(x)a

If p(x)p60 on t then (8.7) may be solved for DNu in some neigh-

bourhood of T. Moreover, we may apply Dv to the resulting equation

for an infinite number of new relations of the form Dmu= linear ex-v

pression in derivatives of lower order, allowing recursive calcu-

lation of all Dvu on F. Clearly this allows construction of a uni-

que solution of (8.5) by its Taylor series, assuming that an analy-

tic solution exists. The recursion may be used for estimates pro-

ving convergence of the Taylor series and existence of an analytic

solution, if the coefficients are analytic. This sketches a proof

of the Cauchy-Kowalewska theorem mentioned initially. Moreover the

importance of the concept of characteristic surface is clear from

the result below, the proof of which is evident, after the above.

Proposition 8.1. The function p(x) of (8.9) vanishes identically

on t if and only if r is a characteristic surface. We get p#0 on

t if and only if I' is nontangential to any characteristic surface.

To provide more detailed information on characteristic surfaces

and bicharacteristics we return to some examples, mainly of sec.4.

a) L = t (of (4.1)) : The characteristic equation

(8.10) cpIx2 = Lj=1 'pIx2 = 0

has only constant real-valued solutions, since it requires TIxmo.

Real characteristic surfaces do not exist. However, A has constant

(i.e., analytic) coefficients. The Hamiltonian system (8.4) may be

regarded as a set of ODE for complex-valued analytic functions.

There exist solutions of (8.10) in the n complex variables x1,...,

xn, defined in a domain of T . Setting such q constant will given


surfaces in Cn, called complex characteristics. Notice that the

Helmholtz operator (4.2) has the same principal part, hence the

same real or complex characteristics as the operator A .

b) L=at-A, in the n+1 variables x0=t, x=(xl,...,xn), (i.e., (4.3))

The characteristic equation is (8.10) again. But we now have one

more variable t=x0. Thus (8.10) requires that cp(t,x) is indepen-

dent of x, but it may depend on t. It follows that the characteri-

stic surfaces all are hyperplanes of the form t=const. Complex

characteristics of more general kind may be constructed, of course

In sec.4 we solved a modified Cauchy problem, of the form

(8.11) Lu = f , t a t0 , u = tV at t = t0 ,

for f, WE Co. Only one condition at t=to (instead of 2) is needed

to make the solution of this characteristic Cauchy problem unique.

The real characteristics of L all are multiple characteristics:

For c(t,x)=t-t0 we have cpjx (1,0), hence a2(glx)=a2101110=0-

c) L = at - A , (i.e., ex'le (4.4)) , again discussed in n+1

independent variables (t,x) . The characteristic equation is

(8.12) WIt2 = W1x2 = =1 T1. 2

Being only interested in the surfaces T=const., for (WIt,WIx)#0

we may reduce the number of variables: (8.12) implies WIX 0 unless

fIt#0. Hence T=0 may locally be solved for t. Writing t=J(x), get

(8.13) 2 2j=1 JIx, = JIx = 1

This equation is often referred to as the equation of geometrical

optics, or the eiconal equation. The bicharacteristic strips (so-

lutions of (8.4), in this case) are straight lines x=at+b with

constant a,bE Rn , Ia1=1, and constant . The base curves of the

bicharacteristic strips are the light rays of geometrical optics.

The Cauchy problem (8.5) is well posed for the wave operator

along non-characteristic surfaces (cf.sec.4 and ch.7.). Again the

Klein-Gordon operator (4.5) has the same principal part and charac-

teristics as the wave operator, and a similar Cauchy problem.

d) L is an elliptic operator, with Coo-coefficients. That is, we

have as xE cz, E n, O. Again the characteristic equa-

tion (8.3) implies cplx=0 assuming T real-valued. Real characteri-

stics do not exist. Complex characteristics again may be defined

only if the as are analytic in xl,...oxn .

e) L = P(D) is a constant coefficient hyperbolic operator (cf.

VII,!), with respect to some given real vector h# 0. As a conse-

0.9. Lie groups and Lie algebras 45

quence of VII, prop.1.5. the principal part PN(U) also is a hyper-

bolic polynomial: We have PN(h)#0, and the algebraic equation

for fixed real , has N not necessarily distinct real

roots X., j=1,...,n . The characteristic equation is of the form

PN((plx)=0. For simplicity let h=(0,...,0,1), as always may be ach-

ieved by a linear transformation of independent variables. Then

the characteristic equation decomposes into the N equations

(8.15) TIxn = tj(roIx1,...,(' Ixn-1) ,

with t =the real solutions of

0 for _In the presence of multiple roots we cannot expect that the

tj depend smoothly on 1If smoothness can be arrangedthen each of the N equations (8.15) will give its own family of

characteristic surfaces and bicharacteristics. Thus one may expect

N different types of characteristics for an N-th order expression.

The N different types of real characteristics do exist if

the N roots are all distinct, for=(1 0)#0. Then

the t. will be smooth. In this case P(D) is called strictly hyper-

bolic (cf. VII,2). We shall see in ch.7 that the Cauchy problem

(8.5) is always well posed, even for variable coefficients, if L

is strictly hyperbolic. For constant coefficients the Cauchy pro-

blem of a general hyperbolic L is well posed (cf. [Gai], [Hri]).

9. Lie groups and Lie algebras, for classical analysts.

Let us discuss the relationship between a Lie group G and A,

its Lie algebra, using a form suitable for analysis minded readers.

Generally, G may be represented on some GL(1N) (we will not show

this, although the tools are developed). For simplicity let G be a

Lie subgroup of GL(1N), as will be true for all our applications.

A path ip(t), Itl<rl , in G then is a path of NxN-matrices together

with its derivative V'=di,/dt. For i,(0)=e=unit of G , and gE G ,

the path gV(t) starts at g and (gi(t))'=0'(t) with matrix product.

Let y:U- G, or explicitly, y=y(u), y(0)=e, be a charted neigh-

bourhood of e, with an open set UC 1v, v=dim G, OE U. For a gE G

the function yg=gy(u), uE U, defines a charted neighbourhood of g.

The 'partial derivative' ylu is defined as directional deri-7

vative along the coordinate line uk=const, k#j, uj=t (i.e., of)

Similarly for ygluj .


In these coordinates the tangent space at e=y(O) (i.e., the

corresponding Lie algebra A) consists of all a=(a3) E &v, repre-

senting the matrix Aa = 2aJylu (0). We get 1-1-correspondencesj

Av= A H {Aa:aE Av} H {a(g):aE &v} with the vector field a(g) def i-

ned by using the same aE my in the local coordinates yg at g :

a(g)=jajygluJ

(0) . a(g) defines a global folpde La on G - i.e.,

a(yg(u))=La(yg(u)), with La, in coordinates y(u), near e, given by

( 9 . 1 ) La D Y.A30 (u,O))Y(u)a/aul, where 9(u.v)=Y(Y(u)Y(v)),1 j j

similarly for general g with yg . Again A **IL a:aE Av} , so that

aE A appears in 4 forms, as a v-vector (a3), a matrix Aa , or a

folpde La on G , or again as a(g)=2ajygluj (0)=gAg .

For aE A define a smooth curve V(t) in G as solution of

(9.2) 3 = a(p(t)) , t near 0 , V(O) = e

Locally, (9.2) constitutes a system of v ODE's of order 1, with

real C"-coefficients. Its Cauchy problem (9.2), even at to instead

0, is uniqely solvable. In fact, (9.2) amounts to y1'=Aa , V(0)=e,

with the above Aa, and matrix multiplication. This is trivially

solved by the exponential function y,(t)=exp(Aat)=FòAajtj/j! . By a

continuation argument get yi(t)=exp(Aat)E G , and (9.2) for tE I

In particular, {yi(t)} defines a subgroup of G. We summarize:

Proposition 9.1. For each tangent vector aE A=Te(G) there is a i-

parameter subgroup P(t)=Pa(t), tE 1, defined as unique solution of

the Cauchy problem (9.2), with a(g)wLa of (9.1).

Corollary 9.2. The connected component GeCO of the unit element e

is generated by all the above 1-parameter subgroups, and even by

{Va(t): aE A , 0st5s} , for any fixed s>0 .

Indeed, any gE Ge may be connected to e by a path r , and F

may be replaced by a 'polygon' with sides translates of 'lines'

Va(t), Ostsr's , for fixed aE A. This shows that, indeed, g=

Va1(E1)Va

2 (E2) .... VaM(Em) ,

for finite M and sjss.


Corollary 9.3. We have

A t(9.3) Va(t) = e a = A,,3/j! , Aa = Va.(0)_EalY luj(0)

AThe subgroup Ge is generated by all e at, aE A, O<tse>O

Completing the structure of A, we define a bracket operation:

Consider the commutators [Aa,Ab]= AaAb-AbAa and [LaLb]=LaLb LbLa

of the matrices and (matrix-valued) first order expressions, resp.

Get y(u)y(v)=y(9(u,v)), near u--v=0, with 0(u,v) of (9.1). Differen-

tiating this conclude that [y lu (0),Ylu (O)]= 2jcjlYlu (0) , with1 r r

xjl = 9rlu v (0,0)-9rluv (0,0) . Accordingly,

7 1 1 j

(9.4) [AaAb]=Ac , c = (cr)=( is 1ajbl)

again belongs to the tangent space of G C GL(IN) at g=e. Next,[LaLb] is a first order expression: LetipdI()=11aO v (u,0)Y(u)

jl k

The matrices pj(u), ql(u) commute for uE U, hence (9.1) yields

(9.5) [LaLb]= j71(Pjglluj-gjplluj)au1

At u=O get pj=aJe qj=b3e , 0(O,v)=v , hence 0 (0,0)=Olk Thus1Ivk

{q+}lu.(0)={ij }Ylu.(0)+E{bkPllvku,(0,0). We get

(9.6) j(pjgllu-gjplluj) _ 7,(ajbl-bjal)Ylu=b1Aa-alAb

with our above Aa, using the symmetry dlluvk=01lvjuk, at u=v=O.

Accordingly, [La,Lb]Y(0)=I(b1Aa-a1Ab)Ylu1-AaAb AbAa [Aa,Ab]=LcY(0)

The argument may be repeated at general gE G , using Yg(u) instead

Y(u). One finds that [La,Lb]Yg(0)=Lcyg(O) , for all gE G , with c

of (9.5) . Accordingly Ac **Lc

, under our 1-1-correspondence.

This induces a bracket operation in the Lie algebra A, given by

(9.7) [a,b] H[Aa,Ab]= Ac H[LaLb] = Lc , c of (9.5) .

[.,.] has the usual properties, skew-symmetry, and Jacobi identity.

Next we assume given a Lie matrix algebra - i.e., a linear

subspace A of some L(IN ) containing all of its (matrix-) commuta-

tors [A,B]=AB-BA, A,BE A. Assume dim A = v . Prop.9.1 and cor.9.2


then suggest a direction of approach for generation of a corres-

ponding Lie group G - a Lie subgroup of GL(in) : G should contain

all matrices eAt, tE &, AE A, and all their finite products. It

should be closed under matrix multiplication and inversion. Here

we are only interested in connected Lie groups (i.e. might get on-

ly the component of 1 of a larger group). In fact we will look at

the minimal such set - all finite products of eAt, AE A, tE k

Observe that Vt = eAteBt , for A, B E A , no longer defines

a 1-parameter group, unless A and B commute. For general A, B we

will prove the Baker-Campbell-Hausdorff formula: For small Iti ,

(9.8) Vt = eC(t) , where C(t) E A

is a convergent power series in t (cf. lemma 9.5, below).

As a first step in this direction:

Proposition 9.4. Vt satisfies the differential equation (""'="dtit)

(9.9) Vt' - (A+eAtBe-At )Vt ,

where the coefficient A+eAtBe-At E A is a convergent power series.

Proof. (9.9) follows trivially by differentiation. Also, the

family Bt = eAtBe-At satisfies the differential equation

(9.10) Bt' = adA Bt , B0 = B ,

with the linear operator

(9.11) adAB = [A,B] , adA E L(L(&N) )

Therefore,

(9.12) Bt eadAtB = L4=Q. (adA) JB = B+[ A, B] t+[ A, [ A, B] ] 22 +... ,

showing that Bt E A , hence A+Bt E A , q.e.d.We will prove (9.8) in the following form.

Lemma 9.5. For A , B E A , define C=C(t) = log(eAteBt) by setting

C(t) = X(t)-X(t)2/2+X(t)3/3 t...+(-1)kX(t)k/k t....(9.14)

with X(t) = eAteBt-1 = (I+At+A22,+...)(1+Bt+B22l+...) -1 .

Then we have a convergent power series expansion

(9.15) C(t) = log(eAteBt) _ 1C.t3, Itl< (log 2)/(IIAII+IIBII)

where C(t) and all coefficients Ci belong to the Lie algebra A


In particular,

(9.16) C1= A+B , C2= 2 A, BI , C3= 2[A,[A,B]]+ 2[B,[B,A]]

The coefficient Cj is a finite linear combination of terms which

are applications of (together j-1) operators adA or adB to A or B.

Proof. We use a proof of D.Djokovic [Dj] (we owe to G. Hochschild)

Let C(t)=log(eAteBt) , defined as composition of

the logarithmic power series and X(t)=' 1XjtJ . Note that the

matrix coefficients cj1 of C(X) = log X = > 1(-1)iXJ/j are con-

vergent power series in the N2 complex variables xjk , where

X=((xjk))j,k=1,...,NIwhenever IIX(t)II<1. The latter holds for

eIIAIItellBIIt_1=e(IIAII+IIBII )t_1<1 , i.e., (IIAII+IIBII )t < log 2 .

Thus cjl are complex differentiable functions of the N2 complex

variables xjk , which in turn are power series of t . This implies

complex differentiability of the function C(t) for t as in (9.15).

So C(t) is a convergent power series , and (9.15) follows. We are

only left with showing that the coefficients Cj belong to A . The

first coefficient (cf. (9.16)) is easily verified by direct compu-

tation. For the general case we will show that C(t) solves an ODE

((9.18), below) which leads to a recursion for the Cj.

We calculate that (eC)'e C = = 1=0 N(-!k,)l(Cm),Ck

( 1)n-m 1 (Cm),Cn-m = I -II(n)(Cm),Cn-m

(n-m)'m. 1 namn m

x-n n-m n m n-m m I k m-k-1X00 1I_Oh C'C Thus=ten=1 n!=1(-1) (m)(C )'C Here, (C )'=

-k=0(_1)n m-1(m+1)CkC,Cn-k-1the inner sum equals Tn

= min=01

7t1-

1

=0CkC'Cn-k-1=k(-1)n-m-l(m+n

1) _CkC'Cn-k-1Zk

, where Zk =

n-k-1= (_1)n-1-k(n-1), by induction. We get T ==0 7 k n

4_01 CkC'Cn-k-l(-1)n-1-k(nk1 ) _ (ad C)n-IC' whence

(9.17) ' 1n!(ad C)n-IC' = (eC)'e-C = Vt'Vt 1 = A + eCBe-C


adusing (9.9) and eC=eAteBt=Vt. With (9.12) we get eCBe-C= e C B

The left hand side of (9.17) may be written as f(adC)C'

with the power series f(ey)= 1n-1/n! =(ex-1)/x . Notice that

is a power series too. Applying it to (9.17) yields the

desired differential equation

ad(9.18) CO = g(adC)(A + e C B)

Expanding both sides of (9.18) we get Cl- ' 1(n+l)Cn+ltn

while the right hand side may be written in the form

g(L1 tkadck)A + h( tkadC

k)B , h(x)=g(x)ex

The coefficient of the k-th power of t , at right, is a finite

linear combination of a finite number of applications of adC toj

A or B , where js k . In fact, the sum of all j in each such term

equals k . We know that C1E A . If we assume by induction that

C1,...,CkE A, it follows that (k+l)Ck+l E A , hence Ck+IE A Also

Ck+l must be a linear combination of k applications of adA and adB

to A or B , as stated. The proof is complete.

Corollary 9.6. Let All... ,A, be a basis of A (as a linear space),

and let A(u)= "J=1Ajuj , ujE 1 . There exist v power series

fk(u,v) k=1,...,v in 2v variables u,v convergent for IuMvI« 0

with some e0>0 such that fk(u,v) = uk+vk + ... , k=1,...,v , and

(9.19) eA(u)eA(v) = eA(f(u'v)) ,juI,IvI<s0

.

Proof. As in the proof of lemma 9.5 we conclude that eA(u)eA(v) _

eC(u,v)where C(u,v) is a convergent power series in u and v as

IIA(u+v)ll<log 2 . Lemma 9.5 then implies that C(u,v)E A for suffi-

ciently small Iuj,IvI. Hence we may write C(u,v)=A(f(u,v)), where

again fk(u,v) are convergent power series. It then is evident that

fk(u,v) = uk+vk + higher powers. Q.E.D.

From the given linear subspace A of L(EN) we now define a

group G - G(A) as the collection of finite products of matrices

eA , with A E A , where the group operation is matrix multiplica-

tion. Clearly G contains I = e0 . We intend to show that G is a

(v-dimensional) Lie-group. Moreover, the tangent space of G at


its unit element I equals the Lie-algebra A . And, vice versa,

if we depart from a general Lie-subgroup G of GL(&N), then define

A as the linear space of all directional derivatives of curves

in G starting at I , and then, with this A, defining the above

group G(A) , we get G = G(A) .

First let us establish a coordinate chart for a neighbour-

hood of the identity I E =-G . With the notations of cor.9.6 define

(9.20) Y (u) = eu' A, +u?. A2 ...+u, A,= A(u) ul<s0

where EO>O is kept fixed. Then y(u) defines an invertible map of

the ball BED {lul<E} onto a set UIC G containing I, and y(0) = I

For general GE G similarly YG(u) = y(u)G defines a map of B. onto

a subset UG C G with G=YG(0) E UG . Introduce a topology on G by

using all the "balls" yG(BE) 0<ESE0, GE G , as a basis. Then each

set UG is an open neighbourhood of G . Let UG fl UG, Then

eA(u0)=eA(v0)Z , Z=GG'-1E G , for some u0 , v0 E BE . Using cor.

0 09.6 we get Z = eA(f(u-v)) , assuming E0 properly chosen.

Notice that f(u,v) = 0 can be solved for v near u=v=0 , giving

a function v=v(u) , v(0)=0 , since the Jacobian flv(0,0)=I is

invertible. It follows that also flu(u0,-v0) is invertible, so

f(u,-v)=f(u0,-v0) may be solved for v giving an analytic map v-

v(U) , v(u0)=u0 , provided that u0, v0 are sufficiently small,-

i.e., that e0 is chosen sufficiently small. Conclusion:

Z = eA(f(u0,-v°) =eA(f(u,-v(u)) =eA(u)e-A(v(u))for u close to uO.

Or, YG() = YG,(v(u)) for small lu-u0l showing that the map

YG'-1°YG : YG 1(UG(l;JG') -> YG'-1 (UG"JG, ) is analytic. Thus indeed,this imposes a manifold structure (and a topology) onto G making

it a Lie Group. Every finite product Thai , BiE A may be connected

to I, by the path 17etBi, Osts1, hence G is connected. Evidently,

the tangent space of G at I (i.e.,the space of all directional de-

rivatives of curves in G starting at I) coincides with A .

Thus, indeed, we established a 1-1-correspondence between the

connected Lie subgroups of GL(2N) and the Lie subalgebras of L(&N),

Chapter 1. CALCULUS OF PSEUDO-DIFFERENTIAL OPERATORS

0. Introduction.

In this chapter we deal with the details of pseudo-differen-

tial operator calculus. We follow a presentation in a lecture of

1974/75 [CP], inspired by the local approach of Hoermander [Hr:],

dealing with operators on Rn, for didactical reasons. Replacing

asymptotic expansions of [CP] by Leibniz formulas with integral re-

minder of 1.5 is an improvement we learned from R.Beals [Bi] who

uses 'weight functions' more general than our (x)()ml of (3.2).

Still asymptotic expansions are needed, and will be studied in 1.6.

We will discuss 4 different representations of ido', referen-

ced as a(x,D)=a(M1,D), a(Mr,D), a(M1,Mr,D), and a(Mw,D), the first

two corresponding to the left and right multiplying of Kohn and

Nirenberg [KN], and the others to a representation of Friedrichs

[Fr3], and the Weyl representation.

The reader who dislikes the infinitely repeated formal dis-

cussions has our sympathy. For other approaches to the same sub-

ject cf. ch.7 where the 4do's of certain symbol classes are ident-

ified as operators on H=Lz(&n), smooth under action of certain Lie

subgroups of U(H). Or else, cf. [ C1] and [C2], where regular andsingular elliptic boundary problems are approached with tools of

C*-algebras, avoiding entirely the ado-calculus.

The calculus presented generalizes formal calculus of diffe-

rential operators. We get a collection of Frechet algebras contai-

ning differential operators, with formulas for product and adjoint

like Leibniz formulas, containing generalized inverses (so-called

Green inverses) of their elliptic and hypo-elliptic operators, as

seen in ch.II. The algebras are 'graded': Each of their operators

has a differentiation (m,) and a multiplication (m:) order.

We will get the same Fredholm theory in Sobolev spaces as*

provided abstractly, using C -algebras, in [C,],III,IV.

1. Definition oftpdo's.In this section (and occasionally later on) we will write a(M)

for the multiplication operator u(x)-a(x)u(x). More generally, for

52

1.1. Definition 53

a pseudo-differential operator generated from a 'symbol'

E C00 (1e3n), with ten , we will write a(M1,Mr,D) to indicatean order of operation: Multiplication corresponding to the x-(y-)

variable is carried out to the left (to the right) of the differen-

tiation D of the a-variable, hence "M " (hence "Mn') "(cf. cor.2.3).1

Introduce the space ST of all C'(&3n) satisfying

( 1 . 1 ) 0(()x(k).((x,Y))X(1)IaI+I1Isk, IYIs1

with nondecreasing x(k),X(l) , k,1=0,1,..., such that

(1.2) j) = limj.(?(j) - j) = -0°

Here, and in all of the following, we use the abbreviation

(z) = (1 + IzI2)1/2 , z E 1em

(particularly ((x,y))=(1 + x2 + y2)1/2) Another useful shorthand:

(1.4) Ox = (2n)-n/2dx , 0 _ (2n)-n/2d

measures over ten. The functions of ST are called symbols. For aE

ST define a linear operator A=a(M1,Mr,D) called pseudodifferential

operator (t,do) with symbol a by setting, with integrals over ien,

(1.5) (Au)(x) = J4j y u(y)

The precise meaning of (1.5) is clarified in thm.1.1, below.

Theorem 1.1. The right hand side of (1.5) is well defined, for uE

S, xE &n, in the following sense: The integral f ¢y exists as impro-

per Riemann integral, for >en. It supplies a C(2")-function

I. For fixed xE &n, I(x,.) E L'(&n), and f A defines a function

v=AuE S=S(&n). Moreover, the map A:S-'S is continuous S->S.

Proof. Clearly Jyiy u(y) exists as descri-

bed, since the exponential is bounded, while

is of polynomial growth in y,for fixed x, so that

a useful -estimate one uses the identity

(1.6) ei (x-Y) = (x-Y)

m=0,1,2,..., with the Laplace operator Ay = 32 . A partial inte-J 7

gration for fdy is applied over a finite ball. The boundary terms

54 1. Calculus of pseudodifferential operators

die out, as the ball tends to In, by (1.1) and uE S. This yields

(1.7) -2m JAlY

In the following we shall say that a given expression is 'in

the span' of another given set of expressions, if it is a finite

linear combination with complex coefficients of these expressions.

For example, the integrand of (1.7) is in the span of

(1.8) ei (x-Y)(Da a)(x,Y,t)u(R)(Y), Ial + I1I s 2m.

Note that we get

(1.9) Pap

using (1.1) and the decay of uE Sfr, where q may be arbitrary. Thus

(1.10) I(x,T) = 0( ( )K(2m)-2m J((x,Y))"(0)(Y)-q dy ).

A substitution y = (x) y- , dy = (x) ndy brings 'f 11 in (1 .0) to

(1.11) (x) ,(0)+nffJ(Y)1(0)(1+(x)2lyl2)-q/2s c(x)X(O)+n,

c=rr1J(Y)%(0)-q

Therefore

(1.12) (0)+n

and, with (1.2), it now is evident that

(1.13) J a P1

exists as an improper Riemann integral for all x.The estimate

(1.14) x(2m) - 2m < -n-Ial

holds for large m. It insures that I(x,.)E L' , for all fixed X.

For a=o this proves the first assertion of the theorem.

Let us clarify the dependence of (1.12) on the function u

Lemma 1.2. We have

(1.15) II(x, )I s cllullk ( )K(2m)-2m(x)X(o)+n0

with c depending on m and aE ST, but not on uE S, where the norms

(1.16) IIulik = suPlalsk,xE R"' II( x)ku(a),LOD

of S are used.

This lemma is clear, since Iu(a)(x)IsIIuIIk(x)-k, xE In, Ialsk,

1.1. Definition 55

estimates u(a) by IxI-q, as the only way, u enters (1.12).

Let us prepare a few other simple lemmata for thm.1.1.

Lemma 1.3. For a E ST we get aaSY=DaDYDYaE ST and ST, for

any polynomial P=P(x,y,l;). With x,a. of(1.2) for a, corresponding

functions for aaPy and P- a are is (j+ a + 3 I) , ?. (j+ Y I) , and is (j+L) ,X(j+M), resp., with the -degree L and (x,y)-degree M of P.

The verification of Lemma 1.3 is left to the reader.

Lemma 1.4. The integral at right of (1.5) defines a Cc°-function

of x denoted by v . Moreover, the derivatives v(a) may be calcula-

ted under the integral signs of (1.5).

Proof. Notice that is in the span of

(1.17) a + y =

By lemma 1.3 we have aDXa(x,y,l;)E ST. For an induction proof one

only must show existence of the first derivatives and differentia-

bility under the integral sign for 101=1. Let qE L'1L (B), supp cp

CC & . Using (1.12) and Fubini's theorem (twice) we get

.Jdxjq(xj)JdjdY

(1.18)

fDO

X= -iaxj

where we set cp [a,t] , the characteristic function of [a,t]. Theinner integral at right of (1.18) may be calculated explicitly. A

differentiation of (1.18) for t will give the desired result.

Lemma 1.5. For aEST and every multi-index y we have

fdjdy u(y)(1.19)

r

_ (-1)YfdjdY u(y)

Proof. By the second identity (1.6), (1.19) is a matter of partial

integration, carrying the differentiation from the exponential

to But the interchange of fd and fdy is generally im-

possible, so that we will require the argument, below.

1) It suffices again to assume jYl = 1, by lemma 1.3.

2) A conclusion as in lemma 1.4 may be used to verify that

fdjdyaj{u(Y)ei(x-Y)a(x,Y,U)}=fda jfdyu(Y)eit(x-Y)a(x,Y,)


3) The right hand side of (2) is zero, because boundary inte-

grals vanish, by (1.12). Thus (2) yields (1.19) for IyI=1, q.e.d.

Proof of thm.l.l (continued). Let v = Au, u E S, as defined by

(1.5). Lemma 1.4 implies

(1.21) xav(p)(x) = f41JoY

where the right hand side is in the span of

(1.22) 6+s=a, a+t=(3.

Using Dxel (x-Y) = in (1.22) we may integrate by

parts, in the inner integral, to obtain expressions in the span of

fd(1.23)

6+s=a, a+T=p, X+µ+v=a

We also applied lemma 1.5 and another partial integration Note

that IF.-µIskEIs IaI, and ITI+IvIS ItI+IaI=lpi Hence lemma 1.2 and

lemma 1.3 give an estimate of (1.22) by

(1.24) fdAccordingly xav(p)(x) also is bounded by (1.24). Applying this for

all IaI s 1 , for some integer 1, we get

(1.25) Iv(p)(x)I s cO(x)n+k(1)-lllulik

,

)x(2m+IPI)-2m

Here m , 1 are integers, but m must be chosen large enough

to insure existence of the integral in (1.25), by (1.2). Also, co

and k depend on the choice of m,l and p, but not on u. Using (1.2)

it follows that for 10 there exists k, c0 independent of u with

(1.26) IIAuII10

s 0OIIulik , u E S

Clearly this amounts to continuity of a:S-'S and thm.1.1 is proven.

2. Elementary properties of y,do's.

Theorem 2.1. Let

(2.1) (u,v) = f n u(x)v(x)Ax , u,v E S

Then we have, for all a E ST ,

(2.2) (a(M1,Mr,D)u,v) = (u,a(Mr,Ml,-D)v) , for all u,v E S ,

1.2. Elementary properties 57

with A"=a(Mr,Ml,-D), the 4do with symbol a

Proof. Clearly a-(=- ST as aE ST. The statement is a matter of inte-

gral interchanges. Get L1(&2n), vE S, by lemma 1.2.

Hence (v,Au)

foJoyfpix ... = f oyfRi f41x ... _ (u, A" v) ,

for A = a(M1,Mr,D) , with a 3-fold application of Fubini's theorem

and a substitution (y,x,-l;) of variables, q.e.d.

A" will be called the distribution ad joint (or dual) of the

operator A, (to be distinguished from the 'formal Hilbert space

ad joint' A*, formed with the inner product (u,v) = f uvdx ). Note:

(2.3)*

A = a(Mr,Ml,D)

Since A=a(M1,Mr,D) is strongly continuous S-'S, by thm.1.1,

an extension B : S' - S' of A : S - S is given by

(2.4) (Bu,c)) = (u,A (p) u E S' , (P E S ,

with (u,cp), the value of uE S' at TE S. In other words, we may re-gard the ',do either as operator S-S or, S'-S'. Thm.2.1 gives AC B.

Generally we will not distinguish in notation between the ope-

rators A and B , but will regard a 'pdo as an operator A:S'-'S' with

AIS mapping to S. In fact, 'pdo's more or less will be regarded

like differential expressions in [C2],II with domain to be fixed.

Theorem 2.2. ST is an algebra of complex-valued functions under

pointwise addition and multiplication, with (real) involution a-a;

Proof. Evidently ST is closed under "+", """, and scalar product.

Note that DaDyDY(ab) is in the span of (2.5), by Leibniz' formula:

(2.5) (Da'DR'DY'a) (Da"DR11DYlob) , a'+a"=a , l'+0"=(3 , Y'+Y"-Y .x y i x yIn (1.1) we may assume the same x and X, for a and b. Using (1.2)

write x(j) = j - a(j) , ?(j) = j - t(j) , o(j)and estimate the products of (2.5) by

t(j)

(2.6)(x) IaI+ IRI-a(la' I+IR' I )-a( ICE" I+IR")( (x,y)) I? I-t( IY' I)-t( IY"I )

Then define al(j)=Min {a(1)+a(j-l)} , t'(j)=Min {t(1)+t(j-l)},lsj le. j

We get the estimates (1.1) for c with x'=j-a', X'=j-t' instead of


x,X. In particular we also get x' -> 00 , %' - 00 , q.e.d.

Corollary 2.3. The symbol algebra ST contains all a of the form

(2.7) ba(x)ca(y),aasNwhere ba , ca E COO(&n) satisfy the estimates

(2.8) A a(x) = 0((x) P) , Dc(x) = D((x)p) x e &n ,

with a constant p , independent of x , a , R . Then we get

(2.9) A = a(Ml,Mr,D) = ba(M)Daca(M)

a differential operator.

Proof. The Fourier transform u^ = Fu and its inverse u" = F lu

(2.10) u" () = J91x u°(x) = JF1 u E S

satisfy (c.f. 0,(1.22)).

(2.11) FDjF-1 = Mj , j = 1,...,n.

Theorem 2.4. Let a,b E EST be independent of y and x, respectively.

Then the 1pdo's A = a(Ml,D) , and B = b(MrD) may be written as

(2.12) (Au)(x)= =fdye

for u E S , with the Fourier transform (2.10).

Proof. The first formula is clear, using (2.10). Also,

(2.13) (Bu)(x) _f4eix

,f

JP1y a '

where the inner integral is in L1(&n) , by lemma 1.2 (In this

case I is independent of x). Hence we may write (Bu)(x)=I"(x). We

know that Bu E S . Therefore (Bu)^=(I`)^= I , as proposed, q.e.d.

Theorem 2.5. For a,b EST let We have

(2.14) c EST , and c(Mi,Mr,D) = a(Mi,D)b(Mr,D)

Proof. From Theorem 2.2 we conclude that c EST. Using (2.12),

A(Bu)(x)=ff

u(y)

= (c(Mi,Mr,D)u)(x) , u E S , q.e.d.

For a symbol a E ST consider the bilinear form

1.2. Elementary properties 59

(2.15) (v,Au)= (v,a(Mi,Mr,D)u)=

for u, v E S . We have seen that f9Ixfpig fyiy=f9I JJTx9Iy .Note that a

temperate distribution k E S'(&2n) is defined by

(2.16) (k,w) = J JoxPzy w E S(&2n)

By the techniques of sec.1 we get

(2.17) (k,w) = J<) -2mfP1xPlyei (x-Y) (1-Ox)m(aw)

For large in this is in the span of

(2.18) (x,Y) , IPI+IYIs2m,

where (1.1),(1.2) give existence of the

inner

integral J. But J is

(x,y)-continuous and O(((x,y))%(0)). Thus (J,DXW) define DXJE S'.

We shall call k, defined by (2.16), the distribution kernel

of A=a(M1,Mr,D), aE ST. Note kE SI(1e2m) is uniquely determined by

(2.19) (v,Au) _ (k,v(gi) , u,v E S .

Formally, with distribution integrals, we may write

(2.20) (Au)(x) = fk(x,y)u(y)piy , (A"u)(x) = fk(y,x)u(y)yiy

Proposition 2.6. We have

(2.21) k(x,y) = kA(x,Y) = a`'(x,Y,x-Y) = a'(x,Y,a)Ia=x-Y

in the following precise sense: The inverse Fourier transform a-

of with respect to its third (its 2n+1-st...3n-th) argu-

ment is a distribution in S'(13n) which may be written as a con-

tinuous family pa =a3'(x,y,(x-y)+a) of distributions inS,(R2n),

(2.22) (P(Y,V) = fdxdypO(x,Y)V(x,Y) , 'VIE S(R2ri)

in the form

(2.23) (8'',q)) = fdo(pa,V(,) , 'W0(x,Y)=T(x,Y,o+x-y) , AE D(13n)

with a distribution integral in (2.22) defining p0EC(&n,S,(R2n))

while (2.23) is a Riemann integral with continuous integrand, of

compact support. Then the distribution kernel k coincides with the

distribution p0 of (2.22) for a=0 .


Proof. Clearly a family p0E C(&n,SI(it2n)) is defined by setting

(2.24) (P0,V) = )'W(x,y) , PE S(&2n)

Indeed, we get pa in the form (2.17) with a replaced by aei6

Then a look at (2.18) (with this a) shows that (p(,,p) EC0(ln)

We get fdo(p6,ypa)=

,(P)

confirming (2.23), where we have used a linear substitution of in-

tegration variable, and some Fubini-type interchanges. Q.E.D.

The distribution kernel's singular support will be of inter-

est in the following. In sec.3 we introduce restricted symbol clas-

ses with the property that the kernel k=ka for such a symbol a has

singular support contained in the 'diagonal' {(x,y)E1e2n:x=y}.

For general a e ST this is not true: Consider

E ST. Then sing supp ka coincides with the set {(x,y):Ix-yI=1}, as

follows from prop.2.6 , 0,lemma 4.1 and 0,(4.42).

In the following let ST1={aE ST: aaa=0} be the set of aE ST

independent of y. Following a general convention the class of all

pdo's A=a(M1,D) with symbol in a class P will be denoted by OpP .

3. Hoermander symbols: Weyl ydo's; distribution kernels.

For differential operators A of the form (2.9) it appears

that there are many representations as y,do's, due to the fact that

Leibniz' formula may be used to convert a product Daa(M) to a sum

of products aa,(M)Da' , and vice versa. In fact, it is clear that

a differential operator (2.9) always may be rewritten in the form

(3.1) A = b(M1,D) = b1(Mr,D) ,

with unique symbols bE ST and b1E ST depending on x, only.

The same will be true for a iyido A=a(M1,Mr,D), if a satisfies

stronger inequalities. In fact, a generalization of the Leibniz

formulas will be developed, either involving a Taylor expansion

type reminder, or an asymptotically convergent infinite series, as

discussed in sec.'s 4 and 5. Later we will use only one represen-

tation A=a(M1,D) for all y,do's (also written as A=a(x,D)), follo-

wing a general convention. But it will be convenient to have more

1.3. Hoermander symbols 61

general forms available. The restricted symbol class SSC ST below

will have other desirable features: The distribution kernel kA of

A, for an aE SS, (cf. sec.2) will have sing supp kAC {x=y}. OpSS

will be an algebra. Also, as seen in 11,5, a 'pdo in OpSS will not

increase the wave front set of a temperate distribution..

Apart from the representations A=b(M1,D), and A=c(Mr,D) resul-

ting for A=a(M1,Mr,D)E OpSS, a third representation, of the form

A=e((M1+Mr)/2,D)=e(Mw,D) is useful, with a third symbol e , and

e(M1+Mr)/2,D)=f(M1,Mr,D) with e(Mw,D) will

be called the Weyl representation of a pdo. b(M1,D) and c(MrD)

are referred to as the 'left (right) multiplying representations'.

Let SS denote the class of functions a E COO(R3n) such that

(3.2) DaaDyDya = o(()m,+S(IaH+IRI)-p' I?I(x)m2-P2IaI(y)m3-P3IRI)

with real constants mj,pj,i satisfying

(3.3) 0 < pj s 1 , j=1,2,3, 0 s S < pl .

Also, by SSm'p'S we denote the class of all aE ST satisfying

(3.2) for a given m=(mt m2 ,m3) , p=(pi p2 ,p3) , 6 , where only thefirst condition (3.3) will be required, and no longer S<p3. Thus,

Ssm'p,5 in general will not be a subset of SS.

Let 4h and 1phm,p,S denote the classes of a(x,U)E SS or SSm,p5S'

(i.e., a is independent of y). Clearly,for yah and yadm,p,S, the con-

stants m3, p3 are redundant: Any m3 a0 and any P3 E= 1 may be chosen

since aaa=0. We then will set m3 =0, p3 =1 , or omit m3 , p3 , writingm=(ml,m2) and p=(pl,p2) as 2-component vectors only,ignoring the

trivial estimates (3.2) for 5#0. We will mostly be concerned with

y*, since it will be found that OpSS = Opih.

We also define

fl {yi1'm,P,S : m E 12 } = s(12n)

}P,6 _ - U {'m,p,S : m E It2

Hoermander [Hr2] has introduced a class SP,, of local symbols

consisting of C00(axln)-functions, with a domain 11C In using ine-

qualities similar to (3.2). He defines SP,s by the estimates

(3.4) E In , x E K

for all a, on all sets KCC 11. A iydo A=a(x,D), similar to a(M1,D)

is defined for C0(11)-functions u by the first formula (2.12).


Accordingly we shall refer to symbols in the classes

SP,6 , SS , q* , SSm'P'b , Vh,,p,b as Hoermander type symbols

From (3.2) it follows that

(3.5) Im, I+Im: I)

recalling that pj Z 0. This implies (1.1), with K(k)= m1+bk, ?(k)=

Im2I+Im3I . Moreover, (1.2) holds whenever 6 < 1 . Accordingly,

(3.6) SS CST , SSm,p,o CST , as 6 < 1 .

Lemma 3.1. Let a E SSm'P'b , b E SSm,P1'S', then we get a + b ESSm 'p- '6- , a b E SSm 'P^ '6^ , where

(3.7) mj=Max{mj,mj}, 6-=6-=Max{6,6'}, mj=mj+mj, pj=pj=Min{pj,pj}

Moreover, U {SSm'P'b:mj E 2}=SS001P'6 is an algebra under pointwise

addition and multiplication. All statements remain true if SS , in

all expressions, is replaced by ih.

The proof uses Leibniz' formula. It is left to the reader.

Remark: Note that SS and bah are not algebras. In particular the

condition 6 < p1 needs not to hold for sum or product of two sym-

bols with summands or factors of different p and O.

Thm.3.2 below is central for the calculus of ydo's. Its proof

will be given after some preparations, discussed in sec.5ff.

Theorem 3.2. Let a E SSm'P'b, with m,p,b satisfying (3.3) . There

exists a unique b E ohm .P b- and b1 E1yihm .P-,s- with

(3.8) m1 = ml , m2 = m2+m3 . P1 = P1 . 6- = b , p2 = Mini P2'P31

such that (3.1) holds for A=a(M1,Mr,D). Moreover, there also ex-

ists a unique b2E 0 m with m , p- , b- of (3.8) such that

(3.9) A = a(Ml,Mr,D) = b2((Ml+Mr)/2.D)

Observe that c=b2((x+y)/2,1;)E ST, whenever b2 EE hm,p,b, 6<1, by an

estimate similar to (3.5). Thus c(M1,Mr,D)=b,((M,+Mr)/2,D) is well

defined, but need not be in SS. We will find, later on, that, with

m,p,b satisfying cdn.(3.3), the classes {a(Ml,D)}, {a(Mr,D)}

{a(Mw,D)}, where a E Vhm,Pb, are identical..

Note that, for C = a((M1+Mr)/2,D), we get from (2.3) that

(3.10) C* = a((M1+Mr)/2,D)

1.3. Hoermander symbols 63

Note that, for C = a((M1+Mr)/2,D), we get from (2.3) that

(3.10) C* = a((M1+Mr)/2,D)

This points to one of the advantages of the Weyl type ipdo: If the

symbol a is real then the operator C above is formally selfadjoint

On the other hand the 'left multiplying type' a(M1,D) has other

advantages: If a polynomial in l;, the resulting differ-

ential operator a(M1,D) is in the conventional form -coefficients

at left from the differentiations. Recall, for Weyl type ?4do's,

(3.11) a(Mw,D) =a((M1+Mr)/2,D) =c(Ml,Mr,D),

The restricted class SS produces useful distribution kernels:

Theorem 3.3. Let a E SS. Then the distribution kernel k E S'(&2n)

of A=a(M1,Mr,D) has its singular support contained in the diago-

nal {(x,y)E R2n: x=y} = O of R2n.Moreover, for fixed x (fixed y),

the distribution k(x,.) (k(.,y)) equals a function in S(ln) out-

side any neighbourhood of x=y , and uniformly so for xE K (yE K)

K any compact set (in fact, we have (y)alaXaYkIscaRYE as xE K

Ix-YI-F ((Y)(1laXaYkISCapYE'

as yE K , Ix-YIZE))-

Proof. Referring to prop.2.6, and the discussion in [C,],II, note

that a(x,y,.) E M hence & (x,y,.)E Sos (cf.[C,],II,thm.4.3). This

implies the statement. Offering details (and independence from00 2n[C,]), let wE CO(I ), w=0 near e. Then supp w has positive distan-

ce from O, and Ix-yI -mE C00(supp w). By (1.6), with a partial inte-

gration, one may bring (2.16) to the form

(3.12) (k,w) = J0JfR1x91Y

From (3.1) we get

(3.13) 4ma(x,y,l;) =O((,)m, -2mp, ((x,Y))m:+m3 ) , m

If m is large enough then the integrand will be L1(R3n), so that

the integrals may be interchanged, for

(3.14) (k,w)= fkwoxgly , k(x,y)= fs

(3.14) has convergent Lebesgue integrals, if m is large.

Moreover,as m gets larger and larfrger, the formulas

(3.15) aaayk(x,Y) = f41 aaaP els(x-Y)A a(x,Y,U)/Ix-YI2m

give the derivatives of k(x,y) (k is independent of m; the inte-


that sing supp k C 0 . Very similar arguments will show that

y alaIayklscaRYE, as x E K , jx-ylze , etc. Q.E.D.

4. The composition formulas of Beals.

Proposition 4.1. For aE C0*(&2n) , bE C0OO(&3n) we have the formulas

(4.1) a(Mr,D) = p(M1,D) , b(M1,Mr,D) = q(M1,D)

with p,q E S(&2n) given by

(4.2) iyrla(x-Y,t-j), q(x,t)=JPlY9ll1e

Proof. Focus on the first relation (4.1), typical for the second.

For u E S and a E C0OO(I2n) we get

(4.3) a(Mr,D)u(x) = Jdyaz'(Y,x-Y)u(Y)

with '"y" indicating the inverse Fourier transform with respect to

the 2-nd (set of)variable(s) of by integral interchange.

Write ay(y,x-y)=aa(x-(x-y),x-y)=c(x,x-y), i.e., c(x,z)=a'(x-z,z)=

fg eil'za(x-z,T;). Clearly cE Cam(&2n), c=0 for large lxj, c(x,.)E S,

for fixed x. Hence cE S(B2n); we may define p(x,U) fgtzc(x,z)ell;z

E S(>Q2n), and get c(x,z)=py(x,z), c(x,x-y)=pr(x,x-y). Accordingly,

(4.4) a(Mr,D)u(x) = J41yp (x,x-Y)u(Y) = JR (x-Y) u(Y)

Clearly the right hand side of (4.4) equals p(M1,D)u(x). Also,

(4.5)

which becomes (4.2), after an integral substitution -t=rl , q.e.d.

Proposition 4.2. Let A=a(M1,D), B=b(M1,D) with a,bE C0"O(&2n). Then

Then C=AB=c(M1,D), A*=a*(M1,D) with c(x,i;), given by

)e-iyq(4.6)

a*(x,s) =

In prop.4.2 A* denotes the formal Hilbert space adjoint of A,

as in (2.3), here of the form A*=a(Mr,D). Thus the second formula

follows from (4.1) and (4.2). For the product AB write

(4.7) (ABu)(x) = Jo,t OYglrlgz eirl(y-z) u(z)

1.4. Beals composition formulas 65

We may write a 4n-fold integral, the integrand being L1(R2n). Use

(4.8)

and continue (4.7) as follows.

f(4.9) = e1T1(x-z) u(z) aDenoting the inner integral by c(x,il), (4.6) follows, q.e.d.

In the trivial case of CO symbols formulas (4.1),(4.2),(4.6)

achieve the transition M1 **Mr

, and the calculation of the symbol

of operator product and adjoint. We will show that the formulas re-

main in effect for symbols in ST and STS if the Riemann integrals

in (4.2) and (4.6) are replaced by a type of distribution integral

we call a finite part - it resembles a concept of Hadamard [Hdl].

Let us still look at the Weyl-representation for Co°-symbols.

Clearly (3.10) gives a(Mw,D) =a(Mw,D), a formula like (4.6), but

much simpler - it does not involve integrals. But the formula for

the symbol of a product, now involves a 4n-fold integral:

Proposition 4.3. Let a,b E C"(&n) , and c E C0OO(I3n). We have

(4.10) C = c(M1,Mr,D) = P(Mw,D) Q = a(Mw,D)b(Mw,D) = q(Mw,D)

where p , q E S(R2n) are given by the formulas

(4.11)

q(x, )=f (x-Y+z/4, -rl+t/4) b (x-Y-z/4, -n-t/4) a-i ('1z-tY) .

Proof. For (4.10)i depart from (4.3), writing

c' (w+z/2,w-z/2,z) , with w=(x+y)/2, z=x-y. Defining p(w,l;)=d'(w,l;)

=JFizd(w,z)e 12t, get Cu(x)=fslyp (x ,x-Y)u(Y)=P(MW,D)u(x). Again

pE S(1Q2n); interchanges are trivial, confirming (4.10)i-(4.11)i

For the second formula one writes

(4.12)Qu(x) = (ABu)(x) = fr(x,z)u(z)ylz ,

r(x,z) =

Again write r(x,y)=s(w,v)=r(w+v/2,w-v/2), w=(x+z)/2,v=x-z,and get

Q=q(Mw,D), with along lines used

before. Then show that q assumes the form (4.11) after an integral


substitution. No trouble with integral interchanges. qE S follows.

Now we start with the discussion of a more general singular

integral called the finite part (integral) since it resembles Had-

amard's finite part (cf. also [Ci],II).This will be a distribution

integral using partial integration, based on identities like (1.6)

(4.13) a-iY'"

=(Il)-2N(1_Ay)Ne_in

_ (Y)-2M(l_01)Me in

valid for all nonnegative N,M. Using (4.13) and some (legitimate)

partial integrations, the first formula (4.2) assumes the form

)Nein

(4.14)f

= aN(z,t)=(l-OZ)Na(z,t)

Using the other identity (4.13) we get, similarly,

(4.15) f 9jYPTqe 1(Y) -2M(l-0I)M(('il)

Here the expression (1-0I)M((,1)-2NaN(x-y,t-rl)) is in the span of

(4.16) IaI+IPI s 2M .

If M and N are chosen sufficiently large then the right hand side

of (4.15) is meaningful, as a Riemann or Lebesgue integral, even

if we only require that aE ST1. Indeed, the expressions (4.16) are0((l)-2N(x_y)X(2M)( _l)x(2N)) , with x,a, of (1.1), so that the in-tegrand of (4.15) is 0((Y) -2M( x_y) a. (2M) (11) -2N(1; _,1) x (2N)). We assu-me xa0, x.20 in (1.2): they may be replaced by Max{0,x} , Max{O,a.},

also satisfying (1.1), ( 1. 2 ) . Thus (x-y) X=0((x) k+(y) X); similarly

for -i. Existence of the integral (4.15) follows from (1.2).

The integrals in (4.2)2,(4.6),(4.1l) are treated similarly,

using (1.1) with identities like (4.13), and partial integration.

Note that all these integrals are of the form

(4.17) fdsdaeisw(s,a) ,

with an integral over &2m, and wE If W(=- Cm(12s) satisfies

(4.18) asaaw(s,Q) = 0((s)X(IaI)(a)%(jPj)) , X(k)-k + -oo , k - oo

we define the finite part(integral)p.f.f dsdae's w(s,a) as follows:

Use identities like (4.13) and formal partial integrations as abo-

ve until an integrand in L1(R2m) is reached. Then define p.f.f as

the Lebesgue integral of that integrand. That is,

1.4. Beals composition formulas 67

p.f.fdsdae'saw(s,a)

(4.19)=fdsdaeisa(a)-2M(1-Os)M((s)-2N(1_A(3)Nw(s,a){

=f dsdaeisa(s)-2M(1-A0)M((a)-2N(l_A5)Nw(s,a))

for sufficiently large M,N, where it yet has to be shown that this

definition is independent of N,M and the choice of first or second

expression,at right.The latter will follow from the lemma,below.

Lemma 4.4. Let w E ST1(&m) (i.e., satisfy (4.18)), and let

(4.20) wj(s,(J) = w(s,(J)Xj(s)Xj(a) , j = 1,2,... ,

with Xj(s) = X(s/j) , where X E C00(e) equals 1 in Isi s 1

If the two expressions at right of (4.19) are denoted by I(w) _

IN,M(w) and JN,M(w) = J(w) , respectively, then we have

(4.21) limjIN,M(wj) = IN,M(w) , limjooJN,M(wj) = JN,M(w)

for all sufficiently large N and M.

Proof. If wj is substituted for w in IN,M (or JN,M) then Leibniz'

formula may be used to obtain I(wj) , for example, as sum of terms

2Masaaw(s,a),IaI+IYI52N,IPI+IOIs2M.

If N,M are chosen such that the integral IN,M(w) exists,i.e.,

(4.22) ?(2N) - 2N < -n , X(2M) - 2M < -n ,

then each integrand is of the form X(a)(s)X W )(a)O(((a)(s))-n)

This implies that all integrals tend to zero, except for a=4=0

using that cja)(s) = j-IaIX(a)(s/j) . On the other hand, the inte-

gral for a = 0 converges to I(w) , since cj(s)- 1 , sE &m , q.e.d.

The lemma below will be useful to differentiate finite parts

under the integral sign.

Lemma 4.5. For a symbol w E ST1 the function

p.f.f¢y

is C-(&2n) , and we have

h(P)(x, ) = p.f.fOTY> 1iyyl

The proof is a calculation: Just use (4.19), with M,N suitable


for both above finite parts. Then confirm that the differentiation

may be taken into the Lebesgue integrals,and also commutes with

the two Laplacians there. Observe w E ST1 implies that

STS for arbitrary X, IA E & , x° , °E n .

Lemma 4.6. If in prop.4.1 we require only aE STI , bE ST , then

the integrals in (4.2) still exist as finite parts, and define

Co functions p,q , with all derivatives of polynomial growth:

N(4.23) p(R)(x, ) , a,P)

Moreover, we have (4.1) satisfied in the sense that, for u E S

Proof. For a,b E Co we may differentiate p,q of (4.2) under the

integral sign, getting p,q in the span of the expressions

(4.24) J91y91rle l a(R)(x-y,

-r0 ,J91y911le-11aXayaOb(x1y1U)

, y+S=a.

For general aE ST1, bE ST again construct sequences

bj= bxj(y)xj(U), as in lemma 4.4. Let aj be substituted for a, and

the partial integrations of the finite part be executed in (4.24).

As j-.oo get limits with integrands =0(((Y) (r;) ) -2N ( (x-y) ( -rl)%(2N)) ,

in the first case, and inthe second, letting k=Ial+l0I. We were setting N=M, and x(k)=X(k),

as always in the following. We get, with r = 2X(2N+1cI+IOI),

(4.25)

and the same formula for q. Here pj, qj denote the p,q for aj,bj.

The estimate O(...) in (4.25) also holds for pj, uniformly, since

the aj satisfy uniform estimates (1.1). Clearly (4.1), in the form

above, is true for uE S and aj, bj, pj, qj. For j-- get the desi-

red relation for p and also for q, by a similar conclusion, q.e.d.

We leave it to the reader to confirm that the corresponding

statements are true for prop.'s 4.2 and 4.3. Summarizing:

Theorem 4.7. If the symbols a,b of prop.4.1, symbols a,b of prop.

4.2 , and a,b,c of prop.4.3 are chosen in ST1, ST ; STI , STI

ST1, STI, ST, respectively, instead of in C" then the integrals

in (4.2), (4.6), (4.11) still exist as finite parts. They define

Co functions p(x,l;), res-

pectively, which are of polynomial growth in together with* *

all their derivatives. Moreover, (4.1) , AB=c(M1,D) , A =a (M1,D)

1.5. Leibniz formulas with integral remainder 69

and (4.10) still hold for u E ES in the sense of lemma 4.6.

By silent convention we omit the finite part label at the

integrals and write the composition formulas for symbols (which

we shall call Beals formulas) in their forms (4.2), (4.6), (4.11),

even for symbols in ST and ST1.

Note that we are not getting the estimates (1.1) back for

the symbols p,q, etc. Instead we obtained the weaker estimates

(4.23) of polynomial growth. It is of interest to ask for subsets

of ST1, for example, characterized by stronger inequalities, such

p, c, a q again satisfy the strongerthat the composed symbols

estimates. As a comparatively large such class we introduce

(4.26) ,pt = frt. _ {aE (x) rtL' ),for all a,(3}

Here the pair m = (mi,mz) will be called the order of the symbol

a , and Ptm C pt will denote the class of all symbols of order m.

Theorem 4.8. Let 1I 11 "'r, 4% denote the classes of all a(Ml,D),

a(Mr,D), a(Mw,D), respectively, with aE y,t. Then we have qT1 qTrW =qT=OPVt, with transition formulas between representations of

AE W given by thm.4.7. Moreover, LI' is an algebra under operator

multiplication containing its Hilbert space adjoints. The order m

of AE LPl' (defined as order of its symbol) is independent of the

representation used. Orders add when operators are multiplied. Thm

4.7 gives formulas for symbols of adjoints and products.

For a proof we check earlier arguments. The stronger condit-

ions imply the algebra property. Details are left to the reader.

5. The Leibniz formulas with integral remainder.

The Leibniz formulas express a product P(D)a(M) of a differen-

tial polynomial P(D) and a multiplication a(M) by a C° function a

as a finite sum of terms a.(M)P.(D), with multiplication and dif-

ferentiation in the other order. Similarly a(M)P(D), (P(D)a(M)) .

We have a(x,D)b(x,D)=c(x,D), (a(x,D))*=a*(x,D), for differential

operators a(x,D), b(x,D), with finite sums

c(x, ) =(5.1) 0

7,(-i)'01/9! a(g(x, ))e

Other formulas express p,q of (4.2), or of (4.11), in that case.

In the following corresponding formulas will be derived for


-Pdo's. We focus on symbol classes 0m,P,B of Hoermander type.

Actually the formulas follow for symbols in ST or ST1, and

the tool of derivation is the ordinary Taylor formula with inte-

gral remainder. There will be a finite sum, as in (5.1), but also

a remainder term, as in Taylor's formula. Unless the symbols be-

long to SSmp,b, 6<pi, however, the formulas may be of little va-

lue, since the remainder does not decay. The formulas will be cal-

led Leibniz formulas with integral remainder.

We recall Taylor's formula (with integral remainder):

(5.2) I OsNa(O)(x,U)(-T1)O/O! + PN ,

1

pN (N+1) I (-r1)0/0!P0,N ' PO,N =IOI=N+1 0

Substituting (5.2) into (4.2), (4.6), (4.11) one obtains the fol-

lowing typical collection of Leibniz formulas. Other formulas, for

similar symbol transitions, follow just as easily.

Theorem 5.1. Let a,b E ST1, c E ST , and let the symbols arl(x,),

cl(x,) , cw(x,) , pl(x,) , pw(x,) , a*(x,) be defined by

arl(Ml,D) = a(Mr,D) , cl(Ml,D) = c(Ml,Mr,D) = cw(MH,D) ,

(5.3) pl(Ml,D) = a(Ml,D)b(M1,D) , pw(Mw,D) = a(Mw,D)b(Mw,D) ,

a(Ml,D) = a (M1,D)

We have the following Leibniz formulas (with integral remainder):

(5.4)arl(x,) = Lj=0(-iaxa)Ja(x,)/7! + (N+1)PN

1

PN(x,) = di(1-i)NIN!p.f. 91YF1Tle-iyr!(-iaX a N+1a(x-Y,- Tj);0

(5.5)cl(x,) =-J=II/j!{(-iay a )N+1c(x,Y, )}x=y + (N+1)PN ,

1

PN(x,U) = dt(1-s)NINIp.f.f plyAe-iyr!(iay a N+1c(x,x-Y, -T1);0

cw(x, )=`7=01/j! (ia ay)Jc(x+Y/2,x-Y/2,)j + (N+1)pN(5.6)

J

1

PN = di (1-i)/N! p.f.fP1YAe_iy1(idya N+1c(x+y/2,x-Y/2,-il);

0

p1(x,5)= 01/j! (-ia ay)1(a(x,)b(Y,i1)jx=Y,=1 +(N+1)pN(5.7)

1.5. Leibniz formulas with integral remainder 71

(5.7) 1 _pN (1-t)N/N! di p. f. ly"(-iay

0

pw(x, )= =0, ay-aX a,l)J(a(x, b(Y,Y1) Ix=Y, +pN,(5.8)

1 N 21(zrl-yt) i N+1M -t) dt e (2(asaK-ataa>> a(s.a)b(t,K),pN0

M=N+1, s=x-y, a finite part integral.

(5.9) a (N+I)pN ,

with PN as in (5.4) , with a replaced by a

Note that the term pN has been used in all formulas,to denote

the remainder, without implying equality in different expressions.

Proof. In each case the equation at once is confirmed formally, by

substituting (5.2) into (4.2), (4.6), (4.11), and using that

(5.10) iy = ay(1 )" = (2x)n/2ayO(Y)

in the distribution sense. A partial integration must be used with

(5.11) (ax ay)1/1! = I aXay/8e 1

For cw we must apply (5.2) onto the product ab, in 2n variables.

The detailed derivation is technical and repeats the same steps

over and over. Focus on (5.7), the others follow similarly. Using

(5.2) in (4.6)1, (with c(x,x) called pl(x,x)) we get the terms

(5.12) p.f.fgypiri e-i"1(-1)e/O!

For a,b E C0*(&2n) this is easily transformed into

(5.13)

,=(-i) 10 91Y9lT1 ay(e

=(-i)I0I/O!JP1YPlrl

with N sufficiently large, but with the n-partial integrations

reversed to the status M=O. This still gives a well convergent

integral, as long as a,b are Co . Recall that bN = (1-Ax)N b

Here the ii-integral may be evaluated. It gives the Fourier

transform ((Y1)-2N)^=((,q)-2N)' =EN, the unique fundamental solutionin S' of (1-A)-N. An expression for EN in terms of modified Hankel

functions is known (cf. (0.24) for N=1; the method works for gene-


ral s>0.) EN decays exponentially, as lxI-goo. Without

Hankel functions one confirms easily that EN equals a function

in S, for 1xIa1. (c.f. [C1],II). Hence (5.13) is continued as

(5.14)=(-i)'°1/91

(-1)'01/01

confirming the first expression of (5.7), using (5.11), and for

Cp functions only. For the remainder term, we look at

fp.f.f91y51rl a-iY110

(5.15) _ 91Y91r1 aiyrl (1-z)Ndia(e)(x,

0

1(l-i)NdiP1YP1r1

0

as a matter of Fubini type integral exchanges. Write (-rl)°e iyr) _

(-i)IOIaye-1Y11 , and carry out a partial integration, for

n r

0

(5.16) iai [l IT f _,.... ,a%

0

completing the discussion of (5.7), for a,b E COOO

(0)(x-Y,) ,

For general a,b E ST1 one now may use sequences aj, bj E CO, as

in lemma 4.4, and arrive at (5.7) by passing to the limit j -> -,

using lemma 4.4 and a formula similar to (4.25). Details are left

to the reader. In particular the occurrance of the factor r, in

the argument of a , does not influence the derivation.

6. Calculus of Vd2o's for symbols of Hoermander type.

The Leibniz' formulas with integral remainder of sec.5 may

be of little use, unless we restrict symbols to the class SS of

sec.3. For symbols in SS or h the formulas of thm.5.1 imply cor-

responding asymptotic expansions of arl' cl' cw' p1' pw' a*The key is an estimate of the remainders pN of (5.4)-(5.9).

Theorem 6.1. Let a E E - =0 S, b E y>hm, 1, 61 , c E SSm",p","Pwith the 3 sets of paramaters satisfying (3.3). Then the remain-

ders pN of (5.4) through (5.9) satisfy the following estimates:

1.6. Calculus of pseudodifferential operators 73

For (5.4):

oPPPl(

)m.-(N+1)(p.-b) (x)mz-(N+l)pz )

For (5.5): =of( )m."-(N+1)(p'"-b") (x)m2"+m3"-(N+1)p3°`.

For (5.6): o(()m. "-(N+1)(P' "-b") X)M2 "+m3 "-(N+1)p-)

p-=Min{p2 ",p3 J"} .

For (5.7): 0f(l;)m1 +m1 '-(N+1)(P.-b') (x)m2+m2'-(N+1)p2)

For (5.8): pN(x,U) =

pj = Min{pj,p'} , b^ = Max{b,b'}

For (5.9): Same as for (5.4)

Moreover, these remainders will be symbols of Hoermander type,

of the order indicated by the above formulas. That is,

PN E A'm-(N+1)(p-be'),p,b ,e'=(1,O) for (5.4) and (5.9),

PN E 4*(MI "-(N+1) (P "411) M2 "+m3 "-(N+1 )p3 "), (P' P ),b ' for (5.5),(6.1)

pN as for (5.5), with p3" replaced by p" , for (5.6) ,

PN E 4*m+m'-(N+1)(p-b'e'),p^,b^ ,for (5.7) and the same with 6'

replaced by b^,for (5.8) .

Here we must require that b^< p,^, in case of (5.7) and (5.8).

Then both orders of pN will tend to --, as N - - .

The proof proceeds similar as in sec.4. In each case we sup-

ply estimates, uniformly in s, of the integrand di only. For exam-

ple, to estimate PN of (5.4), it suffices to consider

p.f.fdydrl a-h 1a(6)(x-Y,x-T1)

(6.2)

=fdydTl e

again with aN (1-AX)N a, for sufficiently large M,N. Use (3.2) for

(6.3) =olf dy(x-Y) mz -pz 1111 (Y) -2M fdll( -iTl) m. -p. 19 1+6 101(,n) -2N)

.

Here we also employed the estimate

(6.4) ((Tl)-R)(a) = o((Tl)-R-Ia

which is easily derived. Then apply the well known inequality


(6.5) (x) s(x-Y)s = 0((Y)IsI) ,

valid for sE 1, to estimate (6.3) by (x) MI -PZ 10 I ( ) m, - (PI -6) I 0 I

confirming the estimate for the remainder pN of (5.4).Similarly for the other 5 estimates.Also we may apply Lemma

4.6 to the remainder of (5.4), for

)(6.6)pN(R) f 0(1 )N/NIf I1YAe iYn(-ia - a N+1a(a) (x-Y, -TI

Then the above estimate may be repeated for

(6.7) PN(a) = 0(()m,-(N+1)(P'-b)-P IaI+OIl1(x)m2-(N+1)p:-PZ IPI),

which confirms (6.1) for the first remainder. Again similarly for

the other five remainders, q.e.d.

Theorem 6.2. Under the assumptions of thm.6.1 the symbols arl'

cl, cw, pl, pw, a* of thm.5.1 are in yah , provided that, for

the product symbols pl , pw we still require that 6^ < p,

(which is automatically true if (m,p,S) = (m',p',S') ) .

More *precisely, we get arl, a E 0m,p,S Cl,cwEyhm ,P-,s

with m =(m, ,m2+m3) , P =(PI ,P2 ) , p2 =Min(p2 ,P3) , P1 , Pw EApm+m',p^,S^ with p^ , 6A as in thm.6.1.

The proof of thm.6.2 is a consequence of (6.1).

Note that thm.6.2 contains the (so far unproven) thm.3.2 as

a special case. Indeed, the symbol b of thm.3.2 is given by our cl

while b1 may be obtained by writingc(Mr,Ml,D)=(c(Ml,Mr,D)*

b1(M1,D), taking adjoints. b2 coincides with ow of thm.6.2.

For uniqueness of b, b1, b2 note that a ,do A=a(M1,Mr,D) vanishes

if and only if its distribution kernel (2.16) vanishes. For

set w(x,y)=u(x+y)v(x-y), u,vE S. (2.16) yields

f dsdteitta(s,t )u(s)v(t)=f dsd; a(s,, )u(s)vr'* (( ) =0 for all u,v E S.This implies a=0. Thus the Weyl representation is unique. Similar-

ly for the left (right) multiplying representations.

Similarly thm.6.2 implies the following result.

Theorem 6.3. Each sethoo,p,b forms an algebra under operator mul-

tiplication, containing its Hilbert space adjoints, whenever p,S

satisfy (3.3). The algebra product a°b may be defined by anyone of

the representations a H a(M1,D), or, a H a(Mr,D) , or a H a(Mw,D).

That is c=a°b satisfies c(Mx,D)=a(Mx,D)b(MM,D), x=l or x=r or x=w.


The proof, in essence, is an application of (5.7), (5.8), and

(5.9), as modified by thm.6.2. Details are left to the reader.

In the following we tend to use the left multiplying repres-

entation A=a(Ml,D), and then will use the more conventional nota-

tion A=a(Ml,D)=a(x,D). For a class X of symbols let OpX={a(x,D):

aE X}. For a class Y of ,do's let symbY be such that Op(symbY)=Y.

For a symbol aE 11hm,p,s , and a sequence of symbols aj, j=0,

1,2,..., ajE yt j , m+ =(mi,m2), mk-.-oo, as j-oo, k=1,2, we willm ,p,S

say that a has the asymptotic expansion =0 aj , written as

(6.8) a =0

aj (mod Vh_.) , (or simply "(mod S)"

if both orders of the symbol pN=a-e=0 aj tend to -oo, as N- (that

is,more precisely, if rNE h N , with µN=(µN,µ2), whereµ ,P,5

j- , k=1,2). It is seen at once that µJ=mj+1 is the best possible

choice for µj . Also we must have m=m° .

Using this terminology we find that all formulas (5.4)-(5.9)

imply corresponding asymptotic expansions For example,

(6.9) a (mod 0_,O)

as aE 0M OP 16, 6<p3, and similarly for the other 5 formulas.

It is clear that the asymptotic sums in (6.8) or (6.9) are not

in general convergent infinite series. For future applications it

is important that, for an v sequence aj of symbols of order mi--00,

an asymptotic sum may be constructed, by the lemma, below.

Lemma 6.4. Let ajE j=0,1,..., where p,5 needs not satisfy

(3.3), but is independent of j , and where mJ=(mi,m2) , with

(6.10) mk>mk>mk> ... , as jaoo k=1,2

There exists a symbol a Eh m = m0 such thatm,p,S

(6.11) sN = a - 4=0 aj E yh N N=0,1,2,... .

M ,p,b

Proof. Indicating first the idea of the proof, let

(6.12) s 1/2 , o )=l , as jxI+1fl z 1 .


Assume wE C°°(1e2n) , and let x=1-wE CD . Assume Os w,x sl in &2n

For an increasing seqence {tj} , tjE I j-- , to be

determined later, define

(6.13) w(x/tj,/tj)

Conclude that w.=0, as /2. Since t.->oo the sum in (6.13)

is finite near Thus is well defined. We get

(6.14) a - f-laj = -Lj=oxjaj + > N wjajJ=O

where the first term at right is C0(22n), and satisfies (3.2) for

any choice of parameters m,p,S. For the second term we get

(6.15) IwjajIscj(1;}ml (x) sup{( )8 i(x)0, is JxJ+1i 12ttj/2}

with 9S =md -m 9'Z =mz -lZP . By choosing tj sufficientlylarge the supremum in (6.15) can be made less than 2-3/cj . Usingthis in (6.15) , and then substituting (6.15) into (6.14) we get

(6.16) a - Lj=pai = 0(()mi (x)

Writing (6.16) for N+1, we notice that the highest term aN , at

left is of the same order as the right hand side. Thus we get

the first estimate (3.2) contained in (6.11).

To get the other estimates we improve the choice of tj. For a

given a,4 and N let No=Max{jaI+I3I,N+1}. Write =a-`zX- j=0a. , and,I- J

(6.17) aXo sN = 0

Observe that o.E yah0,p,6, for every p,S with (3.3), and 0=(0,0).

Thus also a.w4E1yah j and (6.15) follows for aaap(wja.) insteadm ,p,S

of-w a. with c., mN-1, mz-1 replaced by cj,a,,ml-1+Sjaj-piipi

13 3m2 -p2jal , respectively, with a constant cj,a,p depending on j,

a,s only. The first sum in (6.17) is CO and satisfies (3.2). The

second sum is finite; its terms satisfy the proper estimate. To

control the last sum, tj must be chosen according to

(6.18) (x)6'Z s 2-j, as jaj+jPjs j.

This amounts to finitely many estimates, for each t, hence the

selection is possible. In each term of the last sum we get I-1+1131

sN0sj, by construction of N0. Thus get the proper estimate,q.e.d.

Lemma 6.5. Let {bj1:1=1,2,...} be sequences with the assumptions


of lemma 6.4, for j=1,2,... Also, let bj1E 1hm-(j+1)µ,p,S'

µaO, so that the asymptotic sums bj = 2;=0 bjkE 0 m_jIl,p,6 again

form a sequence of symbols with the assumptions of lemma 6.4. Then

(6.19)-7=0 0 bjk = =O4=0bk, j_k (mod O_

where infinite sums are asymptotic. Equality in (6.19) is under-

stood only up to a term in yah_..

Proof. All symbols bjl ,j+l=k , are in the same class yahm-kµ,p,S'

hence their sum is in that class. Thus the asymptotic sum at right

is meaningful. Denote the left and right hand side of (6.19) by b

and b' , and write bN = fj,k=ob.k , b'N =71

b 1 . Note thatj+lsN j

bN - b'N is a finite sum of symbols of order m-Nµ hence itself

is of that order. Moreover, b' - b'N is of order m-Nµ , while b-bN

=b - 4 4=0bj + =0(bj - Vk=0bjk) again is of that order. Hence

b-b' _ (b-bN) + (bN_b'N) + (b'N-b') is of order m-Nµ , for all

N = 0,1,2,... . It follows that b-b' E 0_0. , q.e.d.

Lemma 6.6. An asymptotic expansion (mod S) of the form (6.8), with

a E yhm,p,o , aj EOyh , m = m0 , etc. may be differentiatedmj,p,s

term by term . That is, (6.8) implies

(6.20) a(P)=Oaj(a) (mod S

The proof is evident: All remainders rN are symbols, their

derivatives also are symbols, of orders tending to --, as N-- .

Problems. 1) A polycylinder C=S'x)en-l is described analytically by

its universal cover 1n=2x1n_ 1={ (x, , x° ) : x, E 1, x° Etn- l } :Write C°°(C) )={uE C°O(&n) : u periodic (2n) in x, }, S(C)={uE COO(C) : aX u(x, , . E

S(kn_l),j=0,1,..,uniformlyin x}. Show that a tpdo A=a(x,D)E OpVt0

with periodic (2n) in x, may be regarded as a map S(C)-

COO(C), given by Au(x)= _ooel3xl

x° )uj

2nwith u j(x° )= 2a

fa-ljx'u(x,,x° )dx, and the Fourier transform

0

""" with respect to x° . 2) The operator A=a(x,D) of pbm.1 may be

regarded as a 4d o on &n-1 with operator valued symbol, acting on


functions on Rn-1 with values in C'°(S'): For1n-1 define

A(x° ,t") :COO(S' ) ,C00(S' ) by A(x" t" )u(9)=eljea(O,x" , j , where

2nuj= e-1 u(9)dO. Then write v(x")=u(.,x") , and Av(x")

f oweix"t" )v ( "),for VE S(Rn-1,COO(S' )) S(C). Show that,

for a symbol aE 9pto (and even for aE STS ) this operator A mapsS(2n-1,C"(S'))

+ S(IIn-1,C,(S')), and continuously so when we equip

S(C)=S(1n-1,C°°(S')) with the Frechet topology of all norms

II(x")ku(a)Il 0 , k=0,1,..., aE IIn. (Recall that must be 2n-L

periodic in xi . ) 3) Investigate linear operators C00(S' as

B=A(x" of pbm.1: Let Bu(xi )=;. fdyi el3(x' -y' )bj(X, )u(y,) , with

a sequence of functions b.(x,) , j=O,t1,t2,..., bounded in j and2n-periodic (and C°°) in xi . First assume that bj(xt) is indepen-dent of j for j>>1 and j<<-1. Then B=b , (M)+b2 (M)H+K, with the Hil-

2nbert transform H on S' (i.e., Hu(xi )= dy, cot(x, -yi )/2 u(y,)

0

with a principal value integral) and with an operator K of finite

rank. Next assume that the limits limj.apj(xi)=Pt(xi) exist. Try

for conditions still insuring the above form, whith more general K

4) Investigate the operators Bi 2 -DX 2)-1/2 and B2 =Dx Bi inthe sense of pbm.3.

7.Strictly classical symbols; some lemmata for application.

For many discussions, notably elliptic and hyperbolic theory

it proves unnecessary to carry along the complicated classes of

Hoermander developed for hypoelliptic theory. Let us introduce

,pm = pm,(1,1),0 , UCr = U m , '_O, = n "m ,

setting p1 = p2 = 1 , 6 = 0 . Symbols in y,cO, , and corresponding

operators in Opi,coo will be called strictly classical . Note that

(7.2) 'm C VIM C VSM C yitm ,

with the classes pt of (4.26) and 1,1 ps of ch.VIII.

The lemmata below are simple consequences of our calculus of

1.7. Strictly classical symbols 79

Iyido's in sec.6. We are preparing them for later application.

It is clear that the classes Vtm and VCm , for finite at, are

Frechet spaces under the collections of semi-norms, respectively,

(7.3) sup {ja(")(x, )fi(x)-m2 x, E&n} , a,l EIIn

and

(7.4) sup a(P(a))(x,

)I(x)-mZ+IaI( a,p EIIn

Similar sets of semi-norms for the classes t*m,p,b of sec.3 are

suggested by the estimates (3.3),of course.

Recall that a subset M of a Frechet space is called bounded

if each semi-norm of a defining set remains bounded on M. Clearly

a bounded set of Pm is bounded in Vtm For a bounded set M of "m

the set aE M} is bounded in ,tm -lal -ICI. Abounded set

in cur (Vtm ) is bounded in 4Cm, (Vt.,,) whenever j=1,2.

Lemma 7.1. For bounded sets A={a}C cm, B={b}C yiam, the collection

(7.5) C {c E 1ycm+m'° c(Ml,D)=a(Ml,D)b(Ml,D) , a E A , b E B}

is bounded inm+m and the sets A* ={a*} of symbols of adjoints

and P = {p E i1cm : p(MrD) = a(M1,D) a E A } are bounded in wcm

Similarly , if ,c is replaced by it , in all the above.

Lemma 7.2. Let a E pcm , b e iycm, . Then we have , with e

a(M1,D)b(M1,D) - (ab)(M1,D) E OpyrCm+m'-e'

(7.6) a*(M1,D) - a(M1,D) E Opcm-e

a(M1,D) - a(Mr,D) E Opicm-e

Also, if A = a(M1,D) , B = b(M1,D) then

(7.7) [ A, B] = AB - BA E Opyicm+m'-e

Moreover, if a, b range over bounded sets of lycm and VCm respec-

tively then the symbols of the expressions in (7.6) and (7.7)

form bounded sets of Vcm+m'-a, V. m-e' 'c m-e, Vcm+m respectively.

In fact, this is correct if we do not necessarily require that AC

yicm, and B C Vcm, are bounded, and a E A , b E B , but only thatrespectively, for the 4 relations in (7.6), and (7.7),

(i ) {a0} bounded in tpCm-e, , and {b°} bounded in 1 cm-e. ,


(ii) {a(8)} bounded in 4cm-e

(iii) {a(9)} bounded incm-e

(iv) {a(e)} bounded inVcm-e'

, and {a(0)} bounded in Vcm-e2 ,

and {b(e)} bounded incur,-e,, and {b(0)} bounded in gym,-e2.

For (i) ,...,(iv) the conditions are required for all 101=1 .

Remark 7.3: Using the 1-1-correspondence a « A=a(M1,D) between yucm

and Opyscm, for x=c,t, one may transfer the Frechet topology of the

spaces 1xm onto their corresponding sets Op1xm. Then we may speak

of bounded sets in Optcm , for example.

