The Technique of Pseudodifferential Operators [First edition.] 0521378648, 9780521378642

Table of contents :
TABLE OF CONTENTS......Page 8
0.0. Some special notations, used in the book......Page 14
0.1. The Fourier transform; elementary facts......Page 16
0.2. Fourier analysis for temperate distributions on In......Page 22
0.3. The Paley-Wiener theorem; Fourier transform for general uE D'......Page 27
0.4. The Fourier-Laplace method; examples......Page 33
0.5. Abstract solutions and hypo-ellipticity......Page 43
0.6. Exponentiating a first order linear differential operator......Page 44
0.7. Solving a nonlinear first order partial differential equation......Page 49
0.8. Characteristics and bicharacteristics of a linear PDE......Page 53
0.9. Lie groups and Lie algebras for classical analysts......Page 58
1.1. Definition of pdo's......Page 65
1.2. Elementary properties of ipdo's......Page 69
1.3. Hoermander symbols; Weyl pdo's; distribution kernels......Page 73
1.4. The composition formulas of Beals......Page 77
1.5. The Leibniz' formulas with integral remainder......Page 82
1.6. Calculus of 1pdo's for symbols of Hoermander type......Page 85
1.7. Strictly classical symbols; some lemmata for application......Page 91
2.0. Introduction......Page 94
2.1. Elliptic and md-elliptic Vdo's......Page 95
2.2. Formally hypo-elliptic pdo's......Page 97
2.3. Local md-ellipticity and local md-hypo-ellipticity......Page 100
2.4. Formally hypo-elliptic differential expressions......Page 104
2.5. The wave front set and its invariance under yxlo's......Page 106
2.6. Systems of ,do's......Page 110
3.1. L2-boundedness of zero-order do's......Page 112
3.2. L2-boundedness for the case of 6>0......Page 116
3.3. Weighted Sobolev spaces; K-parametrix and Green inverse......Page 119
3.4. Existence of a Green inverse......Page 126
3.5. Hs-compactness for ftpdo's of negative order......Page 130
4.0. Introduction......Page 131
4.1. Distributions and temperate distributions on manifolds......Page 132
4.2. Distributions on S-manifolds; manifolds with conical ends......Page 136
4.3. Coordinate invariance of pseudodifferential operators......Page 142
4.4. Pseudodifferential operators on S-manifolds......Page 147
4.5. Order classes and Green inverses on S-manifolds......Page 152
5.0. Introduction......Page 157
5.1. Elliptic problems in free space; a summary......Page 160
5.2. The elliptic boundary problem......Page 162
5.3. Conversion to an &n-problem of Riemann-Hilbert type......Page 167
5.4. Boundary hypo-ellipticity; asymptotic expansion mod av......Page 170
5.5. A system of fide's for the Vj of problem 3.4......Page 175
5.6. Lopatinskij-Shapiro conditions; normal solvability of (2.2).......Page 182
5.7. Hypo-ellipticity, and the classical parabolic problem......Page 187
5.8. Spectral and semi-group theory for ado's......Page 192
5.9. Self-adjointness for boundary problems......Page 199
5.10. C*-algebras of tpdo's; comparison algebras......Page 202
6.1. First order symmetric hyperbolic systems of PDE......Page 209
6.2. First order symmetric hyperbolic systems of fide's on n .......Page 213
6.3. The evolution operator and its properties......Page 219
6.4. N-th order strictly hyperbolic systems and symmetrizers.......Page 223
6.5. The particle flow of a single hyperbolic pde......Page 228
6.6. The action of the particle flow on symbols......Page 232
6.7. Propagation of maximal ideals and propagation of singularities......Page 236
7.0. Introduction......Page 239
7.1. Algebra of hyperbolic polynomials......Page 240
7.2. Hyperbolic polynomials and characteristic surfaces......Page 243
7.3. The hyperbolic Cauchy problem for variable coefficients......Page 248
7.4. The cone h for a strictly hyperbolic expression of type e?......Page 251
7.5. Regions of dependence and influence; finite propagation speed......Page 254
7.6. The local Cauchy problem; hyperbolic problems on manifolds......Page 257
8.0. Introduction......Page 260
8.1. ,do's as smooth operators of L(H0)......Page 261
8.2. The 11DO-theorem......Page 264
8.3. The other half of the gbbO-theorem......Page 270
8.4. Smooth operators; the V -algebra property; 'Wdo-calculus......Page 274
8.5. The operator classes 'S and 'IVL , and their symbols......Page 278
8.6 The Frechet algebras y"x0, and the Weinstein- Zelditch class......Page 284
8.7 Polynomials in x and ax with coefficients in 'TX......Page 288
8.8 Characterization of qtX by the Lie algebra......Page 292
9.0. Introduction......Page 295
9.1. Flow invariance of V10......Page 296
9.2. Invariance of Vsm under particle flows......Page 299
9.3. Conjugation of Optpx with eiKt , KE Opy)ce......Page 302
9.4. Coordinate and gauge invariance; extension to S-manifolds......Page 306
9.5. Conjugation with eiKt for a matrix-valued K=k(x,D)......Page 309
9.6. A technical discussion of commutator equations......Page 314
9.7. Completion of the proof of theorem 5.4......Page 318
10.0. Introduction......Page 323
10.1. A refinement of the concept of observable......Page 327
10.2. The invariant algebra and the Foldy-Wouthuysen transform......Page 332
10.3. The geometrical optics approach for the Dirac algebra P......Page 337
10.4. Some identities for the Dirac matrices......Page 342
10.5. The first correction z0 for standard observables......Page 347
10.6. Proof of the Foldy-Wouthuysen theorem......Page 356
10.7. Nonscalar symbols in diagonal coordinates of......Page 363
10.8. The full symmetrized first correction symbol zS......Page 369
10.9. Some final remarks......Page 380
References......Page 383
Index......Page 393

