Introduction to the Theory of Linear Partial Differential Equations (Studies in mathematics and its applications)

Introduction to the Theory of Linear Partial Differential Equations STUDIES IN MATHEMATICS AND ITS APPLICATIONS VOLUM...

Author: Jacques Chazarain | Alain Piriou

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Introduction to the Theory of Linear Partial Differential Equations

STUDIES IN MATHEMATICS AND ITS APPLICATIONS VOLUME 14

Editors: J. L. LIONS, Paris G. PAPANICOLAOU, New York R. T. ROCKAFELLAR, Seattle H. FUJITA, Tokyo

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK OXFORD

INTRODUCTION TO THE THEORY OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS

JACQUES CHAZARAIN and ALAIN PIRIOU Professors at the Universite de Nice

English version edited, prepared andproduced by TRANS-INTER-SCIENTIA P. 0.Box 16, Tonbridge, T N l 1 8 0 Y , England

1982

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK OXFORD

North- Holland Publishing Company, I982 All rights reserved. N o part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any f o r m or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN 0 444 864.520

Translation of: Introduction a la Theorie des Equations aux Derivees Partielles Lineaires Bordas (Dunod), Paris, 198 I

Publishers: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD Sole distributors f o r the U.S.A.and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 5 2 VANDERBILT, NEW YORK, N.Y. 10017

L i b r a r y 0 1 Congress Cataloging in Puhlication I)ata

C h az a r a in , J a c q u e s . I n t r o d u c t i o n t o t h e t h e o r y of l i n e a r p a r t i a l d i f f e r e n t i a l equations. ( S t u d i e s i n mathematics and i t s a p p l i c a t i o n s ;

14 T r a n s l a t i o n 9 f : I n t r o d u c t i o n a l a f h i o r i e des Gquations aux d e r i v e 6 s p a r t i e l l e s l i n e a i r e s Bibliography: p. Includes index. 1. D i f f e r e n t i a l e q u a t i o n s P a r t i a l . I. P i r i o u , Al ai n . 11. T i t l e . 111. S e r i e s . QA374.C4513 1982 515.3'53 82-8226 I S B N 0-444- 86452-0 (Else v i e r )

.

PRINTED IN T H E NETHERLANDS

TABLE

FOREWORD

OF

CONTENTS

................................................

xiii

............. .........................

Chapter 1 : D i s t r i b u t i o n s a n d o p e r a t o r s

. 2.

Spaces of distributions

1

Convolution and Fourier transformation of distributions

................................... 3 . Singular spectrm of a distribution ............. 4 . Operators and kernels ........................... 5 . Operators and support properties ................

.

1 1 13 21

23

30

6

Differential operators with constant coefficients

33

. 8.

Operators and distributions on a manifold

.......

44

7

9

.

Operators and kernel distributions on a manifold

59

Regular open subsets of lRn and manifolds with boundary

62

........................................

.

10

.................. ............ Homogeneous distributions ............... Fourier transform of eiXx / 2 ............ Fundamental solution of the Cauchy-Riemann operator ................................. Fundamental solution of the Laplace operator .................................

Additional notes and exercises 10.1

10.2

10.3 10.4

10.5

10.6

Paley-Wiener-Schwartz theorem

Fundamental solution of the heat operator -v-

71 71 72 73

74 74 76

vi

CONTENTS

10.7

Fundamental solution of the Schrgdinger operator

10.8

Fundamental solution of the wave operator

10.9

............................... ...............................

Inverse image of a distribution

........

77 78 80

....... 1. Dirichlet's principle ......................... 2. The spaces Hs(mn) and HS (X) ................ loc s S 3. The spaces H (X) and Hloc(X) ..................

104

4.

112

S o b o l e v s p a c e s and a p p l i c a t i o n s

Chapter 2 :

.................. 5 . Application to the Dirichlet problem .......... 6. Sobolev spaces and regularisation ............. 7 . Additional notes and exercises ................ 7.1 Poincar6's inequality ................... Trace theorems. spaces HS (X) 0

7.2 7.3 7.4 7.5

7.6

Invariance of H

S

under diffeomorphism

...

....................... ..................... inequalities ................

Strict inclusions

Lax-Milgram theorem Hbnander's

.

1

. 3. 4.

2

83 85

122

126 131

131 132

133 133 134

Local solvability for differential operators of principal type with real principal symbol

135

Symbols. o s c i l l a t o r y i n t e g r a l s and st a t i onary-phase theorems

139

........................

Chapter 3 :

83

............. Introduction .................................. Symbols ....................................... Elliptic symbols .............................. Asymptotic expansions of symbols ..............

139 140

149 152

CONTENTS

.

vii

................. 6. Various generalisations ....................... 7. Oscillatory integrals ......................... 8. Integral operators associated with a phase and an amplitude .................................. 9 . Stationary-phase theorem ...................... 10. Additional notes and exercises ................ 5

Topology on the symbol spaces

10.1 10.2

10.3

10.4

Symbols with uniform upper bounds with respect to x in Bn

............ Oscillatory integrals and decomposition into plane waves ....................... Lacunas of the fundamental solution of the wave operator ......................

. 2. 3. 4. 5

.

6.

182 182 183

184

Stationary-phase theorem for a surface integral

188

...............................

196

...

216

Pseudo d i f f e r e n t i a l o p e r a t o r s

Symbolic calculus of p.d.0.'~ on a manifold Elliptic p.d.o.'s

........ 10. Friedrich's lemma and generalisations ......... 9.

167

187

............................. 8. P.d.0.'~ and Sobolev spaces .................... 7.

159

......... Definition .................................... A characterisation of p.d.0.'~ ................ Symbol of a p.d.0. ............................ Algebra and symbolic calculus of p.d.o.'s ..... P.d.0.'~on manifolds .........................

Chapter 4 : 1

'157

Fundamental solution of the wave operator for space dimension equal to 1. 2 or 3

...........................

10.5

155

Elliptic complexes and Hodge's theorem

191

205 207 211

221 227 243 250

viii

CONTENTS

.

11

................ P.d.o.'s with uniform symbols .......... P.d.o.'s with uniform symbols and dependent on one parameter ............. Second proof of the estimate (7.6.1) of Chapter 2 ........................... Convolution with certain homogeneous distributions ..........................

Additional notes and exercises

256

11.1

256

11.2 11.3

11.4 11.5

Endomorphisms of complexes and traces

11.6

Cotlar-Knapp-Stein lemma

11.7

Calder6n-Vaillancourt theorem

Chapter 5 :

............... ..........

Elliptic boundary-value problems

.....

..................................

.

Introduction

2

.

Regularity of the potential at the boundary

3

.

The Calder6n projector

1

4.

..

...

........................

...................................... 5 . Examples ...................................... 6. Additional notes and exercises ................ 6.2

6.3 6.4

6.5 6.6 6.7 6.8

Example of a p.d.0. which does not satisfy the regularity property of Theorem 2.5

............................ Cauchy's formula and the Hilbert operator ............................... Weakening of assumption (2.1) .......... Local regularity in the neighbourhood of a point of an

264 264 264 266 267

271 271

Application to elliptic boundary-value problems

6.1

261

....................... Global regularity ...................... Converse of the regularity theorem ..... Case of systems which are elliptic in the sense of Douglis-Nirenberg ......... The oblique derivative problem .........

275 290 295 302

307

307

308 310

310

310 311 312 316

CONTENTS

6.9

Chapter 6 :

.

1 2

.

. 4. 3

ix

Elliptic boundary-value problems with interfaces

........................

320

....................

325

Evolution equations

The Cauchy-Kovaleski and Holmgren theorems

....

325

Necessary condition for the Cauchy problem to be well posed

337

Hyperbolic operators with constant coefficients

344

Hyperbolic Cauchy problems with variable coefficients

359

.................................

.................................. 5 . Parabolic Cauchy problems ..................... 6. Semigroups of operators and applications ...... 7. Additional notes and exercises ................ 7.1

Counter-example to the property of finite propagation speed in the non-differential case

....................

7.2 7.3 7.4 7.5 7.6

Generator of a strongly continuous group

....... . Wave equation with Neumann boundary conditions ............................... Equations of the Schradinger type ....... Further discussion of Theorem 6.9

Energy inequality for the wave equation

............. 1. Introduction .................................. 2. Operators and spaces used ..................... 3. The uniform Lopatinski condition .............. 4. Energy inequalities ........................... 5 . Construction of the symmetriser ...............

Chapter 7 :

M i x e d h y p e r b o l i c problems

378 384 402 402 402 403 404 405 407

409 409 411 421 440 448

CONTENTS

X

6

.

S o l u t i o n of t h e problem without i n i t i a l conditions

.................................... 7 . S o l u t i o n of t h e mixed problem ................. 8 . F i n i t e propagation speed ...................... 9 . A d d i t i o n a l n o t e s and e x e r c i s e s ................ 9.1 Spaces eyt H ' ......................... Y 9.2 Energy i n e q u a l i t y i n t h e s c a l a r case .... 9.3 Algebraic p r e l i m i n a r y t o E x e r c i s e 9.4 ... 9.4 P o s i t i v e symmetric systems .............. 9.5

Example of a boundary-value problem f o r a f i r s t - o r d e r d i f f e r e n t i a l system s a t i s f y i n g t h e uniform L o p a t i n s k i condition

...............................

Chapter 8 :

. 2. 3. 4.

1

5

.

Microlocalisation

......................

General p r o p e r t i e s of t h e s i n g u l a r spectrum (WF)

...................... .........................

460

478 483

491 491 493

494 495

499

501 501

The fundamental theorems

508

The c a s e of manifolds

523

..... ................

The c a s e of p s e u d o - d i f f e r e n t i a l o p e r a t o r s

534

A d d i t i o n a l n o t e s and e x e r c i s e s

539

5.1 5.2 5.3

5.4 5.5 5.6

Conclusion of t h e c a l c u l a t i o n of w F ( G ( x ' ) 0 Y ( x n ) )

...................

539

Conclusion o f t h e c a l c u l a t i o n of WF'(v(x-y)) for v E a'(iRn)

540

Construction of a d i s t r i b u t i o n whose s i n g u l a r spectrum i s a d i r e c t i o n i n T*(X)\O

540

.........

.............................. I n t r i n s i c c h a r a c t e r i s a t i o n of t h e s i n g u l a r spectrum ....................... Trace c a l c u l a t i o n s ...................... The Lefchetz-Atiyah-Bott formula ........

541 543

545

CONTENTS

Bibliography

Index

....................................

...................................................

xi

549

555

This Page Intentionally Left Blank

FOREWORD

Since t h e f i f t i e s , t h e t h e o r y of p a r t i a l d i f f e r e n t i a l equations has undergone c o n s i d e r a b l e development , y e t it i s c e r t a i n L y one of t h o s e f i e l d s where w e f i n d b o t h t h e l a r g e s t number o f r e s e a r c h a r t i c l e s and t h e s m a l l e s t number o f works of s y n t h e s i s . The s p e c i a l i s t can come t o terms w i t h such a s i t u a t i o n , b u t t h e student who d e s i r e s an i n t r o d u c t i o n t o t h i s t h e o r y f i n d s it r a t h e r a handicap.

Thus, i n t h e p r e s e n t book, which i s not a

t r e a t i s e b u t an i n t r o d u c t i o n , we aim t o p r e s e n t a r e a s o n a b l y l a r g e range of r e c e n t methods:

pseudo-differential operators,

o s c i l l a t o r y i n t e g r a l s , s t a t i o n a r y phase expansion, m i c r o l o c a l i s ation

....

We show how t h e s e methods permit t h e s o l u t i o n o f

numerous c l a s s i c a l problems i n an e l e g a n t and g e n e r a l manner: e l l i p t i c e q u a t i o n s , boundary problems, e v o l u t i o n e q u a t i o n s , mixed h y p e r b o l i c p r o b l e m s , . . .

, We have a l l o t t e d c o n s i d e r a b l e

room t o t h e e q u a t i o n s of Physics because, h i s t o r i c a l l y , t h i s has been t h e o r i g i n of important problems i n p a r t i a l d i f f e r e n t i a l equations and remains t o t h e p r e s e n t day t h e p r i n c i p a l s o u r c e of interesting questions.

A s r e g a r d s t h e form of t h e book, we have endeavoured t o g i v e very d e t a i l e d p r o o f s i n such a way a s t o r e n d e r t h e work accessi b l e t o a r e a d e r a t post-graduate l e v e l .

With t h i s i n mind, we

have sometimes p r e f e r r e d t o r e s t r i c t t o some degree t h e g e n e r a l i t y of c e r t a i n results i n o r d e r t o avoid o b s c u r i n g t h e ideas- under-

-xiii-

xiv

FOREWORD

l y i n g t h e t e c h n i q u e s b e i n g used.

Each c h a p t e r b e g i n s , i n

g e n e r a l , w i t h a b r i e f d e s c r i p t i o n of t h e s u b j e c t m a t t e r and t h e methods which w i l l b e developed t h e r e i n , and ends w i t h supplements i n t h e form of e x e r c i s e s t o g e t h e r w i t h h i n t s f o r s o l u t i o n ( a n a s t e r i s k i n d i c a t e s t h o s e which w i l l b e u t i l i s e d i n t h e t e x t which follows).

For t h e convenience of t h e r e a d e r , t h e main e x t e n s i o n s

a r e l i s t e d w i t h i n t h e t a b l e of c o n t e n t s .

i s deliberately restricted:

The b i b l i o g r a p h y given

anyone w r i t i n g on such a s u b j e c t

r e f e r s t o such a l a r g e number of s o u r c e s t h a t an e x h a u s t i v e l i s t of t h e s e would be unwieldy.

It seems t o us t o be p o s s i b l e t o use t h i s book a t two l e v e l s ; on t h e one hand t h e f i r s t f o u r Chapters ( D i s t r i b u t i o n s and Operat-

ors; Sobolev Spaces; Pseudo-differential

Symbols and O s c i l l a t o r y I n t e g r a l s ;

O p e r a t o r s ) can s e r v e as t h e b a s i s f o r an

introductory post-graduate course.

On t h e o t h e r hand, t h e l a s t

f o u r Chapters ( E l l i p t i c boundary problems; Mixed h y p e r b o l i c problems;

Evolution equations;

M i c r o l o c a l i s a t i o n ) a r e aimed more

p a r t i c u l a r l y a t Ph.D. s t u d e n t s and a t r e s e a r c h workers who w i l l f i n d c o n t a i n e d t h e r e i n numerous r e s u l t s f o r m e r l y d i s p e r s e d amongst s p e c i a l i s e d j o u r n a l s .

CHAPTER I

D I S T R I B U T I O N S AND OPERATORS

1.

SPACES OF DISTRIBUTIONS The c o n s i d e r a b l e expansion o f t h e t h e o r y of p a r t i a l d i f f e r -

e n t i a l e q u a t i o n s d u r i n g t h e l a s t twenty y e a r s h a s , i n l a r g e measure, i t s o r i g i n i n t h e development by L . Schwartz of t h e theory of d i s t r i b u t i o n s .

T h i s t h e o r y c l a r i f i e s and u n i f i e s

s e v e r a l concepts which were p r e v i o u s l y r a t h e r u n c l e a r , l i k e f o r example t h e concepts o f :

weak d e r i v a t i v e , g e n e r a l i s e d F o u r i e r The t h e o r y of d i s t r i b u t i o n s i s

t r a n s f o r m , fundamental s o l u t i o n . nowadays u n i v e r s a l l y known;

however, f o r t h e convenience of t h e

r e a d e r , we s h a l l h e r e summarise t h e elements of t h i s which will The most s t a n d a r d of t h e p r o o f s w i l l b e

prove u s e f u l l a t e r .

o m i t t e d , t h e s e b e i n g a v a i l a b l e i n SCHWARTZ [11, HORMANDER [11,

TREVES [ll. Letting

c1

.. , a n )

= (a1,,

E

W n be a m u l t i - i n d e x , we p u t

with

= - l-

D

j

I.\

=

a,

+

...

+ a"

9

1

a! = (a,!)x

a

i ax

...

a

x (wn!);

2

DISTRIBUTIONS AND OPERATORS

i f x = (xl,

...,

n x n ) E IR we p u t 1x1 = (x:

We l e t X be an open s u b s e t o f IRn

, and

+

(CHAP. 1)

... + x2n )1 / 2 .

we suppose m

E

IN u { +

3.

m

m

We denote by C ( X ) t h e space of complex-valued f u n c t i o n s whose d e r i v a t i v e s of o r d e r l e s s t h a n or e q u a l t o

m

a r e continuous.

The f a m i l y of semi-norms

where K i s a compact s u b s e t o f X and

j

i s an i n t e g e r such t h a t A subset

0 5 j S m y e q u i p s C m ( X ) w i t h a Frgchet-space t o p o l o g y .

B of C m ( X ) i s s a i d t o b e bounded i f , f o r any semi-norm p have

Sup cp E B

p (cp) K, j

c

+

OD

K , j y we

.

The s u p p o r t of a f u n c t i o n cp

E

C o ( X ) i s by d e f i n i t i o n t h e

s m a l l e s t c l o s e d s u b s e t o f X o u t s i d e o f which t h e r e s t r i c t i o n o f

i s z e r o (or a l t e r n a t i v e l y , t h e c l o s u r e of t h e s e t of p o i n t s of X where Q i s n o n - z e r o ) , and we denote it by supp

C i orBK)t h e

.

If K i s a

o r b K ( X ) (or more con-

compact s u b s e t of X, we denote by C:(X) cisely

cp

c l o s e d t o p o l o g i c a l subspace of C m ( X ) formed

by t h e f u n c t i o n s w i t h s u p p o r t i n K.

The v e c t o r subspace o f

m

C ( X ) formed by t h e f u n c t i o n s w i t h compact s u p p o r t i s denoted by

C t ( X ) or b m ( x ) (or more c o n c i s e l y & ( X )

when m =

+

m);

e q u i p it w i t h t h e i n d u c t i v e l i m i t t o p o l o g y of t h e C"(X)

K

v a r i e s over t h e f a m i l y of compact s u b s e t s of X.

a l i n e a r mapping from

c:(x)

we when K

In p a r t i c u l a r ,

i n t o a t o p o l o g i c a l v e c t o r space F i s

Q

3

SPACES OF DISTRIBUTIONS

(SEC. 1)

m

continuous i f and only i f i t s r e s t r i c t i o n t o each C,(x)

i s contin-

ous, s o it i s s u f f i c i e n t t o b a s e o u r r e a s o n i n g on sequences. bounded s u b s e t o f $ ( X )

i s a bounded s u b s e t of C:(X)

A

for a certain

compact s u b s e t K .

A distribution

DEFINITION 1.1 :

continuous linear form on

c ~ ( x =)

u

on the open s e t X i s a i . e . a linear form u

&(XI,

on B(X) such t h a t f o r any compact subset K C 2 0, j

IN with \ u ( y ) I

E

C

5

The i n j e c t i o n of & ( X )

p

K,j

there e x i s t s

(Q) f o r a l l Q

Bm(X)

into

c X,

E

i s continuous, and a l s o

o f dense image ( c o n s i d e r a r e g u l a r i s i n g sequence, i . e . of f u n c t i o n s p

where p

E

&(IR")

I p ( x ) dx = 1;

i s such t h a t

r

%

+

=

Ju(y)

u in

&'Im(X)

of

a sequence

K o f t h e form

we c o n s i d e r t h e r e g u l a r i s a t i o n s u

u,(x)

&,(XI.

pk[x-y)

Bm(X)i f

k

Bm(X)i s

+

dy

+

k = u

*

pk o f

suppose u u

E

Bm(X);

d e f i n e d by

and we can e a s i l y show t h a t

a).

Thus t h e d u a l t o p o l o g i c a l space

i d e n t i f i e d w i t h a v e c t o r subspace of 8(X)

( w e i d e n t i f y a continuous l i n e a r form on

% . m( X )

with i t s r e s t r i c t -

ion t o & ( X ) ) .

a m ( X ) i s termed t h e space o f d i s t r i b u t i o n s on X

of o r d e r 5 m.

I n p a r t i c u l a r t h e d i s t r i b u t i o n s of o r d e r 5 0 are

measures.

For example, f o r a

E

X , we d e f i n e t h e Dirac measure

4 8 Q

(CHAP. 1)

DISTRIBUTIONS AND OPERATORS E$'(X)

a E

a

a t the point I Let Lloc(X)

B0(X).

< 6 a , Cp > =cp(a) f o r

by

be t h e space of c l a s s e s o f f u n c t i o n s

which a r e i n t e g r a b l e over any compact s u b s e t of X for t h e Lebesgue measure dx = dxl f dx

E

,&''(X)

.. .

dx

n

i n X.

For f

E

1 Lloc(X),

t h e measure

i s i d e n t i f i e d w i t h t h e d i s t r i b u t i o n ( a l s o denoted

f) d e f i n e d by

by

I n p a r t i c u l a r we s h a l l c o n s i d e r t h e c a s e where

i s t h e Heavi-

f

s i d e f u n c t i o n Y i n IR d e f i n e d by Y(x) = 1 i f x 2 0 , and by Y(X)

=

o

if x < 0.

We now i n d i c a t e w i t h o u t proof s e v e r a l t o p o l o g i c a l p r o p e r t i e s

of a ( X ) ,

a'(X )

which w i l l be u s e f u l l a t e r on.

n a t u r a l t o p o l o g i e s on 8 . ' ( X ) :

t h e topology

ence, and t h e s t r o n g d u a l t o p o l o g y ,

"e,, of

There a r e two

rso f

simple converg-

&(XI, i . e . t h e t o p o l -

ogy of uniform convergence on t h e bounded s u b s e t s o f &(X)

(since

a bounded s u b s e t o f b ( X ) i s r e l a t i v e l y compact, t h e t o p o l o g y o f uniform convergence on t h e compact s u b s e t s of 8 ( X ) c o i n c i d e s with

rb). When we

equip a ( X ) with topology

rb,t h e

strong

d u a l of ,#( X ) i s i d e n t i f i e d a l g e b r a i c a l l y and t o p o l o g i c a l l y w i t h

(X).

Furthermore,

a(X)i s

t h a t t h e bounded s e t s of

rb,and t h a t

&'(X)

a b a r r e l e d s p a c e , which means a r e t h e same f o r

Ts

and

t h e y c o i n c i d e w i t h t h e e q u i c o n t i n u o u s s u b s e t s of

"es

We t h e r e b y deduce t h a t

,@'(X).

5

SPACES OF DISTRIBUTIONS

(SEC. 1)

yb c o i n c i d e

and

on t h e

I n p a r t i c u l a r , i n t h e c a s e of a

bounded s u b s e t s of #(X).

sequence of d i s t r i b u t i o n s , convergence i n t h e s e n s e o f convergence i n t h e s e n s e of

b

c o i n c i d e , and we t h e n speak of a

convergent sequence of d i s t r i b u t i o n s .

,

from a m e t r i z a b l e space i n t o #(X) depend upon t h e t o p o l o g y (

Thus, i f A i s a mapping

t h e c o n t i n u i t y of A does n o t

r sor "e,) w i t h

which we e q u i p #(X).

m e same i s a l s o t r u e i f A i s a l i n e a r o p e r a t o r from

a'( X) , where

zs and

B ( Y )i n t o

s i n c e t h e c o n t i n u i t y of

Y i s a n open s u b s e t of IRp

A i s e q u i v a l e n t t o t h a t of i t s r e s t r i c t i o n t o each

BK(Y).

This

b e i n g t h e c a s e , we s h a l l simply r e f e r t o a continuous l i n e a r i n t o #(X).

o p e r a t o r from &(Y)

,&(X) i n t o &(X)

I n p a r t i c u l a r , t h e i n j e c t i o n of

i s c o n t i n u o u s , and we s h a l l s e e t h a t it i s o f

s e q u e n t i a l l y dense image ( s e e Remark 2 . 3 ) .