Proof. Clearly (7.6) and (7.7) are consequences of thm.6.1, noting

that Wcm '*m,e,0' On the other hand all statements about boundedness of sets follow by just going again over the proofs leading to

thm.6.1, keeping in mind the boundedness of sets in Frechet spa-

ces. For example, regarding boundedness of the set C of (7.6) it

is sufficient to show that the constant of O(.) in thm.6.1, regar-

ding (5.5), as well as the corresponding constants for the deriva-

tives pN(a) of thm.6.2 of that remainder, all depend only on the

bounds of the Frechet norms of a and b, in their bounded sets.

This needs only be done for one N, say, for NO. Or even, one

could estimate the first expression (4.6) of the Beals formula for

the product symbol, in its finite part integral form (4.19), of

course, using the principles of the proof of thm.6.1.

Actually, going over that estimate again it becomes evident

that indeed only bounds of expressions (6.2), for the symbols

a and b, and corresponding bounds for the explicit symbols

(7.8) nm(x, ) = mI (x) m2E '4'am

and existing definite integrals over in

(7.9) f(x) -rdx , r > n

as well as binomial and multinomial coefficients occur.

We will not go over all the details again .

Chapter 2. ELLIPTIC OPERATORS AND PARAMETRICES IN In

In this chapter we focus at the main feature of our algebras

OpWhp6

: They contain (generalized) inverses of some of their ope-

rators - called (md-)(hypo-)elliptic. For historical reason such

an inverse is called parametrix or Green inverse. Speaking algebra-

ically, we deal with inverses modulo some 2-sided ideal, either

the class', of integral operators with kernel in S(12n) (K-parame-

metrix), or the operators of & with finite rank (Green inverse).

Integral operators with kernel in S(&2n) are compact opera-

tors of L2(&n), as well as in every Sobolev space (cf.III,5). Thus

a K-parametrix (Green inverse) of A will be an inverse mod K(H)

(a Fredholm inverse) as well, under proper assumptions, giving nor-

mal solvability of the equation Au=f (cf. [C,],App.Al).

For Oplhm pS

'elliptic' usually will be 'md-elliptic', amoun-

ting to ellipticity 'at and at jxI=co'-i.e., "d-" and "m-"

ellipticity. This is parallel to discussions in algebras of singu-

lar integral operators ([Ci],[Cz]): For operators of order 0 'md-

elliptic' simply means that the symbol is #0 at 1x1+1V_o .

An md-elliptic operator of order m will have a K-parametrix

of order -m , and existence of a K-parametrix of order -m is in

fact necessary and sufficient for md-ellipticity of AE Op'hm,p,S.

However, it is possible for a non-md-elliptic operator to have a

K-parametrix of order >-m. One such class of operators - the for-

mally and-hypo-elliptic operators - is studied in sec.2.

The ellipticity concept can be localized, in the x-variable

as well as in the 5-variable, and 'local parametrices' may be con-

structed under various assumptions. In particular, (formal hypo-)

ellipticity (for all ) over an open set 92 C Rnimplies hypo-ellip-

ticity in the proper sense: All distribution solutions of Au=f are

Coo(sl) whenever fE D' is Coo(ci) (cf. sec.3 and 4). For proper discus-

sion of such facts we introduce the wave front set WF(u) of WE D'

(sec.5), and show its invariance under spplication of ado's.

In sec.6 we discuss (left)(right) md-ellipticity of a matrix

of pdo's, together with certain partial inverses.

81

82 2. Elliptic operators and parametrices

1. Elliptic and md-elliptic pdo's.

Historically a parametrix is an integral kernel with a sin-

gularity similar to that of a Green's function, which can be used

to invert a differential operator, up to an integral operator with

continuous or smooth kernel (c.f. E.E.Levi [Lvl], D.Hilbert [Hbl]).

Hoermander [Hr2] used the term 'parametrix' for a local inverse up

to an infinitely smoothening operator, for a ipdo. Here we will use

a type of global parametrix with similar properties. Let A,B be

lpdo's with symbol in ST. We shall say that B is a K-parametrix of

A if we have AB-lE Opyh-o. , BA-1E Op%ph-,O. Clearly then A and B are

K-parametrices of each other. We speak of a left (right) K-parame-

trix B of A if only the second (first) condition holds.

A symbol aE Phm,p,6C 1ph will be called md-elliptic (of order

m) if (i) for all sufficiently large and (ii)

the function b=1/a, well defined for large equals some

symbol in Vh-m,p,S' for large jxI+1g1. Then the yxio A=a(x,D) also

will be called md-elliptic. We will see that md-ellipticity is a

property of the operator, insofar as, for an md-elliptic A=a(M1,D)

=b(Mr,D)=c(Mw,D) all 3 symbols a,b,c will be md-elliptic.

Proposition 1.1. A symbol a E 1phm,p,8 C Vh is md-elliptic

if and only if, with constants c,c'>O, we have

(1.1)I

a c(a)m (x)m2 , as IxI+IsI a c' ,

Proof.If a is md-elliptic we get b=1/a for large and bE

-m,p,S' hence (x)m` ), using I,(3.2). This implies(1.1). Vice versa, if a symbol satisfies (1.1) then

a(x, )

C w-1, as w-0, as defi

nes a CC°(&2n)-function satisfying the first estimate I,(3.2). By

lemma 1.2, below, it satisfies all estimates I,(3.2), q.e.d.

Lemma 1.2. Let Oy6aE COO(Sc). Then (1/a)is in the span of

(1.2) {[IIi (a(,CE3j)/a)]/a : Ya3 = a , f3 _ (3 }

This lemma follows by induction.

Proposition 1.3. For aE iphmIp,SC 0, let A=a(M1,D). Let A also be

given as A=b(Mr,D)=c(Mw,D)=d(M1,Mr,D), with symbols b,cE lphm p,g

and dE SSm 'p 'b C SS. The following conditions are equivalent:

(i) b is md-elliptic ; (ii) c is md-elliptic ; (iii)

is md-elliptic (of order m ). (iv) A is md-elliptic.

2.1. Elliptic and md-elliptic operators 83

This follows from calculus of do's: The symbol a may be ex-

pressed as an asymptotic sum of b or c or d and their derivatives.

We get modulo terms of lower order.

Thus (1.1) holds for all or for none of these functions, q.e.d.

Proposition 1.4.(a) AE Opyah is md-elliptic if and only if A* is.

(b) The product AB of two md-elliptic operators AE Opwhm,P,a,

B E OpVhm,,P1b , again is md-elliptic (of order m+m')

Again this follows from calculus of ,do's.

Recall that a differential operator A =IaJSN

ac`Da is called

elliptic at x0 if its principal symbol

(1.3) a aNaa

is #0 for x=x0 and all E Rn with In that respect notice

prop.1.5 below, which also motivates our terminology.

Proposition 1.5. A differential operator A=a(x,D)E Opy1hm,P,S ,

m=(N,m2) is md-elliptic (of order (N,m2)) if and only if

(1.4) a p(x)m" , for all x , and all 1

and also, with suitable constants p, c>O,

(1.5) Ia(x,l;)I a for all x,g with lxi a c .

Proof. Note that 'jxI+ii large' means that 'either lxi is large

or is large' (or both). The first points to (1.5), the second,

for a differential operator A = a(M1,D) , translates into (1.4),

using that the terms a l < N , all are N-1( x) m,) .

By (1.5) md-elliptic differential operators are elliptic in

&n. (x)-m-A even is 'uniformly elliptic' in ien. (1.4), in effect,

is another ellipticity, with variables x, reversed.

Theorem 1.6. An operator AE OptNhm.P.b , admits a K-parametrix in

Optph of order -m if and only if A is md-elliptic.

Proof. If A and B are K-parametrices of each other then the rela-

tion AB=1 and the uniqueness of the left representation implies 1=

modulo terms of order -Ee, e=(1,1), E>O. Thus it fol-

lows that jabi a1/2, as ac, for large c. Accordingly, jal

a2 b ap( )m' (x)m' , I x I + I I ac, p>O, using that b is of order -mhence, m'). Thus A is md-elliptic (order m). Vice

versa, let A=a(M1,D) be md-elliptic. Using a recursion and I,lemma


6.4 we will construct a K-parametrix for A.

First, let b1 be a symbol such that ab1=1 for large

by definition of md-ellipticity. If B1=b1(M1,D) then AB1-1=e1(x,D)

where e1 is of

order Next set b2=e1/a, for large correc-

ted to be C00(1 2n). Setting B2=b2(M1,D), get AB2+E1=e2(x,D)=E2 of

order -2a, where, again, we used lpdo-calculus for AB2. Together,

we get A(B1+B2)=1+E1-E1+E2=1+E2. It is clear now how to iterate:

Next set b3=-e2/a, and B3= b3(M1,D), AB3+E2=E3=e3(M1,D), with e3

of order -3o, and A(B1+B2+B3)=1+E3, etc. Let B =7, Bj denote an

asymptotic sum. Then B-f=1Bj is of order -(N+1)a, by definition.

Also A(-1Bjj is again of order -No. Thus AB-1 is of or-

der -No, for every N = 1,2,.... It follows that AB-1 must be of

order --. Or, B is a right parametrix of A. Similarly construct a

left parametrix B" of A. Get B =B AB=B (mod Opph-0), q.e.d.

Finally, in this section, we show that Op4h-o,05(2n) =

Lemma 1.7. The class Opih_.0 of ado's of order -oo coincides with

the class 'f of all integral operators

(1.6) Ku(x) = fk(x,y)u(y)gIy , u E S

with rapidly decreasing kernel k E S(&2n)

Proof. We have seen before (cf. I,(2.16)) that the distribution

kernel k of a yxlo is given as

(1.7) k(x,y) = a4'(x,x-Y) =

For aE ih-oo S(le2n) (1.7) is a well defined integral. The Fourier

transform and b(x,y)- b(x,x-y) are isomorphisms S -S , q.e.d.

2. Formally hypo-elliptic yxlo's.

In sec.1 we found that a ipdo A of order m has a K-parametrix

of order -m if and only if it is md-elliptic. It is possible, how-

ever, for A E C Wh to have a K-parametrix in OpWhm 1P'6 P'S

with mj a-mj without being md-elliptic.

Let aE iphm,P,S

C Wh. We shall say that a (or A=a(x,D)) is

2.2 Formally hypo-elliptic operators 85

formally and-hypo-elliptic if for we have (i)

and (ii) a=O((l;) (x)m2 ), with some m'=(m1 ml ), and (iii) forall a,(3 the functions a((cc)/a satisfy the estimates

(2.1) a(a)/a = of( )-P'

In this case we shall call m'=(m1 m' ) an -inverse order of a

Lemma 2.1. Let a E m,PSC ph . The symbol a is formally and-hypo-

elliptic if and only if (i) a and a(c)/a are (ii)

the functions of (i), for all a,(3, extend to symbols in 0m',p,S

and Vh , respectively, where, (3ma P,S

(2.2) ma',p =

Proof. Trivially the condition of the lemma is stronger as that of

the definition, because (2.1) is only the first I,(3.2), defining

hma,p,p'S. We must show that also aXa satisfy I,(3.2).

This follows from the lemma below easily proven by induction.

Lemma 2.2. For all a,(3,x,X we have in the00

span of the products

(2.3)=1(a( )/a) , 3=163=a+k=103=(3+x , r=1,2,...

Again formal md-hypo-ellipticity is independent of the choice

of left (right) multiplying representation.

Proposition 2.3. Let A=a(M1,D)=b(Mr,D) with a, b E 0m,P,6C Vh

Then A (or a) is formally md-hypo-elliptic if and only if b is

Proof. From formula I,(5.4) we conclude that

(2.4) j(i)191/9!

a is formally md-hypo-elliptic then the asymptotic sum

(2.5) c = '01/e! a(e)/ae

is well defined, assuming that 1/a has been suitably modified for

I xI +I I < r!, to be C (&2n

) This follows from lemma 2.1. Moreover,

the highest term in the sum (2.5) is identically 1, implying that

the symbol c is md-elliptic of order 0 , by prop.l.1. One confirms

easily that ca = b (mod iph_,O). Indeed, from (2.5) we get

(2.6) c - /0! a(0)/aE 1ih_(N(P'-S),Npz),P,S


This may be multiplied by the symbol a , using lemma 2.1, for

(2.7) ca a(9) Em-Np+NSe',p,S

N=0,1,2,....

In other words, we get ca = b (mod 1ph_,,) , using (2.4), as stated.

Using I,lemma 6.6, (2.4) may be differentiated, for

(2.8) 8(i)I0I/0l a(6+a) (mod

Then, repeating the above conclusion we see that

(2.9) b/a , a/b E 0 0,P.o b(P)/a E'iph

Again, the functions (2.9) must be modified on some KCC I2ninde-

pendent of a,P, before they are symbols as stated. (2.9) implies

1/b = (1/a)(a/b) E hm',P,S b(R)/b=(b(a)/a)(a,b) E 0 «,(3m ,p,S

so that b is formally md-hypo-elliptic. Vice versa, if b is formal-

ly md-hypo-elliptic then a may be obtained as asymptotic expansion

in terms of the derivatives of b. The above conclusion may be re-

peated to show that also a is formally md-hypo-elliptic, q.e.d.

Proposition 2.4. (a) AE OplphMIP16C Op1ph is formally md-hypo-ellip-

tic if and only if A*, its adjoint, is. (b) If A=a(M1,D),B=b(M1,D)

E UmOpylhm,p,8C Opyah are formally md-hypo-elliptic then so is AB.

(c) The formally md-hypo-elliptic symbols of (b) form a group un-

der pointwise multiplication, for each p,S. (d) An md-elliptic sym-

bol of 1phmIP.SC 0 is formally md-hypo-elliptic, inverse order -m.

Theorem 2.5. A formally md-hypoelliptic A=a(M1,D)E Oplphm,p,S ad-

mits a K-parametrix B=b(M1,D)E Opom"p'S, for suitable m'a-m.

Proof. As in the proof of thm.1.6 we focus on construction of a

right K-parametrix. First set b1=a, and B1=b1(x,D). By

assumption b1E 1phm,,PS, and the product formula yields.

e

(2.10) AB1-1=E1=e1(M1,D) , e1= Ie ,0(_i) a(e)bl(0) (mod

By lemma 1.2 the term a(e)bl(0)=a(e)(1/a)(e) is in the span of

{(a(())/a)rI(a('j) /a) : YaJ=6} . Lemma 2.1 yields a(e)bl(e)E

Vh-Np+NSe',p,S' I0I=N. Accordingly el E y't2_p+Se',p,S has orderless than (0,0), as in thm.1.6.

Next, in the recursion, define b2=-el/aE lphm'-p+Se',p,g Try

to verify that B2=b2(M1,D) satisfies AB2=-E1+E2, with E2=e2(x,D),

e2E 1ph_2p+6e' , p , S Indeed we get

2.3. Local md-ellipticity 87

(2.11) AB2+E1=e2(M1,D) , e2 = 10 f0a(0)(el/a)(0) (mod h-,0)

A term a(e)(el/a)(e) for 101 = N is in the span of

(2.12) { (a(e)/a) el(a) II.(aWj)/a) : a + I P3 = N }

again by lemma 1.2. Using lemma 2.1, confirm the proper conditions

e2. Evidently the recursion of thm.1.6 works again, q.e.d.

3. Local md-ellipticity, and local md-hypo-ellipticity.

Recall that (formal) md-(hypo-) ellipticity of a symbol aE

y,h , is a condition for large If a is modified in a set

KCC R2n these conditions are not disturbed.

A parallel, though slightly different concept of md-ellipti-

city introduced in [Cy l],for the operators of some C -subalge-

bras of L(L2(&n)). Similar algebras are studied in [C2] for gene-

ral noncompact manifolds c. Their operators also have symbols now

defined over the boundary of a compactification of the cotangent

bundle T*S1 - for 11=1n for (x + =o, at the 'boundary' of R2n

The generators of such an algebra are yxio's as studied here.

In [C1], ch.IV, a compactification Pnxien of RnxIn=R2n appears.

The symbol of an operator A in the algebra, called A, for a moment

is only defined at the boundary ffi=d(1QnxPn)=Pnxpn\R2n of PnxPn.

Then AE A is called md-elliptic if its symbol is #0 on M. md-ellip-

ticity is necessary and sufficient for AE A to be invertible modu-

lo operators of finite rank. The algebra A is generated by the spe-

cial pdo's a(M) and b(D), where a,bE COO(R2n) are bounded and have

a(a)=o(1), bO)=o(1), as lxl-oo, for all a. len just is the smallest

compactification such that all such a,b extend to C(P).

This result on C*-algebras of pdo's offers some guide lines

on the question of local md-ellipticity. Within A we speak of md-

ellipticity of AE A in a subset MC ffi if its symbol is #0 near M .

For fixed p,S consider AP,b, the closure of p'hO,P,b withinCB(R2n).

The maximal ideal spacePP,6

of this function algebra is a compac-

tification of R2n - again the smallest to which all functions of

Vh0,P,6 extend continuously. We set ffiP,b=dPP s=PP,b\R2n again.

Proposition 3.1. A symbol aEm,P,6

is md-elliptic if and only if

(the continuous extension to 2P,s of) the function b(x,l;) =m, (x) mZ 00,P,S does never vanish on the set fP,6The proof follows from (1.1) and prop.1.4, and md-ellipticity


of (gy)m' (x)m'= Vhm,p,b, using ospjst and Sao .

Remark: A result of III,1 below implies that OpiihO'p's, for bsp,

is an adjoint invariant subalgebra of L(L2(&n)). The results on

A sketched above, may be also derived for the norm closure Ap'o of

OpPh0,pb,

assuming pj>O, 0<6<p, (cf. V,10, for pj=1, 6=0).

Following the above lead, a symbol aEm,p,b will be called

md-elliptic in a set MC 2p'b if (the continuous extension to &p,b)

of is #0 on M.

We reformulate this definition, avoiding reference to values of b:

Definition 3.2. Vhm,p,b is called md-elliptic in MC ffip'b

if there exists a neighbourhood N of M in Bp,b with (1.1) in

Kfl it2n for every KCC N. A symbol aE 0m,p,b is called formally md-

hypoelliptic in M if (i),(ii),(iii) of sec.2 are valid in Kfl I2n

KCC N, with N as above. Again we call m' the inverse order of a

The first part of def.3.2 is equivalent to our first defini-

tion of local md-ellipticity, because a continuous function is 340

at a point p if and only if it is bounded away from 0 near p.

Def.3.2 still contains a reference to MC ffip'bC Bp'b. Also,

Bp,b is of Stone-Cech type - its subsets are too general for our

purpose. It seems practical to get restricted to a special kind of

subsets M : The functions s(x), Ph0 p,b generate a subalge-

bra B of Ab,

for p sl, ba0. The maximal ideal space of B is a

compactification of &2n again- it equals BnxBn, with the directio-

nal compactification Bn of Rn (cf. sec.5 or IV,1, below). Bn con-

sists of &n and the ' infinite' set { oac° : x° E Rn x° I =1 } . The func-tion s=(sl...sn)(x) provides a homeomorphism I B1={xE Rn,IxI<1}.

3n is characterized by the property that s(x) extends to a homeo-

morphism B"H Bi={jxjs1} . The inclusion is an isomorphism,

its dual is a surjective map t:B]BnxBn .

Henceforth we will apply def.3.2 only for sets MC ffip'b of

the form M=L_1M' with M'E a(BnxEn) . Then we may choose the neigh-

bourhood N of def.3.2 in the form N=L-1N' where N' is a neighbour-

hood of M' in BnxBn. ffir,d does not enter the estimates of def.3.2.

Thus we refer to M', N' only, i.e., in def 3.2 replace ffip,b by

M°=a(BnxBn), changing notation from M',N' to M,N. An aE h 0,p,6needs not to extend to C(I0), but for symbols of differential ope-

rators and their parametrices we have that property in W below:

I° = a (BnxBn) = IQnxaBn U aBnxaBn U aBnxln

the first term at right denoting xE ien, E aBn}=w. The set

WC ffi° will be called the wave front space, in view of an applica-

2.3. Local md-ellipticity 89

tion in sec.5. If M is a subset of W then we will drop 'md-' and

refer to a symbol (formally hypo-) elliptic in M. For M of the

form M=XxaBn, with XC 2n the operator A=a(x,D) and its symbol a

are called (formally hypo-) elliptic in %, in agreement with con-

ventions for differential operators. For more general MC W another

notation is in common use: Such sets are called non-characteristic

sets of the operator A and symbol a (cf. [T12], [Tr1], [Hr3]).

The union Xp BnxaBn of the first two terms in (3.1) and Ms=

asnxien, the third term, are called the principal and secondary

symbol space, resp. For MC ffis we speak of m-(hypo-)ellipticity. In

[CP] a symbol aE Vhm p,sC Vh (formally) md-(hypo-) elliptic over

Htp BnxaBn was called (formally) d-(hypo-) elliptic. "md-(hypo-)

elliptic over Mp "just amounts to the conditions of (formal hypo-)

ellipticity in some aol , instead of a set IxI+IIaq.

If a symbol is md-(hypo-) elliptic over M C M°=a(BnxBn), for

an open set M, then one may expect a local parametrix for the cor-

responding lpdo A=a(x,D). We mainly will consider the case MC W, no-

ting that the case MC ffis may be treated similarly, reversing x, .

Instead of a local parametrix as in [Hr:] we construct a global

i,do inverting mod C00in some set M of (formal hypo) ellipticity.

Let be a symbol in Vhv,p,sC ph, for some vE &2. A sym-

bol aE 0 m,p,6 will be called formally md-hypo-elliptic with res

pect to q if for xI+I xj , some fl>0, we have (i) ay60 in supp cp

(ii) TE 1hv+m',p,6 '(iii) the restriction of a(R)/a to the set

supp T , extends to a symbol in VhR

,

m ,P,S

with ma'P of (2.2). If m'=-m we will call a md-elliptic with res-

pet to T. Clearly a (formally) md-(hypo-) elliptic symbol is just

a (formally) md-(hypo-)elliptic symbol with respect to T=1.

Theorem 3.3. Let Tr= lphv P16 , and let aE lPhm,p6be (formally) md-

(hypo-)elliptic with respect to T. Then we have ejE hv-m',p,S

j=1,2, where m'=-m in the elliptic case such that A=a(x,D), Ej=

ej(x,D), (D1=cp(x,D)=y(M1,D), (D.2

=cp(Mr,D) satisfy the relations

(3.2) AE1 = (D1 + K1 , E2A = 02 + K2 , K1,K2 E Op'tih-oo .

Proof. 'Elliptic' just means m'=-m, hence needs no special conside-

ration. We look at E1 only, taking adjoints for E2. With E2=B we

once more go through the construction of the right parametrix of

thm.1.6 (or 2.5). Set b0=a, getting b0E v+m',p,S' by (ii). Write

(3.3) AB0=(1+c0(x,D) , c0=J J11


where the asymptotic sum c0E hm+v+m'-p+6e',p,6' The next correc-

tion will be b1=-c0/a. Dividing (3.3) by a we get bl as an asymp-

totic sum of terms in the span of

(3.4) (a(0)/a)(a)(9) 101 Z1

It follows that bl E 0v+m'-p+Se'Iand that

(3.5) AB1=-c0(x,D)+C1(x,D) , c1=- j(-i)101/0!a(0)(c0/a)(9)

This then leads into the recursion bN =-cN-1/a , BN = bN(x,D)

(3.6) ABN =-cN-1(x,D)+cN(x,D) , CN 1.f-1(-0101/0!a(0)bN(0)

N=1,2,... . We show that, successively, bN, cN are of proper order

with B. well defined, while the orders of C tend toI N

As instrument in this proof we use the lemma below.

Proposition 3.4. Each symbol bN ,N=1,2,... , is an asymptotic sum

of terms in the span of the expressions

(3.7) nj=!(x((33/a) : LjOal=L40PJ=101

where 101 a N , for bN .

Clearly prop.3.4 yields bNE 0v+m'-N(p-Se')' hence cN abN+l

E v+m+m'-N(p-Se') has orders tending to -oo. The asymptotic sum

0bN b E E O v+m',p,6 is well defined and gives the desired e1=b.

Thus thm.3.3 is reduced to prop.3.4.

We prove prop.3.4 by induction: It is true for N=1. (3.6) gives

(3.8) bN+l = - j(-i)101/9!(x(0)/a)bN(e) .

If bN is as in prop.3.4 we may reorder the double series by I,

lemma 6.6, and get an asymptotic sum of terms in the span of

(3.9) (a(e,)/a)bN(e') 19112:1

where bN runs through all terms (3.7) , for 101=N. Each term (3.9)

is of the form (3.7) , for 191Z N+1 , q.e.d.

We now apply thm.3.3 to a locally (hypo-)elliptic symbol:

Proposition 3.5. Let the symbol aE yMIP' 6C h be (formally hypo-)

elliptic in an open set MC W, and let TE yihv, p , 6 , T=_0 for (x,) O_ Kwith KCC NC BnxEn , as in def.3.2. Then the symbol a is (formally)

2.4. Formally hypo-elliptic expressions 91

md-(hypo-) elliptic with respect to q (so that thm.3.3 applies).

Also apply thm.3.3 to the (formally) d-(hypo-) elliptic case:

Proposition 3.6. If aE 0m,P,b is (formally) d-(hypo-)elliptic,

then it also is (formally) md-(hypo-)elliptic with respect to any

function where xE C (&n ), x=1 for sufficiently large

x=o in a spere ICI s rl, for sufficiently large i1>O.

The proof for both prop's is immediate. It already was seen

that (formal) d-(hypo-)ellipticity amounts to (formal) md-(hypo-)

ellipticity for large IxI only, not for

Problems. 1) Discuss md-ellipticity on a cylinder in the setting

of 1,6, pbm's 1-4. Require the operator family

C00(S') invertible for i xA I + I ° I large; also, in the norm (III II of

L2(S' ), that ')II= O((x^)mz-Pz MI-pt IaI+olII), a,(3E IIn, as in I,(3.2)-(3.3). Discuss construction of a K-parametrix

B with 1-AB, 1-BAE4V (!fit as in 11,0, but with L(L2 (S' ))-valued ker-nels). 2) Show that the abovetf(with operator-valued kernel) con-

tains noncompact operators of L2(S'x&n-1), but that KE Xwith ker-

nel in K(L2(S')) is compact. 3) For mutual K-parametrices A,B, as

in pbm.1, a K(L2(&n-1))-valued symbol may be defined for each, AB=

I+K and BA=i+Kz, K.E. If and only if these symbols are inverti-

ble the operators A and B are Fredholm operators of L2(S'x&n-1)

4) The generators [Cz ],VIII,(2.2) of the C *-algebra C there -i.e.,

on our polycylinder C=S'xRn-1, the multiplications by a(x)E COO(e)

(2n-periodic in xi), and the operators A , ax A , axzA, ..., A =

(1-A)-1/2 (on C) all are pdo's as in pbm.1 (verify!). Discuss the

relation of pbm.3 above with [Cz],VIII,thm.2.6.

4. Formally hypo-elliptic differential expressions.

Let a be a domain of &n, and let

(4.1) L = a(x,D) =IaTsN

aa(x)D'

be a differential expression with Co coefficients : aa E C'(S2). We

shall call L (or its symbol )formally hypo-elliptic if for

every compact set K C c there exist constants Mi, i, p, such that

(i) a(x,l;) 30 0 , as x E K , ICI Z 11 ,

(4.2) (ii) as x E K , ICI a 'n ,

(iii) a(() /a as x E K , ICI a 1 .

(ii) is superfluous if L is locally of order >0 - i.e., L is


of order Nx near x, with 0<NxsN, aa(x)#0 for some laI=Nx. Then

NXP) , by (iii).

This yields (ii) locally, with M,=-Nxp, because aa(x) is bounded

away from zero, locally, for suitable a. This also shows that M,=

-p Min Nx s-p usually may be assumed negative :

For a constant coefficient L=a(D) of order >0 formal hypoel-

lipticity is equivalent to hypo-ellipticity in the sense of the de-

finition in ch.0, sec.5. This was shown by Hoermander [Hr1],p.101.

On the other hand there exist hypo-elliptic expressions

with variable coefficients which are not formally hypo-elliptic.

Here we use our parametrices of sec.3 for the following:

Theorem 4.1. A formally hypo-elliptic differential expression is

hypo-elliptic. If M, of (ii) can be chosen <0 (as true for L of

locally positive order) then L-X is hypoelliptic for all XE M .

Proof. The second statement follows from the first, because for

M,<0 we get

liml l uniformly on KCC

a , for

the theorem is best looked at in the light

of invariance of the wave front set of sec.5, below. We actually

will use thm.5.4, below. Assume uE D'(cz'), fE C00(SZ'), for some

open Sz' C S2, and let Lu=f in Sz' . We must show that uE COO(Sc') . Fix apoint x0E Sz', and balls B(x0,6) , B(x0,26) with center x0, radius

6, 26, closures C Sz'. Let TE Co, T=1 near x0, supp T C B(x0,6)

Define a global extension a as follows: Let

XE C0"O([0,25)), x=1 in [o,6], x=o in [2O,00), 0stx(t)s2O. Define

(4.3) a(x0+(x-x0)x(Ix-x0I) , ) , for x, E in

(4.4) a0E PcN.O C Vh(N,0),(P,1),O ,

and that a0 is formally d-hypo-elliptic in the latter space, with

inverse order M=(M1,0). By prop.3.6 a0 is formally md-hypoelliptic

with respect to a suitable X(U), x=0 near 0 , x=1 near -. By thm.

3.3 construct EE OphM,O

with Ea0(x,D)=x(D)+K, KE Opih-o,. Clearly

a=a0, as xE B(x° ,6), hence cpa = cpa0, for all assuming theleft hand side zero whenever a is undefined. Accordingly

(4.5) cpf = cpLu = Lcpu + [cp,L]u = L0((pu) + [cp,L0]'yru

with another cut-off p, supp V C B(x,26), =1 in supp [cp,L0]0

2.5. The wavefront set 93

Note that yfE CO(&n)C S, while puE E'(&n)C S'. Thus guE HS, for

some s, and h=[q,LO]VuE Ht, for some t. Clearly, h=O near x°, 0

since cp=1 near x , causing the commutator to vanish. We get x in

the complement of sing supp h . Hence x° sing supp Eh , using

thm.5.4, below. Applying the operator E to (4.5) we get

(4.6) epu+ (1-x)(D)qu + Kcpu = E(pf - Eh .

Here EcpfE S, since cpfE S. Also KfuE S, since TuE S', and K has or-

der (--,-oo) . Also, w=1-xE CO(&n)C S, thus w(D)cpu f 9Tyw' (x-y) ((Pu) (y)E S, the distribution integral existing, due to a E S, puE S'.

Also Eh is C" near x°, hence Tu is C" near x0. Since T=1 near x0

we find that also u is C" near x0 . Since this construction may

be done for every x0 E fz' , the theorem is proven.

We made extensive use of hypo-ellipticity in [C,],II,IV.

Hypo-ellipticity of L-?. , especially of L.-ti is required to make

the Carleman alternative work, as well as in general the construc-

tion of e.s.a.-realizations in [C:],II.

5. The wave front set and its invariance under.Vdo's.

We define the wave front set WF(u) of a distribution uE D'(sz)

over a domain cC &n as a subset of the wave front space W=OT(fa) ,

already discussed in sec.3, for the special case ft=&n. For a gene-

ral fcC &n we define 1(fz)=fzxaznC &nxa16n=0r(&n) .

It is common practice to interpret this space as T=fzxSn, i.e.,

(5.1) T = W(c) = {(x,i;) : x E sz , E &n , ICI = 1 } .

Similarly, for distributions on a differentiable manifold a one

defines LT(n) as the "co-sphere bundle", i.e. the bundle of unit

spheres in the cotangent space, with respect to a suitable smooth

Riemannian metric. It is clear then that the unit vector of a

pair with x E =-n just indicates the direction tx , as t - - ,

where the real point (x,«1) is to be found. In other words, for a

manifold sz the wave front space is properly defined as the subset

W of the symbol space $1(R) , as defined in [C2] ,VI, p.161.One defines WF(u) as a subset of W by its complement WF(u)c=

W\WF(u), as follows. Specify two types of cut-off functions called

cp(x), and respectively. For x0E sz a cut-off cp is a C func-

tion, cpa0, y=1 near x°. For a "direction" o0E a>sn given by °E &nwith 1°1=1 a cut-off is a C(Zn) -function, "0, x=1 near

x=0 outside some neighbourhood of °, and, 'Wl&nE C"(&n). Also, x


equals some homogeneous function of degree 0 for large One

where x=0 near 0,may think of a C(I)-function

s=1 near oo , µ=1 near l;°We define: (x0, 0)E WF(u)c (i.e., is not in WF(u))

if there exist cp , ,p , such that, with the Fourier transform 11^61 ,

(5.2) E S

Note that Tu admits a natural extension to I , zero outside S2, a

distribution in S' , hence has a Fourier transform in S'.

Proposition 5.1. We have WF(u)c if and only if there existcut-offs cp,1p, near x and resp., such that the ipdo C=1p(D)g(x)E OpipcO'-OO takes u into S ip(D)cp(x)u E S .

The proof is evident,since C=F-11pFg, and because the inverse

Fourier transform F-1 is an isomorphism of S onto itself.

From the definitions of WF(u) and s.s.u we conclude:

Proposition 5.2. The singular support s.s.u of uE DI(n) isthe projection of WF(u) to the first component Sz of W=S2xa18n

(5.3) s.s.u = {x: there exists F; such that (x,l;)E WF(u)} .

Indeed, if xE (s.s.u)c, then a cut-off cp can be found with

cpuE C0C S, so that (Tu) E S, -i.e. every WF(u)c. Vice versa

if for some x every Wf(u)c, then (cu)Ê S (i.e., guE S) fol-

lows for suitable T: The unit sphere is compact, and for 11;1=1 we

can find ((p"u)îp"E S. A finite collection 1pj=,pj has supports cove-

ring the sphere, so that 1pj=vj/yvj is a partition of unity. Let (P_

be a cut-off near x with cpp =cp for all j (i.e. T0=1 on supp

(p- ), then ((p-u)^1pj E S, by prop. 5.3. Hence (cp u) (Jipj)E S. But

it is possible to arrange for w--Y, y,jal outside a large sphere, and

have (o homogeneous of degree zero there. Adding a suitable C,(iQn)-

functions x we get w+xa1 in &n, and ((x+x)-1E T, the space of

Coo(len)-functions with derivatives of polynomial increase. Thus

(cp u)^ w+ E S, hence (y- u)^ E=- S, since 1-w/(w+x)=x/(x+w)E Co, and

((p u)Ê C. Hence cp uE S, uECoo

near x, or xE (s.s.u)c, q.e.d.

We have used the proposition below, also useful later on:

2.5. The wavefront set 95

Proposition 5.3. If we have ((pu)^tpE S , with a pair of cut-offs

q, 1p and a distribution u, and if qi cp=cp, for a cut-off qi , then wealso get (qf u)^ V E S .Proof. We know that ((pu)^,p E S is equivalent to p(D)q>(x)u E Sand similarly for qi instead of T . Write

(5.4) 'cp(D)cp u = ip(D)cp(x)gi u = qi ip(D)p(x)u + [1p(D),gi (x)]pu ,

where the first term,at right, is in S . For the second term we

use the asymptotic expansion of the calculus of ydo's : We get

(5.5) p[W (x),cV(D)]=p(x,D), p=- X01i9lp(0) (0)(x) (mod S)>0

Each term of p is =0, hence p(x,D)E O(-oo). The last term of (5.4)

also is in S, due to p(x,D)*=-[1p(D),gi (x)]cp(x)E O(-oo). Q.E.D.

Theorem 5.4. Let u E ES' , and let A = a(x,D) be a ido with symbol

in some 1phm,p,s , with the usual inequalities 0sO<p1, 0<p2. Then

(5.6) WF(Au) C WF(u) , and s.s.Au C s.s.u .

Proof. We are using As and O(s) of ch.III, here, for convenience.

It is sufficient to prove the first inclusion (5.6), by prop.5.2.

We must show that WF(u)c implies WF(Au)c. For (x,t)

E WF(u)c we have 1p(D)gp(x)uE S with suitable cp,ip. It is found that

(5.7) p(D)cp(x)Au = AV(D)cp(x)u + [C,A]u , C = 1p(D)cp(x) ,

where the first term at right is in S, since A takes S to S. uE S'

is in some Es, sE 22, since S'=Uts. [C,A] is of lower differentia-

tion order than A (and of multiplication order -o).

In order to get the desired property for Au , we will iterate

the proceedure as follows. Construct sequences of cut-off func-

tions <p0, W1' ..., (V0' V1' ) such that q<p0=p0' T0T1=T1'

Tj(pj+1=,pj+1 ''' (and 'VO--V0' VOy'1-V1 " Prop.5.3. implies

(5.8) Cju E S , j = 0,1, ... , where Cj = 1pj(D)gpj(x) .


(5.9) C0C =CC0 =C0 (mod O(-oo)), CjCj+1= Cj+1Cj= Cj+1 (mod O(-0°)).

Proof. We have CCO = 1p(D)cp(x)tpO(D)gO(x) = ip0(D)cp0(x) ++ 1p(D)[cp(x),VO(D)]q)O(x) Hence it is enough to show that

the last term, called V for a moment, is in O(-oo) . But we have


(5.10) V*=z(x,D) , S),

where again all terms of the asymptotic sum vanish identically

Accordingly V* E O(-oo) , hence V E O(--) . This shows that CC0= C

(mod O(-oo)) . Similarly for CiCj+1 . On the other hand,

C0C = V0(D)T0(x)V(D)T(x) = 140(D)cp0(x) + VO(D)[TO(x),V(D)]T(x)Let W be the last term. Consider the asymptotic expansion of W*

W =p(x,D) , le (mod S)

which again implies W* E O(-oo) since all terms vanish, q.e.d.

Using prop.5.5, we may write

CNAu = CN-1CNAu (mod S) = CN-1ACNu + CN-1[CN,A]u (mod S)

(5.11) =CN-2CN-I[CN,A]u (mod S) = CN-2[CN-1'[CN,A]]u (mod S)

_ ... = [C,[C0,[C1..... [CN-1,[CN,A]]] ...]]u (mod S) .

Here the N+2-fold commutator has multiplication (differentiation)

order -- (ml-(N+2)(pl-O)) . Accordingly the right hand vector of

(5.11) is in HM'c., for every M, as N gets large. Thus CNAuE HM'co,

as N>NO(M). The q)k,Vk may be constructed such that (Pk=1 in a fixed

neighbourhood N of x (and similarly for Vk and a neighbourhood N'

of ) . Thus we can find cut-offs cp" , 1p" with cp" qpk=f" "Vk=V" , forall k . Using prop.5.5 we thus get

(5.12) C Au = C"CNAu E HM',, , for every M C Au E S = fl Hs

This completes the proof of.thm.5.4.

Corollary 5.6. Let AE Opyih be formally md-hypo-elliptic with res-

pect to a symbol q' . Let uE S', and let Au=f, and cp(x,D). Then

(5.13) WF(f) C WF(u) , and WF(4i) C WF(f) .

Morever, if q) (i.e., if A is formally md-hypo-elliptic),then

(5.14) WF(u) = WF(f) .

The proof is an evident consequence of thm.5.4 and thm.3.3.

2.6. Systems of operators 97

6. Systems of ipdo's.

It is often convenient to consider a vxµ=matrix

(6.1) A = ((Ajl))j=1,..,v,l=1,..,FL , Ajl= ajl(M1,MrD)

of pdo's where we generally assume that aj1 E SSm'P'b C SS, m,p,b

independent of j,l. Introduce a mat-

rix-valued symbol and write aE SSm'P'b, and A=a(M1,Mr,D). In the

present section, we will use all symbol class notations, such as

ST, yah, hm,p,d also for vxµ-matrices to indicate that the entries

are in the corresponding symbol class. The operator A of (6.1)

will act on spaces of µ-vector-valued functions or distributions.

All formulas of the pdo-calculus of ch.I remain valid, except

those involving a commutator, such as I,(7.5).

For md-ellipticity we focus on square (µxµ-) matrices of Idols

We call A=a(x,D)E Opyahm,p,b (and its symbol a) md-elliptic if

(6.2) E011m,P,6

is md-elliptic. A symbol aE SS (and its operator A) will be called

md-elliptic if is md-elliptic.

Lemma 6.1. A square-matrix-valued symbol aE 1hm,p,b is md-elliptic

if and only if for with some T>0 , the matrix

a square-matrix-valued symbol bE

Vh-mp,b such that as

Trivially the Hilbert space adjoint of an elliptic pdo and

the product of 2 elliptic pdo's are elliptic again.

We can expect a theorem like thm.1.6 for square matrices,

and even a more general result for rectangular matrices. For the

latter let us define as follows: If for a pair of (vxµ- and µxv-)

symbols aE hm,p,b , bE Vh-m p,b ,resp., we have (for some 1>0)

(6.3) 1 = ((6j1))j11=1,.,v' for all rj

then we will call a right md-elliptic and b left md-elliptic .

We then must have v:5µ. a is elliptic if and only if it is left and

right elliptic. b is then called a right inverse symbol of a, etc.

Theorem 6.2. Let a vxµ-symbol aE m,p,bbe (right) (left) and-el-

liptic, and let bE yh-m,p,b be a (right) (left) md-inverse symbol.

There exists a (right) (left) K-parametrix E=e(x,D), eE 4h-m,p,6'

of the pdo A=a(x,D) such that

(6.4) E'U (-m.-p.+6,-m2 -p2 ),p,6 '


Vice versa, if a (right) (left) parametrix of A of order -m

exists then A (and a) will be (right) (left) md-elliptic.

We prove thm.6.2 like thm.1.6, constructing a right or left

K-parametrix. If both exist they coincide mod O(-o3), as usual.

Many 'hypo-elliptic' arguments work as in the scalar case.

For matrix-valued symbols there is another interesting case

to consider: Assume the matrix of constant rank for large

and that, moreover, aE yahmp,, allows a decomposition

(6.5) a = a0+a1 , a1E 0m',p16 , m'<m ,

where a0 allows a 'partially md-inverse symbol' b0E 11h-,,,,b , in

the sense that is a gxv-matrix with

(6.6) for large

where is a vxv-projection matrix of the same rank as

and i.e.,

Clearly (6.6) and rank p0=rank a0 imply im

p0 projects onto im a0, hence 1-p0 vanishes on im a0, i.e,

(1-p0)a0=0. Introduce Then get g0-q02=

b0a0-b0a0b0a0=b0a0-bOp0a0=b 0(1-p0)a0=0, so that also is a

projection. Evidently rank g0(x,t)=rank p0(x,t), im p0 = im b0 .

Generally, if b=b0+b1, where b1E yh-m',p,S, with -m'<-m , we

will call the symbols a and b partially md-inverse to each other.

Theorem 6.3. Assume that aE and bE Vh-m,p,S are partially

md-inverse symbols of each other (where p>0 , pl>S). Let p=ab ,

q=ba , and let A=a(x,D) , B=b(x,D) , P=p(x,D) , Q=q(x,D) . Then

(6.7) AB - P = r(x,D) , BA - P = s(x,D) ,

where the symbols are vxv-matrix-valued and µ.xµ-

matrix-valued, respectively, and are in`Vh_Ee,p,S ,

for some e>0

The proof is calculus of ,do's. We cannot expect better para

metrices making r or s of order -- , with improved B, etc.

Matrix-symbols also occur in VI,IX,X, under different aspects

Chapter 3. L2-SOBOLEV THEORY AND APPLICATIONS.

0. Introduction.

In this chapter we consider ipdo's as linear operators of

a class of weighted L2-Sobolev spaces over in . We specialize

on L2-spaces and neglect LP-theory, because 'ydo's in general are

not continuous operators on LP-Sobolev spaces, for p#2 . To be

more precise, general L2-boundedness theorems for xdo's are true

for A=a(x,D)E OpiPh0,p,6 , assuming pa0, 0s6!-.pi, 6341, but corres-

ponding LP-boundedness statements are false, except for p'=1.

There is an extensive theory in LP-spaces of Sobolev and other

types (cf. Beals [B 41, Coifman-Meyer [ CM] , Marshall [ Mri ] , Mura-matu [ Mml] , Nagase [Ng, ], Yamazaki [Ymi ]).

In sec's 1, 2 we prove the L2-boundedness theorem, for 6=0,

and 0<6s1 , respectively. This result often is quoted as Calderon-

Vaillancourt theorem. In sec.3 we look at weighted L2-Sobolev

norms. Our class of spaces Rs=HS1,s: is left invariant by the Fou-

rier transform, just as many of our ydo-classes. A y,do of order

m=(mi,m2) is a bounded map Hsa Hs-m , for every s. For every m E

12 an order class O(m) is introduced - the operators S-S extending

to operators in L(Hs,Hs-m ) for all s . 0(m) is a Frechet space

under the norms of L(Hs Hs-m) ; O(0) and O(-)=U O(m) are algebras.

A pdo of order m belongs to O(m).

A refined Fredholm theory holds for (formally) md-(hypo-) el-

liptic Ado's. Such an operator admits a Green inverse- the equiva-

lent of the integral operator of the generalized Green's function.

This is discussed in sec.4. In sec.5 we prove that ado's of nega-

tive order (m,<0, mz<0) are compact operators Hs-CHs, for all s.

1. L2-boundedness of zero-order y,do's.

We refer to the class Vt0=lh(0,0),(0,0),0 of symbols here,

as introduced in I,(4.26), and discuss the following result.

Theorem 1.1. An operator A=a(M1,D) E Oppt0 is bounded in H=L1(1n).

More precisely: The restriction AJH maps and belongs to L(H).

99

100 3. L2-Sobolev theory

This result is quoted as the Calderon-Vaillancourt theorem

(c.f.[CV1]). For independent proofs c.f. [CC], and VIII,thm.2.2,

below. The very short approach below is due to Beals [B3].

Proof. Construct a partition of unity for &n, as follows. With the

cube Qr {Ixjl<r,j=l,...,n} choose OsgE CO(Q2), T>0 in Q312. Define

cpa(x)=cp(x-a), aE 2n, and Wa a(ncpa)-1/2. Clearly then

(3E2

(1.1) Va E CD({xE &n:lx-aI<2,j=1,...,n}) a1Va2(x)=1 , Va s 0

Given some symbol a E iVt0 we define

(1.2) Aa = aa(x,D)

We first show that AaE L(H). Since the Fourier transform is unita-

ry we may show Ba FAaF1E L(H). But for aE ipt0, supp aC {Ixlsp}

FAF-lu(11) = J41Xe ixr1JR1(1.3)

fol u(U)J41x Joie as ( -T1.S) u(S)=

with "I" with respect to x written as "u", and trivial integral in-

terchanges. The integral operator at right of (1.3) is L2-bounded

by Schur's lemma We get the operator bound

(1.4) sup f,Jdtla``(t-q,t)I : ,T1 Elen}

In particular,

( -''1) 2Nav( -l1, ) = J x a(x, ) (l-Ax)Neix( -Tl)

=J1xaN(x, )eix( -q) , with aN = (1-Ax)Na

which implies

(1.6) D((-rl)-2N IIaNIi

hence

(1.7) IIAIIL2 s c IIaNII , 2N > n ,

where c depends only on N , n and p .

We need an improved estimate for the operators

(1.8) Aap = Aa'Wl3(M) = aap (M1,Mr,D) aap=Va(x)WR(Y)a(x,)

3.1. L2-boundedness of zero-order operators 101

Taking the Fourier transformed operator A^ for A=a(M1,Mr,D) again,

with a(x,y,l;) vanishing for (x,y) outside KCC &2n , we get

(1.9) Aû=FAF-lu(T1)= fl-ixilJFTKely'{u(x)

Integrals interchange readily, as uE S. Therefore,

(1.10) A^ U(T1) = f91K U(K) fo1 alk

With the methods used to derive (1.6) one finds that

= IIaNNII Loo

aNN = (1-Ax)(1-Ay)Na

Note that (1.11) implies

(1.12) f IIaNNIILOO

0((K_,,) -2N)

Here it is important to observe that the "O(.)-constant" depends

only on n,N, and the volume of the compact set K, the latter since

an L1-norm has to be estimated by the corresponding Lc norm. Also,

-2N(-K) -2N= f -(T1-x)) -2N, and f ((t) (2s-t) )-2Ndt

ItIfIsl + ItIJIsl = I1+I2 , where 12

c(s)-2N, c=f(t)-2Ndt

For the integral I1 the inequality implies 12s-tI22Isl-ItI2lsI

so that the integrand is bounded by (s)-2N(t)-2N , i.e., we get

the same estimate as for 12

. This shows that indeed (1.12) holds.

Next we transform the operatorAa(3

, as follows.

(1.13)Aapu(x) = f91 JR1Y((1-A )

M eii;(x-Y)u(Y)

=am (M1,Mr,D)u(x) , am )Map CEP

where the partial integration is justified as in sec.l.

Proposition 1.2. We have supp aap contained in the set

(1.14) Ix-als2n , IY-PI12n}

Moreover,

(1.15) Ilaa(3NNIILo 5 c(a-R)-2M max :Ia152M, ItI52N}


where the constant c depends only on n, M. and N .

Indeed, (1.14) is trivial, by construction of the Va , while

(1.15) is a calculation, using that (x)(K)/(x)=0(1) for all x

Combining (1.12) and (1.15) we get

(1.16) IlAapllL2 s c (a_I)-2M[ [a]]2N,2M ' 2N > n ,

with

(1.17) [[a]]k,l = max {Ila(a)IIL : Itlsk , lalsl}

where c depends only on n,N,M. Choose the set K as product of the

balls with center a,(3, radius 2n, its volume independent of a,(3.

For u,vE CO(Rn), ua eau, va=1av, 2M>n, we get

(1.18)l (u,Av) l= I S(ua,AaRvt) I S cl

aIIualIlIvnII(a-R)-2Ma

s c2{ Iluall2 ;Ilvpll2} 1/2 = c2llull llvll

with inner product (.,.) of H, where cj=cj[[a]]2M,?3N

j=1,2, with cj' depending only on n,M,N. In (1.18) we used (lull =

XIIuaII2, by construction of Va, and Schur's lemma in discrete form

for 12-boundedness of (((a-(3)-2M)), 2M>n. Clearly (1.18) implies

thm.1.1: Set u=ukE CO(Rn) where uk-> AvE S in H.

Moreover, we also have proven the useful corollary, below.

Corollary 1.3. There exists c>0 only depending on n such that

(1.19) IIa(x,D)IIL2(,n) s c[[a]]NN, N=[n/2]+1 with [[.]] of (1.17).

Revisiting our above proof we notice that the same arguments

apply for an A=a(M1,Mr,D) with aE SS0000 0 '

(i.e., m=0, p=0, 6=0).

The Aa, are of this form anyway. Instead of (1.16) , (1.17) we get

(1.20)

with

IIAaPIIL2 s c (a-p)-2M [[[a]]] 2N,2N,2M

(1.21) [[[a]]]j,k,1 = maxLOO

:IaIsj,ItIsk,IkIsl} .

Thus we have

Corollary 1.4. All operators A=a(M1,Mr,D) with a E SS0,0,0

are

L2(len)-bounded. Moreover, we have

(1.22) IIAIIL2 s c[[[a]]]N,N,N ' N = [n/2]+1 .

3.2. L2-boundedness for positive delta 103

2. L2-boundedness for the case of O>0 .

Note that thm 1.1 gives L2-boundedness of all do's in the

Hoermander classes 00.p,0 , with pj z 0 , but that it does not

apply to the case S >0 . The theorem below asserts L2-boundedness

for general m=0, and pjz0, plab, OsS<1. It was first proven in

[CV 21. Other proofs may be found in [ Ka2] , [ CM] . We discuss aproof after a scheme in [B3] which again is relatively short.

Theorem 2.1. A pdo A=a(x,D) with symbol a Eiih06e',6'0<6<1

induces a bounded operator of H=L2(&n) .

Proof. We use cor.1.4 and the following decomposition. For a par-

tition wj, j=0,±1,±2,..., like Va but for n=1, i.e.,

(2.1) wj(t) = w(t-j) , supp w C {It Is2} , yjwj2 = 1 , wja 0

set xj=wj(log Clearly xj has support near i.e,

2j. Since we get xj=0, as j<-1. The partition of

unity Exj2()=1 has supports in concentric spherical shells:

(2.2) supp xj C {2(3-2)/b sq) s 2(3+2)/s} .

For A=a(x,D) with a satisfying our assumptions, we write

(2.3) (u,Av)=lj1(ujAjlv1). Aj1=xj(D)Axl(D). uj=xj(D)u, vj=Xj(D)v.

We apply Schur's lemma once more to show that the matrix

((2-EIj-1I)) is 12-bounded, and prove an estimate of the form

(2.4) IIAj1IIL2 s c2 EIj-lI , j,l E Z , for some E>0 .

Then the argument following (1.18) will give boundedness of A

We prove (2.3) for Ajl"=FAj1F-1 instead of Aj1 again. Here

Ajl"=Pjl=pji(M1,MrD) with Get pj1=O unless

(2.5) 27-2s(x)6s2j+2 , 21-2s(Y)O 21+2 .

Repeating once more the partial integration of (1.13) we may write

Pjl=pjl(Ml.Mr,D) . pj1=(x-Y)-2M((l-O )M pjl)(x.Y. )

(2.6) _ where e(x,Y)=(Y)26/(x-Y)2

and bM(x, ) = q)-21 (1-Ox)Ma(x,U)


Notice that bME 00,Se',b

again, for every M=O,1..... .

Proposition 2.2. We have IIbM(-D,Mr)xi(Mr)II s c , with c indepen-

dent of 1 , for each fixed M=0,1,2..... .

Applying the Fourier transform we may prove IIbM(x.D)xl(D)II

s c instead. Note also that E O,6e',6 ,so that

E 00'6e' .b .

Proposition 2.3. The family gl(x,T)=c1(2-1x,210 is bounded in Vt0

Indeed,

(2.7)g1((3)(x, )=

21(lal-IRI)ci(R)(2-1x.210

= 2l(lal-1pl)0((210 s(IPI-lal))

with 0(.) independent of 1. Also, x1(21 )=0 except if (210 6 dif-

fers from 21 by a factor between 1/4 and 4. Since qi contains the

factor x1(21 ) we conclude from (2.7) that ql(Q)=O(1), with 0(.)

independent of Thus cor.1.3 gives the uniform L2-bound for

gl(x,D)=Q1 . Note that Q1=VibM(x,D)xi(D)Vl-1, with unitary V1:u(x)

2-In/2u(2-lx), so that IIQIII=IIbM(x.D)xl(D)II, proving prop.2.2.

Applying prop.2.2 for M=0 get IIAj1IIsc, c independent of j,l.

For (2.4) it suffices to look at Ij-lIZk0 with suitable fixed k0.

Proposition 2.4. For Ij-llak0=5S , and s= 6(1-6) we have

(2.8) e(x,y) s c 2 EI3-lI, as (x,y) belong to the set (2.5)

with c independent of j,l.

For the proof we use the estimate

(2.9) (x-Y) z I (x) - (Y) I

easily derived by an elementary calculation. We get

(2.10) O(x.Y) s (Y)2S/I(x)-(Y) I2= v. 41/Ivz 23l-'U, 21r' I2,

with vj satisfying 4svjs4 and rj=1/6>1. For 2iI3-1IZ32, i.e., li-li

a5S=k0 the right hand side of (2.10) is bounded by

c41/4TIjsc41(1-3)=c4-hllj-lI, as j>l, and by c4l/41l=C4-(r1-1)lI7-ll,

as j<l. Hence we get the statement with s=21-2=2(1-6)/6 .

Regarding the proof of thm.2.1 note that Pji has distribution

kernel OM(x,y)k(x,y), with the kernel k of The

latter symbol defines a uniformly bounded operator family, while,

3.2. L2-boundedness for positive delta 105

in supp k , we have (2.8), hence it seems that (2.4) should follow.

For a real proof however we argue as follows: First show that

(2.11) a as yO/O s c ( Y) - 3 (x-Y) -k- I a lk+j=1R1

Indeed, for IaI+It1=1, writing aaayv=j for a moment, we get a(3=

(log O)ias=26 (log( y) )iap-2 (log(x-y) )kR in the span of P and -SSlap

with T=( y) , S=(x-y) . For any " , t , and I E + I rl =1 we get ( t3 )ts 11 =E,P+rj + (p4ap)(R where we may set R=S,T,O. An induction argu-

ment gives in the span of rlK(a3,p3) im(y),63), where K(a,3)=

Sap, L(a,f3)=TaP, with product over all partitions a=jai+ ,3

31+76 3 , with multiindices aj, (3 j , y j , Finally note that aaT=O

as a#0. Thus we may assume ly3 =0, 2Ia3I=IaI. Let YIP3I=k, Y1631=1

and note that (x) (01') /(x) =0 ((x) - I a I) , as used earlier. Hence theabove products are O(S-IaI-k T-1), where k+l=I(3I, implying (2.11).

Now we again recall the above unitary map V1. Look at (2.6),

for M=1 , describing a symbol of Pjl , rewritten as

(2.12) p 1(x,y, ) =

The operator Wj1=V1-1Pj1V1 then has symbol

(2.13)

From prop.2.3 we get describing a bounded set

in Vt0. Notice that of (2.13) describes a bounded set

with respect to the norm (1.22), while a[[[w.,]]]N,N,Nsc2 E13-ll,

so that cor.1.4 gives (2.3). Indeed, axayaywjl is in the span of

(2.14) 1U(v) , x+µ=a , X+v=P

where -(x,y)=e(21x,21Y), wj(x)=X(21x),,21x)

Note that aaaRE/E=21(Ia'1+1p1)(aaaI0/O)(21x,21y). For boundedness

in supp wjlxwe show that 21/(21y) yand 21(21x-21 y) are bounded, by(2.11). In supp wjl get 21/(21y)sc21/21/6sc. Also, 21/(21x-21y)s

c21/12j/6-21/61, as near (2.10). With similar argument (and 1=1/6)

we get sc2l(1-r1) for 1>j, andsc21-13,

as 1<j, so sc in both cases

In other words, the first factor of (2.14) is bounded. We al-

ready know the last factor bounded, since riE Vt0. For the second


we get 6Xw=2IIL I1O((21x)-IRI)s121R1(1-hj), bounded for 1>j, but not

for 1<j. The third factor is bounded by c2 61j-1I. Together we get

the estimates for (2.4), via cor.1.4 , but only for 1>j .

For j<l observe that prop.2.2 does not involve the support

restriction (21x)6, 2j, while prop.2.3 remains intact if that res-

triction is replaced by (21x)s21-5 a jxjs /4-5-4-1 . This means

that we may replace the factor wj(x) in wjl by wj(x)0(x), using a

cut-off z=0 for jxI2:2-6, =1 near 0. For j<<l we get wj=wj6. Repla-

cing wj in (2.13) by wj6 we get a new wj1 with also the third fac-

tor of (2.14) bounded for j<l. The old wjl will now be wj(x)wjl

giving the yido wj(M)Wjl , i.e., the correct old Wjl. The estimates

after (2.11) also remain intact with the new x-restriction.

This completes the proof of thm.2.1. We also get

Corollary 2.5. Under the assumptions of thm.2.1 there exists a

constant c depending on n and S only such that for 0<6<1 we have

(2.15) IIAllsc IPI)IILOO

: IaIs2,I,Is[Z]+1}L

with the L2-operator norm IIAII .

The proof is evident. Note that a result like cor.1.4 is

easily derived for the case 6>0, following the above guide lines.

3. Weighted Sobolev spaces; K-parametrix and Green-inverse.

Define the spaces Hs Hs1 ,s2 ' for s=(si ,s2 ), as the classes

(3.1) He = {uE S' : ns(x,D)u E H

where H = L2(&n) , and ns(x,) _ (x)sZs'E acs . For u,vE Hs we

introduce a norm and an inner product by

(3.2) (u,v)s=(nsu,nsv)L2 , Hulls=lInsulIiz , 11$ = ns(x,D)

Note that the strictly classical 'pdo Its is invertible, as an

operator S -> S , or S' -> S' , with inverse

(3.3) s1= nS = n_s(Mr.,D) = (D)-sl (M)-s2 E Op4c_s

In particular 11s is md-elliptic of order s , and,

(3.4) ns-1 - r1s E OPVC-s-e

by calculus of ipdo's. From (3.2) conclude that II5:H5-*H is an isome

try Hs - H. In particular, Hs is a Hilbert space, for all sE It' .

3.3. Weighted Sobolev spaces 107

We refer to the spaces Hs as (weighted L2-) Sobolev spaces

over &n. In the special case s2=0 we obtain the ordinary (unweigh-

ted) L2-Sobolev spaces, commonly known as Hs - now with sE I .

First we discuss properties of unweighted Sobolev spaces, for

the moment also called Hs sE & : For details (and a more general

discussion involving L1'-norms) cf.[Ci],III, or [GT]. Write

(3.5) As=(D) s , Hs={uE S' :AsuE H} , (u,v)s=(Asu,Asv) , hulls=IIAsull

1) Using Parseval's relation (0,(1.8)) we may write

(3.6) Ilulls2 = JR1 Iu^2s

with the Fourier transform u^=FuE LIloc(&n), where f exists as a

Lebesgue integral. In particular,

(3.7) He = {u E S' : uÊ L2(&n,(,)2sdU)}

2) The spaces Hs form a decreasing set of subspaces of S'

containing S, as s increases from -- to +- . Moreover, hulls is an

increasing function of s, either on &, or on a half-line (--,a)

3) For s=k , a nonnegative integer, we get

(3.8) Hs = {uE S' :u(a) E H for all lalsk}

with distribution derivatives u(a). The norm hulls is equivalent to

(3.9)

(3.10)

llullk = {a1sk

llu(a) ll2} 1/2l

This follows from (3.6) :

llullk = Jdfslu^ l2(1+7)2k = [aJfd2alu" () 12l afsk

(k) Ilu(a) ll2ask awith the multinomial coefficients (k) = k!/(a1(k-lal)1) .)

4) We have Sobolev's lemma : For s > n/2 the space Hs

is (compactly) imbedded into the space CO(&n) of all continuous

functions over &n vanishing at infinity. We have the inequalities

(3.11) HullLOO

s csllulls , cs = ( .J(S)-2sd )1/2 ,

and

(3.12) Iu(x)-u(Y)lsss(lx-Yl)Ilulls , 6s=(2Jsin2(1t/2)( )-2sd)1/2


Moreover, if even s >n/2+j , for some integer j >0 , then we

have a compact imbedding Hs-C03(&n) , defined as the space of all

uE CO such that also u(a)EE CO , for all lags j , and inequalities

of the form (3.11), (3.12) are valid for all u(a) , lalsj .

Sobolev's inequality ( 3 . 1 1 ) follows by showing that the Fou-

rier transform u^ is LI for u E Hs, s>n/2 (a matter of Schwarz' in-

equality). (3.12) is derived by estimating the Fourier transform

(cf.[C1],III ). The compactness of the imbedding is a consequence

of the Ascoli theorem: Equicontinuity of Hs-bounded sets follows.

The imbedding Hs-'CO., as s>n/2+j, follows from (5) below.

5) For u E Hs we have u(a)E Hs-lal , and

(3.13) Ilu(a)Ils-IaI 5 llulls

This is a consequence of the inequality

(3.14) Ilu(a)lls- IaI=JIu^ JlU.

using that 5

6)The spaces

(3.15) He = (1{Hs : sE &} , H-,, = U{HS : sE &} ,

have simple locally convex topologies (cf.[C1], ch.III). The space

H. is contained in COOO(&n) , but contains S , while H-. contains

E', but is contained in S'. All inclusions mentionned are proper.

7) Hs and H-s are mutually adjoint under the pairing

(3.16) (u,v) = (Asu,A-sv) = fuvdx , uE Hs , vE H-s

defined by the second expression, involving the inner product of

H=L2, using that As:H5-. H and A-s:H-s-a. H define isometries.

Returning to weighted Sobolev spaces H5--H5I,sz

of (3.1) we

note analogously the following properties of Hs

(i) We have

(3.17) Hs = {uE S' : II cull<co} = {uE S' : II s*ull<o0} ,

with the operator s*=(D) s' (M) 52 =ns(Mr,D). Moreover, the norms

(3.2) and (3.18) below are equivalent.

(3.18) llulls = Il s*ull = II(D) S' (M)S' U11

This follows by calculus of ado's : ss*-I and s* s-1 be-

long to OpWcOC Op'pt0, They are L2-bounded, by thm.1.1, so that


(3.19) csllulls s Ilulls s csllulls

(ii) We have

(3.20) Hs D Ht , as sjstj , j=1,2 ,

although the norm Ilulls not necessarily increases, as sj increases.

Moreover, the imbedding Ht - H. is compact, whenever i>sj

(3.20) follows from the estimate

(3.21) Ilulls s Ilulls tz 5 1/CS. tz Ilulls. tz s ct/cs. ,tz Ilullt .

using (3.2) and (3.19). or, Ilulls=ll sull=llLt(rc1ns)ull=llrc1 sullt .

where Kst I11 E OpyPcs-t is a bounded (compact) operator of Ht,due to 17tKst1 '= $sr2 S-t(sltrlssl )= s-tLst, where LstE OpipcOCL(H), r,=( x) (D) E K(H), as rj<0 (cf. thm.5.1).

(iii) For s1=ka0 , k an integer, we have

(3.22) Hs = {uE S' : u(a) EL2(&n,(x)2szdx),

1-15k}

and a norm equivalent to Ilulls is given by

llI(x)szu(a)l

(3.23)la>sk

For an integer sz=j2-0 Hs consists precisely of all uE S'

such that xauE Hs =Hs.,0 (with the unweighted Hs,) for all lalsj.

A norm equivalent to Ilulls is given by

(3.24)ka ki

Ilxaulls

(iv) For sE &2, and k=(k. kz ), with integers k. , kz ,

(3.25) %+k = { uE He : xau (0 ) E Hs , as l a l sk. , l l l skz }

and Ilulls+k is equivalent to

(3.26) 2 llxau(R)ll'II lsk. , Ialskz

(v)(Sobolev's lemma): For s.>n/2+j we get a compact imbedding

(3.27) Hs -> {uECJ(&n) : (x)szu E Cok(&n)} .

(vi) Hs and H-s are mutually adjoint, under the pairing

(3.28) (u,v) = ( Su,II sv) = ,fuvdx , uE Hs , vE H-s

where (I'su,II sv) (with (.,.) of H) defines (u,v), uE He, vE H-s.


Note that n =rI , and that (3.28) is independent of s

That is, the continuous linear functionals on Hs precisely

are given as v- (u,v), with some uE H-s and (.,.) of (3.28).

This follows because S :HS- H and lI s:H-s.* H are isometrics.

(vii) We have

S = fl {Hs : sE 12} , S'=U {Hs : E N2} .

Indeed, (v) gives fl HSC C°°(&n). Combined with (3.25) we get

fl HSC S, the opposite inclusion being evident from (3.1). On the

other hand, (3.28) implies that U HsC S', with (u,(P)=(u,g), gE S,

uE Hs. The opposite follows from [C,],I,(6.3) (or [Schwl]).

(viii) The space S is dense in every space Hs .

Indeed, a sequence um-u may be generated, for uE Hs by setting

um=xm*((Omu), X m nx(mx), wm=w(m), with suitable cut-off's x,wmThe spaces Hs will play a central role. They allow a descrip-

tion of topologies on pdo-algebras in terms of L2-norms.

Theorem 3.1. For a symbol a E tptm the corresponding yxlo A=a(x,D)

is a continuous operator H. Hs-m , for every s E &2 . Similarly,

for a E0m,Se',o ,O<6<1 , A is continuous from Hs to Hs-m

The proof is simple, due to our preparations: Since 1Is is an

isometry H5-'H, one must show that s-mA $1 is L2-bounded. By I,thm.

4.8 this is an operator in Op,t0, due to s-mE Op1cs-mC Opy)ts-m

sIE Oppc -5C Opy,t-s. Hence the desired Lz-boundedness follows from

thm.1.1. Similarly for aE using thm.2.1 instead.

We now define a type of order classes: An operator A:S-S,

with the property of thm.3.1 (i.e., A admits continuous extensions

As:Hs->Hs-m, for all sE 8:2, and given mE &2 ) is said to have order

m. The class of all operators of order m is denoted by O(m).

Trivially AE O(m) admits a continuous extension AO:S'-IS' with

As=A0IHS. Thm.3.1 now may be expressed by stating that

(3.29) OP hm1p,6 C O(m) , for all M(=- &2 pa0 Ssp1 ,0s6<1

We also define

(3.30) O(_) = U{O(m) : mE I2} , O(-ao) = fl{O(m) : mE 1e2}

The proposition below is trivial.

Proposition 3.2. The classes 0(oo) and O(0) are algebras under

operator multiplication and 0(-a) is an ideal of both, O(0), O(co).

Moreover, we have


(3.31) O(m)O(m') C O(m+m') .

The class 0(0) has the natural locally convex topology, in-

duced by the collection { IIAIIs : sE 1Q 2} of all operator norms

IIAIIS = sup{ IIAuII : IIuOIss1} .

Proposition 3.3. The above topology is a Frechet topology. An

equivalent countable system of semi-norms is given by

(3.32) { IIAIIk : kE Z2} .

Moreover, 0(0) , under this topology, is a Frechet algebra, i.e.,

it is complete, and the algebra operations are continuous.

This follows from the Calderon interpolation theorem (in

[C1],III the corresponding is shown for unweighted spaces Hs) .

Proposition 3.4. The class O(-oo) of all operators of order -co

coincides with the class yad-,, , and, again, with the class ' of

all integral operators with kernel in S(&2n)

Proof. We have seen in II, lemma 1.7 that ih-00 is the class of all

integral operators with rapidly decreasing kernel. Also, as a con-

sequence of thm.3.1, get Op h- =Opit C O(-co). For the converse

let AE O(-cc). Observe that Aapa,,,=MDRADR'Ma'E O(-co) for all a,13,a

a',4'. We must have A:S'->S, since A takes each Hs into f7ft=S, and

S'=l.Ht. In particular A:H-.S, hence Au(x)=fkx(y)u(y)dy, uE H, with

some kxE H, for xE >en, by the Frechet-Riesz theorem. Similarly,

(3.33) Au(x) _ (kX,u)s , sE 12 , xE &n

with some kS E HS . Using (3.2) one concludes that

(3.34) Au(x) = fIlsfsix(y)u(y)dy , u E S

with a distribution integral, where the temperate distribution

(3.35) kx = ensks E H_s , s E R2

must be independent of s , hence will be in It follows that

(3.36) Au(x) = fk(x,y)u(y)dy , k(x,y) = kx(y) .

But the same conclusion also applies to all AaPa,s, . It follows

by integration of (3.36) for large a, 3, a', I' that k must have

derivatives of all orders, for x, y , and that


(3.37) ±ifxaYa'a 4'k(x,Y)u(Y)dY

where the kernel has the same properties. Thus kE C0O(R2n), and

xaya'aOay'k(x,y)=O(1) , i.e., we conclude that kE S(12n), q.e.d.

Recall the concept of K-parametrix of a y'do of II,1. We now

recognize that a K-parametrix B of A is an inverse modulo O(-oo).

The concept is meaningful for arbitrary AE O(oo) .

Finally we introduce a stronger type of inverse: Two opera-

tors A,BE O(oo) will be called Green inverses of each other if

(3.38) AB-1 , BA-1 E F ,

where F C O(-oo) denotes the class of operators with finite rank.

Notice that F is a 2-sided ideal of the algebras O(oo), 0(0).

Clearly a Green inverse is a special kind of K-parametrix. In sec.

4 we will show that an operator admitting a K-parametrix also

admits a Green inverse. Clearly, if A E 0(oo) admits a Green inver-

se then it is a Fredholm operator of S' as well as of S , since

the Green inverse acts as a Fredholm inverse in both spaces

(cf. [C1],app.A1, thm.2.1).

More generally, if AE=-O(s) and BED(t) are Green inverses of

each other then r=s+ta0 follows. In case of r=0 - i.e., s=-t -

we have A:%-'Hm-s,B:Hm-s-'Hm continuous, for every mE &2 , so

that again A and B are Fredholm inverses of each other, as maps

Hm -HM-S . Hence then also A is Fredholm as a map Hm m-sFor general r=s+t>0 we may regard A as an unbounded operator

from Hm to Hm+t with domain dom A = Hm+r.Then B:Hm+t Hm is

bounded, and we get BA C I+F , AB C 1+F' , with F,F' E F .

This implies that the closure Ac of A:Hm+r dom A -.Hm+t is a closed

unbounded Fredholm operator from Hm to Hm+t' for every m. We shall

discuss details in sec.4, below (cf. [CI],app.A1, sec.6).

Remark 3.5. Note that A must have the same Fredholm index as a

map S-S , S' ->S' , Hm-*Hm-5 (or dom A = Hm+rC Hm -Ht )

Problems. 1) For the t,do's on the cylinder of I,6-II,3,pbms, if

IIA(a) (x° IIL2 (S' )=0(1) , all a,(3, show L2 (S' x&n-1 )-boundedness ofA. 2) Define weighted L2-Sobolev spaces on the cylinder, using the

2:tnorms dxi fdx° (x°) 252 I (1-A)s' uI 2=IIuIIs, with properly defined po

0

wer of 1-A=1-aX,-A6. 3) Discuss H5-boundedness of y,do's as in (1).


4. Existence of a Green inverse.

In this section we will show that also a Green inverse exists

if only an operator AE O(co) admits a K-parametrix. We have intro-

duced the concept of Green inverse in sec.3 as that of an inverse

modulo the ideal F C O(-co) of operators of finite rank.

All results, below, (especially thm.4.1, thm.4.2, cor.4.3)

are valid not only for a Wo with complex-valued symbol but even

for a matrix-valued pdo A =((a.k(x,D)))j,k=l,...,v,although we

only discuss the scalar cases. Proofs extend literally.

Actually, it is practical to prove a stronger result: An A

with Green inverse has a finite dimensional null space ker A C S

There is a complement T of ker A in S' and an operator B OE 0(-)

such that B0Iim A inverts AIT . Here the order of BO may be

chosen equal to the order of B . Also we may choose T as the

ortho-complement T= {uE S':(u,g)=0 for all TE ker Al , with the

pairing (u,q) of (3.28). With this construction B0 has the proper-

ties of a special Fredholm inverse ([C1], p.259). In particular,

(4.1) 1 - BA = P , I - AB = Q

with projections P,QE F onto ker A and ker A*, respectively. Also,

P, Q E 0(0) are orthogonal projections in H=H0 .

For a proper setting of (4.1) let A* be the adjoint of AE

O(o) under the pairing (3.28). That is, A :Hm-s-> Hs is the opera-

tor satisfying (Au,v)=(u,A*v) for all uE Hs, assuming that AE 0(m)

C 0(oo), with (.,.) of (3.28) . This defines A* for each Hs. All of

them agree on S. By continuity they must agree wherever they are

jointly defined. Thus we obtain A* as a map S'->S' with continuous

restrictions S-S, %_s-H-S. Then AE O(m) amounts to the condition

(4.2) s-mA S1 GL(M) , for all sE &2

For the adjoint A* this translates into

(4.3) sl*A*ls-m = h1 sA*rcls E L(H) , s E H2

Or, replacing s by m-s, rIs-mAs

*rf lE L(H), i.e., A*E O(m). It fol-

lows that AE O(m) a A*E O(m), including the cases m=tom. Clearly,

"*" has the properties of an involution 0(00)-O(oo). For a 'tpdo A

this adjoint is the formal Hilbert space adjointA*

of I,(2.3).

Theorem 4.1. Let AE O(m) have a K-parametrix BE O(m'). Then the


operator A , as a map A:S' -> S' has the following properties.

(i) ker A C S , ker A* C S;

(ii) dim ker A < ao , dim ker A* < oo ;

(iii) in A = If E S' : (f,T) = 0 for all T E ker A* }

(iv) For fE He with (f,q)=O for cpE ker A* all solutions u

of Au=f are contained in Hs-m,. There exists ps>0 such that

(4.4)IIff1s=IIAuIIskpsllulls-m,, whenever uE Hs-m (u,cp)=0 for gE ker A

with ps independent of u and f.Proof. For uE S with Au=O get (1+K)u=BAu=O, where KE O(-ao). Thus

u=-KuE S, since an integral operator with kernel in S(12n) maps

S'- S. Similarly B*A*=1+K'*, hence ker A*C S, confirming (i). To

show (ii) note that ker A C ker(1+K). But ker(1+K) is closed in

He since uE Hs, (1+K)u=0 implies uE S. For an infinite orthonormal

set ujE ker(I+K) - with respect to (u,v) of H - the sequence Kuj

has a convergent subsequence, in L(H), K being compact in H, its

kernel in L2(12n) makes it a Schmidt operator. But uj=-Kuj cannot

have a convergent subsequence, as an orthonormal set. Therefore

dim(1+K)<oo , implying dim A <oo . Similarly for A* , proving (ii).

For (iii) we first observe that, trivially, every f E S' ,

f=Av , satisfies (f,g) _ (Av,y) = (v,A*T)=0 , for all g E ker A*

In order to show the converse we will first establish (iv) and

show that a solution u E Hs-m, exists for fE Hs, satisfying (4.4).

To clarify the relation with the inner product of H we assu-

me that m'=0 . This is no restriction: For general AE 0(m) , BE

O(m') observe that (As* )(II m,B)=1+K , (rl m,B)(A1,)=1+K" , with

K"=r1m,K's,-lE O(-co). Also Au=f ,)(II m,u)=f , and ker A* =

ker (,)*

, and IIuIIs-m,-IIn mulls . Thus all terms may be trans-

lated to the case m'=0: Replacing Ar ,E 0(r), r1 m,BE 0(0) with A,B

we get a pair A,B of mutual K-parametrices with AE O(r), r=m+m'a0,

and must solve Au=f under the corresponding conditions.

Now let fE S' be 'orthogonal' to ker A*. We must have fE Hs

for some s, and now will proceed to show that uE Hs orthogonal to

ker A exists solving Au=f, where u and f satisfy (4.4) (with m'=0)

A further reduction of the statement is useful: we may assume

s=0: Au=f ea( $A Sl)( $u)= sf, where rlsu, sfE H--H(0, while IIsAsl

E O(r) has K-parametrix sB s'E O(0). Also, (f,g)=0 for gE ker s*

a (sf,p)=0 for V=II scp satisfying (sA sl )III sA*ffsrl_s(p = 0Thus we now are reduced to an operator equation Au=f in the

Hilbert space H , where we may regard A as an unbounded operator


with domain dom A = Hr , r=m+m'a0 . The operator A satisfies

AB=1+K , BA=1+K' with K,K'E O(-oo) . Let us next prove (4.4), or,

(4.5) IIAuII a pllull for all uE Hr don A with u 1 ker A* .