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London Mathematical Society Lecture Note Series. 202

The Technique of Pseudodifferential Operators H.O. Cordes

Emeritus, University of California, Berkeley

AMBRIDGE

UNIVERSITY PRESS

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

Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521378642

© Cambridge University Press 1995

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission 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 0.1. The Fourier transform; elementary facts 0.2. Fourier analysis for temperate distributions on 0.3. The Paley-Wiener theorem; Fourier transform for

1

3

In

general uE D'

0.4. The Fourier-Laplace method; examples 0.5. Abstract solutions and hypo-ellipticity

9

14

20 30

0.6. Exponentiating a first order linear differential operator

31

0.7. Solving a nonlinear first order partial differential 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 1.0. Introduction I.I. Definition of pdo's

52 52

52

1.2. Elementary properties of ipdo's

56

1.3. Hoermander symbols; Weyl pdo's; distribution kernels

60

1.4. 1.5. 1.6. 1.7.

The composition formulas of Beals The Leibniz' formulas with integral remainder Calculus of 1pdo's for symbols of Hoermander type Strictly classical symbols; some lemmata for application

Chapter 2. Elliptic operators and parametrices in Mn 2.0. Introduction

64 69 72

78 81 81

2.1. Elliptic and md-elliptic Vdo's 2.2. Formally hypo-elliptic pdo's

82

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

87

2.4. Formally hypo-elliptic differential expressions 2.5. The wave front set and its invariance under yxlo's

91

84

93

Contents

viii 2.6. Systems of ,do's

Chapter 3. L2-Sobolev theory and applications 3.0. Introduction

3.1. L2-boundedness of zero-order do's 3.2. L2-boundedness for the case of 6>0 3.3. Weighted Sobolev spaces; K-parametrix and Green inverse 3.4. Existence of a Green inverse 3.5. Hs-compactness for ftpdo's of negative order

97

99 99 99 103

106 113

117

Chapter 4. Pseudodifferential operators on manifolds with conical ends 4.0. Introduction

118 118

4.1. Distributions and temperate distributions on manifolds

119

4.2. Distributions on S-manifolds; manifolds with conical ends 4.3. Coordinate invariance of pseudodifferential operators 4.4. Pseudodifferential operators on S-manifolds 4.5. Order classes and Green inverses on S-manifolds Chapter 5. Elliptic and parabolic problems 5.0. Introduction 5.1. Elliptic problems in free space; a summary 5.2. The elliptic boundary problem

123

129 134 139 144 144

147 149

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

186

Chapter 6. Hyperbolic first order systems 6.0. Introduction 6.1. First order symmetric hyperbolic systems of PDE

189

196 196 196

6.2. First order symmetric hyperbolic systems of fide's on

n .