We can g e n e r a l i s e t o d i s t r i b u t i o n s c e r t a i n o p e r a t i o n s on functions.

I f Y i s an open s u b s e t of X and i f u

ine the restriction u C ulYg cp

> = < u,

distribution u ion a u

E

a'

P(x,

uI

f o r a l l cp

>

#(X)

8 ' ( Y ) of

u E

Ip

> = (-

D) = ,

F

111~1

u

E

U,

C"(X) i s t h e d i s t r i b u t acp

> for all q

E

&(XI.

i s t h e d i s t r i b u t i o n on X d e f i n e d by

c u,

a ( x ) Da

Ilcrcm a c o e f f i c i e n t s aa E c"(x),

The product o f a

&(Y).

and a f u n c t i o n a

aa u of

fl(X), we def-

t o Y by

a ' ( X ) d e f i n e d by e au, cp > = c

The d e r i v a t i v e

dxn

for

cp E

i s s a i d t o be continuous w i t h r e s p e c t t o

xn w i t h v a l u e s which a r e d i s t r i b u t i o n s w i t h r e s p e c t t o x' and we d e f i n e i t s " s e c t i o n a l t r a c e on {x = O}" n

yo u = U(0).

We n o t e t h a t

aa' U.

with t h e function u

yo

IJ

E J'(R"I)

by

a ax ,t u i s t h e d i s t r i b u t i o n a s s o c i a t e d If U

E Ck(R, ,p1(RW1))

we s a y t h a t

i s of class Ck w i t h r e s p e c t t o xn w i t h v a l u e s which a r e d i s t -

r i b u t i o n s w i t h r e s p e c t t o x'.

05 j 5 k

, a,j

u

We n o t e t h a t , for

i s t h e d i s t r i b u t i o n associated with t h e

CONVOLUTION AND F.T.'s

(SEC. 2 )

function



for a l l

Q

E&

The p r i n c i p a l p r o p e r t i e s o f c o n v o l u t i o n a r e summarised i n t h e following :

utions.

u,v are two convoZvabZe d i s t r i b -

Suppose

PROPOSITION 2.2: Then =

u * v aa(u

*

s ~ p p(u

v + u

=

V]

*

aa c

v)

u

*

SUPP u

u

=

v

+

* aa

supp v

v

, where

the right-hand

side i s cZosed.

(2.2.1.)

u

(TX

*

;1(Y1

v

E C" mad

=

REMARK 2.3:

If u

E

a', we

towards

u

in

(u

Y

*

v)(x)

=

, where

4X-Y)

Let p

j

be a r e g u l a r i s i n g sequence ( s e e (1.1.1)). m

can show t h a t t h e C

8';

suppose

xj

functions u n p

j

converge

i s a sequence of t r u n c a t i o n

CONVOLUTION AND F . T . ' s

(SEC. 2 )

f u n c t i o n s as i n ( 1 . 2 . 2 ) ;

Pj

*

E&

( X . u)

J

we can show t h a t t h e f u n c t i o n s

converge towards

sequential density of & i n

2.4

15

u

i n &'

.

This proves t h e

B'.

FOURIER TRAnTSFORMATION:

If

f E L1 = L1(Rn)

,

its

A

F o u r i e r t r a n s f o r m i s denoted by 9 f where f is d e f i n e d by

Let us now d e f i n e a subspace o f 'L

which i s i n v a r i a n t under

Fourier transformation:

We denote by 8 the space of functions

DEFINITION 2.4:

cp

E

'h(V1

C"( IRn ) such t h a t , f o r any i n t e g e r h,

supRn (1

=

1.1

+

1x1')~

1

aa cp(x)l

the quantity i s finite.

o

is

Ih

called the space o f functions which are rapidly decreasing, along with a l l t h e i r d e r i v a t i v e s .

We e q u i p 8 w i t h t h e topology d e f i n e d by t h e f a m i l y o f seminorms q h ( h

E

IN ) .

the following :

8 i s a Fr&chet space.

We r e c a l l without proof

16

(CHAP. 1)


THEOREM 2.5:

The Fourier transformation 'jl e s t a b l i s h e s a

topological vector isomorphism of 8 onto i t s e l f , with inverse ~ r 's a t i s f y i n g

(T'g l ( x 1

where $ 5 = ( 2 ~ ) -d ~5

.

=

I

eiy*c g(c1

for g E 8

cp, Q E 8 , we have

If

(2.5.1 .)

If we have cp E

B , D e f i n i t i o n ( 2 . 4 . 1 ) f o r + shows t h a t it e x t e n d s

i n t o an e n t i r e f u n c t i o n on C

n

d e f i n e d by

We s h a l l now show t h a t t h e behaviour of

@

i n Cn d e t e r m i n e s t h e

convex envelope o f t h e s u p p o r t o f c p .

To do t h i s , we d e f i n e t h e i n d i c a t o r f u n c t i o n IK of a compact s u b s e t K of R n by

For example, i f K i s t h e b a l l B w i t h c e n t r e 0 and r a d i u s R , we

The i n d i c a t o r of K d e t e r m i n e s , for

ach hyperpl

t h e s m a l l e s t c l o s e d h a l f - s p a c e which c o n t a i n s K .

direction,

Consequently,

t h e Hahn-Banach theorem shows t h a t IK d e t e r m i n e s t h e convex n

envelope K of K and IK =

%.

17

CONVOLUTION AND F . T . ' s

(SEC. 2 )

We can s t a t e t h e f o l l o w i n g :

THEOREM 2 . 6 :

(Paley-Wiener).

Let K be a convex compact

subset of B n . An e n t i r e f u n c t i o n U on Cn i s the Fourier transform of a function belonging t o &K i f and only i f , f o r a l l N,

there e x i s t s CN such t h a t

6=5+i7l E C Proof of n e c e s s i t y .

n

.

For any m u l t i - i n d e x a , t h e e q u a l i t y ( 2 . 5 . 1 )

implies t h a t

which g i v e s t h e upper bound.

and ( 2 . 6 . 1 ) f o l l o w s from t h i s immediately.

Proof of s u f f i c i e n c y .

The upper bounds ( 2 . 6 . 1 ) imply t h a t U(c)

i s of r a p i d d e c r e a s e ;

it i s t h u s t h e F o u r i e r t r a n s f o r m of a C

m

f u n c t i o n Q , g i v e n by

(2.6.2

.) I n o r d e r t o show t h a t supp Q c K , i t i s s u f f i c i e n t t o v e r i f y

t h a t i f # { K we have cp(xo) = 0 .

By Hahn-Banach,

there exists

18

(CrnP. 1)

DISTRIBUTIONS AND OPEFATORS

a hyperplane which s e p a r a t e s x

I n o t h e r words, t h e r e e x i s t s

q

0

0

and t h e compact convex s e t K . E

lRn \ 0 and c

E

lR

such t h a t

A f t e r a l i n e a r change of c o o r d i n a t e s , w e can assume rlo

= (0,..., 0, 1); we put 5 =

( E l ,

6,).

For t > 0 , Cauchy's theorem allows t h e i n t e g r a l ( 2 . 6 . 2 ) t o be r e p l a c e d by

Taking N = n + l , w e deduce from ( 2 . 6 . 1 ) and ( 2 . 6 . 4 ) t h e upper bound

0

Since ( 2 . 6 . 3 ) i m p l i e s xn t e n d s t o 0 when t -+

+

m;

> IK(qo), t h e hence

cp(xo]

right-hand s i d e o f ( 2 . 6 . 5 ) =

0

.

I t can e a s i l y be seen t h a t t h e i n j e c t i o n s

B C>

8 &>

caD

a r e continuous and of dense image ( t o prove

t h a t & i s dense i n 8 , we proceed by t r u n c a t i o n u s i n g f u n c t i o n s

CONVOLUTION AND F.T.'s

(SEC. 2)

x

xj(x) = x(x/j), where

Ix I

B(IRn)i s such t h a t x(x) = 1 f o r

E

S is

Thus t h e d u a l s'of

1).

S

vector subspace o f

19

8' c o n t a i n i n g

n a t u r a l l y i d e n t i f i e d with a

d.

8' i s termed t h e space o f

tempered d i s t r i b u t i o n s .

For u

E

O'and

, and

u atq,

E 8 , we a g a i n denote by t h e v a l u e of

Q

for the pair 8 ,8'

analogous t o t h o s e of t h e p a i r b

we have t o p o l o g i c a l p r o p e r t i e s

, s'

.

Formula ( 2 . 5 . 1 ) p e r m i t s

t h e F o u r i e r t r a n s f o r m a t i o n t o be extended t o 8 ' :

For u

DEFINITION 2.7:

Fu

=

G

E

u by< SJ,$I

E 8 , of

Fourier transformation

81, we define the Fourier transform

>=

c u, Sg > f o r a l l J,

E

The

8 .

3 e s t a b l i s h e s a topo ZogicaZ vector isomor-

phism of 8 , (equipped with one of the two topologies

y so r

'%

onto i t s e l f . .

I n t h e c a s e o f d i s t r i b u t i o n s w i t h compact s u p p o r t , we have:

2.8:

THEOREM

If u

E

&, then Zu is the Cm f u n c t i o n

defined b y

[

~

t

=j

for

g E R"

3 u extends the e n t i r e f u n c t i o n on Cn defined by

[su)(G)

=

for

5 E R"

.

3 u i s a sZowly increasing f u n c t i o n i n B n J i . e . there e x i s t C

and u such t h a t

)


20

d

More generaZZy, i f B i s a bounded subset o f

u such t h a t (2.8.1) is vaZid f o r aZZ u

E

( C H A P . 1)

we can f i n d C and

B.

We point out merely that the upper bound (2.8.1) derives from the upper bound of the linear form u on C

m

*

by a semi-norm p

K,!J,

furthermore, we can employ an identical semi-norm for any u because a bounded subset is an equicontinuous part of

E

B

6.

We have the analogue of Theorem 2.6 as follows:

THEOREM 2 . 9 :

(Paley-Wiener-Schwartz). Let K be a convex

compact subset of IRn.

An e n t i r e f u n c t i o n U i s the Fourier trans-

form of a d i s t r i b u t i o n of

i f and onZy i f there e x i s t C and

N such t h a t

We shall not have to make use of this theorem and we postpone a proof until Exercise (10.1).

The principal properties of Fourier transformation in 8'are summarised in the following theorem:

(SEC. 3 )

21

SINGULAR SPECTRUM

Suppose we have u

THEOREM 2.10:

(2.10.1 .)

S(U

*

3(Da

v)

=

U)

=

E

$'and v

E

&'

.

Then

FU.KJ

Fu

2 3 e s t a b l i s h e s a linear homeomorphism of L and we have Parseval's

formula:

Formula ( 2.10.2 ) shows t h a t

, by

Fourier transformat ion , t h e

d i f f e r e n t i a l o p e r a t o r P(D) i s transformed i n t o t h e o p e r a t o r of m u l t i p l i c a t i o n by P([).

This p r o p e r t y makes F o u r i e r transform-

a t i o n a b a s i c t o o l for t h e s t u d y o f d i f f e r e n t i a l o p e r a t o r s .

REMARK 2.11:

Sometimes convolution products can be d e f i n e d

without a s u p p o r t c o n d i t i o n . just as in (2.2.1), (u

u n v

E

C"(IRn),

*

nus, for u

v ) ( x ) = C U,

E

"cx

and we can show t h a t u n v

S' and v

>. E

E

S

, we

put,

We have

8' and t h a t

formula ( 2 . 1 0 . 1 ) remains v a l i d .

3.

SINGULAR SPECTRUM OF A DISTRIBUTION From t h e d e f i n i t i o n o f t h e s i n g u l a r support of u

E

8 ' ( X ) , we

22

(CHAP. 1)


have t h e equivalence

x

0

4

1

s i n g supp u

t h e r e e x i s t s an open neighbourhood V of

xG such t h a t f o r a l l a

we have au

E

E

1

a(V)

&(V)

and by F o u r i e r t r a n s f o r m a t i o n , t h e c o n d i t i o n ciu E & ( V ) i s equiva l e n t t o t h e e x i s t e n c e f o r a l l N of a c o n s t a n t C such t h a t

Iasll

(3.1 .I.)

5

c

(1 +

ICll4

5

for

F?

E

I n t h e l a t t e r form, we can r n i c r o l o c a l i s e t h e concept of s i n g u l a r support b y d i s t i n g u i s h i n g t h e di r e c t i o n s have an upper b o w d o f t h e t y p e ( 3 . 1 . 1 . )

A point ( x o , g o )

DEFINITION 3.1.

5

i n which we

. E

X x ( IRn

-

n o t within the singuZar spectrum o f the d i s t r i b u t i o n u

0

1) is

E

&'(X)

i f there e x i s t s a neighbowhood V o f xo and a neighbourhood W of

5' such t h a t (3.12.)

for a21

c1 E

IG('T?,)I

B(v) and a22 5

c

7

N there e x i s t s C such t h a t

for

T

2

1 and 5

E

w.

The singuZar spectrum o f u i s a closed subset o f X x ( En - { 0 I ) denoted by m u . This concept w a s i n t r o d u c e d and i n v e s t i g a t e d by HORMANDER [ 5 l (under t h e name "wave-front s e t " ) f o l l o w i n g t h e i n t r o d u c t i o n by SATO of t h e concept of t h e " s i n g u l a r spectrum" of a hyperfunction ( s e e for example SATO 111).

See a l s o

DUISTEWT [l] and GUILLEMIN-STERNBERG [11 f o r t h e concept of a s i n g u l a r spectrum. Knowledge o f t h e s i n g u l a r spectrum determines t h e s i n g u l a r s u p p o r t because

4)

(SEC.

rx (WFu) = s i n g supp u where

(3.1.3.)

'* x x m n \ 0 ) 3 x , & ) c - - , x

77 X

In f a c t , i f x (xo, x

0

4

0

4

E

x.

s i n g supp u , it i s c l e a r t h a t

g o ) 4 W F f~o r a l l g

E

zn - i o I .

r,,(WFu), we a s s o c i a t e w i t h a l l g

/El =

23

OPERATORS AND K2RNELS

Conversely, i f

Bn

-

{ 0

and

0

i n which we have an

E

A

1 neighbourhoods Vt and Wi o f x

i n e q u a l i t y of t h e t y p e (3.1.2).

0

1

such t h a t

If we e x t r a c t a f i n i t e c o v e r i n g

t h a t w i t h t h i s choice o f V we have t h e c o n d i t i o n ( 3 . 1 . 1 ) and consequently x

0

4

s i n g supp u.

EXAMPLE 3 . 2 :

I
I

=

WF(6,)

< Y,

lcy(a)l,

=

{a] x

emix5 >

cy

=

and WF(Y] = {O] x

(R" \ 0)

(R \

0)

OPEWTORS AND KERNELS. n Suppose X , Y a r e open s u b s e t s of lR , Bp; we i n v e s t i g a t e con-

tinuous l i n e a r operators f r o m B ( Y ) i n t o

EXAMPLE

4.1

:

a' ( x ) .

Let K be a f u n c t i o n i n C"(X

x Y).

Then

t h e i n t e g r a l o p e r a t o r A which a s s o c i a t e s w i t h each $ E ~ ( Y t) h e f u n c t i o n A$ d e f i n e d by (ASHXI

i s continuous from.f?(Y)

=

\

K(x9 Y ) $ l Y I dY

i n t o C"(X).

Moreover, A extends i n t o

. L

a l i n e a r o p e r a t o r A f r o m c ' ( Y ) i n t o C"(X)

when w e p u t

U

( A v ) ( x ) = < v , K(x, . ) > f o r v m

'H,j

E

&(Y).

i s a semi-norm on C (X), we have

More p r e c i s e l y , i f

(CHAP. 1)


24

and s i n c e t h e f u n c t i o n s 3

c4

X

K(x, . ) d e s c r i b e a bounded s u b s e t of OI

m

C ( Y ) when x d e s c r i b e s H , we s e e t h a t A i s a continuous l i n e a r

o p e r a t o r from

& (Y)

r b ) i n t o C"(X).

(equipped w i t h

More g e n e r a l l y , l e t us c o n s i d e r a d i s t r i b u t i o n K

E

With it we a s s o c i a t e an o p e r a t o r A = Op(K) which

r ( X x Y).

i s continuous and l i n e a r from .&Y)

by p u t t i n g

< A$, ep > = c K, ep e3 6 > f o r a l l J, E

(4.1.1.)

&(X

x X)

.

1

x

E

Bcxl

X 1

We denote by 6D t h e d i s t r i b u t i o n on

be t h e d i a g o n a l of X x X.

< 6,,, i >

X x X d e f i n e d by

. a ( y > , rp E

Let D = d i a g ( X x X ) = { (x, x)

EXAMPLE 4 . 2 :

0 E

i n t o &'(X),

@ [ x , x ) dx

L

for a l l

Then it i s c l e a r t h a t Op( 6 ) = I d e n t i t y .

D

EXAMPLE 4 . 3 :

Suppose we have T

E

&'(En).

With it we

n n a s s o c i a t e t h e d i s t r i b u t i o n on IR xJR denoted, by an abuse o f n o t a t i o n , by T(x

< T(x

- y),

-

y ) and d e f i n e d by

i > =

( t h u s 6D = 6(x

-

< T(x),

Then, Op ( T ( x

y)).

convolution w i t h T: $.

@ ( x+ yly)

+

T

*

-

> dy

i E J(F? x R")

9

y ) ) i s t h e operator of

$.

We l e a v e t o t h e r e a d e r t h e t a s k o f v e r i f y i n g t h a t t h e sing-

ular s u p p o r t of a d i s t r i b u t i o n o f t h e form T(x

-

y ) i s equal t o

t h e set

(4.3.1 .]

C

(x, y) E

R" x R"

I

(x

- y>

E s i n g supp T I .

4)

(SEC.

OPERATORS AND KERNELS

25

We now r e t u r n t o t h e l i n e a r mapping K + A = Op(K) from &(X

x Y ) into&@(Y),

B'(X))

d e f i n e d by ( 4 . 1 . 1 ) .

This mapping

We s h a l l now s e e t h a t it i s

i s i n j e c t i v e i n view of Lemma 1 . 3 . surjective: THEOREM

4 . 4 (Sehwartz's kernels theorem)

tinuous linear operator A f r o r n & ( Y ) unique d i s t r i b u t i o n K

E

&'(X

:

For any eon-

i n t o &'( X) , there e x i s t s a

x Y) such t h a t A = Op(K);

K is

ealZed the kernel of the operator A. PROOF :

The b i l i n e a r form

B[x)

X

B(y) 3

(Cp,

$1 ->

< A$,,cp > E c

induces a l i n e a r form K on t h e a l g e b r a i c t e n s o r product

ax)

@

afY).

(X x Y ) we have

E x p l i c i t l y , for 0 N

(4.4.1 .] where N

(4.4.2.) i s a decomposition o f @,

and

(4.4.1) i s

independent o f t h e choice

of t h e decomposition of 0 .

To conclude, it f o l l o w s from Lemma 1 . 3 t h a t it s u f f i c e s t o show t h a t K i s a c o n t i n u o u s l i n e a r form on topology induced b y a ( X x Y ) .

ax) f23 aY) f o r

the

L e t t i n g H , L be compact s u b s e t s

o f X , Y , we show t h a t t h e r e e x i s t s a semi-norm p such H x L,m that

(4.4.3.)

I
I

5

C p,,

L,m ( 9 )

,

f o r CP E f l y ) w i t h supp 0 c H x L .

be compact neighbourhoods o f H , L ; by t r u n c a t i o n ,


26

we can always w r i t e ( 4 . 4 . 2 ) w i t h (pj €

BH,(X)

(CHAP. 1)

$ j EBL [y) 1

if

supp

I

,

H x L

C

m

Suppose we have f E Co(X) e q u a l t o 1 on H1 and we p u t H2 = S ~ P Pf ; a3

l i k e w i s e suppose we have g

E

Co(Y) e q u a l t o 1 on L1 and we p u t

c o n s i d e r i n g t h i s i n t e g r a l as an i n t e g r a l o f a f u n c t i o n w i t h values i n b

A I)j

=

( Y ) , t h e c o n t i n u i t y of A p e r m i t s us t o w r i t e

.i;d l .,

fis

eiY.7)

f(x1 eix**

.

i j ( s > A! ,

Likewise, by w r i t i n g

we o b t a i n

where

The c o n t i n u i t y o f

< A$,

[cp, I)) &%(X)

x &,(Y)

(p

A i m p l i e s t h a t t h e b i l i n e a r mapping

>

i s s e p a r a t e l y c o n t i n u o u s from

i n t o C;

a n a l y s i s , it i s c o n t i n u o u s .

t h u s , u s i n g a r e s u l t from f u n c t i o n a l

A l s o , t h e r e e x i s t s C , h , II such

that

I n p a r t i c u l a r , it f o l l o w s from t h e d e f i n i t i o n of F t h a t t h e r e

(SEC. 4 )

27


e x i s t s C such t h a t

c

IF(%, S)l

(4.4.5.)

for

(5, 7 ) E R”

By l i n e a r i t y , we deduce from

< K,

[4.4.6. ]

@

>

+ ] 5 l I a (1 +

x RP ,

( 4 . 4 . 1 ) and ( 4 . 4 . 4 ) t h a t

F(5,

=

m

We choose an i n t e g e r

(1

1) k,a) 8 5 8 s

such t h a t

h+&m

c

-

I

;

fwp)

4

i n t e g r a t i n g by p a r t s i n t h e i n t e g r a l d e f i n i n g 9, we o b t a i n :

which a l l o w s us t o m a j o r i s e

( 4 . 4 . 6 ) and o b t a i n ( 4 . 4 . 3 ) by v i r t u e

of ( 4 . 4 . 5 ) .

4.5

TRANSPOSE AND ADJOINT OF A LINEAR OPERATOR: The t r a n -

spose ( r e s p . t h e a d j o i n t ) of a c o n t i n u o u s l i n e a r o p e r a t o r A : ay)+ a ( X )

3 (X) i n t o &‘(Y) (4.5.1.)

C tA

i s t h e l i n e a r o p e r a t o r tA ( r e s p . A*) from

d e f i n e d by

9, d( >

=

C cp, A $

>

( c A* 9, for

(P

T >=

c

Cp,

5 >)

E 4 x 1 , $ E ay’)

We immediately have:

PROPOSITION 4 . 5 :

*

The operators tA and A are continuous

.

Linear operators from B(x)i n t o 8 ( ~The kernel of tA


28

(CHAP. 1)

(resp. of A*) is deduced from the kernel K of A (resp. of

i?) by

We have

interchanging the r o l e s of the variables x , y. * I

t(tA)

and (A )

= A

= A.

The operator A is continuous from &(Y) i n t o C"(X) if and only if t A extends i n t o a continuous operator from

(X) i n t o 8(Y) (both

equipped with the topology 'C,,).

For example, i f A i s a d i f f e r e n t i a l o p e r a t o r

,aE

c o e f f i c i e n t s i n X , t h e n tA i s t h e

w i t h C"

rn ).(,a

f ( a a ( x ) . ) and A* i s t h e

d i f f e r e n t i a l operator

IaE

rn D

d i f f e r e n t i a l operator

I& 4.6

*

~

~

.

~

.

rn

REGULARISING OPERATORS:

DEFINITION 4.6: A continuous linear operator A : R(Y)

->

&(XI

i s said t o be regularising if it extends i n t o

a continuous operator from &'(XI i n t o C " ( X ) , equipped w i t h

where &'(Y) is

Zb).