Suppose this inequality is false. Then there exists ujE Hr ,

uj 1 ker A , IIujII=1 , with IIAujII o , j- . But we get

(4.6) (lull s II (1+K' )uhl + IIK'uhI = IIBAuII + IIK'uII s cllAull +IIK'ulI ,

for all uE Hr . Since K'E K(H) , as used before, there must be a

convergent subsequence of K'u (we assume K'uj convergent itself).

It follows that j-K'ul11_0, j,l- . Thus,ujE Hr, uj-u (in H), 11u11=1, Auf. 0. Hence uE don Ac , with the clo-

sure Ac of the unbounded operator with domain Hr, as discussed.

But we also have IIAu jlls+r_' 0, Ilu-ujlls- 0 for ss0, s+rs0, where nowA:Hs+r_Hs is continuous, so that u is in the null space of A:Hs+r

H. It was seen that this null space is independent of s, hence

=ker AC S. Since uj-u in H and uj I ker A in H it follows that u

1 ker A, hence 1=(u,u)=11u110=0, a contradiction, proving (4.4).

The adjoint A*E O(r) also may be interpreted as unbounded

A*:dom A*=Hr-> H. For this operator we have

(4.7) (u,Av) =(A*u,v)

for all u,v E dom A = domA*

- Hr ,

by definition of A*. In other words, A and A* are in adjoint rela-

tion in the Hilbert space H : The Hilbert space adjoint AA of A ex-

tends A*. We get ker AA =ker A*. Indeed, ">" is trivial. Let fE

ker AA : fE H, (f,Av)=O for all vE Hr . Then A*f=0 wth A*:H-H_r . OruE ker A*C S, ker A* independent of s, so that ker AC ker A*.

We conclude that Ac is an unbounded closed Fredholm operator

of the Hilbert space H since (4.6) implies that in Ac is closed.

It then is well known that in Ac =(ker A)1 (cf.[C,] app.A1,6,7).

Notice finally that Ac equals the restriction to don Ac of

the operator A:H-H-r. Indeed, ujE Hr don A, u j-*u (in M), Au j- Acu(in H) implies ujE H uj-u (in H) , Auj-> Acu (in H-r), hence

Acu = Au , since A:H -, H-r is continuous. This completes the

proof of (iii) and (iv) of thm.4.1. Q.E.D.

Theorem 4.2. An operator A E=-O(m) admits a Green inverse Bo if and

only if it has a K-parametrix B. If BEO(m'), then also B0E O(m').

Also, B0 may be chosen in such a way that (4.1) is valid.

Proof. A Green inverse is a K-parametrix. Thus we must construct


a Green inverse whenever a K-parametrix exists. Applying thm.4.1

we define an operator Bo by setting B u=0 for uE ker A* , and B =0 * 0 0(AI(ker AO)1)-1

in in A =(ker A )1 . Every u65' is in some Hs, and

we have a unique decomposition u=v+w, vE ker A, w 1 ker A. Thus

the above defines a linear operator S'-'S'. Clearly B0Au=u for u 1

ker A, and B0Au=O for uE ker A. But AB0u=u for uE in A =(kerA*

and AB0=0 for uE ker A* . In other words, (4.1) is valid.

On the other hand, (4.4) implies

(4.8) IIA lulls_m, s

P

IIuIIs for all uE H. , u 1 ker A*PS

for the abstract inverse A 1 of A:{uE Hs+m: u I ker Al -> Hs

Actually, we get (4.8) for Ac-1 with the closure Ac of the unboun-

ded operator A:dom A =Hs+m-r=H

s-ml+Hm , and Ac-1 is defined in all

of {uE H : u I kerA*}.

By our definition of B (4.8) implies

* 1s ) is= JIBOVIIs_m,SpsIIvII -csllulls, where u=v+w, wE (ker AIIBOuIIs_m

the above unique decomposition, and where we used that Ilvllss cOIuIIS

uE Hs, c independent of u. Indeed, with an orthonormal base

k (in H) of ker A* we have w=YWj(Wj,u) , v=u-w , hence IIv1Is

s IIuIIs+ JIlrojIlsllrojII_slluIIs , so that c=1+7,Ilrojllsllacjll_s may be chosen.

Clearly Q= 7,cp j) ((pj and P=j) (Vj (with an orthonormal base {W j}

(in H) of ker A ) belong to 0(0) . The proof is complete.

Corollary 4.3. The operators B and B0 of thm.4.2 differ by an ope-

rator in O(-oo)=Opph_oo only. Thus a Green inverse of a (formally)

md-(hypo-)elliptic tpdo always is in the same symbol class Vt'm,p,6

of ch.Il as the K-parametrix constructed there.

Proof. Indeed, that Green inverse is a special K-parametrix, and

we observed earlier that, as inverse modulo the ideal O(-oo), a

K-parametrix is unique up to an additive term in O(-c) .

We conclude this section with the remark that a Green inver-

se of a pdo in essence has properties very similar to the integral

operator with kernel equal to the generalized Greens function of

a boundary value problem: It constitutes an inverse modulo opera-

tors of finite rank; its distribution kernel has singular support

at the diagonal x=y only (as a consequence of I, thm.3.3).

Problems. 1) The operator B: COO(S')-Coo(S') given by u.yeij°bj(O)uj

3.5. Hs-compactness 117

with uj=,1nf a liWu(p)dp , as in I,6,pbm.3, is L2 (S' )-bounded if weassume that =0(1) (in j and 0 , for each k=0,1.... . [De-

rivatives of all orders are not required, but how many are needed

for a proof similar to that of thm.1.1?] 2) Investigate the C*-sub-

algebra of L(LZ(S')) generated by all operators B as in (1). Show

that its finite Fourier transform is the subalgebra A of 12(Z) gen-

erated by the shift operator and all diagonal matrices

with bounded coefficients. 3) Does X have compact commutators?

4) Answer the same questions if the condition of pbm.1 is modified

by requesting that 0((j)-pl) in 0 and j , with the finite

difference Vaj=aj-aj_1 , and some p>O , for all (some?) 1 and k.

5_HS-compactness of ipdo's of negative order.

We shortly discuss an often used compactness result.

Theorem 5.1. A (matrix of) 'tpdo's A=a(M1,Mr,D), with aE SS and or-

ders mj satisfying m,<0, m2+m3<0, is a compact operator Hs- Hs,

for every sE &2. Moreover, for general mj, A: Hs Ht is compact

whenever m, <s, -t, , mz +m3 <s2 -t2 . Especially, A=a(x,D)E 0pOm,p,sp,>0, pz>5, is E K(HS,Ht) whenever mj<sj-tj , j=1,2.

Proof. Observe that Q,=(x)_6(D)_F is compact H-'H, as E>0 (cf. [ Ci ]III,lemma 8.1, for example - but other proofs are known. Details:

The operator (D) -E may be written as convolution (x) -E ° * , where

(5.1) pE(x) =(x)-E (x) = cE'vlxl(E-n)/2K(E_n)/2(Ixi)

,

with a constant cE,n, and the modified Hankel function K0. (5.1)

is verified using techniques as in ch.0, sec.4, for construction

of the fundamental solution of 0+kz . Note the function (5.1) is

L'(1m) The kernel gE(x,y)=(x) EpE(x-y) may be approximated in the

sense of Schur's lemma by Schmidt kernels qE E Lz(&2n). Then (i)

the operators Q£, QEu(x)=fgE(x,y)u(y)dy , are compact, from H to

H . Second, we have lIQE-QElI - 0 , j->oo , by Schur's lemma, so that

QEE=- K(H), since K(H) is closed under uniform operator convergence.

With Lz-compactness of QE we get the statement at once: Com-

pactness A:HS-* Ht means compactness of sIAllt=(1 IAnt+Ee)QE:H4H'

where the first factor is bounded, by III,thm.1.1 (or thm.2.1),

assuming the inequalities of thm.5.1, by calculus of pdo's. Thus

indeed A: Hs-Ht is compact if these inequalities hold. Q.E.D.

Chapter 4. PSEUDO-DIFFERENTIAL OPERATORS

ON MANIFOLDS WITH CONICAL ENDS

0. Introduction.

In the present chapter we will focus on pseudo-differential

operators on differentiable manifolds. We assume either that 11 is

compact -then our theory will not differ from others - or that Q

is a noncompact Riemannian space with conical ends.

While the Fourier transform and, correspondingly, the concept

of Fourier multiplier a(D) = F-la(M)F is meaningfull only for func-

tions or distributions defined on ten , the kind of ydo's we intro-

duced has a natural environment on a type of differentiable mani-

fold, to be studied. The reason: Our ipdo's of ch.2 are invariant

under a type of coordinate transform (discussed in sec.3) while

Fourier multipliers do not have this property.

In sec.1 we discuss distributions on manifolds. A special

type of 'S-manifolds' is preferred, allowing the definition of a

class S(c) of rapidly decreasing functions. The linear functionals

on S(cz) will be our temperate distributions. For simplicity we

will consider only manifolds a allowing a compactification W to

which the Co structure can be extended - making r° a compact mani-

fold with boundary. In essence then S(a) will be the class of

functions over n vanishing of all orders on an .

In sec.2 we will introduce 'admissible' charts, cut-off's,

partitions, as well as admissible coordinate transforms, all

designed to give S(c) similar properties than S(8n) . In particu-

lar a Riemannian metric is introduced, making n a space with coni-

cal ends. In sec.3 we prove invariance of pseudo-differential ope-

rators under admissible coordinate transforms.

Sec's 4 and 5 generalize the calculus of pdo's to spaces with

conical ends. In particular, we again get md-elliptic and formally

md-hypo-elliptic operators, defined by their local symbols.Results

of ch.'s II and III regarding K-parametrix, Sobolev spaces, order

classes, Green inverse, etc., all generalize almost literally.

Similar results, on coordinate invariance as well as and-el-

118

4.1. Distributions and temperate distributions 119

liptic operators on manifolds, were discussed by Schrohe [Schrj]

in somewhat different setting. Regarding application to differen-

tial operators we point to [CDg1] , [C2] where similar C*-algebras

of singular integral operators on L2(a) are considered, with cor-

responding results, but abstract proofs.

1. Distributions and temperate distributions on manifolds.

In ch.0 we were discussing distributions on an open subset

n of In . Presently, we first will extend the distribution concept

to differentiable manifolds, compact or not.

For a C00-manifold a of dimension n (always assumed para-

compact -even with a countable atlas) the space D=D(a)=C00(n) is

well defined, and the convergence cpj-0 in D(O) just as well: One

requires that (i) supp cpjC K, with a set KCC sz independent of j.

We get KC U . , a finite union of charts gill and Tj=Folroj , with a

partition of unity Twl = 1 in K , supp w1C Szl , w1a0 , w1E C000

(n)

One also requires (ii) that wlrpj- 0 in D(SZ1), as j->0 in the coor-dinates of the chart Szl , for 1=1,...,N.

We define a distribution uE D'(n) to be a continuous linear

functional over D(n) , where 'continuous' means that

(1 . 1) (u,(Pj) -> 0 whenever y j - 0 in D(n) .

With this definition a conceptual difficulty arises if we attempt

to interpret a function fE LIloc(a) as a distribution. For S2C in

we defined (f,(p) ffgdx , taking advantage of the existence of a

distinguished measure on 2n - the Lebesgue measure as Haar measure

of the group In . For a manifold a there no longer is such a dis-

tinction, but we may construct a positive Co measure dµ, locally,

dµ=xdx , O<K(=- Coo , on a paracompact manifold n , and then define

(1.2) (f,P)=f fq)dµ , cpE D(SZ)SZ

for fE LIloc(R),establishing the analogous imbedding L1loc+ D'

This dependence on prior choice of a measure may be avoided

by defining distributions as linear functionals on a properly topo-

logized space of signed C0(SZ)-measures - expressions of the form

cpd1t, yE D(Sc), in our terminology. But we are tied to the use of

Sobolev norms anyway, where we use a distinguished measure. Hence

120 4. Operators on manifolds with conical ends

we follow our habit of [Cz], and always assume given a pair {sl,dµ}

of a manifold st and positive C-measure dR on sl, so that dµ, dis-

tinguished by other uses in our theory always is on hand. Then we

may use (1.2) to define the imbedding Llloc-' D'(a). Often ds=dS

will be the surface measure of a Riemannian metric on sl .

Most concepts obviously extend, such as restriction of a

distribution to an open subset, (singular) support, definition of

Lu, for a differentials expressions with smooth coefficients, etc.

A differential expression L on SZ (with CC coefficients) is

defined as a linear map L:C00(sl) - C°°(n) with Lu= sa u in deriva-

tives dX of the local coordinates and C coefficients aa for u

with support in a single chart. The adioint differential expres-

sion is defined as the expression L*:C00(9l) - Cw(c) satisfying

(1.3) fLuvd s = f uL*vdµ , for all u,v E C00(n) with uv E CO(sl).

The dual expression L" of L is defined by setting L f=L'f, fE D .

For a general distribution uE D'(sl) one then defines Lu by setting

(1.4) (Lu,(p) = (u,L c)) , for all qp E D(a) .

One derives the familiar facts on E'(11) , the space of com-

pactly supported distributions, on regularization, and once again

may introduce the standard topologies on D' and El (cf. [C1],I,6).

For an extension of S(c) and S'(n) we introduce a type of

manifold looking conical at - .

The main feature of fE S=S(&n) is the behaviour of f for lar-

ge lxi. In 11,5, looking at wave front sets, we introduced a type

V of directional cut-off and regarded f belonging to S in a coni-

cal sector only if ifE S for suitable such ' .

A general change of coordinates can destroy the property of

a function u to belong to S (or 1puE S). Accordingly, a special

structure on sl, allowing only special transforms as

admissible coordinate changes is required before a space S=S(s1)

can be defined, with general properties of S()Bn). Such structures

were introduced by Schrohe [Schr1,2] and were called SG-structures

We only look at a restricted class of such n. First define

the diffeomorphism s:1en- B1, B1=B1(0)={yE 2n:IyI<1}, by setting

(1.5) y = s(x) = x/(x) , x = s-1(y) = y/ 1- y , = t(x)

Introduce [ y] = 1- y Z , then,

4.1. Distributions and temperate distributions 121

(1.6) Y = x/( x) , x = Y/[ Y] , [y] = 1 /( x)

Proposition 1.1. The map

(1.7) u(x) - v(Y) = u(s-1(Y)) = (uas-1)(Y)

defines a bijection S(&n) - COOO(B1).

In prop.1.1 COOO(B1) denotes the class of Cx(B1)-functions vani-

shing with all their derivatives at aB1.

Remark 1.2. A function fE COCO(B1) has a natural extension to in

-

f=0 there. The extension v is CQ(&n), with supp vC Bi=(B1)clos.

Vice versa, for vE COO(e) with supp wE BC we have wIB1E

Proof. If vE CO"(B1) then v and all its derivatives vanish of in-

finite order at lyl=1, by Taylor's formula. Thus we get

(1.8) v(a)(y) = 0(((1-1y12)k) , for all k=0,1,... , and all a

For a=0 this implies u(x)=v(y)=O([y]k)=O((x)-k), k=0,1.... .

But u(a)(x)=aa(v(x/(x))) is a linear combination of terms

(1.9) v(R)( )II ay

X

k(xjk/(x)) , IRlslal , lYk = «

by induction. Get aX(xj/(x))=O((x)-IY I) , easily confirmed. Eachterm (1.9), hence ua is 0((x)-k) , for all k, i.e., uE S

Vice versa, let u E =-S and v(y) = u(y/[y]). Note that

(Y/[Y])2=1+Iy12/(1-IY12)=1/(1-IY12)=[Y]-2, hence (Y/[Y])=[Y]-1

(1.10) u(a)(Y/[Y]) = O((Y/[Y])-k) = O([Y]k), for all k .

For a=0 it follows that v(y) E CO(B1) . For arbitrary a one again

uses the chain rule to express v(a)(y) as a linear combination of

(1.11) u(P)(Y/[Y])nay j(Y1 /[Y]), IPIslal, j=a ,

similarly as (1.9). Here we get aY(yl/IYI) = O([y]-21Y1-1), andy

(1.10) still implies all terms (1.11) to vanish, hence v(a) =0

at IYI=1 . Thus v E COOO(B1) follows and prop.1.1 is proven.

The above suggests the concept of an S-manifold as the inte-

rior a of a smooth compact manifolds no with boundary. Then define

the space S=S(S2) of rapidly decreasing functions on sa by setting

S(a)=CO'(1 )= {all uE C°(Sto) vanishing of infinite order on an}.no

is a compactification of Sz, just as >sn of &n in II,(3.1).


aSt0 is called the infinite boundary of a; its points at infinity.

We will insist, in the following, that a0 be a compact C-

manifold with boundary. sz0 will be useful for function classes

other than S(c). On the other hand, for defining S(a) this is not

required. For example we might allow sz0 as a cartesian product of

finitely many manifolds with boundary, or allow 110 to have corners

vertices -i.e., at some boundary points we have charts in such Car-

tesian products with Ij=I or I.=le+=[0,oo) , depending on j.

Notice that 2 xln=R2n is (should be) an S=manifold, but the

product sz0xsz0=BixBi no longer is a C* manifold with boundary. As

in prop.1.1 one shows that we get a bijection S(&2n) H COOO(B1xB1)

as well, induced by the map sxs:R2n - BlxBl

Thus, for a-I n we could use as compactification nl =Bi (in

2n dimensions) or szO=BixBi. We choose the first and keep it fixed,

for the following (note the exception at the beginning of sec.4).

Before discussing temperate distributions on a general S-ma-

nifold a we look at the transfer of the Frechet topology of S(In)

onto CO(B1) under the bijection of prop.l.1. The seminorms

(1.12) Ilullk = sup{II(x)ku(a)IIL°°(Rn) : Ialsk} , k=0,1,2,...

generating the topology of S , are equivalent to the'seminorms

(1.13) vk(v) = sup{II[y]-kv(P)II o, : I1Isk} , k=0,1,2,...L (B.)

in the sense that

(1.14) Ilullks ckvl(k)(v) , and vk(v) s ckllulll(k) , k=0,1,2,...,

with ck, 1(k) independent of u,v. This is confirmed looking at the

proof of prop.l.1. On the other hand, since all v(p)(y )=O as Iy I

=1, Taylor's formula with integral remainder (I,(5.20)) implies

(1.15) v(y) = (N+1) 71 O J1(1-i)Nv(0)(y°+i(y-y° ))diIOI=N+1 0

For O#yE Bi let y° = y' , so that I (y-y° )6 I s y-y° j16 I s [y]K. Then

(1.16) v(y)= Max{IIv(0)II :I8I=N}.O([y]N+1) , N=0,1,2,... .L (Bi )

Similarly for derivatives v(a). Combining (1.14) and (1.16) we get

Proposition 1.2. Under the bijection S(zn) H CO (B1) of prop.1.1

the Frechet topology of S (induced by (1.12)) is equilent to that

of COOO(B1) as a subspace of COO(B?), induced by (with Ck=Ck(B?))

(1.17) IIvII k = Max{IIv(a)]IL°°(B)

: jalsk} , k=0,1,2.... .

4.2. Distributions on S-manifolds 123

As a consequence of prop.1.2 a temperate distribution uE S'

=S'(1n) defines a distribution vE D'(B1) , by (v,V)=(u,cp) , whereyE D(In), V(y)=q(x). We get an injective map S'(&n)->D'(BI). Actu-

ally, v defines a functional on the subspace CO"O(B1)C C00 (Bc1)

extending to w:COO(B?)-T , by the Hahn-Banach theorem. More precis-

ely,v:C000

(B1)-'T is bounded with respect to some Ilvil k Then Hahn-

Banach, applied for the B-space Ck(B7) gives a continuous functio-

nal on Ck whose restriction to COO(B?) gives the desired w

In turn we may interpret w as a distribution zE D'(1Rn) with

supp z C B 1 by setting (z , (p) _(w, c)(Bc) , qE D(e). Thus v may be ob-tained as restriction v=z1B1 of a distribution zE D'(&n). Vice ver-

sa, a restriction v=zIB1 of ZED' (,n) clearly transforms to uE S'.

The last arguments may be repeated for a general vE CO"O(S)C

C (fl0 ): The space of continuous linear functionals on CO (SZ) under

the Frechet topology ofC00(SZ°) coincides with the space DA(11) of

restrictions z I Sz of zE D' (SZ, ) , with any Co manifold Sz' M Sz° .

Then we define S'(SZ), the space of temperate distributions

on Sz by setting S'(c)=D°(SZ). We will work on some details in sec.2.

Later on we will introduce ipdo's on S-manifolds.

2. Distributions on S-manifolds; manifolds with conical ends.

We return to the discussion of S-manifolds as interior n of

a manifold with boundary c° . In order to make the space C00O(St)

look like S(n) we will use a special kind of chart only. First,

instead of relating the charts of n° to coordinates defined in

open subsets of a half space, we use charts (for Sz°) of the form

(2.1) of :U° - of (U°) C Bi

with a homeomorphism of between the open sets U° C sz° and oi' (U°)Then an S-admissible chart of sz is of the form

(2.2) w:U - w(U) C &n , where U=U°flsz , w= s-1.(0 1U)

with an sz°-chart {U0,0} as in (2.1), and s- =t of (1.5) . The S-

structure on Sz is induced by an atlas of S-admissible charts.

An interior chart UCC Sz of the noncompact manifold Sz is S-

admissible, since we may define U°=U, of =so (o. A general chart w:U

-.w(U)C ien is S-admissible if and only if the map s,w:U- B1 extends

to a diffeomorphism of :U° - V° between open subsets where UC U° CSz° , sow(U)C V° C B1 , and where U and so w(U) are dense in U° , V1,


respectively, and U°n sz=U. For a given S-admissible UC sz the chart

U° C fe is given as U° = U U (Uen ac° ), with the closure U0 in n' ofU . In particular, {U°,&} is uniquely determined by {U,w} .

Note the 'shape' of a neighbourhood of an infinite boundary

point p°E ast° in 'admissible coordinates': It contains a 'lense'

{ ly-y° l<e , I y < 1 } , with jy° j =1 , y° image of p°, in B1-coordinatesof sz°. With s-1 this goes to NE={j -y°kss}. After a rotation as-

sume y°_(1,0,...,0). The neighbourhood system NE={(1-{x2+ ss2}

is equivalent to ME={1 sa, se}, where x=(x,,x ). ME is the(x) I

intersection of the solid hyperboloids

(2.3) Xi 2 2t%' (1+p2) , Xi >0 , %= 2' E F E

with p=lx 1, and,

(2.4) p2s62 (1+xi2 ) , 6=tea(fig.2.1). In other words, the set NE is contained in cut-off cone

(2.5) CC yo ,T1={xE In: IxI>l IXI

Y° 1s11 } , 11>0

where we may choose r1==1(E) 40 as e-0 . Vi Ice versa, any set Na con-tains a CCy°,,, , for a smaller 1>0. This describes the neighbour-

hoods of our infinite points. The sets (2.5) form a base at p° .

An S-admissible coordinate transform w,X-I:x(U n V)->w(U n v),

for S-admissible charts w:U- In , x:V' In , must be of the form

( 2 . 6 ) x=w,X-1=s-1°x° , s , with x° = w ° °X° -I :X° (U° n V ° ) - of (U° n V°)

a diffeomorphism between open subsets of B1For investigation of maps of this form we focus on the Jaco-

bian matrix P(x)=ay)/ax=((ay,/axl))j,1=1,...,n of a global coordina-

te transform q=t°rp°s, t=s-I with some diffeomorphism W:Bi HBi .Write I=((6jl)), pqT=((pjgl)), for p,qE In, (j=row, 1=col.-index):

(2.7) at/ax --[y(I+t(x)t(x)T) , as/ax = x)(I-s(X)s(X)T)

It follows that

P(x)=((x)[ V(s(x))})-I(I+t(V(s(X)))t(v(s(x)))T)T(I-s(x)s(x)T)

with 111=arp/as(s(x)) . Here A.(x)=y = 1 , hence

(2.8) P(x) _(1Si )1/2(I+t('V(s))t('U(s))T)11J(s)(I-ssT)


Taking determinants we get

(2.9) JT(x) = Idet P(x)I = II-(s )3/2Idet'(s)I

using that det(I+t(x)t(x)T)=1+t(x)2=11x2. Therefore is (ex-1 2

tends to) a smooth positive function on B1. Moreover, 11-ss=

1 distance(s,I) E C'(Bc) (with t=aB?) follows, ause1+ vp gistance(V(s),1') 1 1

p: B1 « B1 must map r to r . (Near s° with Is°1=1 introduce local

coordinates v=(v1,...,vn), setting vl=dist(s,1'), with coordinates

2,...,vn of IS on 1'. In such coordinates the diffeomorphism V is

represented by an n-tuple of funtions V.- , where dist(V(s(v)),r)=

(v). Clearly V1-=0 as v1(s)=0. Thus ''1-/v1E C , and it isclear that this quotient is >0 , since yr is a diffeomorphism.)

This result may be extended, as follows.

Proposition 2.1. The function Q(s)=Pos-1=P(t)=(acp/ax)(-"), sE

B1, extends to a Cco(B?)-function, nonsingular for all sE B1

Proof. First of all, the scalar factor up front in (2.8) is C"O(B1)

and the matrix q(s)(1-ssT) has entries in C"O(B1). Introducing an

orthonormal base SO ,...,s`" of &n varying smoothly with sE B1 near

some point of r such that s°=Is , we may write

(2.10) I- ssT = (1-Is12)s°s°T + L1 sjsjT

hence

(2.11) W(s)(I- ssT) = (1-Is12)(w(s)s° )s°T + _i(tY(s)s. JT

Applying the matrix t(V(s))t(V(s))T=1-s

V(s)?Q(s)T to the termsV -7

at right of (2.11), the first term gives aCC'(B1)-matrix,

using

that 11,,(x)2 E COO(B1). The same is true for the other terms, since

q(s) must map tangential vectors at sE t to tangential vectors at

p(s)E r. or, p(s).W(s)s3(s)=0 as IsH=1, since p(s) is normal to r

at V(s) . Also, q(s) = ('(s).'(s)s3(s))p(s) is Coo near IsI=1.

Thus the quotient 1-V s also is Coo near IsI=1 . Q.E.D.

There exists a finite S-admissible atlas. Indeed we may choo-

se a finite atlas of the compact c° and construct the cor-

responding charts {U3. ,w.} of (2.2). Moreover, for a partition of

unity P,9=1 subordinated to (with supp XOC UO) a correspon-3 J 7 J 1

ding partition subordinated to {u wj} is defined by %j=(X9j )osj, 3

Notice that, in admissible coordinates, we have SSO,(1,1,1),0'

i.e.,%a)=0((x)-Ial) for all a, (T extended 0 outside a(U) in3 3

len). The same is true for the restriction x=x°In of a C (St°)-func-


tion x° with support in a single chart U''. We will call such {X

and x an S-admissible partition of unity, and cut-off function,

respectively. (Often we drop "S-admissible" for "admissible").

With an admissible partition {Xj} and finite atlas {Uj,wj}

we define a Riemannian metric on st by setting

(2.12) ds2 = 71j%jdx2 ,

with the Euclidian metric dx2 of wj(Uj)C ten, in each summand.

From prop.2.1 it is evident that an admissible coordinate

change cp:Rn -10 converts the Euclidean metric dx2 of Rn into a

metric of the form 2 hjl((s(x))dxjdxl where hjlE Cm(B?) , and the

matrix ((hjk)) is defined and positive definite still at asz°. Thus

in admissible coordinates, the metric tensor ((gjl)) of (2.12) has

the same properties - gjkos-1 extends as a Cc'-function to the infi-

nite boundary of the chart, and the matrix is >0 there.

Proposition 2.2 For each infinite boundary point p° and admissi-

ble chart p°E in there exists a cut-off-cone CC Y,

,)C w(U)

y°=w°(p°) such that (i) the Riemannian distance d(p,q) of the

metric (2.12) is equivalent to the Euclidean metric in CCy"Yl , in

the sense that c , x-x° l s d (p, p° ) s c x-x° l for all x, x° E CCyo

and corresponding p,p°E sz ; (ii) the Riemann space fl (with metric

(2.12)) is complete; (iii) after a linear transform the metric is

approximately Euclidean, insofar as gjl(x)=gjl+sjl,

=const. in CCy° , with E jl->0 as (iv) gs 1 is C , andthe metric is positive definite, even at an' .

A complete Riemannian space n with above properties - e.g.,

st admits a compactification 9?°, a compact Co°-manifold with bounda-

ry; a finite atlas of n exists with charts derived from charts of

st° , in the sense of (2.1),(2.2); in a neighbourhood of a point of

an' =n°\n the metric of sz has the properties (i),(iii),(iv) of prop.

2.2 - will be referred to as a space with conical ends. Note that

this term is used synonymously to 'manifold with S-structure' or

'S-manifold', 'S-space', in the following sense: An S-manifold be-

comes a space with conical ends by introducing a metric of the

form (2.12) (or any other Riemannian metric with properties (i)-

(iv)of prop.2.2 - called a conical metric). Vice versa, a space

with conical ends is just an S-manifold together with a conical

metric, not necessarily of the form (2.12). In the following we

always assume a space with conical ends, and then choose the sur-


face measure dµ=dS (=det((g]k))1/2)dx) of the conical metric ds2=

= jjkdxjdxk as for our distributions and Sobolev spaces

In particular we introduce the Hilbert space H=L2(SZ,dµ) .

As a side remark it may be mentioned that the quasi-Euclidean

metric ds2 has a corresponding Beltrami-Laplace operator A . Using

the comparison triple {Sz,1-A,dµ} in the sense of [C2],V one may

generate L2-comparison algebras with complex-valued symbol, con-

taining a core of Wo's of the kind to be studied below.

Remark. For a Riemannian manifold sz introduce d(x)=d(x,x°) , xE S1,

and d(%)=sup{d(x):xE X}, X a subset of Sz , with the geodesic dis-

tance d(x,x°) and a fixed point x° . An end Sz' of a is defined as

a subdomain of Sz with compact as1' and d(i1)=- such that no decompo-sition n' =SZ, Uiz exists with a, M bounded and d (Sk, ) =d (SL )=w (thinkof the two "ends" of a cylinder). (Actually, two such Sk`, called

S1' , all , define the same end if c' \SZ" and Sz"\SZ` are bounded.)Such an end will be called conical if (a suitable) Sk' is isometric

to an end of an S-manifold, as above.

It is possible that some ends of a Riemannian space Sz are

conical, others are not. Also, one may consider cases like the

cone x1=V(x2+...+xn , x1>0 , having the concial end{x1>11, but

a conical tip {x1<1}. Our theory, below, will not apply then. But

the conical end and conical tip may be separated, using algebra

surgery, similar to that in [Cz],VIII, as not to be discussed here

(cf. the problems of chapters I, II, III, IV).

We summarize, below, also discussing some trivial additions:

a)Temperate distributions and rapidly decreasing functions

are introduced for a noncompact manifold S1 which is the interior

of a compact manifold 91° with boundary, where acz°=0 is permitted,

giving the special case of a compact s . We define S=S(S1)=CO00 (S1° ),

S ' =S' (Sk) =D° ( c ) _{ uE D' ( c ) : u=v Sk with vE D' (Sk` ), with an S1' = }b) S(Sk) and S'(n) assume their conventional looks only in

special coordinates: An S-structure, making 91 an S-manifold is in-

troduced by declaring certain charts, coordinates, coordinate chan-

ges etc. as admissible. Admissible charts, atlantes, cut-off's,

partitions of unity, all are obtained from charts,..., of nO,

using the map s(x)=- : In-B1 of (2.1), (2.2), ... .

c) A neighbourhood base of an infinite boundary point in

admissible coordinates is of the form (2.5).

d) An S-manifold sk possesses a distinguished type of Rieman-

nian metric ds , called conical metric. In admissible coordinates2


ds2 has properties (i)-(iv) listed in prop.2.2. A manifold with

conical ends is defined as an S-manifold c with distinguished

conical metric. Write d(p,q) for the geodesic distance, and d(p)=

=d(p,p°) with a fixed point p° and variable p in the following.

The distinguished measure dµ on S2 is introduced as the surface

measure of the distinguished conical metric. We have dµ=x(s(x))dx

in admissible charts, with dx in local coordinates and KE C,(U°).

Also, globally, we have dp.=(d(p))n-ldv with a positive Co measure

dv on the compact S2° , and (d (p)) = 1 +d (p )1e) We have

(2.13) S(S2)= {uE C00(St): (Xu)ow lE S(len) for all w,X} ,

and

(2.14) S'(S2) = {uE D'(S2): (?u). (x lE S'(ien) for all w,X} ,

where, in each case, w:U-w(U)C In and X denote an admissible chart

and subordinated cut-off function (e.g., supp ?C U)

f) The space

(2.15) LpI01(a) = {uE Llloc(n): d(p)-kuE L1(SZ,dµ) for some k }

is naturally imbedded in S'(S2) by

(2.16) (u,) = J u(pdµ , TE S(n)92

g) Define L(S2) as the space of differential expressionsL:C00 (11)->Coo(S2) such that in every distinguished chart w:U->w(U)C In

L is represented by a PDE Maaaa with as of polynomial growth -that

is, Xaaow l E T(&n), with T(In) of[C,],p.28, the space of S'-multi-

pliers (cf. [Schwi]) for every admissible X, , supp XC U. Then

L:S'(SZ) -> S'(S2) for all LE L , and such L is a contionuous map.

h) We have

(2.17) S(a) = {uE C00(92): Lu=O((d(p)) -k) for all LE L , ka0}

i) The topology of S(S2) is the Frechet topology of C00(92°)

it may be generated by all seminorms II(d(p))kLull00

where k=0,1,..,L

and where L ranges over L(c) (or a suitable countable subset). In

S'(S2) we use the topology of weak convergence. (Note that the

inductive limit topology based on S'=USAS is available as well.)

4.3. Coordinate invariance of operators 129

3. Coordinate invariance of pseudodifferential operators.

It is trivial that Apdo's on in transform into pdo's on in

under a linear change of coordinates, of the form x=mx'+p , where

PE &n and m=((mjk))j,k=1,..,n is invertible, mjkE &

That is, if Au=a(M1,Mr,D)u=f, where u,f E S(ien) , a E ST , and

u(mx'+p)=v(x') , f(mx'+p)=g(x') , then we get

(3.1)

where ImI=Idet ml, for a moment. With =m t=(m 1)t, we get

(3.2) g(x')=

ff

Joxlfol t 'MY')

Or, g=b(M1,Mr,D)v, where a(mx+p,my+p,m generally is

in the same symbol class as

For more general local coordinate changes and local ivdo's a

we refer to Hoermander [Hr2]. A very elegant proof may be found in

[Fr3]; its idea seems due to Kuranishi (unpublished?). Using this

technique Schrohe [Schr3] proved a result for a class of global

transforms and pdo's on In of our general kind.

Here we will use the same 'Kuranishi trick' again, for a sub-

collection of OpST. We only admit coordinate transforms T:1n-1n

with the property that sog0s-1= V extends to a diffeomorphism

B +.B1 . Clearly then the diffeomorphism T:2n-jkn extends to a hom-

eomorphism En+ En. The homeomorphism may be used to carry over the

manifold structure of Bi to the compactification in of &n . Thus

we may regard En as a compact manifold with boundary. The above

type of coordinate transform is precisely the class of homeomor-

phisms EnH En preserving this manifold structure.

Given such g:1n-2n and a 'pdo A=a(M1,Mr,D) , aE SSm,P,b let

again f=Au, withffu,fE S. Let g(x)=f(g(x)) , v(x)=u(g(x)). We get

(3.3) g(x)=fob

where Jg(y)=Idet((aTj/ayk))l .

Let us split the expression at right of (3.3) into two parts

by inserting a partition 1=x(s(x)-s(y))+w(s(x)-s(y)) under the in-

tegral signs, where co(z)=0 for lzIse, w(z)=1 for jzj2-2e>0. Writing

g=g1+g2 , correspondingly, we first consider g2. First look at g2

in the old coordinates. With f2(x)=82(6(x)), 6(x)=g 1(x), we get

(3.4) f2(x)=JRiJRye1(x-Y)w(s(6(x))-s(6(Y)))a(x,Y,)u(Y)


With the inverse function t= l=s.0.s-l we have s.O=l s , hence

(3.5) f2(x)=fo1

We use the identity The inte-

grand vanishes near x=y. An N-fold partial integration gives

f2(x)=fO1 foy.il(x))

(3.6)

with aN(x,Y,) = (-A

It is clear that this partial integration is legal, similar as the

ones performed earlier. The function 1, as a diffeomorphism B1+B1,

satisfies an inequality It(x)-t(y)Iaplx-yI , with some p>0 . Hence

the integrand in (3.6) vanishes forp

sp . In other words, we may assume that Is(x)-s(y)Iap .

Observe that s-l(x)=t(x)= x , where [x]=V . We get

(3.7) T(x) = ((tilxl)) =l 3((6il+titl))

=(I+t)(t)

implying T(x) to be real symmetric, T(x)a , xE B1. Notice that1XI

t(x)-t(y) = fd-r UT-x))) ={ fdtT(x+t(y-x)}(y-x) = S(y-x),

where SaMin{ th , [ y] 1. We get 1 =(t (x)) , hence

(3.8)It(x)-t(y)Ia(Min{(t(x)),(t(y))})Ix-yl, x,yE B1 , or,

Ix Y1Min{(x),(Y)} s(x)1s(Y)I ,

x,y E&n

Thus we may assume that

(3.9) Ix-YI-2N 5 (g)2NMax{X2N(),x2N(Y)}

, fi(t)= I

Assuming aE SS., P'6 , p1>O, and N sufficiently large the

integrals f dgf dy in (3.6) may be interchanged, for

f2(x) = f9iyu(y)k2(x,y) , with integral kernel

(3.10)

fk2(x,Y)= Ix-YI

where N is arbitrary (sufficiently large), and where fd exists,

since the i-order of aN is m1-2Np1 < -n , as N gets large.


Proposition 3.1. The kernel k2(x,y) of (3.10) as well as its

transform k2(c(x),cp(y))J(P

(y) = k2IT(x,y) under the coordinate

change x -* T(x) , are inS(&2n) . Hence the operator K2 defined

by K2u=f2 , and its transform under g are qdo's of order

Proof. The integral in (3.10)2 may be written as a4(x,y,x-y).

Clearly vaN(x,y,z)E Coo(&3n\{z=0}). Indeed, aN(x,y,z)=

(- I2M)aN+M(x,y,z) where aN+M decays better as M increases, so

aN+M admits more and more z-derivatives, as M-- . Similarly,

(3.11) aUdyaZaN(x,Y,z)=O((x)m`(Y) ) , as Izk s>0, x,y,zE &n,

with 'O(.)-constant' depending on N, but not on the mj. Therefore,

(3.12) aaayaN(x,Y,x-Y)=O((x)m:(Y)m') , as x,yE &n , Ix-YIae>0

For the factor k3 in front of the integral in (3.10)2 we get

(3.13) aaayk3(x,y) = O(?.21(x-y) Max{?.11(x),?J1(y)})

where, of course, ri=3N may be choosen arbitrarily large. Here

(3.14) Max{X1(x),X1(y)}=?1(x)Max{1,( F1} s kY'(x)(x-y)11

using the well known inequality. Accordingly,

(3.15) aXayk3(x,y) = O(X1(x)X1(x-y)), for all ri>0

Combining (3.12) and (3.15) we indeed get k2E S(&2n)

The remainder of prop.3.1 then is a consequence of prop.1.1 and

prop.2.1 The map (x,y)-(s(x),s(y)) takes S(12n) to CO-(BixB?),

which is preserved by (x,y)-(lp(x),i(y)) . The Jacobian determinant

defines a C00(RO)-function J(os-1 , by prop.2.1. Q.E.D.

After dealing with the part of A=a(M1,Mr,D) belonging to

w(s(x)-s(y)) we now turn to the other part,

(3.16) gi(x) = foI (Y)v(Y)

where X=X(s(x)-s(y))=0 for js(x)-s(y)ja2E . Write

1

(3.17g(x)-(P(Y)= 0 di{atcP(Y+i (x-Y))}= M(x,Y)(x-Y)

where M(x,y) = dx(aq)/ax)(y+i(x-y))0

If x and y are close then clearly M(x,y)zt aye/ax which is an inver-

tible matrix. We have (T(x)-cp(y))= Then an inte-


gral substitution Mt(x,y) , for fixed x,y, close together, may

be used to convert (3.16) into the form of a ipdo-integral again.

Having employed our partition we may assume now that

js(x)-s(y)js2s , where s>0 is arbitrary. The question is whether

this condition is sufficient to guarantee a global such integral

substitution, valid for all x,y, with js(x)-s(y)Is2E .

Write M of (3.17) as M(x,y)= dtQ(s(y+t(x-y)) with Q(s)=0

a(p/ax( s ) E C'(B1) (by prop.2.1). The curve {s(y+t(x-y)):t E &}=T

is a half-ellipse with center 0 and vertices atX_Y

and 7

z= vector to the point of the line {y+t(x-y)} closest to

easily seen. The arc {s(y+t(x-y)):Ostsl} connects s(y) and s(x) on

r. It is contained in the ball Is-2(s(x)+s(y))Is2ls(x)-s(y)Iss.

The COO(BI)-function Q(s) is uniformly continuous and invertible

Thus indeed E may be chosen small to insure invertibility of M

for all (x,y) as js(x)-s(y)ks2E.

Carrying out the integral substitution as indicated yields

(3.18) (x,y, )v(Y) , with

a detM(x,y) a(y(x),T(Y),M X(s()-s(Y)) .

The following observation about the new symbol a" is useful.

Proposition 3.2. For any function b(s)E COO(B?) the composition

c=b,s is a symbol in SS0(1,1,1),0 More precisely, we have

(3.19) c(a)(x) = 0((x)-ja1) , for all a .

Indeed, this is a consequence of the chain rule, and the

fact that s(a)(x)=O((x)-kaI) , for all jal .

Looking at the -derivatives of a of (3.18) it appears that

the parameter pl remains unchanged under our coordinate transform.

An x- (or y-) derivative may land on the M t(x,y) inside the

t-argument of a , or anywhere else. In the first case we get a

factor 0 x1 1-p,(( ) in addition to the already existing ones.

In the other cases we get a factor 0((x) -p: (l;) S ) or 0((x) -(All this for an x-derivative).

It follows that a E SSm,p,, with 6°= Max{S,1-pi}. To verify

this we use prop.2.1 when differentiating for the x in g(x): acp/ax

is of the form needed in prop.3.2. Also, bounds of the form

0<cs(q(x))/(x)sC are easily derived. Summarizing:

Theorem 3.3. Let a E SSm,p,6 , with p.>O , and pjsl , j=1,2,3.


Assume that cp:Rn->1¢n with inverse 9:&n-&n has the property thats,(P,s- 1

(with s(x) (X = ) , a map extends to a diffeo-VTI T-7

morphism B?H B1 of the closed unit ball B1. Then the linear opera-

tor A" = T(PIATT , with A=a(Mi,Mr,D) and T u(x)=(u.T)(x)=u(T(x)), u

E S, is a pdo again with symbol a E SSm,P,

66 =max{6,1-p.}. Ups

to an additional term in S(22n) the symbol a" is given by (3.18).

Usually we will tend to apply thm.3.3 for operators of the

form a(x,D) , a(Mr,D) , a(Mw,D) , with aE ih . These are special

cases of A=a(M1,Mr,D) in thm.3.3. However, thm.3.3 then will give

an A =a (M1,MrD) which must be converted to b(x,D), c(Mr,D),

e(Mw,D), using I,thm.6.2, assuming that S°<p. - i.e., 6<p, and

1-p. <p. ap. >2. We therefore have

Corollary 3.4. Let aE tphm1 p s with pz >O , p. >2 , S<p. . Then thecoordinate transform of thm.3.3 takes each of the operators a(x,D)

a(Mr,D) , a(Mw,D) into an operator of the same form with symbol

in 111hm, p , 6A , S° = Max{ O , 1-p. } . The new symbol, up to terms inm'<m , is given by a with a" of (3.18).

Problems. 1) For n=1, consider the transform t-x=q(t)=et, a diffeo-

morphism & .&+=(0,o) (and a group isomorphism). 'p(t) and its in-

verse t=6(x)=log x ) do not satisfy the cdn's of thm.3.3, but they

take A=a(M1,Mr,D) to a i,do A if only supp KxKxl, KCC P

( KCC R+ in case g). 2) The Mellin transform is defined as M=FTT

with the Fourier transform F and Tu(x)=u(cp(x)), c of pbm.1. Exp-

ress M and M like Fourier integrals. Show that M is the Fourier-

Plancherel transform of the group &+: Define c=a © b =fa(y)b(y)ay,

for a,bE L'(I,dx), and get Mc=(Ma)(Mb). 3) Consider R2*=R2\{0}as

Riemann space with metric ds2=y,(Sjk+ni nk)dxi dxk , ni=xj/Ixl (a co-

ne Z isometric to {y=jxj}C 13 ={(x.,x2,y)} ). Show that this cone

is mapped conformally onto the cylinder {-oo<t<oo, Os9s2n} with me-

tric 2dt2+d62 by the map x=(xl,x2)- (t,8), where t=loglxl, O=arg x

arg x = arc tan(x2/x1). 4) Use the diffeomorphism between cone and

cylinder of pbm.3 to install a natural class of pdo's on Z. In par-

ticular, the pdo's should be the global coordinate transforms of

the "cylinder pdo's" introduced in II,3,pbms 1,2. They should be

"local tpdo's", in the sense ?.At is a t,do for X,pE C00(Z). Try fora concept of K-parametrix, for md-elliptic operators to be defined

especially with an operator-valued symbol at the conical tip.


4. Pseudodifferential operators on S-manifolds.

We start our discussion with a generalization of I, thm.3.3.

Proposition 4.1. For the special S-manifold S2=1Qn let 0,?. be admis-

sible cut-off functions such that0°=00s-1

and 7v°=X s-1 , extended

to B1, have disjoint supports. Then, for any AE Op1Uhm we have 0A?.

E Opph-00 , an integral operator with kernel in S(12n).

Proof. By I, thm.3.3 the distribution kernel k(x,y) of A has singu-

lar support at x=y only, thus the kernel K(x,y)=k(x,y)0(x)X(y) of

0A?, is C'(3k2n). In fact, k equals a function in S for xE KCC &n

and large lyl, and vice versa. Moreover, looking at I,(3.15),

using that Ixkklx-yl. 1y1"Ix-y1 , as xE supp 0 , yE supp k ,

IxI , lye >> 1 , we indeed find that xyy6aadyx(x,y) = 0(1), for

all a,(3,y,S - i.e., xE S(12n) , q.e.d.Now assume that 92 is a Riemann space with conical ends, in-

terior of S2° , a compact manifold with boundary, as in sec.2. We

will use admissible cut-off functions now, defined as 0=01os, with

supp 01 C U° with a chart o :U° -w° (U° )C B1 of S2° , and s of (1 .5) .The above relation between the cut-off 0 and the chart U

will be expressed by writing 0 ® U . The same notation will be

used for another relation: Writing

(4.1) 0 (> O X , or , 7v m 0indicates that 0=0°,s , X=X°,s , where 00 , X° are cut-off's of

S2° with support in a chart U° , and 00=1 near supp X° .

The following is evident: Just look at s? instead of sz

Proposition 4.2. For a given admissible cut-off function 6-:* U we

may construct an infinite sequence 0=0 0<10 01 ® 02 ® ...<XJ

U.

First define the class e =LS-- LS-,,(S2) of integral operators

(4.2) Ku(x) = J k(x,y)u(y)dµ(y) , uE S(S2)

(or uE S' (92), with a distribution integral) with kernel k(x,y)E

S(S2xS2) = COOO(S2xc2) (note the remark above (2.12)). We will regard

dt as the class of gdo's of order --, as in case of S2=ien .

A pseudodifferential operator A on n then is defined as a

continuous linear operator A:S(S2)-> S(92) such that, given any admis-

sible cut-off function X and chart w:U-Ten , with 7`GQ U, there ex-ists A,=a,(x,D)E Optphm p S , KXE r with as x w(U), and

(4.3) A(Xu) = KXu + (AX((Xu)0w I))°w , uE S(S2)

where we assume m,p,S independent of k and w:U->&n .

4.4. Operators on S-manifolds 135

Note, we assume m=(m. ,mz) , p=(p.,pz) , S independent of X ,U.As usual, m1E &, Ospjsl, OsOsp., but for a local calculus of ipdo's

and coordinate invariance we require pj>0, 1-pisO<p.. The class of

such yxlo's for a given m,p,b will be denoted i LSm,p,O. Write LSm

=U LSm,p,s, union over p,b with p>O, 1-p.sO<p., enabling local cal-

culus and coordinate change. Write LSO.,p1s=U LSm,p,s , U LSD LSOO

= LS. We shall see: =LS-ooflLSm. The special case p=e=(1,1), O=0

will be focus of interest. We thus define LCm LSm,e,O'LC. =U LCm.

Note that (4.3) is of strictly local nature, insofar as ku

as well as (AX((Xu)ow 1)ow have support in a single admissible

chart U. Thus it is practical to regard w in (4.3) as an identifi-

cation of the points of U with those of w(U) - i.e., as the iden-

tity map, which may be omitted in writing. Then (4.3) reads

(4.3')

Note, we also get

(4.3")

A(Xu) = AX(a.u) .

A(au) = Kku + k° AX(Xu) ,

with any other admissible cut-off ? o , %< 0 ko Q U . Indeed, we get

X< 0U9 with the admissible chart Uo={Xo=1}into U =wI Uo . By defin-

ition we get (4.3'), possibly with another but ak=O for

x Uo, hence (4.3") follows. On the other hand, if (4.3") holds

for some X Oko Q U we may use the symbol in

place of to get (4.3'), for equivalence (4.3') ca(4.3")

From now on we always assume AE LS - i.e., O<pj, 1-pi2-.6<p.

If we have both XQ U, 0<) U, for two cut-off's X,O , then

A),O=a),(x,D)XO+KXO=a0(x,D)XO+KOX, showing that (ak-a0)(x,D)OXE(f .If X(p)*O, O(p)*O , for some pE ft° , then prop.4.1 implies

existence of an admissible cut-off x with x=1 near p such that

(4.4) S(12n)

Clearly (4.4) expresses a uniqueness property of the local symbol.

Note that LSm,p,5(&n)=OpVhm,p,s , assuming p.>O, 1-p.sb<p.

First let A=a(x,D)E Opiph. For X.ZJ?o Q U , w=idI U , construct%QXo<CU. Then (1-Xo)A?E(`',by prop.4.1, hence (4.3"). Similarly

for a general w extending to an admissible coordinate transformNn .,in

. One then must use cor.3.4. For general admissible w:U-&n

first cover UC 2n by a finite collection U Uj such that w IUj is

approximately linear, hence wlUj extends to an admissible map In

ea&n . Then, with an admissible partition 1=Eµj , p,j<EoUj , let

%j=%Rj . Get AJ%j=ak (x,D)k j+KX , and sum over j, for (3.3). For7 7


AE LS" p,6(1Qn) choose U=& n, w=id. to show that AE Vh m,p,6,For general sa we clearly get LS-,O(sz)C LSm,p,6(a). For general

AE LSm and finite admissible partition of unity EX j=1, % <E)Uj get

(4.5) Au=2 AXju=y, XA), (Xju)+Ku , uE S , KE LS-00 aj<F)k I<E)Uj

just by repeated application of (4.3). Vice versa, a yxlo AE LSm

may be constructed by assuming AX =AjE OplyIhmIP.b in (4.5) as arbi-7

trary i,do's. Before confirming this let us prove the following.

Proposition 4.3. For admissible cut-off's 0=0°os , ?=1,.°os, if

(supp 8° )fl(supp x, )=Jd, then OA), E LS-00 , for every AE LSm .

Proof. Construct a finite admissible atlas and partition of unity

Eoj=1 , %.<10 U. , wj:Uj_wj(Uj)C In such that for each j either

Ujfl supp or U9fl supp 0° = jd . With this partition write

OA?. 29(A), 29(0A), A +K )k

The sum may be extended over j with Oflsupp o, 0, else the fac-7

for 0X vanishes. In the left-over terms we get sinceJ

then k jk = 0 , so that 6AX = E0Kj). E LS-00, q.e.d.For {Uj,wj}, A. j, as above, and general AjE OpVhm,p,S write

(4.6) A =3 3

Let us prove that A E LSm, p , s . We trivially get A: S (SZ) ->S (c) conti-nuous. To confirm (4.3) let A., X. be as in (4.3"). For each j the

product U j is an admissible cut-off with supp(.Aj)C Ufl Uj . The

coordinate transform w o : w . (Ufl u w(Ufl u.) is of the formx=wow 1 = s-1x° os with a7diffeomorphism x° :w.( U° fl u .) -* of (U° fl u .)

7 0 J J 7We may assume the partition wj refined so far that x° differs

little from a linear map. Thus we may assume x° extendable to a

diffeomorphism B .Bi . Hence x extends to a diffeomorphism In E-,&n

Thm.3.3 (or cor.3.4) may be applied to transform Aj to the coordi-

nates of the chart U , resulting in a 'do Aj° E OpPhm,p,S . To be

precise, in the formulation of (4.2) we have

(4.7) (A1((X), ju) "wjl))owj = (Aj°((AAju)oW ))ow

with properly extended diffeomorphism wowjl , but (%Xju)owil has

its support within wj(Ufl u.). Using prop.4.1, write the right hand

side of (4.7) as (0 1Aj° ((..ju)ow 1))ow + Kj K1E LS-.(&n), withan admissible cut-off 0j, supp 0 j C U. By prop.4.2 the integral

4.4. Operators on S-manifolds 137

operator Kj has a kernel of the form v1(x)u2(y)kj(x,y), with admis-

sible cut-off's vj, supp v.C U. Returning to the notation of (4.3)

-(4.3") it follows that indeed each (X9AXjX) transforms to a term

of the form (4.3"), proving that A of (4.6) belongs to LSm.p.S.

Clearly a differential expression LE L(n), L(c) of sec.2,(g)

defines a pdo LE LS , for some N (the order of the

expression) provided that the local coefficients as in admissible

coordinates satisfy aa(P)(x)=O((x)f-P= I(3I) for all a,3As for differential operators a global symbol in general is

not defined, for an AE LSm,P,s although we have a well defined

local symbol ax is not unique, since

(i) any term c(x,l;)E S(12n) may be added, and (ii) is more

or less arbitrary for x outside supp X. For a differential opera-

tor LE L(n) the local symbol coincides with the polynomial a(x,i;)=

where in local coordinates L=Eaa(x)Da, D=iax

It is clear that we will seek properties of operators in LS

similar to those of Opo in earlier chapters. We summarize corres-

ponding facts in thm.4.4. Proofs are straight extensions, and will

not be discussed in detail.

Theorem 4.4. The classes LSmP.S

for form algebras for

each given p?0, 1-pi <S<pj . More generally, we have

(4.8) LSm,P,S.LSm',p,S C LSm+m',P,S,LCm.LCm,C LCm+m'

.

One finds that LS-,Ois an ideal of LS0 , LSOO, LSmPS , m=0,0

.

All spaces are invariant under the involution "*" defined by the

Hilbert space adjoint (with respect to (u,v) f 1vdµ ).

Calculus of 'ado's holds locally: For AE LSm,P.SC LSm , BE

LSm,,P.S the local symbol cx of C=ABE LSm+m,,P,S is expressed by

the asymptotic Leibniz formula (of near any point pE St°

with x (p) x0. The symbols* ax,ax of A E LSm.P.SC LS and its adjoint

A are related by an asymptotic I,(5.1)3 (with x= ). In detail,for

admissible X.J?o ®U, near pE c° with X(p)x0, a....... of (4.3"),get

(4.9) cx=

9

i6

a )bx(e) ax=

6

Kl;9

{xax}(9) (mod h-,O)

where "f=g (mod (Z) near p" means (f-g)xE(; with a cut-off x near p.

Local symbols are unique: ax=aO (mod()near X(p)xO*O(p), cf. (4.4).

The left multiplying representation of (4.3") may be replaced

by Weyl (or right multiplying) representation without changing LS.

As next important point we shall look into existence of a

parametrix, for A E LS . Here A,BE LS are said to be parametrices


of each other if AB-1 , BA-1 E LS-00 . This problem is quickly res-

olved, using our local parametrices of 11,3. For a finite admissi-

ble partition 1=1k let as in prop.4.2.Given AE LS write AR=K j+A11 µ , µ=X. . Assume Ap to admit a left K-

parametrix B. with respect to the symbol X. , so that BJA11=A..+Kj

Write Aµ Aj . Notice we also have .X2A . = X, X. + K j , sinceE Op,ph-co , by prop. 1 . 1 . As a consequence, ? BjXjA =

.B .X .A .k3 + .1B .??A(1-X3 ) _ c +K , using prop. 4.3 with A=T,?333 J J J J J J J j j JX=1-X.. Assuming all left parametrices B. to belong to the same

m'2:-m, it follows that B=l?.jBj?j E Alm,,P,b is a left

K-parametrix of the operator A E LS . Involving adjoints to possi-

bly convert right into left parametrices, we have proven:

Theorem 4.5. Let AE LSm . For an admissible partition X>,.j

=1 , ?.i 61 ®Uj with admissible cut-offs ej and charts w:Uj-.&n ,

let the local operators A. of (4.3') admit left (right) K-param-j

etrix in 4*m',P,B , (with inverse order m') with respect to Xj ,

for all J. Then A admits a left (right) K-parametrix of order m'

A pdo A E LSm,p,6 C LS is said to be md-elliptic (of order

m) if for every admissible cut-off 6 ®U , w:U -> ien an admissible

chart, there exists Oo (E>6 , Bo o` U , such that the local symbol

a0, is md-elliptic (of order m) with respect to 0 .

Theorem 4.6. An md-elliptic 'do AE LSmp S C LSm admits a parame-

trix B E LS-m,P,6 . Vice versa, if AE LSa,P'6 C LSm admits a K-pa-

rametrix in LS-m,P,b then it is md-elliptic.

This theorem is an immediate consequence of thm.4.5, as far

as sufficiency of the condition is concerned. For necessity one

must use local calculus of pdo's .

Theorem 4.7. Suppose AE LSm,P'OC LSm has the property that, for

some m'a-m and some admissible finite partition 1=EXj , %j<0ej®U. , with admissible ej , wj:U.-in , the local operator A0 of

J

(3.3) is formally md-hypo-elliptic (of inverse order m') with res-

pect to %j. Then A admits a K-parametrix BE LSm,,P's of order m'

This theorem is an evident consequence of thm.4.5.

Problems. 1) The operators B: COO(S' ) -C"O(S' ) of III,4,pbm.1 are ydos

in LS(S'), under proper assumptions on the sequence bj(0) of perio-

dic functions. 2) Consider y,do's A with operator-valued symbol

A(x,i), where for fixed x, , is a pdo in LCo(B), for a

4.5. Order classes and Green inverses 139

smooth compact manifold B. As in 1,6, pbm's 1-4 and 11,3, pbm's 1-

4 such operator should act on S(C) , C=Bx&n , properly defined. We

should get a t,do-calculus, a K-parametrix construction, and, gene-

rally, results analogous to those in the problems mentioned, inclu-

ding those of C in [Cz],VIII. 3) Reflect on a theory of 'ado's on a

more general Riemannian manifold st which has conical and cylindri-

cal ends both, as well as conical tips: In a subdomain sic contai-

ning a single cylindrical end, stc should identify with a neighbour-

hood of the right end of the cylinder of 1,6, pbms.1,2,3 (or of a

more general cylinder MxRk , M a compact manifold). Near a conical

tip nz should identify with a neighbourhood of the tip x=0 of the

cone Z of sec 3,pbms.3,4 (or a more general such cone (0,o)xM, M

compact). There should be an operator-valued symbol at a cylindri-

cal end as well as at a conical tip, but not at a conical end (cf.

[CDg]). 4) Show that the operators A of pbm.2 are tpdo's on the non

compact manifold C=Bx&n insofar as, for cut-off's (o,x with support

in a chart UCC st , we have wAx a ipdo on &n . 5) For a distribution

uE S'(ft) , 12 a manifold with conical ends, define the concepts of

WF(u) - the wave front set - and ZF(u), looking at ZF (Xiu) of

11,6 and VI,7, with an admissible 1=1k , xi <s)Uj , wi:Uj->&n

A wave fron space W(c) may be introduced such that WF(u)C W(st).

Show that W(R) may be interpreted as the cosphere bundle S*(c)

i.e., the bundle of unit spheres in the cotangent space, with

respect to any (conical) Riemannian metric on st . What would be

the corresponding space Z(a) , for ZF(u) ? .

5. Order classes and Green inverses on S-manifolds.

On an S-manifold sZ introduce the weighted Sobolev norms

(5.1) HAS = {lllxiulls} 1/2 uE S' (st) , s=(s ,s2 )E &2

with l=? a given fixed finite admissible partition of unity sub-

ordinated to an admissible atlas {U,}. To be precise, we abbrevia-ted IIxiu]IS=II(xi IIns((), ju)owj1)H1 2 n). The (L2-Sobolev-)

L (Rspace Hs=Hs(s2) consists of all uE S'(c) with finite norms Ilull s

Generally, when writing hulls we imply h1uh1s<oo , i.e., uE Hs .

Again we regard the maps wj as identifications, hence kiu

as functions on &n (extended 0 outside wi (Ui)). Notice that

*(5.2) hlxiuIhs = s Sk.u) , as uE Hs


with the pairing III,(3.28), or, with the inner product of L2(in),

as uE H2s, where Qj=Xjrlss

E Oppc2s . Hence we get

(5.3) 11ulls=(u,P2su) , P2s=DcjQjkj , xj=dµ/dx , for all uE H2s

with the inner product (u,v) f uvdµ of H=L2(n,dt). Clearly P2s is

a ido , P2sE LS2s,e,0 = LC2s , as follows by comparing (5.3) and

(4.6). Moreover, P2s is md-elliptic of order 2s, as confirmed by

looking at the local symbol: At each point it is a finite sum of

transforms of the non-vanishing symbols of Qjkj, all of them local-

ly md-elliptic and nonnegative, so that they cannot cancel each

other, looking at formula (3.18) . It follows that P2sP-2t and

P-2tP2s both are md-elliptic of order 2s-2t, by calculus of pIdols.

From estimates III,3,(3.19),(3.21) we conclude at once that

(5.4) HtC Hs , and 1Iu11ss cs,tllullt , as s s t , uE Ht .

Also one at once confirms III,(vii), i.e.,

(5.5)S(cz)=

SHS(n), S'(c) = UHS(c) ,

as well as formulas similar to (3.22) through (3.27) in ch.III

In detail, we have

(5.6) Hs={uE S' : kiu E Hs(in) , j=1,...,n} .

This validates III,2,(i)-(v),(vii) for Hs(n), just as for n=ten

Similar arguments imply that, again, S is dense in every Hs.

It should be desirable to obtain an isometry H ->H , just as

s of (3.2) for 92=1n. Note that P2s:dom P2s = H2s* H=L2(n,dg) may

be regarded as an unbounded hermitian positive definite operator

of H , assuming sja0 . It is found that the positive square root

of the Friedrichs extension of P2s (or its continuous extension to

Hs ) provides this isometry. Instead of engaging in an argument to

show that this operator is a yxlo in LCs we prefer to use the opera-

tor PsE LCs instead. While Ps generally is not an isometry we will

show that it is an isomorphism Hs H H at least.

Indeed, it already was seen that Ps:Hs - H . In particular,

(5.7)IIPsu112 = 11 D,j'/2s/2 a'jull25 cI11ns/2 s/2Xju112

s cy'llsXjull2 = cllulls , uE Hs

using III, thm.1.1 on ns/2s/2 sl E to show that Ps:Hs-H is


continuous for every s. As md-elliptic operator Ps admits a K-para-

metrix Q_S, a i,do in LC_s. If uE H, Psu=O, then 0=Q_sPsu=u-Ksu, Ks

E LS_00, =::,, u=KsuE S =*- (u,Psu)=7'11 s/2Xju112=0, ju=0 u=0, showing

that Ps:HS-H is 1-1. In H let fE H be 1 to im Ps. Then (f,PS(q)=0,

cpE S 0=(f,PSQ-sp)=(f,(1-LS)i) , i E S , where LSE LS_Q. Hence

(f-LSp,i)=0, iE S f=LspE S b0=(f,Psf) =7' IIns/2kjfII2 f=0 . It

5H5 defines anfollows that P5:H5-H has dense image P5. Then Ps1:P-'

unbounded operator from H to Hs with dense domain.

Proposition 5.1. For WE 12 we have Ts=PsP-5E LC0 . TS is bounded

in H=L (S,dµ) and has a bounded inverse Ts :H-H. Moreover,-1E

LC0 is a yido as well, and thus is a special K-parametrix of T5.

Proof. We already noted that TsE LC0 is md-elliptic, hence has a

K-parametrix RSE LCO . As t,do's of order 0 , TS and RS are L2-

bounded. For example, IITsufl=112rs, j?ju+KuUISCIIuII+jIITS' jull , where

III,thm.1.1, or III,thm.2.1 may be used to show IITB'jullscIlull

Next confirm that ker TS =0, and that im TS is dense in H,

where we mean the map TS:H-H. Indeed TSu=PSP_SU=O yields uE S, us-

ing the K-parametrix RS, then P-SUE S=>P-Su=0 u=0, so TS is 1-1.

Similarly, using Rs, show that (f,Tsp)=0 for cpE S implies f=0, hen-

ce im TS is dense. Existence of the (L2-bounded) K-parametrix RS

amounts to existence of an inverse mod K(H). Thus TS is a Fredholm

operator (cf.[C1],App.Al,thm.4.8), it has closed range. Since ker

TS=O and im TS is dense in H , TS is invertible, Ts1E L(H). But we

have R5T5=1-L5, with K5, LSE This implies RS=

T5 -TB KS=Ts -L5Ts . Or, TS =RS+Ts Ks, and T5 =Rs+L5T5 . Substi-

tute the second into the right hand side of the first: Ts1=Rs+RSKS

+LSTs1K5. Here RSE LSO, while R5K E LS_,0. The third term also bel-

ongs to LS_0 , since we get IILsTs KsullmSClIIT51Ks1II5C2IIKsuII5c3IIUIIm'

for all m,m' . It follows that Ts E LCO , q.e.d.

Using prop.5.1 we show that P E L(H H) admits a continuous

inverse. Indeed, T5=PsP_S implies Ps1= Ps5Ts1 where we regard the

right hand side as a composition of Ts 1E L(H) , and P_5E L(H,H5).

To confirm the latter note that IIP-sulk=2Iln 5(XjP-5u)II2=

7,11n s( KkA`ks/2s11 /2?.ku)112 The point is that, for fixed j, all


terms r1,2Ps/2?.ku must be taken to the coordinates of the j-th

chart, using thm.3.3. and only within a neighbourhood of supp XjC

Uj. They will give operators of order s , insuring L2-boundedness

of each term of the sum. Hence IIP_Sullsscllull, i.e., P_5E L(H,Hs),

implying PS1E L(H,HS) . So, indeed, we have the result, below.

Proposition 5.2. The operator PS of (5.3) are 'ado's in LSS . There

exists a y'do Q_SE LS_S , acting as an inverse -i.e.,

(5.8) PSQ_S = Q_SPS = 1 .

We have Q_S:H->HS continuous, i.e., PS

: HS« H and Q_S:R HHS define

isomorphisms between Hs and H . With constants cs , cs>0 we have

(5.9) c5IIPsuII s (lulls s csllPsull , u E RS

Using the isomorphism PS we can prove HS-boundedness of ipdo's:

Theorem 5.3. A i,do AE LSm satisfies

(5.10) llAulls_ms cSllulls , for all u E HS , cs independent of u.

Proof. We have llAulls_ms cfl(Ps_mAQ_s)Psufl , where PS_mAP5 E LSO is

L2-bounded, as seen above. Thus IlAulls_msclIPsullsc'llulls, q.e.d.

In particular thm.5.3 insures that the pdo's PSE LSS and

Q_SE LS_S are operators of L(Ht,Ht_S) and L(Ht,Ht+s) respectively,

for every tE !2 , not only for t=s or t=O , as known earlier.

Next we again introduce a pairing between RS and H_5 by

(5.11) (u,v) = (PSu,Q_SV) _ J uvdµ , uE Hs , vE H_S .

sz

Here PSu, Q_sE R, hence the middle term in (5.11) is well defined,

as inner product of elements of H. It is clear also that (u,PSU)=

=(PSu,u), (u,Q_Su)=(Q_su,u), as uE S, so that (5.11) is meaningful.

With such preparations we reintroduce order classes, setting

(5.12) O(m) = {AE L(S(c)) : PSAQ_S E L(Hm,H) , for all SE &2} .

Or, O(m) consists of all AE L(S) such that for all sE 12 A extends

to a continuous map Hs'Hs-m. Then, clearly, LSmC O(m) for all in.

We again define O(±ao) by III,(3.30), and get LS_00=0(-oo). The con-

cept of K-parametrix is meaningfull not only for y,do's, but also

for general A E O(ao). A Green inverse of AE O(w) is defined as a

K-parametrix BE O(oo) with 1-AB, 1-SAE O(-oo) of finite rank.

We now can repeat every line of argument of 111,4, showing


that a Green inverse - and even a special Green inverse, with all

properties III,(4.1), exists if and only if a K-parametrix exists,

for AE 0(00). This holds, because all arguments there are abstract,

using only the system of Hilbert spaces Hs, the pairing (5.12),

and the operator s, here represented by Ps :

Corollary 5.4. All statements of III,thm.4.1, thm.4.2, and cor.4.3

are also valid for operators of our present order classes 0(m).

Moreover, they still hold for operators between sections of admis-

sible vector bundles (the latter denoting the restriction to St of

a C"o-vector bundle over ? , where evidently Hs-norms and Hs-spa-

ces and pdo's may be introduced for sections, in the same way).

As a final remark, we mention without detailed proof that

a system of equivalent norms is given by

(5.13) 11ulls = II(d(x)) S' (1_0) sh /2u11 , u E S(n) ,

with the Beltrami-Laplace operator A of the distinguished conical

metric (or any conical metric): With constants cs, cs we have

(5.14) csliulls s Ilulis s cs11u11s , u E S

In particular, the system of spaces Hs and the order classes

0(m) are independent of the partition of unity and atlas chosen.

As an argument leading into (5.13), (5.14): For s.E Z, s,>O,

(5.14) follows immediately. For other st>O use an interpolation

argument, as for the proof of III,prop.3.3. For si<0 one uses cal-

culus of [do's in combination with the above.

Let us not forget to mention that 0(0) is a Frechet algebra

again, with topology induced by the operator norms in L(Hs). Use

Calderon interpolation as in III,prop.3.3 to show this.

Problems. 1) Connecting to the problems of IV,4, introduce order

classes and Green inverses for ,do's on manifolds with all three,

conical and cylindrical ends, and conical tips. Note, there will

be a difference in technique of constructing a Green iverse, once

a K-parametrix exists: md-ellipticity is not enough ([C;],VIII).

Chapter 5. ELLIPTIC AND PARABOLIC PROBLEMS

In this chapter we take up the lead of ch.0, sec.4, with re-

gard to elliptic and parabolic problems. There we applied the Fou-

rier-Laplace method to free-space problems of elliptic equations,

and to evolutionary half-space problems of the (parabolic) heat

equation, all with constant coefficients. We covered Dirichlet

and Neumann problems in a half-space, for elliptic equations.

With the tools developed in I, II, III, IV we now can give a

similar "Fourier-Laplace treatment" to much more general variable

coefficients elliptic and parabolic problems. This may be done in

"free space" (that is, in &n , or on a smooth compact manifold n

or on a noncompact n with conical ends - but without the presence

of boundary points). Such results are special cases of theorems on

Green inverses of pdo's already discussed, but they will be summa-

rized (in more general form) in sec.1, below. If Sz is compact we

need ellipticity, else md-ellipticity of the operator. Not only

(md-) elliptic operators on a complex-valued function but even

maps between crosssections of vector bundles are considered.

Note that there is a different approach - a functional analy-

sis approach - to these theorems, not using ado's at all. Elliptic

theory, in its beginnings, was developed for 2-nd order equations.

Such 2-nd order theory is of dominating importance for many physi-

cal applications. The Laplace, Helmholtz and Schroedinger operator

each has its own well developed theory.

One finds that virtually all results we state for &n can be

reached by focusing on the C*-subalgebra A of L(LZ(&n)) generated

by the multiplications a(M):u-au , aE C(Zn) (cf.II,3) and the ope-

rators D.(1-A)-1/2=5., defining S.=s.(D)=F-Is.(M)F as Fourier mul-l 3_1/2 J 7 7

tiplier or else (1-A) as inverse positive square root of the

unique self-adjoint realization of 1-AaO, A=a ([Cl],III,IV).7

Similarly, in case of a general Riemannian manifold with co-

nical and cylindrical ends, and conical tips (cf. IV,4,5, and the

problems of IV,4) we may generate such C*-algebra fromD(1-A)-1/2

with the Beltrami-Laplace operator A of n under the metric discus-

144


sed for such manifolds, with DE D# , and a(M) . aE A# , with sui-

table classes A# of functions and D# of FOLPDE's on a . Such ap-

proach to elliptic theory is discussed in detail in [C1] and [Cz].

Here we look at this in sec.10, only to clarify some questions

comparing both the pdo- and the C*-algebra approach.

The C*-algebra approach is of importance not only for ellip-

tic but also to hyperbolic theory (of ch's VI and VII):Conjugation

with the evolution operator eiLt of a first order hyperbolic equa-

tion defines an automorphism of the above algebra A . The dual of

this automorphism will be a Hamiltonean flow in symbol space deter-

mining propagation of singularities, as in Egorov's theorem.

In sec.2 we start focusing on the general elliptic boundary

problem for smooth boundaries under Lopatinski-Shapiro type boun-

dary conditions, but only for compact szii' CC &n, for simplicity.

First we discuss common facts, such as reduction to the case of

a homogeneous boundary conditions or a homogeneous PDE.

In ch.0 we used a reflection principle to convert a Diri-

chlet or Neumann problem over a half-space into a free-space pro-

blem. Similarly, for a general boundary problem, here we will ex-

tend all functions from MX into all in , and the elliptic opera-tor to an md-elliptic operator on in (sec.3). The boundary problem

will become a 'Riemann-Hilbert type problem' involving (genuine)

distributions. We get a type of distribution best classified as

'multi-layer potentials'. In potential theory one uses a single-

or multi-layer ansatz, using the same kind of distribution for sol-

ving a boundary problem. Here we find that every solution necessa-

rily is a sum of a Co function and certain multilayer potentials.

Actually, in sec. 4, we discuss a result we call boundary

hypoellipticity: Just as uE C°° for fE C"o follows for a distribut-

ion u solving Lu=f for a hypo-elliptic L , inside a , we will show

that, if f admits a certain asymptotic expansion near the boundary

so must u satisfy the same kind of expansion, under proper assump-

tions on L (i.e., hypo-ellipticity, and that r is non-characteri-

stic for L).This will be very useful not only for the elliptic

problem but also for parabolic problems, later on.

In sec.6 we then discuss existence and uniqueness (again in

finite dimensional degeneration- i.e., normal solvability) of the

elliptic boundary problem, if the boundary conditions are of 'Lop-

atinski-Shapiro type'. Of course we use the "multi-layer ansatz".

We only look at the simplest nontrivial case: A single even order

equation. But the generalization to operators (of even or odd or-

der) between vector bundles should be fairly evident; also exten-

146 5. Elliptic and parabolic problems

sion to boundary problems for subdomains of an S-manifolds should

involve no new ideas, under proper assumptions on a and r .

The multi-layer ansatz (for a homogeneous PDE) will convert

the boundary conditions into a system of pdo's on r. Ellipticity

of that system will amount precisely to the L.-S.-conditions. Act-

ually, for existence and uniqueness considerations we need two dif-

ferent such elliptic systems, one n/2xn/2-system, the other nxn .

In sec.7 we turn to the parabolic initial-boundary problem.

A parabolic problem is evolutionary - we assume an equation au/at=

Lu with a differential operator L under initial conditions at t=0.

The evolution operator U(t)=e1tL is well defined as a semigroup.

For a parabolic problem the operator a/at-L is hypo-elliptic. For

existence of the semigroup we assume standard results, such as the

Hille-Yosida theorem. Boundary-hypo-ellipticity (sec.4) will be

usefull to secure C00-solutions instead of distribution solutions.

An efficient use of results such as the Hille-Yosida theorem

in sec.7, will require investigation of R(X)=(L-X.)-1 of an ellip-

tic operator, as will spectral theory of elliptic PDO's. Thus, in

sec's 8-9, we look at spectral theory and R(?.) , for elliptic L.

We only consider the cases a--in , s2 = compact manifold , n =

manifold with conical ends, nCC 1n a domain with smooth boundary.

Extensions to (i) systems of equations (ii) differential operators

between sections of vector bundles of equal dimension, (iii) exte-

rior boundary problems, (iv) Riemann-Hilbert type problems are

fairly evident, but are left for the reader to deploy.

Existence of a Green inverse for operators defined with boun-

dary conditions is discussed as auxiliary result (thm.8.3), exten-

ding results of sec.6. Mainly we focus on compactness of the resol-

vent, under proper assumptions. Results on selfadjointness and dis-

sipativity of differential operators (making thm's 7.3, 7.5 appli-

cable) are discussed under various assumptions (thm.8.5 for n with-

out boundary, thm.9.1 for LYE 1n with boundary). In all cases pro-

per deployment of the results is left to functional analysis - in-

sofar as the spectral theorem , the Hille-Yosida theorem, or exi-

stence of an orthonormal base of eigenfunctions for self-adjoint

operators with compact resolvent is not discussed in detail.

In sec.10, finally, we discuss the C*-subalgebra A of L(H),

H=L2(1n), generated by our algebra OpCo of ch.I. The point: A°

A/K(H) is a commutative C*-algebra isometrically isomorphic to

C(ap), with the boundary aP of a certain compactification 1 of 12n

A corresponding result holds for the norm closures in LZ(st) of

LSO.p.61

6<p2, of IV,4, with similar proof, not discussed here.

