A general calculus of pseudodifferential operators

A GENERAL CALCULUS OF PSEUDODIFFERENTIALOPERATORS

RICHARD BEALS

Contents:

Introduction 11. Weight functions 52. Weight vectors 83. Classes of symbols and operators 104. The calculus of pseudodifferential operators 125. L-boundedness 176. Weighted Sobolev spaces 207. Global operator theory 258. Localization: the operational calculus 279. Localization: weighted Sobolev spaces 3010. A priori estimates and hypoellipticity 3211. Parametrices 35Bibliography 40

Introduction. Pseudodifferential operators have become an indispensabletool in the study of linear partial differential equations. In principle, the largerthe class of pseudodifferential operators for which one has a usable operationalcalculus, the wider the range of applications should be. In this paper we developsuch a calculus for very general classes of operators, and include a few applica-tions for illustration. We begin with a very brief survey of some of the existingtheory and applications; the articles of Calder6n [11] and Nirenberg [39] containmuch additional information.

Calder6n and Zygmund [14] applied a calculus of singular integral operatorsto the study of elliptic equations. Calder6n [9], [10] then used this calculus toobtain the first very general theorems on uniqueness in the Cauchy problemand existence of solutions for systems with variable coefficients.The calculus of singular integral operators was refined and recast into a theory

of pseudodifferential operators by Kohn and Nirenberg [32], Seeley [45], Unter-berger and Bokobza [53] and HSrmander [25]. In this form the theory dealswith operators whose symbols a(x, ) have asymptotic expansions as sums ofsymbols which are positive homogeneous in of decreasing orders; it includesparametrices for elliptic operators. This theory has been used to establish

Received October 4, 1974. Revisio received November 22, 1974. During the preparationof this research the author was partially supported by the National Science Foundation underNSF grant GP28148.

RICHARD BEALS

hypoellipticity and prove subelliptic estimates for problems which are notelliptic, and to prove estimates of Carleman type; see, for example, Kohn andNirenberg [33], HSrmander [27], Egorov [18], Oleinik and Radkevic [43], Kohn[30], Treves [49], and Egorov [17], Treves [51], Menikoff [37], [38]. The theoryis also an essential tool in the theory of local solvability of equations of principaltype: see Nirenberg and Treves [40], [41], Treves [52].H6rmander noted explicitly in [26] what had been implicit in H6rmander [24]

and Egorov [16]. First, the proofs for the calculus of pseudodifferentil operatorsdepend not on the homogeneity ssumptions noted bove, but only on the corre-sponding estimates for the derivatives of the symbol:

(1) DDa[ C(1 +for an operator of order m. Second, these estimates may be weakened signi-ficantly’

(2) ]DDa] < C(1 + ])’-’+’ 0 < < p < 1

Third, the larger class of operators defined by such symbols contains parametricesof some nonelliptic operators, such as the heat operator and the operator

Ixl + 1. Unterberger and Bokobza [54] treated operators such as thelatter in a different way, by introducing symbols homogeneous in of variableorder re(x); in terms of estimates, such symbols satisfy

(3) IOD,Oal C,(1 + I}[)(-’’[log (1 + I}1)1 ’’.The theories described so far are local in character. H6rmander [26] showed,

however, that an operator whose symbol satisfies the estimates (2) with m 0is a bounded operator in L(R"). Kumano-go [34] carried through the wholecalculus globally in R". Calder6n and Vaillancourt [12], [13] introduced somepowerful techniques in proving L-boundedness with m 0 in the borderlinecases0 < 1.Viik and Gruin [551, [56], Gruin [20], [21], [221, [231, and Kumano-go and

Taniguchi [36] developed a global calculus which allows for polynomial growthas x and permits a global treatment of operators like - + ]x] and itsinverse. The estimates (2) are replaced by estimates of the form

(4) ]D=D/al CnX(x, })’-"’=’+’z’, 0 < 1,

where X is a suitable function, e.g. 1 + [}l + Ixl One must also introducecorresponding weighted Sobolev spaces with norms like

iul,,, II [xl

where I[ II is the L-norm.The global calculus just described was introduced to allow one to study

certain hypoelliptic equations by considering pseudodifferential operators with

PSEUDODIFFERENTIAL OPERATORS

operator-valued symbols" it is necessary to have some global operator theoryfor these operator values. Some of the same kind of hypoelliptic equations havebeen studied from a different point of view by SjSstrand [46], [47], Boutet deMonvel and Treves [7], [8], and Boutet de Monvel [6]. These authors introduceclasses of pseudodifferential operators and weighted Sobolev spaces which arenot defined by estimates like (2), (3), or (4); however, they can be defined byestimates like those below: see [2].

In what seemed a very different direction, Fefferman and the author [3] usedthe CalderSn-Vaillancourt boundedness theorem and a localization in phase-space cotangent bundle to extend the Nirenberg-Treves local solvabilitytheorem to the C case. Implicit in this work are the rudiments of a calculus ofpseudodifferential operators whose symbols satisfy estimates of the form

(5)

The calculus is made explicit in [4], [5], and carried further in [2], where theappropriate weighted Sobolev spaces are introduced.

Estimates of the form (5) appear to be natural to any calculus of pseudo-differential operators. Therefore it is also natural to seek a theory with minimalassumptions on the functions , , and . The assumptions made below maynot be minimal--see the discussion in sections 1 and 2--but the theory developeddoes contain (simultaneously) all the calculi discussed above. The presentationof the theory is meant to be relatively complete and self-contained, if concise.There is some overlap with [5] and [2], but even locally the present assumptionson the weight functions , are weaker.No attempt at completeness is made in the examples and applications. The

applications sketched here are to the proof of hypoellipticity and the constructionof parametrices for operators such as those with symbols:

1 + ip(x)x2k: + p2(x), p(O) O, Pl real;- p(x)x2k2 - p2(x)x- + pa(x) - p(x), p(0) > 0;

x -ip(x) - p(x). + pa(x), p,(O) > O.

The first three are generalizations or specializations of operators considered byEgorov [18], Treves [49], [50], Baouendi [1], HSrmander [27], Gruin [21],SjSstrand [46], [47], [48], Boutet de Monvel and Treves [7], Boutet de Monvel[6], Kannai [28], Zuily [57], Kumano-go and Taniguchi [36]. Our viewpointis rather different" it is that once an appropriate pair of weight functions orweight vectors , is chosen, then the treatment of such an operator is a straight-forward problem in the pseudodifferential operator calculus.We work in R with variables x (x, x, x.) and dual variables

(1, , .). The pairing x. is denoted simply by x. We set

4 RICHARD BtALS

In L L2(R’) with the usual inner product (,), we have the Fourier transform

() f e-u(x) dx,

A multi-index is an n-tuple of nonnegative integers, a (a,, a=, a,). ThenI[ , and H , ’. Differentiations are denoted

Oi O/Oxi Di -iOn,

If a(x, ) C (R), then we denote(a) xaa 0 D

We let (R), $ $(R), nd (R) denote the usual spaces of testfunctions, with ntidul spces *, $*, nd * respectively. Thus, for example,8 is imbedded in * by setting

f](u) (], u) ](x)u(x) ax, 1 , u .We shll make frequent but tcit use of the following observations. If ]

C (R) is positive and R, or if ] is nowhere 0 nd is n integer, then nyderiwtive D (]) is linear combination of terms

Therefore if a C (R X R) stisfies inequalities

if follows that b a stisfies

Also, if g C(R) is real and h C(R*), then any derivative D(h(g)) islinear combination of terms

h’’() H D(’’, E () .=1

In particular, suppose a C (R X R) satisfies (6) and suppose h D(R)is 1 near 0. Then the functions

b(x, ) h(N-a(,, )), N > O,stisfy estimates

with constants C.’ independent of N.


1. Weight functions.

Assumptions. We say that a pair of positive continuous functions , edefined on R R R {(x, ()} is a pair of weight Junctions if there arepositive constant c, C, such that

(1.1)

(..2)

(1.3)

_<C;

_> c;

c _< (x, )(y, 7) -1 _< C and c _< (x, )(y, 7) -1 _< C

if ]x Yl <- c,(x, ) and I 71 -< cq(x, );

R(x, O) _< C (x) c, where R q-l.,

c <_ R(x, )R(y, 7) -1 <_ C if I- 71 cR(x, )// and

x Yl <- cR(x, )R(y, 7)-/.

We say that , are localizable if for each compact C R, there is a constante > 0 such that

(1.6) _> e() on XR.Remarks.1.7 The functions , should be regarded as assigning to each point (x, () of

the phase space (cotangent bundle) two characteristic units of length" (x, () inthe x-direction and (x, () in the -directions. Important quantities in thetheory will not change greatly in magnitude over such distances. In particular,(1.3) says that the units of length themselves do not vary too rapidly over suchdistances. Assumption (1.1) is natural with this interpretation. Assumption(1.2) is a nondegeneracy condition related to the condition t _< in [26]. As-sumptions (1.4) and (1.5) are technical conditions used to control the variationover greater distances; in practice they seem easier to satisfy than (13), but it ispossible that weaker, or different, conditions would serve as well.

1.8 Weight functions in the sense of [4] are weight functions in the presentsense, though not conversely. However the present assumptions are not so muchweaker than those of [4] as first appears; see Proposition 1.16 below.

1.9 Suppose , R are positive C functions on R. Then and OR- willbe weight functions (with 1/2) if (1.5) holds and also:

(1.10) c _< R _< Cq,;

(1.11)

Conversely, if , are weight functions with 1/2 in (1.5), then there is anequivalent pair satisfying (1.10) and (1.11); see Propositions 1.16 and 1.29 below.

Examples.1.12. Lete ((), (),0 < < p < 1, # 1. (The condition < o

is forced by (1.2) and the condition # 1 is forced by (1.5).)

6 RICHARD BEALS

1.13. Let 1, ) I(,, 2, ,-1)1 -t- (,1z2. These are not weightfunctions in the sense of [2], [5]" assumption (1.5) is weaker than assumption (iii)of [2], [5].

1.14. Let 1, () + (x). Again these are not weight functions in thesense of [2], [5]; (1.1) and (1.4) are weaker than (i), (iii) of [2], [5].

1.15. Let k be a "basic weight function" in the sense of Kumano-go andTaniguchi [36]. Let k’,= k-, where 0 _< ti _< p <_ 1, ti 1, p 0.

Consequences o] the assumptions. Suppose , are weight functions. Weadopt the following policy on constants: unless otherwise stated, constants c, C,i, vary from statement to statement, but depend only on constants previouslychosen.