We have t h e f o l l o w i n g c h a r a c t e r i s a t i o n :

THEOREM 4.7:

An operator A i s regularising if and only i f

i t s kernel KA is i n C"(X PROOF:

x

Y).

We saw i n connection w i t h Example

4 . 1 t h a t t h e condition

4)

(SEC.


29

To show t h a t it i s n e c e s s a r y , we c o n s i d e r t h e

is sufficient. mapping

x

x Y

3

(x, y )

->

Y)

, . [ a

By h y p o t h e s i s , t h e f u n c t i o n a i s C

=

[A 6 y ) ( x )

E

C

.

m

separately with respect t o

x and y , and t h i s b e i n g s o t h e theorem a r i s e s out of

LEMMP,

PROOF:

4.8:

The function a

We begin

by showing t h a t

is in C"(X

x

Y ) and KA = a.

i s continuous.

a

5

7'.

Furthermore, t h e c o n t i n u i t y o f A i m p l i e s t h e e x i s t e n c e o f Q'' > 0 such t h a t

(4.8.2.)

sup Ix-xo)I

I(A 6 y ) ( x )

T'

- [A

6

)(x)l

I

I Y - YoI The c o n t i n u i t y o f

5

when

YO

a

5

7"

.

f o l l o w s from (4.8.1) and ( 4 . 8 . 2 ) .

I n o r d e r t o prove t h a t

a

is C

m

,

it i s s u f f i c i e n t t o reason

30

(CHAP. 1)


i n l i k e manner for t h e f u n c t i o n s

. I f we denote by A

regularising; multi-index

1

by c o n s t r u c t i o n , we have for any y

E

Y and any

6

which proves t h a t A = A i o n s of t h e t y p e 6 " ) Y

REMARK

a , it i s t h u s

t h e operator with kernel

4.9:

1

s i n c e l i n e a r combinations o f d i s t r i b u t -

6

a r e dense i n

Theorems

Y).

4.4 and 4 7 a r e i n f a c t a p p l i c a t i o n s

of t h e t h e o r y of n u c l e a r s p a c e s ;

for t h i s approach, s e e f o r

example TREVES [11.

5.

OPERATORS AND SUPPORT PROPERTIES

PROPOSITION 5 . 1 :

K~ E P ( X X Y ]

, we

If we l e t A be an operator w ith kernel have f o r v

E

C;(Y)

there e x i s t s such t h a t

PROOF:

(x,y)

y

E

E

supp K~

supp v

S i n c e supp v i s compact, t h e s e t supp KA

0

supp v i s

(SEC. 5)

OPERATORS AND SUPPORT

c l o s e d in X. those u

supp u

E

31

< Av, u > = 0

We now have t o show t h a t

for

C”(X) which a r e such t h a t 0

n

means t h a t

(supp KA

supp

0

[supp u

4

=

@

now, t h i s c o n d i t i o n

;

supp v) fl supp KA

x

=

, hence

g

the

result.

We denote by p [ r e s p . p

X

Y

) t h e p r o j e c t i o n of X

x

Y on X

(resp. Y ) .

We say t h a t the operator A with kerneZ

DEFINITION 5.2:

KA E R’(X x Y ) i s proper i f the projections an d

a r e ;sroper.

p y l s ~ p pKA

p x l ~ ~ pKAp

We have

Suppose A i s a proper operator;

PROPOSITION 5.3:

i ) For any compact subset K

Y, the s e t

c

supp KA

i s a compact subset of X and consequentZy A sends 0IK,

0

K

then =

K’

a K (Y) t o

(x).

i i ) For a n y compact subset K 1

c

X, there e x i s t s a compact subset

K of Y such t h a t : [SUPP

v

n

K

=

6)

[SUPP

AV

n K*

=

ei).

Consequently, the operator A transforms a ZocalZy f i n i t e suni of functions i n &(Y)

i n t o a ZocaZZy f i n i t e swn o f d i s t r i b u t i o n s i n

32

( C H A P . 1)


&' (X). Furthermore, i t extends uniquely i n t o a continuous l i n e a r mapping from C " ( Y ) i n t o &'(X)

PROOF OF i ) :

which again s a t i s f i e s (5.1.1).

We have by d e f i n i t i o n

=

K'

n

pX tP;'c~l

supp K ~ I

so t h i s i s a compact s u b s e t of X because

i s proper. pylsupp KA

PROOF OF i i ) :

K =

I f we p u t

py [ p i ' ( K ' )

'n

, this

supp K,,)

is

a compact s u b s e t o f Y and, by c o n s t r u c t i o n , it i s c l e a r t h a t i f supp v does n o t i n t e r s e c t K t h e n supp Av does not i n t e r s e c t K'.

Let Cp,

(9,) be a

p a r t i t i o n of u n i t y on Y by means of f u n c t i o n s

Ecp1

For v

m

E

C ( Y ) , we have a l o c a l l y f i n i t e sum

; t h u s t h e sum E 'pav U l o c a l l y f i n i t e and we denote it by Av. v

A['?,

=

v)

is also

cy

We have t h u s d e f i n e d an

e x t e n s i o n of A i n t o a l i n e a r mapping, a g a i n denoted by A , from into a ' ( X ) .

C"(Y)

The c o n t i n u i t y o f t h i s e x t e n s i o n f o l l o w s immed-

i a t e l y from t h e d e f i n i t i o n o f t h e s t r o n g t o p o l o g i e s .

The unique-

n e s s comes from t h e d e n s i t y o f C"(Y) i n C " ( Y ) . 0

F i n a l l y , by u s i n g (5.1.11,we have for suppAv C U SUPP A ( ( P ~ tljCU [SUPP KAosupp a cy

ve

v

E

C"[Y)

:

V ) = S U P ~KA

0

s ~ p pv

For t h e s i n g u l a r s u p p o r t w e have

PROPOSITION 5 . 4 : KA

E

.P(X x Y j

.

Let A be an operator with kernel We aesume t h a t A extends i n t o a continuous

(SEC. 6 )

33

D.O.'s WITH CONSTANT COEFFICIENTS

D'(x) such t h a t

operator from &(Y) i n t o A(C:(Y))

c"[x>

C

(5.4.1.)

We then have t h e incZusion

,

s i n g supp A V

s i n g supp K A

C

o

.

v f &"Y) PROOF:

Suppose we have x

hand s i d e o f

(5.4.1);

([x,}

x

0

,

s i n g supp v

which does not belong t o t h e r i g h t -

i n o t h e r words,

s i n g supp v)

n

s i n g SUPP KA

#

=

Since s i n g supp v is compact, t h e r e e x i s t open neighbowhoods U , V of xo and s i n g supp v such t h a t

n

(U x V) Suppose w e have ec s i n g supp v. s i n g supp.

Since

E

C"(V) 0

We w r i t e

AV

K AJU x

v

=

s i n g supp KA

,

=

.

@

i d e n t i c a l t o 1 i n t h e neighbourhood o f

v =

QZ/

+

(I-crJv

, and

s i n g supp A ( W ) s i n c e

we have

E c~(Y)

(I 0 such t h a t

For Q f i x e d , t h e d e f i n i t i o n of r ( Q )shows t h a t t h e r e e x i s t s some

e

E

IR", l e i s 1, such t h a t Q(ze)

(62.2.)

2

c,

,

N(Q)

z

E c

1.

,

= I

.

We t h u s deduce t h a t t h e r e e x i s t s some neighbourhood o f Q i n which I

(6.2.2) remains t r u e w i t h t h e c o n s t a n t C2 = C 2 1' For a l l 5

E

lRn

P6(6) = P ( 5 e x i s t s a 2'

5

, we

d e f i n e t h e t r a n s l a t e d polynomial

+ 5 ) , and

and a neighbourhood V

IP(T)+ z e$l

(6.2.3.)

t h e preceding d i s c u s s i o n shows t h a t t h e r e

5

2

subcovering V,

k '

1

o f 5 such t h a t we have

"pT))

From t h e c o v e r i n g of IRn by t h e V compactness o f t h e s p h e r e s

5

5'

I

Izl = 1 m a g E v

we c a n , by making use o f t h e

= const

which i s l o c a l l y f i n i t e .

, extract

a countable

We l e t ( p , - ) be a n

p a r t i t i o n of u n i t y s u b o r d i n a t e t o t h i s covering and we put

.

Since t h e terms of Q which a r e o f maximal degree 'k are i n v a r i a n t w i t h r e s p e c t t o t r a n s l a t i o n , there e x i s t s some

Bk =

€!

C > 0 such t h a t

5'

(SEC.

6)


37

which when combined w i t h ( 6 . 2 . 3 ) g i v e s ( 6 . 2 . 1 ) .

We d e f i n e E by p u t t i n g , f o r

Cp

E Cz

:

I t can e a s i l y be shown t h a t E i s a d i s t r i b u t i o n by u s i n g ( 6 . 2 . 1 ) and t h e immediate i n e q u a l i t y

F i n a l l y , Cauchy's formula shows

The e x i s t e n c e o f a fundamental s o l u t i o n E shows t h a t any o p e r a t o r P(D)

#

0 with constant c o e f f i c i e n t s i s l o c a l l y solvable

i n d i s t r i b u t i o n space i . e . f o r any f E b' , t h e e q u a t i o n P(D)u = f admits a t l e a s t one s o l u t i o n

u

a

E

*

f

E B'

.

We can a l s o

38

(CHAP. 1)


-

investigate the global solvability of P(D), i.e. the surjectivity of the linear mappings 3

u

~ ( x )3

u

C"(X)

depending on the open subset X

c

IRn

.

PfD)u

E C"[X)

p(oju

E

a(x1

For these questions, the

treatises of RORMANDER [11 and TREVES [21 can be referred to.

6.3

HYPOELLIPTICITY OF OPERATORS WITH CONSTANT COEFFICIENTS: After the questions of existence, a further

important question concerns the investigation of the regularity of the solutions. For example, in which cases do the solutions of

P(x, D)u = f have no singularities other than those of f?

We

once again have the inclusion sing supp P ( x , Dju

DEFINITION 6.3:

C

sing supp u

, which leads us to:

A d i f f e r e n t i a l operator P(x, D) with coeff-

i c i e n t s i n C"(X) is said t o be h y p o e l l i p t i c i n the open subset X

if (6.3.1 .)

sing supp P ( x , D) u =

sing SUPP u

9

For operators with constant coefficients, we have:

PROPOSITION 6.4: i)

The following assertions are equivalent:

The operator P(D) is h y p o e l l i p t i c i n TRn

6)

(SEC.

The operator P(D) admits a fundamenta2 s o l u t i o n which

ii)

is

39

D.O.’s WITH CONSTAMT COEFFICIENTS

ern i n I R ~ -

COI.

The operator P(D) admits a parametrix ( i . e . a distrib-

iii)

ution F E d

such t h a t P ( D ’ J F

- do

E Cm(R”))

, which is

Cm

i n IR” - { o } .

PROOF:

i

The i m p l i c a t i o n s

iii

prove t h a t

=$

i

ii a iii

=$

are t r i v i a l .

We now

L e t t i n g F be a p a r a m e t r i x of P(D) , t o

,

8

within t h e m u l t i p l i c a t i o n of F by a f u n c t i o n from

t o 1 i n t h e neighbourhood of 0 , we may assume t h a t F

identical E

this

& I ;

allows us t o w r i t e

(6.4.1 .]

P(D),F

For any open s u b s e t X ion u E & ( X )

c

do

=

,

+

where

e E :C

,

B n , we need t o show t h a t i f a d i s t r i b u t -

E

P(D)u

satisfies

Cm(X)

then u

m

E

C (X).

For

t h i s , it i s s u f f i c i e n t t o show t h a t f o r any open s u b s e t X1 such that

yl c

c X,

we have u

m

E

equal t o 1 i n t h e neighbourhood o f (6.4

2.)

We have

X1.

g

g E C”(X,)

Suppose we have a

C (Xl).

=

n &‘(R”)

TI;we

P(Qu)

Co(X)

put

s

s i n c e a u = u on t h e open s u b s e t

By convolution o f e q u a t i o n (6.4.2)

The k e r n e l d i s t r i b u t i o n F(x

m

E

- y) of

w i t h F , we o b t a i n

t h e o p e r a t o r o f convolution

40

(CHAP. 1)


(h .3A) an& iii),

by F satisfies, f r o m

sing supp F(x

- y>

C

Diag(R" x

R"')

.

By applying Proposition 5.4, we can thus deduce that sing supp (F

*

g]

C

sing supp g m

The right-hand side of (6.4.3) is therefore C u

ently

. in X and consequ1

is also.

EXAMPLE 6.5: that the operators

The formulas (6.1.3), (6.1.4), (6.1.5)show

aar -

I

A

-A

a,

I

are hypoelliptic.

We now give an important class of hypoelliptic operators.

DEFINITION 6.6:

A d i f f e r e n t i a l operator P(x, D) with Cm

c o e f f i c i e n t s i n an open subset X X

c

IRn is said t o be e l l i p t i c i n

if i t s principal symbol does not vanish on

X x

R"

-

{ 0 ]

.

We shall prove in Chapter IV that elliptic operators are hypoelliptic; for the moment we prove in particular the following:

PROPOSITION 6.7:

If P(D) is an e l l i p t i c operator with

constcmt c o e f f i c i e n t s , then P(D) i s h y p o e l l i p t i c i n IRn

PROOF:

.

We shall now construct a parametrix F of P(D) which will

be Cm in lRn - 101.

If m

is the degree of P(D) , the assumption

of ellipticity is equivalent to

(SEC. 6 )

41


By homogeneity, we deduce from t h i s t h a t

1

\P,({)

,

151"

c

2

and by choosing R s u f f i c i e n t l y l a r g e , we have t h e lower bound

[6.7.1.)

Ip(5)l

;151m

2

I f we have 6

W

E

C

, identical

5 R , t h e lower bound

, 151

5 E R"

for

151

t o 1 for

(6.7.1)a l l o w s

-1

2

2

.

R

and z e r o f o r

us t o d e f i n e

. This i s a p a r a m e t r i x of P(D), s i n c e we have

P(D)F

-

ho

=

- 1)

s'(e(5)

E 8

The r e g u l a r i t y o f F away from t h e o r i g i n w i l l f o l l o w from

LEMMA 6.8: xa F

For any multi-index

E c~(R")

PROOF OF THE LEMMA:

with

a

q = rn

L)e have

+ 1.1

-n- 1

.

The e q u a l i t y

(6.6.1 .) a e

shows t h a t i n t h e c a l c u l a t i o n of D (-),

S P

t h e t e r m s which i n v o l v e

the d e r i v a t i v e s of t h e function 8 provide a C a

x F.

m

contribution i n

It i s t h u s s u f f i c i e n t t o c o n s i d e r o n l y t h e d i f f e r e n t i a t -

(CHAP. 1)


42

i o n s which r e l a t e t o

-.1P

We can show by r e c u r r e n c e on la

1

that

f o r 151 2 R we have

=

Di (&)

a l i n e a r combination of terms of t h e form

(6.8 2.1

Furthermore, for any m u l t i - i n d e x

6 we have

c o n s e q u e n t l y , t a k i n g i n t o account (6.7.1) each t e r m on t h e r i g h t hand s i d e of ( 6 . 8 . 2 ) i s bounded above by

151

2 R.

These upper bounds imply t h a t t h e i n v e r s e F o u r i e r t r a -

nsforms a r e c e r t a i n l y i n C q ( I R n ) because r a b l e for

O(I{l-m-lal) for

151

2

I ~~q-m-~cul

R.

This lemma shows t h a t , f o r any i n t e g e r

E Cq(Rn)

1x12p F

F E Cm(Rn

-

{ 0

i s integ-

1)

with

q = rn

+ 2p

p

,

-n-1

;

hence

.

For o p e r a t o r s w i t h c o n s t a n t c o e f f i c i e n t s , H O W D E R h a s given a . c h a r a c t e r i s a t i o n o f t h e h y p o e l l i p t i c i t y of P(D) i n terms of t h e polynomial P ( 5 ) which i s e x p r e s s e d i n t h e f o l l o w i n g theorem:

(SEC.

6)

43


The following a s s e r t i o n s are equivalent:

THEOkEM 6 . 9 :

The operator P(D) i s h y p o e l l i p t i c

i)

I f we have

ii)

iii)

N = {

6E

C"

There e x i s t s 6 > 0 and C

I

> 0

PROOF OF i)

+ ii):

If F

E

for

such t h a t f o r any multi-

5 E R"

for

, then

P(6) = 0 }

,

151

2

c

.

&' is a parametrix of P(D), and we

Put P(D) F

go

=

+

,

0

where

0

Eb

,

then by Fourier transformation this equality becomes

P[Cj.i(S)

(6.9.3.)

1 +

=

361

for

From the Paley-Wiener theorem, there exists C such that

(6.9.4.)

where A

= x

When 5

E

N, ( 6 . 9 . 3 )

SUP E SUPP

c m

1x1

0

be written

6 E c".

44

(CHAP. 1)


which, having r e g a r d t o ( 6 . 9 . 4 )

(1 and p r o v e s

+ 1 ~ 1 ~ 5)

c

,

implies

eA l I m

6\

6EN

for

(6.9.1).

The proof o f i i )

a , iii) i s b a s e d on p u r e l y a l g e b r a i c p r o p e r t -

i e s f o r which we r e f e r t h e r e a d e r t o HORMANDER [11 (Chapter IV).

i) i s proved by c o n s t r u c t -

F i n a l l y , t h e implication iii) i n g a p a r a m e t r i x F of P(D) i n t h e form showing t h a t t h i s i s CcD i n IRn

F

=

g’(#)

- (01, proceeding

and by

i n an e n t i r e l y

analogous manner t o t h a t o f t h e proof o f P r o p o s i t i o n 6 . 7 ;

this

i s p o s s i b l e due t o t h e upper bounds ( 6 . 9 . 2 ) .

I n t h e c a s e o f o p e r a t o r s P(D) which a r e n o t h y p o e l l i p t i c , we make use o f q u i t e g e n e r a l theorems t o d e s c r i b e t h e s i n g u l a r suppo r t of

C61

ANDER

7.

u

We r e f e r t o t h e monograph o f HORM-

f o r these questions.

OPERATORS AND DISTRIBUTIONS ON A MANIFOLD

7.1

x

from t h a t of Pu.

:

For

X

+

v

CHANGE OF VARIABLES I N DISTRIBUTIONS:

Let

Y be a diffeomorphism between two open s u b s e t s of I R n .

1 E LlDC(Y) , the

function

u = v

o

x

=

x*

V

1 E hoC(x)

9

(SEC. 7 )

OPERATORS ON A MANIFOLD

considered a s a d i s t r i b u t i o n , s a t i s f i e s f o r

Dx (F) = the

J(rl)

=

of v a r i a b l e s

‘y

E

+

, the

formula (7.1.1)d e f i n e s a d i s t r i b u t i o n

a(X)

denoted by

x x.

v and c a l l e d t h e i n v e r s e image o f If

an open s u b s e t Z o f IRn and i f show t h a t (X,

O

of d i r e c t image

X)* w

x+

REMARK 7.1: 0

J a c o b i a n determinant o f t h e change

)c

under t h e diffeomorphism

cp

:

x = x-’(y)

v E &(Y)

If now u

cp E C r ( X )

< u,cp > =

(7.1.1)

where

45

x-’

i s a diffeomorphism o f Y o n t o

w E &(Z)

* * X (X, w ) .

=

under

x1

x

v

, we

can immediately

We d e f i n e t h e o p e r a t i o n

a s b e i n g t h e i n v e r s e image under

.

x -1

The p r e s e n c e of t h e f a c t o r IJI a l o n g s i d e

i n (7.1.1)shows t h a t it i s n e c e s s a r y t o c o n s i d e r

the t e s t functions measure cp(x] dx

.

Q

n o t as f u n c t i o n s b u t as C

m

d e n s i t i e s of

This remark w i l l b e fundamental t o t h e d e f i n -

i t i o n o f d i s t r i b u t i o n s on a m a n i f o l d .

7.2 C

m

DIRAC MEASURE ON A HYPERSURFACE:

If

S

is a

h y p e r s u r f a c e o f an open s u b s e t X of Fin, t h e E u c l i d i a n s t r u c t -

ure of

mn

induces on S a Riemannian s t r u c t u r e , and we denote by

doS t h e a s s o c i a t e d volume element on S.

46

(CHAP. 1)


We d e s c r i b e a s t h e Dirac d e l t a f u n c t i o n on S t h e d i s t r i bution 6

S

&(X)

E

< tis, cp >

d e f i n e d by

Is

=

dOs(X1

Q(X)

m

More g e n e r a l l y , l e t f : X +IR be a C

grad f ( x )

S= { x E

x

=

x

..., af

(z(X]t

I

f[X)

=

0 ]

axn

.

putting

< b(f'l, cp >

=

m

n'

hypersurface of

b ( f ] E & (X) on

i,l*l

For example, i f f ( x ) = x

on t h e s e t

Then S i s a C

and w e d e f i n e t h e d i s t r i b u t i o n

8x1

mapping such t h a t

f 0

-(X))

Q E

9

d5&4

S by

cp E

9

w.

we once a g a i n have t h e d e f i n i t i o n of

t h e d i s t r i b u t i o n 6( x = 0). n

I f % i s a diffeomorphism Y

*

+

X such t h a t ( %

f ) ( y ) = yn,

t h e n it can e a s i l y be shown t h a t a d i s t r i b u t i o n % * ( S ( f ) ) i s precisely the distribution S(y = 0 ) .

n

It i s t o obtain t h i s

t r a n s f o r m a t i o n law t h a t w e need t o i n t r o d u c e t h e f a c t o r /grad f

1

i n t o t h e d e f i n i t i o n of 6 ( f ) .

For a thorough s t u d y of

t h i s t y p e o f d i s t r i b u t i o n , w e r e f e r t o GELFAND and. SHILOV c11.

7.3

CHANGE OF VARIABLES I N DIFFERENTIAL OPERATORS:

X be an open s u b s e t of Eln and P(x, D ) =

,@E

aa(x)

D@

Let be a

m

m

d i f f e r e n t i a l o p e r a t o r of degree m w i t h c o e f f i c i e n t s i n C The f o l l o w i n g i s a c h a r a c t e r i s a t i o n of t h e s e o p e r a t o r s :

(x).

7)

(SEC.

47


Let P be a continuous Zinear mapping from

PROPOSITION 7 . 3

Then P is a d i f f e r e n t i a 2 operator of

Cz(X) into Cm(X).

degree m if and on2y if f o r aZZ rea2-va2ued functions and any function a

.I

[7.3.1

E

C"(X) 0

R 3

+j

e-iTq ~ [ eiTp)[x) a

Im

with respect t o

The n e c e s s i t y i s obvious.

PROOF:

= x.%

with

5 E Rn

C"(X)

the expression c.--------*

is a poZynomiaZ of degree

cE

T

f o r every x i n X.

Conversely, l e t u s t a k e

and put

m e- i T x.5

(7.32.)

P(a eiTxgj

=

ak(a,

x , 51

,

Tk

k= 0

g j of

The c o e f f i c i e n t ok[a, x,

Tk

is a

m

c

f u n c t i o n o f x and E;

since t h e same i s a l s o t r u e o f t h e l e f t - h a n d s i d e .

5 and we can show (by r e p l a c i n g 5 by t

t

E

IR -

0

T

Moreover,

by t . r w i t h

1 ) t h a t ok i s a homogeneous f u n c t i o n of degree k w i t h

respect t o 5 ; consequently, it i s a homogeneous polynomial o f degree

k with respect t o

5 s i n c e it i s C

m

for

l e t K be a compact s u b s e t of X and we have a 1 i n t h e neighbourhood of K , for u

E

&,(X),

E

5

E

.$'(X)

lRn.