5.1. Elliptic problems in free space 147

1. Elliptic problems in free space; a summary.

In the present section we summarize (and trivially extend)

results on elliptic equations, previously discussed in the more

general context of pseudodifferential equations. "Free space" is

interpreted as "no boundary" - a compact manifold is a free space.

Theorem 1.1. Given an elliptic differential operator

of order N, on a compact C"-manifold n (of dimension n) without

boundary. That is, in coordinates of a chart UC St, for uE Cp(U)

(1.1) Au(x) = aa(x)Da , aaE Cm(U)IaIsN

with

(1.2) a a 0 , as xEU

Assertion:(1) The

differentlia_lN

equation

(1.3) Au=f , f e C00(n) ,

is normally solvable, for uE C°°(s2) . That is, (i) the homogeneous

equation Au=O admits at most finitely many linearly independent

solutions; (ii) there exists a solution uE Cm(n) of Au=f if and

only if f satisfies finitely many linear conditions of the form

(1.4) fndu ju = o , j=1,...,ja .

with any positive Cm measure dµ on a and certain C"(f)-functions

Vi [In fact, the Vj may be chosen as a basis of ker A* , with the

adjoint A* of A with respect to the inner product f.uvdµ] .

(2) The operator A:C0O(St) * CO'(S) admits a special Green inv-erse G E LS-N = LS(-N,0),(1,0),0 , the latter as in IV,4, with

(1.5) GA=1-P , AP=I-Q ,

where P is a projection onto ker A, annihilating some (arbitrarily

chosen) complement of ker A , and Q a projection onto an (arbitra-

rily chosen) complement of im A annihilating im A .

Proof. The compact manifold a is a special case of S-manifold stu-

died in ch.IV - there are no conical ends at all. Since there is

no infinity, the classes LSm,p,s are independent of m2, p2; We

thus write LSm,,p,6 1

and LCm in case of p,=1 , b=0


Also, since there is no infinity, md-elliptic means the same

as 'elliptic' - i.e., cdn.(1.2) above. Thus the existence of a

K-parametrix in LS-N follows from IV,thm.4.6. The special Green

inverse G may be constructed as in 111,4 (cf.IV,cor.5.4). Once we

have G , the normal solvability (1) is immediate.

Thm.1.1 generalizes immediately to the case of an elliptic

differential operator A:I->Y , with X,Y the spaces of C'° sections

of two vector bundles E,F over sz with dim E = dim F = ro . We will

not discuss details of the proof of thm.1.2, below: They are just

formal extensions of earlier discussions.

Theorem 1.2. Given an elliptic differential operator A:I-Y , with

X,Y as described above, A of order N , on a compact Cc manifold a

of dimension n without boundary. That is, in local coordinates of

a chart UC Sz , for uE I with supp u C U , we have (1.1) , where

now aa(x) are roxro-matrix-valued, while (1.2) is replaced by

(1.6) det 0 , xE U ,=1 .

Then the assertions of thm.1.1 hold again, as follows:

(1) The equation

(1.7) Au=f , fE Y ,

is normally solvable , as u E I - that is, dim ker A <oo

codim im A <oo .

(2) There exists a special Green inverse G , locally an

roxro-matrix of pdo's in LC_N , such that (1.5) holds.

Proof: See the remarks above.

If the manifold cz without boundary no longer is compact,

but, at infinity, still is conical - in the sense of ch.IV , then

our theory of earlier chapters yields results of similar structure

but only if (i) the space C"O(sz) is replaced by a smaller space -

X(n) still containing all ofC000

(n) ; (ii) the coefficients as of

(1.1) and their derivatives satisfy growth restrictions at 0

(iii) we supplement the ellipticity condition (1.2) (or 1.6)) by

a condition at infinity (m-ellipticity).

A useful choice for I(c) is the space Ham, a fl(Hs (tz) : sz =a}for any aE 1. Then, to get a normally solvable operator A:X-X , we

may choose aa(x) as restrictions to U of functions in the symbol

class 0o,e,o ,whenever the chart U gets near a point of c0 .

(That is, the full local symbol a a belongs toasN

5.2. The elliptic boundary problem

VC(NIO) .) Finally, instead of (1.2) we ask for

(1.8) x as rl , with some c,rl>O

149

That is, A must be ad-elliptic of order (N,0) .

More generally we may regard operators A:X->Y , %=Hm'a(n)

Y=HOOT(S), or even let %,Y be spaces of smooth sections of S-admis-

sible vector bundles of the same dimension ro, with components in

Hoo, x , x=a,t, again. Let us summarize this in thm.1.3, below, again

a consequence of IV,thm.4.6, and 111,4 (i.e., IV,cor.5.1).

Theorem 1.3. Given an md-elliptic differential operator A:X->Y

with the spaces %, Y of Co sections on two S-admissible vector

bundles E, F, dim E =dim F =ro, with components in HOO,a and Hoot

respectively. Let A be a differential operator , A E LC(N,x),

where x=a-r , and let A be md-elliptic of order (N,K) . That is,

in the coordinates of an S-admissible chart UC St, we have (1.1),

(1.9) aa(P)(x) = 0((x)x-IPI) , for all P ,

and (with the ro xro -matrix norm 1 . 1 , and some c,rl>0)

(1.10) Ia(x, )I=I a asasNThen the differential equation

(1.11) Au=f , fE Y ,

is normally solvable, for uE % - i.e., dim ker A <oo , codim im A

< oo . There exists a special Green inverse GE LS (-N,-x) such that

(1.5) holds, with projections P,Q as described.

Problems. Consider the paraboloid P:xi=x:2+x32 and the one- and

two-shell hyperboloids B+:xI2-x:2-x32=t1 in &3 . Let A be the Bel-

trami-Laplace operator of the Riemannian metric induced in P and

B+ by the Euclidean metric of &3. 1) Show that the natural S-struc-

ture of 23 discussed in IV,1 also induces an S-structure in B. :An

atlas of admissible charts may be obtained by restricting admissi-

ble charts of 1R3 to B. .) With this S-structure, 1-A is md-ellip-

tic. 2) Discuss the corresponding facts for the paraboloid P .

2. The elliptic boundary problem.

In this section we start a discussion of the elliptic boun-

dary problem on compact domains with smooth boundary.


Consider a compact subdomain SEC 1n with boundary r=anC &n

where r = U rj is a finite disjoint union of compact smooth n-1-

dimensional submanifolds of &n . Let A=a(x,D) , Bj=bj(x,D), j=1,..

..M, be differential operators of orders N, Nj, with smooth coef-

ficients, defined near rtlf and near r , respectively. Assume com-

plex (globally defined) coefficents, for the moment, - i.e.,

A = aa(x)D" , B. = ba(x)Da'(2.1) IaI-N J

where aaE C°O(N) , baE C"(Nj) , ttU'CC N , Ft N j, N,N1C in

For given functions fE C0O(c&f) , PjE C°'(r) one seeks to finda function uE C°°(nLX) satisfying the equations

(2.2) Au=f , xE Stld' , Bju=pj , xE r , j=1 , ... ,M .

The problem of finding u for given f, Tj (or the discussion of

existence and uniqueness of such u) is called a boundary problem;

the first relation Au=f is called the differential equation, while

the relations Bju=gj , required for xE T only, are called the

boundary conditions of the problem.

We recall the definition of COO(nLr) and C°°(T) : uE C°°(c f)extends to a C0O(N)-function for some czLfCE N . Similarly for r .

Example 2.1. Consider the Laplace operator A=A = 4=1a2 (i.e.,

J

N=2) , with M=1, B1=1 (i.e., B1(u)=u). The boundary problem (2.2)

with this choice of A and Bj is called the Dirichlet problem of

the Laplace equation. It is well known in potential theory, and

has many physical applications.

Example 2.2. Problem (2.2) with A=A as in Ex'le 5.1, but B,=BM=I,=

Ivjax (where v=(v1,.... Vn) denotes the exterior unit normal of r)

is called the Neumann problem of the Laplace equation. It has many

applications as well. More generally, for

(2.3) B1=BM = av+h , hE c°O(T) ,

(one often assumes h(x)>0 ) one gets Hilberts boundary problem of

the third kind, known for problems of heat conduction for example.

Physical applications are not restricted to 2-nd order equa-

tion problems. For example, the case of N=4, A=A 2, under various

choices of Bj (such as B1=1, B2=BM av ) describes a loaded elastic


plate over n , under various restraints at the boundary 1' .

Often a different problem is of interest, of the form

(2.4) Au=f , xE In\S2 , B ju=gyp j , xE r , j=1 , ... , M, uE B., ,

where now A and f are defined (and Coo) in the noncompact closed

set In\S . Here Bco denotes a linear set of functions defined near

infinity, containing all uE C*(In), so that "uE Boo" amounts to a

(set of) conditions at infinity. Also, to arrive at a well posed

(or normally solvable) problem one must replace the condition

fE C00(ln\n) by a stronger one "fE C00(In\R)rW00 with some space I

satisfying the above conditions for B0. .

(2.4) is often called an exterior boundary problem. As exam-

ple consider A=A , B1=BM l , Boo {uE Coo(ln): u(x)= O(IXI)}, leading

to the exterior Dirichlet problem of the Laplace equation.

The examples given all use an elliptic expression A=a(x,D).

Here we used A=D and A=D2 . Other examples, using a general ellip-

tic A=ajk(x)axjaxk +A1 with Al of first order, ajk(x) real, C"

((ajk(x))) a positive definite nxn-matrix, are easily quoted.

For the moment we shall focus on (2.2), also referred to as

interior problem, assuming that A is elliptic in Said' . A fortiori,

we assume that A extends to an md-elliptic expression on In :

A=a(x,D) , a={cIN} , where c= cas E tc

(2.5)

1aIsNa N,m2

is md-elliptic of order (N,m2) , for some m2 EE I

(2.5) will be convenient but is not essential. It trivially holds

after restricting A to any small ball, hence will disappear if

the theory is drawn up for general manifolds with boundary, as we

shall not do here. Note that, under (2.5), interior and exterior

boundary problem both are meaningful, once B.0 and %00 are fixed.

If, for a moment, we admit an empty domain n=y! , then the

exterior problem fits into our earlier theory of md-elliptic

problems on In : Choosing B. = Bs - for any given fixed s=(si,s2),

problem (2.4) becomes normally solvable if we require in addition

that fE Bs-(N,m2) -%.. Accordingly we tend to think of similar spa-ces Boo , %00 , also for non-empty Sa and r .

One will expect that the Bj must satisfy certain conditions

before solution of (2.2) or (2.4) can be attempted. A (somewhat

complicated) set of conditions called Lopatinski-Shapiro condit-

ions - will be introduced below. Then we will prove normal solvabi-

lity of (2.2) : u exists for all f, Tj satisfying finitely many


For f=cpj=O the space of solutions is finite dimensional.

In other words, with an elliptic A and a system {A,B1.... BM}

satisfying the Lopatinskij-Shapiro conditions, the linear map

(2.6) AxB1x...xBM : C00(Sllf) -> C"0(aLf)xC00(r)x...xC00(T)

defined by u - (Au,B1u,...,BMu) is a Fredholm map.Apart from existence and uniqueness we will pursue a quest-

ion parallel to hypo-ellipticity, as in II,thm.4.1. For open SIC In

if uE D' (n) has CuE CO(n) then uE C00(12), by hypo-ellipticity of C.In the line of solving boundary problems, a similar result,

with Slid' instead of sI would be useful. Of course, D' (Slid') is notdefined. Using ideas of Melrose [Me1] we replace D' (SII.[') by a spa-ce of extendable distributions. Then this result indeed extends,

since existence of certain asymptotic expansions is preserved.

We shall refer to this as boundary hypo-ellipticity.

As already mentioned, a boundary problem (2.2), in its ab-

stract setting, amounts to the problem of inverting (or Fredholm

inverting) a linear map of the form (2.6). However, (2.2) proves

equivalent to either of the two simpler problems arising if one

either assumes f-0 on s or Tj-O, j=1,...,M , on F. That is, either

(2.2') Au=O , xE Slid' , Bju=Tj , xE 1, , j=1 , ... ,M .

Or,

(2.2") Au=f , xE Slid' , B ju=0 , xE I' , j=1,. .. ,M .

Clearly, if (2.2) is (normally) solvable then so are (2.2') and

(2.2") each. Vice versa, let (2.2') be (normally) solvable. Assume

Condition S: The differential equation Av=f admits a solution vE

Cm(nid') for each fE C00(slld') . Moreover, a unique suchv can be assigned to every f, such that v=Xf becomesa linear operator (In other words, A: C00(nLr) - C°°(SllF)admits a (linear) right inverse).

Given fE Cx(nLr) and a solution u of (2.2) we define w=u-v. Then

w solves (2.2') with ypj replaced by pj-Bjv. Since (2.2') is normal

ly solvable we get a solution w if only cpj-BjXf satisfies finally

many linear conditions, translating into finitely many linear

conditions for f,cpj . Clearly also the null space of (2.2) coinci-

des with the null space of (2.2'), hence it is finite dimensional.

Next let (2.2") be normally solvable. Assume

Condition BS: For an arbitrary selection of TjE Coo(r) there exists

a function vE C (slid') such that Bjv=Tj on r , and, again,


a linear right inverse of v - (Bjv) may be constructed.

Then, for given f,qj , and a solution u of (2.2) set w=u-v

again. Then w satisfies (2.2") with f replaced by f-Av. A solution

w exists if f-Av satisfies finitely many linear conditions, etc.

When are cdn.S or cdn.BS satisfied? For cdn BS there is a

simple answer: Observe that the normal derivatives of order 0,1,..

... of a C'O(szl.d')-function v may be arbitrarly prescribed, and thatthe equations Bv=c)translate into a system of differential equa-

tions for (vO,...,vN-1) , where vj=O vjT on the boundary r . Cdn.

BS simply says that this linear system can be solved for every

(Pi, and, moreover, that a linear right inverse can be constructed.

One may check with the examples and find this trivially satisfied.

For cdn.S we use that C is md-elliptic hence has a special

Green inverse G (III,thm.4.2): CG=1-P, P a finite dimensional pro-

jection onto some complement of in C, annihilating in C=(ker C )l

Let gE C0(1 n) extend f to &n : g=f near c&E', gE CO. Clearly

(2.7) Pz= Dj(ajz) , zE S' ,

with a basis {a.} of ker C , C" the distribution adjoint of 1,3,

and a basis {(3j? of the complement of in C, bi-orthogonal to {aj}.

Try to select Rj such that (3j=O near s F. We only must construct

a set {y j } of linear independent functions yJ.E S such that yJ.=0 inB. ={ I x I s,q} , for some sufficiently large rl , and that no linear combination y of yj satisfies (y,al)=0 for all 1.

Let us first assume that we have "unique continuation" of

solutions of Cu=o, a property common to all second order (and

much more general classes of) elliptic equations (cf. [As1], [Cu],

[Hr3],ch.28). We will not discuss such results here. In particular

our present use is not essential for discussion of L-S-theory here

Under unique continuation no solution a of C a=0 may vanish

outside BI, except a=0. Thus a basis of ker C is linearly inde-

pendent in X,1=&n\BI : The matrix ((JX aja1dx))=Z is nonsingular.I

For a suitable cut-off x with supp x C X1-£ X=1 in X1 and yj=Xaj

we get (((yj,al))) close to Z , thus nonsingular as well.

Thus yj give the desired functions.

For P of (2.7) with aj , (3j as constructed and the correspond-

ing Green inverse G let w=Gg. Then Cw=CGg=g-Dj(aj,g)=f for x near

szld'. Thus a v solving Cv=f near nL-r exists: Set v=w in N(szLf)Moreover, the above construction has given a right inverse of A


as required, and cdn. S always holds, under above general assump-

tions, assuming that C"u=O has the unique continuation property.

Remark 2.3. Note that the last argument is superfluous if C has

a fundamental solution -i.e., a special Green inverse which is

also a right inverse. Examples 1 and 2 above satisfy this condit-

ion - the fundamental solution cnlx-y,2-n (or c2log1x-yl) is known

A transformation of the general (2.2) to a form similar

to (2.2') will be essential, in the following. Since the above

discussion undesirably depends on unique continuation we note the

proposition, below, tailored to fit the proof of thm.6.3.

Proposition 2.4. Pick any basis yl,...,yR of a complement of in C

(We may choose the yj as CO(ln)-functions.). With the above Green0

00inverse G of C let z=Gf', f' an extension of fE C-(nLf) to CO(In).

Then u- =u-zlnLr transforms (2.2) to the following problem:

(2.8) Av=y , xE Snl,r, B,P=1l j=(,j-Bjz , xE r ,

with yE M=span {(3IfLf}, a fixed finite dimensional space, with I3

of (2.7). We may arrange f->f' such that is a linear operator.

The proof follows our discussion around (2.7).

3. Conversion to an In-problem of Riemann-Hilbert type.

We enter the discussion of solving (2.2) (or (2.4)) by

setting up relations for the extended functions v , g of u and f

(3.1) v=u in StLF , g=f in nLr v=g=O in In\ (Stld') .

For a function zE LIloc(Stlr) we denote the "zero-extension to ln"

by z" , so that z" =z in c&f, z" =0 in In\S2\r. Thus g=f" , v=u .

We have Cv-g=O in a and in In\S2. In general v,g are discon-

tinuous on r, although clearly v,gE Lpol(In)C S'. Looking at Cv-g=

h (interpreting the derivatives of C as distribution derivatives),

(3.2) Cv-g = h E E'C S' , where supp hC r .

A distribution with support in a single point aE In is known to be

a finite sum of derivatives b(a)(x-a) (Schwartz, [Schwi],thm.35)

Similarly, a distribution with support on a smooth hypersurface

locally is a finite sum of distributions o ,rE D'(In) of the form

(3.3) (6k,r,cp) = (V,avqIr) gE D(n) , with some 1NE D'(r)

5.3. A Riemann-Hilbert problem 155

where av denotes the k-th normal derivative at r (l.c.,thm.37).

Therefore h of (3.2) must be such a sum.

Actually, for an uE C0O(nLF)rW , we get such a representation

for h explicitly, as a result of Green's formula. To be precise we

introduce (t,v) as new coordinates on s near r , t denoting the

foot point of the perpendicular from x to r , and v the distance

from x to r , with negative sign in the interior of sl . For a suf-

ficiently small neighbourhood NT of r the footpoint t of x is uni-

que, so that (t,v) are useful coordinates. Then ak simply denotes

the k-th partial derivative for v, - derivative along t=const.

Let gE D(ln) be a testing function, and let uE C00(cZLf), Lu=f,

and v, as above. For the distribution derivative Cv=c(x,D)v we get

(c(x,D)v,(p) = (v,c(Mr,-D)f) = Juc(Mr,-D)cpdxS2

The right hand side may be integrated by parts, for

(3.4) (Cu ,q) = ffcpdx+

jaj+1PjSN-1

Here yap are COO(T)-functions depending on the order of partial in-

tegration, dS denotes the surface element on r . Perhaps it is

useful to introduce the coordinates (v,t) in the second integral:

(3.5) (h,q)=(Cv-g,<p) _IaI+1P +j+ksN-1 T

We still may integraterate byy g parts along r using that COO(r)

Accordingly, all t-derivatives may be assumed on apu . We proved:

Proposition 3.1. Given uE C'm(Rlf), f=a(x,D)u=Cu . Let v=u , g=f-,as above. Then, with differential operators P1 of order 1 on r

(3.6) h, (p) = I f(PN_1_k_.avu)avpds , wE D(in)j+ksN-1 T

where the P1 have smooth coefficients.

Note that (3.6) amounts to

(3.7) h = =Ub k ,T , V PN-l-k-javu , 6.,T of (3.3)kkehence it gives the desired decomposition of h explicitly.

,6,TRemark 3.2: For a distribution TEE DO(in) of the form T=Yj

with SN I, of the form (3.3) and a finite sum, the distributionsJ,

Sv T E D'(T) are uniquely determined (as long as we define av byJ,

means of the above coordinates (v,t)). Indeed, for any given inte-


ger ka0, and cpE D(1'), a function P°E D(kn) may be constructed such

that jk on 1' , j=0,1,... . Then clearly

(3.8) ('!'k,(p) (T,q°)

showing that indeed the ,yk , hence the 6k are fully determined.

Proposition 3.3. Suppose for any u, fE C0O(fz(X) and their zero-extensions u-, f- we have (in the sense of D'(len))

(3.9) Cu = f" + h , supp h E T .

Thus Au=f in fz, and h coincides with the distribution (3.7).

Indeed, we trivially have Cu =f in the open set fz . Since u,

fE C"(fz) derivatives in Cu In are classical, hence we may write Au

=f in n. Taking closure in fzld' get Au=f in c&f. By above construc-tion, get (3.7) for h, with the precisely determined yak, q.e.d.

The discussion shows that, for u,fEC00

(nLr) , the PDE Au=f

is exactly equivalent to (3.9) with h of (3.7). For a boundary pro-

blem (2.2) we must solve Au=f, hence (3.9), for given f, where uE

C"(fzlf)' VO' V1' ... VN-1 E C"(F) are to be determined.

The right hand side of (3.9) belongs to S'(2n) . Therefore

(3.9), for given y,j , may be solved with the methods of ch.II (or

V,thm.1.1). The md-elliptic operator C admits a Green inverse G .

We get CG=1-P where P may be chosen as in (2.7). The distribution

(3.10) v = Gf- + Gh

will solve Cv=f-+h if and only if P(f-+h)=0 (i.e., (aj,f"+h)=0

for a basis aj of ker C-). Then the general solution (in S') is

(3.11) v = Gf" + 'j_IGo + ,jwj , tyjE C" (r)Vi

with X jE C, and a basis wj of ker C. For arbitrarily given f, +yo,.

'VN-1 one will not expect such v to vanish outside fall' - i.e.,

v is not a zero-extension of some u. In view of the above our pro-

blem of solving (2.2) thus appears reduced to the following:

Problem 3.4. For given fE C"(c&E'), T .E C" (r) determine w0" "VN-1E C"(F) such that (i) (f"+h, a) =0 aE ker C" , and that, for

suitable X. the function v of (3.11) satisfies (ii) v=0 in In\a\T;

(iii) vIfz extends to a function uE C"(r&F), so that v=u ; (iv) we

get Bju=Tj, xE t, j=1,...,M - i.e., u satisfies the cdn's of (2.2)

Actually, v of (3.11) is a temperate distribution only, now

to be examined for its properties. Evidently we have wjE S . The

5.4. Boundary hypo-ellipticity 157

other two terms involve the operator GE OpVc-N,-m2 , of different-

iation order -N< 0. G sends Hs to Hs+(N,m2),as we know. From II,

2.5 we know that the ado G leaves wave front sets and singular

supports invariant. Also, the distribution kernel k(x,y)=g,(x,x-y)

g=symb(G) is equal to some function in S(&n), for fixed y and lar-

ge lxi (and uniformly so for yE RCL 1n). We have proven:

Proposition 3.5. v of (3.11) is C'(kn\r) . Also, v is equal to a

function in S(In) for sufficiently large lxJ .

Indeed, sing supp(f"+h)C r, and Gf" f 121. k(f +h)dy, nLr(= 1n.We will make a detailed discussion of Gf" and G6 ,1 uj near

their singular support r in sec.4, below, and then return to pbm.

3.4. In essence one finds a system of pile's for the y,j of pbm.3.4.

The vj will be called multilayer potentials (induced by G) . This

notation is suggested by considering examples 2.1, 2.2 again. For

C=A use of the fundamental solution G of 0,(4.12) instead of the

Green inverse is natural. The two distributions vo , v, are known

in potential theory as single and double layer potentials. They

are used to solve the Dirichlet and Neumann problem for C=A .

With a result of sec.4, we get an improvement of prop.3.5:

Proposition 3.6. 1) For fE C00(c&f) we have w=Gcp" E CN-1 (Rn) . More-over the N-th derivatives w(a) , JaI=N , jump accross r, but are

otherwise smooth: For every a the restrictions of w(a) to 12 and to

Rn\(f2LF) extend to functions in C00(c&F) and COO(In\12), respectively2) For 'tpE C°°(r) , and j=O,...,N-2 the multilayer potential

vj=GSA is a function in CN-2-j(Nn). For all derivatives (a) the

restrictions to n and &n\(nu) extend to functions in CO0(nLr) andin C00(&n\ft) , respectively, j=0,...,N-1. (That is, all derivatives

of order s N-1-j may be expected to jump on r) .

Proof. We will get C(Gf") = f" + s' , Cvj = 6j + s" , s',s"E S.

Then thm.4.4 implies the statement; details are obtained by compa-

ring the particular asymptotic expansions of f" and fi,.

4. Boundary hypo-ellipticity; asymptotic expansion mod av

In this section we focus on matters of boundary hypoellip-

ticity, as mentioned in sec.2, and used in prop.3.6. With notat-

ions of sec's 2 and 3 focus on PDE's of the form (3.9), i.e.,

(4.1) Cv=f" +2_0Sj

where vE D'(1n). We will mainly be interested in the case p=N-1


fE C"(MX), 1V E C"(r), attempting to solve pbm.3.4, but results

will be interesting for general p and ii.E D' (r) and fE C"(n) with

f- E Llloc(&n), as will be assumed henceforth. Define the spaces

(4.2) Z ,=j wE D' (&n) : supp wC nil , wE C" (n) } , Y %szm l loc(&n

)

Let us use the coordinates (v,t) introduced in sec.3, where

vE & , T E=- r . They are valid only in some neighbourhood N of r .

Since r is compact N contains some set NE=IExr , IE={jvjsE} , E>O.

Here the neighbourhood NE of r may be regarded as a subset of the

cylinder n'= &xr as well. With a cut-off function x , supp x C IE,

x=1 near 0 , and w=xv , equation (4.1) takes the form

(4.3) Cw=f"x+ fj=o(-)javs(v)e)Vj+ a. , XE CO({va0}), a.=O as v<E/2,

with the Dirac measure 6 on &={-oo<v<oo} , and the distribution ten-

sor product (We were using the fact that v(=- C"(&n\r), by hypoellip-

ticity of C). We get v=w near r={v=0}. The behaviour of v near r

may be studied by looking at w of (4.3), while (4.3) may be regar-

ded as a PDE on the cylinder n' . (Assume C extended to Ill , for

example by freezing the coefficients: In these coordinates we get

(4.4) C = Lj=oCN-j(v)ay

with PDE's {Ck(v) : vE [-e,E]}, of order sk. Extend Ck(v) to I by

setting Ck=Ck(±E) as vaE and vs-E, resp., smoothened near tE .)

We work with (4.3) and n° = {v>0} C sz' =&xr instead of (4.1) and n.Correspondingly, %I° and Yre are defined using &xr instead of &n

Clearly CO(v)= c0(v,t) of (4.5) is a function. By ellipticity

of C we have c0(v,t)#0 on n'= &xr . This condition alone - in fact

only the property of r being noncharacteristic for C , i.e.,

(4.5) cO(0,t) # 0 (on all of r) ,

is sufficient for our present discussion. We assume C hypo-ellip-

tic (not necessarily elliptic) apart from (4.5). By assumption f

is C"(nld') . It admits a Taylor expansion in the variable v :

(4.6)

q 7

f(v,r) _ =0(ayf(O,t))j! +Jrq(v,t)V q

rq(v,t) _ (aq+lf(O.t))(vq dx , q=0,1,... , vaO0

For the extended function f" we get an expansion valid in a 2-si-

ded neighbourhood of r={v=0} by introducing xj(v), j=0,1,...

(4.7) xj(v) = 0/j! , as va0 , xj(v) = 0 , as v<O .


For N=0,1,2,...,

(4.8) f- (v,t) _ oavf(O,t)xj(v) + rq (v,t ) vE T , Bilks

The xj(v) are homogeneous of degree j in v . (4.8) may be

expressed as an asymptotic expansion mod av (at v=0), in the sense

of [C,]1II,3 : For fixed t=t° E T we have

(4.9) f (v,t°) _ =0 ayf(O.t° )Xj(v) (mod av) , at v=0

In details, the terms of (4.9) are homogeneous of degree pj=j-+ 00

j=0,1,... , (XjE Hj"', speaking in terms of [C.],II,(3.3) ).

We have r-25_, av(p(O,t° )xj=rq E Cq(St`) by well known pro-

perties of the remainder rN . Equivalently, for N=0,1,2,... there

exists M such that rk E CN for all kxM . The terms of (4.9) are

Lpol hence also are homogeneous distributions.

In [C,],II,3 we were discussing more general such asymptotic

expansions, allowing arbitrary homogeneity degree pj - positive

and negative, and even complex - but with certain restrictions for

degrees pjE Z , pjs-n , and allowing certain non-homogeneous terms

for integers pjkO . Here it is practical to extend in a somewhat

simpler way: For j=1,2,... define the distributions

(4.10) X_j = avc0 = av-1b

Observe that then, for k=0,1,...,

aYXi - Xj_k , jE Z , vkX (k+ )IXk+j jZ0

(4.11)

vkX_j= (_1)k _lk i Xk_j , j>k , =0 , jsk

All xj , also for j<0 , are homogeneous distributions of degree J.

We shall say that a distribution T E %., allows an extended

Taylor expansion (mod av) at v=0, (abbreviated "ETE") written as

(4.12) T = 0pX? yj (mod av) at v=0

where pE Z , and yjE DI(T) , j=p,p+1..... , if for every M=0,1,...

there exists NO=NO(M) such that yjE CM, near v=0, as Na

N0. The largest such p will be called the order of the expansion.

Actually we always will require that yjE C°O(T) , as ja0 , while

the yj , j<0 will be allowed as general distributions in D'(c)

This definition proves useful at once: (4.3) may be written as


Cw = Ixj® y. (mod av) at v=0 , with(4.13)

yj=avf(O,t) , ja0 , as j<0

It is evident that the 0-extensions of C'O(aLr)-functions

are characterized by the nature of their asymptotic expansions:

Proposition 4.1. A distribution g(=- D'(a') with supp g C {v2:0} and

sing supp g C {v=0} is Coo({vz0}) if and only if there exists func-

tions yj(t)E C°°(r) such that

(4.14) g='.0 xj®yj (mod av) at v=0

Then we have yj=avg(0+,t) , c G G

Proof. One direction was shown above. Vice versa, if the expansion

(4.14) exists, then it is evident that gj{v>0} extends to a

CN({vZO})-function, for every N , hence is C00({vz0}) , q.e.d.

Proposition 4.2. 1) An extended Taylor expansion may be differen-

tiated term by term - for v as well as t , of arbitrary order.

2)The product of , where wE C00(al) and f allows an extended

Taylor expansion, possesses an ETE with terms explicitly determi-

ned by Cauchy multiplication of the expansions of f and of =(oIc0 )"

using (4.11). (Note that of has an ETE , by prop.4.1.)

The proof is left to the reader. A consequence of prop.4.2:

Proposition 4.3. If u allows an extended Taylor expansion then so

does Cu , where C is any differential expression as in (4.4).

The matters of boundary hypo-ellipticity, as well as prop.3.6

now are settled by proving the following.

Theorem 4.4. Let C of (4.4) be hypo-elliptic with smooth coeffici-

ents. Assume c0(v,t) of order 0 ( i.e., a function), satisfying

(4.5), but the Ck(v), k>0, are differential expressions in t only,

of arbitrary (finite) order, with coefficients depending smoothly

on i and T. If, for uE DI(al) we have Cu=fE %,o , and if f allows

an ETE of order p at v=0 , with smooth negative coefficients, then

so does u. Moreover the expansion of u is of order p+N .

Proof. Assume we have supp u C {v2:0} and

(4.15) j=p x3Otp j (mod av) at v=0

As a first step we will construct a distribution of the form u0=


xp+N® Yp+N such that the extended Taylor expansions of Cu and Cu0

have the same term of order p. By (4.11) we get

a Taylor expansion Ck(v) ' YICkj)(O)vj/j: . Hence

(4.16) CN-j(v)avuO = xp+j®CN-j(O)Yp+N+ ... (mod av) at v=0

Taking the sum (4.4) one finds that

(4.17) Cu0 = xp0(c0(0't)Yp+N) + ... (mod av) at v=O

To get the same lowest order terms of (4.15) and (4.17) we choose

(4.18) Yp+N = wp/c0(0't) .

(Note (4.5) - i.e.,T is noncharacteristic for C).

After constructing u0 we form w1=u-u0. Clearly the expansion

of Cwt is of order p+1. We may repeat the process to construct u1=

xp+N+1® Yp+N+1 such that Cwt and Cu1 have the same lowest order

term, so that C(w1-u1)=C(u-u0-u1) has an expansion of order p+2.

By iteration we get u0,u1,..., such that C(u-u0-...-udyl) , for each

M, has an expansion of order p+M .

Finally notice that, formally, the infinite sum u0+u1.....

is well defined. Write uk =

(4.19)

=p+k+Nx j0 Y j . Then

IZ=Ouk= cp+N®Yp+N+ xp+N+10(Yp+N+Yp+N+I) +...

the factors of xl at right being well defined finite sums. Setting

(4.20) Yp+N+j= Yp+N+j+Yp+N+j-1+ "' + Yp+N , j=0,1,... .

we may write the formal sum (4.19) as

(4.21) 2k=Ouk = V=p+Nx j0 Y j

We show that (i) there exists vE D'(n'), supp vC {va0} with

(4.22) v = r=p+Nxjo Yj (mod av) at v=0

and that (ii) we have

(4.23) C(u-vq)E Cq , q=p+N,p+N+1,... , where vq =p+Nxjo Yj

Constructing v of (4.22) means writing v=Dcjxj0 Yj , with suita-


ble cut-off's xj(v) near v=0. With a technique similar to that of

I,lemma 6.4, we show that xj may be chosen to converge weakly in

D'(S2'), and to satisfy (4.22). Then (4.23) follows easily.

It follows that C(u-v) has an ETE with all terms vanishing.

This means that C(u-v)EC00(c'). By hypo-ellipticity of C this imp-

lies u-vE C"(S2° ), implying that u=)j0 yj (mod av) at v=0. Q.E.D.

Corollary 4.5. Every solution vE D'(Rn) of equation (3.11), with

fE CO0(Stld'), vjE C" (r), has vIS1 extending to a C00(Stld')-function.The proof is evident.

Remark 4.6. Prop.3.6 follows from thm.4.4 and remarks following it

Remark 4.7. The technique may be used as well for nonsmooth Cv=z=

f"+` j_16v assuming that f,'j belong to suitable Sobolev spaces.7

Remark 4.8. Thm.4.4 is of local nature:If its conditions hold only

for a subregion R of r , then (4.15), valid only for (v,x)E 1QxR

implies a corresponding ETE for u , also valid only in RxR

Indeed, all discussions of the proof extend literally.

5. A system of pde's for the y,j of problem 3.4.

We return to our pbm.3.4, the general elliptic boundary pro-

blem. After sec.4 we know that vE D'(len), satisfying (3.11) is

C"()en\t) and has interior and exterior limits for all aav on F

If v is to satisfy cdn's (i)-(iv) of pbm.3.4, then the exterior

(interior) limits must vanish (satisfy the boundary conditions.

In the present section we will analyze these conditions, and

convert them into a more explicit form. We shall see that a system

of pde's on r results, which is normally solvable, under certain

conditions - the Lopatinskij-Shapiro conditions already mentioned.

First of all, let us assume cdn.S (of sec.2) satisfied. In

other words, then it suffices to focus on (2.2') - i.e., assume

f=O in SZId' . This simplifies (3.11): Up to the additional solution

D,,jwj of Cu=O, v of (3.11) must be a sum of multilayer potentials.

Thus, clearly, the well known multi-layer Ansatz from potential

theory appears to be justified. We have proven:

Proposition 5.1. Every solution u of the boundary problem (2.2')

5.5. A system of operators 163

is representable (mod ker C) as a sum v= i of multi-layeri

potentials, with certain viE Coo(r) . That is, u is the continuous

extension of v l Sa to Said' , up to an additional term in ker C .

Remark 5.2. Note that the behaviour of v in the outside of QLr is

unessential. If z is a solution of an exterior boundary problem,

using the same C and r , but possibly different boundary operators

Bi , (but with f=0 in &n\(Mr)) , then the LIloc(R")-function

w=u (in 12) , w=z (in &n\(RLr) ) , undefined on the null set r

solves Cw= 4_i63 , with certain y,iE C"(Sa) , hencei

(5.1) v = e4_oGbj + w , wE ker C , p1E C°O(r)

J

This suggests that -possibly- a solution u of (2.2') has man rep-

resentations as u=vin, with v of (5.1), and different sets of 1yi.

To prove existence of u - a solution of (2.2') - it is enough

to find Vi such that v of (5.1) (with certain w) satisfies the

boundary conditions (2.2'), using derivatives from the inside of

r . In details the Vi must satisfy (with interior derivatives)

(5.2) ",J=IBkGOI +Bkw = (Pk , k=1,...,M_ i

Then u , the zero extension of u=vl will also satisfy (5.1)

but with redefined Vi , as we have derived initially in sec.3.

Moreover, u-, of course, will satisfy the additional boundary

conditions (in addition to the interior conditions (2.2))

(2.2)1 avu Li='(-iDv)kGbj +avw= 0 , xE G , k=O,...,N-1J

with derivatives from the exterior - since u =0 outside Said'

Thus, for uniqueness, we may show that the N+M conditions (2.2)

(with yi=0) and (5.2)t, imposed on N functions Vi , imply Vi=0

We now will attempt to express terms of the form BGOV , with

a boundary operator B - such as Bi of (2.2) or B=av of (2.2)1

with interior or exterior derivatives, in the form BGb,=QW , where

Q must be studied. Q will be a pdo, and (2.2'),(2.2)1, after modi-

fication, systems of pdo's. Fixing some y,i, (2.2') alone is conver-

ted to an elliptic system with Green inverse. Thus we will settle

existence and uniqueness (prove normal solvability of (2.2')).

The manifold structure of r , and the fact that r=ac implies

existence of a coordinate transform, for each xoE r , mapping a


neighbourhood Nx° of x° onto a neighbourhood of 0 in the half space

ie+={xja0}. r is covered by a finite number of such neighbourhoods.

Using a partiton of unity {xj} subordinate to such covering {N j}x

we split: 1 = DCjV , supp Kjy C Nxj , and write v=auk , vk=Bb lp.

Our above coordinate transform (for N j chosen sufficiently small)x

may be extended to a global transform meeting the requirements of

IV,3. In the new coordinates vk transforms into a linear combina-

tion of multilayer potentials with respect to the half space &+

with boundary y given by {x,=0} = &n-1 . That boundary is noncom-

pact. However, we now may assume the density 1V to have compact

support. G, of course, will transform into a 1,do of the same pro-

perties. Returning to our old notations, we now will have

a--jx=(v,t )E &n:v>0} ,I'=&n-1={x=(v,t )E &n:v=0} , 1E CO (r)

(5.3)

v =

G is a special Green inverse of CE OpVCN,ma , an md-elliptic

differential operator of order N . For our boundary conditions

(2.2) we also need derivatives Dav , IaIsN-1 , where

(5.4) Dav = (-i)kDaGDv(6(v)QV(t)) , 'WE C0(17)

For the calculation of (5.4) we approach O(v) by a 6-family:

Choose any family 6E(v)E=- E>0 , with as E-0 , weakly

in D'(&) - for example 6E()=6i(E), E>0, with 6,E C'(&), fbtdv=1.

By continuity of the tensor product and the operator B we then get

Dav = (-i)kDaGD;(O(v)QV(t) =(-i)k(5.5)

limE-0TE

TE= DaGD1(SE (v) P=DUGDiE Opwc(IaI+k)el -m

Clearly 6E0 1V E CD(1n) C S(e), hence (with x=(v,t)=(x,,x°))

(5.6) TE(xI ,x°)= JtfoKeiT(x° K)14(tJove ivxi6e. (v)p(x.(v.t))

with p=symb(P), 6,.=Fourier transform of 6E . We have written _

(v,t)=(l,,t), y=(y,,K), and used I,(1.5) with some Fubini-type in-

tegral exchanges, due to 66Ê S, p of polynomial growth.

Introducing the n-l-dimensional symbol (with parameter m)

QQ

5.5. A system of operators

(5.7)PE

(XI ,x" J9iveix, v be. (v)p(Xi ix" 'V'V

we get TE= PE (x, ,x° ,Dxs )1. We get 6E^ S^ =, as E-0, hence expect

(5.8) x° . ' ) =f dveix, vp(x, XA 'V' V )=K111? (xl x° ,xI

possibly with a distribution integral. Actually, (5.7) amounts to

RE=p(x, x° D X, )O E , for fixed x° implying the convergence(5.8) for each fixed in S'(&). Formally we thus expect

(5.9) ikD v(v,t) = Yo (v,t.D,C)1U , PO (v,t. ' ) =

with the inverse Fourier transform ,3' for the 3-rd argument.Let us analyze the special form of

Proposition 5.3. (and its derivatives for x, of arbitrary

order) may be written as. homogeneous asymptotic sums

(5.10) p(x, ) - Lj=0pj(x, ) (mod ICI) at

in the sense of [Ci],II,def.5.2, and uniformly so for xE K CC in

any K . Here the pj are rational functions of with coefficients

smooth in x , homogeneous of degree -N-j+Ial+k in 1;. Specifically,

e.=(1,...,0)(5.11)

rj(x,U)/cN+1(x,U) , rjE Wcjm-je+(Ial+k)e,

where the rj are polynomials in of degree sNj-j+Ial+k with coef-

ficients in VcO'jm -j , and where denotes the principal

part polynomial of (i.e. the homogeneous part of degree N).

Asymptotic convergence mod ICI at 1_oo means that

(5.12) 0( as RZRo (q),

uniformly for xE KCC In , all K , and with all -derivatives.

Proof. G is the operator B constructed in II,thm.1.6, up to an ad-

ditional term in 0p5(1Q2n). In the notations used there,

b = bj (mod pc-.) - where b0=c (large IxI+

IL;I ) while, for j=1,2,.., the bj are recursively defined :

(e)(5.13) bj+1=Dl6bj(9)c

,K0=(-i)I()I/01 for 1:5101:5N, =0 otherwise


for large again. At the moment we have x varying over a

bounded set, so, 'large a'large ICI ' . For our present

notation we use g=symb(G) and replace bj by gj in (5.13).

Proposition 5.4. We have (for xE KEE In , ICI large

(5.14) gjE span{iII(c(aj))/c) : j s YIajl = IIPJI s jN}

The proof is immediate, using (5.13) and induction.

Notice that a term at right of (5.14) is of the form

jjII(c ( R j) /c) : 7,I aj I =j 113 j I =k . Such a term is a symbol in Vc-m-ke

m=(N,m2), e=(1,1). Asymptotic convergence of Igj , in our case,

means that as RaR°(q) and xE K, hence coincides

with asymptotic convergence mod ICI, as defined. Terms of (5.14)

are rational functions of l; with coefficients coo in x. Denomina-

tors are powers of By reordering the expansion we get

(5.15) g =L1,=0Pj/e7+1

(mod ICI) at p0=1 , PjENj_j,Nmz_j

where pj are polynomials in of degree s Nj-j. Write c=Lj=0cj,

with homogeneous cj of degree j. The md-ellipticity of c implies

(5.16) IcN(x,° ) I zp>O, for all 11;11=1 , xE K .

We get

(5.17)c cN(x, )

{1+cN"(x, °)11-1 +...+cN(x, °) ISI-N}-1

cN(x, IF

where the yj are homogeneous of degree 0 . Uniform convergence of

the series 1+y1II-I+... in (5.17) for large ICI and xE K is evid-

ent, using the geometric series +y 1-y+y2-+... . Again, a study

of the coefficients yj, from expanding yj=(C -...CNshows that

(5.18) c = c Oqk/ck degree gkskN-kN

The series converges uniformly for xE K and large ICI with all

(term by term) derivatives. A fortiori we have asymptotic conver-

gence mod Ii;I. Similar expansions result for all powers c-I . Sub-

5.5. A system of operators 167

stituting (5.18), etc., into (5.15), and reordering we get the

desired expansion (5.12), for the case k=a=0 .

For general P=DaGDX we use I,(5.7), giving the finite sum

(5.19) P=Y 1 )101_

In other words, the symbol of P=DaGDX is a finite combination of

symbols g (x,U) . Since we know that (5.12) may be differen-

tiated term by term, and since multiplication by a power of

does not change the asymptotic convergence, nor the structure of

the expansion terms, we get (5.12) in the general case. In parti-

cular the term is as stated, as a comparison shows. Q.E.D.

Here we return to formula (5.9), so far instituted only for-

mally. Substituting from (5.10), taking the Fourier trans-

form "I" term by term, we must deal with the distribution integral

(5.20) Nr2_npJ f dxeivxpj(v,T,K, ° )

where pj = rj/cN+1 is a homogeneous rational function of degree

k+Ial-N-j in the variables For fixed the pj are

rational functions of the single variable K . Their poles, in the

complex x-plane, are independent of j - just the roots of the

denominators cN+1 i.e., of the polynomial equation

(5.21) 0 .

Since C is elliptic there are no real roots of this equation, by

(5.16). Assuming first that (5.20) is a Lebesgue integral, one

will tend to evaluate it by a complex method: Assume v#0 - i.e.,

x is not on F. Depending on the sign of v we will integrate over

a semi-circle in the upper or lower half-plane: For v>0 (v<0) the

function eiync=eiVRex.e-VImKwill decay as Im K -> oo (Im x -> -00)

Let the roots of (5.21) be called x1 . Then it is clear that

(5.22)f

K, as v>0,

- mx <0 ResK as v<0.1 1

So far we assumed a Lebesgue integral. However, in general, the

term pj=rj/cN+l is a linear combination of terms where

°e/cN+1 is L1, or at least improperly Riemann integrable, as v#0.

Thus (5.22) holds for these terms, while ".xq" amounts to "Dq"

under Fourier transform. Differentiation for a parameter may be ta-


ken under the residue. Therefore we get (5.22) in the general case

Looking at (5.9) we will be interested in the two limits

(5.23) ikD v(±0,t)=limva0ikD v(v,t) , as y»0 and ip<0 , resp.

With (5.22) we get the following result.


(5.24) ikDav(t0,t) = n±(t,D.)i , iE CD(ien-1)

where is smooth in for all and satisfies

(5.25) n±(T'b° ) = j=Onj'+(T,S°) (mod I ° I ) at °=oo

with

(5.26)III, >0 Resx pj(0't'x' °)

1 1

n7'-(t'°) _ Y'Imx <0 Resx pj(0't,x,°)1 1

for TEE K= ten-1 , all K , uniformly in t , and for partial deriva-

tives of all orders. In particular, the nj'*(t,°) are smooth in

as 1;°340 , and positive homogeneous of degree -N+Ial+k-j+1

(5.27) Resxlpj(0't'x'V) = JIx-x11=E pj)dx

with small a>O .

Proof. Note that the roots of (5.21) are continuous

in as long as x°340 . Moreover, these roots are homogeneous

of degree I in " : p>O .

Using this, and that pj are homogeneous of degree -N-j+Ial+k we

find that the pj'±(t,t°) are homogeneous of degree -N-j+lal+k+1

in ° , by an integral substitution in (5.27). Furthermore, the

nJ'$(t,°) are smooth in t,t° as well, since they may be represen-

ted by an integral over a countour containing all roots in the

given half-plane, where the countour may be locally kept constant.

As for the asymptotic convergence in (5.25), the remainder

Pq = p-2gp . =0 ( 1 1 _ R ) as qZq° (R) , and I x I -R={ I ° 1, +xz } -R/2

hence pq (p-2gpjyp = O(fdKltl-R) = O(jt°j-R+1) . Similarly we may

argue for partial derivatives.

Finally, regarding smoothness of we look at the ex-

5.6. Lopatinskij-Shapiro conditions 169

pansion (5.15) (which, in fact, is mod S(R2n), not only mod ICI).

That is, the remainders are O(()) not only 0(1 1_R) , so that

the "a"-remainders are arbitrarily smooth. Regarding(pj/cj+1)0,

The same residue calculus may be applied - but remember, the poly-

nomial may have real roots, in a bounded set IKI51 . But

we really have xpj/cJ+1 with x=O as lxlsq . Thus the path of the

Fourier integral may be laid clear of those roots , generating a

remainder involving an integral over a C°° function on a compact

countour - aC00-function. For fypj/cJ+1 , along a path y clear of

the roots of c -but coinciding with I for jxI>21 we may repeat the

above residue argument to get smoothness of the term. Q.E.D.

6. Lopatinskii-Shapiro conditions; normal solvability of (2.1.

It is time now to return to the two systems of equations

(5.2) and (5.2)1 seen to govern our boundary problem. Writing

(6.1) Bk bk(t,Dx)= k ba(t)Dxa , k=1,...,M , NksN-1

in our boundary coordinates of sec.3, using (5.9), (5.10), (5.11)

and (5.24),(5.25),(5.27), equations (5.2) assume the form

(6.2) k=0glk(t,Dt)Vk = W1-B1W 1=1,...,M

where 1 ba(t) n+'k,a(t, °) .

aIsN(We write n' of (5.24) as n`'k,a .) In view of (5.25) the glk aresmooth in and have asymptotic expansions

(6.3) glk = 1=O(mod I ) at °

k kglkj=(-i) ba(t)nJ'+'I sN

Here the nj,t,k'a are given as sums of residues, of the pJ,=pj,k,a

in the upper and lower half-plane, respectively. For every j,k,a

and the poles of are the roots of

Looking at (5.11) we first seek an explicit form for glk0

(6.4) glk0- mx1>OResxl(-ix)k( Ia sNl ba(t)(x. °)a)/cN(O.t.x. °),

simplifying to

(6.5) fy dx(-ix) kbl(t,(x ixi °

+


with a positively oriented countour y+=Y+ in Im x >0 , sur-

rounding all poles there. Similarly we get

(6.6) glkj =fy+dx6(-ix)kIIalsN1 a(t)(rj,k,a/cN+l)(O,i,x,

with the polynomials rj=rjk,a in =(x,°) of (5.11). Note, the

rj,k,a are of degree sNj-j+Ial+k. Thus the polynomials in the nume

rator of (6.6) are of degree s Nj-j+k+N1 . Arguing as for prop.5.5

we find that the glkj are (finite) sums of homogeneous functions

with largest degree -N-j+k+N1+1 . After reordering we thus get a

homogeneous asymptotic series, and the following result:

Proposition 6.1. The system (5.2) of boundary conditions trans-

lates into a system of pseudodifferential equations

(6.7) =OGlkk = p1-B1w , 1=1,...,M

with pdo's Glk on r . After a local coordinate transform near a

boundary point onto the coordinates (v,i) of sec.3 the symbol

allows a homogeneous asymptotic expansion

(6.8) glk(t, °) _ 7j Oglkj(mod 1) at 1;°=oo

where glkj are homogeneous of degree -N+N1+1-j+k in ° , and

smooth in t . Moreover, the highest order terms are given by

(6.9) glk0(':' ° ) = fy+°+x n)

+ N

Here bl, 1(t, °)= 1bla(T)l;a is the principal part polynomial

N

of -'1( )- 1 °) In (6.9) we used the notation n=(1,0,..,0)

=e,= interior unit normal. is the above countour.

For existence of a solution of (5.2') we must find density func-

tions V0 .. VN-1 solving the system (6.7). Here we leave the term

B1w , wE ker C , undetermined, noting just that they are Coo(r)-

functions, belonging to a finite dimensional space.

In matters of uniqueness we also will translate the system

(5.2)1 into a system of 4do's. A similar procedure may be applied:

We now replace B1 , 1=1,...,M, by Da , 1=0,...,N-1 , and interior

by exterior derivatives. We get the following:

Proposition 6.2. The system (5.2)1 is equivalent to a system of

pseudo-differential equations of the form

(6.10) `k=OHlkPk = D1 w , 1=0,...,N-1


withtpdo's Hlk on I' . Locally, in the boundary coordinates of sec.

6, the symbol hlk of Hlk allows a homogeneous asymptotic expansion

(6.11) hlk(i, ) _ Ylj 0 hlkj (mod °) at °=o

where hlkj is homogeneous of degree 1+1-N-j+k. Specifically,

(6.12) (-i)kf7 dK k,1=0,..,N-1.

To prove uniqueness - or rather the finite dimension of the

null space - we must show that (6.7) and (6.10) together, imposed

on a set of density functions Vj , imply Vj = 0 (allow only finite

ly many linearly independent solutions). Again the right hand side

of (6.10) is C00(I') and belongs to a finite dimensional space.

We now are equipped for a result on the boundary problem

(5.2)' . For simplicity assume N even, and M=N/2 . We impose

Condition L-S: (The Lopatinski-Shapiro conditions) (i) In the

boundary coordinates of sec.3, for every &n-1 , the equation

(6.13) 0 , KE C

admits exactly k=N/2 roots 1=1,...,N/2, in the upper

half-plane Im K >0 , counting multiplicities.

(ii) For every 0A°E In-1 the polynomials

(6.14) b1, 1(t,l;°+Kn) , 1=1....N/2N

are linearly independent mod c:+ M =nImK >0 (K K1)1

Theorem 6.3. For an even N the elliptic boundary problem (5.2)'

under Lopatinski-Shapiri conditions, is normally solvable.

Proof. The point is that, under cdn.L-S, (a) we have the matrix

of symbols of rank N/2 ,

and (b) the matrix of symbols of rank

N/2 as well , and (c) both matrices together of rank N .

More precisely, under cdn.L-S, the polynomial c+(K) is

of (precise) degree M=N/2. By the Euclidean algorithm we may write

(6.15) (31(K)=b1,N1(t,°+Kn) = a1(K)cN(K)+pl(x) degree pl<M

with uniquely determined polynomials pl(K) of degree <M . cdn.(ii)

of L-S means that the M polynomials p1,...,pM are linearly inde-

pendent. Looking at (6.9) we find that


(6.16) glkO - fy+dK(-1K)kpl(K)/cN

since the integrand involving al(x)c+(K) is regular inside y+, so

that the integral is 0. Linear independence of the pl means that

(6.17) pl(K)=f4_lP1jKJ'

P=((plj))1=l'..,M,j=1,..,M-1 invertible.

Focus on the matrix ((glkO))1=I,..,M,k=O,..1M-1=Z . To show that

Z is invertible we must show that

(6.18) Y=((JY dK KJ+l/cN(O,t

is invertible, since, essentially, Z is the product of Y and P

But, indeed, Y of (6.18) is invertible: Suppose not, then there

exists a polynomial 9(x) of degree < M , not =O , such that

Jy dK 6(K)?(K)/cN(K) =0 for every polynomial X(x) of degree <M.

However, 9(x)/cN(x) must have poles inside y+, since

ahas poles

N

of total order M there, while s is only of degree <M , hence can-

not cancel all poles. If x0 is a pole of O/cN

then set T,=II(K-xl),

the product running over all other poles of i/cN . This choice of

X(x) is permitted, but the integral cannot be zero now - it will

give the residue of a holomorphic function with a (genuine) simple

pole at xO . Thus a contradiction results.

This verifies (a) above, giving a corner of the matrix of

rank M . Note that the same argument shows that any MxM-section

(6.19) ((g for R=0,1,..,M,1k0 1=1,..,M,k=R,...,R+M-1 ,

is nonsingular , for all T 1;° 340. Also, note that (b) followsfrom a similar argument: just note that c-(K)=-Zmx <0(K-xl) also

1

is of degree M, and that our matrix P above resembles the matrix

(6.12). Regarding statement (c): Suppose we have

(6.20) Ik=0glkOzk 0 1=1,..,M and "k=0hlkOzk 0 1=0,..,M-1

Repeating the above argument, this means existence of a polynomial

X(K) of degree <N (with coefficients (-i)kzk ) such that

(6.21) J dKX` 0=0 , j=0,..,M-1 , and J dx KJ=O , j=O,..,M-1y+ 0N y- cN

Thus it follows that X/cN is regular inside both curves y+ . Since

a non-zero polynomial of degree <N cannot cancel all the poles of


1/cN it follows that %mo, i.e., z0=...=zN-1=0 . Note we only used

half of the matrix hlkO'and have shown that the square matrix

(6.22) (g1kO " ..'gMkO'hOkO " ..'hM-1k,O)k=o,..,N-1

is nonsingular for every t°#0 .

For the proof of thm 6.3. it now will be necessary to convert

the nonsingularity of our matrices (6.19) and (6.22) into ellipti-

city of corresponding systems of equations. In that respect is

should be noticed that the principal symbols glk0 and hlk0 are

homogeneous in t° , but not of the same degree.

To remedy this we use the operator A=PL-'2 , with P=Ps of IV,

(5.3), on r , with its inverse A71=Q'2 , both i,do's in LCt1 , as

discussed in IV,5. Again, we might choose instead AA' as (inverse)

square root of the second order elliptic differential operator

(6.23) L = jw(I-0)w ,

with (1-0) in local coordinates, and a subordinated partition{w}

of a finite atlas on r . This choice indeed will work, but will

require an additional effort we tend to avoid.

In (6.7) and (6.10) introduce

(6.24) Vk=A kwk , k=O,...,N-1 .

lAlso, multiply the 1-th equation (6.7) by AN-1-N, and the 1-th

equation of (6.10) by AN-1-1. After this modification we arrive at

a new pair of systems oftpde's, now all of order 0, and in the new

unknown functions wk. The nonsingularity of (6.19) and (6.22) now

indeed imply that (a) the modified system (6.7) is elliptic of or-

der 0; (Ii) the modified system (6.7) and the first M equations of

the modified (6.10) again form an elliptic system of N equations

in N unknown functions. Green inverses may be constructed.

Conclusion: (1) Equations (6.7) are solvable if cpl-Blw satis-

fy finitely many linear conditions - i.e. the cpl satisfy finitely

many conditions, since the B1w are finite dimensional.

(2) Equations (6.7) and (6.10) together , with zero right

hand sides, have at most a finite dimensional solution space.

Again since the B1w are finite dimensional this means that there

can be at most a finite dimensional set Vk solving both (6.7)

(with (p k=0) and (6.10). This completes the proof of thm.6.3.

Problems: !)Set up a generalized set of Lopatinski-Shapiro condi-

tions working for an elliptic system of R equations in R unknowns,


under proper boundary conditions, just as (5.2) . Then generalize

thm.6.3 to this case. Hint: Virtually all discussions of the four

preceding sections generalize immediately. However, it now may be

practical to leave a part of cdn's L-S in a form using matrices of

complex integrals. 2) Same as (1) for an even order PDE on a comp-

pact C -manifold R with boundary. 3) Same as (1) for a PDE mapping

between sections of vector bundles over a of equal dimension (This

is completely formal). 4) Same as (1) for an exterior problem:

(6.25) c(x,D)u=f , outside r , Bju=cpj on 1' , if , f- E=- S,(ln) ,

where again if f" denote the zero-extensions (=0 in a, this time).

5) Same as (1), for a Riemann-Hilbert type problem:

(6.26) c(x,D)u=f in In\r , u,fE S'(ln) , Bju=(pj , j=1,...,M .

where now Bj = Bj,+ +Bju_ with.Bj$ containing interior (exterior)

derivatives only, resp. 6) Same as (1) for a noncompact domain S

with "conical boundary r ", in the following sense: The homeomor-

phism s(x)=7 of IV,1 maps MT onto a submanifold of B1 with clo-

sure a submanifold with boundary of B?. (Then the cpj must satisfy

suitable conditions at -. For example, in a chart of an infite

point, after a coordinate transform onto a piece of a half space we

might require yjE $(ln-l) - or only S'(ln-1) ? . Also, of course,

we then must require if , f- E S,(ln) .

7. Hypo-ellipticity, and the classical parabolic problem.

We now want to focus on some problems with features similar

to those of the heat equation - example (4.3) of ch.0. Generally

such problems will be of the form of an abstract Cauchy problem

(7.1) au/at = L(t)u + f(t), t>0 , u(0) _ q) ,

where L(t):X -* % denotes some family of linear operators, depen-

ding on t , and where f(t) is a functions of t taking values in X.

Also, cpE I , and the solution u(t) takes values in % .

The theory of abstract equations of this form, in a vector

space 8 with topology is well investigated ([Kal] , [ Fal] ). onespeaks of an evolutionary problem. The abstract differential equa-

tion (7.1) often is called an evolution equation.

Some formal comments on evolution equations: First consider

the case of a homogeneous equation -i.e. f.0. If a unique solution

5.7. The classical parabolic problem 175

exists for all qE X then a family {U(t):ta0} of linear operators

U(t):X-X is defined by setting U(t)T=u(t) with the solution u at t

for the initial value cp . Actually such a family exists for the

initial-value problem at an arbitrary point T instead of 0:

(7.1) T atu= L(t)u , t>T , u(T)=q .

Let U(T,t) , tit be the corresponding family. Then, for general

f(t) defined for ta0 , the solution of (7.1) formally is given by

t(7.2) u(t) = U(T,t)W + U(T,t)f(T)dr

0

It depends on properties of the family U(T,t) to be derived

whether the integral in (7.2) exists, and a derivative atu(t) is

meaningful. However, assuming that this is true, and that conven-

tional rules hold, it is confirmed at once by formal differentia-

tion that u(t) of (7.2) satisfies (7.1).

The family {U(T,t) : t2T} is called the (family of) evolu-

tion operator(s) (or solution operators).

For constant L(t)=L and f(t)=O the solution formally may be

written as u=exp(Lt)q. We get U(T,t)=eL(t-T). The collection

{U(t)=U(O,t)=eLt : tao} , then has the properties of a semi-group:

(7.3) U(0)=1=identity , U(t)U(s)=U(s+t) , s,t2:0 ,

as follows from the fact that then U(T,t)=U(t-T) .

In the cases considered here % will be a space of functions

or distributions on some domain or manifold sa , and L(t) a family

of elliptic differential operators on 11, of the types studied in

sec's 1-6: 2 is a "free space", as in sec.!. Or else, n 3 in has

smooth boundary -then we define L(t) by a boundary problem (2.1)".

That is, L(t) is an unbounded operator L(t):dom L =Y - %=COO(n),

where (for a domaih 11cc Wen with smooth boundary t) , we define

(7.4)Y = dom L = {uE X: Bju=O , j=1,...,M} ,

L(t)u = A(t)u , as u E Y ,

with a system {A(to),Bj}, for a fixed to, of PDE's fitting (2.2)"

Generally, A(to) is assumed elliptic. But stronger condit-

ions are needed. First we assume that at A(t) (as a PDE in (t,x))

is hypo-elliptic. Second, we require conditions insuring (7.1) to

have a unique abstract solution in some Hilbert or Banach space A.

Such solution u(t)=u(t,x) then will turn out to be smooth, either

by hypo-ellipticity or by cor.4.3, because it proves a distribut-


ion solution of the PDE atu-A(t)u=f (under proper boundary cdn's.

There are well developed abstract tools available, for this

2-pivot approach: For example we have a detailed abstract semi-

group theory (cf. [HP] , [ Yo 1 ] , [ Ka 1 ] , [DS,]). Also, an abstracttheory of evolution equations [Ka5]. For simplicity we stay with

semi-groups - i.e., L(t) of (7.1) is independent of t.

Definition 7.1. A strongly continuous semi-group of a Hilbert or

Banach space I is defined as a 1-parameter family {U(t):tZO} of

U(t)E L(I) which is strongly continuous in t (i.e., limt-toU(t)u=

U(to)u in I , for every uE I , toZO), and satisfies (7.4).

The Hille-Yosida-Phillips theorem [DS1],p.624, states that a semi-

group always is linked to a closed linear operator A: dom A -I

dom A C I , called the infinitesimal generator of {U(t)j, where

(7.5)Au=dU(t)u/dt(O) = limt..0,t>0((U(t)-U(0))/t) ,

for uE dom A = {all WE X for which the limit exists} .

If AE L(I) is a bounded operator, then we have

(7.6) U(t) = Lj=O(At)3/j! = exp(At)

Then U(t) is uniformly continuous in t - and even extends to an

entire function of t, defined on T. In the general case (7.6) no

longer makes sense, although one still might like to write U(t)

=exp(At). The theorem describes the class of closed operators oc-

curring as infinitesimal generators of a strongly continuous U(t):

Theorem 7.2. (Hille-Yosida-Phillips) The class of infinitesimal

generators of C0-semigroups coincides with the class of closed ope-

rators A such that (i) the resolvent R(X)=(A-X)-l exists for a

half line {X>Xo}, (ii) for a constant M>O independent of we get

(7.7) IIR(X)JIJ s M(X-Xo )-j , a,>ko , j = 1,2,... .

Then we have

(7.8) U(t) = limj->Oo(1-tA)-j = t>0

and, vice versa,

(7.9) R()) f dte-ktU(t)dt , Re X >ko0

For TE dom A we have u(t)=U(t)cpE C1([0,00),I) .

5.7. The classical parabolic problem 177

One will recognize well known formulas for the exponential

function as basis of (7.8) and (7.9). Note also that, under (7.7),

formula (7.8) defines the resolvent in an entire right half-plane.

Looking at (7.1) we will focus on the following two cases:

Case (F) : L=A=a(x,D) with an md-elliptic differential ope-

rator on a manifold sz with conical ends. (or a compact manifold st

- then A needs only to be elliptic) .

Case (B) : L is an operator of the form (7.4) involving a

domain S with smooth boundary, an elliptic differential expression

A=a(x,D), and Lopatinski-Shapiro-type boundary expressions Bj .

In order to qualify thm.7.2 we will look at (7.1) in the case

where % is a Hilbert or Banach space. Choose X=H=L2(sz), in either

case. Then (7.4) defines an unbounded operator of H - in case (B),

but this operator is not closed. Its domain is dense in H, and its

adjoint L* exists (its domain containsC000

(n), dense in H, hence is

dense). Thus the closure L** exists. We may look at the problem

(7.1') u(t)E CI([0,oo),H) , au/at = L**u , ta0 , u(0) = tp E Y .

Similarly, in case (F) , we introduce the operator L by

(7.10) dom L = CW(a) , Lu= Au , u E dom L

Again, dom L is dense in H, dom L* contains C00(st), thus is dense;0

the closure L** exists, and we may look at (7.1') with this L**

Now, suppose the closed operator L** of H satisfies the con-

ditions of thm.7.2. Then indeed a solution of (7.1') is given in

the form u(t) = U(t)q , U(t) = exp(L**t) . Indeed, (7.3) implies

(7.11) au/at(t) = limh,0,h>0 U h)-1 u(t) = L**u

in particular, because (7.8) implies U(t): dom L** -, dom L**

In other words, for cpE Y=dom L C dom L**, u(t)=exp(L**t)T be

longs to C1([0,00),H), and solves (7.1'). Moreover, then even u(t)

of (7.2) with cpE dom L**, fE C([0,00),dom L**) (under graph norm

of dom L**) is C1([0,00),H) and solves

(7.12) atu = L

**u+f , t2-0 , u(0)=c ,

by a calculation. Particularly this holds for yE Y, fE C([0,00),Y).

Also one shows uniqueness of such u as a solution of (7.12).

We first focus on case (F). A solution u of (7.12) also is a

distribution solution of the PDE au/at-Au=f in (0,oo)x11 . Indeed,

(7.13) fu(-app/at-A"V)dxdt= f (atu-L**u)4dxdt=(f,tp) , y,E C0-((0,00)xst).


Thus, if fE C'([O,oo)x12) and at-A is hypoelliptic, get u(t)=u(t,.),

where u(t,x)EC00

((0,o)xn): u is a classical solution of (7.1).

In fact, we can put to work our boundary hypo-ellipticity

theorem (thm.4.4): The solution u(t) of (7.1') is CI([0,00),H).

It follows at once that v=u°=u , tZ 0 , = 0 , t<O , satisfies

(7.14) (at A)v=6(t)0 u(0)+f = 6(t)O cp + f ,

in the sense of distributions on $x12 . Clearly the differential

expression C=at A satisfies (4.5) (with n+1 variables v=t , i=x).

Using thm.4.4 it follows that v admits an ETE of order 0 , hence

it follows that uE C00([0,o)xc) . We have proven thm.7.3, below.

Theorem 7.3. In case (F) assume the differential expression C=at A

hypo-elliptic. Let the closure L** of the differential operator L

defined by A in the domain dom A = C0 (St) satisfy the assumptions

of thm.7.2. Then there exists a solution uE C00([0,o)x12) of the

Cauchy problem (7.1), for general pE CD(c) , fE C'([0,co)xf)

supp fC [0,00)xK, KCB n . The solution is unique if a is compact.

Proof. After the above, we only comment on uniqueness: The solut-

ion of (7.1') indeed is unique, by thm.7.2. If 11 is compact then

u(t)=u(t,.) of thm.7.3 also solves (7.1), hence must be unique.

Remark 7.4. It is well known that such solution needs not to be

unique if a is non-compact. Of course, the condition at x=- that

u(t)EE C1([O,00),H) , H=L2(St) , insures uniqueness as well.

Now let us look at case (B). If the semi-group U(t)=exp(L**t)

exists we again get a unique solution of (7.12), for TEC00(Stl.d') ,

fE C0o([0,o)xnLF) , in the form (7.13). Using a localized version

of thm.4.4 (cf. rem.4.8) it then may be concluded, as in the free

case, that u(t,x)E C0o([0,co)x(S& ')) - assuming that (i) at A is

hypo-elliptic, and (ii) A is noncharacteristic along r :

Theorem 7.5. In case (B), assume that L** is infinitesimal genera-

tor of a C°-semi-group. Assume (i) and (ii) above. Then (1.1) ad-

mits a unique solution u(t)=u(t,x) E C00([0,o)x(SZId')) , for every

qE C'(OUG), fE Coo([0,oo)x((&F)), q and f(to ,.) satisfying the boun-dary conditions for all to (-= [0,-) .

Finally we ask: Will our results on elliptic problems, of

sec's 1-6, give control on cdn.(7.6) for L** above: Clearly (i),

(ii) of thm.7.2 only involve (L**-X)-I

= R(?) , for L**. Elliptic

theory will supply a Fredholm inverse, even a (special) Green in-

verse for L-? : In case (F) we need md-ellipticity of A-X, for

5.8. Spectral and semi-group theory 179

case (B) we need A elliptic and cdn's L-S for Bj (cf. thm.8.3).

Still, looking at a variety of examples one finds that

md-ellipticity of A-% alone in case (F) or ellipticity and cdn.

L-S for case (B) though it might be helpful, is not enough.

The insight of the present section is the fact that (i) hypo-

ellipticity of C=at A and (ii) the non-characteristic property for

r with respect to A (or (0,00)xr with respect to at A) reduces the

Cauchy problem (for case (F) and case (B)) to a study of the resol-** -1vent (L -a.) , inverse of the closure of L with respect to H.

Commonly one refers to an evolutionary problem (7.1) with hypo-

elliptic at A and well defined semi-group U(t)=exp(L**t) as a para-

bolic problem. Often one requires also that U(t) is analytic (cf.

Yosida [Yol], also Fattorini [Fal]). As application we mention:

Example 7.6.

(7.15) au/at=k=1

a/axj(ajk(x)au/axk) + f , t20 , u(0,x)=g ,

either in free space (xE &n) or in a bounded domain f with smooth

boundary r. In the latter case one will need a Dirichlet condition

(u=0 on r) or another self-adjoint boundary condition. Assume the

symmetric matrix ((ajk(x))) real and >0 for all x (and smooth). In

case s--&n one assumes that L-% (of (7.10) is md-elliptic of order

(2,0), for a.>0. This requires ajkE 1)c0 , i.e.,

(7.16) ajk(a)(x) = O((x)a') , x E In , all a

and A uniformly elliptic in In .

Under such assumptions the operator C=at -A indeed satisfies

cdn's (i) and (ii) . Moreover, L** then is self-adjoint (admits a

spectral decomposition) and semi-bounded below (sec.8,9). Thm.7.2

is applicable for A=L**. Hence thm.7.3 (or thm.7.5) apply.

8. Spectral- and semiaroup theory for pdo's.

In this section we shortly consider spectral theory of an

unbounded closed linear operator L:dom L-> H with domain dom L C H

of a Hilbert space H. Generally, L will be a realization of a dif-

ferential expression A in the well known sense, where A=a(x,D), a

(locally) of the form (1.1), is given either on In , or on a com-

pact manifold it, or on a noncompact a with conical ends, or on a

subdomain a CC In with smooth boundary as in sec.2.

In each case we set H=L 2(n) - but methods will apply to prop-


erly defined L2-Sobolev spaces as well. A realization of a PDE A

on a is defined as a (closed linear) unbounded operator L of H ex-

tending the "minimal operator" Lo (with L. u--Au as uE dom Lo =C00(a) )For aCC &n with smooth r we assume L the closure (in H) of an ope-

rator of the form (7.4) (that operator is called L' here). Equat-

ion L'u=f (for uE dom L') then corresponds to the boundary value

problem (2.2") , while L=L' . In all other cases we set L=LO

i.e., L is defined as the closure of the minimal operator Lo .

We only consider the elliptic case: For a compact a or the

case of an aCC &n with boundary assume A elliptic and A, B1.... BM

Lopatinski-Shapiro, respectively. For a noncompact St (with conical

ends) we assume A md-elliptic (of order m=(N,m2)), mz20, and the

coefficients of A such that AE LSm,p,0 , with p,=1 , p2 >0 .

The spectrum Sp(L) is defined as complement of the resolvent

set Res(L) _ {XE 0:(L-X)-lE L(H)J. Control of Sp(L) (and of spec-

tral theory of L) depends on criteria for existence of (L-%)-

None of our above criteria of elliptic theory gives existence of a

precise inverse of a pdo, but we have a generous supply of results

giving a Green inverse. Let us first omit the case cc in with T,0

Note that, for m2>0 in case of conical ends (in general for com-

pact 2) not only do we have A md-elliptic, but we even have A-X

md-elliptic, for every XE M . Thus a Green inverse Gk of AX=A-X

exists for every XE T . Moreover, Gk may be chosen such that

(8.1) (A-X)GX=1-PkerA.,GX(A-X)=l-PkerA.*

with finite dimensional orthogonal projections PkerX of H

Theorem 8.1. In case of 12=ten, or n compact, or n noncompact with

conical ends assuming that m2>0, if at least one XoE $ exists with

(8.2) ker(L-X0)=0 , im(L-X0) dense in H ,

then L is of compact resolvent. That is, Sp(L) is a discrete set,

at most countable, with no finite cluster point. The resolvent

R(X) = (L-X)-1 is a compact operator of H , for every XE Res(L)

Proof. First we note that ker(L-Xo)=ker(A-Xo)={u6S'( °°:(A-%o )u=0}.

Indeed let LX=L-X. Clearly LXu=O means existence of ujE C- with uj

->u, Auj->0, as j--. Convergence in H implies convergence in S'.oAl

so AX:S'->S' is continuous. Thus Aku=O in S' follows. Since OE C

and LX is hypo-elliptic we conclude that uE Coo. Moreover, from III

thm.4.1 we know that uE S, since A% is md-elliptic. Vice versa,

every uE S with AXu=O belongs to ker LX. By our assumption it fol-


lows that ker Axo={0}. Hence PkerA =0, and (8.1) yields GXoAXo=1.,o

Specifically, for u=AXo -Iv we get GXo V=(G%O A%o )AX -Iv=AXo -Iv,vE im(Ao -Xo ). Since in LXo is dense the same holds for im(Lo =Xo ) ,since LXo is the closure of Lo -Xo . Therefore (L-Xo )-I is bounded.Its domain is H (since it is closed). Moreover, R(Xo)=(LX

)-1=Gxo

in all of H. Moreover, GX,E O(-m) is a compact operator of H, by

III,thm.5.1. Thus indeed L is an operator of compact resolvent,

and the remaining statements follow from standard functional ana-

lysis (cf. Kato[Kal], 111,6,8 or [DSI]) . Q.E.D.

Corollary 8.2. Theorem 8.1 holds also for fCE &n, r#0: If A is el-

liptic and (A,B1,...,BM) satisfy cdn. L-S, then (8.2) for at least

one Xo implies L to be of compact resolvent.

Proof. This depends on the construction of a 'Green inverse' for

the operator L' (i.e., the operator L of (7.4)). The term Green

inverse should be redefined for operators acting on a bounded

domain of &n : We will take this in the sense of the well known

generalized Greens function, defined as integral kernel of a spe-

cial Fredholm inverse of an operator of the form (7.4).

Theorem 8.3. Under the general assumptions of sec.2 let A be ellip-

tic. Assume that (A,BI,...,BM) satisfies cdn.L-S. Then there

exists an integral operator G. : C°O(s LF) ' dom L' ,

(8.3) Gxu(x) = fa

gk(x,y)u(y)dy , u(=- C'(fl )

such that

(8.4) GXLXu=u-PXu , uE dom L1, LkGkv=v-Q%v , vE C00(slld')

where P., QX are projections of finite rank onto ker L' and a com-

plement of in L' in C0O(nLF), and where the integral exists as a Le-

besgue (or improper Riemann) integral. We call gX a generalized

Green's function of L We get gX(x,y)E C00 (Slxcl\{ (x,x)E Slxsa}). At x=y gX has the same singularity as the distribution kernel of the

Green inverse of C-X, C as in (2.5), as in sec.3. Calling that ker-

nel h (x,y), for a moment, we have gX-hXEC00(S1xSa). Moreover,

GX

: C (Slid')- dom L' extends to a compact operator H+H.

Proof. We may assume X=0, since A and AX=A-X have the same princi-

pal part, hence AX is elliptic ((AX,Bj) is L-S) if and only if A

((A,B.)) is. Given a uE dom L' solving L'u=f - i.e., a solution003

uE C (Said') of (2.2"). Let C be the md-elliptic expression of sec.3


extending A to Rn, and let G be a Green inverse of C, as in sec.3.

With the procedure of sec.3 we convert (2.2") into (3.11). Assume

ker C = ker C"=O for simplicity. Again we are only interested in

achieving equality in (3.11) inside a , and it suffices to set

VN/2= " 'VN-1=0 , then solve the system (with interior derivatives)

(8.5) `j/U-1 BkGO = Bkf" k=1,...,M=N/2j

With the procedure of section 6 - using the ipdo's A defined near

(6.23) we set Vj=A iwj , and multiply the k-th equation (8.5) by

N-1-NA 1

. One obtains an elliptic system of order 0 for the wj

N-1-N(8.6) Hw = -(A B1Gf")1=1,...,N/2

N-1-N1With a Green inverse K of H we get cu=-K(A B1Gf") +Pw , P of

finite rank. Here K will be a ydo of order 0. Again assume P=O,

just to clarify the idea of proof. In detail, we get

u = Gf" + /2-1 GSA , xE n , where(8.7) 4=0 Vi

N- 1-N'W j = 4/2A7 jKjkA k[ BkGf" ] T, int

where Klk are certaintpdo's on r , of order 0 .