PROPOSITION 1.16.

(1.17)

(1.18)

(1.19)

LetR ,-. Then

cR-/ <_ <_ C, cR/" <_ <_ CR;

c < < c(<> + <x>);c(x)-C(})-x

_, where e > O.

Proof. Assumptions (1.1)and (1.2)imply

_c and c-2

_ - R

_C.

Assumption (1.3) implies (x, )

_C(x, 0) unless [1 - c(x, ), so

(x, ) _< c(<> + (x, 0))

_< c..(<> + R(x, 0))

_< c<> + C<x>.Similarly,

R(x, )+ <_ C(([) + <x)C),and this inequality, together with the first part of (1.17), gives (1.19).

PROPOSITION 1.20.

(1.21)

then

(1.22)

and

(1.23)

Proof.Then

[- nl - c{R(x, )+1/2 + R(y, n)+/} and

[x Yl - c{R(x, ) + R(y, v)} max {R(x, )-/, R(y,

(y, r) __< [(y, )(y, )](x, ))2

(y, ?)’

_[(y, V)(y, v)l(x, ).

Let (x, ), . (y, ), and define ; and R accordingly.. ,,R. (.)(R.R )R1-1<_ C(9.)(RR )1.


By (1.5), RRI-1

_C. This proves (1.22) and the proof of (1.23) is similar.

COROL,ARY 1.24. Let a M’, where M and m are real. I] (1.21) issatisfied, hen

(1.25) a(x, )a(y, )- C{ (x, )(x, ) + (y, )(y, )}c.

The inequality (1.25) is important in the proof of the composition theorem forpseudodifferential operators, Theorem 4.1.

Equivalence o] weight ]unctions; smooth weight ]unctions.Suppose f and g are positive functions defined on the same set. We write

] g if ]g- and g]- are bounded.If , are weight functions and , are positive continuous functions such

that and , then , are also weight functions. We say that theyare equivalent to , . Equivalent weight functions will define the same classes ofpseudodifferential operators and weighted Sobolev spaces.

PROPOSITION 1.26. I] , are weight ]unctions, there is an equivalent pair ,such that ]or each a, :(1.27) () (") cO-"-;(1.28) [()(") c,-"’-’.

Prool. Choose I. (R) with I 0, l(t) 1 if [tl c, f(t) 0 if It 2c. Let

{(Y, v)(Y, v)}-" gv dy.

By (1.3), the integrand is supported where (y, ) (x, ) and (y, )(x, ). Therefore . The derivatives of may be estimated in a straight-forward way: an x-differentiation introduces a factor (y, v)- which is (x, )-on the support of the integrand, etc. The function may be defined in a similarway.

PROPOSITION 1.29. I] , are weight ]unctions, there is an equivalent pair ,satisIying (1.27) and (1.28) such that R - satisfies

(1.30) R()(") c.R(R+/ + )-"(R-/ + )-Proo]. Choose , as in Proposition 1.26. Let G + R +/, g +

R-/ GR- We claim that if c is small enough, then

(1.31) Ix y cg(x,) and - v] cG(x,)

implies

(1.32) c g(y, v)g(x, )- c- and c G(y, v)G(x, )- c-.

8 RICHARD BEALS

In fct, by (1.5), if (1.31) is true then R(x, ) R(y, 7). Therefore (1.32) is trueunless

G(x, )

_C max {(x, ), (y, 7)}

and similarly for g. But these inequalities and (1.31) tken together imply that

q)(x, ) (y, 7) and (x, ) (y, 7).

This implies (1.32), a contradiction.Now let be as in the proof of Proposition 1.26 and set

Rl(x, ) ff R(y, )](g(y, )- Ix y I)](G(y,

g(y, )"G(y, )n dy d.-1This function satisfies the estimates (1.30), and we may replace by R1 to

complete the proof.

We say that pir of weight functions is smooth if it stisfies the inequalities(1.27), (1.28), nd (1.30).Note that (1.27) and (1.28) with la -[- tl 1 imply (1.3), while (1.30) with

la -[- ! 1 implies a slightly weaker version of (1.5), with the condition onIx Yl being

x Yl <- cR(x,

2. Weight vectors.

Assumptions. We say that a 2n-tuple of positive continuous functionsO, , ., , , is a pair of weight vectors if there are

positive constants c, C, such that for all :(2.1) ,.

__C,

(2.2) ,i >_ c,

(2.3) c ;(x, }));(y, )-’

_C and c

_;(x, })(y, )-

_C if each

< <(2.4) R(x, O)

_C (x} c, where R -1;

(2.5) c

_Ri(x, )R(y, )-’

_C if each

and ]10 1 <- cRy(x,

(2.6) ((x, )

_C((x, )c, all

(2.7) with r(x, ) min (x, )(x, ),(., )(y, q)-

_C[(x, ) --[- (y, )]c and

q(x, )(y, )-’

_C[-(x, ) -[- -(y, )]c

if lx y[ and I[ vl satisfy the conditions in (2.5).


We say that weight vectors , are localizable if for each compact a C Rthere is a constant e > 0 such that

(2.8) (x,) >_ e() on a X Rn,allj.

Remarks 2.9. We interpret ), as assigning to each point of phase spacecharacteristic units of length ; in the x-direction and ; in the -direction.The interpretation of (2.1)-(2.5) is the same as that of (1.1)-(1.5) for weightfunctions. The condition (2.6) is clearly unnecessary for weight functions, and(2.7) is also superfluous in that case: (1.22) and (1.23).

Examples.2.10. When, are weight functions in the sense of.section 1, we may take, ,allj.2.11. Let 1, ,- [(1 n-i) + (n)1/2, n+ + ,_ + ], These weight vectors are particularly adapted to the

discussion of second order parabolic equations and their parametrices.

Consequences o] the assumptions.Suppose , are weight vectors.

PaOPOSTION 2.12. Let R -. Then ]or each j

(2.13) cR-/ E C, cR(2.) c , c(() + (x)) .Pro@ The proof of (2.13) is the sme s that of (1.17). Let rain

The proof of (1.18) shows

c c(() + (x)),

while (2.6) shows E Cc.PROPOSITION 2.15. Suppose M R, m R and

a M MI] the estimates on x y] and in (2.3) are satisfied, then

(2.6) a(x, )a(y, )- C{(x, ) + (y, )},where C C(M, m) and rain.Pro@ This is consequence of (2.7).

Equivalence o] weight vectors; smooth weight vectors. If , re weight vectorsnd , is 2n-tuple of positive continuous functions such that # nd, 11 j, then , re lso weight vectors. We sy that they re equivalentto , .The following two propositions, nd their proofs, prllel Propositions 1.26

nd 1.29 nd their respective proofs.

10 RICHARD BFLS

PROPOSITION 2.17. I] 9, b are weight vectors, there is an equivalent pairb, such that ]or each

(2.18)(a)(2.19) [() c,-"-.

PROPOSITION 2.20.satis]ying (2.18) and (2.19), such that R - satisfies(2.21) R() (")] c,R(R+/ + )-"(R-/ + )-.

3. Classes of symbols and operators.

Orders. Suppose , is a pair of weight functions. Let 0(, ) denote thespace of continuous real functions defined on R with the properties"

(3.1) IX(x, }) X(y, v)l C if ]x y g c(x, ) and l} v (x, });

(3.2) for some real K, k, m,

c() e-%- C().0(, ) is a real vector space. For each real K, } it containsPROPOSITION 3.3.

the ]unctions(K, k) K log -4- k log ,

(K) K log R.

Proo]. The first statement is an immediate consequence of the definition.If ), (K, k), then (3.1) follows from (1.3), and (3.2) is true with m 0.

Classes o[ symbols. If X C 0(, ), we let S S,., denote the space ofsmooth complex functions a on R such that for each pair of multi-indices a, f:

(3.3) n,.(a) sup e-l"lll[a()(")[ <We say that such a function a is a symbol o] order .Note that if 9, , , , 0(I,, ), 0(, ), and ex e", then the

corresponding spaces of symbols coincide.

PROPOSITION 3.4. S is a Frechet space with respect to the seminorms in (3.3).II a S’ and b S, then ab Sx/ and a() I1 a is any multi-index, S- C S.

Proo]. Each of these statements is an immediate consequence of the defini-tions, Leibniz’s rule, and the assumption _> c. (We take (a,as defined in Proposition 3.3, to interpret S-(" ’).)PROPOSITIO 3.5. I] 0(, ), there is a symbol a S such that a e.Proof. We proceed as in the proof of Proposition 1.26. Choose ] )(R)

with ] >_ 0, ](t) 1 if Itl

_c, ](t) 0 if Itl >_ 2c. Let

PSEUDODIFFERENTIAL OPERATORS 11

a(x, ) JJ eX(")](e(y,

{(y, 7)e(y, 7)}n dy 47.Examples.3.6 If (), e (), and k m log (), then S is a uniform version

of HSrmander’s class S, in [26].3.7. If k is a "basic weight function" as in [36], he, e k, and

m log , then S" is the class S.. of [36].3.8. Suppose p is a smooth positive function on R with all derivatives

bounded. Let (), (1 -f- log (>)--1, % p(X) log (). Then S-(’) isessentially a uniform version of the class S(p; It) of Unterberger and Bokobza [54].

Classes o] operators. Suppose O(q, ) and a Sx. Let

(3.9) a(x, D)u(z) f e""a(x, )() d’, u

It follows from (3.3), (3.2), (1.17), (1.18), and (1.19) that

(3.10) [a(x,Therefore the integral (3.9) converges absolutely.

PROPOSITION 3.11. I] a S, the operator a(x, D) is a continuous mapping]rom $ to $.

Proo]. Suppose u . By (3.10) and similar estimates for the derivativesof a, we may differentiate under the integral sign in (3.9) and conclude thatv a(x, D)u is smooth. Moreover

e (x>-(1-so we may integrate by parts in (3.9) and get

{a(x, D)u(x){ f (x)-’e’(1 A)a(/, )fi() d’

< C<x>- f (<x> + <>)’:<>- d

for any positive integer N. This shows that v is of rapid decrease. Finally,

Dv(x) a(x, D)u(x) + a(x, D)Du(x)

where a. i-10a/Ox S-’. By induction, each derivative of v is of rapiddecrease. The argument shows that seminorms of v can be estimated in termsof seminorms of a and of u, so the proof is complete.

Let 2 2. be the space of operators from $ to g which are of the formA a(x, D) for some a Sx. The next proposition shows that the corre-spondence between symbols and operators is bijective. Therefore we maytopologize 2 as a Frechet space with a a(x, D) a topological isomorphism.