If we

identical t o

w e o b t a i n by

inverse Fourier transformation

By c o n s i d e r i n g t h i s l a t t e r i n t e g r a l as an i n t e g r a l w i t h r e s p e c t m

t o 5 w i t h v a l u e s i n C (X), t h e c o n t i n u i t y o f P. i m p l i e s t h a t

P.1

[XI

=

O

l,i.5 .

e-bsP(a e h c j

and by v i r t u e of ( 7 . 3 . 2 ) , we o b t a i n

c(5)

48


( C H A P . 1)

so that

n

E

%

T h e r e f o r e , P i s given by a d i f f e r e n t i a l o p e r a t o r of degree

m

=

(Pu)fx)

(7.3.3.)

ok[a, x, Dx)u

,

for

u

k=o

when it a c t s on f u n c t i o n s from .8 K ' i v e sequence of compact s u b s e t s K

j

By c o n s i d e r i n g an exhaustand a s s o c i a t e d f u n c t i o n s a

2'

t h e e q u a l i t y ( 7 . 3 . 3 ) shows t h a t t h e d i f f e r e n t i a l o p e r a t o r s

m Ok('j,

piece together into a d i f f e r e n t i a l

x, Ox)

k=o

m

o p e r a t o r of degree

w i t h c o e f f i c i e n t s i n C"(X); t h i s

concludes t h e proof o f t h e p r o p o s i t i o n .

The e x p r e s s i o n ( 7 . 3 . 1 ) p r o v i d e s a new means of c a l c u l a t i n g t h e p r i n c i p a l symbol o f P . t h a t t h e c o e f f i c i e n t of P,(x,

dcp(x)) a[x)

i f we p u t dq(x

0

.

T

m

I n f a c t , L e i b n i z ' formula shows i n ( 7 . 3 . 1 ) i s given by

I n p a r t i c u l a r , i f we have x

) = 5, and assume t h a t a ( x0 ) =

e f f i c i e n t o f rm i s given by P (x

m

0'

0

E

X, and

1, t h e n t h e co-

' 0 ) .

If % : X + Y i s a diffeomorphism between two open s u b s e t s of Bn, and P i s a d i f f e r e n t i a l o p e r a t o r of degree define the transported operator v

i.e.

E

C"(Y)

)& P o f P by

x:

m

on X , we


(7.3.4.1

X*P

Proposition 7.3 shows immediately t h a t

m

operator of degree

e-iT

t(YOI[,

p)

T

m

=

(ei7e ) ( y o )

Ix,

where x

0

=

P is a differential

no)

E

Y x l R n , we c a l c u l a t e

i n t h e polynomial

where $ i s such t h a t d $ ( y o ) =

(73.5.)

x+

In order t o calculate i t s

on Y .

p r i n c i p a l symbol a t t h e p o i n t ( y o , the c o e f f i c i e n t o f

*

X * o P o X

=

49

P I m (yo'

e-iT

n0. TI,)

p(ei7

$(YO)

e

x I (x-'

0

(Yo)

We o b t a i n

t

=

TI,)

~ l ( ~ , )

x-1( y o ) .

We now g i v e a b r i e f o u t l i n e of t h e g e n e r a l i s a t i o n of t h e preceding d i s c u s s i o n t o t h e c a s e o f systems of d i f f e r e n t i a l operators.

$I d i f f e r e n t i a l systam ( q , r ) o f degree

m

consists

of an o p e r a t o r o f t h e form

where t h e c o e f f i c i e n t s a r e m a t r i c e s a,(x) m

C

dependence on x i n X.

.

C(Cq, C r )

with

It i s c l e a r t h a t such an o p e r a t o r

d e f i n e s a continuous l i n e a r mapping from

Cm(X; C r )

E

m

c0 (x;)'C

into

P r o p o s i t i o n 7.3 g e n e r a l i s e s immediately t o t h i s

case, it b e i n g s u f f i c i e n t t o t a k e t h e f u n c t i o n

a

in

Cz(X; C q ) and t o demand t h a t t h e e x p r e s s i o n (7.3.1)be a polynomial i n T w i t h c o e f f i c i e n t s i n . ' 6

We d e s c r i b e a s t h e p r i n c i p a l symbol o f P at a p o i n t

'1

DISTRIBUTIONS AND OPEmTORS

50

(x, 5 )

E

X

x

(CHAP. 1)

n

IR t h e l i n e a r mapping

The c o e f f i c i e n t o f .rm i n ( 7 . 3 . 1 ) i s e q u a l t o P,(x,

dq(x)).a(x),

which p r o v i d e s a new d e f i n i t i o n o f ? (x, 5 ) and a l l o w s us t o

m

prove t h a t ( 7 . 3 . 5 ) remains t r u e .

7.4 dimension C

m

DISTRIBUTION ON A MANIFOLD:

Let X be a manifold o f

n (we s h a l l assume i n t h i s book t h a t a manifold i s

and c o u n t a b l e a t i n f i n i t y ) .

-

We e q u i p C"(X) semi-norms '9

w i t h t h e t o p o l o g y d e f i n e d by t h e f a m i l y o f

p ( x , q),where cy

charts X : X 3 U

--L

U

x

d e s c r i b e s t h e f a m i l y of l o c a l

c R" o f X , and p m-)

of semi-norms d e f i n e d on C ( U ) .

describes t h e family

By u s i n g a p a r t i t i o n o f u n i t y ,

we can show t h a t it a c t u a l l y s u f f i c e s t o c o n s i d e r o n l y t h e semi-norms a s s o c i a t e d w i t h an a t l a s of X and we can show t h a t

C"(X)

is a Fr6chet s p a c e .

With an obvious n o t a t i o n , we d e f i n e as i n s e c t i o n 1 t h e t o p o l o g i c a l v e c t o r s p a c e s C;(X)

=DK(x),c:(x)

=D(x).

If

7

i s a complex v e c t o r bundle o f rank N on X, we denote by C"(X;

3 )' t h e

space of C

m

s e c t i o n s of

t o p o l o g y as above by r e d u c i n g t o of l o c a l t r i v i a l i s a t i o n s of 3;.

3

and we e q u i p it w i t h a

='* N C (U; C )

= (Cm(z))N

by means

(SEC. 7)


R ( X ; 3)

We define s i m i l a r l y t h e spaces

ax; ?I

and

K

.

s)

q x ;

=

51

In o r d e r t o d e f i n e d i s t r i b u t i o n s on a manifold X , we s h a l l need f i r s t t o d e f i n e t h e bundle of d e n s i t i e s on X.

mapping from

A

n

E \ 0 i n t o Q: such t h a t

~ ( h w )= 111

('7A.l.I

,

W(W)

In p a r t i c u l a r , i f vl,..., v

1 E R\O

for

w €An E \ D

and

i s a b a s i s f o r E and i f A

n

we have

~(Av,, A

(7.4.2.)

...

A Av,)

More g e n e r a l l y , i f we have a order a on E i f

is a

n

A d e n s i t y w on a v e c t o r space E of dimension

(7.4.1)i s

= E

ldet A 1 w(v, A

R, we say t h a t

L)

...

E

GL(E),

A v ),

.

i s a density of

s a t i s f i e d when we r e p l a c e IhI by

j h l a ; we t h e n need t o use l d e t

A1

a

(7.4.2.).

in

The s e t o f d e n s i t i e s of o r d e r a on E c o n s t i t u t e s a complex vector space o f dimension 1 which we denote by /El

IRnI1

=

[ a

...

dxl

.

a'

dxn

I

a

E

C

.

f o r example

By m u l t i p l i c a t i o n , w e

define t h e b i l i n e a r mapping

[7.4.3.)

IEI,

X

and t h i s shows t h a t

]El, 3 IE

€3

i d e n t i f y t h e d u a l o f lEla w i

I f X i s a manifold of dimension

XI

spaces / T above x c 1

x

n , t h e family o f v e c t o r

arranges itself naturally into a vector

.

(CHAP. 1)


52

bundle o f rank 1, termed bundle of d e n s i t i e s of o r d e r

:

I n f a c t , i f X,,

X and w r i t t e n Q ( X ) . a

i s a c h a r t of X , it d e f i n e s a c h a r t AX and hence a b i j e c t i o n

$

:

X 3

U2 7

Ixl I a

5

Rn

C

x

1

3

u1

C

cl =mn

: An'TU1 P

ncy(ul)onto R,(cl]

of

on

a

u

,N

U1 x

An TG,

c.

If

i s another c h a r t o f X , formula

( 7 . 4 . 2 ) ( w i t h ldet A l a ) shows t h a t , i n t h e common open s u b s e t ,

151a

t h e coordinate t r a n s i t i o n function m u l t i p l i c a t i o n by Idet(X 1

o

'

()XI

la)-'

jJ@ .

o

Let p be a d e n s i t y o f o r d e r a on X , i . e . a C

Qu(X); i t s r e s t r i c t i o n c

of o r d e r a on U

.

j'

PjFj)

.

with a . ( ) 3

a "

E

p

I U:J i s

t r a n s p o r t e d by

we denote t h i s by p

*.(:.)

=

C (U. ) J

J

.

J

constitutes

I x.1 J

m

U

section of i n t o a density

and we put

j

ldz.1"

J

The t r a n s f o r m a t i o n l a w of t h e bundle

(x)

i m p l i e s t h a t i n t h e open s u b s e t U1 n Up, we have I -1 cy (7.4.4.) = ddX1 0 $

i2

a

-I

+,I

)I

Formula ( 7 . 4 . 4 ) shows t h a t we can speak o f real-valued d e n s i t i e s and of s t r i c t l y p o s i t i v e d e n s i t i e s .

To l e s s e n t h e n o t a t i o n a l

burden, we put

c"(x; nJx))

=

c"[lx

cy

I

.

1

c"(

Since t h e d e n s i t i e s o f o r d e r 1 a r e measures on X , we can i n t e g r a t e them; having r e g a r d t o ( 7 . 4 . 3 . ) , t h i s allows us t o d e f i n e t h e b i l i n e a r mapping


53

r

when a

+ p

= 1. cy =

The p a r t i c u l a r case

0

,

p

= 1 m

define continuous l i n e a r forms on C o (

shows t h a t C"(X)

1x1)

functions

which l e a d s u s , i n

accordance w i t h Remark 7.1, t o s t a t e :

DEFINITION 7.5:

A d i s t r i b u t i o n on the manifold X i s a

continuous linear f o r m on the space C z ( 1 on X.

1x1 )

of d e n s i t i e s of order

&(x)

The space of d i s t r i b u t i o n s i s denoted by

More generally, i f

3 i s a comples vector bundle over X , we

describe as the space of d i s t r i b u t i o n a l sections of of continuous linear forms on

denote i t by

.

&[x;

.

f$(XI 8 3 1

C:(X;

S the space and we

( '3%denotes the dual bundle of

9). We have t h e i n j e c t i o n of

f(x;

9)

in

& ( x i s")

d e f i n e d by

the p a i r i n g

Taking i n t o account ( 7 . 4 . 3 ) and

(7.4.51,we

can show t h a t

the dual of t h e space C z ( X ) i s i d e n t i f i e d w i t h t h e space of d i s t r i b u t i o n a l s e c t i o n s o f Q,(X).

When X i s equipped w i t h a

s t r i c t l y p o s i t i v e d e n s i t y 1-1 (we can always c o n s t r u c t one o f t h e s e by using a p a r t i t i o n o f u n i t y f o r X ) we a r e a b l e t o i d e n k i f y t h e s e spaces, s i n c e we can use t h e t o p o l o g i c a l v e c t o r isomorphism


54

( C H A P . 1)

We now give a b r i e f o u t l i n e of t h e p r i n c i p a l p r o p e r t i e s of d i s t r i b u t i o n s on a manifold, l i m i t i n g o u r s e l v e s , f o r s i m p l i c i t y , t o t h e case

3=

c

.

a(X)

We have f o r t h e p a i r

,

PIX)

the

same t o p o l o g i c a l p r o p e r t i e s as t h o s e mentioned i n S e c t i o n 1, f o r t h e c a s e where X i s an open s u b s e t o f I R n .

.

P(X)

We have t h e continuous i n j e c t i o n of L1 ( X ) i n t o loc we l e t V be an open s u b s e t of X , a d i s t r i b u t i o n

u E S(X)

d e f i n e s by r e s t r i c t i o n t o C"( IVI) a d i s t r i b u t i o n

UIV

0

If

E P(V1

The same proof a s was used i n t h e case o f an open s u b s e t of lR

P[v)

shows t h a t t h e spaces

form a sheaf on X.

*

n

The concepts

of s u p p o r t and s i n g u l a r s u p p o r t g e n e r a l i s e i n t h e same way t o t h e c a s e of a manifold:

t h e space &'(X) of d i s t r i b u t i o n s w i t h compact m

support i s i d e n t i f i e d w i t h t h e d u a l of C (

1x1 ) .

--

We can a l s o g i v e a d e f i n i t i o n o f d i s t r i b u t i o n s w i t h t h e a i d of l o c a l c h a r t s .

:

Letting

be a c h a r t of X, t h e n i f

x 3 u

Ly

u E

b(x)

CR"

uLy

t h e diffeomorphism

p o r t s it i n t o a continuous l i n e a r form

w

XLy

[u\u)

x,

on

U i . e . i n t o a d i s t r i b u t i o n Z,on

r,

U,.

xB i s a n o t h e r c h a r t , w e have t h e c h a r t t r a n s i t i o n formula f o r t h e r e s t r i c t i o n s of 5,and

xLy[uen up)

and

ii t o t h e open s u b s e t s

%WLyn up'

B

trans-

If

7)

(SEC.

OPEMTORS ON A MANIFOLD

55

(7.5.2.) c

(x,)

Conversely, if

i s an a t l a s o f X , i n each open s u b s e t U, we N

u

take a d i s t r i b u t i o n

E &(x)

E

&(E@)

such t h a t (7.5.2) i s s a t i s f i e d

Then, t h e r e e x i s t s a unique d i s t r i b u t i o n

f o r a l l a,B. U

u *

such t h a t

N

=

Xcr(Ucy)

4

,

In fact, the distribut-

"@

ions

a

l i f t into distributions u

her i n t o a unique d i s t r i b u t i o n

cy

E &(UCY)

u E P[X)

which p i e c e t o g e t by v i r t u e of t h e

sheaf p r o p e r t y .

DIFFERENTIAL OPERATORS ON A MANIFOLD:

7.6

f o l d of dimension

n

, Proposition 7.3

DEFINITION 7 . 5 : cm(X);

degree

i f f o r a l l real-valued

a

E

l e a d s us t o s t a t e :

Let P be a continuous linear mapping from

~"(x) into 0 m

If X i s a mani-

we say t h a t t h i s i s a d i f f e r e n t i a l operator of rg E C"(X)

and alZ

c~(x), the f u z e t i o n

is a polynomial of degree

m w i t h respect t o

T

for a l l fixed x

i n X.

This d e f i n i t i o n shows t h a t t h e r e s t r i c t i o n P

IU

of P t o an open

subset U o f X i s a d i f f e r e n t i a l o p e r a t o r on U.

Let

X

:

X 3U

P

GCW"

be a c h a r t of X, and P a

56


( C H A P . 1)

m

m

l i n e a r mapping from C (X) i n t o C (X); we d e f i n e t h e l i n e a r oper0

ator P

m

X

,-

e

from C o ( U ) i n t o C (U) by p u t t i n g it

With t h i s n o t a t i o n we have:

PROPOSITION 7.7: C I ( X ) i n t o C"( X)

.

Let P be a continuous l i n e a r mapping from

Then, P i s a d i f f e r e n t i a l operator of degree m

i f and only i f f o r any chart P

X

x

from an a t l a s of X the operator

i s a d i f f e r e n t i a l operator of degree m.

The n e c e s s i t y f o l l o w s from P r o p o s i t i o n 7 . 3 .

PROOF:

The s u f f -

i c i e n c y i s immediate by u s i n g a p a r t i t i o n o f u n i t y s u b o r d i n a t e t o t h e c o v e r i n g o f X by c h a r t s from t h e given a t l a s .

We can d e f i n e t h e p r i n c i p a l symbol o f a d i f f e r e n t i a l o p e r a t o r P of degree

x

morphism

m

on X .

If

x

c

U

:

-+

U i s a c h a r t of X, t h e d i f f e o -

*

a l l o w s us t o d e f i n e on t h e c o t a n g e n t space T U a homo-

geneous f u n c t i o n Pu by l i f t i n g t h e p r i n c i p a l symbol o f

m

x

P.

The e q u a l i t y ( 7 . 3 . 5 ) shows t h a t t h e f u n c t i o n s P" combine t o g e t h e r

m

m

into a C

%

f u n c t i o n on T X.

T h i s f u n c t i o n i s denoted by

P m ( x , 5 ) and i s c a l l e d t h e p r i n c i p a l symbol of P ; geneous of degree

m

with respect t o

By c a l c u l a t i n g w i t h

6

E

T

*

X

it i s homo-

X.

a c h a r t , we can show t h a t t h e c o e f f -

i c i e n t o f -cm i n t h e polynomial

(7.6.1)i s

given by

(SEC. 7 )

57


The concept o f a d i f f e r e n t i a l system g e n e r a l i s e s i n t o t h e concept of a d i f f e r e n t i a l o p e r a t o r a c t i n g upon t h e s e c t i o n s o f v e c t o r bundles. rank

Suppose

r

and

g

'? and

9

a r e complex v e c t o r bundles of

above X .

7.8: A continuous linear mapping

DEFINITION

P : c ~ ( x ,$1 +~ " ( x , Q) is a

cp E c"(x)

of degree m if for a l l real-valued a

E

is, f o r all x

x,

E

a polynomial of degree

with coefficients in the fibre

3

x

and a l l

F) the function

C:(X;

Let

differential operator

(F, Q)

: U -9

and

C,

i? CR"

f

uxcq

I

X

-

i?

u

with respect to

above x .

3

and g

f

s/

Iu

1

I

u

;

L

ExC'

y

E

-

Taking P t o be a continuous l i n e a r mapping from

C"[X, pf,g

6)

,

T

be a c h a r t above which t h e bundles

admit t r i v i a l i s a t i o n s

-

N

Sx of $

m

1

CI(x; b)

into

we d e f i n e by t r a n s p o r t a continuous l i n e a r mapping

~ l from C ~ ( Cq)

into

C""(r;

C')

by p u t t i n g :

58

( C H A P . 1)


17 .8.2.)

=

pf,g

PIUj

g*

0

f*

With t h i s n o t a t i o n , P r o p o s i t i o n 7.7 g e n e r a l i s e s immediately by r e p l a c i n g t h e operators of type P

X

t h e r e i n by o p e r a t o r s o f t h e

t y p e P f , g a s s o c i a t e d w i t h a covering of X by open c o o r d i n a t e

9 and 6 are t r i v i a l i s a b l e .

p a t c h e s above which t h e bundles

(s,$)

If P i s a

d i f f e r e n t i a l o p e r a t o r o f degree

$ and

by u t i l i s i n g l o c a l t r i v i a l i s a t i o n s of

which depends only on dq(x)

and a(x).

f u n c t i o n i s homogeneous of degree

5

dcp(x) =

we can show

c o e f f i c i e n t o f -rm i n ( 7 . 8 . 1 ) i s a f u n c t i o n

t h a t f o r x fixed,’the

iable

, then

m

m

Furthermore, t h i s

w i t h r e s p e c t t o t h e var-

and i s a l i n e a r mapping from

into X

QX

w i t h r e s p e c t t o t h e v a r i a b l e v = a ( x ), and we denote t h i s by Pm(x,

c)

t h i s i s by d e f i n i t i o n t h e v a l u e of t h e p r i n c i p a l

;

symbol o f P a t t h e p o i n t ( x , 6 )

EXAMPLE 7 . 9 : and n-1;

we c o n s i d e r t h e bundles

k *

,

exterior differential

C:(x,

(7.8.1)i s e- i T Q ( X )

k be an i n t e g e r c o n t a i n e d between 0

Let

S = (A T x j 8 c

from

T* X.

E

fl

Q

= (hk+’ T*X) e3

d

c

over X.

The

d e f i n e s a continuous l i n e a r mapping

6)

i n t o C”(X,

.

In t h i s case, the function

given by d(eiTrp.e)[x)

consequently

d

is a

=

(a,Q)

i 7

dcp(x)

A

a(x)

+

da(x)

9

d i f f e r e n t i a l o p e r a t o r o f degree

(SEC. 8 )

KERNEL DISTRIBUTIONS ON A MANIFOLD

1. The c o e f f i c i e n t o f a t t h e p o i n t (x, 5 ) (A

k

*

Tx X) 2' 3 C

E

T

T

*

shows t h a t t h e p r i n c i p a l symbol of

*

Tx X ) 8 & d e f i n e d by l e f t e x t e r i o r

m u l t i p l i c a t i o n by t h e one-form

8.

d

X i s t h e l i n e a r mapping from

(Ak+'

into

59

it.

OPERATORS AND KERNEL DISTRIBUTIONS ON A MANIFOLD

L e t X , Y be manifolds equipped r e s p e c t i v e l y w i t h s t r i c t l y p o s i t i v e d e n s i t i e s y and v. t h e r e s u l t s of S e c t i o n s

4

With any d i s t r i b u t i o n continuous l i n e a r o p e r a t o r

We now show how t h e d e f i n i t i o n s and

and 5 g e n e r a l i s e t o t h i s s i t u a t i o n .

K E a ( X x Y) A :

, we

C:(Yl

can a s s o c i a t e a

J(X)

by p u t t i n g

Conversely, we have t h e k e r n e l s theorem:

THEOREM 8.1: A : C:(Y)

->

KA E P I X x Y)

PROOF:

P(X)

Any continuous l i n e a r operator i s defined by a unique kernel d i s t r i b u t i o n

such t h a t we have (8.1.1).

L e t (Ua) and ( V ) be a t l a s e s o f X and Y .

B

rjith t h e s e c h a r t s , Theorem

By t r a n s p o r t

4.4 shows t h a t f o r any a ,

f3 t h e r e


60

e x i s t s a unique d i s t r i b u t i o n

Ku,B E p ( u ,

(CHAP. 1)

such t h a t

X

Making use of t h e bundle p r o p e r t y , we deduce from t h i s t h a t t h e r e e x i s t s a unique d i s t r i b u t i o n KIUu x VB

Kac,B

=

K

f o r any a,@.

E

p [ X x Y)

such t h a t

By p a r t i t i o n of u n i t y , we t h e r e -

by deduce t h a t K s a t i s f i e s ( 8 . 1 . 1 ) .

REMARK

8.2:

I t i s c l e a r from (8.1.1) t h a t t h e k e r n e l KA a,

:

A

of an o p e r a t o r

Co(Y)

d e n s i t y p chosen on X.

x

" : X + X, 0

A :

C:(y]

does n o t depend on t h e

Furthermore, i f we have diffeomorphisms

Y

: Y -+ Y we d e f i n e t h e t r a n s p o r t e d o p e r a t o r

-

ICI

-J"X)

H

&();

o b t a i n e d from A by A = X,.A.e*. N

We s p e c i f y

N

a p o s i t i v e d e n s i t y v on Y ;

we can immediately show t h a t t h e

w

k e r n e l K- o f A s a t i s f i e s

A

c)

where K i s o b t a i n e d by t r a n s p o r t i n g K under t h e diffeomorphism A A Ie , X I : x x Y -X - x zY and where f i s t h e cm f u n c t i o n on Y d e f i n e d by 8+ v =

ry

f. v

.

I n p a r t i c u l a r , a change o f d e n s i t y on Y m a n i f e s t s i t s e l f on t h e k e r n e l K t h r o u g h m u l t i p l i c a t i o n by a s t r i c t l y p o s i t i v e C of t h e second v a r i a b l e .

m

function

(SEC.