6.3. The evolution operator and its properties

200

206

Contents

ix

6.4. N-th order strictly hyperbolic systems and 210

symmetrizers.

6.5. The particle flow of a single hyperbolic pde

215

6.6. The action of the particle flow on symbols 6.7. Propagation of maximal ideals and propagation of singularities

219

Chapter 7. Hyperbolic differential equations 7.0. Introduction 7.1. Algebra of hyperbolic polynomials 7.2. Hyperbolic polynomials and characteristic surfaces 7.3. The hyperbolic Cauchy problem for variable coefficients 7.4. The cone h for a strictly hyperbolic expression of type e°

226 226 227 230

235 238

7.5. Regions of dependence and influence; finite propagation speed 7.6. The local Cauchy problem; hyperbolic problems on manifolds Chapter 8. Pseudodifferential operators as smooth operators of L(H) 8.0. Introduction 8.1.

223

241

244 247 247

,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

8.5. The operator classes 'S and 'IVL

261 ,

and their

symbols 8.6

8.7

265

The Frechet algebras y"x0, and the Weinstein-

Zelditch class

271

Polynomials in x and ax with coefficients in 'TX

275

Characterization of qtX by the Lie algebra Chapter 9. Particle flow and invariant algebra of a semistrictly hyperbolic system; coordinate invariance 8.8

279

of OpWxm.

282

9.0. Introduction

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

293

296

Contents

x

9.6. A technical discussion of commutator equations

301

9.7. Completion of the proof of theorem 5.4 Chapter 10. The invariant algebra of the Dirac equation 10.0. Introduction

305

10.1. A refinement of the concept of observable 10.2. The invariant algebra and the Foldy-Wouthuysen transform

314

10.3. The geometrical optics approach for the Dirac algebra P

310 310

319

324

10.4. Some identities for the Dirac matrices 10.5. The first correction z0 for standard observables 10.6. Proof of the Foldy-Wouthuysen theorem

329

10.7. Nonscalar symbols in diagonal coordinates of

350

10.8. The full symmetrized first correction symbol zS 10.9. Some final remarks

356

334 343

367

References

370

Index

380

P R E F A C E

It is generally well known that the Fourier-Laplace transform 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 logically 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 solvability) 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 operator 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 manifestation with respect to such operator algebras: conjugation with

Preface

xii

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 'distribution 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 invariance of ido-algebras under conjugation with evolution operators, and do not touch the type of oscillatory integral and further discussions needed to reveal the structure of such evolution operators as Fourier integral operators. In terms of Quantum mechanics 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 theory 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 conical tips or cylindrical ends, where the 'Fredholm-significant symbol' 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 University 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 transforms of temperate and nontemperace distributions. In sec.4 we discuss the Fourier-Laplace method of solving initial-value problems 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 initialfor boundary problems. Sec.6 is concerned with the operator eLt ,

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 miniintroduction 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

(b)

(x) = (l+Ix12)!/2

(2n)-n/2

,

ox = xndx1dx2...dxn = xndx

(1+1

,

!

12)!,2

,

.

etc.

0. Introductory discussions

2

(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 compoJ

nents

ulxi xi

z

'

For a function of (x,i;)=(xl

..

'.'fin)

"

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 a xa= 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 continuous 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, distribution spaces D', E', SO. (g) LP-spaces: For a measure space X with measure dµ we write Lp(X)=Lp(X,dµ)={measurable functions u(x) with luIp integrable} for lspY

.

.