12 RICHARD BEALS

PROPOSITION 3.12. Given o R,, there is a sequence (uj) C S such that]or any a S and any x R’,

(3.13) a(x, o) lim e-Xa(x, D)u(x).

Proo]. Take v )(Rn) with v()d’ 1, and take u. with ;() v(j( o)).The case o] weight vectors. Suppose , 1, , 1 ,are

weight vectors. We define 0(, ) as the space of continuous real functionssuch that

(3.14) I’(x, ) (Y, )1 -< C if each Jx- y] <_ c(x, )

and each ]; nl -< c(x, );

4. The calculus of pseudodifferential operators.

The composition and adjoint theorems. Suppose , are weight functions orweight vectors. The corresponding classes of pseudodifferential operators areclosed under composition.

THEOREM 4.1. I] A a(x, D) is in 2 and B b(x, D) is in 2 then ABis in 2/. The symbol a o b o/AB has the asymptotic expansion

(4.2) a b (a!)-la(")b(,)

in the sense that ]or any finite set J o] multi-indices,

(4.3) aJ J

(3.15) for some K, ]c Rn, m R,

c < eX-- _< Cr,where again r(x, ) min .(x, );(x, ).The preceding results all carry over to this case. We let

(K, ]) K, log . + lc; log

(g) (K, -K) K; log R;, K

These functions are in 0(, ). The space S is the space of smooth a such thatfor each a, ,(3.16) n,.(a) sup e-X"ola(Then Propositions 3.4 and 3.5 remain valid for weight vectors. If athen a(x, D) is defined by (3.9), and Propositions 3.11 and 3.12 carry over.Finally, x is defined as before.


In the case of weight functions we may replace (4.3) by the simpler version

(4.4) a o b-ai<N

Consider the sesquilinesr psiring on

(4.5) (u, ) f u()/() dx,

which induces an imbedding ofIf A" 8 -- 8 is continuous, we denote by A* the corresponding adjoint mappingin 8*. Our classes of pseudodifferential operators are closed under the takingof adj oints.

THEOREM 4.6. I] A a(x, D) is in 2), then the restriction to 8 o] A* is alsoin 2). Its symbol a has an asymptotic expansion

(4.7) a*(x, ) (a!)d()

COROLLARY 4.8. I] A 2), it has a unique continuous extension mapping8" to 8".

Pro#. Let B be the restriction of A* to $. Then B* extends A. Since $ isdense in $*, the extension is unique.From now on we shall consider A a(x, D) as being defined on g*, by identi-

fying it with its extension.Theorems 4.1 and 4.6 are proved at the end of this section.Asymptotic expansions. Suppose , are weight functions or weight vectors,

and suppose (;)o is a sequence in 0(, ) such that

(4.9) 0 _< ; ;+1 _< ,;.-1 , j > 1:

More generally, we may assume

(4.10) ),+ <- i + ci

(4.11) Xi Xi+ < d(X_ X) + c,

where c; and d; are positive constants. Note that (4.10) implies

(4.12)

THEOREM 4.13. I] ()o is a sequence in O(q, ) which satisfies (4.10) and(4.11), and i] ai Sxi, then there is a symbol a Sx such that a ai in thesense that

(4.14) a [ ai Sx.i<N

Proo]. Let

bi(x, ) h(ei exp {h;_(x, ) i(x, )})ai(x, ),

where h C(R), h(t) O, <_ 1, h(t) 1, > 2, and e; > 0 is to be chosen.

14 RICHARD BEALS

Thenb; a; except where ; 1- ; < log-e On this set

D(._, ) + C+ E+Therefore

b- aSx.Now b 0 except where ex tex-’. Since a Cex, the series a bwill converge nd be dominated by ex provided

Ctit- o 2-.Similarly, if is small enough for j N, then

la,, I e, + I1 Nnd

l{a- b,},, ICe, II+IIN.Thus can be chosen so that a is s desired.

Proo# o] the composition and adjoint theorems. Consider the ease of weightfunctions; the ease of weight vectors is similar, but notationlly more complex.Theorem 4.1 follows easily from the next lemm.

LEMMA 4.15. Suppose b(x, y, , n) (R) satisfies estimates:

(4.16) IDD,bl Cee’+’,--,where X(x, n,), , (z, n,), (Y, ), (Y, ), n, + t(n ),and O 1. Lt(4.17) a(x, ) ff e’-"-)b(z, y, , ) dy dn.Then a satisfies the estimate

(4.8) tal Ce+,.where the constant C depends only on the C (and not on t).

Proo]. We my assume that , are smooth. All the constants below mybe chosen to depend only on the C (nd on , , X, ). Set

Vo (x, ), o (x, ), , (x, ,), , (y, ),

Ko exp IX(z, ) + (z, )], K exp [X + ],

R -’ R Ro + R + R, r (Ro-’ + R- + R-) -’

o+ + %, o++,h exp [i(z y)(n )],

ao= +Rolx-y[+Ro-’[-nl,a= +l-yl+r-’l-nl.


Let L be the differential operator

Lu g-l(1 RA, -r-lA)u.

Then L , and we may integrate by parts in (4.17) to get

(4.19) a(x, ) f ]b,(x, y, , ) dy d,

b, (L)b,

where N is a positive integer to be chosen. Recall that R C and R-C; see (1.16). Therefore

(R’/D,)"b + (r-1/D)"b C.(R/r)’"/K

and similarly with g in place of b and K. Therefore

(4.20) Ib[ ,Cg-(R/r)K.

We shall estimate a by writing (4.19) as the sum of integrations over threesubregions, a a, + a + a, and using (4.20) in each subregion. We take a,to be defined by integration over the region

--1(4.21) x-- y + - -- vl c.

If c is chosen so small enough, then in this region we have , K Ko, etc.Therefore in this region (4.20) becomes

(4.22) [b] Cgo-Ko.

We choose N > n. Then the right side of (4.22) may be integrated over allof R" to give the estimate [al] CKo.

Let a be defined by integration over the region

(4.23) -1 Ix- y[ + - [ c,

(4.24) R-r-/ v + R-r/ x y c.

If c is small enough, then (4.24) and (1.5) imply that R R Thereforewe also have

(4.25) g C()MKosee (3.2) and (1.25), which explain the term (). Now (4.23) implies

so (4.20) becomes

C M-(4.26) b] go go.We choose N > n W M and integrate the right side of (4.26) over M1 space.

16 RICHARD BEALS

Finally, let a8 be defined by integration over the complement of the regions(4.21) and (4.23), (4.24). Over this region,

> R Ix- + r-’ ,I(R/r)R[R-r x -Yl "-R-2R- I- 1’]

>_ c,(R/r)R2 >_ c2n,

If e > 0 is small enough, then,

Alsog >_ c(R/r)R’g .K <_ CRMKo.

Therefore (4.20) becomes (since g >_ go)

(4.27) Ibl <_ Cgo-gRM-VKo.We take N > n/e, N >_ M/e and integrate (4.27). This completes the proof.

Theorem 4.6 follows easily from the next lemma.

LEMMA 4.29. Suppose b(y, , 7) )(R’) satisfies estimates

IDD,"bl < C.e-I"l-I1

where X X(y, v), , (y, v), (y, v), v + t(v ), and 0< 1. Let

a(x, ) f e(-)("-)b(y, , v) dy dv.

Then a satisfies the estimate

a[ Ce,where C may be chosen to depend only on the C, and not on t.

The proof is essentially the same as that of Lemma 4.15, but simpler.

Proo[ o Theorem 4.1. Suppose first that the symbols a and b have compactsupport. A simple calculation shows that AB has symbol

(a b)(x, ) f ha(x, v)b(y, ) dy d’v,

where again h h(x, y, }, v) exp [i(x y)( })]. By Lemma 4.15 (with0),

aob] < Ce+

S-seminorms of a and the S"-seminorms of b.where C depends only on theThe identities

(v })h -Dh, (x y)h -D,h


allow us to integrate by parts after differentiating a o b and conclude that

Dx(a o b) (D,a) b + a o (Dxb),

D(a o b) (Da) b + a o (Dab),

and similarly for higher derivatives. Therefore Lemma 4.15 also yields appro-priate estimates for derivatives. The estimates are uniform for a and b inbounded subsets of S and S", respectively, so we may approximate a given pairof symbols by a sequence of pairs having compact support and conclude thata o b S+.To derive the asymptotic expansion, we may assume once more that a, b have

compact support. Consider the Taylor expansion

a(x, q) ., (a!)-la(")(x, )(v ) -[- aN(X, ),

where

aN(x, ) N _, (a!)- a(")(x, -t- t(7 ))(v )" dt.

An integration by parts and use of the Plancherel theorem show

a o b _, (a!)-la(")b(,) + N(a!)-’r,[al<N [al=N

where

r.(x, )

By Lemma 4.15,

f ha(")(x -t-- t(7 ))b(.)(y, 7) d’7 dy dt.

[r,[ <_ Ce+’-("’"),where C depends only on the appropriate seminorms. Derivatives of r arelinear combinations of terms with a similar form, and may also be estimatedby Lemma 4.15. This completes the proof of Theorem 4.1.

Proo] o] Theorem 4.6. Again, suppose a has compact support. Directcalculation shows that A* has symbol

fa (x, ) ha(y, 7)dy d’7.

The argument now proceeds as in the proof of Theorem 4.1, with Lemma 4.29used in place of Lemma 4.15.

5. L-boundedness.

Suppose , are weight functions or weight vectors.A" 8" -o 8".

If A recall that

18 RICHARD BEALS

THEOREM 5.1. I] A 2 (), then A" L --. L continuously.

The proof of this theorem uses results of Cotlar-Knapp-Stein [29] and ofCalderon and Vaillancourt [12]; we refer to these papers for the proofs.

THEOREM 5.2. (Cotlar-Knapp-Stein) I] (A) is a sequence o] bounded opera-tots in a Hilbert space and"

(5.3) ]A] M, all ;(5.4) JlA**A,II + IIAA**I[ M IJ- ][- lot I- It[ N;then A cotverges strongly to an operator A with

THEOREM 5.5. (CalderSn-Vaillancourt) I] a C (R) satisfies(5.6) a()(") C, all a, ,then the corresponding pseudodifferential operator a(x, D) is continuous in L,with norm depending only on the constants C,

Proo] o] Theorem 5.1. We shall assume that , are smooth weight functions;the proof in the case of weight vectors is only notationally more complex.

It is sufficient to consider a symbol a (R) provided we show that thenorm of A a(x, D) depends only on the S()-seminorms of a. The constantsbelow can be chosen to depend only on these seminorms.