8)

61

KERNEL DISTRIBUTIONS ON A MANIFOLD

F i n a l l y we n o t e t h a t i f t h e manifold Y i s n o t endowed w i t h a d e n s i t y , we s t i l l have a k e r n e l s theorem as l o n g a s w e seek K i n t h e space 8( (X x Y ;

no(xj

Q

me

Q, [Y) j

method of proof

used for Theorem 8 . 1 allows u s t o prove t h e f o l l o w i n g :

A continuous l i n e a r operator

THEOREM 8 . 3 :

A : C I I Y ' ) - . P ( x ) i s regularising i f and only i f

its kernel KA is

i n C"(X x Y). m

P(x) I

I f A i s a continuous l i n e a r o p e r a t o r from c 0 ( y ) i n t o

t h e d e n s i t i e s p and v a l l o w u s t o d e f i n e l i n e a r o p e r a t o r s 'A and

fl ( y ]

A* from C z ( X ) i n t o

REMARK

8.4:

, by

putting

P r o p o s i t i o n 4 . 5 g e n e r a l i s e s word f o r word t o

t h i s case.

THEOREM 8.5:

We assume t h a t the manifold Y i s compact.

Then i n t h i s case any regularising operator A :

c"(Y)

PROOF:

c"(x)

i s a compact operator.

Since A extends c o n t i n u o u s l y t o

f o r t i o r i i n t o a continuous o p e r a t o r from

$(yI

c0 ( Y )

8

into

it extends a m

c ( x ) , giving

62


(CHAP. 1)

t h e commutative diagram

I t , i s t h u s s u f f i c i e n t t o prove t h e f o l l o w i n g :

LEMMP,

j : c"(Y)

PROOF:

8.6:

->

I f Y is a compact manifozd, then t h e i n j e c t i o n

c"(Y) is compact.

We need t o show t h a t t h e r e e x i s t s a neighbourhood

Y

of 0 i n C"(Y) which is r e l a t i v e l y compact f o r t h e t o p o l o g y induced by Co(Y).

Let ( V . ) be a f i n i t e c o v e r i n g o f Y by open J

c o o r d i n a t e p a t c h e s . We d e f i n e

2/

t o be t h e s e t of f u n c t i o n s

m

v

E

C (Y) which t r a n s p o r t under each c h a r t t o f u n c t i o n s

having t h e i r f i r s t d e r i v a t i v e s bounded by 1.

By c o n s t r u c t i o n ,

m

i s a neighbourhood of 0 i n C ( Y ) composed o f e q u i c o n t i n u o u s functions;

t h u s it i s r e l a t i v e l y compact f o r t h e t o p o l o g y of

CO(Y).

9.

REGULAR OPEN SUBSETS OF IRn AND MANIFOLDS WITH BOUNDARY

We i n t r o d u c e i n t h i s s e c t i o n some c o n c e p t s which w i l l b e u s e f u l i n t h e s t u d y of boundary problems.

(SEC. 9 )

Let X be an open subset of IRn with clo-

DEFINITION 9.1 : sure

5 and

with boundary

ax

subset o f D n i f

ax.

E

ax,

We say t h a t X i s a regular open

i s a hypersurface o f D n and i f X i s locaZZy

situated on onZy one side of x

63

MANIFOLDS WITH BOUNDARY

ax.

This means t h a t f o r any

there e x i s t s an open neighbourhood U of x i n # c

diffeomorphism

x

:

U

+-

U of

and a

CI

u

onto an open subset U of IR" such

X'

ax

hr

u

X

n

X

In t h e preceding s i t u a t i o n we s a y t h a t

?

a t t h e boundary.

t h e hypersurface

ax

We n o t e t h a t

xlU

x

i s a l o c a l c h a r t of

ax

is a local chart of

a t x.

If X is an a r b i t r a r y open s u b s e t of Eln, we denote by

64


~ " ( 2t)h e

space of f u n c t i o n s cp

E ~"(ic) such

( C H A P . 1)

a"? E c0ljT)

that

Equipped w i t h t h e f a m i l y o f semi-norms

f o r any multi-index a .

I a@ q ( x ) 1 , where K is an a r b i t r a r y

com-

la/ 5 j p a c t s u b s e t of

?

If K i s a compact s u b s e t of

Frgchet space.

a,(?)=

and j is an a r b i t r a r y i n t e g e r 2 0 , C"(?)

2,

is a

we denote by

t h e (FrBchet) subspace o f C m ( z ) composed o f t h e

C",?)

f u n c t i o n s which a r e z e r o i n X \ K , and by C z ( ? ) =

a(?) the

inductive l i m i t of

8K ( 2 ) when K v a r i e s over t h e f a m i l y o f com-

2.

We h e n c e f o r t h assume t h a t t h e open s u b s e t

pact subsets of

X i s r e g u l a r ; we s h a l l s e e t h a t we can d e f i n e

C"(2)

n of t h e r e s t r i c t i o n s t o X o f Cm f u n c t i o n s inIR

.

a s t h e space More p r e c i s e l y ,

we have :

PROPOSITION 9 . 2 (SEELEY) :

-

There e x i s t s a continuous

Linear extension operator p : c m ( x )

C m o R n ) whose r e s t r i c t i o n

t o .8(X) i s a continuous Linear extension operator (again denoted by p ) p : S(X) +

PROOF:

x

R"+ = {

extension.

smn),

We begin w i t h t h e s p e c i a l c a s e

x

E a"

I

xn

>0 }

,

i n which we use a SEELEY

More p r e c i s e l y , f o r cp

-

E

m n C (IR+), we d e f i n e p ( Q ) by

(SEC. 9 )

x

where

65

W I F O L D S WITH BOUNDARY

i s e q u a l t o 1 i n t h e neighbourhood of 0 and

C;(B)

E

where (A ) i s a sequence o f r e a l numbers such t h a t

k

(9.2.1.)

j E KI

for a l l

>-:

Xk 2 k j

i s an a b s o l u t e l y

k z l convergent s e r i e s w i t h sum ( - 1)j ,

It i s immediately e v i d e n t t h a t t h e o p e r a t o r p s o cons t r u c t e d matches t h e r e q u i r e m e n t . I n t h e c a s e where

of lRn of

ax

charts

, we

c o n s i d e r a l o c a l l y f i n i t e c o v e r i n g of a neighbourhood

i n Bn by ( r e l a t i v e l y compact) open s u b s e t s U

xj

of

x a t t h e boundary.

Suppose we have 'p, Q ,

+

vj

=

j 21 Suppose $ E . j

supp

i s an a r b i t r a r y r e g u l a r open s u b s e t

X

cpj.

E

We p u t

Cm(Uo) and Cpj E

1

C:[uj)

m

c,(uj)

such t h a t

i n t h e neighbourhood of

i s such t h a t $

For c p ~C " ( ? ) ,

3

?.

= 1 in

rp = rp,

we have

Q

+

'pj rp

J We extend

i n t o @ = p ( ~ )E Cm(R")

, where

iP=@,+cPj

of l o c a l

j Uo = X \ (U Uj) . J

.

by p u t t i n g Q0

i s o b t a i n e d by e x t e n d i n g

j p ',

cp by 0 o u t s i d e o f U

0

and where

J

It is immediately obvious t h a t t h e o p e r a t o r p s o c o n s t r u c t e d meets t h e r e q u i r e m e n t . sequence

It remains t o prove t h e e x i s t e n c e of a

Xk s a t i s f y i n g ( 9 . 2 . 1 ) .

Cramer system

With N

E

N , we c o n s i d e r t h e

66


(CHAP. 1

A c a l c u l a t i o n o f Vandermonde determinants shows t h a t

2-j

Since t h e s e r i e s

converges, t h e r e e x i s t c o n s t a n t s C

j s l and c > 0 such t h a t

(1

+

2-3

5

c

J=1

,

and

We have t h e upper bounds

and s i n c e

Letting N

1 1 2a-l 2j - 1

-t m,

for R

2 1, we t h e r e b y deduce

we o b t a i n

Consequently ( 9 . 2 . 1 ) follows from ( 9 . 2 . 2 ) , ( 9 . 2 . 3 ) and ( 9 . 2 . 4 ) . We now consider t h e r e s t r i c t i o n o p e r a t o r

dR")-9 m)

It i s c l e a r l y continuous, and P r o p o s i t i o n 9.2 shows t h a t it is

(SEC. 9 )

By t r a n s p o s i t i o n , we o b t a i n an i n j e c t i o n o f t h e

surjective. dual

67


.PR)

M) i n t o

of

We i d e n t i f y a continuous

ax)w i t h i t s image under t h i s i n j e c t i o n ,

l i n e a r form R on

with t h e d i s t r i b u t i o n

c

(92.5.)

.P[R").

u, ip

u

o n B n d e f i n e d by

>

=

a(~,,]

@ E

for a l l

i.e.

~(R"I,

We t h e n have:

PROPOSITION 9.3 :

By way of the i d e n t i f i c a t i o n ( 9 . 2 . 5 ) ,

the dual P ( y ) o f @) coincides with

J!+'($")

.

This e q u a l i t y

X

is algebraic m d topological, both of these two spaces being

r,,.

equipped e i t h e r with T s or with dual

&'R) of C"R]

coincides with

In the same manner, the

EL(R") X

PROOF: i f supp @

u i s d e f i n e d by ( 9 . 2 . 5 ) , we have < u, Q > =

If C

.

R"\

u E 8. (R")

and t h u s

iT

.

0

Conversely,

we have

(9.3.1.)

Suppose we have

LEMMA:

u E

Ir_(R")

and

ip

E 4R")

X

such t h a t @ = 0 i n X.

Then

< u, @ > = 0 .

This lemma shows t h a t if u E Ir_(R"),

L on

f l ) by

we d e f i n e a l i n e a r form

A

putting

a((p) = < u,

i s an a r b i t r a r y e x t e n s i o n of

cp E

9

>

.

Ed

.Pg) = E[R") X

(z) .

.

4R")

.@i)

Making use o f P r o p o s i t i o n 9 . 2 , w e have and t h u s .f,

@ E

where

a(cp)

=

This proves t h e a l g e b r a i c e q u a l i t y The t o p o l o g i c a l e q u a l i t i e s f o l l o w from t h e

,

68

(CHAP. 1)


It now remains

c o n t i n u i t y of p and o f t h e r e s t r i c t i o n o p e r a t o r .

f o r us t o prove Lemma (9.3.1): through a p a r t i t i o n of u n i t y , we n can a s b e f o r e reduce t o t h e case X =lR+.

xn

@,(x) =

of

@(XI,

+

x, we have

< u,

>

0 since

< u,

@

=

REMARK 9.4

for all Q E

:

fl)

@

6

> ;.Q @e

E

> 0 , we put

i s z e r o i n t h e neighbourhood

; we t h e r e b y deduce t h a t

e

@

in

&(B(R")

when

E

-f

0.

We have t h e continuous i n j e c t i o n

,

with t h e function fo innn \

; since 0

e)

For

By means of ( 9 . 2 . 5 ) , f E

1 Lloc@in)

i s identified

d e f i n e d by f o = f i n

2, fo

= 0

X.

REMARK 9 . 5

:

&(XI

i s s e q u e n t i a l l y dense i n

& F] : by

a p a r t i t i o n of u n i t y we reduce t o t h e s e q u e n t i a l d e n s i t y of &(XI

in

&L'[R")

n i n t h e case X =B+. I n o r d e r t o e s t a b l i s h

X t h i s d e n s i t y , we proceed by r e g u l a r i s a t i o n ( s e e 1.1.1)by

n choosing p w i t h support i n B+.

REMARK 9.6 : manifold M ,

?

When i n D e f i n i t i o n 9.1 we r e p l a c e #

by a

i s c a l l e d a manifold w i t h boundary imbedded i n M.

A l l t h e preceding d e f i n i t i o n s and p r o p e r t i e s extend t o t h i s case

without d i f f i c u l t y .

(SEC. 9)

69


We now g e n e r a l i s e t o t h i s s i t u a t i o n t h e r e l a t i o n s h i p s b e t -

x be

Let Y be a manifold and

ween k e r n e l s and o p e r a t o r s .

a

manifold with boundary imbedded i n M: we assume t h a t p o s i t i v e densities

and v a r e s p e c i f i e d on M and Y ; we n o t e t h a t

!J

i s a manifold w i t h boundary imbedded i n M

x Y

from Lemma 1 . 3

and P r o p o s i t i o n 9.2

fi

s e q u e n t i a l l y dense i n

a(iTx

X X Y

E ( ~ ( Y ) , i~ (X))

d

into

p)

is

We have

, afl)=

(M x Y’)

Y) =

and t h a t ,

f l ]8 &Y)

,

.

x Y)

x Y

P(M) and t h e space

iT

of continuous l i n e a r o p e r a t o r s from B(Y)

i s a subspace of

.

fi [M’))

.C(b(Y),

By v i r t u e of

Lemma (9.3.1),it can immediately be shown t h a t t h e o p e r a t i o n of passage t o t h e k e r n e l ( s e e Theorem 8.1) induces a b i j e c t i o n of

L[B[y),

t r u e f o r a l l cp

A

.&fl

f l [ x ) ) onto

E c(&fl),

E

G ),

.s([Y))

the kernel

JI E B(Y)

A

K Ea[Y x

.

NOW

0

suppose

J!(Y)

r : B(M)

; if

ion o p e r a t o r , we have

and t h a t formula (8.1.1) i s

x Y)

is the restrict-

, and

r € E(B[M), & [Y))

we c a l l

M) of A o r t h e k e r n e l of A ; we can show

a s above t h a t t h e o p e r a t i o n o f passage t o t h e k e r n e l i s a b i j e c t i o n of

e[fi),

.s([(v))

Y)

and t h a t it i s for all

x, 7 be two manifolds with boundary

imbedded

If E 4 Y )

Finally, l e t

b ( Y x

>

again c h a r a c t e r i s e d by:

cp€*),

onto

c

A?,

$V

>=c

K, $ v 63

q#h

.

i n manifolds M, N equipped w i t h p o s i t i v e d e n s i t i e s

u,

v.

70

(CHAP. 1)


A E L(fl),

Suppose we have

fl))

d

;

we d e f i n e t h e k e r n e l K

of A by c o n s i d e r i n g A a s a continuous l i n e a r o p e r a t o r from

m)

fi [M)

into

, or,

which amounts t o t h e same t h i n g , by

c o n s i d e r i n g t h e k e r n e l of t h e o p e r a t o r A

o

r E C[aN],

KEJJ XXTi

Pfl)) .

The above shows t h a t

( M X N ) ~ E [ M X N ) = P( M X N) X x N X V

By

x

u t i l i s i n g Lemma (9.3.1),we can a g a i n show t h a t t h e o p e r a t i o n of passage t o t h e k e r n e l is a b i j e c t i o n o f

i",

q#>

,

We p u t

Show t h a t

and deduce from t h i s t h e upper bound

(2.9.1).

Conversely, l e t U be an e n t i r e f u n c t i o n i n C"

b)

T = rl[U

i n g (2.9.1). We p u t

sequence (see (l.l.l)),w i t h

)

I R" supp p

.

C

Let p

{ 1x1

p u t Tj = p j

*

T. Using Theorem 2 . 6 , show t h a t

c { x

I

dist.(x,

HOMOGENEOUS DISTRIBUTIONS:

t h e two open s e t s 7Rn o r E n \

let

T

x

morphism

E

0 ,

be given by t r a n s p o r t i n g T under t h e d i f f e o -

.&((n)

x

geneous o f degree

x

be a r e g u l a r i s i n g

J

Tj supp T c K .

10.2

.

K) 5 1 ]

j

satisfy-

c---)

m

(with m

.

E

x

E c

We say t h a t T i s homo-

) i n R i f TA = Am T f o r a l l

> 0.

a)

Show t h a t i f T i s homogeneous of degree

i s homogeneous of degree m - la1

Suppose

T

E S*[R")

, then

Da T

.

Show t h a t Da 6 i s homogeneous of degree

b)

m

-

n

-

la1 i n IR"

i s homogeous of degree

m.

I

Show

(SEC. 1 0 ) A

-

t h a t T i s homogeneous o f degree

c)

73

ADDITIONAL N O T E S

T E

Suppose we have

- m

n

(Rn)

A?'

i n IRn.

T

such t h a t

lRn\CO

m

is C

i

and homogeneous.

IR"\lo 3

is

Cm [for

T = UT

function Log 1x1. degree

-1

Suppose p v

-

E

X

1, and t h a t x pv

5 > 0 , and t o i w f o r 6

.I

( * ) 10.3

1 = X

identical t o 1 in the

(1

- U)T

]

i s t h e derivative of t h e

1 is X

odd and homogeneous o f

Deduce from t h i s t h a t t h e

1.

I

f o r a > 0 we have

?U

U c--t Tu = e--CY x2/2

I

.

6U4'2Ie'Re

(r

2

0

,

CY

E

E S'(R1 Show t h a t

Show t h a t t h i s

f 0 , by

Deduce from t h i s t h a t for

2

X

for

We r e c a l l t h a t

> 0 and continuous f o r Re c1 2 0 .

e q u a l i t y remains t r u e f o r

a-'.

i T

--2 i1n (pv; 1 - in 6) .

2 FOURIER TRANSFORM OF e ihx '2.

c1

.

< 0 , and t h a t

F l y

holmorphic f o r Re

3

and t h a t

A is t h e f u n c t i o n e q u a l t o X

Show t h a t t h e mapping

main branch of

+

Jl(CR)

Show t h a t pv

Fourier t r a n s f o r m of pv

(10.2.1

C:(IRn)

c1 E

neighbourhood of 0 , w r i t e

d)

T E 8'[R")

Show t h a t

1R \ {O}, t h e F o u r i e r t r a n s f o r m of ei X x /2 is

taking the

is

74

(CF99. 1)


10.4

FUNDAMENTAL SOLUTION OF THE CAUCHY-RIEMA"

We c o n s i d e r i n IRd

t h e Cauchy-Riemann o p e r a t o r

2 Let Q be a r e g u l a r bounded open s u b s e t of IR

a)

OPERATOR:

.

By app-

l y i n g S t o k e s ' formula, show t h a t

for

u,

(+y

,

E C'(5)

an

z = x + iy,

b e i n g given i t s canon-

i c a l o r i e n t a t i o n as t h e boundary of a compact s e t .

b)

Suppose we have zo

u(z) =J& *

E

Q.

By a p p l y i n g (10.4.1)for

a f t e r having removed from Q a d i s k w i t h c e n t r e

zn and w i t h r a d i u s € , t h e n by l e t t i n g

E

t e n d t o 0 , show t h a t

dx dy

0

c)

Deduce from t h i s t h a t

of t h e operator

a az

.

-

1 is E = nz

a fundamental s o l u t i o n

6

d)

By o b s e r v i n g t h a t E i s homogeneous of degree

10.2.,b ) , show t h a t

10.5

-

1 (see

=

FUNDAMENTAL SOLUTION OF THE LAPLACE OPERATOR:

c o n s i d e r i n IRn t h e Laplace o p e r a t o r

A=k-a2

We

,

j=i ax:

a)

L e t R be a r e g u l a r open s u b s e t o f I R n .

Suppose we

ADDITIONAL NOTES

(SEC. 10)

have

u, v E C’[?i]

, at

75

l e a s t one o f t h e two b e i n g assumed t o

have compact s u p p o r t .

By applying S t o k e s ’ formula t o t h e ( n

-

1) - form

show t h a t

Thus, deduce t h a t

n

=

grad u. grad v dx

an

where N i s t h e i n t e r i o r u n i t normal t o

and do i s t h e Riemannian

an (IRn b e i n g equipped w i t h i t s c a n o n i c a l E u c l i d i a n

density on structure).


b)

(10.5.1) f o r show t h a t

v

Suppose t h a t n 2 3 and

n= {

x E

R”

I

1x1

>

c]

E

4f]

,

.

By a p p l y i n g

and by p u t t i n g r = 1x1,

76

(CHAP. 1)


where un-l

is t h e a r e a o f t h e w i t sphere of l R n .


E = (2

- 1nlurr-l

~

x

l

- i s~ a

fundamental s o l u t i o n o f A when n 2 3.

c)

I n t h e c a s e n = 2 , proceed i n a similar manner by

-n+2 r e p l a c i n g 1x1 by Log 1x1 , and show t h a t

is a fundamental s o l u t i o n of A d)

Show t h a t , for n

2

Log 1x1

.

-

= - 1

3, we have

(use

I %I2

10.2, b )

10.6

1

E =

FUNDAMENTAL SOLUTION OF THE HEAT OPERATOR:

c o n s i d e r i n IR

n +1

We

the heat operator

W e put

a)

(with

T = 0

defines a distribution

+ iy, E E

y

0.

e-at

t -y(t) 0

c)

x)

2

0 }

,

Show t h a t t h e F o u r i e r

is

1 a+io


and, by u s i n g 1 0 . 3 , t h a t

i . e . t h a t E is t h e ( l o c a l l y i n t e g r a b l e ) function

10.7

FUNDAMENTAL SOLUTION OF THE SCHRVDINGER OPERATOR.

'We c o n s i d e r i n En+'

t h e Schradinger operator

. a ) Show t h a t t h e formula

( where fixed;

cp E B(A"+')

?

=

Q

- iy

,y

c0

) defines a distribution E E S ( R w 1 )

and t h a t t h i s d i s t r i b u t i o n does n o t depend on t h e number y > 0

78


chosen.

Show t h a t E i s a fundamental s o l u t i o n o f P, and t h a t supp

b)

(CHAP. 1)

E c { ( t ,x ) E Rn+'

I

t

2

0 ]

.

By proceeding as i n t h e p r e c e d i n g e x e r c i s e , show t h a t

and t h a t

(it w i l l be noted t h a t t h e function

10.8

FUNDAMENTAL SOLUTION OF THE WAVE OPERATOR:

We

c o n s i d e r i n Bn+l t h e wave o p e r a t o r

a)

Show t h a t t h e formula

t h i s d i s t r i b u t i o n does n o t depend on t h e number

y > 0 chosen.

Show t h a t E i s a fundamental s o l u t i o n o f P and t h a t

SUPP

Et

where

c

E C

[t, x)

E R"

By o b s e r v i n g t h a t P(7,

b)

79

ADDITIONAL NOTES

(SEC. 10)

Show t h a t

"'w

=

Et(x)

t

2

0

3

.

5) =

*

E)(;l tn-?

=

I

f o r t > 0 , and t h e r e f o r e

that

A

Show t h a t E, e x t e n d s i n t o t h e e n t i r e f u n c t i o n

c )

Show t h a t I z

lU(c)I SUPP

5

El c

1 1

C 1.l

= 15 I and t h a t / B 1 5

elT1 and 5 13.

IqI.

Thus deduce t h a t

t h e n b y u s i n g 10.1, t h a t


80

(CHAP. 1)

By u s i n g (10.8.1),i n f e r t h a t

(10

a 2.1

Supp

E

{ (t, x )

c

I

E R"+'

1x1 I t

.

3

F u r t h e r p r o p e r t i e s of E w i l l be seen i n E x e r c i s e 1 0 . 3 o f Chapter

111.

10.9 f :

R"

-

R

u = f

*

We d e f i n e u = v

v E >([R") 0

# 0 when f ( x ) = 0.

, the

E

.

We s h a l l now

v

under

E C"(P( \ 0)

i n v e r s e image of f

f on t h e open s u b s e t

a l o c a l diffeomorphism

Zn,

v

be such t h a t

neighbourhood of a p o i n t xo

f(x) into

Let

be a Cm mapping such t h a t f ' ( x )

v E &'((R)

Let define

INVERSE IMAGE OF A DISTRIBUTION:

-1

( Rn

\

0)

.

f

.