L(X) (i) Classes of linear operators (X= Banach space) (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) differential equation (or "expression"). FOLPDE (or folpde)= first order linear partial differential equation (or "expression"); 'pdo=

pseudodifferential operator. (n) Integrals need not be existing (proper or improper) Riemann or Lebesgue integrals, unless explicitly stated, but may be distribution integrals By this term we mean that either (i) the integral 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 improper 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 adjoint vely. For a symbols

AT=A*,

in case of a matrix A=((a.k)), respectia* (or a+) may denote the symbol of

as specified in each section. (p) supp u (sing supp u (or s.s.u)) denotes the (singular) support of the distribution u. the adjoint 'tpdo of a(x,D)

,

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)

with x =x. =

(1.2) Note that u^

(1.3)

x E &n ,

u^(x) = J

an existing Lebesgue integral. Clearly,

u^ (x) I s

IIu11L1

=

n9lxlu(x)l

is uniformly continuous over In : We get

u^ (x)-u^ (y) I s 2f¢t s NIX-111 llullL1 +

I

R1

where the right hand side is 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 + The latter gives 100 s

(11vIILW+ 11v' 11L00)

,

f1OjZbcos(NO)wl0(O)dO).

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

6

0. Introductory discussions

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^-u1^IIL2=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^

f (u^ -z)cpdx=0 for

,(p)=f z pdx for all cpE Cc". This yields

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:

xo+b (1.16)

JgIxv^(x)eixxo sin 6x

v(x)dx =

(20)-I

xo -b

Assuming only that v (1.17) limn-0(26)-nfQ

v^ both are in LI

,

xo ,s

v(x)dx =f9lxv^

(Actually, our proof works for n=1 be extended to all xo

,

,

and general n

,

it follows indeed that (x)eixxo=

xo .

> 0 only

(v^ )I (xo ), xoE 2n. ,

but can easily

One must replace the deri-

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

)

Indeed,

0.1. The Fourier transform

7

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

,

(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^)

have u^

,

u" E CO(1Qn)

relation (1.8)

.

.

are defined for all u E L1(&n)

,

and we

For u E L1(In)1L2(In) we have Parseval's

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'

(1.21)

w^ (

let w = u*v

.

Then we have

E &n

) = u^

Proof. We have fpiye

fgixe

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.

(1.22)

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

u(a)^

h

E &n

.

8

0. Introductory discussions

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

,

where fE L1(&n), Dx =-iax j

P(D) =

ja sN

ax

a Da

one might attempt to find solutions by

,

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 noror u(4 LL, in practical applications. However, as to be discussed in the sections below, the Fourier 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).

mally (1/P)(4 L', or f (4 L'

,

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 a_ ax 2

2) For general n obtain the Fourier transform of

,

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

5)The Riemann-Lebesgue lemma states that f^E 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 transform 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 changing its value at boundary points.

,

possibly after

0.2. Fourier analysis for temperate distributions

9

2. Fourier analysis for temperate distributions on &n 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 continuous if (f,gj)-.0 whenever (pj-0 in D. The latter means that (i) fj K independent of j, (iii) E D, j=1,2,..., (ii) supp cpjE KCC &n ,

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 locally 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 (f(a),(P)

( 2 . 2 )

= ( - 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)

pED.

,

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 distribution 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 closed set E (intersection of all closed sets E) such that f=0 (such C00 that f is ) in the complement of E .

The concept of Fourier transform can be generalized to distributions 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.

0. Introductory discussions

10

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

- the derivatives of y decay faster than every power Note that, equivalently, we could have prescribed that for every 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)

CB , or CO

,

or L2

,

or is o(1)

,

or is

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 bijective linear maps S -S , inverting each other. Proof. Using repeated partial integration and get

fdxe

=iIaI+ISIa

(2.7)

hence

(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 (2.5)

,

for u E S

.

11

Therefore the right hand side is in CO

,

so that u^E S

,

for

again by the equivalence (2.5). Thus we get u^ E S for all u E S Similarly for """ . Also, the Fourier inversion formula holds for u E S and the left hand side of every a,(3

,

.