The second term in (8.7) is of the form Zf with an integral

operator Z on tt with C0O(ttxa)-kernel : Let xE a. For fE C0(n),i.e

f=0 near r, get f " E C00(&n) , a'Gf" _ JOa g(x,y)f (y)dy, with the

distribution kernel g of G. The integral exists as Lebesgue (or im-

proper Riemann) integral. a g is C" except at x=y, where it has a

singularity of order N-n-Ial. The A3 may be written as j-th order

differential operators with 0-order yido's as coefficients - for

N-1-Nexample A=A1P1, Ps of IV,(5.3). Hence Kk1A 1B1Gf =27kaaaGf

with 0-order ydo's Jka , and a sum over laIsN-1 . For xE T , yE

supp f the kernels ax (x,y) are C00(T) . Hence we get

k N-1-N1(8.8) A Kk1A B1Gf"(x) = Jvk1(x,Y)f(Y)dy I vk1E C (tuts)

Combine this with

f(8.9) GO (x) =(-1)J aj

r vYg(x,Y)W.(Y)dSY

to conclude that the second term in (8.7) is as described.


An operator Ku(x) f k(x,y)u(y)dy withk(x,y)=O(Ix-yIE-n)

is bounded as a map L2 (St)- L2 (I'), whenever E>2, by a variant ofSchur's lemma (We leave the little exercise to the reader). Conclu

N-I-Nsion: A bounded map L2 (.Q)-L2 (t) is given by f-' A kKklA 1B1Gf- ,

and also by the maps f- Vj defined by the second line of (8.7).

Finally look at the maps L2 (I')- V (a) induced by 'y» GS, def-ined explicitly by (8.9). We may assume ksN/2-1<N-1, since our sum

in (8.7) only extends to these cases. For JalsN/2 look at aaGSkk-=

jl.dSyaXavg(x,y)V(y) where the kernel also has a singularity of or-

der N-n-k-Ialal-n. These operators are of the form adjoint to tho-

se L2 (SZ)- L2 (I') used earlier. Thus they are bounded as well, bythe same Schur-type argument. Assuming Na2, defines a boun-

ded operator of H, for every k of the sum (8.7). The operator even

is compact, by III,thm.5.1, since VX is bounded, for every first

order differential expression V (i.e., we may write X as a finite

sum D j(LjX) with QjE LC-1).

The kernel of G has the corresponding properties, hence the two li-

nes of (8.7) define an operator of the form (8.3) with all proper-

ties of thm 8.3 (for ?=0). Let us next admit P#0 in (8.6). P

may be chosen as an orthogonal projection (in L2(n) onto a space

%C COO(T) with dims<oo, by standard elliptic theory applied to H. We

still get (8.7) if only the right hand side of (8.6) is 1 X. This

amounts to f 1 vj, j=1,...,R, with v1E COO(c&f). With the orthogo-

nal projection Q in H onto span{vj} introduce F=Z(1-Q), Z the ope-

rator of (8.7). Confirm that L'F=1-Q, while F of course still is

compact in H and has a kernel just like Z. In particular F is a

right Fredholm inverse of V. But we saw in sec.7 that L':Y-X is

Fredholm. The space ker L' consists of C°° functions. We get a left

Fredholm inverse W of L' such that WL'=1-PkerL" Then get WL'F=W-

WQ=F-FPkerL " hence F is an H-bounded Fredholm inverse of V. The

special Fredholm inverse satisfying (8.4) differs from F only by

additional Fjcj) (Xj , Kit XjE COO. Theorem 8.3. is established. (We

still assume ker C =ker C-=0 , but its removal is a technicality.)

Proof of cor.8.2. We get II(1-Px)uUI=IIGxLx(1-Px)uIIs cIIL'(1-P,)uII

uE dom L' . Taking closure it follows that

(8.10) IILull a cliull , for all u 1 ker L' , uE dom L


This implies ker L = ker L' C COO(c& '). Clearly we may take closure

in (8.4) - i.e. substitute L' by L there. Similarly as in thm.8.1

we conclude Gko Lko u=u, uE dom L, G. v=L,,o v, vE in Lko . Lko -1 isclosed hence im Lko=R, R(X)=Lk-1 exists (compact) at k=Xo. Q.E.D.

Definition 8.4. The operator L is called hermitian if

(8.11) (u,Lv) = (Lu,v) for all u,v E dom L

L is called formally dissipative if

(8.12) (u,Lu) + (Lu,u) s 0 for all uE dom L

Both conditions may be verified for Lo (in case s2=&n, St com-

pact, n with conical ends) or for L' (in case 12CC &n) only: (8.11)

((8.12)) holds for dom L if and only if it holds for dom Lo - or

dom W. Indeed, L is the closure of either L. or L', respectively.

In the cases without boundary these conditions may be expres-

sed by direct reference to the differential expression A : 'Her-

mitian' means that A=A* (with the Hilbert space adjoint A* of A).

'Dissipative' amounts to (u,(A+A*)u)z0 for all uE C"(cZ) i.e., the

minimal operator of the expression A+A* is positive. Such expres-

sion A will be called self-adjoint (dissipative) , respectively.

For a boundary problem L' we also need the above conditions

for the expression A . But, in addition, a postulate on the boun-

dary conditions results: Certain self-adjoint or dissipative boun-

dary conditions (with respect to a given expression A) are to be

defined, such that L' is hermitian (dissipative) if and only if

A is self-adjoint (dissipative) and the boundary conditions are

self-adjoint (dissipative) with respect to A .

For the general case of an L-S-system (A,B the discussion

with self-adjoint (dissipative) expression A the discussion of

general self-adjoint (dissipative) boundary conditions is quite

complicated, and will not be attempted. A simple example will be

the case of the Laplace operator L=0 : Self-adjoint L-S-conditions

with real coefficients all will be of the form

(8.13) bo avu+b, u=0 , xE T ,

with real-valued bo, b, , bo 2 + b 2 =1We exclude the case n0Z &n for the remainder of sec.8. A result

similar to thm.8.5 for boundary problems is found in sec.9.

Theorem 8.5. Assume 12=&n , n compact , or n noncompact, with coni-


cal ends. Let A be md-elliptic of order (N,mz) with N>0 , m2a0 .

Assertion:1) If A is self-adjoint then L is not only hermitian but

even a self-adjoint operator of R=L2(f): It admits a unique ortho-

gonal spectral measure (cf. [C 2],I,thm.3.3) If mz>0 then L is of

compact resolvent. It admits a complete orthonormal set of eigen-

functions {cpj}j=1,2,... , with (pE S(c) .

2) If A is dissipative then L is infinitesimal generator of

a semi-group Ut = etL E L(H) , tZO, in the sense of thm.7.2.

Moreover, the operators Ut are contractions : we have

(8.14) IIUtII = 1 .

Again, for m2>0 L is of compact resolvent.

Proof. The assertion is equivalent to showing that for X=tia (for

?=E>O) the equation (A-?)u=O has no nontrivial distribution solu-

tion u with uE H . But we know that for md-elliptic A+X every

such u belongs to S . Note that A-A , for X=tEi (?.=E) indeed is

md-elliptic for small s>0 due to m:a0 , and since we get

real (or Re for self-adjoint (dissipative) A .

On the other hand, we clearly get SC dom L , hence (L+1,,)u=O . Thus

(8.15)0 = Im((L±Ei)u,u) = e1IuJI2 u = 0 , for A self-adjoint,

0 = Re((L-a)u,u) s-EIIu1I2 u = 0 , for A dissipative.

This argument shows that indeed im(LtEi) is dense in H for A self-

adjoint (since every uE H, u I im(L±Ei) will be a distribution

solution (in HC S') of (A;ir)u=0 u=0) . Also (Ltai)u=0 u=0

by an argument as in (8.15), and, moreover II(LtiE)ulla£IIuII implies

that R(;ai) = (L.ai)-l E L(H) exists. By a well known result (cf.

[C 2],I,cor.2.3, or [Ka,],V,thm.3.16) this implies self-adjointness

of L in H - i.e., L has the desired spectral resolution. If m:>0

one may apply thm.8.1 with ?.o=±Ei: L is of compact resolvent. This

implies discrete spectrum of L , and the orthogonal set {Tj} .

On the other hand, for dissipative L we similarly get exi-

stence of R(E) _ (L-E)- E=-L(H) , and, IIR(E)IIsE-1 . This implies

(7.7) with M=1 , by a simple additional argument. Hence thm.7.2

applies, and we also get (8.14) and the compact resolvent - the

latter for m2>0 only - from thm.8.1 again, with Xo=a . Q.E.D.


9. Self-adiointness for boundary problems.

We discuss an extension of thm.8.5 for boundary problems,

for some examples like the cdn's (8.13) for L=A, and the self-ad-

joint case. These matters are very technical. Perhaps we point out

the high lights and leave complicated calculations to the reader.

For an A as in sec.2 , nCC &n , consider the sesquilinear form

(9.1) Q(u,v) = f dx(Auv-uAv)dx , u,vE C"O(r&f)

In coordinates (v,t) (sec.2) for u,vE CO'(nUP) let uA=(aju)1=0..N-1

v°=(ajv)j=O,...,N-1' for xE r. By partial integration get

(9.2) Q(u,v) = (Qu°,v° )I. , where (u° ,V°) =f dS u°Tv°r r

with an NxN-matrix Q=((gjk(t,at)))j,k=O...N-1 of differential

expressions Qjk gjk(t,at) in t . Here Qjk 0 for j+kaN, and Qjk is

of order N-1-j-k for j+ksN-1. Specifically the 'anti-diagonal

terms' Qjk=gjk(t) : j+k=N-1 , all are functions. They are bounded

away from zero if A is elliptic. Note iQ is a formally selfadjoint

expression on r , we get Q(v,u) Q(u,v) , u,v E CO0(c f) , hence

(9.3) (Qu° vA )I. = -(u° QvA for all uA vA E C°°(r) .

For simplicity consider the case where all other Qjk are of

order 0 as well (A=A is such a case). Also assume that the bounda-

ry operators Bj all involve normal derivatives only - i.e., in the

coordinates (v,t) they may be written as a set of M=N/2 linear

conditions on the vector u°, without any differentiation for t, of

the form bjTu°=0 , xE G, j=1,...,M. (Again this holds for (8.13).)

Then Q=((gjk(t))) is an invertible matrix of functions (as

lower right triangular matrix with nonvanishing antidiagonal). The

boundary conditions Bju=O , j=1,..,M, of (2.5 translate into

(2.2)4 u°E Zx , for all x E r ,

with a linear subspace Zx of TN (all uA with bTu°=0) , at xE r

The property of a given fE C0O(ar) to satisfy the condition

(9.4) (f,Au)=(Af,u) for all uE Cw(sz1J') with Bju=O, j=1,..M=N/2,

translates into a condition of the form

(9.5) f°(x)E (QZx)1 = Zx , for all xE r

5.9. Self-adjointness for boundary problems 187

Clearly (9.5) may be described by 'adjoint boundary conditions'

(9.6) Sjf° = 0 , j=1,...,N/2

again. Then L (defined by (A,Bj)) is hermitian if and only if the

boundary conditions (9.6) are equivalent to the conditions Bju=O.

Theorem 9.1. Assume A elliptic and (A,Bj) to satisfy cdn. L-S .

Let the differential expression Q of (9.2) be of order 0 (i.e.,

an invertible NxN-matrix ofCOC

functions on r) , and let the

boundary operators Bj only contain normal derivatives. Assume

L of sec.8 hermitian. Then L is self-adjoint - it has a spectral

resolution - and is of compact resolvent as well. There exists

a complete orthonormal basis of eigenfunctions.

Proof. We argue similarly as for thm.8.5: Given fE H with

(9.7) (f,(L'-z)u)=O , for fixed zE C , all uE dom L' .

Self-adjointness of the closure L=L'** follows if we can show that

(9.7), for z=±i, implies f=0, ([C2],I,cor.2.3, or [K1],V,thm.3.16)

Now (9.7) implies that (A-z)f=0 , xE 1t , in the sense of distribu-

tions. Since A is hypo-elliptic conclude fE C00(f2). Let f- and C be

as in sec.3. We get (C-z)f- = h , supp hC r . The distribution h

must be of the form h=f=OS , y, jE D' (r) , S of (3.3).j j

In fact, we claim that R=N-1. Indeed, let v(x) - so far only

defined near r - be extended into a C00(&n)-function, also called

v=v(x) . For any uE C00(nUP) conclude that vNu=vE dom L' , since

clearly v°=(av(vNu))=0. Thus it may be substituted for u in (9.7),

implying f dxf(A-z)(vNu)=0 for all uECOO(fLf). In turn, this yields

(9.8) (f-,(C-z)(vN(P)) = f of (C-Z) (vNcp)dx=O for all cpE D'(&n),

with C and the "zero-extension" f- of f (=0 outside f&f) defined

as in sec.3. In other words, we get vN(C-z)f-=vNh=O in D1 (,n)A calculation shows that (,N6 j,,P) =('4', av (vN(P) r) r =0 as jsN-Ibut =j!/N! as jaN . Thus it follows that Vj=0, jaN.

Observe that we obtained a more explicit description of

the distributions ,y j in (3.7), but only in case of fE COO(c2Lf )With our matrix Q of (9.2) we get


(9.9) j=p ,(p) _ (Qfg°)r , q ED'J

as a careful comparison of (3.4) - (3.7) with the above shows.

We were assuming that the matrix Q contains no differentia-

tions but simply is an invertible matrix of smooth functions on r.

On the other hand, we can meaningfully define the vector f° _

=(avf)j=0,...,N-1 also for our above f E H : We get

(9.10) (C-z)f h = =O6`J

J

where C is noncharacteristic on r . With the same iterative method

as in sec.4, conclude the existence of distributions fj E D'(r) ,

j=O,...,N-1 such that, locally, near r , in the coordinates (v,s),

jED'(r)( 9 . 1 1 ) ( c-z)(f"-j

3 = 0j=a

Clearly these fj are uniquely determined. A close examination

shows that then f°=(fj)j=0,...,N-1 , may be substituted in (9.9),

above, under our present assumptions. In other words, we get

(9.12) ((C-z)f-,(p) = (Qf°,q ) r , (pED'(1Rn) .

Introducing some complex conjugates we conclude that

(9.13) 0 = ((C-z)f ,(p)=(f,(A-z)gPIcj) (Qf° q>E D(1n)

Or, in other words, equation (9.8) reduces to

(9.14) (Qf ,u°)=0 for all u° with bjTu° = 0

Or, recalling self-adjointness of the boundary conditions, one

concludes that we must have the boundary conditions Bjf=O satis-

fied for f in the following generalized form: We have

(9.15) Bjf = bjTf° = 0 , j=1,...,M=N/2 .

We also have (A-z)f=0 and may rederive (3.11) in the present case

(with G replaced by Gz). Thus f- again is given by a sum of multi-

layer potentials, only the y,j no longer are C" , but VjE D'(r).

Thus the cuj=n-jy, must satisfy a homogeneous elliptic system

of Wo's. Since ellipticity implies hypo-ellipticity, the Vj are

C00(ct) after all. Hence also the fj are C" , and f E C" (c&F) fol-lows. In other words we get fE dom L'. Then, however, we must have

(9.16) 0=Im(f,(L-z)f) = (Im z)J1f112 f=0, if In z #0. Q.E.D.

5.10. C*-algebras 189

10. C*-algebras of y,do's; comparison algebras.

In this section we start from the following observation. From

III, thm.3.1 we know that Opyic0 is an algebra of bounded operators

on H=L2(Kn) (in fact, a subalgebra of L(RS), for every s=(s1,52)).

Moreover, this algebra contains its (Hilbert space) adjoints, by

1,2 and 1,6. For each s the norm closure of Oppc0 in Hs will def-

ine a C*-subalgebra (called As) of L(HS). Commutators in Optpc0

are in Opyic-e , as we know (cf.I,6). Thus III,thm.5.1 implies com-

mutators to be in K(Hs), sE &Z. It is known that the compact opera-

tors of a Hilbert space I form a norm closed ideal K(X) of L(X).

We claim that our algebra As contains K(Rs) , for every s

Indeed, OptpcOC As clearly contains the operators

(10.1) sj(M) , sj(D) , j=l,...,n, where sj(x)=xj/(x) ,

and sj(M)u=sj(x)u(x). The pdo's (10.1) generate a (unital) subalg-

ebra Cs of As, a comparison algebra in the sense of [C2],V,1. Thus

As D Cs contains K(H5), by [Cz],V,lemma 1.1. (The proof is simple:

An irreducible C*-algebra containing at least one compact operator

#0 always contains K(H) (Dixmier [Dxl], cor.4.1.10). Thus we prove

Cs irreducible and get O#CE Kf1C (details cf. [C.],p.130,thm.1.1.).

Quoting another result on C*-algebras: The quotient of a C*-

algebra by a closed ideal is a C*-algebra again. Accordingly, the

quotient A °= A K(Rs s /

s) is a commutative C*-algebra.

Let us focus on A, = AO. . By the Gelfand-Naimark theorem

([DS2],IX.3.7 or [C.],p320,thm.7.7) A° is (isometrically isomor-

phic to) a function algebra C(ffi) with a compact space X. One will

ask about the nature of the space ffi , and the isomorphism A"-C(ffi).

These facts are intimately related to normal solvability of

the equation Au=f, for AE A. First, by a theorem of Atkinson ([C.].,

p.271,thm.4.8) an operator AE L(H) is Fredholm if and only if its

coset A"={A+K(H)} is invertible in L(H)/K(H). Second, A", as a C*-

subalgebra of L(H)/K(R), contains its L(H)/K(H)-inverses: A"E A"

is invertible in L/K if and only if it is invertible in A', by ano-

ther well known result ([C.],p.322,1.7.15). Hence Au=f (for AE A)

is normally solvable if and only if the continuous function a asso-

ciated to A° by above isometry is invertible in C(ffi): a(m)#0, mE ffi

In other words: For an AE A let us introduce a continuous

function a=OAE C(R), called the symbol of A (relative to A) using

the map A- A- =A/K(R)-.C(I). Then the above chain of arguments gives

Theorem 10.1. An operator AE A is Fredholm if and only if its


symbol aA(m) , defined on It , does never vanish.

As in [ C, ] , [C: ] we call II the symbol space of the algebra AOAsking about the nature of I and the homomorphism E:A-C(D1) we susp-

ect a relation between 'algebra symbol' aA and '1pdo-symbol' a(x,l)

=symb(A) for the generators OptpcO. These questions were discussed

in detail for the algebra C with generators (10.1) by E.Herman and

author, using methods unrelated to tpdo-theory ([CHe1], thm.36 -the

algebra C is called T there - (cf. also the algebra called 4s in

[Ci],p.135/136,pbm's 1-4). Clearly the 4do-symbols of (10.1) are

(10.2) for A=sj(M) , for A=sj(D).

The functions si(x) are continuous over &n. Recalling the

directional compactification Sn of II,(3.1), note that the sj even

extend to functions in C(Bn) - actually, Bn is described as the

compactification of in allowing continuous extension of sl,...Isn

Clearly the ido-symbols (10.2) of the generators (10.1) of C

extend to continuous functions over the compact space

(10.3) BnxBn = n} .

Moreover, by the Stone-Weierstrass theorem, the si(x),sj( ) genera-

to the algebra C(BnxIn) - they strongly separate the space (10.3).

The result of [CHel] regarding C may be stated as follows.

Theorem 10.2. The symbol space ffi_ IC of the algebra C is (homeo-

morphic to) the boundary (cf. 11,3) of the space (10.3), i.e.,

(10.4) ffi°= a(BnxIn)=BnxaIn U aBnxBn = IxI+jtj_.}

Moreover, the symbols of the generators are the restrictions to I

of the continuous extensions to BnxBn of their pdo-symbols:

(10.5) as,(M)=si(x), as BnxBn ,

Accordingly an operator AE C°, with the algebra CO finitely

generated by the operators (10.1), is a Fredholm operator of L(H)

if and only if its pdo-symbol

a 9 > 0 for all 1/11 ,

for some ii chosen sufficiently small.

The last statement was inserted because it may clarify the

relation between thm.10.2 and II,thm.1.6: We may rephrase (10.6)

by stating that AE C°C is ad-elliptic (of order 0). By II,

thm.1.6 this is necessary and sufficient for existence of a K-para-

metrix (hence also a Green-inverse) in Op4co .


On the other hand, AE C° is a finite sum of finite products

of the generators (10.1) hence aA is the corresponding sum of

products of sj(x) and The sj(M) and sj(D) do not all com-

mute, but their commutators are in OpVc-e , hence their pdo-sym-

bols vanish at lxI+11I=c . Since sums and products are finite the

statement aA#O on 111 of thm.10.1 indeed is equivalent to (10.6). In

other words, for AE C° the abstract discussion leading to thm.10.2

gives a very similar necessary and sufficient criterion on existen-

ce of a Fredholm inverse. [In fact, similar considerations for the

Cs guarantee existence of a Green inverse as well if and only if

A is md-elliptic of order 0 (cf. [Ci],p.149).]

Let us now state a result for the algebra A. The symbol class

VCo forms a sub-algebra of CB(RnxIn), the set of all bounded conti-

nuous functions on Rnx1n. The closure (i,co)c (under sup-norm of

CB(RnxIn is a * n n(unital) commutative C -subalgebra of CB(R xR ).

Thus (Vco) c is a function algebra C(P) with the compact space P=

Pe,O of 11,3. The space RnxIn is densely imbedded in P, as the spa-

ce of all maximal ideals mxo o= for given

(x°,1;°)E Rnx1n. Or, P is a compactification of Rnxln. It is known

that a 1-1-corresponce between compactifications of R2nand C*-sub-

algebras of CB(22n) containing 1 and C0(12n) exists: Each compacti-

fication defines a subalgebra - the algebra of its continuous func-

tions. Each subalgebra defines a compactification - its maximal

ideal space. This relation is order preserving. The largest - cal-

led Stone-Cech compactification - belongs to CB(Rn) itself. The

smallest is the 1-point compactification - it belongs to I+CO(12n).

Our P will be 'larger than' EnxEn , since pCo contains the

jgenerators s(x) , sj() of C(EnxEn). The map P->EnxEn is given as

dual of the injection C (EmX 1) -+ ( tWco ) C . (For details cf. Rickart[R1],III,2; Kelley [Ke1],ch.5,p149; [Ci],IV,lemma 1.5, and p308f.)

Theorem 10.3. The symbol space of A=A0, and As, sE &2, is given by

(10.7) 1 = Z(A) = M(As) = at = P\(RnxIn) = fle,0 . (cf.II,3).

Moreover, for A=a(x,D)E Op14)Co the symbol aA equals the restriction

to I of the continuous extension of a(x,l;) to P , for all sE P2

Proof. The last sentence of thm.10.3 describes a homomorphism t:

since C=c(x,D)=a(x,D)b(x,D)-(ab)(x,D)E Op1c-e has c=

symb C =0 at xI+l I=oo, hence tC =0. We tend to show that t extends

to A, with ker t =K(H), hence the induced t` :A/K.C(aP) is an isome-

try. This implies that at and 2 are homeomorphic, proving thm.10.3.


Proposition 10.4. The homomorphism r satisfies the inequality

(10.8) inf{IIA+KII:KE K(H)} Z IITAII , AE Vco

with L2 -operator norm IIA+KII and sup norm IIT All

Proof. Note that ITAI, the restriction to aP of Ia(x,l;)I extended

to P assumes its maximum a=IItAII at some pE P. There exists a sequ-

ence p , E 1n , where Ixjl+I jl-o0 . Either we have

IxJI or or both. We may assume

I 3Is11 are mutually disjoint. Pick pE with

II(vIIL2=1, and let uj=c)j° , u=cp` , go, "= inverse Fourier

transform. Then the uj=eix u form an orthonormal system of H. Let

b(x) be a bounded C00(I )-function. With "=" to be defined write

(10.9)IlbAujll = IIbJF ti IIbfol

First consider the case where Ix3I remains bounded. In the

compactification Enx)gn of It2n will converge to

where I° I =oo , 1x0 I <00 . We have the estimates s

eclx-x°I+clx°-xil+Ej

where the second and third term at right tend to zero, as j-o0

For e>0 pick a ball Ix-x°I< a/c=ry and conclude that

(10.10) a-E-E j , as Ix-x° Is rj , Ej-0 as j-*oo

Accordingly, choosing bE CO({Ix-x°Isi}) , we get lla(x,gj)bullz

(a-E)Ilbujll - Ej . Next, to control "u" , we get

(10.11) s CA O - 0 ,

Furthermore, bAuj=Abuj+Cuj , with CE K(H) . Since {uj} is ortho-

normal it weakly converges to 0, hence Cuj+0 strongly. All in all

we conclude that, for any KE K , we have

(10.12) II(A+K)bujll Z (a-E)llbujll - Ej , Ej- 0 , as j.00 .

Here Ilbu jll=llbull#0. As j- co get IIA+KIIza-E . This for all E>0, hencefor E=0. With "inf" over K get (10.8) so far only if xj is bounded

In the leftover case Ixil-oo we follow the same argument, but


now require a more precise evaluation of the error terms. For sim-

plicity of calculations we discuss only the case n=1. Choose

(p(l;)=1 , as 0 elsewhere. We get u(x)=2S1x x , uj=eix ju.

Now we have

al s

sc/(xj) I x-xj I + E j ,

hence (10.10) now holds with I x-x° I srl replaced by I x-xj I srl(xj)We now will pick b=b. depending on j , according to b.=1 in

Ix-xjls(xj) bjE C0"0({Ix-x3l<Tl(x3)}) , IbjIs1 , insuring that

(10.13) c(xj)-1/2 s Ilbjujll s C(xj) -1/2 , j=1,2.... .

A choice is bj(x)= b((x-xj)/(xj)), bE Co(IxI<q), b=1, IxIs1/2

It is fairly evident that now it suffices to show that

(10.14) Ilbj(x)f O(Ej(xj)-1/2)

and that

(10.15) II[bj,A]ujIl = o(Ej(xj)-1/2) , Ej = 0(1) , as j-oo .

Indeed, we then may introduce v3= bjuj(xJ)1/2 with C>IIvjiI2c>0. One

easily shows that vj- 0 weakly, so that IIKvjll- 0 for compact KRepeating the arguments of (10.12) one gets

IIAvjllz(a-E)Ilvjll-Ej,

hence II(A+K)vjIIZ (a-E)Ilvjll-Ej , again proving (10.8).

Let us offer some details: For (10.14) look at

1f

J

-1

J1-.1 o(.) as Ix-x31si1(x3) , j-ioo1

implying (10.14). For (10.15): We get [bj,A] Z (bj,a)(x,D) with

the Poisson bracket (bj,a)=bjlajx

-bjlxa,, = P(x,l;) .

Clearly IIP(x,D)ujll=llbjlx-"

plral,(x,l;3+rj)11=o(Ej(l;j)) as desired.I

For the error term expressed by "-" we involve I,(5.7) for N=1.

From I,(5.7) and I,thm.6.1 we get the error as IIYj(x,D)ujll with

1

,rTl)blxx(x-y)0

where O((l;j)-2) . Thus the error term obeys the same


estimate, and we get (10.15). For the weak convergence of vj ob-

serve that, for cpE C0(1), we have (v.,cp) = 0 for large j , since

supp v C{ I x-xj I s i(xj) } where 11<1 may be assumed so that I x I a

I xj I -ri(xi) oo , as j-o . since Ilv Its C and Co is dense in H weget I (v j,(P) I s I (v j,fl

I+ CIIro-VII <a with suitable y,E Co . Q.E.D.

To continue with the proof of thm.10.3 observe that (10.8)implies the map t":Opipco" C(at) to have a continuous extension

to A"_.A/K , since the left hand side of (10.8) is just the norm

II A` II. Then t' induces a (continuous) algebra homomorphism AV.

C(aP) . Likewise t itself extends to a continuous homomorphism

and t--rlox with the canonical injectionWe claim that tv defines an isomorphism between Av and C(dk) .

This indeed will establish thm.10.3, since then the dual t°' will

give a homeomorphism I « at , the latter being the maximal ideal

space of C(aP) .

Proposition 10.5. An operator AE Op1co is Fredholm in L(H) if and

only if TO 0 in all of at .

Proof. Clearly TA#O on all of at means that A is md-elliptic

(of order 0) . Such an operator will have a Green inverse BE 0(0)

by II,thm.1.6 and III,thm.4.2 . Clearly, B is a Fredholm inverse

in L(H) as well, so that B must be Fredholm. Vice versa, let AE

Opyico be Fredholm in L (H) . Then we haveS

1 Ails = A + s1 [ A, s]Fredholm too, for all sE &2 since s1[A, s] E O(-1)C K(H) .

Here Ill = ns (x, D) = (x) $2 (D) as in 111,(3.1). Recall thats:Hs- H is an isometry between Hs and H. The above means that not

only is A:H-H Fredholm but also A:Hss

. For clarity let us deno-

te the latter operator by As . Then As:Hs->H5 is Fredholm for all

s and we have A0=A . All operators As have the same index, v(As) _

dim ker As - dim ker AS . However, as s, increases, for fixed s2

one finds that dim ker As decreases while dim ker AS increases

Since the difference V(A5) is constant, they both must be constant

(Here As denotes the adjoint of A. with respect to the pairing

(10.16) (u,v) = (n-s(x,D)u,ns(x,D)v)Lz , uE Hs , vE H-s ,

not with respect to the usual Hilbert space pairing (u,v)S . Thus

we have AS:H-s H-s , in fact, As=a(Mr,D) with a symbol indepen-

dent of s . This explains why dim ker AS increases with increasing

sj.) The fact that ker* ASC Hs, dim ker A. =const then implies that

ker As C S , and ker As C S as well. Both spaces are independent


of s . For a more detailed discussion cf. [C,], ch.III .

It is clear then, that a special Fredholm inverse B of A,

constructed as in 111,4 - i.e., AB=1-Pker A*'BA-1-Pker A ' also

defines a Green inverse of A. Thus, if A:H- H is Fredholm, it fol-

lows that a=symb(A)E 1pco is md-elliptic, hence rA#O on at. Q.E.D.

Finally, in order to show that t" is an isomorphism, assume

that AE A and that VA(p° )#0 for some p°E at . Then t * (p)ac>0 inA A

Np0 , a neighbourhood of p° in at . (Note that clearly t is a

*-homomorphism.)

Observe that t maps onto a dense subset of C(al) (containing

the set "ola1 ) . But a *-homomorphism u:X-Y between C*-algebras

always defines an isometry u°:(X/ker u)-*Y (cf.[Ci],p.323,thm.7.17)

Therefore its image is necessarily a closed subalgebra of Y . In

our case it follows that im t = C(aP) , since im t is dense in

C(a1) . There exists a function XE C(al) such that x=1 near p°

supp X C N(p°) , while Xko . Conclude the existence of XE A with

TX=X . Then the operator XA*A+(1-X) = Z has cZ=Xt * +(1-X)#0 onA A

all of at. There exists a sequence Z jE OpyCo , Z j- Z. Then tZ j+ tZ.

For large j we get tZ ai>0 . By prop.10.5 Z. is Fredholm. There

exists a Fredholm inverse Y j with tY =1/zZ = fltY 11:z 1/rI . Usingj J j

(10.8) we get an inverse mod K(H) , called Wj with JlW111s 2, and

71a Fredholm inverse Vj with JIV1Ils 1 . This implies that Z is Fred-

holm as well (cf. [C,], corollary on p.267).

But if A were compact, so is XA*A compact. Hence 1-X then is

a Fredholm operator. The latter is impossible: Then we get a Fred-

holm inverse BE A and CBz(1-X)=1 on all of at =*O-rB(po )(1-x(po ))

=1 , a contradiction. Thus VA=O implies A compact, and t" must be

an isomorphism. Theorem 10.3 is established for s=0. For general

s observe that As= s1A

s, using the isometry 5:H5-> H . Also,

s1AIIs=A + K , KE K(Ht), for all t, showing that AE Op1NCo has the

same symbol for all s. Q.E.D.

Chapter 6. HYPERBOLIC FIRST ORDER SYSTEMS

O. Introduction.

We now focus on hyperbolic theory. It the present chapter 6

we look at a first order system of pseudodifferential equations on

&n of v equations in v unknown functions, of the form

(0.0) au/at = iKu , K=k(t,x,D) ,

with a vxv-matrix of symbols k=((kjl)), kjlE 1Ce, for fixed t.

In essence 'hyperbolic' means that either the matrix is

self-adjoint, or at least has real eigenvalues, both mod loco .

In sec.1 we discuss a symmetric hyperbolic first order syst-

em of PDE's, using a method of mollifiers, after K.O.Friedrichs.

The case of a ydo K(t) may be treated by a similar technique (sec.

2). Here we use the weighted LZ-Sobolev spaces of ch.III.

In sec.3 we will look at properties of the evolution opera-

tor U=U(x,t). We find that U(i,t) is of order 0, while atagU is of

order (p+q)e. In sec.4 we discuss strictly hyperbolic systems, no

longer symmetric hyperbolic, by the method of symmetrizer.

In sec.5 we discuss a global Egorov type result for a single hyper-

bolic equation, proving existence of a characteristic flow. Actual-

ly our flows are of more general type, called particle flow, using

a generalized principal symbol of K , no longer homogeneous in i .

This flow is related to the family A H U 1AU of automorphisms

of the C*-subalgebra A of L(H) generated by Opilco (V,10), where U=

U(x,t) is the evolution operator: The restriction of the flow to

is the dual of the induced automorphism of A A/K(H). In

sec.6 we discuss the action of these flows on symbols. Sec.7 deals

with propagation of singularities and maximal ideals of A.

General theory of hyperbolic N-th order (systems or) equa-

tions will be reduced to the case of i-st order systems, as alrea-

dy prepared in sec.4. This will be discussed in ch.7.

1. First order symmetric hyperbolic systems of PDE.

We enter the theory of hyperbolic (pseudo-)differential

equations from our discussion in 0,6. There it was seen that a

196

6.1. Systems of PDE's 197

single linear first order PDE of the form au/at=Lu , L=2 bjaxj +p ,

with real-valued bj(x) , essentially is equivalent to an ODE along

a set of curves in (x,t)-space. The curves were called the charac-

teristic curves. They were given as solution curves of the system

of ODE's dt =-b(x)=-(bl(x),...,bn(x))

If u(x) is no longer a scalar, but a vector function, and

the equation a system of equations, then the simple approach of

0,6 no longer works - although the essence of it can be saved if

there is only one x-variable - cf.[CH],II, ch.V, for example. We

regard it as a principal merit of yido-calculus that it will resto-

re the relationship between the PDE and its 'characteristic flow',

i.e. the system of characteristic curves. First we study abstract

existence and uniqueness results.

For a first result, for PDE's only, look at the problem

a tu + iKu = f , t E I , u(0) = y ,

where

(1.2) L = iK = Lj=1a3ax + a0 , aJT(x) = aJ(x) j = 1,...,nJ

is a first order vxv-matrix of differential expressions in x with

real symmetric principal part, and coefficients independent of t.

Let L be defined on &n, and let a.E C,(&n), aj , aj1x1 bounded.

The adjoint expression L* of L is given by

(1.3) L* = -iK* = -L + R(x) , 1(x) =a0T+a0-2aJ

jxj

Clearly it also has bounded coefficients.

We regard (1.1) as an ODE in t with dependent variable u ta-

king values in the Hilbert space H=L2(&n,tv), assuming that f(t)E-=

H , for every t. It then is natural to interpret L as an unbounded

operator of H, defined in a suitable domain dom L C H .

In this setting problem (1.1) suggests that we look at the

family {eiKt} possibly generated by the infinitesimal generator

L=iK. Formally L+L*=(3(x) is a bounded matrix function. Thus L+y

is formally accretive (dissipative) whenever the constant y E I is

large (small). In other words, we have, correspondingly,

2Re(u,(L+X)u) = (u,Lu)+(Lu,u)+2X(u,u) = (u,(L+L*+2?.)u))(1.4)

_ (u,((3(x)+2a.)u) Zo (or s0) for all uE C00(e)

198 6. Hyperbolic first order systems

By V,thm.7.2 the operator L will generate a strongly conti-

nuous group {E(t) = eLt = eiKt :t E 1e} such that

(1.5) dom L = {u E EH : d/dt(E(t)u)lt=0 exists in H } ,

and that u(t) = E(t)u , for u E dom L , satisfies

(1.6) u(t)E C1(R) , du/dt - Lu = 0 , t E le , u(O) = u ,

provided that we can verify the assumptions for all real IXIay .

The t-derivative in (1.6) exists in strong convergence of H.

We get E(t)E=- L(H), and, with operator norm 11.11 of H, M of V,(7.7),

(1.7) IIE(t)11 s MeyI tI , t E & .

Then the inhomogeneous Cauchy problem (1.1) may be written as

(1.8) at(E(t)u(t)) = E(t)f(t) , tE & , (Eu)(0)=p

An integration then solves (1.1) : For f(t)E C(I,H) , TE H, we get

t(1.9) u(t) = E(-t)T + E(t-t)f(t)dt

0

Note that (1.1) and (1.9) are equivalent only if tpE dom L

Theorem 1.1. Let L=L** be the closure of the minimal operator

L0 (with domain dom LO=CO(>en)) induced by the expression L . Then,

under our above formal assumptions on L and its coefficients,

L and K=-iL satisfy all of the above properties, so that the group

E(t) is well defined, and the Cauchy problem (1.1) admits the

solution (1.9) for every f E C(le,H) , TEE dom L

For the proof we need the lemma, below.

Lemma 1.2. Under the assumptions of theorem 1.1 the closures

of the minimal operators of L and its adjoint expression L* are

mutually adjoint unbounded closed operators of H .

(This often is stated by saying that the concepts of weak and

strong L2-solution coincide, under the assumptions given.)

Assuming Lemma 1.2 correct we next notice that

(1.10) Re (LOu,u)I =ZI(Lu,u)+(u,Lu)I s y(u,u) , u E CO(In)

hence for real ? we have

(1.11) (X-y)(u,u) s Re((L0+%)u,u) , u E dom L0

Taking closure we get

6.1. Systems of PDE's 199

(1.12) (k-Y) IIull2 s IIull II (L+X)ull , u E dom L

which implies that (L+X) is one to one and has closed range and

(1.13) II(L+? )ull 2 IX-Y I IIull , u E dom L , for all A>y .

However, in (1.10) we may replace L0 by the minimal operator

of the adjoint expression. Since its closure is the adjoint of L,

we get exactly the same properties , and the estimate (1.13) also

for the adjoint operator L* of L . It follows that also (L+X)* is

is one to one, and has closed range. Accordingly L+% is 1-1 and

onto, and has an inverse R(7,)=(L+k)-1 E L(H) , for all X>y .

Similarly we get existence of R(k)=(L+?,)-1,for all .<-y. For

real ? satisfying IXI>y we get II(L+X)-1IISI-Y .

This implies the

assumptions of the Hille-Yosida theorem, and thm.1.1 is proven.

Proof of Lemma 1.2. Let L and M denote the closures of L0 and L0

respectively. Clearly L and M are in adjoint relation. That is,

(1.15) (Lu,v) _ (u,Mv) , for all u E dom L , v E dom M ,

as follows for u,v E CO(n) by definition, and then for general

u,v by taking closure. In other words, we have L C M* , and are

left with showing that M* C L . Let f E dom M* , i.e.,

(1.16) fE H , (f,Mu) _ (g,u) , for all u E CD(ien)

for some g E H (defined as g = M*f) Then we must show that f

E dom L and that g = Lf In other words a sequence fkE CO(c) must

be constructed such that fk + f , Lfk - g , (in H) , as k ' o .

Introduce the convolution operator JE , E>0 , by setting

(1.17) JEU(x) = JwPE(x-Y)u(Y)dy , gE(x) = E nT((x-Y)/E) ,

with some cp E C"(&n) , supp p C {Ixls 1} , q);0 , jpdx = 1 (P even.

It is clear that fE=JEf+f (in H) as E->0 , for every fE H . Indeed,

for continuous f with compact support one finds

IfE-fl(x)=lfroE(x-Y)(f(x)-f(Y))dylSMax{If(x)-f(Y)Itlx-YIsE}+0, E+0,

while supp (f-f) stays within a fixed compact set, implying that

f,-)-f in H . On the other hand, JE=V£1JlVE with the unitary (dila-

tion) operator VEU(x)=En/2u(Ex) , shows that IIJEII=c is independent

of E . Also C0(In) is dense in H

For fEH satisfying (1.16) let hF--VFfE=tPEJEf, with VF=cPE^^(Ex), as in (1.17). Then h. -)-f in H holds as well. Also, hEE C0

*


for every E>O . To prove lemma 2.1 we must show that also LhF-0g .

But we get LhE _ WELfC + [L,pE]fE , where the commutator [L,iE] is

a multiplication by the matrix Ejaj(x)T4jx (Ex). The latter tendsJ

to zero uniformly on &n. Hence [L,'tp2]fE-*0, and we must show LfE+O.

Let us write a3(x) = ((aJ (x) )) and introduce the vector (P,k

(y)kl E,x

((PE,x(y) )j=l,...,m = (bkj(PE(Y-x)) For the k-th component

of LfE we get (summing over all double indices)

(LfE)k(x) _ J(akl(x)axj+a01(x))rPE(Y-x)fl(Y)dy

(1.18) = J((-ayjagl(x)+agl())(P E:X(Y))fl(Y) dy

=(L*Te,x,f) + Tkf

with

Tkf = J(ayj(aql(Y)-aql(x))+(agl(x)-agl(Y))TE"X(Y)fl(Y)dY

But (1.16) implies

(1.19) (L*(PE,x,f) = (rpE,k,g) = (JEV)k

As e - 0 this converges to g (in H) Therefore the lemma is esta-

blished if we can show that Tf-> 0 in H , as s -> 0 .

For the latter we first note that 1ITHH is bounded for all e >0.

Indeed this follows because the first derivatives of cpE(y) areO(E-n-1)

, while the support is in a ball of radius E and the

factors a(y)-a(x) are O(E) It follows that the integral kernel

tE(x,y) satisfies Schur's condition fltE(x,y)l{ay s c for all x,y.

On the other hand if fE Co(&n),then a partial integration, remo-

ving the derivative from qE(x-y) implies that Tf-+0 in H , as O.

Combining the two facts we conclude that Tf-0 for all fE H, q.e.d.

2. First order symmetric hyperbolic systems of t,de's on ten

In this section we shall work on the Cauchy problem for a sys-

tem of pseudo-differential equations over &n , of the general form

(2.1) au/at + iK(t)u = f(t) , t1 5 t 5 t2 , u(0) = qp .

For convenience we assume a compact interval 0 E I = [tilt 21-

6.2. Systems of operators on Rn 201

We assume that K(t) are matrices of ,do's in the sense of ch.I

(2.2) K(t) = k(t,M1#D) = ((kjl(t.M11D) ))j,1=1,...,v'

where the functions are symbols, for every fixed t.

Again we regard (2.1) as a first order ODE for u taking val-

ues in a space HSO CV of all u=(u1,...,uv) , ujE Hs (of III,(3.1))

For simplicity we write Hs instead of HSO Cv . We assume K(t)E

Opipce , e=(1,1), introducing the class VcV of vxv-matrix-valued

symbols with entries in ipcm. By III,thm.3.1 get K(t)E L(Ht'Ht-e)

for all tE 22. In general we do not get KE L(HS) but may interpre-

te K as an unbounded operator of HS with domain HS+e, e = (1,1).

We consider initial values T E Hs , and require f E C(I,Hs).

The following precise assumptions on the are imposed:

(2.3) }-I+lalk(a)(t'x' ) E C(I,CB(&2n,Lv)) , a, PE=- IIn

and

(2.4) (x) lal E C(I,CB(&2' ,iv)) , a, PE IIn

Here 1v denotes the algebra of complex vxv-matrices,and we write

CB(X,Y) for the space of all bounded continuous functions from X

to Y. Also IIn is the class of all n-multi-indices a = (a1,...,an),

aj E II , II = {0,1,2,3,...} and we have written K*=k*(t,M1,D) for

the Hilbert space adjoint K *(t) , a yido with symbol given by

I,(5.9). Alternately we may express (2.3) and (2.4) by writing kE

C(I,4ce) andk-k*

E C(I,i,cO) using the Frechet topology of tpcm .

Clearly (2.4) implies that the skew-symmetric part 1/2(K(t)-

K*(t)) of K(t) is of order zero hence bounded in every HS . This

motivates the notation "symmetric hyperbolic system" of ipde's. As

a special case we may choose K as a differential operator with the

properties of K in (1.1), (1.2), but K may depend on t here.

In the functional analytical sense the solution of (2.1) no

longer involves a group eiKt ,since now K(t) depends on t. While

abstract existence results (cf. Kato [Ka5]) could be used we find

it practical to use the technique of lemma 1.2 again (cf. [Tl1]).

Theorem 2.1. Under the assumptions (2.3), (2.4) the Cauchy problem

(2.1) admits a unique solution u E C(I,Hs) f1C1(I,Hs-e) , for each

cp E HS , and f E C(I,HS) , where s E II2 is arbitrary.

Remark. As a consequence of uniqueness the solution u of (2.1) is

independent of s, if gE S=fHs, and fE C(I,Hs) for all s. Then u

also takes values in S, and u , atu are continuous in every HS


Proof. The key of the proof is the so-called energy estimate

(prop.2.3) We first shall solve (2.1) for a "mollified" operator

KE(t) instead of K(t) where KE(t) are bounded operators of Hs .

Under such conditions Picard's method of succesive approximation

may be used to solve the Cauchy problem. A limit s - 0, using the

Arzela-Ascoli theorem, will provide the desired solution of (2.1).

As in section 1 we choose a regularizer (or mollifier)

(2.5) is _ ('WE*) = it (ED) , e > 0 ,

with a fixed C000( ,n)-function ii,, supp yi C { l x l < 11, V 2t 0, f ypplx =1,and VE(x) = E-nV(x/e) . Observe that yr E S . We introduce

(2.6) KE(t)=V (sM)K(t)yr(ED)=kE(t,M1,D), '(Ex)yr(e )

Since yr E S, it is clear that the matrix kE has all its entries in

wto, using (2.3). Therefore KE(t) are bounded operators of every

Hs, for s>O, tE I. In fact, we get KE(t)E C(I,HS), for E>O. Thus

the Cauchy problem (2.1) E (with KE instead of K in (2.1)) admits a

unique solution u=u8E C1(I,Hs) , for each e > 0 ([Dd1] ) .

Letting e - 0 we attempt to construct a uniformly convergent

subsequence of u5(t) with limit function u(t) solving (2.1). We

shall need more properties of JE and KE(t) , in that respect.

Proposition 2.2. a) The family J. = V (ED) , 0 e. e s 1 defines a

bounded function [0,1] - L(Hs) , for each s E 12 , which is norm

continuous in (0,1) , and strongly continuous at 0. We have J0=1.

If V is choosen as an even function , then also all JE are hermi-

tian symmetric in the Hilbert space H = H0 .

b) The family KE(t) defines a bounded function [0,1] x I

L(Hs ,Hs-e) which is norm continuous in (0,1] x I , and strongly

continuous in [0,1] x I . We have K0(t) = K(t) , t E I .

Proposition 2.3. (Energy estimate). Let u(t)E C(I,HD) f C1(I,Hs-e)

be a solution of (2.1) 6 for given s , s , y , f(t). There exists

a constant c independent of s,s,q and f such that

(2.7) Ilu(t)IIs s IlaPllse°Itl + e°ltlte-oillf(z

sgn t)Ils ds0

with the norm II.II of H$

A proof, based on the symmetry of K(t) expressed in (2.4), and

some commutator relations of 1,7 is postponed to the end of sec.2.

Note that the energy estimate (2.7) implies that

(2.8) IIuE(t)Ilss c, , 0 s E s 1 , t E I,


with c independent of s and t . Let us first assume that y and f

satisfy the assumptions of thm.2.1 for all s . The uniqueness of

the solution us contained in Picard's theorem then implies that

us is independent of s , and that us E CI(I,HS) , for all s. Using

(2.8) and the differential equation (2.1)s it follows that

(2.9) IIuE(t)Ills IIKE(t)u,(t)Ils+ Ilf(t)Ilss cllus(t)Its-e+ Ilf(t)Its sc.

We also were using prop.2.2. Similarly,

(2.10) II(M)(D)uE(t)Ils s cslluE(t)lls+e s c

all with constants independent of t and s . In (2.10) we used thatthe norm II(M)(D)ulls is equivalent to the norm Ilulls+e (cf. 111,3).

The operator C=((M)(D))-I is compact in Hs, for all s, by III

thm.5.1. Thus each set Cc {uE Hs : IlC- Iullsc} is conditionally com-

pact in Hs: Each sequence ujE Cc has a convergent subsequence.

The family {us : 0sss1} satisfies the assumptions of the Ar-

zela-Ascoli theorem. There exists sj>0, such that us (t) -

u(t)E=- C(I,H5), uniformly in I. Under the present restrictions on

q and f this holds for all s. Using another Cantor diagonal scheme

s may be chosen such that this convergence holds in every H , s

EjZ2, and then for all Hs, sE I . In particular, we get uE C(I,S).

Next one confirms that

u(0) = limjusj(0) = cp , limjuE (t) = f(t) - i lim Ksj(t)usj

(t)

(2.11)= f(t) - iK(t)u(t) in S and every Hs uniformly over I

Since us and its derivative uE converge uniformly the limitj j

function u(t) is in CI(I,HS) for every s and in CI(I,S) Also

uE a u and u solves the Cauchy problem (2.1)o = (2.1)

Now consider the general case of cE Hs, fE C(I,HS), for fixed

s. We may construct sequences cpjE S, fjE C(I,S) such that cpj-*q,

fj(t)-af(t) in Hs, uniformly in I. For example one might choose

(2.12) cpj = tr (M/j)q (D/j)q , fj(t) = 'iV'' (M/j)'W" (D/j)f(t) ,

with i as in (2.5), using prop.2.2. Then, letting uj be the solu-

tion of (2.1) with cpj and fj, it follows that uj-ul solves (2.1)

for cpj-cpl and fj-fl. Thus the energy inequality (2.7) implies

(2.13) Iluj - u11lss c{Ilavj- cpllls+ MaxtE Illfj- f11ls} - 0 , j,l- °°


Thus {uj} is a Cauchy sequence in C(I,H5), and u =lim ujE C(I,H5).

Similarly u(=- CI(I,HS-e) , and u satisfies (2.1), proving existence.

Uniqueness of the solution is a consequence (2.7). Q.E.D.

We now are left with the discussion of prop.s 2.2 and 2.3. We

first discuss a few commutator properties of the 4do's involved,

referring to the machinery prepared in 1,7.

1) The families {Le--V (EM) : 05s51} , and {JE=FLEF}, are

bounded in L(H0) . Both have strong limit 1 , as E-0 , in L(H0)

(Indeed the family of functions IV (ex) : 05651 } is bounded,

and (Ex) = yr(0) = fgIx p(x) = 1 , pointwise, for all x .)

2) The family {K(t) : t E =-I } is bounded in OpVice . (This

follows directly from (2.3).)

3) The family {K(t) - K*(t) : t E I } is bounded in OpPcvO

(Again this is a consequence of (2.4).)

4) The families 1[n jE]/E } , and {[ s,LE]/E}, where 0<e5l,

and s = (M) S2 (D) s' , with fixed s E H2, are bounded in OpPcs-e

Indeed we may apply I, lemma 7.2 onto(ED)](D)-s' (M)-s2 = [(M)S2 Ir (ED)](M) s2

noting that ((x)s') (0) EO, s2 -1' while { (p (Ex) (0)/E : 0sss1} is

bounded in Wsi-1.0' as 101=1. For the latter one confirms that

(2.14) a )s cEl (1+E )-1=O((

with 1 = 101 , for any (o E S . Similarly for LE5) The family {[As,K(t)] : t E I } is bounded in OPiycg

This follows from (2.3) and I, lemma 7.2 again.)

6) The families {[K(t),JE]/E } and {[K(t),LE]/E } where

0 < E s 1 and t (=- I , are bounded in OpipcO . Indeed we again may

apply I, lemma 7.2 using that E ipce2 , k(0

E-=Vice, :0< E s 1} is bounded in Vc-e, for 101= 1.

7) The family {[JE,LE] : 0 < e s 1} is bounded in Opipc-e

(Proof similar to the above.)

Proposition 2.4. We have with the inner product (u,v) s of Hs

(2.15) I(U,K Oses1, tE I, sE &2, uE S,

where the constant cs is independent of e,t,and u.

Proof. Let v=su=(M) '(D) 'u. Abbreviate K(t)=K , JE =J, LE =L, IIs=II:(KEU,u)s - (u,KEU)s =((K-K*)v,LJV) +(Kv,[J,L]v)+([K,J]v,Lv)

(2.16)+(v,[K,L]Jv)+([I],LKJ]u,v)+(v,[LKJ,II]u where [II,LKJ] =

= [ 17, L] KJ + L[ II, K] J +LK[ H, J] ,


(.,.)=inner product of H. We assumed J£=J9 (V even) and used that

(2.17) (lulls = II s'vIls = I1vII2 = 11V110

For the first 4 terms at right we use,in this order,(1) and (3),

(2) and (7), (1) and (6), (1) and (6). For the last two terms,

the commutator split as stated, use (1),(2),(4),and (1),(5),q.e.d.

Proof of proposition 2.3, for u E C1(I,S). First let the solution

u of (2.1) 9 be in C1(I,S). Then µ(t)=llu(t)II, is C'(I,le). We get

(2.18)dµ(t)/dt = µ (t) = (u (t),u(t))8 + (u(t),u (t))s

= -i((K9u,u)s - (u,K9u)s) + 2 Re(f,u) s

Thus (2.15) implies

(2.19)I 1A (t) I s Cellu(t) II$ + 11f (t) 112 + HUM 112

=(cs+1)µ(t) + IIf(t)112 , tL(0) = IIcvlis

The Cauchy problem (2.19), for an ordinary differential inequality

may be integrated in the usual way. It follows that

(2.20) µ(t)e-ct - (Irons s foe ctllf (z ) Ilsd

proving (2.7) for t a 0 . Similarly for t s 0.

In the general case uE C(I,H,) fl CI(I,Hs-e) it no longer is

evident that µ(t) is differentiable since the right hand side of

(2.19) no longer is meaningful. The discussion below shows that µ'

E C(I) with (2.19) still exists. Then (2.20),(2.7) follow again.

Instead of µ we first form µb(t)=(u(t),J,Lbu(t))8, tE I, 6>0.

Abbreviating as in (2.16) again we note that JLu E C1(I,S). The

inner product of Hs is trivially extended as (g,h),=(A,g,A,i ,

gE S' , hE S . Hence we may carry out a differentiation as above:

(2.21)(t) = (u ,JLu)5+ (u,JLu )5=-i((K9u,JLu)5-(u,JLK9u)5)

+ (f,JLu)5 + (u,JLf) 8 .

We get restricted to the case e = 0 since for e > 0 we may

keep 6 = 0 and carry out (2.16), using the extended inner product.

For e = 0 consider the term, with M=H6=JL=J6L6 , v= su=Ihi,

(2.22)(Ku,J6L6u),- (u,JSL6Ku)5 = ((K-K*)v,Mv) + (Kv,[11,M]u)

+([II,K]u,Mv) +(v,[K,M]v)+(v,[MK,f]u) + ([I1,K]u,[II,M]u) ,


by a calculation.As 6-0 the right hand side of (2.22) converges to

(2.23) ((K-K*)v,v) + ([II,K]u,v) + (v,[K,II]u) = p(t) ,

uniformly for t E =-I . Indeed,the second,fourth and sixth term,at

right,tend to zero,using (2) and (4) ,(6) , (5) and (4) ,respecti-

vely. The remaining three terms of (2.22) converge to to the terms

of (2.23) in the order of listing,using (1) and (3) ,(1) and (5),

and (l)and (4) for [MK,rl] = M[K,IT] + [M,rl]K - [K,rIJ in L(HS,H0)respectively. Each convergence is uniform, for t E I

Using this in (2.21) as 6 - 0 we find that

(2.24) limb_.0µb (t) = p(t) + 2 Re (f,u)s uniformly for t E I

Since µb and p'6* both converge uniformly it follows that the limit

µ(t) is C1(I) and lim p'b. = µ Hence we indeed get (2.19) esti-

mating the terms (2.23) with (3),(5). This completes the proof.

3. The evolution operator and its properties.

As a consequence of the existence and uniqueness theorem of

sec.2 we observe that, for every fixed t E I , the assignment

cp - u(t) defines a linear operator U(t): He HS , where u(t) de-

notes the solution of (2.1) for f e 0 . This operator U(t) will

be called the evolution operator (or the solution operator) of the

Cauchy problem . As an immediate consequence of (2.7) we find that

IIU(t)roIIS 5 IIroHe , E Hs , t E I .

Moreover from thm.2.1 it follows that the operator family

{ U(t) :t E I } is strongly continuous in L(Hs) , while the deri-

vative U (t) = dU(t)/dt exists in strong operator convergence of

L(HS,HS-e) and is C(I,L(HS,HS-e)) under the assumptions stated.

Similarly, under the assumptions of thm.1.1, the evolution opera-

tor of (1.1) will define a strongly continuous group in L(H).

With thm.2.1 U(t)=sU(t) exists for every sE &2, and we get

sU(t)q)=s,U(t)T whenever gE He f1 HS,. One obtains a well defined

continuous map S'-3-S', of order 0 in the sense of 111,3. We denote

it by U(t)E=- O(0). Clearly U' exists in S' , and U'E O(e). Also,

(3.2) U(t) +iK(t)U(t) = 0 , t E I , U(0) = 1 = identity.

Somewhat more generally we may consider the Cauchy problem

(3.3) du/dt + iK(t)u = f , t E I , u(t0) = g ,

6.3. The evolution operator 207

with given z0E I. Then I no longer must contain 0. This problem re-

duces to (2.1) if the new variable t+t0 is introduced. Consider

(3.4) v+ iK(t+t0)v = f(t+t0) ,t E I-t0 = [tl-t0,t2-t0], v(0)=q.

Clearly v solves (3.4) if and only if u(t)=v(t+to) solves (3.3).

Thm.2.1 applies to (3.4), which is of the general form (2.1) with

all assumptions satisfied also for the coefficient K(t+t0) and

f(t+t0). Hence for each f E C(I,Hs) there exists a unique solution

u E C(I,Hs) fl C1(I,Hs-e), solving (3.3). Thus get a more general

evolution operator U(t,t): mapping Hs-'H5, sE &z , satisfying

(3.5) atU(t0,t) + iK(t)U(t0,t) = 0 , t E I , U(t0,t0) = 1 .

If K(t) is independent of t we get U(t0,t) = U(t-t0) = e-iK(t-to).

so that U(t0,t) is determined by the group U(t) , similar as in

thm.1.1. For reasons similar as above we have

(3.6) IIU(t0,t)Ils 5 cs , t,t0 E I .

Also U and aU/at are strongly continuous for each fixed to. Regar-

ding dependence of U(t0,t) on t0 we consider the dependence of v

in (3.4) on the parameter t0. We strengthen our assumptions (2.3)-

(2.4) on k=symb(K(t)) by requiring that also kE C'(I,,ae), i.e.,

(3.7) (x)-1+Ial{ E C(I,CB(&2n,Lv))

Let us return to the mollified equation (3.4)E, i.e., (3.4)

with K replaced by KE of (2.6) . The coefficient KE(t+t0) is

C1(I,L(Hs)) , with the set I no {(t,t0) E &2 : t0 , t + t0 E I } .

Classical theory of linear ODE's with bounded coefficients implies

that vE(t,t0) solving (3.4)E for f=0 is C1(I,L(Hs)). Moreover,

atovE=WE solves the formally differentiated Cauchy problem

(3.8) wE + iKE(t+t0)we+ iKE (t+t0)vE = 0 , WS(0) = 0

where vE is given as solution of (3.4). Assuming first again that

q E S , the technique of sec.2 may be repeated. One finds that

d/dt(IlwElls) = O(lIwEII$ + Ilvalls+e) ,using (3.7). This implies

(3.9) IlwEll2 s ecltIIlvell22+e 5 e2CItIIIVP112+e

with c independent of E,t,t0 , and cp . The family {vE(t0,t):0<E51}

is equi-continuous; it has bounded first partials for t and t0 .

It is bounded, and maps into a conditionally compact set. Thus Ek,

k=1,2,..., may be found, Ek-0, vE converging to v(to,t)E C(I,S),k


in every Hs. Also atVF k=-iKEk(t)v£

k(t) converges in S, hence atvE

C(I,S). Moreover, the function z5(t)=(v(tO+S,t)-v(t0,t))/S solves

(3.10) z5*(t)+iK(t+t0)zO(t)=-i(AK/O)v6 , z5(0)=0 ,

with v (t)=v(t0+S,t) , AK=K(t+t0+S)-K(t+t0). For the difference

p=zS-zS, , S , S' >0, with A'K=AK, for 6=6' , we get

(3.11) p +iK(t+t0)p=-i(AK/S (v S-v")+(AK/S-A'K/S')v

', p(0)=0 ,

t+to +Swhere AK/S=S-1 K (x)dx is a bounded family in L(H,,H,_ )

t+tosince K'(t), tE I, is bounded in OpWce. Moreover, AK/6-K'(t+to)-0

in L(Hs'HS-e), by (3.7) and III, thm.1.1. Also, v -v -'0 in each

Hs, since vE C(I,S). The right hand side of (3.11) tends to 0 in

each Hs, as uniformly in Z. Using (2.7) on (3.11) get p-0

as 6,6'-0, in each Hs, uniformly for (to,t)E I, hence ato vE C(I,S)

exists. Taking limits in (3.10) conclude that w = ato v satisfies

(3.12) w + iK(t+t0)w +iK (t+t0)v = 0 , w(0) = 0 .

Clearly (3.12) implies that w is continuous, so that vE C1(I,S).

Finally, if T E HS we again construct T.= LJp E S , Tj - T

in Hs and then will get 11v-v100s -* 0 , 11v -V1 "s-e -> 0 , as in the

proof of thm.2.1, with vi the solution for Tj . Using (3.12) we

also get flw-wills-e - 0 , for wi = at0Vi . Consequently,

(3.13) v(t0,t) E C(I,Hs) (1C1(I,Hs-e)

Similarly u(t0,t)=v(t0,t0+t) , the solution of(3.3), satisfies

(3.14) u(t0,t) E C(IxI,HS) f C1(IxI,Hs-e) .

Thm.3.1, and cor.3.2, below, now can be left to the reader:

Theorem 3.1. Under the assumptions of (2.3), (2.4), and (3.7) on

the symbol of K(t) the evolution operator U(t,t), t,tE I, of the

Cauchy problem (2.1) has the following properties.

(i) U(t,t) : S'->S' is an operator of order 0 ; it continuously

maps S-S, and Hs-Hs, sE ie2. Its first partials atU , atU exist in

strong operator convergence of L(Hs,Hs-e), sE &2. They are E O(e).

(ii) U(t,t) is bounded and strongly continuous over IxI , in

L(HS) , s E H2. atU and a

tU both are bounded and strongly conti-

nuous in L(Hs,Hs-e) s E H2, t0,t E I

6.3. The evolution operator 209

(iii) For t,T,x E I we have

(3.15) U(t,t) = I , U(t,x)U(t,t) = U(t x) , U(t,T )U(t t) 1

(iv) U(t,t) satisfies the two differential equations

(3.16) atU(t,t)+iK(t)U(T,t) =0 , aTU(T,t)-iU(t,t)K(T) =0, t,TE I,

(v) The operator U(T,t) is uniquely determined by properties

(i), (ii), (iii), and one of the differential equations (iv).

Corollary 3.2. The evolution operator U(T,t) is invertible in

S , S' and in every Hs , for T,t E I . Its inverse is given by

U I(T,t) = U(t,t) . Moreover, the family

(3.17) V(T,t) = U 1*(t,t) = U*(t,t) , t,TE I ,

is the evolution operator of the adjoint equation's Cauchy problem

(3.18) V'+ iK*(t)v = g , t E I , v(t) = 1 ,

where "*" means the L2(mn)-adjoint.

For fE C(I,Hs) the unique solution of (3.3) is given by

t(3.19) u(t) = U(T,t)tp + U(x,t)f(x)dic

T

Next we notice that higher T,t-derivatives of U will exist

if higher derivatives of K(t) are assumed. For example, let

(3.20) symb(K) = k E C,(I,4ce) .

Then it is clear that K E C*(HsHs-e

) for all s. Accordingly it

follows from the differential equations (3.16) that atU and a

TU

have first order partials for t and T in L(HsHs-2e) in strong

operator convergence. Thus the second order partials of U for t,T

exist and are strongly continuous in L(HsHs-2e)' Using this in

(3.16) again we find that the third order partials of U exist in

L(HsHs-3e),etc. By iteration one finds the corollary, below.

Corollary 3.3. Under the assumption of (2.3), (2.4), (3.7), and

(3.20) U(T,t) has partials of all orders existing in strong opera-

tor convergence of S and S'. We have ataiUE O((j+l)e). That deriva-

tive is strongly continuous in every L(HSHS-(j+l)e)' sE &2

Remark 3.4. It should be noticed that diffentiability in thm.3.1

and cor.3.3 may be strengthened if we replace (3.7) or (3.20) by

(3.21) kE C'(I,i4ae,)


(or, resp., kE C1(I, yice,) ) , where a°=e'=(1,0) or a°=e2=(0,1)With the stronger assumption get aTU, atUE 0(-e°), existence, con-

tinuity of atU,ayU in strong convergence of L(HS'HS-e)' (for C1),

and atazUE O(j+l)e°),corresponding existence, continuity (for c0o).

Proof by reinspection of (3.16), under the new assumptions.

4. N-th order strictly hyperbolic systems and symmetrizers.

The hyperbolic equations encountered in ch.0 were of order 2.

Correspondingly we plan an existence and uniqueness theorem for an

N-th order hyperbolic system (4.1), below, with a detailed theory

to be worked out in ch. VII . With I=[ ti , tz ] , t. E I, let

(4.1)Lu=1=0(-i)iAN_j(t)(at)ju=f(t), t(=- I, dlu/dt1=W1, t=t0, Osl<N

where Ak(ttt)=ak(t,x,Dx) are k-th order differential operators in x

on &n, with smooth coefficients, and A0(t)=1 . Again (4.1) will be

regarded as an ODE for u=u(t) , taking values in L2(]en) or He(&n).

Using a standard method of ODE convert (4.1) to a first order

system. In correspondence with rem.3.4 use a comparison operator

(4.2) A 1=Ae,-l=nee=ne° (x,D)=(x)(D), a°=e, =(x), °=Z.

Note that Ae,E Opyc_e,. With the new dependent variables

(4.3) uj = AjDtu , j = 0,1,...,N-1

the Cauchy problem (4.1) (i.e., (5.7)) is equivalently written as

(4.4) v + iK(t)v = g(t) , t E I , v(t0) = V

where we have introduced the (column-) vectors

(4.5) v=(uO,...,uN_1)T, g=(0'...,0,iAlf)T, "((p0,...,(-iA)N-1(PN_1)T'

and the square matrix of yido's

0 -1 0 . . . . 00 0 -1 . . . . 0

(4.6) K = A 1 . . . . . . .

0 NO ... 0 N- 1 IAN,A AN_1A'' A, ... , AIAN.

Clearly u=u0 solves (4.1) if and only if v solves (4.4).

K of (4.6) is a matrix of tpdo's. If (4.1) is a single equation for

6.4. N-th order strictly hyperbolic systems 211

one unknown function then (4.4) is an NxN-system. In general, if

(4.1) is an vxv-system, then (4.4) will be an NvxNv-system. We

desire to match the system (4.4) with the assumptions of thm.2.1,

or better, thm.3.1 and cor.3.3 (or rem.3.4), but find it generally

impossible to achieve the symmetry condition (2.4). All other con-

ditions translate into natural conditions for equation (4.1):

We assume that symb(K(t))j1E C"(I,wcea), with the 3 choices

of ea corresponding to (4.2). This means that

(4.7) ECoo(I,1jea)

,j=I,...,N

Or, as a possibly stronger condition we require

(4.8) aa, j(t,x) E C'(I,Vc(0,(N_j)fz )) , f = (fl ,fx) ea

Let us be guided by the case of (4.1), for scalar u(t,x) (i,

e., v=1), being constant coefficient hyperbolic with respect to t

(ch.O,sec.8, ex'le (e), or VII,1). We then think of ea=e', A--%(D),

The symbol a of the operator matrix KO=AK then has the

property that,up to lower order terms, det(a+z)=-7=O ZN_jaj(g)Xj(g)

where PN(t,g) is the principal part polyno-

mial of L=P(Dtlj )= N_j(Dx)Dt. If L is hyperbolic then the roots

of PN(X, ) are real, as O. L is strictly hyperbolic if and only

if they are real and distinct. For ItI=co this happens if and only

if symb(Ko) has real and distinct eigenvalues.

Accordingly, from now on, we focus on a system (2.1), where

(4.9) KO(t) = AfK(t)=k0(t,x,D) + x(t,x,D) ,

with k0E 'CE C°°(I, cve), the vxv-matrix having

v real and distinct eigenvalues , for all (t,x,t)E IxMe,0 ' with

f2 IxI+fi with the compactification Me,0 of 11, 3, ea=(f, ,f2We then will speak of a strictly hyperbolic system of type f=eA .

An N-th order vxv-system (4.1) will be called strictly hyper

bolic of type ea if it leads to a strictly hyperbolic system (4.4)

We will rederive all results of sec,s 2.3.4 when the symmetry

condition is replaced by a weaker condition, using "symmetrizers".

A (global) symmetrizer is defined as a one-parameter family

{R(t)=r(t,Mw,D):tE I} of zero-order 4 do's, r(t,.,.)E C0O(I,ta0),

such that, in the Hilbert space H, we have R*(t)=R(t)ac>0, where

R(t) is md-elliptic, uniformly over I, and that


(4.10) (RK)*(t)-(RK)(t)=K*(t)R(t)-R(t)K(t) E C'(I,OptpcO) .

Here "*" denotes the H-adjoint. The operator R(t), in other words,

is a bounded hermitian and positive definite operator H - H. Since

we use the Weyl representation R(t)=r(t,MM,D) of 1,3, it is no

loss of generality to assume hermitian and zc>0 .

Symmetrizers were first studied by Friedrichs and Lax [FL].

For an application similar to ours cf. Taylor [T1]. We achieve a

simplification of the method of [T1], using prop.4.1, below.

Proposition 4.1. Given a symmetrizer R(t) = r(t,Mw,D) (satisfying

above assumptions). Then the (unique) positive square root S(t) of

R(t) , defined for every bounded positive self-adjoint Operator

H-36 , is a pdo of order 0 again,together with its inverse S-1(t) .

Moreover, S, S-1are md-elliptic, and we get S , S-1E

Coo(I,Optpc0).

Proof. Write S(t)=Af dX\/X(R(t)-X)-1 as a complex curve integralr

with a suitable branch of VT and a curve r in the upper halfplane

surrounding the spectrum of S(t). The resolvent (R(t)-X)-1 is a

Green inverse coinciding with the K-parametrix of II,thm.1.6, mod

O(-oo). In particular, r may be laid such that R(t)-a, is md-ellip-

tic for tE I, ?E r. The terms of the resulting asymptotic expan-

sion of the symbol of the integrand are analytic in X and smooth

in t. The 0-order term yields s0(t,x,U)=2nf r t x%- ,

using a

harmless interchange of integrals. Lower order terms contain

higher powers of r-X in the integrand, but are well defined sym-

bols of proper order. One gets an asymptotic sum j defining a

symbol of S(t) (mod O(-co)). Then all other properties follow.

Notice that prop.4.1 legalizes the transformation

(4.11) S(t)u(t) = v(t) , t E I ,

for the evolution problem (2.1). That problem is taken to

atv+iK"(t)v = g(t)=S(t)f(t) , tE I , v(0) = i=S(0)g ,

K _(t) = S(t)K(t)S-1(t) +iS (t)S-1(t) .

Clearly K"E COO(I,Opyicf) , by prop.4.1. Moreover, we confirm that

6.4. N-th order strictly hyperbolic systems 213

(4.14) K-*(t)-K (t)=S-l(K*R-RK)S-l_i(S-lS'+S'S_l)E C'(I,Oppc0)

using calculus of tpdo's. For a =e all assumptions of thm.2.1, thm.

3 . 1 , cor.3.2, cor.3.3 now hold for K-. We have S, S I O(0), they

map Hs-Hs, and S-S, and S'-'S'. For e°=e we even have the stronger

conditions of rem.3.4. The result is summarized in thm.4.2, below.

Theorem 4.2. Let the system (2.1) of ic1o's satisfy kE Cm(I,Vcv,)

and assume that there exists a symmetrizer R(t), satisfying above

conditions. Then the statements of thm.2.1, thm.3.1, cor.3.2, cor.

3.3 still hold. Moreover, for e°#e, the solution u of (2.1) satis-

fies atuE Hs_je, (not only E Hs-je), as f(t), gE Hs . Also, for

the solution operator U(t,t) we get aiatUE O((j+l)eA)

Now we will show that a strictly hyperbolic system (2.1)

(or (4.1)) of type a°=e, e',or e2 , in the above sense, always has

a global symmetrizer, so that thm.4.2 applies. Indeed, we have

(4.9), where the eigenvalues of are real and distinct

uniformly over in the compactification Ie'0. It fol-

lows that k0(t,x,t;) is diagonalizable for and that

its eigenvalues may be arranged for

(4.15) t-'2(t,x'U) < ... <

Moreover, we must have

(4.16) 2Tt1>0, j=1,...,v-1, f2 IxI+ft krl0.

with a constant rt1 > 0 , independent of t,x,l; . By well known

perturbation arguments for matrices it then follows that the

functions R j are coo for t E I , f2 I x I +fi 11; I ZTI0 .

The point is that the spectral projections pj(t,x,t) of the

symbol defined for f2JxI+f1JlJ 2 rt0, may be extended to

symbols pjE C"(I,tv), and that then R(t)=r(t,MM,D) with the symbol

(4.17) rO+r_e

for a suitable term r_e, will define a symmetrizer. Note r(t,x,l;)

of (4.17) is symmetric modulo lower order. Its first term r0 is in

vertible, since r0u=0 implies pju=0, j=1,...v, hence u=lpju=0. We

have k0=D'lpl,hence r0k0 X jpjTpj , using that pjpl=pjoj1 . Also,

with pTplT=pTSj1 , and k0=q,1p1T , we get 0r0=Xjpj Tpj=r0k0 , for


tE I, If we can show that tpco then we

can hope to use calculus of Wo's , and a proper choice of r_e to

get the desired properties for R(t) .

First the eigenvalues µj(t,x,t) are solutions of the charac-

teristic equation ®(? =0, with ©, a poly-

nomial in x with highest coefficient (-1)v, and other coefficients

in tvc0. Its roots are distinct, by assumption, and are bounded by

the matrix norm of k0(t,x,t) which is bounded above. For the first

derivatives one finds

µj(a)=-(0(a)/©(4.18)

(=1,j'1(µj(t1x1t)-µl(t,x,t))

which is found to be 0(( )-JaJ(x)-JPJ) , using condition (4.16),

and the boundedness of Rj . Similarly, for the higher derivatives,

including t-derivatives, one finds by successively differentiating

(4.18) recursively that t,j(P)= O(x-JaJe'-JPJez(x,U)) , so that

indeed µ.E C (I,'tpa0) . In all the above we were always assuming

f21xI+f. 4 1kn0, and all estimates are derived only for those x,

For eA=e it is clear that any extension of the restriction of Rj

to JxI+ItIX210 to a C00 (Ixle2n)-function will satisfy the same esti-

mates, thus give symbols in C°°(I,tvc0) equal to the µj, for large

IxI+lgl. For a°=e' use a cut-off x(t) , =1 in ItIa2rI, =0 near

111-n} , and use µjx instead of µj . Similarly for aA=ez

For the pj we have a complex integral representation

(4.19) i/2xJ-µj (=Y1' /2

with the constant n1 of (4.16). Here (4.19) may be differentiated

under the integral sign, where the integration path may be kept

constant during differentiation. It is clear at once that we get

p jE C0o(I,Vcyo) , after modification outside J f2 Jx J+f. I t I x2rI} _ E11 .

For e°#e again choose xpj as pj, with x as for µj. We summarize:

Proposition 4.3. For a strictly hyperbolic system of type a°, with

e° = e , e' , ez , and k0E Cm(I ,'grc0) , the eigenvalues R j of k0 aresymbols in C°°(I,'Wa0) , and the spectral projections pj are symbols

in Cm(I,Vcvo), both after suitable modification outside some Ix 2" ,i.

Theorem 4.4. Let equation (2.1) be strictly hyperbolic of type e°

and assume that k0E C0O(I, Then there exists a symmetrizer,

and thm.4.2 is applicable.

6.5. Particle flow 215

Proof. For a°=e let r be given by (4.17) with r-e to be determined

For e°#e add (1-x) with or x(x) to r0 at right of (4.17), res

ectivel Observe that R *p y. O=2(r0(t,x,D)+r0(t,x,D) ) has the proper

form, by calculus of pdo's , while R0 is a bounded self-adjoint

operator of H by III, thm.1.1. Evidently R0 is md-elliptic of

order 0 , and so is RO-y , 0sys1/n , because, for zE &n , we get

(4.20) z r0z = 1ipjzl2 anlIzl2

Thus we conclude that the self-adjoint operator RO(t)-y is

Fredholm, for y<l/n , hence can only have discrete spectrum there,

by 11,4. Moreover, the eigenfunctions of this operator are in S,

hence its spectral projections are operators in O(-ao). Thus we may

define R(t)=R0(t)(1-Py(t)), for example, where Py(t) is the ortho-

gonal projection in H onto the span of all eigenvectors to eigen-

values s y < 1/n . Here y must be selected locally, for tE AC I

such that none of the eigenvalues ever equals y . Then P(t) E

C00(A,4v-,0) , and we may define a global R(t), for all tE I , as

a weighted mean, using a suitable partition of unity 1=(j(t)

This completes the proof of thm.4.4.