Let (f), -1 j < , be a smooth partition of unity on [0, ), chosen sothat 1_ is supported in [0, 1) and 1 is supported in (2-, 2+), j 2, and alsoso that"

(5.7) ],(t) 1 if It 2[ < 2-, j 0;

(5.s) ll,(’)(t)l 5 c,2-, 0.

(Such a partition of unity exists.) Let

g(x, ) ](cR(x, )),

where R - and c > 0 is to be chosen. Since (5.7) and (5.8) are assumed,and , are assumed smooth, it follows that (g) is bounded in S (). Leta ga. Then (a) is also bounded in S (). We want to apply Theorem 5.2to A A, a,(x, D).To prove (5.3) we let K R/. Recall thatK C,K- g C. Onthe

support of a;

Thus

cK <_. K <_ CKi Ki 2.


for some e > 0.

(5.11)

or

Let V; be the unitary operator in L-"Vu(x) K"/u(Kx).

Then B; V*AV is the pseudodifferential operator with symbol

b(x, ) a(K-lx, K).The estimates (5.9) imply

(5.10)

C,o independent of j. By Theorem 5.5, I[B;]I _< M. But lIB;I[ [IA[[, so (5.3)is established.

According to (1.4) and (1.5),

c <_ K(x, )K(y, )-’ <_ C unless lY x[ >_ c(K(x, ) + K(x, 7)K(x, )-’,and

c <_ K(x, ()K(x, 7) -1 <: C unless I,- (I > (K(x, ) + K(x, v))/,

Therefore, if ]j k > C then dk(x, 7)a(y, () 0 unless either

[Y

(5.12) 17 (] > c(K + K.)’K-where K min {K K}.Now

A*Aiu(x) J k(x y) u(y) dy

k(x, y) f ha(z, ()a(z, 7) dz d’(

h exp [i(x z - z7 YT].

We integrate by parts to get

k(x, y) f hbv dz d’(

br (x z}-2V(z y}-N(1 A)N(1 A,)[l 71-(-A)dk(z, )a(z, 7)].

Then

(5.13) Ibl <_ C{(x z}-(z Y)-I(- 7I-I(K + K,,)}.On the support of b if IJ lcl -> C, then (5.12) is true. We rewrite (5.12),with a new e > 0, as

20 RICHARD BEALS

and substitute in (5.13) to get

(5.14) Ibvl <_ C{(x z}-l(y z}-l(- 7}-(K-[- Kk)-} 2N.

If 2N > n, 2Ne _> 3, and 2Ne > n, we may integrate (5.14) to get

Ik(x, Y)I - C<x y>-2T(K + Kk) -3.

This implies the desired estimate for ]]AA }.To estimate IAA*II it is convenient to look at the Fourier transformed

operators"

f(AAk*u) () k(, 7),(7)

where

](, 7) J ha(x, )d(y, ) dx cl’ dy,

h exp !i(-x + x y +We integrate by parts to get

/(, 7) f hb dx d’ dy

b (- }-’(- v}-’(1 A)(1 A)

[ix y-(-- A)(a(x, )d(, y))].

Then

}-I(K +(5.15) b C{(- } (- y

On the support of b. if ]j k] C, then (5.11) is true. Therefore, witha new > 0,

x y c(K + K)(x y}(K)-.We substitute in (5.15) to get

(5.16) b.] C(- }-(- }-’]x y-(K + g)-.If 2M > n, 2Ne > n, and 2Ne 2M + 3, we may integrate (5.16) to get

+This, together with the Plancherel theorem, gives the desired estimate for[AA*]

6. Weighted Sobolev spaces.

Suppose , are weight functions or weight vectors. If 0(, ), we let

H H. span {Aulu L, A 2-’}.


This is a subspace of $*. We give it the finest topology with respect to whicheach mapping A" L --, Hx, A 2-x, is continuous.

If (}, (}, and k m log (}, then H is the Sobolev space Hm.In general, these spaces are again Hilbert spaces.

THEOREM 6.1. Suppose , O((P, ). Then

(a) H () L, topologically;(b) H *, densely and continuously;(c) (HX) * H-x, topologically;(d) A 2 implies A" Hx/" -- H continuously;(e) there is an operator A 2 such that A" Hx/E -- H is a topological iso-

morphism. In particular, H has the topology o] a Hilbert space.

The proof of this theorem depends in part on the following lemma. Forsimplicity, we state it only in the case of weight functions.

LEMM 6.2. I 0(, ) satisfies

(6.3) c(.) _< e"-- _< C()

with Igl + I]1 + Im[ <_ e for suciently small > O, then there are sequences ofsymbols av S, bv S- such that as N ---,

(6.4) aob- 1--,0 and bv o a l O in S ().

Proof. We may and shall assume that , are smooth and that is chosenso thate" S. Takeg (R) withg _> 0andg(t) 1if It[

_1. Let

a’(x, ) g(N-R(x, )).

We know c <_ <_ CR, so on the support of a’:CN < e" < CN.

On the support of the derivatives of a’ we have R N. The smoothnessassumption means

IR()(.) <_ C.R(R+I/. + )-I.I(R-,/..We may assume ti _< 1/2. These inequalities imply

(6.5) {N-D"ar’} is bounded in S-(/), lal 1;

(6.6) {N-Da’} is bounded in S/(/), I[ 1.

Next, let h 1 g and set a" h(N-)e. On the support of a", _> N(so R >_ cN), and

It follows that the a" satisfy (6.5) and (6.6) also; in fact one could take exponents1/2 in place of ti e.

RICHARD BEALS

To handle the region where < N and R > N, we observe that K(@)eR, where P + p /, P- p k. Let

av’"= h(N-1R)g(N-I,;,)R.On the support of aN’",

cN-’ < e"R- < CN’.

Arguing as above for the derivatives of aN"’, we find that the aN’" also satisfy(6.5) and (6.6). Let aN aN’ + a#" + aN’", b aN-. Then the argumentshows also that the bN satisfy (6.5) and (6.6) with replaced by -t. Theoperational calculus then shows that

{N2-"’(aN o bN 1)} is bounded in S <>

and the same is true for bN o aN 1. We only need e < t.Remark. 6.7. In the preceding construction, let pN N’aN qN N-’bN

Supposea Sx. Then

(6.8) qN} is bounded in S-";

(6,9) pN o qN 1 and q o pN 1 --, 0 in S<);(6.10) {N-2’(a o PN PN 0 a)} is bounded in Sx/’.

Proo/ o] Theorem 6.1.(a) Theorem 5.1 implies H C L continuously. The identity operator,

having symbol 1, is in O,,soHO L.(b) By definition, H C 8". Since A - is continuous from 8" to itself,

it is continuous from L to *. Therefore the inclusion is continuous. Theother part of (b) follows by duality once we prove (c).

(c) Suppose (d) and (e) have been proved. If A x, then A* x, soA*: H -- L continuously. Therefore A" L -- (H) * continuously. Thisproves H- C (H)*, continuously. On the other hand, if A is chosen so thatA*: H --, L" is a topological isomorphism, then also A (L) (HX) *, hence(HX) * C H-x continuously.

(d) If U C H is open, we must show A-I(U) open in H+. This is true ifsnd only if B-*A-(U) (AB)-*(U) is open in L whenever B 2-x-". Forsuch B, AB 2-’, so (AB)- (U) is indeed open.

(e) Suppose first that ) --- 0 and that t is bounded below and is as in Lemma 6.2.The boundedness condition implies S-" C S), so H C L. Choose sequencesof symbols (a), (b) as in Lemma 6.2. Let A aN(X, D), Bv bv(x, D).Theorems 4.1 and 5.1, together with (6.4), imply that ABv and BNAv areeventually invertible as operators in Ls. Choose N large and let A AB B Then A" H" --, L is onto, with continuous right inverse B(AB)-.Also, A is injective on L D H, with continuous left inverse (BA)-IB. Itfollows that A" H" -- L is a topological isomorphism. Now B’ L --+ H" hascontinuous right inverse (AB)-A. Because A is a topological isomorphism, itfollows that B is also. Note that we may take adjoints and conclude, as in the


proof of (c), that A*: L -- H-" and B*: H- -- L are also topological isomor-phisms. Therefore this alsoproves (e) when 0, t is small (as in Lemma 6.2),and t is bounded above.Now we want to assume h --- 0 and small, but to remove the boundedness

restriction on t. This restriction was used only to ensure that AN was eventuallyinjective on N". Let e log R, which is bounded below. For small theabove argument applies, so H is a Hilbert space. For small t, H H-v.Now (o) C (H-,) continuously, so BNAN is eventually invertible in H andAN is eventually injective on Hr. This shows that the boundedness assumptionmay be dropped.The argument just given will extend to prove (e) for arbitrary and small

once we know that H is a Hilbert space. Therefore we may induce on M toconclude that H is a Hilbert space provided

[)’1 <- Me log R + C,where e > 0 is fixed and small. Every h 0(, ) satisfies this restriction forsome M. Finally, we may remove the restriction on by a similar induction.This completes the proof.

Remarlc 6.11. To some extent, at least, the space H depends only on h andnot on the weight functions , . For example, if )

_and

_ , and ifx. The corre-, 0(, ) 0(9, k), then clearly S. is contained in

sponding inclusion follows both for the spaces H and for their duals H-x, sothe spaces coincide.

Compact operators and compact imbeddings.

THEOREM 6.11. Suppose a S and suppose that ]or each

(6.12)

Then A a(x, D) is compact as a map ]rom Hx+" to Hx.COtOLLaY 6.13. I] U, 0(, ) and u --) + as Ix] + I[ --* ,

then H" H compactly.

Proo]. In Theorem 6.11, replace u by u h and let a 1.Given a closed set in R", let H be the (closed) subspace of H consisting of

distributions with support in .COOLLtY 6.14. I] U, 0(, ) and u + as Ii uni]ormly

/or x in a neighborhood o] a compact set r, then H," Q H compactly.

Proo]. In Theorem 6.11, replace t by u h and take

a(x, ) g(x), g (R"), g - 1 on a.

Proo] o] Theorem 6.11. Choose g )(R) with g(t) 1 for Itl

_1. Let

gN(x, ) g(N-[]xl + I1 +

RICHARD PEALS

Then (gN)N >_ is bounded in S (). LetaN gva. Thena aN--+0in S"asN -+ o, so AN aN(X, D) converges in norm to A. But aN is in (RZ), so

H + HA $*

limit in norm of compact maps, it is compact.