In t h e

IRn such t h a t f ( x ) = 0 , we consider 0

5

x cc x

x(x)

transporting

of 33"

and we p u t

* 0

Show t h a t t h e d i s t r i b u t i o n s u, u

0

piece together i n t o a distribShow t h a t i f a sequence

u t i o n on l R n , which we denote by f* v.

v

J

E p ( R ) fl C"(R\O)

v -

J

0

in

cw((R\O)

,

m

: vJ-0

i s such t h a t then

f*

v)

=

v

J

-

in

0

&((R)

&(f)

in

Show t h a t (103.1

,I

SUPP(f*

f-l(s"pp

v)

,

, ,

(SEC. 1 0 )

In t h e c a s e where

ADDITIONAL NOTES

v

i s t h e D i r a c d i s t r i b u t i o n i n IR, show

t

that f

v coincides with t h e d i s t r i b u t i o n 6 ( f ) defined i n

Section 7 . 2 .

We s h a l l g i v e i n Chapter V I I I a more g e n e r a l d e f i n i t i o n o f the i n v e r s e image.

81


CHAPTER 2.

SOBOLEV SPACES AND A P P L I C A T I O N S

1.

DIRICHLET'S PRINCIPLE

Sobolev spaces a r e spaces of d i s t r i b u t i o n s corresponding

t o a c e r t a i n degree of r e g u l a r i t y and t h e y have, i n a d d i t i o n , a H i l b e r t space s t r u c t u r e .

They a r i s e i n a n a t u r a l manner i n

the study of boundary-value problems f o r d i f f e r e n t i a l o p e r a t o r s , as t h e simple example which f o l l o w s w i l l show.

We l e t X be a

r e g u l a r bounded open s u b s e t of IRn and we c o n s i d e r t h e f o l l o w i n g D i r i c h l e t problem:

Find

where

g

I;

such t h a t Au - u = 0 i n X ,

u / a x = in We can f o r example assume g

i s given.

begin by s e e k i n g u

E

C2(x).

E

C

2

(ax)

and

P h y s i c a l c o n s i d e r a t i o n s l e a d us t o

consider a q u a n t i t y which w e c a l l t h e energy o f u, d e f i n e d by

Then, we have

83

84

SOBOLEV SPACES AND APPLICATIONS

The above Dirichlet problem admits

DIRICHLET’S PRINCIPLE :

a t most one solution.

u i s a solution, i t minimises the

If

energy E over a l l the functions equal t o g

I n f a c t , suppose we have v Let h = v E(v)

-

- u.

i,

2 Re

E

2 C ( X ) such t h a t v

( g r a d u (x)

U(X)

From Green’s formula

-

(where

au denotes an

normal t o

ax),

and s i n c e h

I ax

I ax

= 0.

-

we s e e t h a t E ( v ) = 0.

m ) d x

I x A u . hdx

t h e d e r i v a t i v e of

-

u

+

E(h)

.

i, 2 -

+

- , h d x

along t h e e x t e r i o r unit

E(u) = E(h) since A u

-

u = 0

Now E ( h ) i s always p o s i t i v e o r z e r o , and

i s zero i f and only i f h

-

=

= g.

I ax

-.

+

g r a d u.grad h dx

on the boundary.

u is a solution:

We have, i f

=

E(u)

(CHAP. 2 )

Thus, E ( u )

i s c o n s t a n t , i . e . zero s i n c e

h

5 E ( v ) and t h e e q u a l i t y

i s equivalent

t o v = u.

O f c o u r s e , t h e p r e c e d i n g argument does not prove t h e e x i s t -

ence of a s o l u t i o n and i n f a c t t h e r e does n o t always e x i s t a solution

u

in

c2(?).

It i s t h e r e f o r e n e c e s s a r y t o seek 2 -

i n a space l a r g e r t h a n C ( X ) .

We observe t h a t E(u) i s t h e

norm a s s o c i a t e d w i t h t h e p r e h i l b e r t s t r u c t u r e on C 2 ( ? ) by t h e i n n e r product (u, v),

=

u

( (grad u

.v + grad

)u ;,

dx

defined

.

(SEC. 2 )

85

HS(IRn) AND H s o c ( X )

SPACES

The D i r i c h l e t p r i n c i p l e l e a d s us t o c o n s i d e r t h e completed

1 H i l b e r t s p a c e , which we denote by H ( X ) , and which, as we s h a l l

1-

see, c o i n c i d e s w i t h t h e Sobolev space H ( X )

H”(K~

=

{ u E L ~ ( x )I

aj

E L*(x)

u

1-s jr n

for

>

,

*

We s h a l l a l s o s e e t h a t we can d e f i n e i n a n a t u r a l manner t h e trace v

=

I ax

ax

of v on

{ v E H”(Y)

I

when v vlax

1-

H ( X ) , and t h a t t h e a f f i n e v a r i e t y

E

g

=

}

i s closed i n

$(XI.

nus,

by analogy w i t h t h e D i r i c h l e t p r i n c i p l e , we can c o n s i d e r t h e orthogonal p r o j e c t i o n

4

that

u

u of 0 onto

c;

we s h a l l show i n s e c t i o n

1i s i n f a c t t h e unique element of H ( X ) which i s =

au-u

0

in

u a x = 9

in

such t h a t

2.

THE SPACES

For

E

2

L

s

E

IR, we denote by HS (IRn ) th e

u E S’(Rn)

space of d i s t r i b u t i o n s

I c12)s’z

.

HSDn)

DEFINITION 2.1 :

[I+

X

n

(R )

.

such t h a t

We equip H S ( E n ) w ith th e in n e r

product

and w i t h t h e associated norm

11

1 Is

defined by

86


-

We c o n s i d e r t h e t o p o l o g i c a l

We s h a l l f r e q u e n t l y p u t H S m n ) = H S .

: 88

v e c t o r isomorphism

(u, ")*

=

.

As u

81

d e f i n e d by we have

s

A

(CHAP. 2 )

v dx

from Theorem 2 . 5 of Chapter I .

HS i s t h u s t h e H i l b e r t space o b t a i n e d by t r a n s p o r t i n g t h e 2 n H i l b e r t space L (B ) by t h e isomorphism icular H

o

2 n = L (IR ) .

9 E IR

Let s,,

R S ;we

have i n p a r t -

be such t h a t s, 5

9 ;it

S

i s immediately c l e a r t h a t HS2 c H S

S

into H

H

and t h a t t h e i n j e c t i o n o f

i s continuous.

=

We put H+

fl

H'

,

s E R H4

=

u

,

8 c Hk

; we have

HS

and, from ( 2 . 8 . 1 ) o f

s E R Chapter I ,

.

&' c H-

upper bound

cy

15

I

I

i s continuous from H

S

From Theorem 2 . 1 0 o f Chapter I and t h e

I I I Icy/

,

a differentiation operator

i n t o Hs-lal.

Da

2 We have seen t h a t Ho = L ;

more g e n e r a l l y we have

For rn

PROPOSITION 2 . 2 :

tributions u

c

B([Rn) such

multi-index a s a t i s f y i n g

E

m y H" is

that

Icy1

D'

rn

S

the space of d i s -

u E L2(R")

,

for any

Moreover, the inner

product i n Hm is equivaZent t o the inner product

PROOF: we'have

-Da u

Suppose E

L

2

u E .B'

; from Theorem 2.10 o f Chapter I ,

i f and o n l y i f

ga ;I

E

L2

,

and i n t h i s

(SEC. 2 )

HS(Bn)

SPACES

AND HS (X) loc

87

e a s i l y be shown t h a t t h e r e e x i s t p o s i t i v e c o n s t a n t s C

such

C

1' 2

hence P r o p o s i t i o n 2 . 2 .

PROPOSITION 2.3

3

continuous;

HS; f o r h

E

E

HS

9

T~ u

4

u

I/f'-,

"11,

=

E

we denote by

IRn, x

-h

CX ,I

HS

In the ease where h = (0,

PROOF:

8

.. .

j '

h

0, h j , 0,

tends t o

-+

0

.. ., 0 ) a.

J

A

S

h

u the

Then

. the d i f f e r -

u in HS-l when

We now show t h a t t h e i n j e c t i o n o f

continuous; s i n c e onto

when

T h U - U

entiaZ quotient

T

I/ul/s

and in

is

c ,HS

the i n j e c t i o n of HS i n t o gr is

u under the t r a n s l a t i o n

image of h'

i s dense i n HS;

Suppose u

continuous.

8

The i n j e c t i o n

:

8 i n t o HS i s

is a t o p o l o g i c a l v e c t o r isomorphism o f

and o f HS o n t o Ho, it i s s u f f i c i e n t t o c o n s i d e r t h e

case s = 0 .

For

Cp

E8

we have f o r example r

6

88

(CHAP. 2)


so that

which proves t h e c o n t i n u i t y which we seek.

The i n j e c t i o n o f HS i n t o %' i s continuous; in f a c t , as p r e v i o u s l y we reduce t o c o n s i d e r i n g t h e c a s e s = 0 and we observe t h a t f o r u

I
t , and t h a t K is a compact subset of Bn s

H ~ ( R)

t

n

1is

.

E.

:

IR with

Then the i n j e c t i o n

compact.

be a sequence i n H

S

k

such t h a t l/uklls < 1.

We

now show t h a t we can e x t r a c t from t h i s a subsequence which i s convergent i n Ht.

Suppose we have

the neighbourhood of K .

u = TU

,

gu(

= (2~)’~

For u

E

*z ,

cp

E dRn)

equal t o 1 i n

S

HK, we have Da

= ( 2 ~-n) Du Cp A

*2 .

51

t o g e t h e r w i t h t h e e x i s t e n c e of a c o n s t a n t Ca such t h a t

As

(CHAP. 2)


96

We t h u s have:

(1

2 4 2

+ 151 )

IDcy

Uk[s)l

ccu

5

for a l l

k and a l l 5

E

IR",

a which shows t h a t t h e sequence (D uklk i s uniformly bounded on any compact s u b s e t o f l R n .

By a p p l y i n g t h i s r e s u l t f o r m u l t i -

i n d i c e s a such t h a t ( a (I 1, and by u s i n g A s c o l i ' s theorem, we s e e t h a t we can e x t r a c t from t h e sequence

%

a subsequence ( a l s o

denoted, w i t h an abuse o f n o t a t i o n , by uk ) such t h a t

(2.10.1)

\

converges uniformly on any compact s u b s e t of

We s h a l l now deduce from t h i s t h a t t h e subsequence u in H

t

.

Suppose R > 0 .

We have

where

f

Writing

we o b t a i n , s i n c e

t-s < 0

:

IIUk

-

2

Uallt

=

k

1,

W

n

.

converges

+

5

,

SPACES HS(iRn ) AND Hy,,(X)

(SEC. 2 )

97

so t h a t I2 can b e rendered a r b i t r a r i l y small i f R i s chosen s u f f iciently large.

For R f i x e d i n t h i s manner, I 1 can be rendered

a r b i t r a r i l y small i f k , R are s u f f i c i e n t l y l a r g e , by v i r t u e o f

(2.10.1).

Let sl, s, s2 be such t h a t s1 < s < s2.

PROPOSITION 2.11:

For any

E

> 0 , there e x i s t s

=

P

i . e . by p u t t i n g

a constant c E such t h a t

1

+ l5l2

, that

i.e. that

1 I S P

VS

+

-(Q-sl

) / ( ~ - 4s,-s

a

P

l/S2-S

By p u t t i n g

h = s

, this

i n e q u a l i t y can b e w r i t t e n

(CHAP. 2)


98

s -s

If ~p

2

1, ( 2 . 1 1 . 1 ) i s s a t i s f i e d s i n c e ( A p )

2 1 s -s

If Xp < 1, ( 2 . 1 1 . 1 ) i s s a t i s f i e d s i n c e ( X p )

2

1.

We now s t u d y d i s t r i b u t i o n s which behave l o c a l l y l i k e elements of

HS

. DEFINITION

suppose s u

E

HS

l 0 C

a'(X)

2.12 :

Let

x be an open stcbset

We denote by H ; o c ( X )

E 7R.

such t h a t

QU

E HS f o r a l l

of I R ~and

the space o f d i s t r i b u t i o n s Q

E

We equip

Cz(X),

( X ) with the topoZogy defined by the f a m i l y of semi-norms

u c ~ I ! ( p u ] /, ~ Q describing C E ( X )

.

I f U i s an open s u b s e t of X , t h e r e s t r i c t i o n o p e r a t o r i s e v i d e n t l y continuous from H

S

S

loc

(X) i n t o H l o c (U).

Furthermore,

we have :

PROPOSITION 2.13 :

( x ) into

i n j e c t i o n of

H~

s > n/2 + k

,

injection.

We have

loc

we have

i s a Frgchet space, and t h e

H;oc(X)

i s continuous.

P(X)

HSl o c (X)

n

,

c Ck(X)

HYoC(X)

=

C"(X)

men with continuous

.

A differ-

s E R entiaZ operator of degree m with Cm c o e f f i c i e n t s i n

x

is

continuous from H'Joe (XI i n t o H~;:(X).

PROOF:

Let K . be an e x h a u s t i v e sequence of compact J

s u b s e t s of X and suppose we have Then t h e sequence o f semi-norms

cpj E u

c--,

f(X)

(IQj

equal t o 1 i n K u(ls s u f f i c e s t o

j'

99

SPACES HS(IRn) AND H!oc(X)

(SEC. 2 )

define t h e topology of H

S

loc(X): in fact, if

,

cp E c:(X)

we

have, f o r j s u f f i c i e n t l y l a r g e ,

wj

cp =

1k9.11,

and

=

lIwj ulIs

C

from Theorem 2 . 6 , C only depending on s and cp is metrizable.

E

all (Pu

k

f(X]

'p

E

, Wk

i s a Cauchy sequence i n H in H

IJ

S

,

%

.

has a l i m i t

and a f o r t i o r i i n

Thus u

E

HS

loc

u in

( X ) and

a', w i t h cpu

\

ulis

Thus H;oc(X)

be a Cauchy sequence i n H

(9

C:[x)

.

Let (\)

possesses a l i m i t

from t h i s t h a t

livj S

loc S

(X) : for

, thus

a' ; it

= u f o r all (9

S

+

follows

u i n Hl o c ( X ) .

The

other a s s e r t i o n s o f t h e p r o p o s i t i o n a r e immediate, by u t i l i s i n g i n p a r t i c u l a r Theorem 2.5.

DEFINITION 2.14 :

We denote by H:omp(X)

th e union of the

spaces H! when K describes the famiZy of compact subsets o f X , and we equip HEomp(X) w i t h t h e i n d u c t i v e l i m i t topology of th e topoZogies of t h e HL.

THEOREM 2.15 :

The i n j e c t i o n s

c"(x)

C z ( X ) i s dense

-

in HS ( X ) and in H:omp(X). loc

H ; (X) ~ ~ and

-

C;(X)

Hs

(X)

COW

are continuous and of dense image and allow us t o i d e n t i f y th e duaZ of H Y o c ( X ) w i t h H-' ( X ) and the duaZ of HEOmp(X) w i t h c omp H;:c(X).

PROOF:

The c o n t i n u i t y of t h e i n j e c t i o n s f o l l o w s immed-

i a t e l y from t h e c o n t i n u i t y o f t h e i n j e c t i o n

R

- HS .


100

q?j a s i n t h e proof o f Theorem 2.13.

Suppose we have f u n c t i o n s If u

if p

S

Hloc(X),

E

it i s immediate t h a t

cp j u

--

i n HS ( X ) ; loc

u

i s a r e g u l a r i s i n g sequence, we have seen (Theorem 2 . 7 )

k

that

(CHAP. 2 )

pk

*

(cpj u)

-

i n H:oc(X).

t h e d e n s i t y o f C"(X) 0

-

P k * U

k

; hence

E

Similarly, i f

i n HS ( X ) when c omp

l-4

-+

k

i n HS when

cpj u

+=

S

HCD,,(X)

1

; hence t h e

4

i n HEomP(X).

d e n s i t y of C"(X) 0

We s h a l l now show t h a t each space HS ( X ) loc

, HLZmp(X)

is

i d e n t i f i e d w i t h t h e d u a l of t h e o t h e r by means of t h e b i l i n e a r form on HS

(X)

< h, v >

,

loc

x

HcZmp(X) where

x

E

d e f i n e d by

u, v-

=I

i s an a r b i t r a r y f u n c t i o n having

C:(X)

t h e v a l u e 1 i n t h e neighbourhood o f supp v , and where t h e l a s t s n s e t of b r a c k e t s a r e t h o s e o f t h e p a i r i n g between H (B ) and F i r s t , it i s immediately c l e a r t h a t < u, v > depends

HdS(Bn).

This proves t h a t 

c o n t i n u o u s l y on u for v f i x e d . does n o t depend on t h e f u n c t i o n

x

chosen, and we t h e r e b y deduce

t h a t depends c o n t i n u o u s l y on v f o r

u fixed.

Conversely, l e t R be a c o n t i n u o u s l i n e a r form on H and l e t

v

be t h e d i s t r i b u t i o n i n

< cp, v > = k(cp)

-

for all

cp E C"(X)

c'(X)d e f i n e d

ip

0

E

have

= < Xip, v

=.

I

J(Xip)

.

x

m

E

C0 ( X )

Then

depends c o n t i n u o u s l y on

Cz(IRn) f o r t h e topology induced by H S @ i n ) .

v E H-'(R*)

by

; suppose we have

e q u a l t o 1 i n t h e neighbourhood of supp u.

S

loc ( X ) ,

In similar fashion,

We t h e r e f o r e

SPACES HS(IRn) AND H 7 0 c ( X )

(SEC. 2 )

i f R i s a continuous l i n e a r form on H

u E &[X)

distribution

jl

E

Cz(X).

If

rp E C(;X)

depends c o n t i n u o u s l y on by H-‘(IRn),

d e f i n e d by

thus

then @

E C:[R”]

p E HS(R”]

REMARK 2.16 :

S

c omp

101

( X ) , we c o n s i d e r t h e

u, jl > = a ( $ ) f o r a l l

@-
= A(cp5)

f o r t h e topology induced

and f i n a l l y

The preceding i d e n t i f i c a t i o n s a r e a l g e b r a i c .

We can show t h a t t h e y a r e a l s o t o p o l o g i c a l when we equip each dual w i t h t h e topology of uniform convergence on t h e bounded subdomains.

We can i n f e r from t h e t h e o r y of p s e u d o - d i f f e r e n t i a l ope r a t o r s t h e i n v a r i a n c e under diffeomorphism of t h e spaces

HYoc(X)

( s e e C o r o l l a r y 8.9 i n Chapter I V ; s e e a l s o E x e r c i s e

7.2 f o r a d i r e c t p r o o f ) :

I)

THEOREM 2.17 :

let

x:

Let U, U be two open subsets of IRn and

c

U

-t

U be a diffeomorphism.

Then

x+

d e f i n e s a topotN

ogicaZ vector isomorphism of HSl o c ( U ) onto HSl o c (U) as we21 as o f

This theorem allows us t o g e n e r a l i s e t h e d e f i n i t i o n of H Y o c ( X ) t o t h e c a s e where X i s a manifold:

DEFINITION 2.18 :

Let X be a mcmifold.

We denote by


102

HSoc(X) the space of d i s t r i b u t i o n s f o r any chart equip HS

loc

E

u

X :

(CHAP. 2 )

> (X)

such t h a t

-u

w

u

of X.

We

(X) with the topoZogy defined by the family of conc

tinuous semi-norms o f x w u i n H Y o c ( U ) ,

x

describing the famiZy

x.

of charts of

If U i s an open s u b s e t o f X , t h e o p e r a t o r of r e s t r i c t i o n t o U i s e v i d e n t l y continuous from H

S

loc

(x) i n t o H'l o c ( u ) .

Let X be a manifoZd.

THEOREM 2.19 :

( Sheaf property of HS (X)). loc

(i)

be such t h a t , f o r a22 x U of x in X w i t h

uIu

E

E

Let

HF(xJ

=

E >(X)

X, there e x i s t s a neighbourhood H:oc(U)

.

Then u

( i i ) Let K be a compact subset of X .

subspace

u

c u E H:~~(x) I

HS (X). loc Then the E

3 of IIuI(~,~

c K

supp u

S

Hloc(X) is hiZbertisabZe, and we denote by

an

I f the manifoZd X i s compact,

admissibZe norm in H;(X), S

we put H ~ ~ ~ = ( Hx'(x). ( i i i )Proposition 2.13

and Theorem 2.15 remain vaZid.

For the duaZity properties, we assume X t o be endowed with a positive density.

PROOF:

u E HYo,(Uj)

such t h a t indices J show t h a t

For ( i ) ,c o n s i d e r an a t l a s

.

X

U j'

-J J 5

U

of X

for any j belonging t o t h e s e t of

Let

x.: U

-

Xw u

E H:oc(U)

,

w

U

be an a r b i t r a r y c h a r t o f X ; we i . e . that

cp & u

E

H5

for

(SEC. 2 )

HSDn)

SPACES

me

r~ E C;[U~

n uj),

X(U

c o n s t i t u t e an open c o v e r i n g

c . )

of U; t h u s t h e r e e x i s t s a f i n i t e s u b s e t F o f J such t h a t

u

c

SUPP

x(u n u j ) ,

Let t h e r e be f u n c t i o n s

j E F

n uj))

E C:(X(U

'pj

.

j E F

i s transported i n t o a function $

j

, cpj cp E

X"

0

n UJ.I),

f c"(x.(U O

i s transported i n t o $

j

X

J

J*

y j y )&

F, hence

Q

x,

E HS

.

n

C:(X(U

U.

Ujj)

and

which belongs t o

u

Theorem 2.17 t h u s shows t h a t y . cp X*

HS by h y p o t h e s i s . E

in the

=

We have Q X, u

By t h e c o o r d i n a t e t r a n s f o r m a t i o n X j

for a l l j

1

j E F

neighbourhood of supp cp

yj CP X , u

=

'pj

such t h a t

J

E HS

LJ

Furthermore, we s e e

t h a t , f o r cp f i x e d , t h e r e e x i s t s a c o n s t a n t C such t h a t , f o r a l l

u

E

HS (X), we have loc

Thus

(2.19.1.) If

of

x.J

u i n HS

loc

(j E J)

Xj

i s an atZas of X , the semi-norms

(Uj) are s u f f i c i e n t t o define the topoZogy o f

' s

Hloc(X). W e now proceed t o t h e proof of ( i i )and ( i i i ) .

We can c l e a r l y S

choose J t o be countable i n (2.19.1),which proves t h a t Hloc(X)

i s metrizable.

Let ( \ )

be a Cauchy sequence i n H

u

X : U+U

any c h a r t s

-

(U) , therefore Hloc

of X ,

x \

x* \

S

loc( X ) ; for

i s a Cauchy sequence i n d

h a s a l i m i t u i n HS (U); i f X lot

x'

is

another c h a r t o f X, we can s e e by p r o c e e d i n g ' t o t h e l i m i t t h a t

ux, = [ X I

o

)(-I)* ux ; t h u s t h e r e e x i s t s u

E

8 (X) \

U

X

= X+ u for any c h a r t

x

of X.

We have

such t h a t


104 i n HS

Uk-

loc

Therefore HS

(X).

loc

U

-

cy

U

K c

(j E F) such t h a t

of X

'pj E C I ( U j )

For

i n t h e neighbourhood of K .