5. The particle flow of a single hyperbolic ide.

In this section we again consider the Cauchy problem (2.1)

but assume v=1, i.e., a scalar problem

(5.1) u +ik(t,M1,D)u = f , t E I , u(s) = cp ,

with a compact interval I , containing t , and a complex-valued

symbol kEC00(I,Pce)

. The conditions (2.3), (2.4) then amount to

(5.2) k1 real-valued, k0E C(I,'pc0)

Under these assumptions thm.2.1, and thm.3.1 apply. For j=0,1 we

assume kjE C"(I,y1aje) and then have cor.3.2 and cor.3.3 as well.

Assume a°=e; the cases e°#e mean stronger assumptions, to be stu-

died later on. We will get back to systems in ch.9.

In this section we focus on the result below, regarded as one

of the focal points. Thm.5.1 will be called Egorov's theorem,

although what we present will be an amended global version of the

result of Egorov [ Egl] .


Theorem 5.1. Conjugation A - At't= U(t,t)-IAU(t,t) with the evo-

lution operator U(t,t) of (5.1) constitutes an order preserving

automorphism of the algebra Op4v of pseudodifferential operators

with symbol in e = yic00 , as defined in 1,7, for all t,% E=- I. In

other words, for A = a(M1,D) E OpVcr , rE 12 , we have

(5.3) At ,.t = U(t,t)AU(t,t) = at,t(Mi,D) ,

with a certain symbol at't E Vcr , for each t,t E I .

The proof of thm.5.1 will be prepared by discussing a few

consequences of the theorem providing hints towards the proper

approach of proof. First consider the case r=0. In V,10 we have

discussed the C*-subalgebra As of L(HS) obtained as norm closure

of Op,a0 in L(Hs). It was seen that the symbol space of As -i.e.,

the maximal ideal space of the commutative C*-algebra As/Ks - is

given as boundary Me'O of 12n in the compactification Pe,O allo-

wing continuous extension of symbols aE VcO. We trivially have

(5.4) U(t,t)KSU(t,t) = Ks , KS=K(HS)

From thm.5.1 we conclude that

(5.5) U(t,t)ASU(t,t) = As

Accordingly there is an induced automorphism of the commutative

C*-algebra AS/KS a C(Me,O). It is well known that every such auto-

morphism of an algebra C(Me'0) of continuous functions must have

the form y(x) - q(v(x)) , with a homeomorphism v: Me,O `-'Me,O

The map v is defined as dual of the automorphism. Thus we have

(5.6) v' = vi't , v': Me,O H Me,0 , t,t E I ,

a family of homeomorphisms induced by the automorphisms (5.5) with

(5.7) at't = aovt,t , as IxI+I I=o , for all aE pcO .

In other words, the map a-at,t of the symbol of A=a(x,D) un-

der the automorphism A->U 1AU of thm.5.1 is given by a certain fami-

ly of homeomorphisms, as far as the values of a at - are concerned.

Perhaps this justifies the idea to attempt construction of

at't over all of 12n from a family of diffeomorphisms

(5.9) vt,t:12n , 12n , t,t E I ,

extending continuously onto the compactification Pe,0 , where

6.5. Particle flow 217

(5.10) V1mt,t vt,t Me,0 '

while we expect U IAUa(aovt't)(x,D). Let us assume that

(5.11) vt 't = identity = id , vi'K

= vt'xovt 't t,x,t E I ,

because the corresponding relations for v' follow from (3.15).

Actually (5.11) points to the fact that vt't is a flow generated

by a system of differential equations (details later on).

Our condition that vt,t extends continuously to the compac-

tification 1e,0 amounts to the statement that

(5.12) a E lc0 aovt't E Vc0

(We shall see that even a E tpcr implies aovE 1Jcr , for all rE 1R2. )An explicit construction of vt't and a proof of thm.5.1

results if we assume that the symbol at't of (5.3) is of the form

(5.13) at = aovt,t + rt,t rt,t E yic-e .

For simplicity of notation assume t=0 , and write

U(t) = U(0,t) . At = AO,t , at = a0,t , vt = v0't , v't = V10't

From thm.3.1 we know that U(t) hence At admits a strong

t-derivative in L(SsHs-e

) Using (3.16) we find that

(5.14) At =U (t)AU(t)+U(t)AU (t)=iUI(t)[K(t),A]U(t)=i[K(t),A]t.

Notice that the commutator [K(t),A] of two Wo's is readily

evaluated in terms of 'pdo-calculus. From I,(5.7) we conclude that

(5.15) i[K(t),A] = (k,a) (Ml,D) + Rt , Rt E Opyic-e ,

with the Poisson bracket

(5.16) (k,a) I j=1(k(e)a(8) - a(8)k(0)) .

Assuming ( 5 . 1 3 ) , and in addition that rt exists and rt E la_e

(5.17) at (x, ) =

a symbol of order -e . It is practical to replace k(t) by

the real-valued k1(t) of (5.2), effecting an error term (k0,a) of

order -e again. Applying the chain rule to the left hand side and

(5.14) to the right hand side of (5.17), with

(5.18) x, E &n


a comparison gives the relations

(5.19) xt =kIN t I, x0=x , 0

modulo terms of order -e again.

We try to satisfy (5.19) precisely, not only modulo -e, obser-

ving that (5.19) gives a Cauchy problem for a nonlinear system of

2n ODE's in the 2n variables This is a hamiltonian system

with real-valued Cm'-coefficients kll , kllx. Locally, for a small

the solution of (5.19) exists uniquely, and depends

smoothly on the initial parameters by basic results on ODE's.

The local solution may be continued into all of II2nxi provided

that we can derive apriori estimates insuring boundedness of

over I, for any given finite x,g. Such estimates will be

obtained in sec.6. Hence for a,1E IIn, aXa t(x,) EC00(Ix&2n) , by standard results (cf. [CdLv], for example).

Note that the same argument for U(i,t) instead of U(t)=U(O,t)

will lead to the same system (5.19) of differential equations, but

with initial conditions at x, not at 0. Thus we get (5.11) as an

consequence. Particularly, vt,T,vt't = identity, so that vt't is

indeed a family of homeomorphisms (even diffeomorphisms) of &2n

onto itself. This family will be called a particle flow of equat-

ion (5.1) Clearly the particle flow is not uniquely determined by

(5.1), since the symbol ki is only fixed up to an additional term

of order -e. However, it is unique at lxI+lgl=oo , as will be seen.

Theorem 5.2. For any given real-valued k1E pce the flow vs't defi-

ned above (following the solution of (5.19) through (at t=

t) to t=t) induces a family a - at't = aovT' t of automorphisms

'm H yicm , for all m. Moreover, we have at,tE C'(IxI'Vcm), mE I

The proof will be discussed in sec.6.

It is evident, after thm.5.2, that the map aa,,t aov,,t also

provides an automorphism of the sup-norm closure of the function

algebra c0 , the maximal ideal space of which defines the compac-

tification 2e,0. The associate dual map of this automorphism will

be a homeomorphism defined as the continuous extension

of v't onto Pe'O. The restriction v'T't to Me,O=ape,0 will be

uniquely determined by the system (5.1). This map coincides with

the homeomorphism v't,t'Me,O-Me,O previously discussed (cf.(5.6)).

Proof of thm.5.l. From (5.11) we get at(vt',0v,'t) = 0 . From the

ODE's (5.19), using the Poisson bracket of (5.16), we get

(5.20) atxt'i + (kl,xt't) = 0 , atgt'T + 0 ,

6.6. The action of the particle flow 219

with (,) of xt"t (x,l;) taken componentwise.For an arbitrary smooth a(x,g) and at.t=aovt,,L we get

(5.21) atat,t(x, ) + (ki,at'z) = 0

as follows by using (5.20) in

(5.22)

)atxt,t +a,, (xt't )at t.tFor a E Vcm let us use the operator

(5.23) Pt't= U(t,t)At't U(t.t) . Ate.,=at,t (x,D) , at,,=acv t',t

Clearly At't EC00(IxI,L(HS,Hs_m)), by 111,3, and thm.5.2. Since

atU(t,t)E C(IxI,L(HS,HS_e)) in strong operator topology, by thm.

3.1, we get Pt't E C1(IxI,L(Hs'Hs-m-e)) in strong convergence and

(5.24) atPt 't = U(t,t ){ (atat,t) (x,D) + i[ K(t) ,At't] }U(t ,t) = Vt

by (5.14). Using (5.15) and (5.21), and calculus of pdo's, and UE

0(0) (by thm.3.1) we find that VtE O(m-e). Also Vt is strongly

continuous in t as a map Hs-Hs-m+e Hence Vt is integrable, and

Jt(5.25) Pt,t Pt,t = U(t,t)At,tU(t,t)-A = doVV E L(HS'HS-m+e)

T

This confirms that at least (5.3) holds modulo an additive term in

0(r-e). To get more precision note that Vt=U(t,t)WtU(t,t), with

Wt E OpVcm-e , hence (5.25) assumes the form

t(5.26) U(t,t)AU(t,t) -at't(M1,D) - dK U(K,t)WKU(t,K)

t

Clearly (5.25') may be iterated. Using (5.26) on Wk expresses

U(K,t)WKU(t,K) as a sum TK+RK, with TKE Opyicr_e, remainder RK in

0(r-2e), as in the integral of (5.26), etc. Thus, for all N,

(5.27) U(t,t)AU(t,t) = at,t(M1'D) + "l=1Tt t + XN

with Tt E Op"cr_Ne XN of order r-(N+1)e Then we get

00l Tt't + X00(5.28) U(t,t)AU(t,t) = at't(M1,D) + Yj

with X°'E Oppc_o,, and an asymptotic sum at right, using I,6. Q.E.D.

A generalization of thm.5.1 to hyperbolic vxv-systems was

discussed in [CE]. We will discuss this in ch.9 for a larger and

more natural algebra of Wo's (called Opts), rather than for Opypa.


6. The action of the particle flow on symbols.

In this section we again look at the particle flow {vtt}

In particular we prove thm.5.2. Assumptions are as in thm.5.1.

In order to simplify the calculations let us write

(6.1) cP=cP(x,)=VP(x,,t,i)=tt(x.)

Writing k1=k , for simplicity, we then have (5.19) in the form

(6.2) f'=kl (f,g) , cp'=-kix(f,g) , f=x , T= at t---r .

Proposition 6.1. The functions f , cp satisfy the apriori-estimates

(6.3) 0<CS(f) /(x) sC , 0<cs( (P) /( ) sC ,

for t,t E I , &n , with constants c,C independent of

Proof. One derives the following inequalities for (f)' , ((P)' :

(6.4)(f)' = f.f'/(f) = kl .f/(f) = O((f)) ,

((p) * = cp cp * /(cp) =-k I x . w/(Cp) = 0((q)) )

Integrating (6.4), under the initial conditions (f)=(x) , ((p)_( )

at t=v we get

(6.5) log((f)/(x)) = O(1) , log(((P)/(l;)) = O(1) .

Since (f) , (q) a 1 this indeed give (6.3), q.e.d.

Note that prop.6.1 gives the apriori estimates required for

continuation of the local solution of (6.2) into all of I , for

any choice of initial values x,l; . As noted before, we then also

get continuity of derivatives aiatf(R) , aial(P(s) of all orders

j,l,a,p , as t,tE I , 1n

Proposition 6.2. For j,1=0,1....., and a,pe ffin we have

(6.6) aiatf (a)=0((x)1-IUI()-IaI) ,

a C°-function consider the composition

(6.7) b(f(x,5),g(x,t))

The derivative c(R) is in the span of the terms

(6.8)

6.6. The action of the particle flow 221

(PXvb)(f. ))njri f03 nlsl

where Iajl+If3 I21 , Ia'lI+IP'lIa1 , Yaj+2q'l=a , 2Pj+D "=p

t

and where the tensors V V b and vectors f(a) , are to bex (PIcontracted in arbitrary order of indices, f with T with 0

This proposition follows by induction.

For the proof of prop.9.2 we also use induction and first

assume j=l=0 and a+t3=1 . Note that the case j=l=a=p=0 is a matter

of (6.3). Differentiating (6.2) we obtain

f(a). = k f(a) + k (a) f(a)=x(a)

(6.9) (3) Ix (1) (0) (1) , as t=t

(P(P) --klxxf(R) - klx(P(3) ,

,(P)_(P)

Here we have written klxf=4Ixjfj , butjfj , etc.

Multiplying the first equation (6.9) by (cp)/(f) we get, as matrix,

(6.10)

(p)/(f)f(R)I

(a)(P -k l X'( f) -k l xi; I I a, I

The 2nx2n-matrix in (6.10), called P1 , has bounded coefficients,

since kE pce. Let the vector at right in (6.10) be called p=(pZ)

The left hand side in (6.10) is not p', but we get

(6.11) P. ._

hence (6.10) yields the linear ODE p'=Pp with a matrix P differing

from P1 only in the upper left nxn-corner. There we find the addi-

tional term -((f.kl,)/(f)2+((p.klx)/(p)2) , also bounded in x,

so that P also is a bounded matrix. We get

(6.12) IPI2'=2p.p' = 2p.Pp = O(IPI2)

This may be integrated for log (IpI/Ip(s)I)=O(It-tI)=O(1). Calcu-

lating Ip(t)I from (6.9) we get (6.6) for j=1=0, IaI+I3I=1 .

Proceeding in our induction proof, assume (6.6) true for j=1=0 and

IaI+I1Isr-1, and consider a pair a,3 with I(xI+I3I=ra2 . We claim

that the vector p at right of (6.10) now will satisfy

klgx

(6.13) p* = Pp + q , p(t) = 0 , q = (qZ )


with the above matrix P , and with q' , q2 in the span of

(6.14) (0) 111=1 (R4)

and of

(6.15) (Qs+IQs'k(f,T))n3=1 f(3) 1Z1=1 T(e)

respectively. In (6.14) and (6.15) we assume that s+s'sr, and

1sIajI+I3jI<r, 1sla'll+IP'lI<r, jaj+Ya'l=a, 2Pj+2P'l=P. Again con-

traction of V V k with the products of vectors is in arbitrary

order of indices. Relations (6.13), (6.14), (6.15) follow by re-

peated differentiation of (6.9), using (6.8).

The induction hypothesis implies that

by direct examination of the terms (6.13), (6.14). Then (6.13)

implies Ip12.

= O(IpI2+IgI2) 1 IP(t)I2=0 . This may be integrated

for completing the proof for j=1=0.

Next we look at t and t-derivatives of f,q . First it is

evident for the vector of (6.13) that also p'=O((x)-IPI(t)1-Iai)

just looking at (6.13). This gives (6.6) for j=0, 1=1. In fact,

we may differentiate (6.13) for t . The right hand side then con-

tains only terms involving the 0-th and 1-th t-derivatives of

f(a and cp already known to satisfy (6.6) Using this, one

finds that also p''=0((x)-IRI(1;)1-IaI) In fact, the proceduremay be iterated to get the same for all atp . This implies (6.6)

for j=0 and all 1=0,1,2,... .

Finally let us look for t-derivatives. In that respect it

is convenient to notice that (f,tpj=(xtt'tt) satisfies the system

(6.16) f'=-k1 (f,q) , (p'=klx(f,(P) ,"l" = "a/at"

Indeed, relations (5.20), (5.21) indicate that, for fixed

the curves vt,t(x°,°) are the characteristic curves of

the single first order PDE (5.21) in the unknown function

This equation differs from (5.17) only by a sign, hence

it follows that vt,t must be the 'reverse flow' satisfying (6.16),

just as vt,t satisfies (5.19) . This clearly implies (6.16).

However, (6.16) implies an ODE like (6.13) with ""ll replaced by

for the same vector p with components ((,)/(f)f(R)=p'

y(a)=pz. It follows at once that (6.6) holds for 1=0, j=0,1,2,...,

for reason of symmetry. But mixed t,t-derivatives may also be est-

imated in this way: For example, writing p'=Qp+r, we get p''=Qp'+

6.7. Propagation of maximal ideals 223

Q'p+r'=QPp+Qq+Q'p+r', where the right hand side consists of terms

already estimated. Hence p''=atatp may be estimated. Similarly for

all mixed t,t-derivatives, completing the proof of prop.6.2.

Note that now the proof of thm.5.2 is a matter of induction.

First of all it follows from prop.6.3 that ai't aovt,t=a(f,(p)6Vcm

for all t,t E I , whenever aE Vcm : All terms (6.8) obey their

proper estimate, using (6.6) . Moreover, also the derivatives

ajd1at't exist in the Frechet topology of "m , so that at'tE

C (IxI,tUcm) . For example,

(6.17) atat,t = alx(f,(P)f' + a, ro' =

using (6.6), prop.6.1, and the estimates for the symbol aE Vcm .

Similarly for all higher derivatives, by induction.

7. Propagation of maximal ideals and propagation of singularities.

We have noted before that the particle flow induces a class

of homeomorphisms v.t:2 1 , of the space 1=2e,o , i.e., the

maximal ideal space of the C*-algebra A=AO of V,10. In particular

vtt is defined as restriction to M=aP of the continuous extension

to P of vtt defined by (5.18),(5.19). The algebra symbol aA oftt

A t=U(t,t)AU(t ,t) is given as

(7.1)('A = 'A°vtttt

for every AE A. Similar for AE As , the Hs-closure of OpycO

It is interesting to note that the above facts immediately

translate into a result concerning propagation of wave front sets

- and, correspondingly, propagation of singularities - under hyp-

erbolic evolution. The key is the following result (cf. [Tl2]).

Theorem 7.1. For any distribution uE 5'(ien) the wave front set

WF(u) (as defined in 11,5) is given as

(7.2) WF(u) = fl(char(A,C): AE C° , AuE S} ,

with the characteristic set of A relative to C of V,10 defined by

(7.3) char(A,C) = {mE W : aA(m)=0} .

Proof. Suppose a(x° #0 for some (x° )E W , A=a(x,D)E CO ,

with AuE C'(An) . Then it follows that, with respect to some cut-offs (Px° (x) 4k-0 () , as in 11,5 , we must have A md-elliptic with


respect to in the sense of 11, 3. By II,thm.

3.3 we then get a local left parametrix Et such that EiA=q(Mr,D)+

Ki , Ki E O(-oo) , Et E Op4c0. It follows that (tk, ((px° u)" )" =E, Au-K, uES , hence W\WF(u) , by II,prop.5.1.SSIn other words, we

have "C" in (7.2). Vice versa, if (x° , °) E W\WF (u) , then thereexist cut-offs as above with i(D)W(x)uE S , by II, prop.5.2. It is

clear then that B=(p(D)g(x)E C° , while does not belong tochar CB . Hence we also get ":" , q.e.d.

Thm.7.1 first prompts us to define an A-wave front set

(7.4) WF(u,A) = fl{char(A,A): AEOp(Nc0 , AuES}

where

(7.5) char(A,A) = {mE W(A): UA(m)=0}

with the wave front space W(A)=i-1W

= b-1(&nxa1n) of A , using the

surjective map L:Me,O - I of 11,3 . Clearly we get

(7.6) LWF(u,A) C WF(u) .

We also may define the larger sets

ZF(u,A) = fl{charmd(A,A) : AE Oprquo: AuE S} , with(7.7)

charmd(A,A) = {mE ffi(A) : aA(m)=0} .

Note that ZF(u,A) (and ZF(u,C) = ZF(u) defined similarly, with

reference to C instead of A) also addresses singularities of uE S'

at x=-. In fact,

(7.8) ZF(u)J WF(u) U WF(u^) .

Theorem 7.2. Consider the solution u(t)=U(t,t)q of the Cauchy pro-

blem (3.3) with f(t)=O . We have

(7.9) WF(u(t),A)= vt(WF(q),A)) , ZF(u(t),A)=v,t(ZF((p,A))

Similarly , for f(t)#0 , the sets at left in (7.9) are contained

in the unions of the sets at right with the sets ZFJ(U(K,t)f(K),A)

=vxt(C JZF(f(K)A)), K between t and t

Proof. Observe that ZF(u(t),A)=fl(charmd(A,A):U(t,t)AU(t,t)TE S}where charmd(A,A)={m:OA 0}={m:OAtt(vtt (m))=0}=vtt(charmd(Att'A) )

Therefore, ZF(u(t),A)=vtt{charmd(Att'A):AttuES}=vtt(ZF((p,A)

Similarly for WF(u(t),A) , and for ZF}(U(K,t)f(K),A). Q.E.D.

6.7. Propagation of maximal ideals 225

Corollary 7.3. Let k0(t,x,l)=symb(K0(t)) of (4.9) allow an asympto-

tic expansion k0 Dc0j (mod ICI) at 1;=-, as in V,(5.10). Assume kj

E Co°. for tE I, 1 1;I=1, xE KG: &n, all K. Then we also get formulas

(7.9) for WF(u) instead of WF(u,A) .

For a proof we only must show that (i) CO in thm.7.2 may be

replaced by the algebra B of all AE OpipcO with symbols having

asymptotic expansions, and (ii) that conjugation with U(x,t) lea-

ves B invariant - assuming the asymptotic expansion on k0 , while

(iii) the particle flow leaves M invariant. Details of these (ra-

ther formal) discussions are left to the reader.

Remark 7.4. A propagation law for the generalized singularity sets

ZF(u) may be derived as well, under proper assumptions on k0 .

Problems. (1) Do the algebras VOS and 'L (cf.VIII,5) have commut-

ators in K(LZ(&n))? (2) For the L(L2(in))-norm closure A of an al-

gebra of (1), give a useful definition of WF(A,A) and ZF(A,A).

Chapter 7. HYPERBOLIC DIFFERENTIAL EQUATIONS

0. Introduction.

A polynomial P(x) = I sNaaxa , aaE E (and the PDE P(D))

is called hyperbolic with respect to a vector hE in if (i) PN(h)#O

(ii) for some real to we have P(t+ith)#0 as t E=- &n, tsto.

The above definition was given by Garding [Gal]. Its impor-

tance for the Cauchy problem of P(D)u=f in the half space x.h2:0 ,

with data given at the hyperplane x.h=0 becomes evident if we try

to apply the Fourier-Laplace method of ch.0,4: Let h=(1,0,...,0).

Taking the Fourier transform in (x2,...,xn) and the Laplace trans-

form in xl will convert P(D)u=f into P(t+ith)u^=f^, where the ima-

ginary part t of the t1-variable must be sto, for some to, in ac-

cordance with the rules of the Laplace transform of ch.O,(3.14).

Thus (ii) insures that we can write as tsto. Spec-

ifically, for f=6/xn, the inverse transform (P( +ith))"=e defines

a fundamental solution of P(D). An analysis shows that (i),(ii) in-

sure existence of e as a complex integral. Moreover, a=0 outside

a cone lxlscx,. This leads to Garding's theorem:

Theorem 0.1. A differential polynomial P(D) admits a fundamental

solution eE D'(&n) with support in a strictly convex cone OE h C

{0} U {x-h>0} if and only if P is hyperbolic with respect to h.

For a distribution e with the properties of thm.0.1 and a

general fE D' with support in the half-space x.hzO the convolution

product u=e*f E D' is always defined, and supp u s {x.hz0} ([CI],I,

thm.8.1,for example), and P(D)u=f. In fact, the noncharacteristic

Cauchy problem of the half space x.ha0 is well posed if and only

if P is hyperbolic with respect to h (written as "PE hyp(h)"). For

a detailed discussion of such results cf.[Hrl].

For PDE's with variable coefficients we do not know results

of similar precision. Existence theorens assume strict hyperbolici-

ty (cf. VI,4) which is not a necessary condition. Still we discuss

some useful algebraic properties of hyperbolic polynomials in sec.

1 below, focusing on the cone h of time-like vectors. In sec.2 we

226


consider strictly hyperbolic (differential) polynomials. The class

s-hyp(h) of strictly hyperbolic polynomials consists precisely of

all PE hyp(h) with only simple real characteristic surfaces (i.e.,

the PE hyp(h) of principal type). "s-hyp(h)" is a property of the

principal part of P, insensitive against lower order perturbations

A PDE a(x,D) with variable coefficients is called s-hyp(h)

at x=x° if a(x°,D)E s-hyp(h). For an a(x,D), strictly hyperbolic

over a domain SIC Rn the vector may depend on x. If is can be cho-

sen constant, we speak of a normally strictly hyperbolic a(x,D).

It is convenient then to assume h=(1,0,...,0) and distinguish the

time direction h. This leads to sec.3, considering an N-th order

vxv-system, looking like an ODE in t with coefficients PDE's in x.

We prove existence and uniqueness for a relativistic and a nonrel-

ativistic normally strictly hyperbolic Cauchy problem.

In sec.4 we simplify the assumptions, essentially by linking

strictly hyperbolic differential operators with the strictly hyp-

erbolic i,do's of ch.VI. In sec.5 we discuss the 'hyperbolic featu-

res' - region of dependence and influence, and finite propagation

speed. In sec.6 we derive a local existence result, and apply it

to hyperbolic problems on a manifold.

1. Algebra of hyperbolic polynomials.

We follow [Hri], discussing some algebraic properties of hy-

perbolic polynomials. No differential operators will get involved.

Proposition 1.1. A homogeneous polynomial P = PN is hyp(h) if and

only if (i) P(h) # 0 , and (ii) for each fixed 1; E Rn the equation

P( + th) = 0 has only real roots.

Proof. Since P = PN the equivalence of conditions (i) is evident.

For a homogeneous P E hyp(h) let P( + rh) = 0 . Using that P is

homogeneous we also get P(aT+ath) = t)h)+i(Im t)a) = 0,

for all real o. Since P is hyperbolic, we get a Im t zr0, for real

a, implying that In t =0. Hence all roots of are real.

Vice versa if all roots are real, we get for all real

t#0 , implying (ii) with t0=0 . Hence P is hyperbolic.

Corollary 1.2. A hyperbolic homogeneous polynomial has real coef-

ficients, except perhaps for a common complex factor # 0 .

Proof. We have Thus must equal

the product of all the (real) roots of hence must be

real, for all real . This implies that has real coef-

228 7. Hyperbolic differential equations

ficients, since all its values, for real l;, are real.

Proposition 1.3. If P is hyp(h) , then P is hyp(-h).

Proof. We have PN(-h)=(-1)NPN(h)#0 , i.e., condition (i) for -h.

Consider the polynomial of the single variable t

(1.1) P(T) = P(l+ith) = PN(ih){tN +

of exact degree N. We must have with the N roots

tj( ) of p(t)=0. Also, pl(t) must be a first degree polynomial in

, i.e., a linear function of 1. We have Re tj ar0 for all , thus

Re pl(t) a Nt0 . For a linear function this implies that Re

const. Accordingly, Re tk = Re p1 - I Re tj s Re p1 -(N-1)t0j7q--

is also bounded from above. Accordinglg we also have p(t) # 0

for all t E Rn, and t s t1 , q.e.d.

Proposition 1.4. The principal part PN of a PE hyp(h) is hyp(h).

Proof. We note that PN(t) = limlal_.,,,, with Q0(t)=a NP(at).

Let qa(t) =Q0(g+ith) , then also lim qa(t)=q(t)=PN(l;+ith) , forall t . Suppose q(t)=0 , for some real t . There must be some com-

plex root to of qa(t) = 0 , 101 > 00 , with lim t a = t . But this

implies aNga(ta) = P(ao+itaah) = 0 , so that a Re ta a t0 . Since

Re to -> t this is possible only if t=0 . Accordingly,as t is real and # 0 . Therefore PN is hyperbolic. (Note that the

highest coefficient of qa(t) is PN(h)(i)N again, hence is indepen-

dent of a . Therefore qa is close to q on some sufficiently large

circle Jtj=a0 , so that Rouche's theorem supplies the root to .)

The converse of prop.1.4 is false: A polynomial P with PNE

hyp(h) needs not be hyp(h). For discussion see the proof of prop.

2.6. Given a hyperbolic homogeneous PN , the polynomial P=PN+R ,

deg R <N , will be hyperbolic, if (and only if), with constant c

(1.2) 7,IR(a)( )I2 s c 1PN(a)(,)12,

(=-ina a

We will discuss this in sec.2.

A polynomial P (or a differential expression P(D) ) will be

called strictly hyperbolic (or Petrovsky-hyperbolic) (with respect

to a real vector h3A0 ) if (i) PN(h)#0 , and (ii) for each E 1n

linearly independent of h , the equation has real and

distinct roots. We use the notation s-hyp(h) instead of hyp(h)

for strictly hyperbolic polynomials or expressions. Note that


"s-hyp(h)" is a property of the principal part PN. The lower order

terms do not enter the definition. We will show in sec.2 that

"s-hyp(h)" implies "hyp(h)". Thus PNE s-hyp(h) has the property

that all P=PN+R , deg R <N , are hyp(h) .

For a polynomial PE hyp(h) , O#hE mn , we introduce h=h(P,h)

as the set of all TIE In such that the polynomial p(X)=PN(rl+?h) has

only negative real roots. Prop.1.1 and prop.l.4 imply that the

roots are real. The highest coefficient P (h) is constant, hence

X changes continuously with q . Thus hC I is an open set. For rl=h

we get p(X)=PN(h)(1+X)N, with all roots ?=-1. Hence hE h , and h

is non-void. Also, h is a cone: If ?.E h, and p>O, then PN(pll+Xh)=

pNPN(rl+hk/p)=0 implies that 2=p(k/p)<O. In terms of special rela-

tivity, for the special hyperbolic expression D2+...+Dn-Di , this

cone h plays the role of the cone of time-like vectors.

Theorem 1.5. For P E hyp(h) the set h = h(P,h) is an open convex

cone. Moreover we have P E hyp(rl) for all rl E h

The proof requires some preparations.

Proposition 1.6. The set h equals the connected component of h in

the open set {TIE &n: PN(rl)#0}gym . Moreover, h is starshaped with

with respect to h : For riE h the straight segment rjh also is in h.

Proof. For rl E h we get PN(q+th)=PN(h) llj=1(t-t j) , and all t are< 0 . Hence PN (rl) = PN (h) n(--c j ) # 0 . We already know that h isopen. For an it on the boundary ah all roots tj still are s 0 , but

at least one must be zero, so that PN(rl)=0 . Hence ah is a subset

of am . To complete the proof it suffices to show that h is star-

shaped, since then it must be connected, and cannot be a proper

subset of the component of m containing h; otherwise some boundary

point of h would be in the interior of m. Let TIE h , and =X.rl+µh,

X,p > 0 , X+µ=1 . Write PN(t+th)=PN(q +(t+µ)h)=XNPN(rl+h(r+µ)/a,).This shows that for a root t of PN(l+th) = 0 the expression t' _

(t+µ)/,% must be a root of PN(q+th)=0 , hence is negative. Hence

also t=1v'-p< 0 . It follows that tEh , so that h is starshaped.

Proposition 1.7. For q G h we have as Re t < t0,

Re a s 0 , E 1n , with t0 for h as in the definition of hyp(h).

Proof. This evidently is true for Re a=0, by definition of hyp(h).

Consider the roots of the polynomial

as Re t < tO . As t varies these roots must vary continu-

ously, since the highest coefficient of the polynomial is indepen-

dent of t . By our above remark none of the roots may cross the


imaginary a-axis so that the number of roots of q(a) = 0 within

Re a < 0 stays constant, as t varies within the half plane Re t

< to. Therefore it suffices to show that q has no roots with Re a

< 0 whenever t is real and ItI is large.

Assume t#0 , and write a = tµ , so that q = 0 takes the form

0 . We know that all roots of PN(h+1x11) = 0 arenegative real, since 11 E Eh . Consider the family of polynomials

rt (µ) (h+1111)) = PN(11)RN + ... . Clearly we getlim -,oo rt (14) = PN(h+µr1) . Accordingly rr (ti) = 0 implies Re µ < 0,since µ must be close to one of the roots of PN(h+µr1) = 0 , as

ItI is large. In other words, for t<0, with large (ti , all roots

a = rr of q(a) = 0 must have positive real part, q.e.d.

Proof of Thm.1.5. Let us apply prop.1.8 for real a,t , where t=ea,

a > 0 , assuming to < 0 . We get

(1.3) 0 , as a < t0/E , Z; E &n .

We know that PN(1l+ah)#0 , due to 11+EhE h , since h is starshaped

and a cone. It follows that P E hyp(11+eh) , e > 0 . But h is open,

hence contains 11E = 11-Eh , for small a>O . Applying the above for

wla instead of 11 one concludes that P E hyp(11) , for every rl E hFinally, to prove convexity of h , let rl1 112E h. It follows

that P E hyp(1l3) , j=1,2 . Moreover, we get h = h(P,11J) , j=1,2,

since h and the 113, all are in the same connected component of the

set m. Since h then is star-shaped with respect to 111, the segment

from 111 to 112 must be in h , so that h indeed is convex, q.e.d.

As an example consider the 'wave operator' a2 -a2 -...-a2xl x2 xn

This operator is hyperbolic and strictly hyperbolic with respect

to the "time-like" direction h=(1,0,...,0) . Indeed, we get

I

I2-1;12. hence

I. The two roots are real and distinct except if t-=0 .

We have P=PN homogeneous, and PN(h)=1#0. The cone h of time-like

vectors is given by as follows from prop.1.6.

2. Hyperbolic polynomials and characteristic surfaces.

In this section we shall investigate the relation between

"hyp(h)" and "s-hyp". It turns out that the set s-hyp(h) consists

precisely of all PE hyp(h) with simple characteristics. Here the

concept of characteristic (surface) must be understood in the sen-

se of ch.0,8. There we already defined the concepts of simple and

7.2. Hyperbolic polynomials 231

multiple characteristic. Presently we look at these definitions

under the aspect of constant coefficient expressions.

A differential polynomial P(D) is said to be of principal

type if PNI (U)#0 for all 0#tE &n . For a principal type expres-

sion all (real) characteristic surfaces are simple. Consider a

surface cp(x)=W(x°), near x0, where PN((PIx(x0))=0 (cf. ch.0,(8.3)).

If h E In is any vector and i; = Tlx(x0) , then the polynomial p(t)

= has a root at t = 0 . If h can be found such that 0 is

a simple root of p(t) then the surface is called simply characte-

ristic at x0. We have given the same definition for characteristic

surfaces in ch.0,8. The multiplicity at x0 is defined as the low-

est multiplicity of the root 0 of p(t) atteinable for various h

For a real root of let so that (pIx t,

and PN(cIx) = 0 . It follows that, for a differential polynomial

P(D), every real root t of the principal part polynomial PN gives

raise to a family x.T;=c of real characteristic hyperplanes, with

normal vector t . Vice versa, for an arbitrary real characteristic

surface t of P(D) the tangent planes, at all points x0E T , and

their parallels, all are real characteristic (hyper-)surfaces.

Now let P(D) be of principal type. Clearly Taylor's formula

yields Hence if is a realroot of PN 0, then we may set h=PNI (U)#0, and get p(t)-rPNI

which clearly has a simple root at t=0 . Vice versa, for an

expression P(D), let at some t#0 . Using Euler's for-

mula for the homogeneous function PN we find that NPN(t)=t.PNIt(l;)

=0 . Thus we get p(t)=O(t2) for any choice of h. Or, the surfaces

with normal t are multiple characteristics. We have proven:

Proposition 2.1. A constant coefficient operator P(D) is of prin-

cipal type if and only if all of its real characteristic surfaces

are simply characteristic.

Elliptic operators (in the sense of ch.0,sec.8(d)) have no

real characteristics, hence cannot have multiple real characteri-

stics. Thus an elliptic operator always is of principal type.

Proposition 2.2. An operator P(D)E s-hyp(h) is of principal type.

Proof. Let PE s-hyp(h), and let for &n. We know that

PN(h)#0 , so that =?.h is impossible. Hence is linearly indepen-

dent of h, and PE s-hyp(h) implies that has a simple

root at t=0. Hence the planes normal to 1; are simple characteri-

stics. It follows that all characteristics are simple, q.e.d.

Theorem 2.3. We have hyp(h)C s-hyp(h). For a polynomial PE hyp(h)


we have PE s-hyp(h) if and only if all real characteristics of

P(D) are simple. In other words, s-hyp(h) consists precisely of

all PE hyp(h) which are of principal type.

We prove thm.2.3 in several steps.

Proposition 2.4. Let P=PNE hyp(h) be homogeneous, and let the po-

lynomial R of degree <N satisfy (1.2). Then also PN+RE hyp(h).

Moreover, if P=PNE s-hyp(h) then PN+RE hyp(h) for every R, deg R

< N , regardless of (1.2). In particular, s-hyp(h)C hyp(h).

Proof. Observe that PN+RE s-hyp(h) implies PNE s-hyp(h), so that

the second statement implies PN+RE hyp(h), reducing the third

statement to the second. To reduce the second statement to the

first, observe that PNE s-hyp(h) must be of principal type, by

prop.2.2. Conclude that PN is homogeneous of degree N-1,

while for all ICI=1. It follows that, with some co>O,

(2.1) aIPNa)( )I2Z IPNI(

)I2. for all E In

However (2.1) implies that (1.2) holds for all R, deg R <N

To prove the first statement it is convenient to write (1.2)

as R< J PN, with the order relation 11<0"" of [Hr1]. Or, F®G means

(2.2) 7, IF(a)(,)I2 s c Ia a

with some constant c, for all E &n. One confirms easily that this

defines a partial ordering compatible with the algebraic structure

of the ring of polynomials. Using Taylor's formula, we get

(2.3) F ( ) = { lIF(a)(,)I2}1/2 s c(T1) F ( + ) ,

for all E In , rl E En , with a constant c(1) independent of .

Accordingly, F®G implies F1. J G,q, for the (real or complex)

translations F,q( )=F( ..... . Evidently F<0 G implies that the

order of F cannot exceed the order of G . Moreover, let G=GN be

homogeneous and F=Lj=1 Fj , with homogeneous parts Fj of degree j,

then F<0G implies that Fj®G for all j. Indeed it follows at

once that also for all t>0, hence that

For a collection t0,...,tN of distinct positive numbers

the van der Monde determinant is different from zero, and we get

(2.4) /k=0 G , j = 0,...,N

Thus indeed, we get Fj <V G , j = 0,...,N

7.2. Hyperbolic polynomials 233

Let PN E hyp(h). From thm.1.5 we conclude that h+Re i; is

time-like for sufficiently small ICI . Accordingly,

as I1Is 2Vn-b >0 , E ten. Using Cauchys estimate lal-times, on the

analytic function f(t)=PN(t+i(h+t)) of n complex variables i;, get

(2.5) IPNa)(t+ih)/PN(t+ih)l s(c/O)lal1max

5

with c independent of . For every fixed t , the polynomial cp(t)

= f(tt) cannot vanish, as ItIs2, assuming Il;i Is O , i.e., IzIsVno.

This gives (p(1)=T(0)r)j=1((tj-1)/s with the roots ti , ltjlz2

of q . Hence the Max at the right hand side of (2.5) is estimated

by I(p(1)/q)(0)Is(1/2)N, as I?Isvno. We have proven the following.

Proposition 2.5. If P=PNE hyp(h), then for all a and E In we have

(2.6) c independent of and a

If P<MOPN then we have seen that Pj<u PN , hence

<ZJ by translation invariance. This and (2.6) implies

cIPN(t+ih)12, t E ,n , j=O,...,N-1

where again c is independent of t . Using homogeneity we get

IP j(g+ith) I=1t I iIP Isclt I j1 PN(t/t+ih) I =clt I j-NILet R = PO+P1+...+PN-1 , then we get

Therefore P = PN + R satisfies the estimate

(2.7) IP(t+ith)l Z (1-cltl-1)IPN(g+ith)I # 0 , as t < -2/c

This establishes prop.2.4.

Proposition 2.6. Suppose all polynomials P with principal part PN

are hyp(h). Then PN is of principal type, and PN is s-hyp(h) .

Proof. It is clear that PN must be hyp(h) , so that 0

has only real roots, and PN(h)#O . If PN is not strictly hyperbo-

lic then for some ? , linearly independent of h, there must be a

multiple root T. of To prove prop.2.6 we will show

that in this case not every P=PN+Q, deg Q sN-1, can be hyperbolic.

We may assume to =0. Then the polynomialhas degree N-k s N-2 .

Now we will show: If, under above assumptions, P=PN+R is

hyp(h) then the polynomial p also must have degree

sN-2. For a general R of degree N-1 or less, this can not be true.


Therefore not every PN+R can be hyp(h), and prop.2.6 follows.

Consider the equation as a polynomial equation

in 2 variables a,t With x=,11 one may write it in the form

(2.8) xr(x,a) = 0 , r(x,o) =

with a polynomial r(x,o) in x, deg r sN-1. At x=0 we get the equa-

tion PN(i;+ah)=0 , which, as we know, has the k-fold root a=0, ka2.

From Rouche's theorem we derive the existence of N roots aj(x),

j=I,...,N , of (2.8), near the roots of for all IxIsO.

These roots may be organized into "Puiseux series" -i.e. power

series in x1/1 , with suitable integers 1>0 . In particular each

of the k roots approaching 0, as K-0 , has an expansion I amxm/1Ms

Let s be choosen such that as is the smallest coefficient #0

Clearly szO , since the series vanishes for x=0 . If 0<s<1 , then

ta(1/i) sast 1-s/1 =I asI It I Yexp{ i(arg as+y arg t )} , where 0<y =1-s/l<1 . For real t we may set arg t =vn , with an integer v . Here v

may be choosen such that arg as+ vyn is not a multiple of n, since

I must be i2. Then we get It Im aj(l/t)I as Itl- . However,

we have Imtaj(I/t)) so that Imtoj(1/s)remain bounded by definition of hyp(h). Thus we get a contradic-

tion unless ral . It follows that a(x)=0(Ixl) , as 1(90 , for at

least k a 2 of the roots aj , while all others are 0(1)

Finally observe that Xj(t)=ra(1/s)-1 are the N roots of

for sufficiently large Itl . It follows that

that f(0) = PN(h)4=1(_? (t)) = O(ItIN-k) . Hence indeed

p -(T) = P(tl;+h) must have degree s N-2 , q.e.d.

Corollary 2.7. If P E s-hyp(h) , then also PE s-hyp(rl) , for every

time-like vector Tl E h .

Proof. The cone h of time-like vectors clearly is determined by

the principal part PN only. Also we know that PN+R=P is hyp(,q) for

every R of degree N-1 or less, because each such P is hyp(h) ,

hence also hyp(q), using thm.1.5. Therefore P E s-hyp(rl) follows

from prop.2.6, q.e.d.

Clearly thm.2.3 is a consequence of prop.2.4 and prop.2.6.

The fact that, for PNE hyp(h) the condition P®PN is implied by

PE hyp(h), was proven by Chaillou [Chu1] and Svensson [Sv1] .

7.3. The hyperbolic Cauchy problem

3. The hyperbolic Cauchy problem for variable coefficients.

235

A linear differential expression L=a(x,D), as in ch.O,(8.1),

will be called strictly hyperbolic with respect to h, O'hE &n, at

x°, if the polynomial in is s-hyp(h). If this holds for

all x°E cC Rn, where the vector h is allowed to change with x°,

then L is called strictly hyperbolic in si. We shall mainly focus

on the case of h being independent of x0 in some domain st. Then we

call L strictly and normally hyperbolic with respect to h , in sz .

We will write the latter property as LE s-hyp(h)=s-hyp(h,st). For a

general set M we say LE s-hyp(h,M) if LE s-hyp(h,st) for open sz M.

In this section we will assume h to be one of the coordinate

vectors. This is no restriction, in view of the fact that "s-hyp"

is a property of the principal part LN of L , while the principal

part transforms like a contravariant tensor of N variables when

new independent variables are introduced. A linear transform x=my

with constant matrix m takes a(x,Dx) into a"(y,Dy) with principal

part polynomial a N(y,rI )=aN(my,m-Try), where m -T=(m -1)T. Choosing

h°=mTh=(1,0,...,0) will make the transformed expression a"(y,Dy)

s-hyp(h°) . In fact, for a more general transform x=m(y) the con-

stant matrix m may be replaced by the Jacobian matrix J=ax/dy .

Then h-=j Th will depend on y. One may use this - under proper con-

ditions - to transform an expression L=a(x,D), strictly hyperbolic

with variable hx, onto one, L which also is normally hyperbolic.

Of special interest will be the case of a second order

strictly hyperbolic expression L written in the form

(3.1) L = ja jkDjDk + lower order terms , ajk = akj .

The principal part polynomial is given by We get

(3.2) with

ajk a common complex fac-

tor). Cdn. "s-hyp(h)" then means that

tor). "s-hyp(h)" then means (i) a2(h)¢O, (ii)

>0, as t#yh. Here (ii) yields for tE N={h}l. Assuming

a2(h)>O we get the form a2 negative definite over N . Hence, up to

a nonvanishing complex factor all second order s-hyp-operators are

of the 'classical' form: The principal symbol matrix ((ajk)) has

one positive and n-l negative eigenvalues.


We now assume h=h°=(1,0) , OE Rn, in Euclidean space Rn+l =

{(t,x): tE &, xE Rn}, and an expression L=a(t,x,D) of the form

(3.3) L=1-1+k:-.N

aa'k(t'x)DaxDt=4=OAN-k(t)Dt' AN-k IaIsN_kaa,kDx

with principal part

(3.4) LN aa.k(tx)DxDt-4=OBN-k(t)Dt, BN_g aa.kDk.IaI+k=N IaI=N-k

Let sz=IxRn, I=[ t] ,t2], and aa,kE COO(R) , for all a,k. For complex-valued aa,k the expression (3.3) is s-hyp(h0) if and only if

(3.5) a0,0(t,x)'0, k=0 has real distinct roots,IaI+k=N

as polynomial in z, for all (t,x)E Sz, and all O E &n, by a simple

calculation. It is convenient to replace the first cdn.(3.5) by

a0O(t,x)=1(=AN(t)), as can be achieved by a normalizing factor.

So far, in this chapter, we assumed complex-valued coeffi-

cients. However, the results, below, are valid for the case of a

vxv-system as well. From now on we assume that aa,k(t,x)E Lv, with

the algebra Lv of all complex vxv-matrices. A vxv-matrix-valued

expression is called s-hyp(h°) if the determinant of its symbol is

s-hyp(h°). For L of (3.5) this means that a0'0(t,x)=identity, and

(3.6) p(i)=det( 7 has Nv real distinct rootsIaI+k=N

for all (t,x) E Rn+1 , 0#

E Rn

This condition ties up with the conditions on L in VI,(4.1).

There we were transforming to a problem of the form VI,(4.4) with

intent to apply thm.4.4, requiring "strictly hyperbolic of type e

or a"'. The latter is identical for xE KCC gn with our present

"s-hyp(h°)" - while at x=w the conditions do not match.

We focus on the local "non-relativistic" Cauchy problem

(3.7) Lu = f , (t,x)E S , atu as t = t0 , j

for given f (defined in Sz) and Tj (defined for t=t0E I), assumed

CO-functions. The function (distribution) u is to be found.

The term 'non-relativistic' expresses that the initial condi-

tions atu = are imposed at a given constant time to. It will be

practical to also consider "relativistic" problems with initial

conditions on a more general surface EC c . We find that the study

of relativistic Cauchy problems is only a matter of a 'space-time-

transform' -i.e., a transformation of independent variable. Recall

the cone h = ht,x of time-like vectors. h is convex, h°E H, and L

7.3. The hyperbolic Cauchy problem 237

of (3.3) is s-hyp(rl) at (t,x), for every 1E Ht,x (cor.2.7).

A smooth hypersurface I C St will be called space-like if its

surface normal is a time-like vector, at each (t,x)E S . Since h

is convex and WE h, a space-like hypersurface has a unique projec

tion to t=0. Thus it is given by an equation of the form t=©(x).

Assume such space like EC S2 given, ©(x)E C°°(len) and O(x)=t° E I forlarge jxl. The relativistic Cauchy problem (for E) is defined by

(3.8) Lu=f , (t,x) E Sz , avu=gj , (t,x)E I , j=0,...,N-1.

Here av denotes the normal derivative to S, av defined as in V,3.

We choose the normal pointing to increasing t. Assume f, WjE Co00

given, defined on Si and E , respectively.

The relativistic Cauchy problem may be reduced to the nonre-

lativistic problem by a coordinate transform. For simplicity we

only discuss the case where the cones ht,x hx are independent of t

(for example, if L or its principal part are independent of t).

Let us consider a transformation of the form

(3.9) t' = t'(t,x) , x' = x , (t,x) E SZ=Ix&n

inverted by

(3.10) t = t'(t,x') , x = x' , (t',x') E S2'

defining a diffeomorphism S2 «12' , which takes the surface E onto

the plane t'=0 (i.e., require that t'(O(x),x)-0 ). One then veri-

fies that the cdn's avu=ypj of (3.8) transform to cdn's of the form

in (3.7), with modified data. On the other hand, the DE Lu=f trans-

forms to L'u=f', (t',x')E c' of the general form (3.3). We get

(3.11) aN'(xaI,te') = aN(xa(xa),Jt"I) , J = ((axa'j/axa1))

where, for a moment, we are introducing the notation xa=(t,x)

xa' = (t',x') , and with aN(xa,ta) = aN(t,x,t,t) . In the Jacobian

matrix J the row index is j , and the column index is 1 . One cal-

culates that Jl;a' = (t'Itt',t,'+t'Ixt') = (0,t') + t'h(t',x'), with

(3.12) h(t',x') _ (at'/at,at'/axl,...,at'/axn)(xa(xa')) .

Thus 0=det aN'(xa',l;a') = det aN(xa(xa'),(0,t'+t'haxa)) has realdistinct roots t' if L is s-hyp(h) for h of (3.12), all (t',x').

Let t'(t,x)=t-8(x) , so that t(t',x')=t'+O(x'), and get h=

=(1,V8(x')) independent of t'. Clearly h is normal to I , as (t,x)

E I. Thus indeed there we get h E ht,x since I was assumed space


like. Now, if ht =h is independent of t then h(t'x')E ht,x' for

all t,x , and the transformed problem is s-hyp(h°), and nonrelati-

vistic. Clearly the set a' contains a smaller sk"=I"xlen.

Note that the same transformation still works if h is time

dependent, if it is assumed that S is not only space-like, but,

afortiori, has its normal at x in the intersection fl(ht,x:tE i}=

=hx. Clearly hx is a convex cone with nonvoid interior, for all x.

Theorem 3.1. Assume L of (3.3) in s-hyp(h0), and, moreover, assume

that the transformed problem VI,(4.4), with the matrix of VI,(4.6)

is strictly hyperbolic of type e or e'. Then there exists a unique

solution u(t,x)E C0O(I,Hs) , for all sE k2, of the nonrelativistic

Cauchy problem (3.7), for every f and ypj E Co .

The proof is an immediate consequence of VI,thm.4.4.

We observed that the condition "strictly hyperbolic of type

e°" coincides with (i) "s-hyp(h°) on all compact sets KC 11 " and

(ii) an additional condition obtainable by modifying L outside an

arbitrarily given compact set KO . If S is a surface as above,

then it follows easily that a transformation x'=x , t'=t-8(x) will

not influence cdn.(ii), since we were setting ©(x)=t° for large

lxi. As a consequence we have the result, below.

Theorem 3.2. Let L satisfy the assumptions of thm.3.1, and let EC

dint be a space-like surface with unique projection onto the plane

t=0, and parallel to t=0, for large I xI. Let T, =Min © , s2=Max e ,

taken over In. Assume that E is not only space-like, but even that

its normal is contained in the cone hX fl(ht,x: (t,x)E M}, with the

"slab" M={z,-t,St-O(x)st:-t2}. Then there exists a unique solution

u of the relativistic Cauchy problem (3.8), defined for (t,x)E M00

and each f, E CO , such that the function u(t-0(x),x)=u (t,x)

satisfies u" EjC0o([i, -t, ,t3 -t2 ] Hs) for all sE N2

The two above results, thm.3.1 and thm.3.2 will be improved

in sec.5, below, after discussing 'finite propagation speed'.

4. The cone h for a strictly hyperbolic expression of type a°.

In sec.3 we already used the fact, that an expression L of

the form (3.3) is s-hyp(h°,K), h°=(1,O), for every K(= Ixxn, when-

ever the first order system VI,(4.4) is strictly hyperbolic of ty-

pe e or e'. Discussing this in detail, note the symbol of the ope-

rator KO AK, K of VI,(4.6), is of the form k0+K0, where the xOE

7.4. The cone 239

C'(I,yrcf_e), while the matrix k0(t,x,t) coincides with the matrix

in VI,(4.4) except in the last row. All elements in rows 1 to N-1

equal 0 or 1, as in VI,(4.6), but the elements of the last row are

(4.1)(n-e°(x,U))N-jaN_j(t.x,U) , j=0,...,N-1

with aN-j lal

in the order listed.

We may choose k0 and x0 above for the decomposition VI,(4.9).

Then the condition "strictly hyperbolic of type e° " implies that

has real and distinct eigenvalues µ1<µ2<...<RN for all

(t,x,t) E IXZe,0 with It was pointed out before that the

functions then must be continuous in this compact set.

Under the present special conditions, the matrix k0 and its eigen-

values even extend to continuous functions over {Ixffi°xZn: W-1.1,

with in of 11,3, and the compactification RA allowing continuous

extension of all functions a(x)E ip0 . Here we assume that

(4.2) aa, j(t,x)E (N-j)(e°-e') as

Accordingly, for every jt°j=1 , the limits limpj(t,x,p°)-Rj(t,x,«x°) exist, and define Nv real-valued continuous functions

over Ix>2i^x{jV°j=1}, assuming mutually distinct values at each

point of this compact space. A calculation shows that tj=

-µj(t,x,° °) are the roots of the equation

(4.3) det( 7v-ltjbl (t,x, ° )) = 0

j+1 =N

where bN- 7 are the principal parts of7 Ian=N-j J

the aN_j , and we have assumed b0=1 again. Also, we have set v=

(x) for e° =e , and u=1 for e° =e' .

Note that (4.3) may be rewritten as

(4.4) p(tu) = det Y, bl(t,x,°)(tv)j = 0j+l=N

with the polynomial p(T) of (3.6) , taken at In other

words, the roots of (3.6) are given as the numbers -vµj

We have proven the following result.

Proposition 4.1. a) Assume that the coefficients of L of (3.3)

satisfy (4.2), and that the equivalent first order system VI,(4.4)

is strictly hyperbolic of type e° , where a°=e or e°=e' . Then


the differential expression L is s-hyp(h°,K) for every KC: Ixin .

Moreover, we even have L E s-hyp(h°,Ix&n) , uniformly, insofar as

the minimum difference between two roots of (3.6) is bounded below

by c° >0 for all 1 ° 1=1 , as e°=e' , and by c° (x) , as e° =e. Further-more, the roots of (3.6) are bounded by a constant Co as a°=e'

and by Co ( x) as e° =e , for all tE I, xE en, I T;° 1=1 .

Remark. Note that a different choice of k0 and x0 in VI(4.9) does

not influence the above derivation, since the limit will

give the same polynomial as in (4.3) .

Note that our knowledge about the roots of (3.6) gives the

following information about the cone h=ht,x of time-like vectors.

Proposition 4.2. Under the assumptions of prop.4.1 the cone ht,x

for L of (3.3), at (t,x), is given by

(4.5) &n , i>-v(x) lµj(t,x,°°ll)

, j=1,...,Nv)

where u(x)=nea-e, (x,i;)=1 as a°=e' , =(x) as a°=e

Corollary 4.3. If a°=e' , the cone ht,x contains a cone {t2Co 1 1}=

=ho for all (t,x)E Ix& . If a°=e then, at least, ht'x contains

the cone hlx1={tZC°(x)1 1} , for every (t,x)E Ix&n . Here Co is

some positive constant independent of t and x .

All above statements are evident, after our remarks.

Now assume that an expression L as in (3.3) is given, with

aa,k satisfying (4.2). For all (t,x)E Ix$n, 1V°1=1, let equation

(3.6) have real and distinct roots, bounded by Co, with differen-

ces bounded below by co, just as in the statement of prop.4.1.

Proposition 4.4. Under above assumptions the system VI,(4.4) is

strictly hyperbolic of type e' .

Indeed, we will just make the same decomposition into k0 and

x0 , as above. The negative eigenvalues ij=-pj then again satisfy

(4.6) 1 (

k+l=N

with aN-j of (4.1) or (3.3). We may wite this as

(4.7) p(t;t,x,y)+ 7, Tk ( )l-lalaa,N-l(t,x)( -)a=0k+l=N laI<1

where denotes the polynomial of (3.6). For 1;=«1°

equation (4.7) assumes the form and the above

assumptions then state that its roots are real and distinct. On

the other hand, the polynomial in z of (4.7) has its highest coef-

7.5. Regions of dependence and influence 241

ficient equal to 1, independent of Thus the roots may loc-

ally be arranged into continuous functions. They are distinct at

the compact set IxX°xaBn, hence they will stay real and distinct

in some neighbourhood Bn, But this is precisely

the statement of prop 4.4. Q.E.D.

5. Regions of dependence and influence; finite propagation speed.

Let us again focus on the Cauchy problems (3.7), (3.8). We

can significantly improve on the existence results thm.3.1 and thm

3.2, insofar as it may be shown that "uniform s-hyp(h°)" is suffi-

cient - it will imply the condition on the system VI,(4.4). Also

the solution u(t) is CD (&n) for each fixed t, not only in S=f1HS

To simplify the assumptions let us assume that the condit-

ions of thm.3.1 are satisfied for all compact intervals I=[ti,tz].

Consider the adjoint L* of our expression (3.3), defined as

(5.1) L* = I Dxa'Dt a,k(t,x)

Ial+ksN

Proposition 5.1. We have L E s-hyp(h0) if and only if L* is

s-hyp(h0) . Moreover, the cones ht'x of time-like directions for L

andL*

coincide, at each (t,x) , and a surface I is space-like

for L if and only if it is space-like for L* .

For L with complex-valued coefficients we know that "s-hyp",

the cones ht'x , and "space-like" are completely determined by the

principal part of L. Cor.1.3 implies that the principal parts of L

andL*

differ only by a nonvanishing complex factor y(t,x): We get

real-valued, for some g(t,x), hence the prin-

cipal part of L* evaluates as aN gb=(g/g)aN. Since "s-hyp" implies

"hyp" this proves the proposition for complex-valued expressions

L. For general matrix-valued expressions we still have aN (aN)T ,

and det aN = (g/g)det((aN)T) follows because det aN is a hyperbo-

lic polynomial. Thus again the proposition follows, q.e.d.

In the following let us assume that L , L* both satisfy the

assumptions of thm.3.1. Consider a region no C Rn+l as in fig.5.1.

In detail, the boundary an0 is piece-wise smooth, and consists of

two parts, E1 and E2, both space like surfaces, with equations

t=91(x), j=1,2.


The surfaces E1 intersect

at positive angle in some

smooth compact n-l-dimen-

sional surface. Moreover,

we assume 81 extended over

all of In, and that the

extended surfaces t=01(x)

(5.2)

Fig. 5.1

t=t, satisfy the assumptions of

thm.3.2. Assume the com-

pact region ito defined by

rte= {(t,x) :xE &n, 81(x)sts82(x)} .

Lemma 5.2. Suppose a solution u of the relativistic or nonrelati-

vistic Cauchy problem ((3.8) or (3.7)) with f.0 is defined in the

compact region rto , and assume that u=ult=...=cat 1u=0 on I, rl9rtoThen u vanishes identically in all of rto .

Proof. Note that the relativistic Cauchy problem (3.8), for L*

rather than L , and for the surface E2, admits a solution v . Here

we assume that Tj=0, but will allow an arbitrary C function f

Existence of v then is a matter of thm.3.2. Note that u and v both

are C (rto) . By a partial integration we get

(5.3) f u fdtdx = fuTL*vdtdx = 0RO

Indeed, we get Lu=f=O, by assumption. No boundary terms may appear

because the Cauchy data of u and v vanish on It fl912o and on E2 (8a ,

respectively. Thus the partial integration gives 0 . Since this is

true for arbitrary smooth f with compact support, we conclude that

u=0 in r)o , q.e.d.

Proposition 5.3. Assume that L, L* both satisfy the assumptions of

thm.3.1. Let E:t=8(x) be space-like surface satisfying the assump-

tions of thm.3.2, and let u be the solution of the relativistic

Cauchy problem of thm.3.2, for the surface I , and for data f,Tj

where fE CQ(IxIn) , for every I , and qvi of compact support on I

Then, if E° is any space-like surface t=01(x) , xE 1n , with the

property that supp f, and supp Tj all lie above I°, in (t,x)-space

then u vanishes identically in the set 8(x)stse1(x) (see fig.5.2).

Proof. This is a matter of constructing "lenses" onto which lemma

7.5. Regions of dependence and influence

5.2 may be applied. First one notices

that, for each fixed t the function

u(t,.) has compact support: For exam-

ple, one may work with a surface E°

carrying a spherical bulge, such as

t=to +Max{ x 2 -2a.x ,b}=& (x), with 0<b<la12, smoothened at the rim, where

b must be chosen sufficiently close

to 1a12 . Using cor.4.3 one finds that

such a satisfies the assumptions of Fig. 5.2

243

the lemma, provided that the maximum slope is sufficiently small.

This does not limit the height such a bulge can 'climb'. The

latter works if a°=e' , while for a°=e a somewhat more complicated

lense must be chosen. Once we have shown that u(t,x) has compact

support for all t, we may examine the solution in successive slabs

tj-lststj , where tj=ej , for small e>0 . The surface t=tj outside

E° , t=tj-1 inside E° t=o' otherwise then, after smoothening, may

be seen to satisfy the assumptions of thm.3.2 for existence of the

solution v for the adjoint problem, as in lemma 4.2. Then an argu-

ment as in the lemma will give u=O below that surface. Q.E.D.

Theorem 5.4. Let L of (3.3) have its coefficients satisfying (4.2)

Assume that L is s-hyp(h°,&n) uniformly, in the sense that the

roots Tj of equation (3.6) are real and distinct for all (t,x)E

In", 1n, while, on 1 1=1, t, ststz , we have Ii - IsC° , Ii j-sll>c° with O<c°, Co independent of Assume E to be a space-like

surface satisfying the assumptions of thm.3.2. Then the relativis-

tic Cauchy problem, for the surface E, and arbitrary smooth data

f,Tj, where yj are compactly supported while (supp f)MxlnO Rn+1

for all I( 1, admits a unique solution uEC0'

(1n+1). Moreover, we

have (supp u)fl(Ixtn) compact for all compact I.

Proof. Let us set a°=e'. Then confirm that L as well as L* satisfy

the assumptions of prop.4.4. Accordingly, the corresponding first

order systems VI,(4.4) are strictly hyperbolic of type e'=e° , and

we have the assumptions of prop.5.3 satisfied. Thus our assertion

follows. Q.E.D.

Remark 5.5. It might be interesting to note that our above discus-

sion will still work in the case where we require that (4.2) holds

for a°=e - i.e., we have

(5.4) aa,j(t,x)E W(N-j)e2 as jajsN-j .


However, we then must require the roots of (3.6) to be bounded by

C. (x) and their differences bounded below by co (x) , co >0 , asIn addition, one will require the same conditions on the

roots of (4.7), valid not only for large but also for all iii,

as long as lxi is sufficiently large. We will not discuss details.

We now may compare our results of thm.5.4 with those achie-

ved by the Fourier-Laplace method in ch.0,4. Note that use of

ydo's has been eliminated, except in the proofs. For a Cauchy pro-

blem as in thm.5.4 we have finite propagation speed: The solution

u(t,x) stays =0 for sufficiently large lxi , as long as t stays

within a compact interval. There is a "region of influence" :

For a change of data - within a compact set K - the "influence" on

the solution u(t,x) will be felt only inside a conically shaped

region, expanding from K towards increasing times. There is a

"domain of dependence": The values of u at or near (t°,x°) depend

only on the values of the data within a conically shaped region

with tip at (t°,x°), expanding backward in t.

For a point (t°,x°) one can define a precise region of dep-

endence RD (t° , x°) and region of influence RI (t° , x°) as the back-ward part and the forward part of the envelope of all space-like

surfaces through (t°,x°). This envelope will have the normal cone

of the cone hto , x° as a tangent cone, at (t° , x°) . Hence, at thatpoint (t°,x°) , it may be decomposed into a forward and backward

part. The regions RD and RI also may be described by the bicharac-

teristics of L, in the sense of ch.0,8, at least for v=1. Or, it

may be linked to the "particle flow" of VI,5,6, generalized to

systems of the form VI,(4.4) - of. ch.IX .

6. The local Cauchy problem; hyperbolic problems on manifolds.

Notice that thm.5.4 implies an existence result for a local

hyperbolic Cauchv problem. Assume first that, in a set a--IxB3 ,

with BY =BY (0)={xE In: I X <Y } , I=[ t1 , t2 ] , we have a differential ex-pression of the form (3.3) defined. Assume that LE s-hyp(h°,S) ,

and let the coefficients aa,j(t,x) be COO(n) . Assume a space-likeintsurface EC n . Then ask for existence of a function u(t,x), def-

ined near I , satisfying Lu=f (near X) and atu_c on E, j=O,..N-1,

where f and yj have support within IxB1 .

To solve this problem we will modify and globalize the ex-

pression L as follows. Letting L=a(t,x,Dt,Dx) , with the polyno-

mial a (t,x) %k , and with a function 6(p)E-=a,k

7.6. The local Cauchy problem 245

C00((0,00)) satisfying i(p)=p as pE [0,1] , 0s9(p)<2 in (0,m) , 0(p)

=0 as pa2 . Then define L° = a°(t,0(jxj) f ,Dt,Dx) , tE I, xE Bn

Observe that L=L" within IxBi . Also the coefficients of L° are

independent of x as jxja2 - we have L° = a(t,0,Dt,Dx) there. In

particular, L° is uniformly s-hyp(h°,Ixen), in the sense of prop.

4.4. We may modify the surface E outside IxBi to obtain IA satis-

fying the assumptions of thm.3.2, at least close to E. Since E=E°

within IxBi we may extend the initial conditions imposed there by

setting Tj=0 , f=0 outside IxBi . In this way we obtain a Cauchy

problem satisfying all assumptions of thm.3.2. The solution exists

and its restriction to IxBi will supply a solution of the local

problem.

Now, for a general set n=Ixn° , where n° CC Rn we may solve

the corresponding problem, using a partition of unity. Assume L

of the general form (3.3), defined in Ixf with an open set SE J

nc , and let it be s-hyp(h°,Ixn-) . Construct a finite covering of

n by balls BE(xk) with the property that B3E(xk)C si , and a par-

tition of unity =1 , supp wk C BE(xk) . The solution of

(6.1) Lu=f , near I atu=4Pj at I j=o,...,N-1 ,

where I C Ixn is a space-like surface, and where supp g C Ixn

supp y)jC n , may be obtained as a superposition of the solutions

of (6.1) for f « wkf , cj H wk(pj , to be regarded first as a pro-

blem within IxB3E(xk). The solution of the latter problem may be

extended zero along E , for a small time interval, because it, also

will vanish within IxBE(xk) outside the region of influence. The

problem within IxB3E(xk) may be translated and dilated into a pro-

blem in IxB3 . We have proven the following result.

Theorem 6.1. For a differential expression L of the general form

(3.3) with smooth coefficients defined in Ix12 , and s-hyp(h°,Ixft )

there, and a space-like surface ECC Ixn , where ncx sz , the

Cauchy problem (6.1) admits a unique local solution, defined in a

sufficiently small neighbourhood of I .

Remark: The local uniqueness simply is a matter of an argument

like that in the proof of lemma.5.2.

Next we notice that the concepts "s-hyp(h)", "s-hyp(h,n°)"

are meaningful for differential expressions defined on a differen-

tiable manifold n. Since we have coordinate invariance we may int-

erpret h as a vector of the cotangent space T*ax° , at a point x°,

and define L to be s-hyp(h) if this is so in any set of coordina-


tes. If we are given a field of cotangent vectors h=h(x)E T*nx

for x E n°, an open set Ca, then we shall define LE s-hyp(h) _

s-hyp(h,nz°) if LE s-hyp(h(x)) , at each xE n' . We then will have

a cone hxC T*nx of cotangent vectors ar x - it is the cone h of

the differential polynomial of L at x, in any set of local coordi-

nates; note that the polynomial is now defined on T*nx, and that

its principal part is defined as well. It determines hx as a sub-

set of T*n . A space-like surface EC n will be defined as an n-1-

dimensional submanifold with normal at xE E contained in hx. Here

we will assume that L is s-hyp(h,n ) with some domain sE C n , and

that XC n° c n- . Then again we may formulate a hyperbolic Cauch

problem: For compactly supported fE CO 0(n) and (fjEC0(E),

N-1 (where L is of order N) find a solution uEC (n') of

(6.2) Lu=f , xE n' , ah(x)u=Wj , xE E , j=O,I,...,N-1

where n' C :re is some (sufficiently small) open neighbourhood of E,

and with the j-fold derivative ah(x)u (for t, at t=0) of the func-

tion u(x+th(x)), in some fixed set of local coordinates.

Theorem 6.2. The solution u(x) of problem (6.2) exists and is

unique, up to the choice of n' .

The proof is by localization, exactly as for thm.6.1.

Notice that we have admitted vxv-systems, in the above

Correspondingly we may allow an expression L mapping between v-di-

mensional complex vector bundles on n , in thm.6.2, and then spe-

cify data f , (pj which are sections in vector bundles.

Clearly the notions of finite propagation speed, region of

influence and dependence RI and RD extend literally as well.

The case of a second order scalar operator again merits spe-

cial attention: This will be related to relativity theory. We have

seen that, locally, up to a nonvanishing complex factor, a second

order expression L is of the form (3.1) , with a quadratic form

of rank n and signature n-2 .

Chapter 8. PSEUDO-DIFFERENTIAL OPERATORS

AS SMOOTH OPERATORS OF L(H) .

0. Introduction.

In this chapter we look at a different way to introduce pseu-

dodifferential operators. From their introduction in ch.I - using

I,(1.5)- they appear as a technical device inspired by the Fourier-

Laplace method. It may be surprising that there is a natural defi-

nition of pdo's at least for some special symbol classes: Certain

algebras of yado's appear as "smooth" subalgebras of L(H). For exam-

ple, the algebraOpCB00

(12n), with CB'(12n)=pto the class of

EC00

(3k2n) , bounded with all derivatives, coincides with the class

of operators AE L(H) which are both "translation smooth" and "gau-

ge smooth". That is, with the family {Tz:zE Rn} of translation ope-

rators TZu(x)=u(x-z), and the family {M:tE ien} of "gauge multipli

cations" Mu(x)=eitxu(x), we get T-ZATZE C(&n,L(H)) and M-,AMtE00 nC (& ,L(H)). Notice that both Tz and Mt are groups of unitary ope-

rators on H, representations of the Lie-group In in U(H).

In terms of ch.I,Oppt0=OpCB00

(12n), a subalgebra of L(H) by

III,1, is the class of translation and gauge smooth operators.

There are other natural actions of Lie groups on H=L2(Hn),

such as the dilations u(x)-u(yx), with y340, or the rotations u(x)

-u(ox) , with a rotation o:&n..&n, or the linear group u(x)-u(mx)

with a nonsingular real matrix m. If we require dilation and rota-

tion smoothness, in addition to translation and gauge smoothness

we obtain a more practical algebra (called Oppso) with a calculus

of '4do's and symbol decay similar to Optpc0. With the linear group

we get a still smaller algebra, called Opy,10 .

Similar symbols were studied by Weinstein and Zelditch [WZ].

We will give a completely independent presentation of these

facts, introducing lpdo's again, without reference to ch.I. In par-

ticular, we will work with symbol classes 1tm, Vsm, and V1m exclu-

sively, in ch.9 and ch.10 below, derived from the natural classes

1pt0, ,s0, and V10 These classes are similar, but not identical to

yicm. They allow a (slightly more complicated) calculus of pdo's.

In sec's 1,2,3 we are concerned with the 'IDO-theorem' des-

cribing the new characterization of Opipt0. In sec.4 and 5 we dis-

cuss the classes Opps0 and Opipl0. Sec's 6 and 7 are concerned with

symbols in Vxm for general m, with calculus of ipdo',s, and decay

247

248 8. Smooth operators of L(H)

properties of the new symbols. Sec.8 deals with corresponding Lie-

algebras and Beals-Dunau-criteria.

An earlier version of our theory was announced in [C<] and

distributed as lecture notes of a seminar at Berkeley in 1984,

but remained unpublished.

1. pdo's as smooth operators of L(H).

Before giving a natural introduction to the concept of pseudo

differential operator we make some observations concerning smooth

ness of continuous linear operators on the Hilbert space H=L2(&n).

ien is an abelian group under vector addition. The characters

of this group are the functions e1 , E &n, the character group

is 2n again. For zE In define the translation operator TzE L(H) by

(1.1) Tzu(x) = u(x-z) , u EH .

For each character we define the gauge operator Mt=eitm by

Mtu(x) = eitxu(x) .

Note that Tz and M are unitary operators of H . Moreover,

(1.3) Tz+z' = TzTz, , Mt+V = M t M V , for all z,z',i;,t' ,

which expresses that z - Tz and - Mt each define a unitary rep-

resentation of the group ien on the Hilbert space H .

We want to emphasize: The differentiable manifold In with abo-

ve group structure, is a Lie group, but the representations (1.1)

and (1.2) are imperfect insofar as z->Tz and -Mt are not smooth as

maps &n, L(H). For fixed uE H the functions Tzu=f z, M u= are con-

tinuous but need not be differentiable ( u(x)=sgn(x)e IxI, for

example). For uE CO we get fZ, 9,E C00(0,H) but not in general.

In fact, it is known that the operator families Tz and M

are strongly continuous in their parameter z or t , but are not

norm continuous: Choosing u E H with support in a ball Jxlsa (or

Ix-x°lss with x°(?-%')=n ) with small e one confirms that

(1.4) IITz-TZIII a , IIM-MII a , as z34z' , K ' .

Similarly, for a given operator A EEL(H) the families

(1.5) AZ'o = TZ-'ATz , AC't = M1AMt , z,t E In

are strongly continuous, but in general not uniformly continuous.

8.1. Smooth operators of L(HO) 249

The pseudodifferential operators within the algebra L(H)

will be defined as those operators A E L(H) for which both fami-

liesAz,O and AO,. are not only norm continuous, but even are00

C

(1n,L(H))

, L(H) being equipped with the norm topology.

Let us denote the class of all such operators by 1I0T. That is,

t1CT = {AE L(H): Az,0 , A0't E Co(&n,L(H))} .

We will show in the following that every operator A E tICT C L(H)

has a unique representation in the form

(1.7) Au(x) = (2n)-nfd

where a complex-valued function in

C00:partials b(a) of all orders are bounded}

called the symbol of A . (1.7) holds for uE C,(ien), but determines

the operator A since Co is dense in H. Even for uE Co the existen-

ce of integrals in (1.7) will be a point of discussion.

Interestingly, the existence of a representation (1.7) with

symbol function aE CB00(1en) characterizes the operator class IICT :

For any aE CB"(le2n) the integrals in (1.7) exist for all uE Co(&n)

and Au(x) of (1.7) belongs to H. The operator A:C0(&n)->H thus defi-

ned extends to a map E L(H). Moreover, A belongs to iIGT

In the remainder of this section we make a few observations,

preparing the detailed discussion of the result mentioned above.

First, observe the 'commutator relation'

T-zMtTzu(x) = eit(x+z)u(x) = e1ztMtu(x)

Or,

(1.9) T-MTZM-l-e1z

Thus the subgroup of the unitary group U(H) of H generated by all

Tz and Mt , z,t E 1n , consists precisely of the operators

(1.10) Ez,t,T = ei((P+xt)Tz, TEE & , z,tE &n .

This group is called the Heisenberg group. We denote it by GH. Get

(1.11)Ez+z',t+t',T+w'-t'z

Thus GH is isomorphic to the group obtained by equipping the space

gh=lenx&nx$1={(z,t,T):z,tE &n,WE $1=&/2nZ} with the group operation

(1.12) unit = (0,0,0) .


Clearly GH with the manifold structure of gh is a Lie group.

Our above condition describing 'WT is equivalent to the following:

For AE L(H) let AZ't = EZ,t,W IAEZ,t,(P . (Note that AZ't

is independent of cp .) Then we have

(1.13) Az,t E C00(gh, L(H)) .

Indeed, (1.13) trivially implies the condition of (1.6).

Vice versa, note that (1.9) implies

(1.14)AZ,t = T-zAO,tTz = M-tAZ,OMt

If (1.6) holds then use that

Az.t,ii =Ao,t,ii

, Az,,tii = IIAz'0- Az,'011

showing that Az, is continuous in z for fixed t and continuous in

t for fixed z , and either continuity is uniform in the other

variable. This clearly implies norm continuity ofAz,t

in both

variables. Similarly (1.6) and (1.14) imply existence of

(1.15)aaAZ,t= T-za AO,tTZ ' aRAZ,t-

in operator norm topology, uniformly in the other variable. Since

aaAO' and aPAZ,O are still C"(&n,L(H)) , we may repeat the above

conclusion and find that both functions (1.15) are in CB(gh).

By a standard argument we then conclude that all mixed deri-

vatives as Az't exist (and are CB(X2n)) as well. Indeed, this

needs to be shown at z=t=0 only, in view of the group property.00

Thus look at XAz,t with a cut-off function XE Co , X(z,t)=1 near 0.

For u,vE H consider Let p(a)=a Zap p. We get

E E =L2(QyQy=I(z,t):Izjlsy, j=1,..,n}, for large y, and

all a,(3. Parsevals relation for Fourier series implies that the

(multi-)sequence {q0'.: O,vE Zn} of Fourier coefficients of p sat-

isfies Oag0,v,v1g0,,u E 12 = L2(Z2n) , for all a,P . Thus also

Oav (1+jOj2+Ivjz)kE 12 for k=0,1,..., and all a,(3. It follows that

p(R)E Hk, for all a,(3, with the Sobolev space Hk--Hk,0 of 111,3. By

Sobolev's lemma we thus get p(a)E Ck(I2n), for all k. Looking at p

in dependence of u, v we get II p (a) ii , 11p(p)11- c ii u li ii vii , with normsin Hy and H, resp., and with c undependent of u,v . Following this

estimate through the above argument - Parseval, Sobolev, etc.,...,

we find that also iip(a) ii k s ciiuii iivli , for each This impliesC

(%Az,t)(R) E L(H) , and even ii(XAz,t)(Pa)iis c . Then one concludes

8.2. The pseudo-differential operator theorem 251

that aZ,PAZ,t=(AZ,t)(a) exist in norm convergence, for all a,1

Thus indeed (1.6) and (1.13) are equivalent.

Secondly, we want to examine existence of the integrals in

(1.7). Note that the inner integral fdy exist trivially, as a pro-

per Riemann or Lebesgue integral, and equals

where u E CO(&n) . More-

over, it is well known that u^ is a rapidly decreasing function.