Determining a norm in Htopological isomorphism of H onto L. The norm ]u[x I]Au] is then ad-missible in Hx, that is, it defines the topology of Hx. The following two proposi-tions give a more practical determination of an admissible norm in many cases.

PROPOSITION 6.15. I] a S is a symbol which is independent of x and ]orwhich a] ex, then A a(x, D) is a topological isomorphism o] H onto L.There is an admissible norm defined by

(6.16) jujx f a(x, )() d’.

Proo]. Let b a- and let B b(x, D). The assumptions imply thatb S-x and that AB BA I. Therefore A is a topological isomorphism.By the Plancherel theorem, the norm (6.16) is [Au]]We shall state and prove the next proposition for weight functions, and then

indicate the necessary modification for weight vectors.

PROPOSITION 6.17. Suppose a, a a S and ]ai] ez. Suppose0(, ) and

(6.18) ce e C()e

]or some m O. Let A a(x, D). Then u is in H i] and only i] u is in Hand each A u is in L. There is an admissible norm in H defined by

ProoJ. We may assume that m is a positive integer. The inequality (6.18)implies S-x D S-", so H" C Hx. Therefore the conditions u Hx, Au Lare necessary, and the norm (6.19) is continuous in H". To prove the converse,we let ( X)/m and show by induction on the nonnegative integer k that

(6.20) u Hx+ u Hx, Au Hx+-",and that the norm defined by

(6.21) Ilu[I.+2 Ilu[l + IIAull+=-2is admissible in Hx+" for any choice of admissible norms in Hx+’-’, lc 0,1, m. This is trivial when lc 0. Onee more the conditions are necessaryfor u to be in Hx+,, and the norm (6.21) is continuous in Hx+,. To prove theconverse, we let b ( lal=)- S-=" and B b(x, D). The assumption6.18 implies v C. By the operational calculus, therefore,

(6.22) I


If u H and each A ;u Hx/k-" C H/(k-1) -’, then the inductive assumptiongives u H+(-1). Then Tu H+ and (BA*)(Au) H+, so u Hx+.Moreover, (6.22) and the inductive assumption imply readily that the norm(6.21) is admissible. When k m, we obtain the desired result.In case q, are weight vectors instead, we must replace (6.18) by the condition

(6.23) ce < e < Cr’ex,where again r min ;,..essentially the same.

The corresponding proposition and proof are

Examples.1. In any case when m log (} 0(, ), the corresponding space is, by

Proposition 6.15, the usual Sobolev space H’.2. If (} -t- (x} and 1, then for any nonnegative integer m, it follows

from Proposition 6.17 that H’) consists of all u L such that (x}rDu Lfor 0 k m, + k m. Moreover, an admissible norm is

liull,.,o) II<x> DaullI1 +k=m

Thus H(’’) is the space H(,) of [21].

(7.1)

THEOREM 7.2.a S and

(7.3)

7. Global operator theory.

Throughout this section we shall assume that the weight functions or weightvectors , are coercive, i.e., that (in the case of weight vectors)

[xl + all j.

Suppose , are coercive and , u O(,I,, ). Suppose

lal ce for Ixl + I[ C.

Then A a(x, D) is a Fredholm operator ]tom Hx+ to Hx, i.e. it has finite dimen-sional kernel, and the range is closed and has finite codimension.

I] also tt ---) A- as Ix[ -4- 11 then as an operator in U with domain Hx+,A has the property that its spectrum is either C or a discrete set, and any point o]the spectrum is an eigenvalue o] finite multiplicity.

Proo]. Choose j C(R2) with j 0 if Ix A- I] -< C and j 1 if Ix[ -4-I1 >2C. Letb ja-l(=Oiflx[A-I] <-C) andsetB b(x,n). By Theorem4.1 and Theorem 6.11, AB-I and BA-I are compact in H for every h. There-fore A is a Fredholm operator.Now assume tt --* o as Ixl -4- I1 -- o. For any complex z, then,

[a z >_ ce for Ix[-{-[[ >_. C(z).

Therefore A zI is also a Fredholm operator. In particular, if z is in thespectrum, it is an eigenvalue of finite multiplicity. If A ZoI is invertible withinverse S" H -o Hx+", then (A zI)S I "4- (Zo z)S. By Corollary 6.13,

26 RICHARD BEALS

Hx+" C H compactly, so S is compact as an operator in H. It follows thatA zI is invertible except for a discrete set of z. This completes the proof.We say that , are strongly coercive if (in the case of weight vectors) there

are c, > 0 such that

(7.) >_ c([xl + [1) , al .LEMX 7.5. II , are strongly coercive, then ]or any X 0(, ),

(7.6) (% Sx-("’") $(R2n), (% Hx- ("’") $(Rn).

Proo]. The first assertion is immediate from (7.4). The second assertionfollows from the first, since a $ implies a(x, D)(52) C $.

THEOREM 7.7. Suppose , are strongly coercive, and suppose a Ssatisfies (7.3). Then ]or each 0(, ), the operator A a(x, D)" H/ -- Hhas kernel contained in $ (hence independent o] ). I1 A is injective (respectivelysurjective), it has a le]t (resp. right) inverse which is in 2-.

I1 --, o as Ix[ + [1 "* , then any eigen]unction o] A in $* is in $.

Pro@ Choose j as in the proof of Theorem 7.2, and define bo ja-1. Defineb inductively by

b,v -bo a!)-:a(’)(bv_,.,)(o,)

Let b S- be a symbol such that b b see Theorem 4.13. It followsfrom Theorem 4.1 that

[a o(" b,.)] 1 J S-(

r<N

Therefore a o b 1 $(R2n), and B b(x, D) is a right parametrix for A, i.e.

(AB I)" g* --) g,

Similarly, A has a left parametrix; therefore B iself is also a left parametrix.Now supposeug*andAu 0. Thenu (I BA)u is in g. If--, o

as Ix[ + Il and z is complex, the operator A zI satisfies the same hypoth-eses as A. Therefore the kernel of A zI in g* is contained in $, also.Suppose A" H /" H has a bounded inverse A-. Then

A-1 B (I- BA)A-I(I- AB) + B(I- AB),

which maps $* to $. It follows that A -1 B is an integral operator with kernelc $(R’).

(A-’ B)u(x) f y)u(y) du, u g

Let r be the Fourier transform of k with respect to the second variable"

r(x, f "]c(x, y) dy.


Then A-1 B r(x, D), and r 8(R) =/’. S. This shows that A-1 2-".Suppose A" H/ -- H is injective. Note that the symbol d of A*A satisfies

[d >_ ce for Ix[-b I1-> C,by Theorems 4.1 and 4.6.and

Therefore, if u 8" and A*Au O, we have u 8

0 (A*Au, u)= ]lAull 2.

Since A is injective, this implies u 0. Now the adjoint of A*A as a mapfrom Hx/" to Hx- is A*A as a map from H-x to H-x-, and these are Fredholmoperators. It follows that A’A: H/ -- H- has an inverse. We know that(A’A)- is in 2-". Therefore A: H/ -- H has left inverse (A*A)-A * 2-.

Finally, suppose A" Hx/ --, H is surjective. Then A* is injective, and Ahas right inverse A*(AA*)- 2-.

Examples.1. Consider the n-dimensional Hermite operator A --A - Ixl. Let

1, (x}-[- (}. ThenA (.o). As an operator inL with domainH(’) {u H Ixl u Ll, A is symmetric and positive. It is Fredholmas a map of H(’), therefore selfadjoint in L. Its spectrum is discrete, theresolvent operator is in (-") and is compact, and the eigenfunctions are in $.

2. More generally, the results of this section contain results of Gruin in [22]and [21, 2].

8. Localization:the operational calculus.

Suppose t is a non-empty open set in R. A pair , of positive continuousfunctions defined on [t R will be called local weight ]unctions if for eachcompact a C 2 there are localizable weight functions defined on R" X R whichcoincide with , on a R.Throughout this section we shall assume that , are local weight functions

on ft R", and shall "reduce" the local theory of pseudodifferential operatorsto the global theory of sections 1 to 6.A continuous linear map A: (2) -- *(t) is said to be properly supported

if for each compact C It there is a compact C 2 such that

suppAuC r if suppuC a;

(suppAu)a if suppu r

If A is properly supported, so is its hermitian adjoint A*; (ft) -- )*(t). If so,then A: (ft) --, *(gt) continuously, and A has a unique continuous extensionmapping 8(t) into *(2).A map A: )(2) -- *(gt) is said to be smoothing if it has a continuous extension

mapping *(t) into (). This is true if and only if A is an integral operatorwith a smooth kernel. If A, B: )(t) --, *(gt), we write A B if A B issmoothing.

28 RICHARD BEALS

Given local, weight functions , , let 0(,, ,) denote the space of continuousreal functions k defined on 2 X R and satisfying (3.1), (3.2) uniformly oneach z R, z compact in t. If k 0(, ), we let

=be the set of smooth functions a on X R which satisfy

a( ()] C(z)eX-’’-’’ on z X R,

for ech compact C . If a S(), then

a(x, D)u(x) f e’:a(x, )() d’, u ()

defines mp A a(x, D)" () (); in fct A" $ (). Again, thecorrespondence between symbols nd operators is bijective.We let 2() 2.() denote the set of properly supported mps A" ()

*() which are of the form A a(x, D) for some a Sx.(The notations 0(, ) nd 2() s used here conflict with the notations of

the erlier sections, nd it would be more precise to write 0o(, ) nd2o.o() here; however no confusion should rise.)

LEMMA 8.1. If a Sx(), there is a symbol a Sx() such that a(x, D)a(x, D), a a S"(), and a(x, D) ().

Proof. Let be a neighborhood of the diagonal in X with the propertythat for each compact C , the sets {(x, y) ;x }, (x, y) ;y }are compact. Choose g 3( X ) such that g 1 in a neighborhood of thediagonal, while supp g C . Let

Au(x) f e(-)a(x, )g(x, y)u(y) dy d’, u ().

The assumptions on g and imply that A is properly supported. The differenceB a(x, D) A is given by

where

Bu(x) f eXb(x, )t() d’,

b(x, ) f e’(-)(’-)a(x, n)[I g(x, y)] dy

Given x t, the integrand in (8.2) vanishes if Ix Yl -< c. As usual, we mayintegrate by parts and replace a(1 g) in the integrand by

lc ix YI-(--AV)[(V )-=(1 Ay)a(1 g)].