I=

j- E ~ ' j '

X j + ( ~ j u)-

loc ( X ) , and

u

Uj

.

vj

= 1

j E F

j

Let t h e r e be f u n c t i o n s

u =

S

L e t t h e r e be a f i n i t e f a m i l y o f c h a r t s

t h e r e f o r e complete. :

( X ) i s a Fr6chet s p a c e .

i s closed i n H

L e t K be a compact s u b s e t o f X ; H;(X)

xj

(CHAP. 2 )

and u

-f

0 in H

9

0

such t h a t

E HE(X),

u S

loc

we have

( X ) i f and only i f

i n HS f o r every j

F: t h i s i s c l e a r l y

E

n e c e s s a r y , and it i s s u f f i c i e n t from (2.19.1)by c o n s i d e r i n g

an a t l a s of X formed by t h e union of

x j and

complement of t h e neighbourhood of supp

an a t l a s o f t h e

qj.

Thus t h e topology

i n H s ( X ) can be d e f i n e d by t h e norm a s s o c i a t e d w i t h t h e i n n e r

K

product

We l e a v e t o t h e r e a d e r t h e t a s k o f g e n e r a l i s i n g t h e o t h e r p r o p e r t i e s s t a t e d i n P r o p o s i t i o n 2.13

3.

THE SPACES HS ( 2 )

and Theorem 2 . 1 5 .

AND H s o c ( ? ) . n

L e t X be a r e g u l a r open s u b s e t of IR Chapter I ) and l e t s E IR

H5(Rn)

-

&(X)

.

(see Definition 9.1,

The r e s t r i c t i o n o p e r a t o r

1":

d e f i n e d by r u = uIx, h a s f o r i t s k e r n e l

S

t h e c l o s e d subspace H @in) o f H S ( B n ) ; we t h u s have an isoIRL1\X morphism of HS / HS onto I m r . we equip H' / H' with the RT1\X Rn\X H i l b e r t - q u o t i e n t s t r u c t u r e , which t h e p r e c e d i n g isomorphism allows us t o t r a n s p o r t i n t o a H i l b e r t s t r u c t u r e on I m r .

This

leads t o :

Let X be a regular open subset of Bn

DEFINITION 3.1 : and l e t s it E

E

s'(x)

VQ denote by H s ( ? )

1R.

a h i t t i n g m extension

with the norn;

11~11,

il~ll,,

= Inf

the space of d i s t r i b u t i o n s

Z in

we equip

H'(IR~).

where

H'IX)

describes the famiZy

of extensions of u i n H ' (I$).

Tne decomposition i n t o an o r t h o g o n a l d i r e c t . sum

H'(R")

H~

=

(R")

8

R"\ X shows t h a t , for a l l

u E Hsfl],

e x t e n s i o n U o f u such t h a t

we have

I/uIls

=

II~ll,

U

'

[Hs (R") Rn\ X

)'

t h e r e e x i s t s a unique

(HS [I?"))' , and also t h a t Rn\X 2 We have H o ( ? ) = L ( X ) and, more

E

generally:

PROPOSITION 3.2 :

Suppose we have m

i s the space of d i s t r i b u t i o n s

The norm i n H-m(?) defined b y

u E P(X)

E

W.

Then H-m(Z)

of the form

i s equivalent t o the norm

u C-- I u

(CHAP. 2 )


106

where the lower bound i s taken f o r a l l the decompositions ( 3 . 2 . 1 ) of u.

-

PROOF:

Suppose we have

F

u

a*(fE)

E fi (X)

of t h e form ( 3 . 2 . 1 ) .

f:

E

obtainecl by e x t e n d i n g f a by 0 o u t s i d e o f

x.

u

We p u t

=

Icy

where

5.m

2 n L (R )

is

The d i s t r i b u t i o n

Y

u i s an e x t e n s i o n of u and, from C o r o l l a r y 2 . 9 . , w e have E- H-m(Rn)

with

Conversely, suppose u that

Fa

/\ulLm= \/Ul\-rn 2

E

r

E

H-m(%)

j

we c o n s i d e r

I,Fm IiF,lo .

H-m(IRn)

such

2

-

IUIErn

We s h a l l now s e e t h a t H1(?) eared i n the introduction.

PROPOSITION 3 . 3 :

Then, f o r

E

, From C o r o l l a r y 2 . 9 , t h e r e e x i s t s

n

L (IR ) such t h a t

compact.

U

~f

we p u t

is indeed t h e space which app-

More g e n e r a l l y :

We assume t h a t X = X?: rn E N

,

HmF)

d i s t r i b u t i o n s u E b ( X ) such t h a t

aty

or t h a t

ax

is

i s the space of 2 u E L (X) for lall m.

The inner product i n

e(?) i s equivaZent

t o the inner product

I

(3.3.1

.I

. . )

For t h e moment we s h a l l denote by

PROOF:

of t h e u

8‘(XI such

E

that

acy u

with t h e i n n e r p r o d u c t ( 3 . 3 . 1 ) .

8(?) t h e space

2

E L (x) f o r J c c 1s m , equipped From P r o p o s i t i o n 2 . 2 , we have

Hmg.‘) C PflT) w i t h c o n t i n u o u s i n j e c t i o n . Conversely, we Frnfl] C HmF)w i t h continuous i n j e c t i o n . can show t h a t This amounts t o p r o v i n g t h a t t h e r e e x i s t s a continuous l i n e a r m n e x t e n s i o n o p e r a t o r from Hm(x) i n t o H (B ) . II)

By l o c a l c h a r t s and

a p a r t i t i o n of u n i t y , it can e a s i l y be shown t h a t it i s s u f f n i c i e n t t o c o n s t r u c t such an o p e r a t o r i n t h e c a s e X =B+. We c o n s i d e r t h e continuous l i n e a r e x t e n s i o n o p e r a t o r

p :

a$) -B[R”)

c o n s t r u c t e d i n t h e proof of P r o p o s i t i o n

9 . 2 Chapter I , or even s i m p l e r , t h e e x t e n s i o n p c o r r e s p o n d i n g t o :

I t i s immediately e v i d e n t t h a t p i s continuous w i t h v a l u e s i n

?(Bn) for t h e t o p o l o g y induced by

Vn-n

H [R+) on

m+).

In

o r d e r t o conclude t h e p r o o f , it i s t h u s s u f f i c i e n t t o prove t h a t

m+)

i s dense i n

Tq)

,

which we s h a l l do by t r u n c a t i o n

Let t h e r e b e f u n c t i o n s

and r e g u l a r i s a t i o n .

xk

and pk as i n

t h e s t a t e m e n t of Theorem 2 . 7 , w i t h i n a d d i t i o n SUPP p

c {

X n I

0

]

.

For u

Epm:)

L e i b n i z ’ formula Shows

108

SOBOLEV S P A C E S AND A P P L I C A T I O € J S

a"[\

that

=

U)

\ a" u +

, where

v

(CHAP.

2)

i s a l i n e a r combinat-

v

ion of terms of t h e form

I

k-1"'

(a"' x) (x / k l a""

'pk = ( u

We now p u t of lcll

0

by 0 f o r xn < 0 .

u

*

p

> ,, k IR+

We have

+

"" =

where u

0

p I

"8

5

1)

.

i s t h e extension

vk E C " q )

and, for

s m;

i t s convolution w i t h p choice of p .

-

a' (U0 7

s i n c e , because

Since

k

a

E

Pk-(a' u

2

i n L (By)

Suppose

s E R.

where i : C;(Rn)

there-

Suppose we have


Hs(T)

x is compact.

c ~ ( R " )i s

-

, and

, we


The f i r s t i n j e c t i o n can be w r i t t e n r

c;(iT)-

3 ,

i n view of t h e i n L2(Fn)

u)"

a"

p-(,

1

xn = 0

pq) .

R with s > t .

i s compact when PROOF:

cy

c

i s supported by

i s continuous and of dense image.

CI@>-Hs(% s, t

*

u)"

in

P R O P O S I T I O N 3.4:

u)"

n

(a"

'pk-u

fore that

(a'

i s supported by { x I 0

can t h e r e b y deduce t h a t

p :

(a'

u

0

i

-Ht(T) 0

p , where

a continuous l i n e a r e x t e n s i o n o p e r a t o r ,

Hs(Rn)

i s t h e i n j e c t i o n , and where

-

r : Hs(Rn)

is the restriction.

H'fl')

Its continuity thus

follows from t h e c o n t i n u i t y o f i ( s e e P r o p o s i t i o n 2 . 3 ) .

r

second i n j e c t i o n can be w r i t t e n

ps

H'(KJ

:

-H'[R")

o p e r a t o r , where bourhood of

2,

x).

(K = supp

x

E

j

E

, where

S

0

and where J : H-:Ht

is the injection

I t s compactness t h u s f o l l o w s from t h e compactness F i n a l l y , suppose u

H S ( B n ) i s an e x t e n s i o n o f

5 i n HS(IRn ).

+

Xp

C"( Bin) i s a f u n c t i o n e q u a l t o 1 i n t h e neigh-

u

E

HS

(x)and suppose

; we know ( s e e P r o p o s i t i o n

2 . 3 ) t h a t t h e r e e x i s t s a sequence Cp, Cp,

D

i s a continuous l i n e a r e x t e n s i o n

of j ( s e e Theorem 2 . 1 0 ) .

%

D

E

Cz(lRn ) such t h a t

i n Hs(X), and hence t h e d e n s i t y o f C"j?)

in Hs(X).

The dua2 of H s ( x ) (resp: of

PROPOSITION 3.5:

h

Ci

> = < u,

v

>

for u

E

H'R) ,

v

the

,

X

.

We begin by showing t h a t < %, v > does n o t depend on t h e

extension

(3.5.1)

K

E H~'(R")

being an arbitrary extension of u i n t o HS ( Bn )

PROOF:

is

H-'(R")

Hs(T)) by

i d e n t i f i e d with the normed space HIS(Bn) ( r e s p : X pairing v

-

u

~p

We can t h e r e f o r e deduce t h a t

klX

= 0

.

I n f a c t , by l o c a l c h a r t s and a p a r t i t i o n of u n i t y , we can


110

n reduce t o t h e case X = IR,,

(CHAP. 2 )

Suppose

U having compact s u p p o r t .

we have a r e g u l a r i s i n g sequence p k ( s e e Theorem 2 . 7 ) w i t h supp p

c

f

xn < 0

so t h a t

< U,

v >=lirn

ion

v

. < pk

pk

We know t h a t +6

U, v

> = 0

*

U

-

U in H

S

,

since the distribut-

i s z e r o i n t h e neighbourhood o f t h e s u p p o r t o f t h e t e s t

function pk " "

u , v > i s w e l l d e f i n e d and t h a t

This lemma shows t h a t

I
)

v

lFl\, IlvIl-,

I 5

IN, ~vC, .

l i n e a r form on H s ( X ) .

r : HSIJ?")-HSfl]

E

so t h a t

Since t h e r e s t r i c t i o n o p e r a t o r

r i s a continuous l i n e a r

0

thus it i s i d e n t i f i e d with a d i s t r i b u t i o n

satisfying
n/2

with c o n t i n u o u s i n j e c t i o n , when

n H;,,[R)

=

C"V)

;

loc

(x)i s

an a h i s s i b l e H i l b e r t norm

H;oc(Z)

I t can immediately be shown t h a t

in HE(?).

HS

+

Ck(y)

C

,

;

k

a d i f f e r e n t i a l o p e r a t o r of degree

m

S

w i t h c o e f f i c i e n t s i n C"(?)

H'l-o~(c?).

lo c (2

i s c o n t i n u o u s from HS

into

BY a p a r t i t i o n of u n i t y , we can c o n s t r u c t a c o n t i n u o u s

l i n e a r e x t e n s i o n o p e r a t o r from H

S

loc

(X)i n t o

(MI,and we can H' loc

t h e r e b y deduce t h e c o n t i n u i t y of t h e i n j e c t i o n

C"P)

-

HYocfl)

t o g e t h e r w i t h t h e d e n s i t y of C:(x)

in

We denote by HZ ( M ) t h e union o f t h e H s ( M ) when K X , comp K d e s c r i b e s t h e f a m i l y o f compact s u b s e t s of and we e q u i p

H:oc(?).

X

(M) w i t h t h e i n d u c t i v e l i m i t t o p o l o g y . When we assume X , comp t h a t M i s endowed w i t h a p o s i t i v e d e n s i t y , arguments analogous

:H

t o t h o s e of Theorem 2 . 1 5 S

d u a l of Hloc(X)

(resp:

and P r o p o s i t i o n 3.5 show t h a t t h e of HZs (M)) i s i d e n t i f i e d w i t h X,comp


112 H

< u, v > = < u, v > .v

E

u

where

loc

3

-

HSR)

'pu E

u

REMARK 3 . 8 :

(x)i s t h e

H[MI I'

I

s' (X)

E

and a l s o t h a t

Let Y be a manifold ( w i t h o r without boundary)

D e f i n i t i o n s 2 . 1 8 and 3 . 6 :

9

N

9

by u s i n g t r i v i a l i s a t i o n s of

CN) = (HSm:))N.

Y.

above

We

) i n a manner analogous t o

HS(Rn;

c o o r d i n a t e p a t c h e s of Y , we a r r i v e a t

above

CN.) = (HS(Rn))N

We l e a v e t o t h e r e a d e r t h e t a s k

of g e n e r a l i s i n g t o t h e c a s e o f s p a c e s

of t h e s p a c e s

and where

I/'pu//, d e f i n e i t s t o p o l o g y .

be a complex v e c t o r bundle o f r a n k

HSm:;

u

space of t h e u

cp E f r X ]

for a l l

n a t u r a l l y d e f i n e t h e space HS ( Y ; loc

4.

E

v

I n t h e c a s e where X i s a r e g u l a r open s u b s e t

we can show t h a t HS

t h e semi-norms

or at

and

i s an a r b i t r a r y e x t e n s i o n of

Hs,,(M)

such t h a t

and l e t

HY~,~)

x,conp

REMARK 3.7: o f E?

u E

of

(CHAP. 2)

H;o,[Y;

3) t h e p r o p e r t i e s

n-1 E R

xn

HYoc(Y) g i v e n e a r l i e r .

TRACE THEOREMS, SPACES H:(X).

We pt

R"

= { x =

(XI,

xnj

I

XI

E R 3

.

We

s h a l l s e e t h a t we can d e f i n e i n a n a t u r a l manner t h e t r a c e on t h e hyperplane { xn = 0 condition s >

i.

1

o f an element

u E Hs(Rn.)

under t h e

-

TRACE THEOREMS , SPACES HE (X)

(SEC. 4)

THEOREM 4.1:

operator y : C:[Rn) ( y ~ ) ( x 'j

-

Let

s E

u ( x ' , 0) extends

B be such t h a t

C;(R"-']

s >

113

f.

The trace

defined by

in a unique manner i n t o a continuous

Zinear operator (again denoted by

y)

from Hs(IRn ) i n t o

HS-'(R"-'

We f u r t h e r observe t h a t t h e change of v a r i a b l e

By i n t e g r a t i n g b o t h s i d e s o f t h i s i n e q u a l i t y w i t h r e s p e c t

t o E ' , we o b t a i n

).

114

(CHAP. 2 )


$2

and hence Theorem 4 . 1 , s i n c e C " ( B n

trace operator y

ya = Y

0

T~

a

h = (0,

2 . 3 shows t h a t t h e mapping fixed ( s >

2 ) , continuous

, we

a

likewise define the

xn = a }

on t h e hyperplane

, where

8

) i s dense i n H S ( B n ) .

given i n W

For a

REMARK 4 . 2 :

Jplt

cs

llYUll+

..., 0, a )

CL

ya u

;

,

we have

*

thus Proposition

is, for u

from IR i n t o H

s-$ ( Bn-l

E

HS(IRn )

).

The d i s t -

r i b u t i o n a s s o c i a t e d w i t h t h i s f u n c t i o n ( s e e S e c t i o n 1 . 5 , Chapter

I ) coincides with

u

i s obviously t r u e for u

u

E

i n f a c t , t h e formula

;

E

C m ( W n ) , and t h e r e f o r e a l s o f o r a l l 0

HS s i n c e b o t h s i d e s of t h i s e q u a t i o n depend c o n t i n u o u s l y on

u ( t h e c o n t i n u i t y o f t h e right-hand s i d e f o l l o w s from t h e domina t e d convergence theorem by v i r t u e o f t h e upper bound

COROLLARY 4 . 2 :

s-j > Hs(IRn)

2.

L e t there by s

Then the operator

yj = y

If u

i n t o HS-j-'(Bn-').

E

E

IR and j

DJ

xn. Hs(Wn),

E

N such t h a t

i s continuous from then

il

fwzction of class Ck with respect t o xn with values i n Hs-k-i ( Bn-l

) for 0 < k

I

j.

is a

(SEC. 4)

TRACE THEOREMS, SPACES Hz( X )

PROOF: Dj

-

The f i r s t p a r t f o l l o w s from t h e c o n t i n u i t y o f

H"j(Rn)

: HS(Rn)

Xn

C1(R,

H3-17(R

HS, t h e f u n c t i o n

E

n-I

7

x -1 y n corollary.

X

-

The l a s t p a r t of P r o p o s i t i o n 2 . 3 shows

We assume j t 1. that, for u

and from Theorem 4.1.

x

n

i s in

yxn u

)) and a l s o t h a t i t s d e r i v a t i v e i s t h e f u n c t i o n

a

u

t h i s i m p l i e s t h e second p a r t o f t h e

;

Xn

(Trace on a h y p e r s u r f a c e ) . Let X be a mani-

COROLLARY 4.3:

fold, Y a hypersurface o f X, and suppose ator y : cz(x]-

s >

2.

Then the oper-

of r e s t r i c t i o n t o Y e x t e n d s wziqueZy

Cz(Y)

into a continuous l i n e a r o p e r a t o r (again denoted by y i from i n t o H:it(y),

H:oc(x)

-

ry

PROOF:

For xo

E

X : U

Y, l e t

U

be a diffeomorphism o f a

neighbourhood of xo i n X onto an open s u b s e t of IRn

such t h a t

r

X ( U ~ Y ) =

For

'PI,,=,

Exculxn=o};

, we

Q1-

Q,

E Co(Uo) =

(90 *

If u

c o n s i d e r cp E

C:(X),

From Theorem 4 . 1 , we have

weput

E

C"(E) 0

we have

]/yo X,(Yu)\Is+ 0

(9,

such t h a t

d.

Let

x be a mani-

Then the operator

c"(ax> of r e s t r i c t i o n t o ax extends uniquely i n t o 0

a continuous l i n e a r operator y from HS (x)i n t o H s ~ ! ( ax). loc We conclude t h i s s e c t i o n by s t u d y i n g t h e c l o s u r e of C"(X) 0 i n HS loc

(x). To do t h i s ,

we s h a l l need t h e f o l l o w i n g :

(SEC. 4)

TRACE THEOREMS

PROPOSITION 4.6:

IR

, and

v

0

Zet j

Let

and t

E

E

IR

Then, f o r

v

,

?.

For

E

2 n L (R )

E f(Xj

,

which depends c o n t i n u o u s l y on

for t h e topology induced by H m ( X ) , i n view o f P r o p o s i t i o n

3.5 s i n c e U

HIm(IRn).

This proves t h a t t h e d u a l of H:(X) is X -m a l g e b r a i c a l l y i d e n t i f i e d with H ( X ) ; t h e t o p o l o g i c a l i d e n t i E

f i c a t i o n i s o b t a i n e d i n s i m i l a r manner.

REMARK 4.11 Propositions

:

Suppose s

E

R i s such t h a t

s >.

i.

4.6 and 4.8 show t h a t t h e l i n e a r mapping

i s c o n t i n u o u s , i n j e c t i v e and o f c l o s e d image.

I t s t r a n s p o s e , which i s a c t u a l l y t h e r e s t r i c t i o n mapping

122

(CHAP. 2 )


i s thus surjective.

By p a r t i t i o n of u n i t y and l o c a l c h a r t s a t t h e boundary,

w e can show t h a t t h i s s u r j e c t i v i t y r e s u l t remains v a l i d when

?,

En i s r e p l a c e d by

IRn-l

by

ax,

and y

j

u by

y (+

k)J

,

u

where X i s a r e g u l a r open s u b s e t o f I?? w i t h compact boundary n and where v i s a f i e l d o f v e c t o r s o n B t r a n s v e r s a l t o

ax.

L e t # be a complex H i l b e r t s p a c e .

R E M K 4.12 :

ax,

t h a t t h e p r o p e r t i e s of F o u r i e r t r a n s f o r m a t i o n g e n e r a l i s e

W e know

without

d i f f i c u l t y t o t h e case o f f u n c t i o n s or d i s t r i b u t i o n s w i t h v a l u e s i n 3$ ; f o r example, w e have P a r s e v a l ’ s formula

for

u, v E g($; #)

.

This allows us t o g e n e r a l i s e word f o r

word a l l t h e preceding results and p r o o f s t o t h e c a s e o f Sobolev spaces o f d i s t r i b u t i o n s w i t h v a l u e s i n

5.

3.

APPLICATION TO THE DIRICHLET PROBLEM

We r e t u r n t o t h e example i n t h e i n t r o d u c t i o n , i n a r a t h e r more g e n e r a l form. compact boundary operator P =

-

Let X be a r e g u l a r open s u b s e t o f En w i t h

ax

A +

and l e t P be t h e e l l i p t i c d i f f e r e n t i a l

V(x), where V

E

C”(?)

is such t h a t t h e r e

(SEC.

5)

DIRICHLET PROBLEM

123

e x i s t c o n s t a n t s c and C s a t i s f y i n g 0 < c 5 V[x) IC

for a l l

x.

X E

THEOREM 5 . 1 :

E

ti4 [ax)

Suppose we have f

E

H

-1 -

(X) m d

Then there e x i s t s a ttvlique u E H 1 R) such

,

that

u, v

for

E HIR]

,

From t h e assumptions made concerning

V ( x ) , ( ( u , v ) ) i s an i n n e r product e q u i v a l e n t t o t h e i n n e r product of H1(

2).

We f i r s t prove Theorem 5 . 1 in t h e p a r t i c u l a r c a s e f = 0. Since C ” ( X ) 0

“u, we put

1 i s dense i n H o ( X ) ,

.I)

= =

0

( 5 . 1 . 3 ) i s then equivalent t o

for all

c u EH’(K)

I yu

4 . 5 , 4.9 and Remark 4 . 1 1 show t h a t v a r i e t y i n H1(X)

1 parallel t o Ho(X),

v

=

E HA[X) g

1

.

Theorems

c i s a closed a f f i n e and t h e r e f o r e t h a t


124

u

E

(CHAP. 2 )

u is the

H1(%) s a t i s f i e s (5.1.1),(5.1.2) i f and o n l y i f

o r t h o g o n a l p r o j e c t i o n ( r e l a t i v e t o t h e i n n e r product ( ( u , v ) ) ) of 0 onto

c,

i s t h e unique element

u

i . e . i f and only i f

which minimises t h e energy E(u) = ( ( u , u)).

of

It remains f o r us t o prove Theorem 5 . 1 i n t h e c a s e g = 0 .

From Theorem 4.9, c o n d i t i o n s (5.1.1), ( 5 . 1 . 2 ) a r e t h e n equivalent t o =

((u, Since f

E

E c:[x)

f o r a l l cp

-1 -

H

( X ) , Theorem 4.10 shows t h a t I

a c o n t i n u o u s l i n e a r form on H (X). 0

q))

=

0 , C 2 0 and

from 10, LJ

0 , such t h a t

2).

into

(SEC.