That is u^(t) = for every k=0,1,2, ... , and the

same is true for the derivatives u^(a) . Since a(x,l;)E CBO'(12n)

is bounded, and lelt(x-y)1=1 , it then is clear that also the

outer integral fd exists as improper Riemann or Lebesgue integral

and that that there is a bound for the integral independent of x

In order to show that the right hand side of (1.7) defines

a function in H we prove that xaAu(x) also is bounded, for every

a . Indeed xa =((x-y)+y)a is a finite linear combination of terms

(x-Y)OYY , 3+y=a , while

Again the inner integral equals (2n)n/2eixt(xYu)^, hence is a func-

tion in S , while has all derivatives bounded. It follows

that a partial integration may be carried out, leaving no boundary

terms at infinity, giving

= 0(1)

Thus indeed xaAu(x) is bounded for all a , hence (1+IxI2)nAu

is bounded, and Au E L2

In fact, integrals of the form (1.7) already have been inve-

stigated in 1,1, under weaker conditions on the symbol

2. The 'MO-theorem.

As prepared in sec.1 we will now prove a theorem linking

the concept of pseudodifferential operator, introduced by formula

(1.7) to the class WT of continuous operators on the Hilbert

space H = L2(1n) which are 'smooth' with respect to the transla-

tion operators, and the 'gauge transforms' u -> eitxu (or, equiva-

lently, gh-smooth). Here we were using the standard representation


GH of the Heisenberg group gh, i.e. GH={Ez,.,(P gh} , with

gh = 2nxlnxl1 ,

$1 = 2/(2nZ) , where

(2.1)H , (Ez,(Pu)(x)=el(T+1'x)u(x-z)

Especially,all translations Tz:u(x) - u(x-z),for a constant vector

z, and all "gauge transforms" u(x) - ei(T+tx)u(x) for constants

gE It , t E In belong to this class.

For A E L(H), a continuous operator on H , we are interes-

ted in the function Az,t : GT -> L(H) obtained by conjugating A

with the operatorsEz,.,T

1(2.2)

Az,t =Ez-

,t,TAEz,t,g

Note that Az,t does not really depend on T since the constants

elq) commute with all of L(H) . Hence Az,t may be regarded as a

function GT'- L(H),where GT' denotes the quotient of GT modulo its

normal subgroup {ei(P :cp E 2} Clearly GT' is isomorphic to 12n as

the composition law (1.12) shows. We may write

(2.3)Az,t EZ1tAEz,t ,

EZ,t=

and henceforth regard Az,r, as a function 12n -> L(H)

We restate the essence of our discussion in sec.1 as a theo-

rem, below. Here, for a general class X of functions over 12n

and class Y oftUdo's we write

(2.4) Op X = {a(x,D) : a E X} , Symb Y = {a : a(x,D) E Y} .

Also, for a single function we will write Op(a)=a(x,D) ,

Symb A = where A= Op(a) = a(x,D) denotes the formal pseudo-

differential operator (1.7) belonging to the symbol a E X .

Theorem 2.1. Let Vt0 = CBCO (I2n) denote the class of all complex-

valued with partial derivatives a5a)(x, ) aXa a(x, ) of

all orders bounded over 12n . Then we have

(2.5) Opi,t0 = {A E L(H) : Az, E Cco(12n,L(H) )} = '1GT .

Here the partial derivatives of Az,t are assumed to exist in the

convergence of the operator norm 11.11 of L(H) defined by

(2.6) IIAII = sup { IIAufl/IIuhl : u E H , u# 0 } , A E L(H) .

The ydo A of (1.7) is defined for u E C0OO(2n) by integrating in the


order stated, and for general u E EH by taking continuous exten-

sion from the dense subspace CO .

With pseudodifferential operator notation we have

(2.7)

using that

E =Z ,t

(eizDu)(x)= u(x+z) .

With (2.7) we may differentiate (2.2) formally, for

(2.8) azjAZ,tIz=t=O = i[Dj,A] , atjAz,tJz=t=0 = -i[xj,A] ,

with formal commutators between the operator A E L(H) and the

unbounded differentiation or multiplication operators Dj and xj

Similarly the higher derivatives are related to iterated commuta-

tors between A , Dj ,and xl This provides a link of thm.2.1 to a

slightly different characterization of more general classes of

pdo's as operators S - S with well defined iterated commutators

with differentiations and multiplications (cf. R.Beals, [B2])

Looking for a proof of thm.2.1 we first will discuss the

fact that, given a symbol a E CB"(le2n), the formal expression

A = a(x,D) of (1.7) defines an operator in L(H). Actually, this is

already done (cf. III, thm.1.1), but we offer another proof here,

with an approach first published in [CC]. (It may be a bit more

complicated, but it fits the present general approach.)

Theorem 2.2. An operator A = a(x,D) with symbol a E pt0= CB'°(1e2n)

is bounded in H = L2(&n) . More precisely: The operator of

(1.7) extends to a continuous operator H- H

Proof. It turns out that we need only boundedness of 'a few' deri-

vatives of the symbol, for this proof. Consider the case of n=1,

for simplicity. The case of general n may be handled analogously.

For m=1,2,... introduce the function

(2.9) ym(t) = e-t tm-1/(m-1)! , as ta0 , = 0 , as t<0 .

One checks easily on the following facts:

Proposition 2.3. We have ymE Cm-2(2)fl C"(2\{0}). The derivative

ym (m-1) still is piecewise continuous with a single jump of magni-

tude 1 at t=0 . We have ym(j)(t) = O(e-JtI/2) , as t#O . Moreover,

(2.10) (at+1)mym = 6 ,


with the Dirac distribution 6 . (In other words, ym is a fundamen-

tal solution of the differential operator (at+1)m ).

For k=0,1,2,... let CBk(&n) be the space ofCk(Hn)

with bounded derivatives up to order k including. Also, define

(2.11) x, Elk .

Corollary 2.4. Every function over &2 with (ax+1)2(6+1)2a

=b E CB(1 ) may be written in the form

(2.12) a(x,t) = f x,t E 2

Then aE CB2(1Q2), with derivatives given by differentiating (2.12)

under the integral sign.

We also introduce the 'pseudo-differential operator'

Q = q(x,D) , where of course q is not in i,t0 , but still the right

hand side of (1.7) with a=q defines a function Qu(x) for every

u E CO(&). In fact, Qu(x) = y2(x) F-ly2Fu(x) , so that

(2.13) Q = Y2(x)Y2(D)= x1y2(x) ((l 1

+ix)Z *)

It is clear from (2.13) that Q E=-L(H) . In fact one finds at once

that Q is an integral operator with kernel xi222)

y2(x)/(1+i(x-y))

E L(& - i.e., Q is of Schmidt class. Moreover, we may write

(2.14) Q=V*U , V=(1-ix)-1(1-ax)-1 , U=(1+ix)(1+ax)Q

where both operators U and V are of Schmidt class. Indeed, V and

U have integral kernels in V(11), given by

(2.15) vx(Y)=1+ix y1(x-Y)' µx(Y)°xi(1+ix)(1+ax)((1+i(x-Y))2Y2(x))

Now thm.2.2 will follow from prop.2.5, below.

Proposition 2.5. For a with b=(1+ax)2(1+a )2aE CB(&2) we have

(2.16) IIAuIIL2 5 cIIuIILz , c= 2n[[[U]]][[[V]]] IIbJIL'(R2) ,

with the Schmidt-norms [[[U]]] =' x ' L

2 , [[[V]]]= IIv(y)ItL

2

Proof. Use (2.12) to write

(2.17)

f dzdtb(z,tj91l q(x-z,-t)eixu. () ,


the integral interchange being legitimate, for uE Cp, hence uÊ S.

The inner integral may be written as q(x-z,D-l;) , using (1.7).

For an operator A=a(x,D), aE lt0 (general n) one finds that

(2.18) AZ'Ou(x) =iz = a(x+z,D)u(x),

where a(x+z,D) denotes the 4do with symbol a(x+z,t),of course,

with the constant parameter z E &n. Similarly,

(2.19)A0, tu(x) =

ffl a(x,D+t)u(x)

Combining (2.18) and (2.19) we get

(2.20) Az,t = a(x+z,D+t)

Clearly (2.20) holds for a=q as well. Thus (2.17) may be written

(2.21) Au(x) = fdzdl;b(z,l;)Q-z,-tu(x) .

This leads into the following estimate for the inner product (.,.)

I(Au,v)I= IIbII

LJI(Q-z,-tu,v)Idzdt,

where

I(Qz.tu,v)I = I(UMtTzu,VMtTzv)I s i{IIUMtTZUll2+ IIVMtTzvII2}.

Let w=TZu, keeping z fixed. Using Parseval's relation we get

fdtlIUMtwII2= fdjdxl(Vx,MMw)12 = fdxfdtlfdyµx(Y)w(Y)eiyl;I2

= 2nfdxdYlFAx(Y)121w(Y)12

Recalling now the meaning of w :

fdzdtIIUMtTZU112=2nfdxdydzlµx(Y)I2Iu(Y-z)12=2n Ilull2fdxdylµx(Y)12

Similarly,

f dzdi;IIVMtTzvII2 = 2nIIvII2fdxdylvx(Y)I2

Summarizing, we have derived an estimate of the form

(2.22) I(Au,v)I s i{S2llull2+ T211vII2} , u,v E Co1

with constants S,T. Here we let v=v run through a sequence in CD

with LZ-limit Au/S2, concluding that IIAuIIs STIIuII , uE CD . Noting


the precise values of the constants S,T we get (2.16). Q.E.D.

The case of general n = 2,3,... follows analogously: Define

and the corresponding operator Q=q(x,D)

We get (2.12) with b(x,l;)=rlj=1{(1+ax )2 (1+a Also, the7 7

U and V are tensor products of the 1-dimensional operators, their

Schmidt norms are products of 1-dimensional Schmidt norms.

Corollary 2.6. In the general case of n dimensions, we have

(2.23) IIAII 2 n s cnsup{ Irf'1{(1+ax )2(1+a )2}a(x,U) I len},L(R) 3= j j

where

(2.24) I2ffdxdyl(1+ix)(1+a){ +ixx-}I2}1/2

+ix x( y )

Next we prove that OP,tOC WT. From thm.2.2 it follows that,

for AE OpVto, all derivatives of Az,t exist in norm convergence of

L(H). For example, the difference quotient

(2.25) VT1,jAz't _ (a(x+z+1e7,D+1) - a(x+z,D+t))/1 , ri # 0

is a 4do with symbol

f1

(x+t%1ej,V 1-1(a(x+1e7, ) - dt alxj3 0

(using (2.20) where z=t=O, without loss of generality). Therefore,

(2.26) (Vq.ja)(R)(x') = O dt

But the uniform boundedness of all derivatives of a(x,t) implies

that each derivative a(R)(x, ) is uniformly continuous over 12n

J1

(2.27) a(x,g) = h Xa(x+th,t) = lhlO(1)0

for example,where 0(1) is bounded for all x,h, E 11 Similarly,

(2.28)a(R)(x,

) = O(Ihl+lll)

Thus the integral at right of (2.26) equals

(2.29) 00111)

with 0(.) independent of Thus (2.23) implies

(2.30) lim1,0 1lV1.jAz.t-

alz(x+z,D+t)II = 0

7

Similarly for all partial derivatives, by induction.

8.3. The other half of the theorem 257

3. The other half of the IM- theorem.

Note that we have proven one half of thm.1.2 :Every pdo in

Opit0 is contained in the class WT. To attack the other half we

must answer the following question: Given AE L(R) with Az,t E C_

how to define a symbol aE ,to such that A=a(x,D) ? We will give a

formula for in terms of A, first in the case A=Op(A)=a(x,D)

E OpyrtOC WT, constructing a left inverse for the map a- Op(a).Again let n=1 first. Departing from (2.12), given aE Vto, we get

ix

with (1+ax) 2(1+a ) 2a(x,i;) ,

writing as product of 2 Schmidt kernels:

(3.2) p(x,U)4dl1 {(1+a,TF+-f 77

In fact, both kernels

(3.3) Y 2

and

(3.4) u(x,rl) = (1+a1){(1-if1)2Y2(x)Y2(11)eixrl}

are L2(12)fl L1(1e2), as easily checked. Using (3.2) write (3.1) as

(3.5) fdxdTju(x,rj) f

using a Fubini-type interchange of integrals. We introduce

(3.6) w(x.Tl) = f

and the integral operators U , V , W with kernels u(x,y) , v(x,y)

and w(x,y), respectively. We get W=VF*V, with the Fourier trans-

form F. Note that wE L2(&2), so that W is Schmidt. For two Schmidt

operators U,V consider the inner product of their kernels

(3.7) trace(VU*) = fdxdyu(x,y)v(x,y) .

It is known that (3.7) may serve as a generalization of the matrix

trace, introducing a "trace" for products of Schmidt operators,

hence the notation. With this notation (3.5) assumes the form

(3.8) a(z,t) = trace((Bz,W)U*) ,


where we used (2.20) for the tpdo B=b(x,D) with symbol b of (3.1).

Next we notice that (2.20) implies

(3.9) B = (1+aZ)2(1+at)2Az.0z=t=O .

This shows that a(z,t) of (3.8) is well defined for a general

operator A E TILT . In fact, for general A E TILT we have

(3.10) Bz, = (1+az)2(1+aO 2Az,t E C00 (I2,L(H)).

The product map (A,B)- AB for AE L(H), BE S(H), is continuous

in A,B, in the norms of L(H) and S(H)= Schmidt class: We have

(3.11) [[[AB]]] s IIAII' [[[B]]] , [[[BA]]]5 IIAII [[[B]]]

Limits may be taken inside the norm, hence a of (3.8) E CB°°(12).

In other words, our left inverse S:A - a (with a defined by

(3.8)) of the map 0: tpt0 TILT (with Oa = Op(a)) is a well definedmap TILT-> 1Ut0.'To show it to be an inverse of 0 we prove that SA=Oimplies A=O. Indeed, then S is 1-1, and S(OS-1)=SOS-S=(SO-1)S= 0,

hence OS=1 as well, so that S and 0 are inverses; S maps WT onto

Tyto and we have the full equivalence of the conditions of thm.2.2.

At this time we return to n dimensions: Define the constant

coefficients PDE P = p(az,at) by

(3.12) p = J=1(1+aZ7)2(1+atj)2 = p(az'at)

For A E L(H) with Az,t E C1(&2n,L(H)) define Bz,tE L(H) by

(3.13) Bz,t = PAZ,t = P(az,at)Az,t

With B=B0,0 one findsothat BZ,t=E-1 BEZ,t ,(i.e.(2.2) holds for B)Z,tWe still have BZ,tE C (& ,L(H)), by (3.13), so that BE TILT. Let

(3.14) b(z,t) = Pa(z,1) = p(aZ,a,)a(z,l)

and note that we get (3.1) again in the form

(3.15) a(z,t) =

with 1P3=1{Y2(xj)Y2(j)}

All earlier arguments of sec.3 may be repeated: With the 1-dimen-

sional (3.3),(3.4) define Un, Vn on L2(1n) with kernels

(3.16) un(x,T1)= j) , E L2(12n)

8.3. The other half of the theorem 259

and an operator Wn with kernel

(3.17) wn(x,1) = 1=1w(x j,rl j) E L2(1e2n)

Again Un, Vn , Wn are of Schmidt class. With an analogue of (3.7)

as definition of trace we then get

(3.18) a(z,1) = trace((Bz,tWn)Un*)

valid for all aE 1ptO , and A=Op(a), with BZ,t of (3.13). The right

hand side defines a map S: CWT -> iitO, a left inverse of 0 .Now we prove that S is 1-1 . Let w(x,l;) E D(&2n) . Then we

claim that a function T(x,g) E D(E2n) can be found such that

(3.19) Jdzdl;ix =

Indeed,using prop.2.3 or cor.2.4 we obtain cp in the form

(3.20)

Let the right hand side of (3.18) vanish identically, for some A.

Using (3.7) we get (writing U for Un again, etc.)

(3.21) =K 0 for all z,l;n

where we write the kernels u(x,1)=u1, v(x,ri)=v1 as families of vec

tors u1, vIE H, depending on the parameter i E &n, using the inner

product (.,.) of H . The n-dimensional (3.2) assumes the form

(3.22)

while (3.21) may be written as

(3.23) 0

using that

(3.24)*EZ'tF = e

iztFEt,-z

Multiply (3.23) by T of (3.20) and integrate: Jdj may be pul-

led out; its inte rand is L1(,2n)g . Thinking of BF* as an integral

operator, our attention will be directed to the kernel

(3.25)

k : Use the definition of Ez't, (3.22) and (3.19) for

260

f

8. Smooth operators of L(H)r

fdzdtp(z,l;)eizte-itxe

(3.26) =

Jdzdt(p(z,t)eix4q(z-x,t-t)

-

To use (3.26) we first assume B to be an operator of finite

rank. Let BF* = DO) (,j , bj , cj E H . Also let

with (real-valued) functions x, iV E D(&n). Then, by a calculation,

dzdtd>1y(z,t)(Ez,tuq,BF*Et,-zvn)

=

(3.27)_ (X,BF*'W)

For general B EW T with (3.21) (or (3.23)) focus on Bj=P1BPj,

Pj denoting the orthogonal projection onto span{q1,...(Pj} with

some orthonormal base {ypj: j=1,2,...} of H . Such Bj is of finite

rank, so that (3.27) holds. The left hand side of (3.27), for Bj,

converges to that with B, hence to 0, (by (3.23)): We get B-B j=

Q1B + PjBQj , Qj= (1 - Pj) . Denote the factor of at left

in (3.27) by X(B) , for a moment. By Schwarz' inequality we get

(3.28) IX(PjBQj)ls [[[u]]]'[[[(1-Rj)V]]],

This expression goes to zero, as j+-, for fixed z,t, and it is

bounded in z,t. The first is true because, letting xj(y) be the

expansion coefficient of the kernel v(x,y) in the orthogonal expan

sion with respect to the base 1P j=Et ,-z*Fp j, we have [[[V]]]2 =

1JIxj(Y)I2dy<-, so that [[[(1-R1)v]]}2 = Lj=nJlxj(Y)12dy 0.

For the boundedness we use that IX(PjBQj)Is [[[U]]}[[[V]]]. Simi-

larly for X(QjB). By Lebesgues theorem the integral at left of

(3.27) (with B=Bj) goes to zero. Hence

(3.29) (x,PjBF PjtV) + 0 j + for all x,'tU E D($n) .

It follows that B=0. This gives A=O, in view of the result below.

Proposition 3.1. For an XE WT, assume that either (1+aZj )XZf,.=O

or (1+a .j )Xz,.=O, for some j, and z=t=0. Then it follows that X=O.

Proof. These relations are true for all z,t, if they are true for

8.4. Smooth operators 261

z=t=0, by (2.2). Using that XZ'tE C(&2n,L(H)) conclude that

(3.30) 0 = f0(1+a )X etdt = X + f0X (1-a )etdt = X- t tej,0 - te3,0 t

by partial integration, from (1+aZ )X0,0 =0. Similarly the other.3

This proves prop.3.1, and completes the proof of thm.2.1.

4. Smooth operators; the V*-algebra property; pdo-calculus.

To summarize our accomplishment in sec's 1 to 3: Let us call

an operator A E L(H) translation smooth if the family Az,O=

Tz-1ATz of (1.5) is C"(&nL(H)) , and gauge smooth if the other

family A0' = M4-IAMB. of (1.5) is Then the essence of

thm.1.1 may be reexpressed as follows.

Theorem 4.1. An bounded operator A on H=L2(&n) is both translation

smooth and gauge smooth if and only if it is a pseudo-differential

operator with symbol CB"(Rn), with Au(x) expressible by

(1.7) for u in the dense subspace D(&n) of H. Here the symbol a of

A is given by formula (3.8), for n=1 (or by (3.17) for general n)

W=(2n)n/2FV , U and V denoting the integral operators with kernel

(3.4) and (3.3) (or (3.15)). The trace is defined by (3.7),

and the (Green's) function yj(t) by (2.9) .

Several comments are in order, to illuminate this result.

First, it is trivial, that the class VT of operators which are

both, translation- and gauge- smooth, forms a subalgebra of L(H),

just using the common rules of differentiation. In fact, the alge-

braT is *-invariant - it trivially contains its adjoints, since

A* = A * , A* = A * , and IIB II; 0 implies IIB*II=IIB II - 0.z,0 z,0 0,t 0, 3 * 3 3The *-algebra 11-UT no longer is a C -subalgebra of L(H), but

it carries a Frechet topology induced by the natural norms of CB00:

(4.1) IIAIIk= IIAz,tIICk= sup{IIayaPAz'tII:z.lE 1en,Ia+SIsk}, k=0,1,2,..

The norms (4.1) establish 1113T as a Frechet space. The limit A of a

sequence A.E 1T has Az,

=lim(A.)z in the norms (4.1), implying,

AZ'. EC-(3k 2n,L) so that AE WT. Involution and algebra operations

are continuous, for similar reason, looking at C0O(1Q2n,L). There-

fore WT , with this topology, is a Frechet-*-algebra.

Note that the same topology may be just as well described

in terms of the symbols: For a E Vt0 we introduce the norms

(4.2) &2n,Ia+PIsk}, k=0,1,..


If A = Op(a) , then

(4.3) aZa Az.t-Op(a(a))z,t , a(a)(z,t)=trace(p(azat)aya Az,tWU*),

by (2.20) and (3.8) . By (2.23) and the formulas of sec.3 we get:

Proposition 4.2. Let A=Op(a). There exists a constant c depending

on n only (expressible in a form similar to (2.24)) such that

(4.4) IIalIks IIAIIks cilPalik, P=rIJ((1+azi)(1+at7))2.

The topologies on 1IGT=Op(Vt0) of (4.1) and (4.2) are equivalent.

As another remarkable fact: If A E qGT possesses an inverse

in L(H), (i.e. if there exists an L2(&n)-bounded operator B=A I

with ABu=BAu=u for all uE H), then we have AIE IIGT as well: We

get AIz,0=Az,0-1E Coo, and may calculate the derivatives in the

usual way. For example, azj(A Iz,O)=-Az,O-1(azjAz,O)Az,O-1

, ... .

We summarize:

Theorem 4.3. The class 1IGT = Op(pt0) is a Frechet-*-algebra under

either of the set of norms (4.1) or (4.2). Moreover, tIGT C L(H)

contains all its inverses (with respect to L(H)).

Algebras with the properties of thm.4.3 were investigated by

B. Gramsch, who uses the notation 'p*-algebra' for a *-subalgebra

of L(H) containing all its L(H)-inverses. In many respects, nota-

bly regarding the Fredholm property of its operators, a 1V*-algebra

behaves like a C*-algebra.

We notice that the resolvent R(a.)= (A-X)-I (in the sense

of H=L2(&n)) is a well defined operator of 4I0T as well, for every

point X of the resolvent set Res(A) , an open subset of C , assu-

ming that A E 407 . In fact, the complex derivative dR(X)/dX _

-R(X)2 belongs to qGT as well. Moreover, for the difference quo-

tient Q = I(R(X+h)-R(X)) we getQz

= h{(Az,t->v-h)-1-(Az,t-%)-1}.One finds easily that limh-OaZaQz, exists, for each z,t, uni-

formly as z,t E 2n. Accordingly, the resolvent is complex differ-

entiable within the Frechet-*-algebra 1IGT, defined on Res(A).

Commonly, for a (complex-valued) holomorphic function T(X),

holomorphic in a connected neighbourhood N of the spectrum Sp(A)

of A, and a simple closed contour r containing Sp(A) in its inte-

rior, one defines q(A) by the 'Dunford integral'

(4.5) p(A) =2n

existing in 11GT since its integrand is continuous. We have proven:

8.4. Smooth operators 263

Proposition 4.4. For A E WT and a function cp with above proper-

ties we have y(A) E=-SGT . The symbol of the resolvent R(X)

E 1VT is holomorphic in Res(A), as a function of ? taking values

in yto with topology of (4.2). The symbol of T(A) is

(4.6) -2n fr

Extensions of prop.4.4 are possible, applying to the case

where the contour r touches the spectrum in a single point, or r

may run through - on the extended plane, or that AE OpVtm, m#0

Also, the same questions will arise for other symbols.

The symbol a of an operator A E 40T plays a role similar to

an integral kernel. In fact, we may write (1.7) as

(4.7) Au(x) = u" = Fu

showing that AF* has integral kernel a(x,U)eix =

The question arises for a composition formula, linking the

symbols a,b of A, B E WT with the symbol c of C=AB, similar as

well known for integral kernels. Such a formula, involving a 'very

singular' integral, called finite part, was discussed in 1,4:

(4.8) c(x,t) = 1Y1dydq

The integrand clearly is not L1(R2n) , and the integral was defi-

ned by a special proceedure (cf.I,(4.19)). Similarly, a formula

linking a=symb(A) with a =symb(A ): We have (cf.I,(4.6))

(4.9) a 1C2f (x-y,t-11)e iyndydl

where again the integral is a finite part.

Different formulas are known for differential operators

(4.10) A = a(x,D) = IasNaa(x)Da , aaE C'(&n)

We know that A of (4.1) may be written in the form (1.7) with

(4.11)

a polynomial in t with coefficients depending on x . Indeed,

(4.12)

whence

(4.13)

(Dx.u)" , (DXu)" (U) = eau" (U) ,J

Au(x) = a(x,D)u(x) = xnfdfdyei(x-y)a(x,)u(y)


A differential operator A of (4.10) never belongs to i1GT,

except, perhaps, if as 0 for a#0 . For differential operators

A=a(x,D), B=b(x,D) with symbols we have composition

and adjoint formulas given by the Leibniz' formula of differentia-

tion, expressed as follows : AB= C= c(x,D) and A =a (x,D) again

are differential operators with symbols given by

(4.14) c = (YT 11 a(Y)(x, )b(Y)(x' ) , a-= (Y 1a(Y)(x, )

where the sums at right are finite, since a is a polynomial in .

We have seen in 1,5, 1,6 that (4.14) generalizes to yado's of

rather general symbol classes, giving Taylor-Leibniz-type composi-

tion formulas, involving as sum like (4.14) but also a remainder

term. This was seen in 1,5 for operators with symbol in it, and

even in ST1. However, a Taylor-Leibniz formula proved to be useful

only if the remainders - given in form of finite part integrals -

become small, and an asymptotic expansion results. In that respect

the symbol class ipt proved to be impractical: we always were wor-

king with yrhMIP16 , where both pj>0 , and 0s <p, .

A calculus of ido's will be possible again for operators AE

L(H) which also are rotation smooth and dilation smooth, in addit-

ion to the translation and gauge smoothness already discussed.

Here we refer to the groups On of rotations o:&n- &n and

of dilations a:1Rn_&n, with an orthogonal nxn-matrix o (of determi-

nant 1) and a a> 0 , of I. Define the groups of unitary operators

(4.15) 00u(x) = u(ox) , of On , Sau(x) _ Vu((3x) , aE B(+ .

An operator AE L(H) is called rotation smooth (dilation smooth) if

(4.16) 00*AO0 E C'(OnL(H)) , Sa*ASa E C'(&+,L(H)) ,

respectively. We denote the class of operators AE L(H) with all

4 above smoothness properties by 4I0S , and will investigate this

class in sec.5 below. Clearly WS C'IGT= Op(it0) , so that i1GSconsists of pseudodifferential operators with symbol having all

derivatives bounded. We will see that the symbols of 4GS have

derivatives decaying at , and stronger so, as the order of dif-

ferentiation increases.

Instead of requiring dilation and rotation smoothness, we

even might require smoothness of AE L(H) with respect to the class

of all maps u(x)-u(gx) , where g=((gjk)) is a real invertible nxn-

matrix (in addition to translation and gauge smoothness). This de-

fines yet another class LIGLC'WTC'WT of pdo's, studied in sec.5.

8.5. Operator classes 265

5. The operator classes 'TVS and WL , and their symbols.

As already observed, the classes 1IGS D WL of sec.4 are con-

tained in 1IGT, thus consist of ipdo's (1.7) with symbol in CB00(12n)

The additional smoothness of Az, will lead to stronger conditions

for symb(a), to be worked out next.

The maps for a general matrix gE GL(&n), together

with the translations u(x)-u(x-z) and gauge transforms u(x)-eiz

of sec.1 generate a larger Lie-subgroup of u(H) , we will write as

(5.1) GL = (g,z,1,(P) E gl}

where

(5.2)Tg,z,;,tu(x) = (det g) 1/2 ei(l;x+(P)u(gx+z)

, u E H ,

gl = {(g,z,1,(p): g E GL(In) , z,i;Ein

,(PE S1}

,

The group operation in GL (or gi) is best decribed by introducingthe linear maps g: ten->ien and X: &n->>e, for a given (g,z,t,(p)E gi C

nZ +2n+1& by setting g(x)=gx+z, T,(x)=i;x+T. Write

(5.3) Tg,z,t,e(x) = T(g,?)u(x) = (det g)1/28i%(x)(uog)(x)

We get

(5.4) T(g,X)T(g',).') = T((g,A)o(g',X')) , T((g,l)-1) = T(9r,X)-

where (g,A.)=(g,z,t,(p)E gl and

(5.5) (g 1,-Xog 1).

(5.6)

For a pair of maps g(x)=gx+z, X(x)=1x+q intoduce the matrix

M = Mg, = 0,g,z '

00 ,0,1

The group operations (5.5) correspond to matrix multiplication and

inversion -i.e., (g,X)o(g',a.') H M'M, (g,X)-1 , M 1, by a calcula-

tion. Thus (g,?.) -M defines an isomorphism between GL and the

Lie-subgroup of the linear group GL (&n+2) consisting of all matri-

ces (5.6). Actually we must calculate mod 2n in the variable cp ,

i.e., work with the quotient of the matrix group modulo the sub-

group of all (g,z,t,cp)=(1,0,0,2kn), to obtain an isomorphism.

Speaking in terms of Lie-groups we find that (5.6) describes


a representation of the Lie-group gl in GL(&n+2) .

We introduce groups GS and gs from the unitary maps used for

definition of '1t3S : GSC GT consists of all Tao,z,t,w , with a rot-

ation o and 0<oE I . Define gs={(a,w,z,t,(p):(ao,z,t,c))E g1}, where

0>0, 0 0=1, det o =1 . A matrix representation of gs is given by

(5.6) again, since gs is identified with a subgroup of gl .

We thus have introduced a chain GT=GHC GSC GLC U(H) of sub-

groups of the unitary group U(H) , where GX , X=T,H,S,L, all are

unitary representations of Lie-groups gt=gh, gs, gi . Note that

gx, x=h,t,s,1, are connected. For each gx we also have the finite

matrix representation (5.6). Clearly, the relation between gx and

its corresponding Lie-algebra (called ax) is described by the dis-

cussion of ch.0,9, using the representation (5.6). It is suggested

that the representation GX of gx on the infinite dimensional space

H is governed by similar principles.

Such principles stand behind the discussion of 4L and WS,

below. - This is why we decided to present sec.9 of ch.0. However,

this central part of Lie-group theory will not be needed, below.

To characterize the symbol classes yr10 and ipso belonging to

IL and 'S we assume that A = a(x,D) is given with a E CBm(&2n),

and ask for the symbol (if any) of the operator

(5.7) Z(g) = Z(g,A) = Rg-IARg , Rgu(x) = u(gx) .

Note that the function ug(x) = u(gx) E S has the Fourier transform

(5.8) ug"(x) = (det g )u(g t) = (det g -)u -t(x) ,

gusing the abbreviation g = g-1, g-t = (g-1)t . Accordingly,

(5.9) Aug(x) = xn(det g

Another transformation of integration variable yields

Z(g)u(x) =Aug(g-x) =(det g-)xn elxg a(g t )d(5.10)

t

With another substitution of integration variable we get:

Proposition 5.1. For A = a(x,D) E WT the operator Z(g,A)=RgARgis in t1GT again. We have

(5.11) a(g-Ix,gt )

Note that the symbol (5.11) indeed belongs to lpt0=CB°

Now we can evaluate linear group smoothness; it amounts to


(5.12) Z(g) E C'(GL(len),L(H))

with derivatives existing in norm convergence. Clearly (5.12)

implies the existence of the partial derivatives

a "nn(5.13) (a

gil

)ll...(agnn ) Z(g111 ...gnn) at gjl=6jl

for all integers ajl 2:0 .

Formally we may execute the differentiations of (5.12) by

just differentiating the symbol. For example, using that

(5.14) ag(g-1)

_-g-lhpgg-I ,

hpq= ((6pi6gj))i,j=l,...,nPq

hence

(5.15)

we get

agjl (gix)mlg=1

= -(hjlx)m = -6jmxl

agj1 (hlj)m = slmmj

agjla(g

(5.16) = 4=1(-alxm(x,U)6jmxl+ a,

m(x,U)slm

j)

Our attention is directed to the folpde's

(5.17) ejl = jal-xlaxj , j,1=1,...,n ,

since (5.16) formally implies

(5.18) agjlZ(1) = (Ejla)(x,D) .

Proposition 5.2. For A=a(x,D) ET , u E S and x E &n we have

(5.19) agjkZ(g,A)u(x) = Lp=lgp.(Epka)(g

where we again abbreviate g 1=((gpq)).

Note that (1.7) may be differentiated under the integral sign

for whatever parameter tihe symbol a might depend on as long as uE

S (hence uÊ S) and the derivatives of the symbol a (for the para-

meter) are of polynomial growth in (x,g), uniformly in the parame-

ter. We will omit the function uE S in (5.19), derive a formal re-


lation for the operators, to be checked for polynomial growth.

We have Rhgu(x) = Rgu(hx) = R9Rhu(x), so that

(5.20) Z(hg,A) = Rh_,g_,AR9Rh = Z(h,B) , where B = Z(g,A) .

With Dk=derivative in the direction of (the nxn-matrix) k we get

(5.21)DhgZ(g,A) = {E{Z(g+Ehg)-Z(g)}}IE.

0

{E{Z((1+Eh)g,A)-Z(g,A)}}IE. 0 = DhZ(1,B)

Setting h=h3k=((Sjp6lq))p,q.1,...,n get DhZ(1,B)=agj1Z(1,B) _

(Ej1b)(x,D), by (5.18). The left hand side is

{£{Z(g+Et)-Z(g)}}IEa 0 = tpgagpgZ(g,A) , tpq (hg) p4 sjpglq

Together we have (summing over indices occurring twice)

(5.22) 6jpglgagPq

Z(g,A) = glgagJq Z(g,A) = (Ejlb)(x.D)

where

(Ej1b)(x,U) =

(g Ix.gtU)gpjp p

= {gr jglp'lraTpa(Y.1) - glrgpjYraypa(Y.TI)} IY=g-' x,rt1=gtt

Summarizing we get

(5.23) (Ejlb)(x, ) = grjglp(Erpa)(g-lx,g 5)

Substituting (5.23) into (5.22) the formal relation (5.19) follows

Going again through the arguments, starting at (5.21), find

the polynomial growth condition satisfied. Thus the formal argu-

ment without u(x) leads to a rigorous proof of (5.19).

Notice that (5.19) may be written in the form

(5.24) agjkZ(g,A) = Lp=IgpjZ(g,(Epka)(x.D))

as a consequence of (5.11). This shows that (5.19) (or (5.24)) may

be iterated to obtain all partial derivatives (5.13) of arbitrary

order. These all exist as linear operators S- S, as AE SGT . More-

over, derivatives (5.13) are linear combinations of b(x,D) with

(5.25) b = (Ejj 11 Ej:l:...)a


finite application of the ejl to a, with coefficients rational in

the coefficients of g, with no further assumptions.

Theorem 5.3. Let the symbol class V10 be defined as the set of all

a E CB00(1e2n) such that all finite applications (5.25) also are

contained in CB"O (&2n) . Then we have GL = Opyi10 .

For WS we state a similar result, using the folpde's

(5.26)1ljl- Ejl-Elj= j)+(xjaxl-xlax,) , j,1=1,...,n ,

100 =lepp = "1n=1,pa,p

Theorem 5.4. Let the symbol class Vs0 be defined as the set of all

a E CB00 (1e2n) such that all finite applications of qjl to a are in

CB'(&2n) again. Then we have WS = Op1s0 .

Before discussing the proofs we note:

Proposition 5.5. Introduce (in addition to (5.17),(5.26))

(5.27) Ep0\p0= axp ' EOp=T10p = app , p =1,...,n .

Then V10 and "0 consist precisely of all aE C00(1 2n) allowing ar-

bitrary finite application of ejl (for i,10) and qjl (for ips0), as

bounded functions on &2n, for j,1=0,...,n with sjl or Yljl defined.

For the proof of prop.5.5 we note the commutator relations

(5.28) [ap0'Ej13 _ -6pltjo' [Eop'EjlI = bpjE01 ,

p,j,1=1,...,n,

verified by a calculation. (5.28) shows that, a product (5.25) may

be written as a combination of products withtop

and Ep0 pulled

out to the right. Thus the condition of thm.5.3 implies that of

prop.5.5, while the reverse is evident. Similarly for thm.5.4.

Proof of thm.5.3. It is clear from remarks around (5.24) that all

derivatives (5.13) exist in convergence of S if only A=a(x,D)E

SGT (i.e., aE tpt0). Moreover, the derivatives are linear combina-

tions of b(x,D) with b of the form (5.25), as noted. Thus they be-

long to L(H) - even to 11GT - if aE ipso, with a uniform bound oncompact regions of GL(&n). This implies that A=a(x,D), aE tps0,

must be GL(&n)-smooth. For example we get

t(5.29) Z(g(t))u - Z(g(0))u = dt(d z (g(i)))u , u E S

0d-C

for a smooth curve g(t). Here the integrand I(i) satisfies III(t)II

sclluII, with L2-norms, and c independent of u and t. First conclude


from (5.29) that Z(g) is continuous in g. Similarly all derivativ-

es (5.13) are continuous as maps GL(&n)->L(H). Knowing this we omit

uE S in (5.29) and divide by t. As t-0 get existence of the part-

ials in norm convergence. Iterating we confirm GL(&n)-smoothness.

Vice versa, if A EEL(H) is GL-smooth, in addition to trans-

lation and gauge smoothness, we get A=a(x,D) with aE wto. As in

sec.1 one shows that T(y,X)-1AT(g,?)=A(g,?.) has continuous mixed

partials for z,t,g. All derivatives (5.13) are translation and

gauge smooth, hence belong to OpWto. Hence a=symb A E 4w. , q.e.d.

The proof of thm.5.4 is analogous to the above. For dilation

and rotation smoothness focus on the derivatives along the curves

(5.30) g(t) = tes ,0 < t < ao

, s=((sjk)) = - ((ekj)) ,

with the matrix exponential function es=l+s+s2/2l+.. . It is known

that a rotation o may be written in the form o=es, with skew-sym-

metric s . Thus, the derivatives (5.13) must be replaced by

a a(5.31) ((at)00(as )12... )Z(tes)

, at t=1 , s=0 , sT=-s12

where only derivatives for sjl with j<l occur. For g=tes we get

(5.32)tatZ(tes) _ ljlgjlagjlz(g) _ ;jlpgjlgpj(Epla)(g x,gtD)

4(Eppa) (g x,gtD) - (T 00a) (g x,gtD) = Z(tes, (T100a) (x,D)) ,

using (5.19). similarly, for j < 1 ,

(5.33) ae Z(es) _ q(ases)Pgag

((g))jl jl pq

For s=0 we get

(5.34) as jeels=0 = hjl- hlj , hpq = ((s 8 ))i NAP

vq µ,v=1,...,n

Accordingly, using (5.18),

(5.35) asj1 Z(es)I5=O = (Eji-Eij)a(x,D) = ('njla)(x,D)

For an analogue to (5.19) we repeat the discussion leading

to (5.23) as follows: A modification of (5.21) yields

(5.36) asjlz(eso,A)Is=O = asjlZ(es,B)Is.0 = (Tljib)(x,D)

with B = b(x,D) = a(otx,o D) . Using (5.23) we get

(5.37)(rljib)(x,U) - (ojrolp(Erpa) - olrojp(Erpa))(o

tx,ot)

8.6. The Frechet algebras

= 0jr0lp(Tlrpa)(o x,ot5)

Accordingly, we have the following substitute for (5.24):

271

(5.38) as.1Z(eso,A)1s=0 = f,p=lojr 1pZ(o,Arp) , Axp=rlrpa(x,D) .

We need the Campbell-Hausdorff formula of ch.0,9, quickly re-

derived as follows here: For sE N(so), a neighbourhood of so write

(5.39) es = e7 (s,so )es°, a. = log(eseso ) = _yn=1 (1-ese-so )m .

Clearly ?(s,so), is well defined (and holomorphic in the coeffi-

cients s jl , so jl ), as long as 1-eses0 1 < 1 . Since a=e8es0 ist

orthogonal we get e =a_X

, hence Xt = log(e-k) _ -% , so that X

is skew symmetric (and real-valued, for real-valued s,sO) . For

fixed so the function %(s,sO) has a local inverse at so . We get

(5.40) asj1Z(es) I s=so = <v(axlLvlasjl)(so ,so )aXµv

Z(e%e$0 ) IX=0

Or, with holomorphic functions xjlRV(s), j<l, t<v ,

(5.41) as.1Z(es)=;,<vKjlµv(s)2r,pltr vpz(es,(Tlpga)(x,D))

Note that (5.41) is the desired substitute for (5.24); It

may be iterated, showing that derivatives of arbitrary order

(5.42) ((as)a12....)Z(es)

12

may be expressed as linear combinations of

(5.43) Z(es, (Tl j, 1, Tj j, 1, ...a) (x, D)

with coefficients holomorphic in sjl. The proof of thm.5.3. now

may be repeated with 1jl instead of Ejl, for a proof of thm.5.5

(Note that (5.32) way be iterated as well). Q.E.D.

6. The Frechet alaebras Vx0, and the Weinstein-Zelditch class.

We already stated that the symbol classes s0 and V10 consist

of locally classical symbols. More precisely we have the following

Proposition 6.1. For any open set QCC In the sets 1yx JQx&n, x=s,1,

of restrictions of symbols aE ix0 to Qx&n={xE are subsets

of the Hoermander class S0,1,0(Q) (cf. [Hrz]). That is, we have

( 6 . 1 ) a(Pa) = 0(( ) - I a I ) , as xE Q CC Q


Proof. It suffices to look at i,s0, since V10C iNs0. Then we get

(6.2) xE Q, aEl,s0.

Since gjla = 0(1) as well, we get

(6.3) 00a = ja Ja = 0(1) , kjla = 7)a = 0(1)

Multiplying (6.2) by l andj and adding we get

(6.4) j = jko0a - Dl?jla =

or,

(6.5) 0(1) , i.e., aI = 0(()-1)7 3

This chain of arguments may be iterated, to prove

(6.6) xE Q .

Starting witha(3)

instead of a we get (6.1), q.e.d.

Let us work out a global result, with the same argument.

Proposition 6.2. For a E C0O(&2n) let

(6.7 ) a00 = 100a , ap0 = np0a = a i xp ' a0q = ll0ga= a I q '

apq = Ylpga , p , q = 1,... , n .

Then there exist symbols y 1 j (=-yp 1 2 , y 2 ) E 2 -.1 such thatpq e -e pq e

(6.8) a 13 n 2jI x q=0 Y pq apq ' a l i = fp, q=Oy pq apq

Here we recall the symbol class Vam , at = (ml,m:) , defined by

(6.9) {a E C°D(12n) : O(( )m1-ICI I(x)M2 -ICI) }

For symmetry reason it suffices to discuss only one of the iden-

tities (6.8). For the second one write (6.2), more explicitly, as

(6.10) apax a=a00+ ?pap0 , ( ptl_ q 1 1 )a=apq+(xpag0 xgap0)

p p p q pThen (6.3) may be replaced by

(6.11)

ppap0)+

pp(apj+xpaj0-xIap0)

by the same derivation. Dividing (6.11) by we obtain

a formula of type (6.8) for alb , noting that the quotients7

8.6. The Frechet algebras 273

xp/() 2 , q/( ) 2 , 2 all are in VCe2 -e, . Q.E.D.As a consequence of prop.6.2 we get the result, below.

Theorem 6.3. For a E 1Vs0 we have the estimates

(6.12) -IRI s j s IaI , for all a,f

with O(.)-constant independent of x, but not necessarily of a,(3

Note that (6.12), for. IaI+ISI=1, is a consequence of (6.8),

since a E 1Vs0 has all apq bounded. For general a,s we use that apq

E rVs0, again, so that we may iterate (6.8). Q.E.D.

The class Zn of all symbols a satisfying (6.12) was intro-

duced by Weinstein [Wel] and Zelditch [Zel]. Thm.6.3 amounts to

(6.13) 1V10 C 1Vs0 C Zn .

Let us observe that all three classes 1Vs0 , 1V10 , and Zn are

symmetric in x and . Therefore in prop.6.1 x and may be inter-

changed. We also get an x-decay of derivatives on compact a-sets.

The inclusions of (6.13) are proper. For example, let n=1 and

(6.14) x ,

where x(t), w(t) E COO(&), x=1 for tal, =0 near 0, w=1 near t=1, =0

in It-11 a 1/2 . Clearly all derivatives of a , for x and t , are

bounded. Moreover, a and all its derivatives vanish, as X is out-

side the interval (1/2,2) -i.e., unless we have xs 2T , and s 2x

both. In other words, we have (x) s and s 2(x) in supp aAccordingly all (positive and negative) powers of are

bounded in supp a . Hence we also have all

bounded. In particular, it follows that aE Z1 . On the other hand

it is clear that talk-xalx = -x sin x , as x= , IxI

a 1 . Thus s11a is unbounded and a E iV10 follows.

Next we focus on topological and algebra properties of rVx0.

Proposition 6.4. The classes tVx0, x=t,1,s, are algebras under the

'composition product' a°b defined by (a°b)(x,D)=a(x,D)b(x,D), as

well as the 'pointwise product' (ab)(x,t)=a(x,D)b(x,t) .

Proof. By definition 1Vx0 are algebras under ",". Pointwise we get

(6.15) E jl(ab) = (Ejla)b + a(Ejlb) , rlji(ab) = (rijia)b + a(rljlb).

By (5.19) or (5.35) a,bE iVxo implies Ejla, Ejib, rljla, rl j1bE 1Vx0

hence Ejl(ab) (or rljl(ab)EWt0. Similarly for iterated Eji. Q.E.D.It is natural to introduce on 1V10 the Frechet topology of


the countable class of semi-norms

(6.16) Ile pNgN ..... eplg1 allLoo =IIaII nK 79=(p1,...' N), K=(g1.... ,gN)I

with epq of (6.1) , for p,q a 1 , and

(6.17) ep0 = 1p0 = ax p ' EOq = 110q = a q , (e 00 undefined)

Similarly, for Vs0, using the jjl instead of the ejl. Or, for Vt0,

using only the expressions (6.17).

On the other hand, for Op*0 GX, one may use the semi-norms

(6.18) IIVaa(x,D)II = IIIIaIIIIa ,

with the L(H)-operator norm II'II , and VaA denoting the partial de-

rivatives of Ag,z,t

, or A6101z,

t , or Az,t , at the unit of the

group. Here a will be a multi-index of the dimension of the group.

Proposition 6.5. The topologies (6.16),and (6.18) are equivalent.

Proof. Consider the case of V10 . We know that

(6.19) VU(a(x,D)) = aa(x,D) , as = IIlaisp q aJ J

with some selection ePjgj , corresponding to the multi-index a

From cor.2.6 we conclude that

(6.20) IIIIaIIIIa = Ilaa(x,D)II s c IRl7ly ISn Ilaa(0)IILoo

where the terms of the sum at right are all norms of type (6.16),

since theejl

include the expressions (6.15).

On the other hand, for =WS we have a=symb(A) given by the

trace formula (3.18). For b=1 e a we get B=b(x,D)=VA with1 pjgj

the corresponding combination VU of ag , az and a. on Agizi. atpq p q

(g,z,l;)=(1,0,0), as follows from (2.20) and (5.19). Use (3.18) for

(6.21) IlbIIL s c IPI7IYIs2nII(Vta)(a)(x,D)II

Again, the terms of the sum, at right, are semi-norms occuring in

(6.18) . This shows indeed that, in case of V10, the topologies

are equivalent. Similarly for is0. Q.E.D.

Theorem 6.6. The classes i1G%, X=T,S,L, are V -subalgebras of L(H),

under the operator product and the topology (6.16) or (6.18).

Proof. Prop.6.4 and prop.6.5 make0(or WX) Frechet algebras.

8.7. Polynomials in x and Sx 275

The adjoint invariance and inverse-closed-property follow from

(6.22) A*g,z,S= (Agrzr0 * I

(A 1)g,z,l; _ (Agrz10-1 .

If A is differentiable in L(H) then so are the right hand si-grzrt

des in (6.22), assuming that A IE L(A), in the second case. Q.E.D.

7. Polynomials in x and ax with coefficients in W% .

So far,in this chapter,we were dealing with zero order pdo's

only. This restriction now will be removed by defining the classes

,x of all polynomials in the 2n variables x, with coefficients

in px0, x=t,s,1. Also, xm, m=(m,,m:), with integers mjz0, denotes

collection of all such polynomials of degree sm, (sm2) in (in x)

For general mE 12 we define Vlm , x=1,s, by setting

(7.1) y)x {a E C00(1 2n) : (x) a(x, ) E px0 }

Note that we have

(7.2) (x) = s0(x) + jsj(x) , s0(x) = (x)-1 , sj(x) = xj/(x)

and a corresponding formula for (l;), where the sj(x) and sj(g) are

in Vc0C *0, for x=t,1,s. Thus for integer mjzO an aE Vxm of (7.1)

may be written as a=ao (x) m'(1;) m' , a polynomial (of proper degrees)with coefficients in wxo, showing that the two definitions agree.

Note that ryxm C 4x , even if mj are not integers, since we

always can write a E rpxm as a = ( a0( )m, -M, (x) m2 -Ma ) ( ) M, (x) Mz

with a0E VxO , and with integers O<Mj2mj, so that the expression

(.) is also in px0. We get tpx=U(Vxm : mE ILI } as well. Clearly theVx are graded algebras under pointwise multiplication; we have

m m' m+m' Moreover, the Frechet topology may be carried to

Vxm using (7.1). As seminorms on *m use (6.16) for ao of (7.1).

The above definition of Vxm uses the 'weight functions' (x)=(x)=(1+Ix,2)1/2 again. The symbols of *m are said to

have multiplication (differentiation) (total) order ma (m,) (m).

Proposition 7.1. The class Vsm consist precisely of all aE C"(&2n)

such that, with xm(x,l) of III,(3.1) i.e.,

(7.3) nm(x,U) = Q ) , m = (ml rm2) E &2

we have the estimates (7.4) for all products 1"1=1I1pJqJ of3

(7.4) (rl =111pj,ja)(x,U) = x,l E &n .


Similarly, '1m is described by (7.4) with Y1 µvreplaced by ERv

Proof. Consider x=s only. If aE Vsm , then an-l=bE VsO. Hence

(7.5) llpga = (1lpgb)nm + (b(npgnm)/nm)nm

Note that npgnm/nm E yicO. Thus Y1pga=O(nm). Moreover, (7.5) implies

that11

pgaE Vs m, since ,cOC s0, and since Vs0 is an algebra. Hence

we may iterate, for (7.4). Vice versa, let (7.4) hold for a. Then

(7.6) 1lpgb _ n-mllpga + b(iipgn-m)/n-m) = 0(1) ,

using that r1 pgn-m/n-m E V00 C Vs0. Apply 1rs to (7.6) for 11rsrJpgb=O(1) , etc. ... It follows that bE s0, so that aE ism , q.e.d.

Proposition 7.2. Let aE 4wm, x=s,1, mE 1k2. Then, for all a,1,

(7.7) E'Wxm+r(e'-e2),as -dal s r s 1PI , r = integer.

Proof. First let JaJ+JPJ=1. Prop.7.1 shows that 1xm is left invar-

iant by arbitrary applications of spq or tlpq' resp., hence

Vxm for aE Wxm. Thus (7.7) holds for r=0. On the other hand, (6.8)

expresses by apq with coefficients inr(e'-e2)

r=t1,

correspondingly. This gives (7.7) for r=t1. Formulas (6.8) may be

iterated, proving (7.7) for arbitrary a,P and r. Q.E.D.

Remark 7.3. Note prop.7.2 contains thm.6.3. Generalized classes

Zn,msm' with Zn,O=Zn , may be introduced, replacing O(.) in

(6.12) by O(nm+r(e'-ez))For a symbol aE Vx, x=s,1, we will define the ipdo

(7.8) A = a(x,D) = xxaaa.P(x,D)DR as

However, one just as well might write the same operator A as

(7.9) A = Db a,R(x,D)xaD( ,

with (other) coefficients ba,P(x,U)E i,x0. Or also in any other or-

der A=F'aa,,(x,D)DPxa=YjtaDRda,P(x,D), etc. This follows from (2.8):

The reason for this fact can be found in formulas (2.8) : For A =

For A=a(x,D)E IWXC IIGT, X=L,S,T, we have [xj,A], [Dj,A]E Opy,x0In fact, for x=s or 1, the commutator symbols have better

decay properties: Using (2.20) we get

(7.10) az1A0,0 = alx.(x,D) , atJA0,0 = al j(x,D) ,

8.7. Polynomials in x and 8x 277

so that

(7.11) [xj,A]=a3(x,D)=ial,j(x,D) ,

satisfies stronger estimates of Weinstein-Zelditch type

(7.12)O(((x)/() )1) , -IRI-1 s 1 s lal

O(((x)/( ))1) -l s 1 s lal+1

Similarly for higher order commutators and higher derivatives.

Hence yw is an algebra under a°b of prop.7.4, since in a pro-

duct of 2 expressions (7.8) one may unite the x- (D-) powers at

left (right). With calculus of ipdo's, below,one also confirms that

vxm°1m'C m+m' , i.e., 1Vx is a graded algebra, under 16" .

Next we look at calculus of pdo,s, for the algebra OpVs, si-

milar as in 1,6 for 0M,P,o We have yixmC y,tm, hence may use the

Leibniz-Taylor formulas I,(5.7) , I,(5.9) for product and adjoint

in Opts. The question remains whether (in what sense) the remain-

ders decay, and about significance of the terms of the expansion.

In view of applications in ch's 9,10 we focus on

(7.13) KA , AK , [K,A] , K = k(x,D) , A = a(x,D) ,

where aE yixm, kE Cm m,m'E &2, x=s,1. We get a Leibniz formula

with controlled remainder, but decay of order not quite as fast.


KA= 4=0

I

j=j

(-i)j/8! (k(e)a(8))(x,D) + RN

(7.14) AK = 4=0l

(-i)J/e! (a(8)k(0))(x,D) + RNe j[K,A] = Lj=1(-i)j/j! (k,a)j(x,D) + RN

where k(8)a(0) ,and a(8)k(0) , and the 'iterated Poisson brackets'

(7.15) (k,a)j =lef

,(e)(k(e)a()-k(0)a(e)) , (8) = j!/e!

belong to Vxm+m'-re'-r'e2' for all r,r'=0,1,2,..., r+r'=j, while

R m+m'+re'+r'e2 ,for all r,r'=0,1,..., with r+r'=N+lNE

Proof. Note that k(e)E 1lcm'-je' ' a(A)E m+r(e' -e2) ,r=0,...,jas lel=j, by prop.7.2. Their products are E 1Vsm+n'-(j-r)e'-re2'

proving the first statement. In the remainder of I,(5.27), i.e.,

_ N+1 1

(7.16) PN (N+1)IAlN+1(01)

P 0,N' Pe,N0

(1-t) Ndx IO,N(x,lj,t)I


with the 'finite part integrals'

(7.17)

I may be written as an improper Riemann integral, namely

(7.18) I =JI1YP1Tje i1(Y)-2M(1_A1)M((Tl)-2M

with aM = (1-Ox)Ma, and the usual formal partial integration.In 1,5, by a somewhat difficult interchange of limits, we pro-

ved (7.16), (7.17), (7.18). Formally this is Taylor's formula with

integral remainder. Moreover, we saw that the derivatives I(a) are

obtained by differentiating (7.18) under the integral sign.

We discuss RN only, noting that RN, RN may be treated analo-

gously. We need an estimate for I. They are in the span of int-

egrals (7.18), with replaced by

(7.19) -rq) a , P'+3" = P

We have aNE Vsm' again. Hence, using estimates (6.9) for k and

(7.7) for aN , we arrive at an estimate for I( by a sum of terms

7.20)fdYdTl((Y)(TI) -tq)m' -N-1-Ia, x-Y)m12-r( )mi+r,

where -la"I s r s N+1+I3"I . Here we can estimaterr

J dy(Y) -2M(x_y) M'2 r

= 0((x) m' Z -r)(7.21)

JdTI(1I)-2M(-zTl)m, -N-1-Ia' I = O((t)mI -N-1-Ia, I )

using I,(6.5). Note (7.21) holds uniformly in t, 0sts1. The estim-

ates remain true if. a'=a"=13'=13"=0. Hence,

(7.22)(a)

O( , : ) , r+r = N+1.22) (P) nm+m,-re-r e

for all multi-indices a,3, giving the corresponding for the symbol

rN of RN . These estimates are weaker than the stated ones. But,

(7.23) rN = 71 (-i)IOI/61k())a(6) + rRN+1:I0IsR

where the sum at right is E Vx m+m,_re,-r,eZ, r+r'=N+1, for all R=

N+1,N+2,... For fixed r,r', r+r'=N+1, let R=N+2j, with j>O. Get R+

1=(r+j)+(r'+j), hence rNEtm+m'-re'-r'ez-je' by (7.22). Thus,

' 1(7.24) 1 =111pjgjr2j E Vtm+m' -re' -r' e2 , as N' s j

8.8. Characterization by Lie algebra 279

Thus all products of sj factors are O(nm+m'-rg'-r'e2)' for 4,

hence also for rN. Since j is arbitrary get rNE Xm+m'-re'-r'el

as required. Q.E.D.


(7.25) Opyxm = {(x)m:A(D)m' : A E WX = Opix0}

and, more generally, for arbitrary sE 12

(7.26) Op*xm = {ns(x,D)Anm-s(x,D) : A E WX }

Theorem 7.6. The space Op*xm is characterized as set of all AE

O(m) such that for all T=Tg,z.,.W

of (5.2) the operator T-1AT bel-1

ongs to the subgroup GX of GL ,

A-T-

AT defining a C *-map gx- O(m)

in the Frechet topology of O(m), as discussed in 111,3. Moreover,

a sufficient condition for AE OpVxm is that (i) AE L(Hs'Hs-m)

(ii) T-1AT isC00(gx,L(HS, Hs-m) for just one sE 12 .

We leave the proofs of prop.7.5 and thm.7.6 to the reader.

8. Characterization of '% by the Lie algebra.

In [B 21 , [ Dn1] , [Ci ] , IV, 9-10 one introduces 'derivatives'of a linear operator AE L(H), as commutators with Dj or M1. If all

such derivatives exist (in suitable topology) then A is found to

have a symbol in a class like CBOO or y't'O,P,b - i.e., A is a pdo.

There is a link between that approach and our present one:

It points to the relation between the Lie group GH and its Lie al-

gebra, as discussed in ch.O,sec.9. If a Lie group G is represented

as group of invertible matrices on some BN, the 'tangent vectors'

at IE GC GL(IN ) form a linear space A of NxN-matrices, containing

its commutators - a representation of the Lie algebra of G. The

connected component of I in G then is the set of all products of

Ae , for AE A , by the Campbell-Hausdorff formula.

Now focus on the 'representation' A -> EZ,t,q)-

of our Lie group gh of (1.11),(1.12) on the «-dimensional space

L(H) . Actually, two representations are involved: We represent

gh (=&nx&nxSl) by (z, ,g)- EZ.t, ei((P+K)TZ=ei(T+Mi;)eizD on H (as

unitary operators). That representation generates the above A->

AZ,t as invertible maps L(H)-L(H). Following the lines of ch.0,9

we formally get a tangent space at (z,t,q)=O for the first repre-

sentation as the linear span (called AT=AH) of the folpde's


(8.1) axj , i , ixj , j = 1,...,n .

The (8.1) are unbounded operators of H. A precise definition

will specify the domain - that of the infinitesimal generator of

iD.t ix tIs 3 } , {elt} , {e j }, respectively. These groups consist of

unitary operators of H, thus their generators are skew-selfadjoint

each having an orthogonal spectral measure.

Notice that the precise domains of the operators (8.1)

are not identical. Linear combinations no longer are skew-self-ad-

joint. However, as easily seen, the self-adjoint relizations of

1, Dj , ixj , and their linear combinations zD+jx+? , are unique.

Declare as common domain of 1, Dj , Mj the space S. Then AH indeed

is a linear space; for GAM the closure (in H) is skew-self-

iD **adjoint, and e is unitary. Using ch.0,sec.6 we get

**(8.2)

Ez,t,W+(z. )/2.

Note that (8.2) supplies an analogue to the Campbell-Haus-

dorf formula for our present infinite dimensional representation.

GH and AH are related exactly as G and A of 0,0.

Our second representation A- Az,t of gh on L(H) supplies an-

other representation of the Lie algebra AM , as the (also unboun-

ded) operators on L(H) , formally given as linear combinations of

(8.3) ad axj

, ad ixj , j=1,...n, with (ad X)A = [X,A] = XA-AX .

(8.4)

Indeed, let then, formally,

dt(eiLtAeiLt) _ -i(ad L)A

Formula (8.4) seems to imply that A- Az't is C1 as a map H2n

- L(H) if and only if (ad aj)A , (ad xj)A E L(H), moreover, that A

Az't is C if and only if finite products of ad ax , ad x1 take7

A to a bounded operator. In turn A- Az' is Co. if and only if AE

OpVto , by the 'tpdo-theorem. Both facts suggest thm 8.1, below.

Theorem 8.1. An operator AE L(H) belongs to Opipto if and only if

the operators (ad x)a(ad D)OA: S -* S' all have their image in H

and extend to bounded operators of H , where

(8.5) (ad x)a=(ad x )a' (ad x2 )a:..., (ad D)A=(ad Di ) P' (ad Di ) P2 ...

Here (ad xj)A= xjA-Axj trivially is a map S-'S'. Similarly

8.8. Characterization by Lie algebra 281

(ad axj)A. Higher order commutators (ad x)aA are sums of terms

IhcATbc, with only one A between products The of x j. Again they areS- S'. Thus the statement is meaningful. The proof will be a veri-

fication of above relation between differentiability and existence

of commutators in L(H). Details are left to the reader.

Clearly the map (g,?.)- of (5.2) defines a un-

itary representation of the Lie group gl on H. Note that the map

is just the restriction to gtC gsC 91 of T above.

For GT , GS , GL the Lie algebras AT , AS , AL are the (un-

bounded) directional derivatives of T(g,X)=T at e=(1,0,0,0)

of gl, in directions allowed by the subgroup. In this way we get

AL and AS as the real linear spans of (respectively)

(8.6) xlax + bjl/2 , ax , ixl , i , j,l = 1,...,n ,7 J

and

(8.7) 4=1xkaxk +n/2 x1ax J-xJ

,az1 ,j<l,ax ,ixl,i ,j,1=1,...,n .J

Again declare S as joint domain of all operators (8.6),(8.7).

The exponentiations eLt, as in 0,6, for LEE AX, define operators of

GS. We again get a Campbell-Hausdorff formula, of the form

(8.8)

and where

exP (iDY

)-T9,z° .V , , where g=eY , z° =ei (y ) z'V

(8.9) DY,z,t,(p = Dxgaxp+16pq) + 2zpaxp +iYpxp + i(

For LE AS we get (8.4) again. The algebras ibGX may be characteri-

zed by the Lie algebras A% , just as in thm.8.1 for X--H :

Theorem 8.3. For X=T,S,L, the class W% is identical with the set

of AE L(H) allowing arbitrary finite application of ad L, LEE AK,

in the sense that II(ad Lj)A:S-S' maps S-*H , and extends to L(H).

The proof is similar to that of thm.8.1, and is omitted.

Problems: (1) Verify formula (8.8) in details, using ch.0,sec.6.

(2) Question: Can you relate the proof of the real Campbell-Haus-

dorff formula (0,lemma 9.5) offered in 0,9, with pbm. (1) above?

Chapter 9. PARTICLE FLOWS AND INVARIANT ALGEBRA

OF A SEMI-STRICTLY HYPERBOLIC SYSTEM;

COORDINATE INVARIANCE OF Oppx .

0. Introduction.

We use the natural symbol classes ipx, x=1,s, in this chapter.

First we ask for invariance of OpVz under global coordinate trans-

forms. Each TVX is invariant under conjugation with TE OX. For a

subgroup these are linear coordinate transforms: For GT get the

translations ezD, for GS the 'similarities' Tao,z,0 (all distances

multiplied by a constant), for GL get all linear substitutions.

Another subgroup gives 'gauge transforms' (conjugation by eitx)

The question about more general coordinate (or gauge-) inva-

riance of iIGX='ThX may be phrased as follows: LIGX is the set of AEL(H) with At eLtAe-LtE C_(R,L(H)) for every LEAX In new coordi-

nates y=q(x) the folpdes LE AX will transform to other folpdes L"

forming a Lie algebra AX-, where Lu=f H L" v=g , v=uo cp 1, g=f o q 1

Clearly eLt transforms to eL t, as solution operator of atu=Lu

atv=L"v. Thus "AE 'IGX" means A tetL Ae-tL E C"(&,L(H)), assuming

that u(x)-'u (x)=u((p(x)) defines an isomorphism of H. Thus, in new

coordinates the property "AE VGX" transforms to smoothness under

certain eLt, with more complicated LE AX-, depending on .

Vice versa, since a coordinate transform is invertib'le, "AE

4GX" (that the transformed operator is in LIGX) may be expressed as

smoothness of eMtAe-Mt for certain M - the transforms of LE AX

back to the old coordinates. For coordinate invariance we must

show that eMtAe_Mt E COO(R,L(H)) for the transforms M of epq or Tlpq.

Observe that our folpdes L=epq (or =ilpq) were either skew-

symmetric, or, at least, have L+L a multiplication of order 0.

We shall require that condition for M, as an assumption on the

coordinate transform cp. Then eMt is the solution operator of a hyp-

erbolic equation atu=Mu , of the form studied in ch.VI.

We always will require ME Opyce M+M E OpVcO so that theexistence theorems of ch.VI are applicable. For a detailed discus-

sion of coordiate invariance cf.sec.4. But observe that coordinate

invariance appears as a corollary of a study of invariance of Vxo

under the particle flows of the hyperbolic symmetric equation atu=

Mu . For eMtAe_Mt E C"O(R,L(H)) we first prove at aovtE C°°(R,VXo) ,

a=symb(A)E ix0 (sec.1 for x=1, sec.2 for x=s). The discussions

282

9.1. Flow invariance 283

are parallel to VI,6, but use a different approach: at solves the

PDE atat (k,a), k=-isymb(M). Repeated application of spq (or r1 pq

generates a system of PDE's. We use this for boundedness in t of

rkpjgjat (or rblp7q at ), and, more generally f p.gja1at, etc. In

sec.3 we apply our result to show that At=eMtAe-MtE CO'(&,L(H)).

We prove this for a rather general class of hyperbolic t'ide's

t plyce, K-K E Under proper assumptions on Ka u=iKu with KE O

we get eiKt Se-iKt=fix, as for Oplyco in VI,5. In fact, Optpx =iKt -iKt iKt -iKt

me Opyucme and even a Ae E C (&,OpVxm), as AE Opyurm

This invariance under particle flows and invariance of

under conjugation is the second topic of the present chapter.

Thirdly, we will look at particle flows for systems - in

sec's 5,6,7. focusing on the algebras Opyix. Here KE Opice will be

a vxv-matrix of pdo's. To include the Dirac equation (ch.10) we

require the system atu+iKu=O to be semi-strictly hyperbolic of

type e° , a generalization of a concept of VI,4: The eigenvalues

of (the essential part of) k=symb(K) must be real, of constant

multiplicity, but need not be distinct (sec.5).

It is no longer true that 1W%v (or Opxv) is invariant undermconjugation with a-iKt. Rather e-iKtAeiKt remains a 4 do in Optxvmonly if A belongs to a certain subalgebra P=P(K) of Opwxv. The exi-

stence of this invariant algebra P gives raise to physical specula-

tions in case of the Dirac equation, to be looked at in ch.10.

A necessary condition for a yado AE OpPxm to belong to P(K) is

that its symbol matrix a=symb(A) commutes with k=symb(K) , modulo

terms of lower order. While investigating invariance of P we will

find one particle flow for each of the distinct eigenvalues

of k(x,l) . The flows satisfy VI,(5.19) with k replaced by X .

For the Dirac equation we get 2 particle flows, for electrons

and positrons, resp. The flows will describe the exact relativi-

stic motion of the particle under the potentials imposed.

For 1-dimensional systems we get a theory parallel to that

of VI,5,6. For v>1 the different parts of the symbol matrix

will propagate along different flows. Roughly, an a(x,t) commuting

with will split into 'diagonal boxes' corresponding to the

eigenvalues of k. Modulo lower order each box propagates along its

flow, undergoing a similarity. For details cf. sec's 5,6,7.

1. Flow-invariance of VI..

In this section we assume a scalar real-valued symbol k=k1E

284 9. Particle flow and invariant algebra

yce. Let 'fit be the flow of VI,(5.18), for this symbol k. In VI,6

we proved that vt:142n-12n satisfies at=aovtE yicm for aE 141cm . Herewe prove the same for i1, under stronger assumptions on k.

From VI,(5.17) and 0,6 we know that at is given as solution of

(1.1) a tat + (k,at) = 0 , t E & , a0 = a ,

with the Poisson bracket (.,.)_(.,.)1 of VI,(5.16). Note that

(1.1) is a Cauchy problem for a first order PDE in Our con-

ditions on k will insure that at solving (1.1) gives a symbol atE

11 , for all tE & , a E 1 .

Assuming WE V10 we know that lk q a is bounded for finiteij

such products. We apply epq (or a finite product) to (1.1), for a

system of equations in a pqt=e pqat , a pgrst=ers epqat , etc. Then we

will derive apriori estimates showing that apgt, ... remain boun-

ded in t,x,l . As a preparation we note the following.

Proposition 1.1. (a) The expressions epq, of VIII,(5.17),(5.27)

may be written as Poisson brackets

(1.2) epga = {epq,a} = Epgl"Ix Epgjx a,,

with the functions

(1.3) Epq = - as p,q 2Ep0

= p , EOq =-xq

(b) For Poisson brackets we have Jacobi's identity

(1.4) (f,(g,h)) = ((f,g) h) + (g,(f,h)) , f,g,h E C"(1n)

Using (1.1), (1.2), (1.3), and Jacobi's identity, we get

(1 .5 ) a tapgt + (k, apgt) + ((E pqk) ,at) = 0 .

Suppose that the folpde Lpq:a - ((Epgk),a) may be written as a

linear combination of E00=1 and the other ejl , with coefficients

in tpc0. In other words, assume that, for p,q=0,...,n, (p,q)O(0,0),

(1.6) =arrr,s=0XpgrsErs 'where ?pgrsE "0

Then (1.5) assumes the form

(1.7)atapgt+ (ko'apgt) + D,pgrsarst =0

'p,q=0,...,n, (p,q)#0.r

Now (1.1) and (1.7) may be thought of as a linear system

of (n+1)2 ODE's for the unknown functions apq , along the charac-

9.1. Flow invariance 285

teristic curves of the common principal part of all equations,

That is, along the particle flow vt , above, we get a linear homo-

geneous system (with Xpgrs=X pgrs°vt

(1.8) rP,

Also we of course have the initial conditions

(1.9) apg(x,U) , at t = 0 .

Since the coefficient matrix ((kpgrs)) is bounded in all variables

x, E 1n , t E It , it follows that apqt are bounded in x, , over

compact t-intervals (cf. the detailed discussion in VI,6). We get

(1.10) y(t) _ 2 1a (t)12 IaIscy , y(0) = E 1a I2= 0(1)pp pg pq Pq

c independent of t,x,g, which may be integrated for y=O(ect). Thus

(1.11) at, axPat, a

qat, Epgat pa qat

xgaxPat' p,q=l,.... n,

are bounded in every compact t-interval, as In .

Clearly the procedure may be iterated, under proper condit-

ions on k. Once more, apply epa to (1.7). Write Epoepgat=apgpot :

(1.12)atapgpat + (k'apgpot)

+

(Epok,apgt)+

rs(Ep(?'pgrs)arst = 0

In the last term we use that XpqrsE pcOC VIO , implying epcpgrsE

VcO . For the fourth term we get

r

%parsersapgt=P'parsapgrst

using (1.6) again. With this, and the relations (1.5), (1.7) get

(1.13) atapgrst+(k'apgrst)+ n7, Xpgrsxxpaanxpat = 0

p,q,r,s=0,.... n, again with X pgrsnxpOE tpcO. Again the 'vector'

(a pgrst)p,q,r,s=0,..n satisfies a system of ODE's in t along the

particle flow with coefficients bounded in Again it follows

that the apqrst are bounded for x,1E &n, tE ICC 1. We even may dif-

ferentiate for t, deriving initial conditions for (atapgrst)t=0

from (1.13), for yet another system with bounded coefficients.

Thus t-derivatives are bounded as well. We proved, for m=0 :

Theorem 1.2. Assume that the real-valued symbol kE Vice satisfies

condition (1.6). Then at aovtE COO(2,i10), for every aE 1lm, mE V.


For general m we use VIII, prop.7.1: i41m is characterized by

rkpjqja=O(nm) , for all such products. For aE V1m , at as above,

again get (1.1) , (1.7) , (1.13), etc. The coefficients k, Xpgrs

Xpgrspa ' etc., are bounded as before. However, the initial values

are Thus, the resulting estimates now will be

(1.14) atapgt' atapgrst ' ... , = O(nm'v-t(x,))

However, in VI,6 we proved that the right hand side is

This completes the proof of thm.1.2 for general at.

The class of all symbols kE e satisfying (1.6) will be den-

oted VXne. Note prop.1.3, below, describing a subset VX eof

of Vane. For another such class see sec.4.

Proposition 1.3. Let *e be the class of kE Vce with

(1.15 ) k (x , ) = gkpgxp q+kd+km , kdE kmE 'yice2 , kpqE a

Then V), eC V%ne The (real-valued) functions of VXe

form a Lie-alge-

bra over M (k) under the Poisson bracket product (kl,k2), contai-

ning the a Jl of (1.3), and we have epq: VX e)XeThe proof is a calculation.

2. Invariance of Vsm under particle flows.

In this section we will extend the results of sec.1 to Vsm.

We introduce analogues to PXe, )Xne: Let We consist of all

(2.1) x00x + kd + km ,

with xjlE T, and symbols kd, km, as in (1.15). Let Vane consist of

all bE Vce with bpq=llpgb satisfying

(2.2) (bpq' ) - 4, q=OY pgrs'1rs ' Y pgrs E VC0

'W?ne is a proper subset of yice: For n=1, 14

and Vkne, as easily seen. Clearly VaeC V)è and yiaeC Vane

Theorem 2.1. Let the real-valued symbol be in ryane, and let

a E ilsm Then, with the characteristic flow vt = of (0.2)

and at = aevt , we have atE C00(1k,4-m).

The proof of thm.2.1 is analogous to that of thm.1.2. Define

(2.3),100

rljl= j,1=1,...,n,7101=-x1,

9.2. Invariance under particle flows 287

similarly than the ejl in sec.1, such that

(2.4) 1jla = (rljl,a) , j,1=0,...,n , r110 = x1 , '1O1

Proposition 2.2. The (real) linear span of the functions 171, as

well as the class yPe , are (real) Lie-algebras under (.,.) .

The proof is a calculation again.

More general symbols in Vane are used in the proof of thm.4.1

For hyperbolic systems in sec.5 we discuss the following:

Proposition 2.3. Let aE ism, r=(p,p), pE &, Then

(2.5) (nra) = Pnr-2e( 7' 71P (11pga) + 1100(1100a))p<q

The proof is a calculation

nr a) = P( x) p-2p ja l - P( ) P(x)P-2D[ja

I, j(2.6)

2 z= Pnr-2e pq(xpxgalxq - gxpal p)

xpgq(xpalx-xgalx + al, - gal, ) + 7, xg g(xpalx -tpal,

pq q p q P pq p pwhich implies (2.5).

For a more general class of functions in "ewe introduce

(2.7) tjl = Ijln-e , j,1=0,...,n .

Proposition 2.4. The class ipn0' of all functions c of the form

(2.8) o(tjl) + cd(x'U) +

with a polynomial O in too' t12 " " ' with complex coefficients,

and cdE yic_e2 , cmE Vc_e , is a subalgebra of 1Uco. For cE V(mo ' we

have b=cneE Vane, and 113 jq jb=nec" , c" E yiane j or all products.

Proof: The algebra property of yian0' (under the pointwise product)

is trivial: Clearly l;j1E qco , hence a polynomial in l;jl also be-

longs to c0 (Note that y,c_e, + ''c_e2 is an ideal of Vco). Now let

(2.9) b = x8O(t) + bd + bm , bd E VC e' , bm E "e2 .

We introduce a multi-index-notation, writing

a(2.10) O(t) _(a,a = 'Yan_Iale rla a = '1=0tj1 J1 , YaE T


with an (n+l)2-dimensional multi-index a = (ajl) . The next obser-

vation is that every application r1N rl b = b" again is of theJ_

-1 pjgjform (2.9) . Indeed Ipgbd and rlpgbm have the same property than

bd and bm, respectively. Also, calling the first term in (2.9) b0,

we get b0 E "e , hence Ip0b0 E ice, rl0gb0 E "e2 . One obtains

(2.11) loon Re= µ(nµe-2e2 nµe-2e') fljlnµe = 0 ' j,1=1,...,n ,

where µ E & , by a calculation. Thus we get

(2.12) 11jlb0 = 4=0 la -vYa(('1jln(1_v)e) rla + n(l-v)e (rljlla)).

For j,l s 1 the first term vanishes ; for j=1=0 this term is in

ice, + 4-e2 , hence will go together with bd+bm . We must focus on

_ Pq(2.13) n(1-v)a nj11a

= (1-v)a 4q=1a'pglaE (rljl,ylpq)

with Epq having (p,q)-component equal 1, all others zero. By prop.

2.2 (rl jl,rlpq) is a linear combination of the rlrs. Thus njlb0 isof the same form as b0 except for an additive term in vice, +yice2 .