Then


where P is fixed, > 0, and we may assume e_< 1. If2Ne > P, then

Since N may be taken arbitrarily large, and since the same argument appliesto derivatives of b, it follows that b and each derivative of b is of rapid decreasein (, uniformly on compact subsets of t. This implies b S". The operatorB is an integral operator with kernel

which is smooth in x, y (integrate by parts again, to prove absolute convergencein (8.3) and its derivatives.) Thus B is smoothing.

THEOREM 8.4. I] A a(x, D) is in 2(ft) and B b(x, D) is in 2(), thenAB is in 2x/"(f). The symbol a o b o] AB has the asymptotic expansion

(8.5) a b (a )- ’a b

in the sense that ]or any finite set J o] multi-indices,

(8.6) a o bJ aJ

Proof. Since A and B are properly supported, so is AB.O.

t!set o f, choose compact o’, C ft such thatGiven a compact

B(b) C ),, A(b,,) C ),,

and such that u 0 on o’ implies Au 0 on o, while u 0 on o" impliesBu 0onot. Chooseg 0(2) withg lonotk.)o’anddeline

A lu gAu, Bu gBu, u

Then A1 2 and B 2" with respect to extensions of , which coincidewith q), on (supp g) X R. Therefore ABI 2+. Our assumptions implythat AB1 AB on ), and also that for any u , AIBtu ABu on o. Itfollows that on o X R the symbol of A IB is determined by A, B, independentof g. Thus the conclusions follow from Theorem 4.1.

THEOREM 8.7. IJ A a(x, D) is in 2x(ft), then the restriction to )() oJ A*is also in 2x(f). Its symbol a has an asymptotic expansion

(8.8) a*(x,

Pro@ A* is properly supported. Given a compact o C , choose o’ asbefore. Choose g 0(f) with g 1 on a neighborhood of 0’. Then A gAis in 2 with respect to a pair of weight functions which coincide with , on(suppg) XR". ThenAl*2x. ButAl* A*on0andAu Auonoforany u . Therefore A* has symbol coinciding on o X R with the symbolof A *, and Theorem 4.6 applies.

30 RICHARD BEALS

The case o] weight vectors. We say that a 2n-tuple , 1 .1 , of positive continuous functions defined on 2 X R" is a pair oflocal weight vectors if for each compact z C G, there are localizable weight vectorsdefined on R R which conicide with , on z X R. The results of thissection carry over without change to the case of local weight vectors.

Finally, Theorem 4.13 on the existence of a symbol with a given asymptoticexpansion carries over easily to the local situation.

9. Localization" weighted Sobolev spaces.

Suppose that 2 is a non-empty open set in R, and that , are local weightfunctions or local weight vectors defined on 2 X R. If is compact, let , beweight functions or weight vectors defined on R2" and coinciding with , onT X R", where r is a neighborhood of . If h 0(,), we may choose

0(, ) so that coincides with , on r R. Let

H, (H.

the subspace of H consisting of distributions with support in . It is immediatethat H, is independent of the choice of extensions I,, h, " in view of Theorem6.1(e), v H if and only if v Au where A .C- and u L. Take gwith g 1 on . Then v H if and only if v * and v gAu for someu L. If A a(x,D), then v =Bu where B b(x,D), and b(x,)g(x)a(x, ) is in S-(2).

Let H(2) be the union of the spaces H, for all compact z C 2, with theinductive limit topology. Let Ho(2) be the space of distributions ] *(2)such that u] Ho() for every u b(12), equipped with the correspondingFrechet topology.

It is clear from the definitions that there are continuous inclusions

(9.1) (2) C Ho(2) C *(2);

(9.2) ;() C Hoo() C b*(2),

with each space dense in the next. Therefore

(9.3) (2)

(9.4)

If A (2), then we know A* x(2) also. Since A* maps (2) to itselfand (12) to itself (being properly supported), we may consider A as being definedon b*(12) and as mapping Ib*(2) to itself and *(2) to itself.

THEOREM 9.5. Suppose , 0(, ) and A 2(2). Then

(9.6) A: HX+(2) ---, H0X(f) continuously;

(9.7) A: HoCX+"(2) ---, H,,0(ft) continuously.

ISEUDODIFFERENTIAL OPERATORS 31

Moreover,

(9.8) (HoX (ft)) * Htoo (ft);

(9.9) (H,oox(ft)) Hc-x(ft).

Proo]. If a C f is compact, there is a compact z a such that A (9,) Cand A*(a),) C ) We may extend , , X, tz from a neighborhood of r, andchoose A1 3" so that A A on ), and A lu Au on if u 0 on r. Itfollows that A :Hx/ --, H continuously, agrees with A on H,x/, and mapsH,x/ to Hx. This proves (9.6), which implies (9.7) (since A is properly sup-ported).

If ] H cX(ft) * and u (ft), we may extend , , X as before and concludethat u (HX) * H-x. Thus ] Ho-X(ft). Clearly Ho-X(ft) C HX(ft) *.The identity map is closed, hence continuous.

Finally, if / HooX(ft) *, then ] has compact support. Extend , , }, accord-ingly. Then ] (HX) * H-x, so ] H,-X(ft). Conversely, it is clear thatHo-X(f) C Hoox(ft) *. The topologies coincide on each subspace H,-x, henceare the same.

THEOREM 9.10. SupposeuniIormly ]or x in a neighborhood o/a compact set a C , then H, C H, com-pactly.

Proo[. Extend , , h, tz and use Corollary 6.14.

Given 0(, ) and compact C ft, we call any norm defining the topologyof H, an admissible norm in H,x. The following result is a consequence ofProposition 6.17, after , , have been extended. We state it first for localweight functions.

PROPOSITION 9.11. Suppose A ai(x, D) is in 3"(), 1 <_ j <_ r. Supposethat ]or each compact (r C ,

[al >_ Ce if x a and I1 >- C.

Suppose also that ]or each a there is an m >_ 0 such that

(9.12) cex <_ e" <_ C()e on a X R’.

Then u ), is in g," i/ and only i/ u H, and each A,u L. I] I[ lxisan admissible norm in H,x, then

defines an admissible norm in

In the case of local weight vectors, the product in (9.12) should again bereplaced by min

32 RICHARD BEALS

Remark 9.14. Let v log , in the case of local weight functions, or logein the case of local weight vectors. By assumption, for each compact

on z R". It follows that for ny , 0(, ), there is an integer N lrgeenough that

H C Hx-.Moreover, for any real s, there is an N such that the Sobolev space H,-’ C Hx-’.It follows that

Hoo’() 8(),

the intersection being tken over ll 0(, ).

10. A priori estimates and hypoellipticity.

Let be a non-empty open set in R", and let , be local weight functions orlocal weight vectors defined on R. Recall that A" *(2) - )*(2) is saidto be hypoelliptic if ] *(2) and A] smooth on a subregion 2 C 2 implies ] issmooth on ftl.

THEOREM 10.1. Suppose 0(., ) and A 2(2). I] any o] the ]ollowingstatements is true ]or some 0(, ), then each o] the statements is true ]or every

0(, ), and A is hypoelliptic"(a)x I u fi)*(2) and Au Ho (12), then u HoX+"(ft);(b)x I] u 8"(12) and Au HX(12), then u HX+’(ft);(c)x For each compact r C 2 and each real s, there is a constant C such that

i u ), then

where II [1 is an admissible norm in H, and I[ ]1- is an admissible normin the Sobolev space H,-.

-k-t(d)x For each open U with compact closure in 2, each real s, and each ] Ho,there is v Hoo-X() such that (A*v ])Iv H,oo"(U).

Proof. It is clear that (a)x (b)(b)x (c)x. Given z and s, let

E {u H,-"; Au Hx()Iand let

(10.2) lul,- ]JAn/Ix /

Clearly, E is complete with respect to this norm. Since (b)x implies Eand since the inclusion map is closed, the closed graph theorem gives (c)x

(c)x (d)x. Let E, be as before, and let F, be the closed subspace generatedby D,, where is the closure of U. If ] Hoo "(2))*, then (c)implies that u -- (], u) is continuous on F, On the other hand, u -- [u, Au]


imbeds F, as a closed subspace of H ( Hx. Therefore there is [w, v] in thedual space H @) H-x such that

(], u) (w, u) + (v, Au), u (U).

Thus A*v- ] -w Hoo pn U.(d)x (c)x Let G, be , with the norm (10.2), where a is the closure of U.The antilinear distribution pairing (], u) is a sesquilinear form on Ho () XG, which is continuous in ] for fixed u. By (d)x it is also continuous in u forfixed ]. As a form on the product of a Frechet space and a normed space, it istherefore jointly continuous. Joint continuity is precisely (c)x.

(c)x (c)+ for any u 0(, ). We have (in the case of local weightfunctions)

c(V )- ev-%- g C().It suffices to consider the case K] + k + m e, for some fixed e > 0. Letbe a neighborhood of a, compact and contained in . Let , be weight

functions coinciding with , on X R. Extendsuch that A A on . It is possible, if e is small enough, to choose sequencesof operators P, Q such that:

Q is bounded in

PQ I and QP.

{N[P, A]} is bounded in

In fact, this follows from Remark 6.7, after reindexing.withg 1ont. Thenproof of Lemma 8.15).

< C.

_< C._< C_< C<

If we take a fixed N(c) for every , (b)x for every h. Suppose u (gt) and Au Ho’.

Choose gon )*, the commutator [g, Pz] is smoothing (see theIf u and s is given, then for large enough N:

]]dgPu]]x + Cy’

[IPaul I + C. l][a, PJul I + C" lu[

> C, this inequality isOnce

again, extend the weight function or vectors, the orders and u, and the operatorA from a neighborhood of supp u. Choose g ff)(R 1) with g 1 near 0, andset q(x, ) g(j-lRl(x, )). We assume, as usual, that , are smooth. Then

34 RICHARD BEALS

the operators Qi qi(x, D) are bounded in 2(o) and, it is readily checked,converge strongly to the identity. Moreover, since q has compact support,Q is smoothing. Let i log /min R;}. The smoothness assumption impliesthat {D"q} is bounded in S--/ for ]a 1, while {Dq} is bounded inS-+/ for ]] 1. Therefore the commutators T [Q A] re boundedset in ’-. Choose N so lrge that u H’- H-. Then

Qu+_(_) Co{QA_(_) + Tu_(_) + u_}

This {Qu} is bounded and weakly convergent to u in Hx+"-(-. Iterating,u Hx+".