,

(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 temperaso that a functe distributions is denoted by S'. Clearly, S DD 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) - 0

Temperate 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' have uID=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) , implying 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 conver7 gence j(a)(x)-0 in I and weCoO(&n) have y.- 0 in S, proving (a). To prove (b), let X(x)EE 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.

0. Introductory discussions

12

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

It is sufficient to prove this for F. Again we need an equiin S' valence like (2.2), now for the property :

j=1..... Then 'ypj- 0 in S Proposition 2.6. Let q E S equivalent to each of the following conditions: .

,

(x)kgpj(a) (2.8)

0

,

or Apj(a) a 0

or (xs(pj)(a)

,

'

is

0

or k=0,1,2,..., in one (and

for all multi-indices a ,

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

,

(xaq)j^ ) (P) IICB_ 0 , implying (pj^ - 0 , For a given u E S'

(2.9)

,

q.e.d.

observe that u^

qES

(u^ ,p) =

, defined by ,

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)

.

ix

(u,cP^) =

(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))

(2.12)

. We have the Fourier inversion formula

(u^ )` = (u` )^

= u , for all u E S'.

E S' Also, for u E S' we have 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

with finite sums and constants caRY b) We have (2.14)

Ixals(x)lal

,

and (x)ksc

xau(1) = Ed

,

a(3Yxa-Yu(R-Y)

,

(xa-Y u) (R-Y)

,

aRY

daRY

k laIsk

(x011 with a constant c k

c) We have

(2.15)

(lull

Lp

s II(x)-kll

Lp

II(x)kull

Loo

,

Ispn/p

d) We have

(2.16)

Ilull

lscll (1+Ixl )n+1u^ II

scIIU^ II L

=C

L

E

L

lalsn+l

II (U

(a)

"o s c

)^ II

L

c

Ilxau^ II

7,

lalsn+l

I

lalsn+l

IN

L

(a) II

L

Problems: 1) Show that the following functionals define distributions 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

Ixarl

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

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

0. Introductory discussions

14

E D'(2), but eax

O. 9) Let Tpol be the class of all

S', as Re a

k aE 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' invariant. 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 distributions (If necessary, show, they are in S'): a) xa, a E 2+; b) S

xo

(P),

E Zn +

;

c) eiax , aE 2n. 11) Obtain (p.v.1)^, for the dis3F

1

tribution of pbm.1. 12) Define a distribution x E S', using the same kind of 'principal-value integral' as in pbm 1. 13) Obtain the Fourier transform of a Calculate (p.v.sln-h X)^ 2n-periodic C00(2)- function a(x). Hint: Use that a(x) has a uni2n

formly 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), and supp (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) compactly 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 that Tj (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 extension 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)

For the directional derivative Vhez

(3.5)

,

E > 0

of ez at z , we get

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

0 in E

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

0. Introductory discussions

16

dily verified for

Continuity of u then implies

limE-0,Ex0wE _ (u,Vhez)

(3.6)

,

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

(3.7)

(u^ ,q)) =

rP(U)) =

(u,e

f91

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 f

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

R n \K

again, with convergence in E. Again, convergence in E just means local uniform convergence with all derivatives. One confirms easily the local uniform convergence in the parameter x since the function e,(x) = e-'x is continuous. Similarly for the x-derivatives, 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 distributions. The result is called the Paley-Wiener theorem. .

,

Tn

Theorem 3.3. An entire analytic function v(z) over is the Fourier 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} .

0.3. The Paley-Weiner theorem

17

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 ">" holds in (3.9). Or, I(pja)(x)Is for all lalsj 2,... implying uniform convergence Tja)(x)- 0 ,j-*oo ,

Ix1sj ,

,

,

and j=1,

a contra-

diction, since 1=(u, c).) does not tend to 0. We get u^(z)=(u,Xzez) x(t)=1

xz=X(IzI(IxI-i)) where xEC00(R)

, x decreasing. It follows that supp X. C } so that (XZez)(a)(x)= O(e'1IImzI+1(z)k). Combining this {IxI511+ 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= ,

t