Hence ihlpjgj b is of the form (2.9). We are left with showing that

(b,.) is a combination of the rlpq with coefficients in Pc0 .

The terms (bd,.) , (bm,.) are handled as those in prop.1.3,

using VIII,prop.6.2. For one of the terms of b0 we get

(2.14) (a (1-v)e 11a,a) = a) + n(1-v)e zpgyla Epgrlpga .

The last term indeed is a combination of the gpga, coefficients

E tpc0. For the first term apply (2.5), with r=(1-v)e. Q.E.D.

Define Vanes nm-eyiane, and yianm'=nm-evane' , for m--Re. Thenyianm consists precisely of all cE 1pcm with c " =rl lpjglc satisfying

(2.15) (C- ..) = jl 1jl . Y jl E m-e

as is easily verified, using (2.5). Moreover we note that Van0

and Van0' are algebras, with the pointwise product. For Van0' this

was part of prop.2.4. For c,d E Van0 one finds that

(2.16) (cd,.) = c(d,.) + d(c,.) = l(cOJl+dY jl)1jl

with evident notations, where we get cbil+dyjl Eilio_e . Similarly

for (cd)" , which gives the algebra property.

Proposition 2.5. Every continuous root of

9.3. Congugation with eiKt 289

(2.17) 0(X) = L=09 XJ ON = 1 , 8j E po 0 ' j=0,...,N-1,

may be extended to a symbol XE yxt0 , with all ThlPjgj

XE V= O, pro-

vided that for large IxI+ItI all roots have constant multiplicity

and their mutual distance is bounded away from zero.

Proof. For a root with constant multiplicity p we can write

(2.18) (2xip)-1f i0'(t)/0(t) ,

where E>O is fixed -say, as half the minimum distance between two

roots. But we have expansions

(2.19) .(6 j , ) _ lY jpg1lpq ' Y jpq E "-e ,j=0,1,...,N

Hence, with polynomials Ypq 2'jpgXJ,

(2.20) (0(t),.) = lYpq(t)Iipq 2YPq'(tApq

Substituting into (2.18) one obtains

(2.21) ( X, ) _ jl1jl , Xjl = (2xip)-1 f tdt (Y 3 1l0-ely jl)/02

with path of (2.18). The path is kept constant during differentia-

tion, the integral being locally independent. For t on the path,

(2.22) Ie(t) I = Iri(X j-t) I a EN > 0 ,

since by assumption the path has distance aE from all roots X.

This allows to estimate Xj1 uniformly in x, , for large IxI+hi.

Since 0 and 0' are Vc0 , but Yj1E IPc_e , we get kjl=O(x-e).

The formulas for Xj1 may be differentiated under the integral sign

for Xj1E tpC-e. Also we get expansions for rhlp q X : Apply rjpq toj

J

(2.18), using estimates as above. Q.E.D.

Corollary 2.6. Let k E tVaxpe . Then we have

(2.23) (k, a) EVsm+(µ-1)e for all a E Vsm , m E &2

The proof is a consequence of (2.5) and (2.15).

3. Conjugation of OptVx with eiKt , K E Op1ce

With invariance of the classes Vxm x=1,s under the particle


flow of kE'W%ne or kE Vane established we now prove the invariance

of OpPxm under conjugation with eiKt . The results will be similar

in form and principle to VI,thm.5.1. Existence of the group {eiKt}

of operators S-S (or or with the spaces of 111,3)

is insured by VI,thm.3.1. It would be easy to carry the results to

the case of a solution operator U(t,t) of u'+Ku=O, with K=K(t).

In thm.3.1 we work with k0E 1 e, focusing on a sim-

ple nontrivial case. Slightly weaker conditions on k will be nee-

ded in sec.4, to prove coordinate invariance (cf. cor.3.4). A res-

ult for systems, requiring the classes Wne is discussed in sec.5.

Theorem 3.1. Let K=k(x,D), where k is of the form k=ko+x, with a

complex-valued xE0 , and a real-valued koEa,Then,

for every A=a(x,D)E '1m , me &2 ,we have Ate-iKtAeiKt E Op1lm

tE I. Moreover, we get Ata(t,x,D) with

(3.1) z E 11-e, n yl-e2

Also, At(-= C00(it,Op1xm)), and atAt=(-i ad K)JA .

Proof. We retrace the proof of VI,thm.5.1, taking special care of

the remainders, due to less perfect pdo-calculus of VIII,7. Let

(3.2) P(t) = eiKtat(x,D)e iKt

where aE Vxm, x=1,s, ata,V t At=at(x,D). Here we use thm.1.2 (or

thm.2.1) to assure that atE yurm

and at(x,D) is meaningful. Note

that this is the only direct reference to x=s or x=1 . We get

(3.3) dP(t)/dt = eiKt(i[K,At] - (k,at) (x,D))e iKt ,

where the operators are S -> S . By VI,thm.3.1 at(eiKt) exists in

L(Hs,Hs-e) , and by sec.1 (or sec.2) atat-(k0,at) exists in yucm

Hence dP/dt exists in L(HS'Hs-m-e), for all s, and (3.3) is valid,

using III,thm.3.1.

Now we use our calculus of ,do's. VIII,prop.7.4 yields

(3.4) [K,A]=cA(x,D) ,

where (k,a)E 1)xm+e-re'-r'e2' r+r'=j, and the remainder RN has sym-bol in Vxm+e-re' -r'e2 ' r+r'= N+1, if the first N terms are used.

It is important that, for our present k, we even get (k,a).EVxm-re' -r' e2 ' r+r' = j-1 , hence RNE Op Xm-rei -r' e2 1 r+r' =N . For x=1,j=1 get 1=-(Egpa) =-cgpaE yrxm, while (kd,a) , (km,a) belongto Xm+e' -re' -r' e2 ' and Vxm+e2 -re' -r' e2 l resp. With r=1 (r=0) getall terms E Vsm, as stated. Similarly for j=1, x=s. For j>l the

9.3. Congugation with eiKt 291

first sum (1.15) gives 0, since Epglxx epgl -qpglxx-q pgI -0. The

other terms in (1.15) go as for j=1. To summarize:

Proposition 3.2. Under the assumptions of thm.3.1 we have

3 3(3.5) cA 4=1(-i)7/j! (k,a)j+rN , rNE m-re'_r,e2 ,r+r'=N,

where the j-th term belongs to Vx m-re,_r,e2, r+r'=j-1.

Applying prop.3.2 for N=1 we get

(3.6) icA (k,at) nvxm_e2At

Note that (3.6)implies )(x))-1/2).In fact,

we get yE 'ptm_e/2. Moreover, we know from sec.1 or 2, and the cal-

eculus of ydo's between y'c and ix that y is continuous as a map

Him-e' (or 4w m_e2 ). Actually, we even get it COO: One knows fromsec.1 or 2 that at is C" , and the asymptotic expansion (3.4) (or

Leibniz-Taylor-expansion (3.5)) may be differentiated arbitrarily

for t. Then prop.3.2 may be applied to the differentiated expan-

sion. Hence we even get y E C00(I,Vtm-e/2 ). Again this gives

(3.7) P(t) = A + JtdieiKty(t,x,D)e-iKt

0

Or,

(3.8) e-iKtAeiKt= at(x,D) - ftdieiK(t-t)Y(t,x,D)eiK(t-t)

0

This is a decomposition like (3.1), but the reminder (second

term) is not in iyido-form. It is norm continuous in L(HS'Hs-m-f)' f

=e', e2, e/2. Also,d't(e_iKtAeiKt)

exists in L(HsHs-m), by (3.8).

(3.8) may be iterated: Noting y satisfies (3.6), and even yE

C00(&,Vxm_f), f=e', e2 , we get, setting

(3.9) (t,x,D)- f dx'eiK(x' x)W(x,,t)eiK(x x0

with a ydo W = w(x',t,x,D) obtained by substituting y for a into(3.7) . Clearly it follows that wE C00(12,1Vxm-re' -r'e2) , r+r'=2using (3.6) . Also, yx E C00(1e2,'wxm_f), f=e' , e2 .

With (3.9), for x=t-t, in the integrand of (3.8) we get

(3.10) e-iKtAeiKt = at(x,D) + zit(x,D) + W1t'

where Wit E L(HslHs_m+re'+r'e2),while the symbol 1;1t is in

C0D(H,1xm-f) , f=e' e2 . An N-fold iteration then yields

(3.11) aiKtAeiKt = at(x,D) + 4=1zjt(x,D) + WN ,


with symbols zjE C"(&,"m-re,-r , e2), r+r'=j , and with a remainder

in O(m-re'-r'e2) , r+r'=N+1 . Moreover, the remainder will have

all derivatives for t of order up to N+1 existing in norm conver-

gence of every L(Hs'Hs-re'-r'e2)' since it is an N+1-fold iterated

integral over a bounded, strongly continuous integrand.

Finally we take the asymptotic sum

(3.12) zt = Lj=lzjt (mod Vt-00)

It exists, because tjtE yet-je/2 follows as above. Then e-iKtAeiKt

- at(x,D)-zt(x,D)=Vt clearly is of order --. Its t-derivatives of

all orders exist and are of order -- again. Thus Vt vt(x,D), vtE

Cl2n(1Q,S(& )). Also all t-derivatives of the asymptotic sum z

-iKtt iKtexist in 4wm-f, f=e', e2. Writing zt+vt we get a Ae

=at(x,D)+z(t,x,D), which amounts to (3.1). In particular, we have

verified all the properties stated for z in thm.3.1.

With differentiability established we clearly get atAt=

-i[ K,At] =-i(ad K)At, where [ K,At] E Opyurm again. This may be ite-rated, to complete the proof of thm.4.1.

It will be useful for sec.4 to reexamine the proof of thm.4.1

for a possible generalization to the case where the first sum of

(1.15) (or (2.1)) is replaced by a linear combination of terms

(x)a(s())T;q , with s(x)=(X) , and a function cE C" (B?), with theclosed unit ball B?C 1n , as in IV. In other words, let

(3.13) k= (x) q +km+kd, kmE '°e2 , kdE yice1 , aqE C"(Bi),

Let V%Ke be the class of such functions. Clearly yueC V*eC yip,' e'

Proposition 3.3. We have V%xeC 'Xne. In fact, not only (epgk,.),

but also (k,.) is a combination of the ejl, with [c0-coefficients.

Proof. It suffices to focus on the case

A calculation shows that epq:w7,,Key Vx1ce . Thus we only must show

that (c, .) =2' a pq . But (al, . ) =aa -26a =aa -ja eq pq q xq lxl q1 xq lxl ql

+Y,alx xlax . Here alx E pco, so look at sax -yalx xlax =(x)-laax1 q 1 q 1 q q

(s()+xl(sla(s)-((x)a(s))Ix1

)axq. Finally, ((x)a(s(x)))lx1=(sl(y)M

+OAlxm(s(x))(slm slsm(x)), so that 2lxl(sla(s)-((x)a(s))lx1)=

m,l- xlalxm(s(x))(6lm °lsm)=(x)2,slall(s(x)) . Q.E.D.

9.4. Coordinate and gauge invariance 293

Looking at prop.3.3 and the proof of thm.3.1: In the case x=

1 we clearly get at well defined again, and E E CC00(&,1V1m) . Also,

(k,at)=r pgspgatE lP1m again, if we weaken the assumptions, requi-

ring koE.tVXxe only. Formula (3.6) remains true as well: We need

(k,a)2=jl

for again. First

observe that alx j=sja(s)+sjalj(s)-sjTalm(s)sm is a sum of function

a(s(x). Thus we may look at Ta(s(x))lx1 gaxla =20(s(x))1xleglaj j

+Y,O(s(x))lxixlaxga J . We get a(s(x))lxl=yalm(s)(slm slsm)/(x)E

1Vc-e2 , hence the first term is in Vlm-e, n vim-e: For the second

term we get p1°lm(s)(blm slsm)=omalm(s)/(x)2 , Using VIII,(6.12)

we get the proper estimate for (3.6) for the second term as well.

We have proven:

Corollary 3.4. In the case x=1 the statement of thm.3.1 is still

valid, if "ko E iV%o " is replaced by "ko E iVXxo " .

Indeed, the iteration and asymptotic sum leading to (3.1) no

longer depend on "ko E 1VXo ".

4. Coordinate and gauge invariance; extension to S-manifolds.

We now will look into coordinate invariance of the two alge-

bras 1I.L and Y14S and more generally, Op1V1, Opys. Given a globaltransform x=q(y) with a diffeomorphism sp:&n- In

, inverted by 1V:

&n- In :We have sp(x(x))=x(W(x))=x, xE &n. Assume (the components

of) p(x) , x(x) are Coo(&n). The Jacobian matrix ((a(P,/axl(x))) _J

J(x) is nonsingular, together with ((axj/axl(x)))=J1(T(x)), xE &n

Consider the substitution operators TT, TX defined by

(4.1) (Te)(x) = u((p(x)) = (uo(p)(x) , (Txu)(x) = u(X(x)) ,

for functions u over In . Assume 0<c, s ldet J(x)Is c: , with con-

stants, to insure that TT TX : H -> H

If a folpde

(4.2) L = Yaj(x)axJ + a0(x) , aj E C00(In) ,j=0,1,...,n


is given, then we get a transformed folpde L" = TgLTx . We get

L = - j(x)ax +a"0(x) , a"0 = T,,a0(4.3) j

a j(x) = Y,(axj/ail)((p(x))al(T(x))

The exponentiations eLt and eL t are related by

(4.4) eL" t = TtPeLtTX

since the PDE atu=Lu transforms into atv=L"v , with

To prove coordinate invariance of iIGL or t1GS we will use

thm.3.1 or cor.3.4, with the ,do K=k(x,D)=-iM , M as L- in (4.3),

p replaced by x=q1, with L=ixq ax xgax +2Spq, in case ofWL.P p

For WS we use L=ixq ax , xpax

-xgax , and Ex3 +Z . We mustpq p J

verify that the transformed expressions satisfy the assumptions.

as outlined in sec.0. We consider S-admissible transforms, as in

IV,3, only: With s(x)=5, s:I -B1, of IV,(1.5), assume that

(4.5) tP = s-lo'ylos x=c 1 = s-lo eos

with diffeomorphisms p:B1 u.Bl , O:B1 a B1 , inverting each other.

Under this general assumption, if it can be shown that T(P LIGXTX C

iIGX , for %=L or X=S , it follows at once that also TTOp7UxmTx C

Opyurm because we already know from IV,thm.3.3 that T(,OppcmTx C

OpVcm For AE Op xm we have B=AII-MEE Optpx0 ='WX. Then TWBTXE tIGX

follows. Also, T BT =(T AT )(T P_ T ), where the second factor

belongs to Oppc mC OpVx_XAlso T P T is inverted by T P

MT E

OpVxm. Hence T(P ATx = (TgBTx)(TTPMTx)Ex

For thm.4.1 below we must focus on the expressions M=Mpq

(4.6) MOq ixq(x) . Mp0= Irojlx (x(x))axJ p j

Mpq 2xq(X)Tjlxp(x(x))axj + ZOpq p,q=1.... n

in case of x=L, and MOq , Mp0 , I Mpq Mqp, I:sp=qsn, for X=S.

Here we first apply IV,prop.2.1: With our assumption (4.5) on 9,

_we have TjIx os-IE C"(B?) , and x=s-IoOos , hence Tjlx (X(x))P p

l )o6)os(x) =a.

(s(x)) , with=TjIx

o(s_0 Oos)(x) =(((Pjlx os- 1pP

9.4. Coordinate and gauge invariance

ajp=(cPjjx os-1)oOE C°'(Bcl)p

295

On the other hand, (x)-1cp(x)= yr(s(x)){(x) 1- V(s(x)) 2}-1 ,

where the second factor may be written as 1,I-S, }1/2

E C"(B?) ,-y(s)2

1

just as in IV,(2.8),(2.9). This shows that Mpq ikpq(x,D)+x0, where

(4.7)k0g

(x)a0gos(x),kpg

are symbols of tVXxe of sec.3, so that cor.3.4 applies. We proved

Theorem 4.1. Let the diffeomorphism q:&n -in and its inverse

function x:&n -In be of the form (4.5) with s(x)=`xl, and diffeo-

morphisms y,:Bl-.Bl, O:Bl -Bl, yp°8=id. Then the transforms T and Tx(4.1) leave i1GL and every Opyrlm, mE &2, invariant. That is,

(4.8) TTi1GLTx=TxtIGLTgi1GL , TT OptUlmTx=Tx0ptVlmTgOpyilm , mE &2

Theorem 4.2. Assume that the diffeomorphisms T,x satisfy

(4.9) p(x)=gx+w(x) , x(x)=g1(x)+v(x) , gE GL (&n) , w,vE "o

Then (4.8) is valid - i.e., TT and TX

leave OpyJlm invariant. More-

over, if in addition we have goo , 0<aE & , o orthogonal, then

also WS and OpVsm are invariant under T(P and TX

.

Next let us turn to gauge transforms. A calculation gives

(4.10) L= eiµ(x)Leiµ(x) = L + i

Pp(x)µlxp(x)

Applying our arguments to the generators ixq, ax , xgax +P p

26pgLpq again we now must prove eMtAE-MtE C00

&,L(A), for AE t1Gx,

where M=MpgLpq+Rpq , tOq 0 , tLp0=iR I xp , µpq I A I XI xp.

Theorem 4.3. Let the 'gauge function' µ(x) be real-valued and let

a (x) E yice2 . Then Op1Jixm, mE &2 , X=S ,I, are invariant under

(4.11) A -* e-'R(x)AelR(x) .

The proof follows from thm.3.1, for m=0. For general m we

still must show that a i L( D) se' E Vcs , sE I . We leave this tothe reader; the proof is not difficult.

Finally, after discussing coordinate invariance, it is clear

that the class OpVlm of natural yxio's may be considered on a mani-

fold with conical ends, just as earlier Op*m'.,s (or OpVcm).


Repeating the discussions of IV,4 with W1m instead of gene-

rates classes called LLm of 'natural pdo's' on an S-manifold 11 .

On the other hand, we do not expect invariance of OpWsm un-

der general S-admissible transforms (4.5), although, evidently,

thm.4.2 implies invariance under local transforms extended as a

multiple of the identity outside a compact set.

Our use of Opps in Dirac theory might profit from introduct-

ion of Op1s(sz) for an n coinciding with &n outside K E 11 .

Problem: Give a definition of Opps(11), for a as above, using a par

tition 1=Exj with x'=1 near -, and also, by smoothness properties.

5. Conjugation with eiKt, for a matrix-valued K=k(x,D) .

In this section we take up the question of extending thm.3.1

to the case of a vxv-matrix K = k(x,D)=((kj1(x,D)))jl1=1,...,v of

of pdo's Kj1 = kj1(x,D) . We shall give preference to the class

yes , from now on, but note that most arguments will also work for

'Pl. Generally we assume kj1E Vae (cf. sec.2). Instead of requiring

(5.1) atu -ik(x,D)u = 0 , u=q at t=0 ,

to be strictly hyperbolic of type a°, as in VI,4, we will accomo-

date the Dirac equation by allowing multiple eigenvalues, but of

constant multiplicity for similarly as in [CD] .

With comparison operators A=nea(x,D), a1=e,e',e2, as in VI,4,

we call equation (5.1) semi-strictly hyperbolic (of type e°) if

(i) We have kea=neako, KE Wcea-e,

with entries of the vxv-matrix kea of the form (2.1):

(5.2)kea jl- ,q=oKpg11P +kdjl+kmjl, KpjE T' kdjlE tee' ' kmjlE e2

where all but kdjl or kmjl vanishes in case of a°=e', e2, resp.

(ii) The vxv-matrix is diagonalizable for all fzIxl+

f1 I Zrl0, sufficiently large q0. All eigenvalues Xjof kea are real, and of constant multiplicity vj (independent ofx,i;), as f2 JxJ+fi ICI arl0

Well known perturbation arguments imply that the X j are Co

functions of x and . The matrices kea and k0=kean-ea have the

same spectral projections, and the eigenvalues of k0 are given

by lAj = kjn-ea . We may arrange for

(5.3) vl+...vP=v, f2 lxI+fi ZtIO

As our third condition, we assume that

9.5. Conjugation for a matrix-valued K 297

(iii) 260 , f2JxI+f1Ifl 2110, j=l,...,p-l

with some positive constant 60 independent of x, .

We only will admit the weight functions ne, mentioned, al-

though others, such as nel+ne2 might be useful as well. Mainly

e°=e' will be of interest, for the Dirac equation.

Theorem 5.1. If equation (5.1) is semi-strictly hyperbolic of type

e° then the solution operator eiKt of (5.1) exists as an operator

in 0(0). Moreover, the partial derivatives (5.4) exist in norm top-

ology of L(Hs,Hs-je,). For every sE &2, and j=0,1,2,..., we get

(5.4)at(eiKt) = KjeiKt E 0(jf)

(5.5) r_e(x,) , r_eE 1Va_e

using the spectral projections to of

f2 xl+f1 I 22rI0 , extended to symbols (with coeffivients) in Vco.The discussion follows the proof of VI,thm.4.4, with small amend-

ments, left to the reader. The point is formula VI,(4.19), i.e.,

(5.6) 1x1+1 V >110,

describing the spectral projections in >qo . Then the

Pi must be properly extended to H2n, and r_e must be chosen to

satisfy the cdn's of VI,4.

Proposition 5.2. The eigenvalues Xe, , f2 IxI+fk Ill>2io , of ke,extend ty symbols X jE lPone (cf. sec.2).

Proof. The eigenvalues µj solve the algebraic equation

(5.7)"J=0

61(x,U)ptj = 0

where the coefficients Aj are polynomials in the coefficients of

k0 . If e° #e then we just get µjE Vc0, Xj=µjne, E Vce, , by an argu-ment as for VI,prop.4.3. For e°=e apply prop.2.4 for 01E 1on0',

hence also µjE Van0', by prop.2.5. Note that the assumptions hold,

by (i)-(iii) above. Thus we get Xj=neµjE vane, as stated. Q.E.D.

The following example shows that the class Oppsm of all vxv-

matrices A=((A )) of ,do's A. E Opts is not invariant under the

conjugation A-*Ate-iKtAeiKt . Letm

(5.8) f -()J ' (0 0)


Then clearly k E °e , and a E But a calculation shows that

\(5.9) e-1KAe1K =( 0 0) ,

Clearly P = p(D) , p( ) = e21(x)

with

E Vt0

p = -21(D)

but p 0 Vs0

On the other hand, as a simple consequence of thm.3.1 we get-iKt iKte As E OpVsm if and are diagonal matrices for

large IxI+IgI. If only a and k commute for large IxI+l I we can

expect (3.8) with an error term Zt (the integral at right) of low-

er order, not necessarily a y,do (cf.also [T13]).

Thm.5.4, below, shows that even the error Zt becomes a 4do

E Op sm-e° ' e° =e` , e2 , if only a suitable 'correction' of lowerorder is added to the symbol a(x,i;) commuting with k0(x,t) (that

is, if we let a=q+z, with [k0,q]=0, and suitable lower order z ).

From now on we always assume that (5.1) is semi-strictly hy-

perbolic of type a°. We shall see that a subalgebra P=PKC Opsv

can be defined, essentially by the property that its Ado's remain

4c1o's, of the same symbol type, when conjugated with eiKt

For mE H2 we define f PKm as the class of all A=a(x,D) E

OpVsm such that At of (5.10) belongs to Opysm, and, moreover that

(5.10)e At = eiKtAeiKt E C-(&,OpjsV) , in case of a°=e.

For e° =e' , or e° -e2 we replace (5.10) a by

(5.10)e° atAt E C (H,OP1S jj = 0,1..... .

Then we define, for a°= e , e' , or e2 ,

(5.11) P = PK =U

PKKmEV

Proposition 5.3. The class P is a graded algebra invariant under

ad iK (as in VIII,(8.4)), i.e., (ad iK)A=at(eiKtAe-iKt)It=o

)

Especially, if AE Pm, BE Pm , , then AB E Pm+m' Moreover, if AE Pm,

then (ad K)A E Pm-(e-e ,)' Moreover P0, with the Frechet topology

of VIII,6, is an (adjoint invariant) i*-subalgebra of L(H).

Proof evident. Clearly B=(ad iK)A exists for AE Pm, and Bt=e iKtBeiKt=_At E C00(H,Ovm-(j+1)(e-e°)

Theorem 5.4. Let (5.1) be semi-strictly hyperbolic of type a°.

(1) For each vxv-matrix-valued symbol q with

(5.12) q E iism , [k0(x,),q(x, )] = 0 . 1x1+1 U Z iii

there exists a symbol zE 1ysm_e such that A = a(x,D) , with a=q+z

9.5. Conjugation for a matrix-valued K 299

is an operator in Pm . That is, in particular,

At = e-itK AsitK E OpVsm, t E & , At = at(x,D)

(5.13)

atat E C'(L,'V)S _7(e-f)) , j=0,1,2..... .

(2) Vice versa, if A = a(x,D) E Pm is given , then there

exists a decomposition (valid for all t E & )

(5.14) at qt+zt ztEVSm_e , )]=0 ,2yl0

Moreover, the decomposition (5.14) may be differentiated for t ,

for the corresponding decomposition atat atgt+atzt of the symbol

aja of ((ad. )iA) . In particular, ajz Vsv

t t iK t t t m-j(e-f)-e(3) If, for any q with (5.12), the symbols z1, z2 both sat-

isfy (1) then z1-z2 satisfies (5.13) for m-e , instead of m .

We shall lay out the proof of thm.5.4 in the remainder of

this section, and will finish it in sec.7, after discussing some

auxiliary results on a commutator-differential equation in sec.6.

The particle flows of the eigenvalues . (x,t) will be impor-

tant. Even in the strictly hyperbolic case, when all eigenvalues

are distinct, we will have to solve a succesive sequence of matrix

commutator equations, for careful alignment of the correction z.

Moreover, for multiple eigenvalues, not only will the symbol flow

along these flows, but, in addition, there will be a 'similarity

action' within the eigenspaces of k(x,T;). For details see sec.6.

For a different interpretation for the Dirac equation see ch.10.

Discussion of (3). Prop.5.3 implies that z3(x,D)=(z,-zz)(x,D)E

E Pm-e . Thus, indeed, z3 allows a decomposition (5.13) for m-e.

Discussion of (2). First let a°=e. For an AE Pm we get At at(x,D),

atE C"(2,ivsv) . Then again (5.1) yields

(5.15) At +i[K,At] = 0 , t E & , AO = A .

(We first get (5.15), applied to uE S , due to differentiability

of a*iKt in L(Hs

H s-e) (thm.5.1). Then, since all operators invol-

ved are ydo's, we get (5.15) as an equation for Ado's.) Using cal-

culus of ,do's we may translate (5.15) into symbols :

(5.16) 0 = at + i[k,at] + (k,at) -i/2(k,at)2 + ...(mod S)

Here the Poisson brackets are formed with matrix multiplication :

(5.17) (a,b)=albbix-bl alx , (a,b)j= Ie=j8'(a(e)b(e)-b(e)a(0))


The commutator [k,at] normally does not vanish. However, the

discussion of VIII,prop.7.4, carries over with the same proofs,

except there will be an extra term [k,a] in the third formula

VIII, (7.14) . Moreover, as in sec. 3 , we have (k, at) jE "m-rei -r' eZ 'r+r'=j-1, under our assumptions on k and at. Thus (5.16) implies

(5.18) [kf,at] = rt E 1Vsm .

We regard (5.18) as a commutator equation for the unknown

matrix function at . Here we know that (5.18) admits a solution.

With Xi and pj as constructed, we get

(5.19) at = L4=lp.33atp.

+ j7lpjrtpl/(kj-kl) = qt + zt

(Just note that at Ppjatpl , and that pj[k0,at]Pl =(1j-X1)pjatpl

leading to pjatpl = pjrtpl/(Xj-X1) ,as j#l , and the solvability

condition (5.20) (which here holds automatically).)

(5.20) pjrtpj = 0 , j=l,...,p .

Notice that (5.19), with qt and zt equal to the first and second

sum, respectively, gives a decomposition (5.14), as desired, using

that p jE 1Vc0 , (X j-T,)-lE 1Vc-e , (possibly after a correction forsmall f:IxI+f,ll;I), using (iii) above. Moreover, since pj , Xj are

independent of t , we may differentiate (5.19) for a decomposition

of the symbol of Bit , Bj=(ad iK)1AE Op1Vsm-j(e-e,) using

(5.10). This proves (2) for e°=e.

Next consider e° =e' , noting that the case e° =e2 handlessimilarly, with x and reversed. Again get (5.16), where now wehave kf=kdE 1Vice, , however. Also, atE 1Vs -e: , by (5.10)e' Hence,

(5.21) (k,at) j E 1Vsm+e' -re' -r'e2 r,r'2-0 , r+r'=j

The proof runs exactly parallel to the proof of prop.3.2. (One

gains the slight advantage of an improved multiplication order

because the term km is missing.)

Now, assuming that k=ke, +K , xE 1'e' as required by (i) ofour e'-semi-strict hyperbolicity, we get (5.18), with rtE

But we now also have Xi E 1Vice, , instead of Xi E 1Vice . Thereforethe second term at right in (5.9) still is a symbol in 1Vsm-eagain, and (2) follows for f=e3 as well.

Discussion of (1). First assume a°=e. For a q satisfying (5.12),

9.6. Commutator equations 301

if z E yism-e exists with A=q(x,D)+z(x,D)E Pm, then get (5.16) for

for at qt+zt. Separating symbols in (5.16) of order m from those

of lower order (with (5.12), and cor.10.5), get

(5.22) i[ kf , zt] + qt+ (kf, gt) + i[ K , gt] = 0 (mod 'Wsm-e, fl yisvm-e2

(In particular we know from (2), already proven, that ztE Vsm-e

Also, the only additional term of (5.16) to be taken into (5.22)

for a relation mod Vsm-e will be -i/2(k,q )2. We will not use this

term, at the expense of a weaker (5.21), resulting in slower asym-

ptotic convergence of the series for at, similar as in thm.3.1.)

Vice versa, we start the construction of a symbol z with the

attempt to solve (5.22) exactly, not mod lower order, assuming qt

E 4wm given. Rewrite (5.22) as commutator equation for zt

(5.23) [kf,zt] = -[K,qt] + i(qt + (kf,gt)) = cpt

Assuming that(Pt

is known, for a moment, (5.23) implies

(5.24) zt = Llplztpl + i j7, pj (qt+(kfgt)+i[K,gt])pl/(),j-kl);dl

This solution is valid only under the condition (5.20), i.e.,

(5.25) pl(gt + (kf,gt) + i[K,gt])pl = 0 , 1 = 1,...,v' .

In sec.6 we will investigate commutator equations of the form

(5.18), with solvability condition (5.20) as in (5.25). In sec.7

construction of zt and the proof of thm.5.4 will be completed.

6. A technical discussion of commutator equations.

In this section we consider an equation of the form

(6.1) [b,w] = d ,

for general vxv-square matrix valued symbols

where b,d are given and w is to be found. Assume that b,

d are C" for R C &2n, and (t,x,t;)E 2xcz Let the matrix b

be diagonalizable, have real eigenvalues, and eigenspaces of dimen-

sion independent of x, . As used before (sec.5) the eigenvalues

A,

3

may be arranged as C"(c)-functions: a.. is a simple root of

q(p-1)(A,)=0, with q(A)=det and the multiplicity p of

X. Hence aq(p-1)(Ai)#0. The implicit function theorem yields )vj

E C°D , locally. The real distinct eigenvalues may be ordered by

size, giving globally defined C"(a)-functions. For X=Xj ,


(6.2) p = i/2nJ (b(x,t) -µ)-1

dµ E C"(n)It-?(x, )1=e

with a=e(x,t) > 0 sufficiently small, is the eigenprojection.

For solvability of (6.1) it is necessary and sufficient that

p(x,t)d(t,x,t)p(x,t)=0, x,t E a , for each eigenprojection p of b.

(Indeed, if (6.1) is solvable, then pdp=[pbp,pwp]=[Xp,pwp]=0. Vice

versa, if the latter holds for all pj=p, then (6.1) is solved by

(5.19) with Xj , pj , j=1,...,p, the distinct eigenvalues an pro-

jections of w (and at replaced by w). The first sum in (5.19) is

an arbitrary matrix commuting with b. All solutions of (6.1) are

given by (5.8).)

We plan to solve (6.1) for d of the form (6.3), with given

(t-depedent) symbols g, d^, and d Assume g=g(t,x,t) commutes

with b(x,t), for all (t,x,g) E lxtz, while d^ (t,x,l), d" (t,x,t) areC"-matrix functions, and (.,.)=(.,.)1 is the Poisson bracket.

(6.3) d = g + (b,g) + [d^ ,g] + d" , g = atg .

For given X , p let T1(x,i;),...,q) p(x,l;) be

a bi-orthogonal pair of bases of the eigenspace S=S(x,l)=im p(x,l;)

(such that 1cp1=6 ill for all (x,t)E t , and that also V j,VE C00(c) )Consider the local matrices Y=((Yjl)), S^=((6jl)), b"=((% 1 )) of

g, pd^p, pd-p, (all leaving S invariant): In detail, y-y(t,x,l;),

6^ (t,x,i) , S" (t,x,l) are defined by

(6.4) pgp- Y lp j 1 , Pd"j,l j

P= 6j1Pjl Pd P sj1

with Pjku = cpj('y u) , u E CV , Vk = , for all (x,t) E tz

Proposition 6.1. The condition pdp=0 with d of (6.3), and [b,g]=0

for all (t,x,t) E 2xf , translates into a differential equation

for the restriction g0(t,x,t) = g(t,x,i;)IS(x,l;) of g onto the

eigenspace S of . , of the form

(6.5) p(gp)'PIS+[v,g0] +d"0=0, d"0 -pdpIS ,

where

(6.6) n , of = at + %Itax - xIxat , v = (pd^P + P(PItbIx)P)IS

Moreover, with the matrices y , 6A, 6" of Y0 , d^

0=pd^ p I S ,6-0

with

9.6. Commutator equations 303

(6.8) a=((e jl)) , O jl=-Vi'(p1+

l , 11111=d/ds.

The restrictions g0 , d^0 ,

d"0clearly are linear operators

of S . Prop.6.1 implies that, under above assumptions, cdn. pdp=O

involves only g0, not the restrictions of g to the other eigenspa-

ces, hence a 'decoupled' system of p2 DE,s results for each eigen-

space of b, where p=dim This is a system of PDE with same

principal part, translating into a first order system of ODE along

the flow of the common principal part, in the sense of sec.1.

Let us first focus on (6.5). In this discussion we again con-

sider the full set of eigenvalues ?1,...,kp of multiplicities pj,

with projections p=pj , and gj=gpj=pjg . From pkgr 0 , k#r, we get

(6.9) PkgrPk = -PkgrPk = 0 ' PkgrlxPk = 0 , k#r .

We have g = 29k ' b = DrPr , hence

(6.10)Pk

Pk = T 1 + T2 '

where

(6.11) T 1 = Pk( -M g) pk = Pk

using (6.9). Also, as j#ky6r#j , using gjlxPk = -gjPklx .....

(6.12) Pk(Pr,gj)Pk = 0 .

Similarly, as j # k = r , using pkpkl =

(6.13) Pk(Pk'gj)Pk = Pkl gj(1-Pk)Pklx =0.

On the other hand, for j = k # r , and j=k=r ,

Pk(Pr'gk)Pk =

(6.14) Pk(Pk'gk)Pk = Pkl(1-Pk)gklxPk - Pkgkl(i-Pk)Pklx

= PkIpkIxgk - gkpklPkIx = LPkpklPklxpk'gk]

Accordingly, using (6.9) again,

(6.15) T2 = Dr 1PkPkIPrlxPk'gk1 = LPkPklblxPk'gk]'


Then (6.10), (6.11), (6.15), and (6.9) imply that indeed (6.5) is

equivalent to pdp=0 with d of (6.3) , and [b,g] = 0.

Next a calculation, not given in details, will show that the

"covariant derivative" Pkgkpk of gk in the space Sk has the matrix

(6.16) Yk' + [01,Yk} , 01 = ((,*(Pr)) _ - (('U'j*(Pr))

with respect to any biorthogonal base {(Pj} , {ipj} of Sk, where Yk

is the matrix of gk . Accordingly, (6.5) is equivalent to (6.7),

with 0 defined by (6.8) , q.e.d.

Clearly the system (6.7) of pk ODE's has a unique local sol-

ution satisfying y=y0, at t=0, defined near each for small

t. Let it be assumed that a = N2n ,but that only a local base pair

yj , V, exists, in subsets St1C 12n , covering 12n. Then solutions

in overlapping sets S2 j, for different bases, remain compatible as

long as they are jointly defined. From the well known properties

of ODE's we conclude existence of a unique solution g =Y_g j of our

matrix commutator problem, assuming a given value at t=0.

Note that d/ds just is the derivative along the Hamiltonian flow

(1.3) of the function X(x,l;) (instead of k0 ).This flow is well

defined for all t if we assume that X E y)Qe, , for example (of.

prop.2.5 ). We then get existence and uniqueness of

for all (t,x,l;) E I2n+1

Theorem 6.2. Let the assumptions of prop.6.1 hold for f21xI+f-II

>rjo , and let bE yiae, , d^ E C'(1t, -e), d- E C"(1,Vsm+e° -e)' Fore°=e assume in addition that the coefficients of b are E Wne. For

e°e even atdÊ CO(&, c(1+1)(e°-e))' atdoE C00(&'y'sm+(1+1)(e°-e))00

Then, for every g E tpsm a unique gtE C (Ii sm) exists such that

for all p=pj of b we have (for large

(6.17) [b,g]=0 , p(g +(b,g)+[d^ ,g]+d" )p =0 , tE & , g=g0 at t=0.

We also have atgt E 1 = 0,1,2..... .

Proof.(Assume large,where needed.) We know existence

of a unique solution g of (6.17).In order to show that g has the

stated symbol properties we note that (6.5) and gj = pjgjpj imply

(6.18) gj + [vj,gj] = pjgj+gjpj-pjd pj,

This may be interpreted as another set of ODE's for the v2 coeffi-

cients of gj. There is a unique solution of (6.18) under the init-2n+1ial conditions of (6.17), defined in N. This must be the same

9.7. Proof of theorem 5.4 305

gj , also solving (6.5).

We have p;=(? j'Pj) E 'cf-e' C'(l.'iVcf_e)Therefore (6.18) may be interpreted as a vzxvz-system of first or-

der ODE's for gj" = govt of the form

(6.19) dgj"/ds + cj(s) , s E R , gj 90Pj , at s=0,

g." being interpreted as a v2-component vector with entries

wile cI-E C00 (&,'1cv2e,_e) and the v2-vector cj has entries in

C (R,1Vsm+eA-e)' By our remarks a global solution of (6.19) exists

for all t. Also, prop.2.5 was used, for the above. It is a matter

of deriving suitable apriori estimates to show that gj"E C"(R,Psm)

hence also gjE C"(R,ism) , etc., proving the theorem.

For example, let us introduce the norm

(6.20) Ia12 = E Iajr(x, )I2(1+IxI2)j,r

From (6.19) one obtains the differential inequality

(6.21) Idlgj1m2/d81 5 i1TIgj"Im Isi s T

with a constant riT=sup with a matrix norm

Then (6.21) may be integrated, using the initial conditions, for

(6.22) IgJ Im S ril(e) , s E R ,

with locally bounded ril(s) . Then (6.19) implies

(6.23) Iatgk

Im-e+e°s T12(s)

One may differentiate (6.19) for t to derive estimates for dgj-/dt

recursively. Also apply (a finite number of) ijl of (2.3) to

(6.18) to obtain a similar system of ODE's for r elri 1 withJp p

coefficcients in ps0 , following the proof of thm.1.2. This will

give estimates for all expressions Id/dsI=lri 1 g-I2m, completing7pp 7the proof of thm.6.2. Details are left to the reader.

7. Completion of the proof of theorem 5.4.

In all discussions of sec.7 a restriction

with sufficiently large rio, is assumed wherever needed.

Returning to the proof of thm.5.4,(1) , we observe that

(5.24), together with [kf,gt]=0, i.e., (5.14) are of the form of


(6.17), so that thm.6.2 applies (presently with a°=e). Thus qt is

determined by its initial value qo, by property [q,ke]=0 and cdn.

(5.20) (i.e., (5.24) for (5.22)). By thm.6.2 such symbol exists,

belongs to C00(L psm). We also get as solution of (5.22),

i.e., in the form (5.23). First set plztpl=0. (5.23) implies ztE

C°O(&,yrs _e) , similarly as in the proof of thm.5.4 (2).

Note however,that at qt+zt does not satisfy (5.16) precisely,

but only mod j=1,2. To improve our choice of zt we will

set up a recursion, for a sequence of improvement symbols. An

asymptotic sum will give the total correction symbol. First let

(7.1) Wt = -e-iKtq(x,D)eiKt + gt(x,D) + zt(x,D)

and note that

Wt + i[K,Wt] = Rt = rt(x,D) , tE 2 , WO = ZO = z0(x,D)(7.2)

rt=zt+i[ x , zt] +(k, zt) -i/2(k, zt) 2+...+(K, qt) -i/2(k, qt) 2+...In particular, we have used (6.22) and [kf,gt] = 0 . Note thatrt E J=1,2 , and z0 E 'Wsm_e

We now may solve the linear inhomogeneous differential equa-

tion (7.2): It follows that d/dt(eiKtWte-iKt)=eiKtRte iKt , hence,

eiKtW a-iKt = -Q +eiKt -iKt t iKt -iKit (S2t + Zt)e = Z0 Oe Rte di

which implies

(7.3) aiKt(Q+ZO)eiKt = Qt+Zt - foe-iKTRt_SeiKT dv

Start the recursion by attempting to write [kf,zt]=irt, with

zt to be found. This is a commutator relation as studied. It can

be solved only if rt satisfies a cdn. (5.20), generally not to be

expected. But recall that the first term of (5.23) was set 0 so

far. We may replace zt by ztzt+zt , where zt commutes with kf but

otherwise is arbitrary. Substituting this into rt of (7.2) we get

rtrt+zt with rt denotingour old rt formed with zt . Here we write rt = rt + rt , with

rt = (K,zt) -i/2(k,zt)2 + "' and improve the above 'recursion

ansatz' by attempting to find zt solving the commutator equation

(7.4) [kf,zt] = in , rt rt+zt +(kf,zt)+i[K,z°] , [kf,zt]=0

exactly of the form (6.1), (6.3) again. For j=1,2 we get rtE

With initial value z0=0 use thm.6.2 to get ztE C"(l1S _e) to sol-

ve the compatibility conditions pjrtpj=0. Then (5.19), with the


first sum set 0, gives a solution of (7.4) of the form

(7.5) zt= PjirtPl/(%j-Xl) E C00(R,Wsm-e' -e)(C°°(!,Wsm_ez _e) .

After fixing the symbol ztE C°°(R,Wsm_ej) we also have a well defi-

ned r2E CO0(R,Wsm-e, j'ez) , j+j'-2 , using prop.5.3.

Set rt[kf,zt1+rt in (7.3), and 'integrate by parts': Write

] -i(k,zt)+(-i)2/2(k,zt) + ...),(7.6) rt = rt -i([k,zt2 2

with rt=rt+i ([ x, zt] +i(k, zt) +-...) EE CO0(R,VPSm-je' _j' ez j+j' =2. Get

fte-iKTRt-T aKt dt toe iKtR _t eiKt dr +if to-iKz [ K, Zt_t ]eiRr

e-iKtZ2eiKt _ Z2 +f to-iKtR2 eiKtd0 t 0 t-t

where we introduced rrt It-zt instead of the old rt. Substitute in

(7.3) (we now have Zt+Z instead Zt but Z0=0 by construction). Get

(7.8) eiKt(Q+ZO+ZO)eiKt = Qt+Zt+Zt+Zt - fe-iKTRt_telKtdt

In particular, we have rt E

Cm(R,Wsm-je'-j'Jezz)

, j+j'=2 , and

zt E Cm(R,VSm_ je, _ j ' ez) j=0,1,2.It is clear that this proceedure may be iterated. In the next

step we seek for a representation of rt as a commutator. Again

this requires (a) redefining the symbol zt of (7.5), adding a sui-

table CO0(R,Wsm-je'-j'e2)-function zt , commuting with ke,, and (b)

in the redefined (7.8), using zt+z2 splitting rt into a lower

order term rt , and another term rt fitting into thm.6.2.

The point is this: The zt again will have to be of order m-

je' -j'ez , j+j'=2 , since it must solve (6.5). The rt will be of

this order, with j+ j' =3 , and the zt, solving [ ke, , zt] =rt^ , will beof order m-je'-j'ez-e , j+j'=2 , which is even lower.

After solving that commutator equation the integral in (7.8)

is treated as that in (7.3) earlier. The splitting corresponding

to (7.6) gives symbols of the proper order, and z3. (going to the

next remainder), is of proper order m-je'-j'ez, j+j'=3. Thus get

(7.9) aiKt(Q+ZO+...+ZN)eiKt=(Qt+Zt+...+Zt) f trite-iKERN_teiKT

where generally

(7.10) zt , rt E COO(R,'WSm_ je' _j I G2 ) ,j+j'=1 ,1=0,1,2.... .


Clearly the integral in (7.9) is E C°°(" Vs _je, _ j , e2) , j+j' =N

Next, the asymptotic sum zt Fzt is well defined, and we get

(7.11) aiKt(Q+Z0)eiKt = Qt + Zt + Wt '

with Wt E O(-a), W0 =0, and Wt -E C'(&,Optpsm), using (7.9). We get

A=Q+Z0E Pm, At=Qt+Zt+Wt . We have proven (1) of thm.5.4, for a°=e.

Notice that we were selecting all the initial conditions for

the successive 'commuting parts' of the zt as 0 . This means that

the decomposition a = q+z0 , for t=0, coincides with the initial

decomposition of thm.5.4,(2). In particular, we get q0 of (5.14)

equal to our initially given symbol q of (5.12) .

On the other hand the lower order commuting parts of the zj

are not all zero,in general. Hence the decomposition qt+zt+wt =at

suggested by (7.11) in general is not of type (5.14). In fact,

for general q E Vsm , the symbol zt + wt may not be of order m-e

but only will be of order m-ej , j=1,2. The reason is, that the

'commuting part' plztpl of zt will have to go together with qt ,

to make a decomposition (5.14) .

This suggests that the commuting part of at is no 'clean pro-

pagation' of q , according to the differential equations (6.5).

The above discussion of thm.5.4 (1), a°=e, works again (with

the following amendments) for eA =e' (and hence also for e ° =e2) .First, zt of (5.23) again is in C°D(&,1sm_e) , but for a different

reason: The right hand side of (5.22) now is in Vsm_e2, but divi-

sion by gets this into Vsm_e again. qt of (5.24) has qtE

)sv_e2 . Moreover, atztE C0'(&,'U)svmle2 _e)' and, atgtE C00(N,Wsm-lee )'by a calculation similar to that in [CE].

The corresponding effect is observed on higher iterations.

For example, the symbol rt of (7.2) now is of one order -e2 better

since zt , and [x,zt] , and (kf,zt) all are better. Therefore the

result follows just as well. Details left to the reader. Q.E.D.

For a possible later application we summarize the special

observations in the proof of thm.5.4 (1) as follows.

Corollary 7.1. For any A E -=PP a decomposition (5.14), valid for

tE 2 , may be constructed as follows: Start from any such decompo-

sition a = a0 = q0 + z0 at t=0 and then set

(7.12) qt = Lj=1 gjtpj,

gjt:Sj -> Sj , Sj = im pj


Here the linear operators qjt of the j-th eigenspace Sj of ks, are

given as unique solutions of the first order system of PDE (7.13),

with covariant derivatives in changing with x and ,

(7.13) Pj((gjtPj)It kjI (gjtpj)Ix+?jIX

satisfying the initial condition

(7.14) qjt = gtlSj , at t = 0

and vj :Sj - Sj are defined by

(7.15) vj = (iPjfPj - PjPIkflxPj)ISj

In "local coordinates"-that is if (p1,...,cpPi

, and Pj is a

local bi-orthogonal basis of the eigenspace S (x,g) , defined and

C" in some open set a of if Kjt = is

the matrix of qjt, with respect to that basis, then, for 1,

(7.13) and (7.14) are equivalent to

(7.16) Kt it + `7I Kt Ix [O 1 ] = 0 , KO = Kl ,

where 0j is the sum of the matrix of vj and ((-(Xj1Vr*) s))

For the qt thus obtained one next writes

(7.17) zt=z0+i jI p.(qt+(kf,gt)+i[K,gt])Pl/(Xj-Xl) , [kf,zt]=0.

Here zo=0, zt lzjtpj, zjt:SjlSj given as solution of

(Pj(C),(zjtPj)))ISj + [vjzjt] +(7.18) 7

+ (Pj(zt+i[K,zt]+(kf,zt)+(K,qt)-i/2(kf.gt)2))ISj = 0with the differential expression of the "j-th particle flow"

(7.19) CX j = at + 7,.jl

ax - kj lxat .

The above holds only for large f2lxl+f,ll;l, but the symbols

obtained, defined for large may be extended to 1e2n

to get symbols in 1Vsm, 'Wsm-e' "m-e' n vsm-ez ' respectively, and,

(7.20) at = (qt+zt) + (zt+wt) ,

a decomposition (5.14), where wtE yrsm_je,_j'e2' j+j'=3 may be det-

ermined up to order -- by a recursion, similar as above.

Chapter 10. THE INVARIANT ALGEBRA OF THE DIRAC EQUATION

0. Introduction. Consider the motion of a charged particle in an

electromagnetic field with potentials V(x), A(x)=(A,,A2,A3)(x)

The particle is thought

as a small charged magnetic field Wsphere of mass m and

total charge e spinning B=curl A

about an axis through

its center of gravity, i electric field

and moving along some

orbit x(t)=(xi x2 x3 ) (t) . E=grad VIn the sense of /

classical Physics the 0/particle experiences

forces from the fields I Fig.1

E and B determining

its motion, once an initial location x° and velocity v° are given.

The spinning charge makes the particle a magnet (generates a mag-

netic moment) which experiences a twist from the fields B (and E

Thus the particle is acting like a spinning top with an angular

momentum trying to turn it.

Assume the velocities large to implicate special relativity.

Let c denote the speed of light.

The motion is described by a pair of systems of ODE. With

location x(t) and magnetic moment j(t) at time t we get (with "x"

denoting the cross product of 3-vectors, also in the following)

(0.1) t-x.

cJ.

= -E + C(x'xB)

as equation of motion for the particle, while j(t) satisfies

(0.2) t M 1-x'2/c2 B"xj = 0

where

(0.3) B" = B + (1/ec)(1+ 1-x'2/c2 )-1x'xE

denotes the magnetic field 'seen by the particle' (even if B=O,

310


the relatively moving electrostatic field E is experienced as a

magnetic field). Clearly E = E(x(t)), B=B(x(t)). We work in terms

of special relativity. x' in (0.2) and (0.3) may be substituted

with x'(t) from (0.1), so that the combination (0.1),(0.2) is a

nonlinear system in the variables (xj(t) , jj(t))j=1,...,3'

Also M is a coupling constant, involving the quotient of

the magnetic moment and the mechanical moment of the rotation.

It is given only after making some assumption about the geometric

distribution of charges and masses of the little sphere.

In somewhat greater precision one will have to also expect an

effect of the magnetic moment on the motion of the particle: If B

changes rapidly there will be different forces acting on north and

south pole of the spinning magnet, resulting in a non-zero combi-

ned force on the particle. We have reason to first ignore this

"Stern-Gerlach-effect", as a lower order quantity.

In terms of quantum mechanics, assuming that the particle

has 'spin 1/2' - such as an electron or a positron (or certain

mesons) , the physical description is very different, given by the

Dirac equation, a system of first order PDE .

The physical state of the system is no longer described by

initial position and velocity coordinates and initial spin, but

rather by a unit vector

1V(x) = (1V1 (x) ,1U2(x) ,1V3 (x) ,1V4(x) )T , x=(xi ,x2,x3)

in the Hilbert space H = L2(t3,M4) of squared integrable 4-vector

functions with complex coefficients. (The inner product is written

as (V, (0) = fF, 7,lVj(ojdx , and the norm as II pII=(V,1 )1/2. )

The (bounded or unbounded) self-adjoint operators acting on

a dense subdomain of H are called observables. For example the x1-

coordinate of the particle is an observable, given by the unboun-

ded self-adjoint operator p(x)->x1 !,(x) with domain {i E H : x1i. E H1.'Measuring' an observable A for a physical state 1V will in

general not produce a precisely predictable result. Rather one

will measure one of the eigenvalues of A with a certain probabi-

lity (assuming that A has discrete spectrum). If {cPj:j=1,2,... }

denotes an orthonormal base of eigenvectors with corresponding

eigenvalues X1,,2,... , then we may look at the expansion

(0.4) 1V = =1aj(pj aj = (g)j,1V) Iaj12 = 11V112 = 1 .

The measurement of A in the state 1V will produce the result

312 10. The invariant algebra of a Dirac equation

>`j with probability jai12 . Observe that the sum of all probabili-

ties is 1, as expected, and that the statistical expectation value

of the measurement is given by

(0.5) Lj=1XjjajI 2 = (yr,AV) , for , E don A

If A has continuous spectrum one can talk about the probability of

measuring a value in a given interval A=[a,b]. This probability is

given by IIEAVII2 with EA the value of the spectral measure on the

interval A . The expectation value still is given by (0.5).

After measuring the observable A , the physical state i' in

general has changed: If the eigenvalue Xj has been measured,then

the new physical state will be the corresponding eigenvector Tj .

A successive measurement of the same observable A will produce

the same result T,j with certainty (probability 1), as the expan-

sion (0.4) for "Tj shows. However, measuring another observable

B will transfer the state into an eigenvector of B, hence a follo-

wing measurement of A no longer will produce T,j with certainty,

except if the operators A and B commute. Then the expansion of Tj

into eigenvectors of B will lead to values of B corresponding to

simultaneous eigenvectors of A and B for which both may be measu-

red in any order, always producing the same result.

The first momentum coordinate pl is given as (the unique

self-adjoint realization of) the differential operator -ir<ax

Clearly thus the observables xl and pl do not commute, we have

(0.6) xipl-pixl = ifi ,

the Heisenberg uncertainty relation: Successive measurement of the

observables cannot eliminate a certain minimal uncertainty.

The total energy of the system is given as such an observa-

ble, denoted by H. This observable has a special significance. It

determines the 'time propagation' -substitute for (0.1) and (0.2).

The law of conservation of energy requires that a measure-

ment of H must produce the same result at any time. The represen-

tion of states and observables by vectors and operator of H is not

unique, of course. The choice has been a matter of convenience as

well as of individual preference.

Time propagation may be described either by the Schroedinger

representation: Observables are constant in time, but physical

states change. Then the change of the state V0 in time will be

determined by the differential-equation-initial-value-problem

(0.7) radii/dt + i Hy, = 0 , 0st<ao , p(0) = Vo .


Or, equivalently, by the Heisenberg representation: Physical

states are constant in t, but observables change, according to

(0.8) A - At = eitHAe-itH .

Observe that U(t)=eitH is a unitary operator, defined by the

spectral resolution, and is the solution operator of (0.7).

The quantum mechanical problem corresponding to our above

classical problem is described by the following energy observable:

(0.9) H = V(x) + mc2Il + c ..1aj(-iaa/axj - 2Aj(x)) .

Here V(x) and Aj(x) are the potentials as above, m,e,c,a are mass,

charge, speed of light and Planck constant. Also R , all a2, a3

are certain constant 4x4-matrices. We are used to work with

(0.10) «j = i(_aoj) j=1,2,3o-IJ

with the 'Pauli matrices' aj and I defined by

(0.11) a1- -i 0) ' a2= to 0J, a3= to-!J , I = to

011

For convenience we also define the 4x4-matrices

(0.12) lij = fa oii' pj = ro3a, , j=1,2,3

j j)

and observe that the following (anti)-commutator relations hold:

(0.13) aj2=1, a1a2=-a2a1=ia3, a2a3=-a3a2=ia1, a3a1=-6163=io2 ,

hence

ajak + akaj = 26jk, pjpk+pkpj = 26jk, Paj+aj(3 = 0

(0.14) ajak + akaj = 2ojk, µjµk+l'kµj = 26jk, fl2 = 1

(3a j= iµj, a1a2= ip3 , a2a3= 141 , a3 a1= ip2

Note (0.15) below, for the formal 3-vector a = (x1'(y 2'°3) ,

and arbitrary formal vectors a,b, components commuting with a

(a) a(aa) = a + iaxa , (aa)a = a - iaxa

(0.15) (b) ((ja)(ab) =

(c) [ (axa) , (b a)] =(axa) (b a)-(b (3) (axa) =2i{ (aa)b-(ab )a} .

The (system of) differential equation(s) (0.7), with H of

(0.9), is called the Dirac equation. Clearly the Dirac equation is


a symmetric hyperbolic 4x4-system, in the sense of VI,2. Moreover,

we shall see that it is semi-strictly hyperbolic of type e' , un-

der proper assumptions on V and Aj . Accordingly our results of

IX,5 apply. We are getting an invariant algebra of ydo-s.

From now on let us assume the potentials V, Aj to be bounded

C ()e )-functions. Note, we exclude singularities like that of a

Coulomb potential. This appears as a serious handicap of the theo-

ry, but may, at the contrary, prove to be one of its major points.

It then follows easily that the differential expression H

is formally self-adjoint, and, moreover, that the minimal operator

(with domain C0(&3)) admits precisely one self-adjoint realization

In other words, there is a unique self-adjoint energy observable

(the Hamiltonian) induced, and our theory is well set.

The above scheme indeed is capable of explaining physical

observations around our model of fig.1 with such amazing accuracy

and detail, that it survived, in spite of the fact that some stran-

ge contradictions or paradoxa also were noted. In earlier work

([CD], [CF]) we attempted to show that our invariant algebra of

IX,5 may explain some of the paradoxa. One might even go beyond

this and ask the question whether the 'perturbation symbols' we

discuss in sec.5 could explain other physical phenomena so far

only derived from quantum field theory or gauge theory.

In sec. 1 we propose a modified observable concept. In sec.2

we discuss a link between invariant algebra and Foldy-Wouthuysen

transform. Sec's 3,4,5 give details of the invariant algebra in

case of the Dirac system (0.7),(0.9). In sec's 5-9 we focus on the

'correction symbol' first for 'standard observables' then in gene-

ral, discussing some finer details.

1. A refinement of the concept of observable.

To motivate an application of the theory of ydo's, let us

focus on the concept of observable introduced in sec.0 . So far,

any bounded or unbounded self-adjoint operator of H = L2(H3T4

(having a spectral resolution) qualifies as observable. On the

other hand, the observables of real interest normally turn out

to be either multiplications (like the x1-coordinate, we mention-

ned) or differential operators - such as H above, or the momentum

observable, given as pl = -iha/ax , etc. . We will meet other1

"standard observables" later on, and mention that multiplication

by a (proper) matrix may occur as well, such as the spin observa-

10.1. The concept of an observable 315

ble, commonly defined as multiplication by pj , j=1,2,3, with pj

of (0.12) .

on the other hand, in many respects the admission of all

unbounded self-adjoint operators as observables appears inconve-

niently large and the question can be asked whether it is useful

to get restricted to a special class of observables.

We choose to suggest a certain condition of continuity or

smoothness on observables, to be discussed now:

Note that for the remainder of this section it is not

significant that physical states are 4-vectors. We might just as

well assume that H - L2(&3) (with complex-valued functions), or,

more generally, that H = L2(&n,,v) , for arbitrary n,v = 1,2,... .

First consider a bounded operator A E L(H) . We propose to

admit A as observable only if it is norm continuous under trans-

lations. In more detail, it is well known that the operator family

Ah = T-hATh , h E lea , with the translation operator Th = eihD ,

(Thu)(x) = u(x+h), u E H , in general is not norm continuous in h

in the Banach algebra L(H) , but only 'strongly continuous' :

For each fixed u E H the family uh = Ahu is continuous as a map

from &3 to H . However, this continuity may degenerate if u varies

over a bounded set of H , it may not be uniform. Example: let

A = multiplication by a discontinuous function a(x) like a(x)

= xl/Ixl1 = sgn(xl) .

If we ask for differentiability in the parameter h the situa-

tion gets worse: A derivative ahj

Ahu as limit of a difference quo-

tient exists only under special assumptions on A and u .

We impose as condition (t) for a bounded observable A that

its family Ah is not only norm continuous, but even norm-Cw : All

partial derivatives ahAh exist in norm convergence.

Observe that this condition may be very natural: Given the

inherent inaccuracy of space measurements an observable can have

little significance, if it changes strongly under very small trans-

lations. (Although of course this may be an idealization just as

the concept of derivative - interpreting velocity as time deriva-

tive amounts to a strong simplification, but the built-in limit

in praxis never can be carried out, but will have to be replaced

by a difference quotient).

Next, we impose the same condition also in phase space:

The Fourier transform

(1.1) Fu(r) = f Pox = (2n)-3/2dx


defines a unitary operator of H 'diagonalizing' the momentum obser

vable (i.e., pj _ -iax is transformed into the multiplicationj

operator u^ (x) xJu^ (x) ).Our quantum mechanical measuring process is invariant under

unitary transforms: If i, is a state and A is an observable, as

above, and if U:H-H is a unitary operator then we get exactly the

same probabilities and expectation values if we replace i, and A by

yf = U, and A" = UAU* , respectively. In sec.0 the position obser-

vable was diagonal (was a multiplication operator). If we apply

the Fourier transform then the momentum operator gets diagonal.

Position and momentum play a dual role in much of the theory.

Hence it appears natural to impose the same "translation smooth-

ness condition" (i.e.,cdn.(-u)) also in the momentum-diagonal form.

A simple calculation shows that this amounts to the following

gauge-smoothness condition (y): The operator family Ah=eihxAe-ihx

is norm-Coo in the parameter h (i.e., Ah E C-(&3,L(H)) ).

[Indeed, (FAF*)h = T-hFAF*Th = FAhF* , as easily checked]

Remark 1.1. Both conditions (z) and (y) (i.e. translation and gau-

ge smoothness together) amount to the condition of norm-smoothness

of conjugation within the Heisenberg group HG : The map g -> A(g)=

g*Ag isC00(HG,L(H)). Recall that the Heisenberg group (or its stan-

dard representation within the unitary group U(H) of H) consists

of all unitary operators of the form g - ei(P eil;xeizD=ei(T+tx)Tz

(PE It, z,t E 13 with operator multiplication as group operation.

At this time we recall the concept of pseudo-differential

operator (qdo) with symbol in CB°°(13):

The space of symbols Vt0 = CB°°(13) is defined as

(1.2) Vt0 = E C0O(16): all derivatives are bounded}

(we mean all partial derivatives in the 6 variables x and and

of all orders). For a "symbol" E Vt0 define the operator

A - op(a) =a(x,D) by settingr

(1.3) (Au)(x) = (2n)-3fd )u(y)

The integrals at right exist in the order stated whenever u E S =

S(13) , the space of rapidly decreasing C'O(13)-functions, and the

function v = Au then is in S again, as seen by a technical but

straight-forward calculation (cf. I,1, VIII,1). Thus A of (1.3)

is defined as a (continuous) map S -> S , whenever a E CB'O = y,toThe significance of the last remarks become clear if we

10.1. The concept of an observable 317

look at the following result:

Theorem 1.2. Every pseudodifferential operator A = a(x,D) with

symbol a E Vto extends to a continuous operator of L(H) (from the

dense subspace S of H), also denoted by A . Moreover the class of

all such L2-bounded pseudo-differential operators precisely coinci-

des with the class of all bounded observables satisfying cdn's (t)

and (y) above (i.e., the class of A with A(g)E C'(HO,L(H))) .

Furthermore, for the latter type of A the corresponding symbol is

uniquely defined by the formula

(1.4) a(z,t) = (2n)ntrace{Q_*P(az,at)Az,t}

whereA2.t=

e_itxeizDAe-izDeitx, and where P(az,at) is

any differential polynomial (constant coefficient differential

expression) in dz

=a1az

and atj with the property that it admitsj j

a fundamental solution q(z,l;) of polynomial growth, and such that

the pseudodifferential operator Q_ = q(-x,-D) is of trace class

in the Hilbert space H .(Such a polynomial is given by p(z,t) -

(l+z2)n(1+12)n , for example, as may be seen. The corresponding

fundamental solution q then is given as the product of Bessel

potentials inverting (1-Az)n(1_A )n).

This result was discussed in [CL]. Actually, it coincides

with VIII, thm.2.1, of course, except for some variations in the

symbol formula.

The conclusion we derive from thm.1.2 is that, with our

above restriction of translation and gauge smoothness imposed on

bounded observables, we automatically arrive at the consequence

that a bounded observable must be a pdo , and that a self-adioint

,do AE Opyito automatically is a bounded observable satisfying the

translation smoothness and gauge-smoothness condition.

Notice that we are lead into a set of axioms selecting bet-

ter observables which seems elegant, but has several defects.

First, most standard observables are not bounded (such as energy,

position and momentum, for example) . Second the class Optpt0 of

ydo's with symbol in0 has severe shortcomings, as far as stan-

dard theory of 'tpdo's is concerned: These lydo's need not to obey

the asymptotic calculus of 'gdo's valid for classes of symbols with

derivatives decaying at .

We address the second objection first: If in addition we

also impose a "rotation smoothness condition" (cdn.(p), below)

and also a "dilation smoothness condition" (cf. cdn.(S), below).


then we get a smaller class of bounded observables, and a class

of 4 do's allowing a global calculus of ydo's, in a form to be

specified.

Condition (p) (rotation smoothness): For an orthogonal 3x3-matrix

o define the unitary operator S0

E L(H) by u(x) - u(ox)=S0U(x)

Then the family A(o) = S*AS0 is norm-Coo over the orthogonal

group 3 of IQ3

Condition (S) (dilation smoothness): For a positive number 6>0

define RS EL(H) by setting Rsu(x) = 61/2u(6l). Then the family

ROARS is norm-C00 over R+

Note that cdn's (p) and (S) are Fourier invariant: They mean

the same, if imposed on the position-diagonalized or the momentum

diagonalized representation, because the Fourier transform leaves

the corresponding subgroups of U(H) invariant. All above cdn's

(T),(y),(p),(S) together may be expressed by using the Lie-sub-

group GS of U(H) of all operators of the form

(1.5) g =ei(peizDeii;XR6SO , T E & , z,t E &3 , o E 03 .

Proposition 1.3. For a bounded observable cdn's (s),(y),(p),(S)

together are equivalent to the property that the function A(g) _

gAg , defined over GS , is C0O(GS,L(H)) , with norm topology used

in L(H) .

We denote the class of operators used in prop.1.3 by IWS

and then have the result below (cf. VIII, thm.5.4).

Theorem 1.4. The class WS precisely consists of all yido's A =

a(x,D) with symbol in the class ls0 , where Vso denotes the class

of symbols in yito such that application of a finite number of dif-

ferential expressions 1jl of (1.6), below, repeated in arbitrary

orders always gives a function of Vto

(1.6) `1oo=xjxj

qjl=( A ,- la +(xi axl-x1ax,) , j,1=1,...,n

One confirms easily that the functions in Vso have the

"classical symbol property" on compact sets: For any compact set

K C In and all multi-indices a , 1 we have

(1.7) aaaPa(x,U) = x, E &n , x E K

There is an analogous condition with x and 1; reversed. In fact,

global such conditions (degenerating at infinity) hold cf. VIII,6.

10.2. The invariant algebra 319

Next we introduce the class ips of all polynomials

(1.8) aa,p E 4s0 .

in the 6 variables x,t and observe that Vs as well as the class

'IS = Opts = {a(x,D}:a E ps} are algebras (cf. VIII,7). For a of

the form (1.8) the operator A=a(x,D) is of the form

(1.9) A = with A,,, = aa,0(x,D) .

However, the commutators [ Dj,A] and [ xj ,A] for A E OpVs , handlerather well (cf. VIII,7), and we also may write

(1.10) A

with different coefficient symbols as . A Leibniz-Taylor formula

as in 1,5 holds in OpVt, with the class 'tVt of polynomials with co-

efficients in ipt0, but the remainder may not be well behaved. On

the other hand, for symbols in is , estimates for the remainder

can be derived, resulting in asymptotic expansions, and a calcu-

lus of i4ido's, (although degenerating for cf. VIII,7).

In the following we shall require all observables considered

to be members of 'IN = Opps . (Note, however, that further restric-

tions will be introduced in sec.2, below.

Introducing the classes Vsm (for m = (ml,m2) , mj=0,1,2,..,)

as the polynomials (1.8) of order m1 in l; and order m2 in x, and

the operator classes LISm = OpVSM , we observe that Its = Um lsm, andthat a characterization of tINm by conditions like our above

'smoothnesses' (t),(y),(p),(6) is possible with the (polynomially

weighted) L2-Sobolev spaces of 111,3, i.e.,

(1.11) Hs = Hs ,s2 = {u ES': (x)s2(D)s1u EH} .

Theorem 1.5. Let s=(s,,s2) be given fixed. The class LISm (for

any m=(mi,m2)) consists precisely of all operators in L(HS'HS-m)

with the norm-Coo properties (u) , (y) , (p) , (6) , all with

respect to the operator norm of L(HSHs-m) instead of L(H)

The proof will be omitted.

2. The invariant algebra and the Foldy-Wouthuysen transform.

As stated above, an observable will be a (self-adjoint) ope-

rator of uN , from now on. But it will be seen later on that the


algebra ills is not in general invariant under time translation.

Assuming states constant and observables to change by conjug-

ation with a-iHt, we need At=eiHtAe-iHt to remain in W for all t

before we can admit A E '11S as an observable. In fact, we tend to

require a norm smoothness of the time translation of an observable

similar as posted for space and momentum translation.

To be precise: For an L2-bounded A we require that

(2.1) (x)3a3At E C'(l,L(H)) , j=1,2,....,

and, moreover, (2.1) is not only required for At , but also for

every g-derivative of At(g) , as in sec.1. (over the group GS

For general A E Wm we require (2.1) (for all the derivatives

mentioned) with L(H) replaced by L(HsHs-m

) , as in thm.1.5.

The above condition will be referred to as 'cdn.(t)'.

Clearly cdn.(t) implies that AtE tI{S for all t. Thus At has a sym-

bol at(x,l) . Equivalently we also may express the same cdn.(t) in

terms of the symbol at . For aE Vsm this takes the following form.

Condition (t'): We have At = eiHtAe-iHt = at(x,D) , with atE s

and (with e2= (0,1) ) we have

(2.2) at at E PSm-jet for all t E & , j=0,1,2,... .

Moreover, if we apply any combination (of arbitrary order) of the

differential expressions 1jl of (1.6) to at we obtain a symbol

bt E Vsm still satisfying (2.2) .

The class of all A E iI{Sm (for a given m) satisfying cdn (t)

(or equivalently (t') ) will be called P Pm (since it depends onm

the 'Hamiltonian' H ). One trivially notes that (for fixed H)

Pm C Pm, as mjsmj' , j=1,2 . The union P = UmPm is a graded alge-

bra, left invariant under conjugation with a-iHt . It will be cal-

led the invariant algebra of the Dirac equation (or, more general-

ly of the Hamiltonian H ).

Let us now analyze the significance of cdn (t). The special

form (0.9) of the Dirac operator is of crucial importance, in this

respect. We know that V(t) = e-itH,F is given as the solution of

the Dirac equation (0.7) with V(0)=4i' . Explicitly we are given

an initial-value problem for the partial differential equation

(2.3) avlat + i{V(x)+mc2s+

Note that (2.3) is a symmetric hyperbolic first order system

of 4 partial differential equations in 4 unknown functions. Its


principal part has constant coefficients. The existence and uni-

queness of the solution of such a system is a well known fact

(discussed in detail in VI,2). We will require as additional

condition that (for j=1,2,3, and all multi-indices a) we have

(2.4) V(a)(x) =O((.)-IaI)

,O((.)-IaI)

Then H = h(x,D) is a pseudodifferential operator in Op",

(e'=(1,0) ) with symbol h and pricipal symbol h" given by

(2.5)

h h-

the symmetric hyperbolic

property of (2.3). Under (2.4) we have hE ipce, so that HE Opyice,On the other hand, the hyperbolic system (2.3) is not strict,

ly hyperbolic: The matrix of course has real eigenvalues,

but they are not distinct. A quick check shows that the eigenva-

lues of are =tI I. For X960 they both are double, hence

have constant multiplicity.

The hermitian matrix-function has the diagonalization

(2.6) +(x, )=1(x, ) =

with a unitary 4x4-matrix and eigenvalues

(2.7) 11=12=X+ , 13=14=X- , X+=V(x)tcfr , where fr m2c2+n2

for all x,1; E &n. ('+' denotes the adjoint 4x4-matrix, 'a*' the

Hilbert space adjoint's symbol - a(x,D)*=a*(x,D), in all of ch.X.)

In 1,7 we defined the class of symbols ypcm, m=(ml,m2)E &2, by

(2.8) "'cm = {aE CO: a(T) = O(( )m1 IOI(x)m2 ITI),6,'rE Z+

with ( ) = 1+ i; 1, a(0) = aeaXa. Now we claim that,under the

conditions (2.4), we have X±E yice,, e' = (1,0) . Moreover, the

entries of T are global symbols in Vc0. In fact the 2-fold eigen-

values Xt have the two-dimensional orthogonal spectral projections

(2.9) a0=me/(m2c2 + a2)1/2. t= /(m2c2+n2)l/2

A calculation confirms that X. as well as the entries of are

in yc0 . Let V1, yr2 be the first two columns of p+ , let V3, yi4 be

l/2the last two columns of p- , and let Tj = V Pj/(1+a0) . Then


(2.10) cp(x,t) =

is the desired unitary matrix satisfying (2.5) and again has coef-

ficients in ta0, by a calculation. We write yicm for the class of

all rxr-matrices with entries in cm, so that pt, VE V00.

In matters of the invariant algebra P it should first be

noted that for a single first order hyperbolic equation we get

an Egorov-type result, saying that cdn (t) (or (t')) holds for

every A E ills (cf. VI,6). For a semi-strictly hyperbolic system of

the type discussed in IX,5 we found that the invariant algebra P

is properly contained in 46 , however. In essence, membership in

P depends on the symbol of A to commute with the symbol of H , at

least modulo lower order terms. Such a result was proven in IX,5.

In order to apply these results we just have to confirm that

the Dirac equation (2.3) is semi-strictly hyperbolic of type e'.

Indeed, we have the explicit eigenvalues (2.7) of the (complete)

symbol of H , and it is readily checked that IX,5,(i),(ii),(iii)

are satisfied, if only the potentials V and Aj satisfy (2.4).

(Note that the classes Voxe .... simplify to just yrce, , as &=e i . )

One finds that most standard dynamical observables (such as

position coordinates xj, momentum coordinates pj=-iaxj

, angular

momentum coordinates (xxp)j , etc. all belong to 'IS .

Surprisingly none of these operators belong to P. They all

need (normally small) additive corrections to become members of P.

To illuminate this fact we proceed stating a pair of theorems.

From now on we will distinguish in notation between complex-

valued and 4x4-matrix-valued symbols: Let Vtm , VSm , etc. be the

4x4-matrices of symbols in Wtm, VSm , etc. Note that (2.4) implies

that E Vae, , hence H E Opy,ce, , e'=(1,0), for h and H of (2.5).

Evidently the algebra P contains the identity, and the Hamil-

tonian H as well as all polynomials in H. In fact, it is clear

that every AE LIS commuting with H - in the sense that eiHtA-AeiHt

=0 , using the operator product of O(oo) , satisfies At A=const.

hence will belong to P . For example, the resolvent R(A)=(H-A)-

can be shown to belong to 1IS-e, , as A is non-real: Clearly the

symbol h(x,l;)-A =g(X-A)q)+ is md-elliptic of order e' - one easily

finds that It has a special Green inverse,

in the sense of III,thm.4.2 (or V,thm.1.3). Since it is invertible

in H - due to self-adjointness of H - it follows that R(A) must be

that special Green inverse. Hence R(A)E 11S-e, , implying that


R(.t)E P_e' . On the other hand, the operator eiHt does not belong

to W , hence not to P either . The operator eiHtx , for O#XE C00

shifts singularities (cf.VI,7) while AX E Oppc , for AE X16

leaves wave front sets invariant, by II, thm.5.4.

The result, below, in essence links the algebra P to the

Iido's with symbol matrix commuting with h(x,l;).

Theorem 2.1. (1) For every AEPm, of the form A=a(x,D), the symbol

a allows a decomposition

4 4(2.11) a = q + z , z E y)sm-e, q E s [h(x,i;),q(x,1;)] = 0

the latter for all x,t E R3

(2) Vice versa, if a matrix qE yes of symbols commutes

with the symbol h(x,l;) , at least for all x,T; ER3 satisfying

xl+Z y > 0 , then there exists a symbol z E ys _e such that

A = a(x,D) = q(x,D) + z(x,D) E Pm .

(3) Suppose A1

, A2 E Pm (which must have a decomposi-

tion (2.11) both) have the same q, but different z (=z1,z2), resp.

Then b=z1-z2 allows a decomposition (2.11) with m replaced by m-e.

Thm.2.1 is just IX,thm.5.4, in case of the Dirac equation.

In the next result we are going to construct a unitary oper-

ator of H and a pseudo-differential operator which decouples the

Dirac equations modulo a term of order --. The fact that this can

be done for finite orders - not order- - is well known. The cor-

responding unitary operator is called Foldy-Wouthuysen transform.

Theorem 2.2. There exists a unitary operator X:H-H of the Hilbert

space H = T4xL2(&3) , which also is a 'pdo with symbol in yPc4

such that

(2.12) X=X(x,D) , X=(p+cu , wE 1c4e , cp as in (2.10)

and that the substitution u = Xv (and multiplication from left by

X 1) brings the Dirac equation to the form

(2.13) au/at+ i(A + T)v = 0 .

Here r is a (4x4-matrix of) ,do(-s) in O(-oo), while A vanishes in

its upper right and lower left 2x2-corners. Moreover, we have

(2.14) symb(A) = diag (mod "_e2 ) , e2 = (0,1)

with diag( ) denoting the diagonal matrix with entries listed.


A proof of thm.2.2 will be discussed in sec.6, below.

Thm.2.1 and thm.2.2 are

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