(b)x for every h (a)x for every h. Suppose u *() and Au HoCX().Let U be open in , with compact closure. Choose A X(U) which differsfrom A on 8*(U) by a smoothing operator. Then A satisfies (b)x on U andAlU HoCX(U). If g D(U), there is an e > 0 such that the commutatorTv A(gv) gA v is an operator in 2"-, v log ][. There is a positiveinteger N such that on U, u Hx+’-(U). Then

A(gu) gA,u + Tu Hx-(-)(U)-- (N-l)so gu Hx-(-’)(U), and u Ho (U). Iterating, u Ho (U).This is true for each such U, so u HoX().

(a)x for every hypoellipticity. In fact, the equivalence of the statements(a)x (d)x implies that (a)x holds in each subregion If Au 8(1)

HoCX(,), then u HoCX+"() (). This completes the proof.

CoaoLnY 10.3. I] A satisfies the conditions o] the preceding theorem, thenA*A is hypoelliptic.

Pro@ Suppose z C is compact and suppose u D. We apply (C)(o)

C((A*Au, u) + lu[

if s is large enough. Therefore

]$ul[, N C’(]A*Au]$_, +Since A*A 2=", this inequality is (c)_, for A*A. Thus the theorem appliesto A*A.

Examples.10.4. Let A be the operator in two variables"

A a(x)D + b(x)x=D= + c(x)D + d(x), a(0)b(0) > 0.


The coefficients are assumed smooth (but not necessarily real). This operatoris hypoelliptic in a neighborhood of the origin. In fact, let Ixl21 -+- I1] q-],.I /" q- 1, R (}, o qR-. Then , are local weight functions. Let, 2 log . Clearly A "(Rn). For any compact z,

is an admissible norm.of A*A shows

Also,

If z is a sufficiently small neighborhood of 0, a check

Ilull. C([IAull + Ilull),

Ilull c,,. Ilull-. + Ilul[,=,

Therefore (C)o is true and A is hypoelliptic. (For other treatments of thisoperator in varying degrees of generality, see Baouendi [1], H6rmander [27],Grugin [21], Si6strand [47], Boutet de Monvel and Treves [7], [8], Boutet deMonvel [6].)

10.5. Let A be the operator in two variables:

A a(x)x,.D" 4- ib(x)D,. -k c(x)D1 q- c2(x)

where the coefficients are smooth, not necessarily real, and a(0)b(0) < 0. Thisoperator is hypoelliptic in a neighborhood of the origin. In fact, let],1 q- (}/", , 1, (I)2 IX21l + (), R2 2

__(}, (2 (I)2R2-’. Then

are loeM weight vectors. Let , log 2 Then A "(R"). For anycompact C R",

If a is a sufficiently small neighborhood of 0, a checkis an admissible norm.of A*A shows

llull. C(IIAull + [lull),Again, this implies (C)o is true, and A is hypoelliptic. (With a(x) b(x) 1,c(x) O, this example is due to Kannai [28]; see also Kumano-go and Taniguchi[36]. Kannai’s result has been generalized to the case of analytic real coefficientsby Zuily [57].)

11. Parametrices.

Throughout this section we assume that f is an open set in R", that q,, arelocal weight vectors defined on fa X R, that is in O(q,, o), and that the operatorA is in 2."(f0 and has symbol a. We shall say that an operator B -"() is

36 RICHARD BEALS

a left (respectively right) parametrix for A if BA I (resp. AB I) is in2(t). This implies that the latter operator is a smoothing operator (see

Remark 9.14). Therefore, if A has a left parametrix B 2-’(t), then A ishypoelliptic.Note that B is a left parametrix for A if and only if B* is a right parametrix

for A*. We now restrict our attention to the existence of a right parametrix.This is a local problem (indeed a microlocal problem).

PROPOSITION 11.1. A "(2) has a right parametrix B 2-() if andonly if for each x there is a compact neighborhood z(x) o] x, containedin , and an operator B -() such that AB I coincides on ) with anoperator in 2 ().

Proo]. Let {g.} be a smooth partition of unity on t such that each supp gis contained in the interior of some z(j) z(x(j)). Choose h. (.) withh. 1 on a neighborhood of supp g.. Let

Bu hB() (gu), u

The difference between AhB,()g and hAB,()g is in (t)" see the proof ofLemma 8.15. Therefore

AB I ., h(AB(,) I)g,--[- _. (Ah, hA)B(,)g

is in (2). The converse is trivial.

Throughout the rest of this section we let 0(, ) be such that for eachcompact C there are c, i > 0 such that

e >_ c(} on X R.Examples: log q) or e log R, > O.

PROPOSITION 11.2. A 2"([t) has a right parametrix B 2-() if andonly i] there is Bo -() such that ABo I 2-().

Proof. Necessity is trivial. Suppose Bo exists, and set T I ABoLet t be the symbol of T,m 0, 1, 2, .... Thent. s-m(). There isa symbol c S () () such that

c-<

all m: see Theorem 4.13.Let B BoC 2-().

We may assume C c(x, D) is properly supported.Then for any m,

AB- I ABoC- I (I- T)C- 1

(I- T)(I+T-b -T"-)- I

-[- (I T)(C- I- ’ T"-’)It follows that B is a right parametrix.


PROPOSITION 11.3. A (2) has a right parametrix in 2-() i] and onlythere are B b(x, D) 2-(), j 1, 2, such that the composed symbols satisJy

o)(11.4) [(a o b- 1) I_ Ce --, where min .; <: Ce

(11.5) la o b2- 11_

Ce-V; where min ;; _> C-’e

]or suigiciently large C.

Proo]. Necessity is trivial. In the usual way we may assume is smooth,and smooth mint .. to get a function r min; ;. such that

Choose g (R) with g(O 1 when It] I, g() 0 when t] 2. Let

bo g(e-r)b + (1 g(e-v))b.

Derivatives of g are supported where e. The operational calculus and(11.4), (11.5) imply a bo 1 S-(). Proposition 11.2 implies that A hasparametrix.

It is easy to check the possibility of satisfying (11.5).

PROPORTION 11.6. There is a symbol b S-"() satisfying (11.5) i] andonly i] ]or some large C(11.7) a] C-le when C and min C-e.Pro@ In the region being considered, if b S-"() then

(11.8) ]aob ab Ce-.Therefore (11.5) implies (11.7). Conversely, if (11.7) is true, then there isb S-"() which coincides with a- in the region described in (11.7). Itfollows from this and (11.8) that (11.5) is true.

Example 11.9. This example arises naturally in the study of the obliquederivative problem and in the general theory of equations of principal type:see Egorov [18], [19], and Treves [49], [50]. Let x (x x’), x’ R-,(, ’), and let A have symbol

a(x, }) } + ixp(x, }) + po(x, ),

where k is a positive integer, p and po are symbols in HSrmander’s classes Sand S respectively, p is real-wlued, and

(11.10) pl(x, ) c ]’ if ]’ C.

We are interested in the existence of a right parametrix in a neighborhood ofx 0, so we may modify p and po, assuming they are independent of x and

38 RICHARD BEALS

for Ix[ large and that (11.10) is true globally. Let

g(x, ) I1]-F xl2k I’1-F <>, i (2/ -F 1)-’,

i (}, i 1, j 2, 3, n.

Then, C are local weight vectors. Letg logg, log (}. ThenaS"(R) and min g/*()*-/*. It follows that (11.7) is true, andwe only need to consider the region where C (}. In this region

(1.11) ,1 c ’1.Write multi-indices as a (a a’). Then

a(,’") S"-"-’"’) if [a’[ + [’[ > 0 or a 2.

Also, po S"-". It follows that if b S-"(R"), then

a o b- [aob + (O,ao)(D=,b)] S-"(R"),where

Thus we want to solve

(11.12)

for a function b(xl x’, ).

ao + ix, p,(x, ).

aob + (O,ao)(D,,b) 1

This is an ordinary differential equation in the caseat hand, but we want to obtain the qualitative properties of the solution by ageneral argument. First, we may transform (11.12) to

(11.13)

where A, is the ordinary differential operator with symbol

a,,(t, r) O,ao(t, x’, )r -F itkp,(t, x’, ).

If A,. has a right inverse which is a pseudodifferential operator then the-1solution of (11.13) is just the symbol a,. evaluated at xl, r 1.

Let V, be the automorphism" V,](t) ](r-it). Then V,-A.,V, has symbola,.(rt, r-). In particular, let

P,,. II-’V,-’A=,,,, ]r]--The operator has symbol

2kp,,. r -F it=l[-’p2(ll st, x’, ) -F irx, O,p,(x, ),wherep= p-O,p. LetI, IrI-Ft=-F 1, 1. Then the operators

(.o) for I1 > 1 It follows from (11 10) and{P,,.} are a bounded set in 2.Theorem 7.7 that for I1 >- C the inverses {Q,,.} exist and their symbols {q,,.}

(-1 ,o)are a bounded set in S,. In fact for ][ large,(a) lal -Ifl


and it follows that() 1-1al I1(11.14) Iq.,,(o, -< C-o’’I’- (min {1 1, (t)})-

with similar estimates for derivatives of q,. with respect to xl and 1 as well.The symbol of A,.- for [(I large is

We extend b to be smooth for Il small. In view of (11.14), b(x, ) b,,.(x. ,)is a symbol in. S-"(R’).The same argument shows that A* has a right parametrix. Therefore A

has a left parametrix and

(11.15) Ilull, c(llAull + Ilull-,), u .Since e" _> c((}, (11.15) contains the subelliptic estimate for A" see Treves [49].The method used for this example applies in outline to the next two examples.

Weight vectors are found, the problem is reduced to a problem in fewer variables,a family of homotheties reduces these problems to a family of elliptic operatorsto which Theorem 7.7 applies.

Example 11.17. In two variables (for simplicity), let A have symbol2k k-1a(x, ) - x p(x) + xl p.(x) - pa(x) + p,(x),

where Re p(0) > 0. Let

r (x, I ,l + Ixl%.l + + 1)

t gl/k()$(l-1/k), (1

((), , 1, u 21ogg, u 6log

Then A has a right parametrix B near x 0 if the ordinary differential operator

L D, + p(O)x + p(O)x-’

has a right inverse in 2.t(-.o), where [] + x[ + 1, 1. In turn,L has a right inverse in this class if and only if there are no nontriviM solutionsu $(R) of the equation L*u 0. (This will be true for all but a discrete setof values of p(0), given p(0).) Cf. [6], [7], [21], [36], [48].

Example 11.18. Let A be the operator of Example 10.5, with the weightvectors , and the order given there. The question of existence of a leftparametrix for A, or equivalently of a right parametrix for A*, reduces againto the existence of a right inverse for

L* d(O)x2- i(O)D.in .(-.o) with ,I, Ixl - 1(21 + 1, k 1. The question then becomeswhether Lu 0 has a nontrivial solution u g(R1). Since a(0)b(0) < 0 byassumption, there are no such solutions and A does have a left parametrix.Cf. [28], [36], [57].

])EFERENCES

1. M. S. BAOUENDI, Sur une classe d’oprateurs lliptiques ddgndrds, Bull. Soc. Math. France,vol. 95(1967), pp. 45-87.

2. R. BEALS, Spatially inhomogeneous pseudodifferential operators, II, Comm. Pure Appl.Math., vol. 27(1974), pp. 161-205.

3. R. BEALS, AND C. FEFFERMAN, On local solvability of linear partial differential equations,Annals Math., vol. 97(1973), pp. 482-498.

4. Classes of spatially inhomogeneous pseudodifferential operators, Proc. Nat. Acad.Sci. USA, vol. 70(1973), pp. 1500-1501.

5. Spatially inhomogeneous pseudodifferential operators, Comm. Pure Appl. Math.,vol. 27(1974), pp. 1-24.

6. L. BOUTET DE MONVEL, Hypoelliptic operators with double characleristics and related pseudo-differential operators, Comm. Pure Appl. Math., vol. 27(1974), to appear.

7. L. BOUTET DE MONVEL, AND f. TREVES, On a class of pseudodifferential operators withmultiple characteristics, Inventiones Math., vol. 24(1974), pp. 1-34.

8. On a class of systems of pseudodifferential operators with multiple characteristics,to appear.

9. A. P. CALDERSN, Uniqueness in the Cauchy problem of partial differential equations, Amer.J. Math., vol. 80(1958), pp. 16-36.

10. Existence and uniqueness theorems for systems of partial differential equations,Symposium on Fluid Dynamics, Univ. of Maryland, College Park, Md. 1961.

11. Singular integrals, Bull. Amer. Math. Soc., vol. 72(1966), pp. 427-465.12. A. P. CALDERSN, AND R. VAILLANCOURT, On the boundedness of pseudo-differential operators,

J. Math. Soc. Japan, vol. 23(1971 ), pp. 374-378.13. A class of bounded pseudodifferential operators, Proc. Nat. Acad. Sci. USA,

vol. 69(1972), pp. 1185-1187.14. A. P. CALDERSN, AND A. ZYGMUND, Singular integral operators and differential equations,

Amer. J. Math., vol. 79(1957), pp. 901-921.]5. CHINoHUNG, CHING, Pseudo-differential operators with nonregular synbols, J. Diff. Equa-

tions, vol. 11(1972), pp. 436-447.16. V. Yu. EoIov, Hypoelliptic pseudo-differential operators, Dokl. Akad. Nauk SSSR, vol.

168(1966), pp. 1242-1244; Soviet Math. Dokl., vol. 7(1966), pp. 808-810.17. --------, A priori estimates for first order differential operators, Dokl. Akad. Nauk SSSR,

vol. 185(1969), pp. 507-508; Soviet Math. Dokl. vol., 10(1969), pp. 368-370.18.------, Nondegenerate hypoelliptic pseudo-differential equations, Mat.. Sbornik, vol.

82(1970), pp. 323-342; Math. USSSR Sbornik, vol. 11(1970), pp. 291-310.19. V. Yu. E(onov, AND V. A. KONDRAT’EV, The oblique derivatives problem, Mat. Sbornik.

vol. 78(1969), pp. 148-176; Math. USSR Sbornik, vol. 7(1969), pp. 139-169.20. V. V. GRU,N, Pseudodifferential operators in R with bounded symbols, Funkc. Anal. Pril.,

vol. 4(1970), pp. 37-50; Funct. Anal. Appl. vol. 4(1970), pp. 202-212.21. ------, On a class of hypoelliptic operators, Math. Sbornik 83(1970), 456-473; Math.

USSR Sbornik, vol. 12(1970), pp. 458-476.22. ------, On the proof of the discreteness of the spectrum of a class of pseudodifferential

operators in R’, Funkc. Anal. Pril., vol. 5(1971), pp. 71-72; Funct. Anal. Appl.,vol. 5(1971), pp. 58-59.

23. ------, Hypoelliptic differential equations and pseudodifferential operators with operator-valued symbols, Mat. Sbor,fik, vol. 88(1972), pp. 504-521; Math. USSR Sbor,fik, vol.17(1972), pp. 497-514.

24. L. HJRMANDER, Hypoelliptic differential operators, Ann. Inst. Fourier Grenoble, vol.11(1961), pp. 477-492.

25. Pseudo-differential operators, Comm. Pure Appl. Math., vol. 18(1965), pp.501-517.


26. Pseudo-differential operators and hypoelliptic equations, Proc. Symp. Pure Math.,vol. 10, Amer. Math. Soc., Providence, R. I., 1966, pp. 138-183.

27. Hypoelliptic second order differential equations, Acta Math., vol. 119(1967), pp.147-171.

28. Y. KANNAI, An unsolvable hypoelliptic differential operator, Israel J. Math., vol. 9(1971),pp. 306-315.

29. A. KNAPP, AND E. M. STEIN, Singular integrals and the principal series, Proc. Nt. Acad.Sci. USA, vol. 63(1969), pp. 458-476.

30. J. J. KOHN, Pseudo-differential operators and non-elliptic problems, Pseudo-differentialOperators, C.I.M.E. Stresa, 1968. pp. 157-165.

31. ------, Pseudo-differential operators and hypoellipticity, Proc. Symp. Pure Math., vol.23, Amer. Math. Soc., Providence, R. I., 1973, pp. 61-69.

32. J. J. KOHN, AND L. NIRENBERC,, An algebra of pseudo-differential operators, Comm. PureAppl. Math., vol. 18(1965), pp. 269-305.

33. ----, Non-coercive boundary value problems, Comm. Pure Appl. Math., vol. 18(1965),pp. 443-492.

34. H. KUMANO-GO, Algebras of pseudo-differential operators, J. Fac. Sci. Tokyo, vol. 17(1970),pp. 31-50.

35. ____m, Oscillatory integrals of symbols qf pseudo-differential operators and the local solv-ability theorem of Nirenberg and Treves, Katada Syrup. on Partial Diff. Eqns.,to appear.

36. H. KUMANO-GO, AND K. TANIGUCHI, Oscillatory integrals of symbols of operators on R andoperators of Fredholm type, Proc. Japan Acad., vol. 49(1973), pp. 397-402.

37. A. MENIKOFF, Carleman estimates for partial differential operators with real coecients,Arch. Rat. Mech. Anal., vol. 54(1974), pp. 118-133.

38. ------, Uniqueness of the Cauchy problem for a class of partial differential equations withdouble characteristics, Ind. Univ. Math. J., to appear.

39. L. NHEN,:aa, Pseudo-differential operators, Proc. Syrup. Pure Math., vol. 16, Amer.Math. Soc., Providence, R. I., 1970, pp. 147-168.

40. L. NIRENIERG, AND l. TREVES, On local solvability of linear partial differential equations, I:Necessary conditions, Comm. Pure Appl. Math., vol. 23(1970), pp. 1-38.

41. On local solvability of linear partial differential equations, II: Sucient conditions;Comm. Pure Appl. Math., vol. 23(1970), pp. 459-509; correction, ibid. vol. 24(1971),pp. 279-288.

42. O. A. OLEINIK, On hypoellipticity of second order equations, Proc. Symp. Pure Math.,vol. 23, Amer. Math. Soc., Providence, R. I., 1973, pp. 145-152.

43. O. A. OLEINIK, AND E. V. RADKEVI(, On the local smoothness of wealc solutions and thehypoellipticity of differential second order equations, Uspehi Mat. Nauk, vol. 26(1971),pp. 265-281; Russia Math. Surveys, vol. 26(1971), pp. 139-156.

44. E. V. RADKEVIC, Hypoelliptic operators with multiple characteristics, Mat. Sbornik, vol.79(1969), pp. 193-216; Math. USSR Sbornik, vol. 8(1969), pp. 181-206.

45. R. T. SEELEY, Refinement of the functional calculus of CalderSn and Zygmund, Proc. Akad.Wet. Ned., Ser. A., vol. 68(1965), pp. 521-531.

46. J. SMSSTRAND, Une classe d’optrateurs pseudodifferentiels caractristiques multiples, C. R.Acad. Sci. Paris, ser. A., vol. 272(1972), pp. 817-819.

47. Une classe d’optrateurs pseudodifferentiels caractdristiques doubles, C. R. Acad.Sci. Paris Ser. A., vol. 276(1973), pp. 743-745.

48. Parametrice for pseudodifferentiel operators with multiple characteristics, to appear.49. F. Tm,:w,:s, A new method of proof of the subelliptic estimates, Comm. Pure Appl. Math.,

vol. 24(1971), pp. 71-115.50. --------, Hypoelliptic partial differential equations of principal type. Sucient conditions

and necessary conditions, Comm. Pure Appl. Math., vol. 24(1971), pp. 631-670.

42 RICHARD BEALS

51. A link between solvability of pseudodifferential equations and uniqueness in theCauchy problem, Amer. J. Math., vol. 94(1972), pp. 267-288.

52. On the existence and regularity of solutions of linear partial differential equations,Proc. Syrup. Pure Math. vol. 23, Amer. Math. Soc., Providence, R. I., 1973, pp.33-60.

53. A. UNTEmERGER, AND J. BOKOnZA, Les optrateurs de CalderSn-Zygmund prdciss, C. R.Acad. Sci. Paris, Ser. A., vol. 260(1965), pp. 3265-3267.

54. Les optrateurs pseudo-differentiels d’ordre variable, C. R. Acad. Sci. Paris, Ser.A., vol. 261 (1965), pp. 2271-2273.

55. M. I. VIIK, nND V. V. GaVIN, Elliptic pseudodifferential operators on a closed manifoldwhich degenerate on a submanifold, Dokl. Akad. Nauk SSSR, vol. 189(1969), pp.16-19; Soy. Math. Dokl., vol. 10(1969), pp. 1316-1319.

56. --, On a class of higher order degenerate elliptic equations, Mat. Sbornik, vol. 79(1969),pp. 3-36; Math. USSR Sboraik, vol. 8(1969), pp. 1-32.

57. C. ZULY, Sur l’hypoellipticitd des optrateurs differentiels d’ordre 2 coecients rels, C. R.Acad. Sci. Paris, Ser. A., vol. 277(1973), pp. 529-530.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CHICAGO, CHICAGO, ILLINOIS 60637

Documents

A general calculus of pseudodifferential operators