7)

135

ADDITIONAL NOTES

e ) 'In t h e c a s e where P has c o n s t a n t c o e f f i c i e n t s and where 52 i s a bounded open s u b s e t of R n , prove t h a t

llpll,

=

l)p*

pun,

and t h a t t h e r e e x i s t s C t 0 such t h a t

Thus deduce t h a t i f P is non-zero,

7.6

t h e r e e x i s t s C such t h a t

LOCAL SOLVABILITY FOR DIFFERENTIAL OPERATORS OF PRINCIPAL TYPE WITH REAL PRINCIPAL SYMBOL

n Suppose Q i s an open s u b s e t of IR ; l e t P = P ( x , D ) be a d i f f e r e n t i a l o p e r a t o r o f degree

m

w i t h Cm c o e f f i c i e n t s i n R.

We suppose t h a t P is o f p r i n c i p a l t y p e , i . e . t h a t :

dt P m b l

51 f

where P

i s t h e p r i n c i p a l symbol (homogeneous o f degree

m

0

for

x

~

n ~ E R " \ O

m ) o f P.

With a E Q, we denote by B ( a , L ) t h e open b a l l w i t h c e n t r e a

and r a d i u s L .

a)

Prove t h a t t h e r e e x i s t s c > 0 such t h a t for

5 E R"

Deduce from t h e p r e v i o u s q u e s t i o n and f?om s e c t i o n

,

7.1 t h a t

136

(CHAP. 2)


t h e r e e x i s t s C t 0 such t h a t

Deduce from t h i s , again by u s i n g s e c t i o n 7 . 1 , t h a t t h e r e e x i s t

C ,C 1

2 0 such t h a t :

We s h a l l a c t u a l l y g i v e l a t e r ( s e e s e c t i o n 1 1 . 3 o f Chapter IV) a simpler proof of ( 7 . 6 . 1 ) . Prove, w i t h t h e a i d of

c)

exists a function L

t o 0 when L 2

+

lly\lwl

(7.6.2.) d)

+-

M ( L ) from

(7.5.1) and ( 7 . 6 . 1 ) t h a t t h e r e

10, L ~ ]i n t o LO,

+

4,t e n d i n g

0 , such t h a t :

5

M(L)

(i/b/\E +

lip*

(pll?)

for tP

c:(e(a~

Ll).

Henceforth we make t h e supplementary assumption t h a t

m are r e a l - v a l u e d .

t h e c o e f f i c i e n t s of t h e polynomial p Prove t h a t t h e r e e x i s t s M

for

Q-

e)

E :C ( s ( a ,

2

0 such t h a t

L)

Deduce from ( 7 . 6 . 2 ) and ( 7 . 6 . 3 ) t h a t t h e r e e x i s t L > 0

and M t 0 such t h a t

(SEC. 7 )

137

ADDITIONAL NOTES

(7.6.4.1 f) With t h e number L chosen i n t h i s manner, suppose f

E

H-(mll(flJ

.

E = { P* cp

We p u t

I

cp € Cz(B(a,

L)]).

Deduce from ( 7 . 6 . 4 ) t h a t

L : E - C

*

g = p cp

*

(f, CPJ

i s a well-defined a n t i l i n e a r form on E , and i s continuous when we e q u i p E w i t h t h e topology induced by L from t h i s t h a t t h e r e e x i s t s

Pu = f i n B ( a , L ) .

2

(B(a,L)).

2 u E L [B(a, L)) such t h a t

Infer


CHAPTER 3

SYMBOLS, OSCILLATORY

INTEGRALS

AND S T A T I O N A R Y - P H A S E THEOREMS

1.

INTRODUCTION.

Let P ( x , D ) be a d i f f e r e n t i a l o p e r a t o r of degree m

C

n c o e f f i c i e n t s i n an open s u b s e t X of IR

.

m with

By F o u r i e r t r a n s -

formation, we can e x p r e s s it i n t h e form:

If w e assume t h a t

P = P(D)

i s e l l i p t i c with constant coeff-

i c i e n t s , we have s e e n ( s e e Chapter I , P r o p o s i t i o n 6.7 ) t h a t we can d e f i n e a p a r a m e t r i x F of P i n t h e form F = (5”)

(#).

L e t t i n g Q be t h e o p e r a t o r o f convolution w i t h F , we l i k e w i s e say t h a t t h i s i s a p a r a m e t r i x of P because t h e o p e r a t o r s P o Q and Q o P - I a r e r e g u l a r k i n g .

-

I

The o p e r a t o r Q i s e x p r e s s e d by

f

and we have proved t h a t t h e f u n c t i o n q = l i k e a homogeneous f u n c t i o n of degree -m; any multi-index a , we have

139

e

behaves f o r 151

.+

more p r e c i s e l y , f o r

+

m

(CHAP. 3)

SYMBOLS AND OSCILLATORY INTEGRALS

140

I;?cy

S(5)1

=

d(l+ \I\) -m-l I. )

When P(x, D ) i s e l l i p t i c w i t h v a r i a b l e c o e f f i c i e n t s , we a r e t h u s l e d t o t r y t o c o n s t r u c t a p a r a m e t r i x Q of P i n t h e form

(12.)

(Qu)(x)

=

where q ( x , 5 ) behaves f o r

+

+ = l i k e t h e function c j ( S ) .

For example, as a f i r s t approximation, we may use t h e f u n c t i o n

We a r e t h u s l e d t o d e f i n e and s t u d y a c l a s s o f such f u n c t i o n s q , termed symbols; t h e o p e r a t o r s a s s o c i a t e d w i t h t h e formula

( 1 . 2 ) w i l l be c a l l e d p s e u d o - d i f f e r e n t i a l o p e r a t o r s .

2.

SYMBOLS Let X be an open s u b s e t of Wn ( n 2 0 ) and l e t N be an

integer degree m

1.

2 E

We d e f i n e t h e space Sm(X

x

N

W ) of symbols of

IR.

DEFINITION 2.1 :

We denote by Sm(X x W N ) the s e t of

complex-valued functions a E

~ " ( x x RN ) such t h a t , f o r any

compact subset K of X and any multi-indices C such t h a t (2.1.1.1

laxP a,a

a(x,

ell

5

cIi +

a, 6

there e x i s t s

I.

w1

for x E K , ~

Ve say t h a t 8 i s the frequency variable.

E

.

R

~

(SEC. 2 )

141

SYMBOLS

EXAMPLE 2 . 2

A polynomial

:

5)

p(x,

F

=

ICY

coefficients a a

For m

E

c"(x)

E

9

with

is in S ~ ( X x B").

+

IR, t h e f u n c t i o n (x, 5 )

(1

+ 1%) 2 )m/2

is in

mn).

Sm(X x

I n o r d e r t o g e n e r a l i s e D e f i n i t i o n 2 . 1 t o c e r t a i n open sub-

-

s e t s of X

m u l t i p l i c a t i v e group R (x, 83

The

we n e x t i n t r o d u c e a number of concepts.

X B N ,

+

N

a c t s on X xIR

N

tel

(x,

,

t E $'

by:

A s u b s e t o f X xIR

is s a i d t o

be conic i f it i s i n v a r i a n t under t h i s a c t i o n .

We d e f i n e t h e conic s u p p o r t o f a f u n c t i o n a

E C"(X x

N

p(

o u t s i d e of which

r\o rc =

)

t o be t h e s m a l l e s t c l o s e d conic s u b s e t

i s zero.

a

If

r

cx

{ ( x , e l c r 1 8 4 0 ] ,and We (x, tell (x, el E r, t 5 I 3.

RN

x

,

we p u t

=

c

Let

DEFINITION 2 . 3 : N X xB

.

We denote by

r

N

a l s o p u t I R ~=IR 0

\ 0.

be an open conic subset of

s m ( r )the s e t of complex-valued functions

a E Cm(r) such t h a t , f o r any compact subset K of

r and f o r a n y

multi-indices a , 8, there e x i s t s C such t h a t : (2.3.1.

we p u t

)

\a! a:

s+"(rI

a(x, =

ell

u $(rI m

I CCI

+

, s-(rI

p \ IwI.1 =

for ( x ,

n s"'(rI m

e) E

K'

(CHAP. 3)


142

m

Sm(I') i s obviously a v e c t o r sub-space of C (I');

a

E

Cm(I')

which i s homogeneous of degree

a function

m with respect t o 0 ,

( i . e . s a t i s f y i n g a ( x , t e ) = tma ( x , e l f o r t > 0, (x, 0 ) A function a

Sm(r).

E

l a r g e i s i n Sum(r).

E

r)

is in

Cm(r) which i s zero f o r 101 s u f f i c i e n t l y

We can immediately show t h e f o l l o w i n g :

c S"(r] c S"'(r) c s+"(r)

We have S'"(r]

PROPOSITION 2 . 4 :

for m < m'. Suppose

a,B

3 e,

6

Suppose a

, b E swl~l(r)

E S"(r)

a

E

cm(r); then

E

E Sm(r \ 0)

a

.

+m

In particular, S

.

(r)

-

b

E

S-OP(I').

then ab E S""'[r)

;

sm(r) if

-

b when a , b

E

I

and only if

i s a commutative a l g e b r a , and S-m(I')

i d e a l of t h i s ; we w r i t e a

a

a E

S"'(r]

+m

S

i s an

( r ) , with

Being a symbol i s a l o c a l p r o p e r t y i n t h e

s e n s e o f open c o n i c s e t s :

Let

PROPOSITION 2.5 :

x

x

N IR

(xol

.

e0)

Suppose E

r\

PROOF: we can assume

r;

0

a

E

c"(r)

, there

r

be an open conic subset

,

I f , f o r m y point

e x i s t s a conic neighbourhood

According t o t h e f i n a l p a r t o f P r o p o s i t i o n 2 . 4 ,

r

we put e =

=

r\

0

el

L e t K be a compact s u b s e t of

101 ; we have

Inf (XI

,

E K

E

> 0.

By h y p o t h e s i s ,

(SEC. 2)

f o r any

r

(xo, go) E 8 ) in

of ( x

0 )

143

SYlmOLS

I

0

t h e r e e x i s t s a c o n i c neighbnurhood V

such t h a t t h e upper bounds ( 2 . 3 . 1 ) a r e

satisfied for (x, 8 )

191

V,

E

By c o v e r i n g K by a f i n i t e

we s e e t h a t we o b t a i n t h e

0'

upper bounds ( 2 . 3 . 1 ) f o r (x, 8)

,

s

2

number of such neighbowhoods V

KC.

E

m

We s h a l l now s e e t h a t symbols of degree

PROPOSITION 2 . 6

N1 R ~ X R , ,J

,

Let f

T,

:

b

PROOF: C

-

__+

el

E srn(F2)I

Let K

1

r. 3

be an open conic sinbset of

,

( j = 1 , 2 ]

(XI

If

Let

:

r2 (Y,

be

wl

and homogeneous of

Cm

degree 2 ( i . e . commuting with the action ofm')

a = b

then

o

E

f

be a compact s u b s e t of

r 1; we

C

1

= (y(X,B),h(x,e))

r e s p e c t t o 8 , and

h

;

N2 , t h e r e 0

exist c, c' > 0

clel I Ih(x, 811 5 cllB1 f o r (x, 8) E K:

thus obtain (x, 9')

la(x, 911 2 C(1

E KY

( 2 . 3 . 1 ) f o r (x, 8 )

.

+

.

We

Ih(x, 911 1" I - C ' ( l + l B l ) m

We now e s t a b l i s h t h e upper bounds C

E

We p u t

i s homogeneous of degree 1 w i t h r e s p e c t

1

for

r2.

have

g i s homogeneous o f degree 0 w i t h

t o 8 ; s i n c e g(K ) i s compact inIR such t h a t

.

srn(r,] ,

f(K1) = K2, w i t h K2 = f ( K ) a compact s u b s e t of f(X,B)

a r e preserved

More p r e c i s e l y :

under c e r t a i n changes o f v a r i a b l e .

n,

0

K1 by r e c u r r e n c e on

/a1 +

161.

We assume

144

(CHAP. 3 )

SYMBOLS AND OSCILUTORY INTEGRALS

them t o be s a t i s f i e d ( f o r any 101

+

.

I p-1

ax j

with Since

,

f

o

3% ( 2 . 3 . 1 ) for

Icy1

m) when

and any

Now, we have

so(rlJ ,

E

b

0

a ‘k

+ 181 5

axj ahk

E

,satisfy

f

p-1

aej

sl(r~)

E

.

s-’(rlI

upper bounds o f t h e t y p e

( f o r degrees r e s p e c t i v e l y

m, m-l), L e i b n i z ’ s formula proves t h a t t h e same i s a l s o t r u e o f

axj

(for degrees r e s p e c t i v e l y m, m - 1 ) .

aej a

thus obtain f o r

We

t h e upper bounds ( 2 . 3 . 1 ) when

The f o l l o w i n g i s a u s e f u l s p e c i a l c a s e of t h i s p r o p o s i t i o n . Suppose

r1 is

an open conic s u b s e t of X

x

IRN and suppose 0

f

is

t h e mapping c o r r e s p o n d i n g t o passage i n t o p o l a r c o o r d i n a t e s i n

: ! R

x

:R 3

e l +->f

(x,

Then i n t h i s c a s e

rl

degree 1,.of

r2

= f

(rl 1

[2.6.1.)

C

a E

f

(x,

ms e IeI) E x

x

sN-,

x

R+

i s a diffeomorphism, homogeneous of

o n t o t h e open conic s u b s e t

X x SN,l

Sm[rlJ

x R+

, and

P r o p o s i t i o n 2 . 6 shows t h a t :

if and o n l y i f a

0

f-”

E

sm(r2)

The two p r e c e d i n g p r o p o s i t i o n s a l l o w us t o g e n e r a l i s e t h e

.

145

SYMBOLS

(SEC. 2 )

d e f i n i t i o n of Sm( r )

.

DEFINITION 2 . 7

manifold

We define a conic m m i f o l d t o be a

:

equipped with a Cw action of the m u l t i p l i c a t i v e

M

-k

group IR such t h a t : (2.7.1 .]

For any h

E

M, there e x i s t s a diffeomorphism

homogeneous of degree 1, of an open conic neighbourhood of A i n M onto an open conic subset

DEFINITION 2.8 :

m

E

t h a t , f o r any h

Suppose we have a conic manifold M and

E

EXAMPLE 2 . 9 :

E

fibre.

B

If

E

a

D

f'

x

satisfying

.

E Sm(r)

i s a v e c t o r bundle on a C

W

manifold,

E \ 0 , t h e bundle E deprived o f i t s 0 - s e c t i o n .

we denote by

+

C w ( M ) such

E

M, there e x i s t s a diffeomorphism

the conditions of (2.7.1)w i t h

For t

r ofnn x IRNa .

Sm(M) denotes the space of f m c t i o n s a

R.

x, u

, we

t

t a k e as a c t i o n m u l t i p l i c a t i o n by

i n each

Then E \ 0 , and any open conic s u b s e t of E \ 0 , i s a

conic manifold.

If M i s a conic manifold, any C

homogeneous of degree

m

W

function,

i n M, i s i n Sm(M).

PROPOSITION 2 . 1 0 :

i) Let M1, M2 be two conic manifolds- and l e t f : MI

_+

then

k$ a = b

be a o

f

C"

mapping of degree 1.

E Sm(M1)

.

If

b E Sm(t$)

,

146

(CHAP. 3)


i i ) Let M be a conic manifoZd, and Zet

and m' with

aj €

S"'(U~)

a

5

m

%

J

(mod. Sm'(Uj E

PROOF:

n uk, we

n U,)

have

f o r aZZ j, k

E

J.

Then

Sm(M) unique modulo Sm'(M) such t h a t

a .(mod. Sm'(U

J

Iuj

. Let there be

5 m' < m

such t h a t , i n u .

there e x i s t s a a

-=

)J € 3

j

Suppose we have

be a covering of M by upen conic s e t s . r n E B

(U

.))

J

f o r a22 j

E

J.

P a r t ( i ) i s an immediate consequence o f PropI n o r d e r t o e s t a b l i s h ( i i ) ,we f i r s t observe t h a t

o s i t i o n 2.6.

+

t h e q u o t i e n t space M' = M / B

i s a manifold and t h a t t h e

u is J j -1 c o n i c , we have U = p (U!); s i n c e t h e U! form an open c o v e r i n g j J J of M', we know t h a t we can t h e r e f o r e f i n d a l o c a l l y f i n i t e sub-

p r o j e c t i o n p: M

-+

M I i s crn; we p u t U! = p ( u . 1 ; s i n c e J

c o v e r i n g , and hence by i n v e r s e image under p a sub-covering of t h e c o v e r i n g of M by t h e U . which i s l o c a l l y f i n i t e i n t h e J c o n i c s e n s e ( i . e . such t h a t f o r any compact K o f M y t h e r e i s o n l y a f i n i t e number of i n d i c e s c o r r e s p o n d i n g t o a non-empty i n t e r s e c t i o n withIR

4-

K).

It w i l l t h e r e f o r e be s u f f i c i e n t t o

prove ( i i )when t h e c o v e r i n g i s of t h i s form. of u n i t y , we know t h a t t h e r e e x i s t a ' j

SUPP a'. c

E

ui

C"(Ml)

By a p a r t i t i o n such t h a t

I C Y ~ = 1 i n M'; by p u t t i n g CY j = CY' j 0 P J J we o b t a i n f u n c t i o n s a j E f ( M ] homogeneous of degree 0 ,

,

w i t h Supp a . c U j J

,

aj = 1 i n M.

3

I

(SEC. 2 )

We p u t

147

SYMBOLS

a =

aj

cyj

(where

c1

j by 0 o u t s i d e U

mjlu

of

.aj]

3 k

E

J we have:

a

=

We have a E Sm(M), a n d , for

,

c:[m

denotes t h e extension

j “j

E m j l u k ak

=

a.1

j J luk j m’ (mod S”’(u,]) and a a %(mod S (Uk)); hence t h e I’k existence of a . The uniqueness o f a modulo Sm’(M) i s

Iuk j therefore

obvious.

We now c o n s i d e r a c o n i c manifold N , and a c o n i c submanifold M o f N ( i . e . a sub-manifold M of t h e manifold N which 4-

i s s t a b l e under t h e a c t i o n o f IR

in M ) .

of t h e p r e c e d i n g p r o p o s i t i o n shows t h a t

If a

&IM

E

Sm(N) , p a r t ( i )

E Sm(M)

.

Conversely , we have :

Let M be a ZocaZZy cZosed conic sub-

PROPOSITION 2.11 :

manifoZd of a conic mrmifoZd N; for a

ii E

S”’[N)

PROOF:

extending

E

Sm(M), there e x i s t s

a.

We f i r s t show t h a t t h e p r o p o s i t i o n i s t r u e when we

r e s t r i c t o u r s e l v e s t o a s u f f i c i e n t l y s m a l l conic neighbourhood ( i n N) o f any p o i n t of M.

By u s i n g t h e diffeomorphism

x

of

(2.7,1),t h e n by t a k i n g p o l a r c o o r d i n a t e s inIRN ( s e e (2.6.1)), we can reduce t o N = Y xIR

+ , where

0

Y i s a manifold.

The pro-

j e c t i o n on Y of t h e sub-manifold M of N i s a l o c a l l y c l o s e d sub-manifold X of Y and we have M = X x lR

+

since M i s conic.

148

(CHAP. 3 )


By taking local coordinates in Y we reduce to the case where N = (U x V ) x

R',

{ 0

M = (U x

R ' neighbourhoods of 0 in IRp, I "

we define a = a(y, z ; t )

1) X ' R

.

where U, V are open a = a(y ; t)

If

E

by

N

a(y,

z ;

E

Sm(M)

,

t ) = a(y, t)

In order to prove that the proposition is true globally, we of N which j are locally finite (in the conic sense) and, as in the proof consider a covering of M by open conic subsets W

of Proposition 2.10, we construct functions

CY

aj = 1

and

j

pj E c ~ ( N )

,

c supp B J. c W jl Bj'l J in the neighbourhood of M. We

homogeneous of degree 0, with supp in supp

cyjl

cy.

; now, we have seen that there with a j cuj*a exist a E Sm(Wj) such that EL -- a j j\wj n M jlWj nM * H H If we put a = p j aj then E Sm(N] and

have a

aj

I

N

a d

,

=c .

N

alM

[B,.ZjIIM

-

aj = a

In applications, we encounter symbols which possess an expansion into homogeneous components in the following sense:

DEFINITION 2.12 : a

E

Sm(M) i s a c l a s s i c a l symbol of degree

f o r any j

E

IN a f w l c t i o n

degree m-j, such t h a t

any i n t e g e r k

2

1.

symbols of degree

a,

(a

-

*

E Cm(M), k-I j=o

aj-

We denote by S:l(M) m.

We say that

Let M be a conic manifold.

m i f there e x i s t s

homogeneous of

) E SWk[~)

for

the space of c l a s s i c a l

(SEC. 3 )

149

ELLIPTIC SYMBOLS

I n t h e c a s e of t h e space X

a E C”(X

t h e space of t h e

X I R

x RN)

It i s c l e a r t h a t t h e f u n c t i o n s a

N

, we

m

denote by Scl(X

x

#)

such t h a t

j

a r e uniquely determined f o r

; t h e f u n c t i o n a0 i s ‘ c a l l e d t h e p r i n c i p a l p a r t ,

a E Szl(Uj

m, of

homogeneous of degree

EXAMPLE 2.13 :

a.

We can show t h a t t h e symbols c o n s i d e r e d

i n Example 2 . 2 a r e c l a s s i c a l , and t h a t t h e y admit r e s p e c t i v e l y a s p r i n c i p a l p a r t of degree m:

3.

ELLIPTIC SYMBOLS DEFINITION 3 . 1 :

rn

E

IR.

Suppose M

We say t h a t a symbol a

i f there e x i s t s b

E

S-”(M)

E

-3

a con:> manifold and

Sm(M) i s e l l i p t i c of degree m

with d e b

E

1

(modulo

s-’(M))

Note t h a t t h i s d e f i n i t i o n only depends on t h e c l a s s o f

S~(M)/S“+’ (MI

a

in

.

From P r o p o s i t i o n 2 . 1 0 , e l l i p t i c i t y i s a l o c a l p r o p e r t y i n t h e s e n s e o f conic neighbourhoods; i n o r d e r t o s t u d y t h i s , we can t h e r e f o r e reduce t o t h e c a s e where M i s an open conic s u b s e t

of

x

XIRE.

(CHAP. 3)


150

X x Bt; a symbol a

r

Let

PROPOSITION 3 . 2 :

be an open conic subset of

s m ( r )i s e l l i p t i c of degree

E

only i f , f o r any compact subset K of

, if and

m

r, there e x i s t s

c > 0 such

that

el\

la(x,

(3.2.1.)

\el"

c

2

(x,

for

PROOF: K1

e] E

We assume t h a t ( 3 . 2 . 1 ) i s s a t i s f i e d ;

c$c

s u b s e t s of

... c r

K.

J

c

...

\el

KC and

2

E E

(u

c2

2

...

2

2

lR+ and t h a t F i s a s u b s e t of

(K5+1<j

)

since

*

be an exhaustive sequence of compact

and we l e t c 1

j 21

C

we l e t

c

2

j

..

be a sequence of

a s s o c i a t e d c o n s t a n t s , i n accordance w i t h ( 3 . 2 . 1 ) . that

-1

K.

J

c c Kj+l

r,

We suppose

and we p u t

.

It can e a s i l y b e shown t h a t A i s c l o s e d , and we t a k e a f u n c t i o n p E C m ( T ) e q u a l t o 1 on A and w i t h support i n B.

n o t v a n i s h on B , we d e f i n e b t o prove t h a t

(x, 0 )

E

r E S-"(r)

I! E c"(r) .

, we

Since

does

We p u t r = ab

-

1;

observe t h a t any p o i n t

T admits a conic neighbourhood To such t h a t f o r E

a

s u f f i c i e n t l y s m a l l , we have To c A , and s i n c e

r

E

> 0,

i s zero i n t h e

neighbourhood of A it s u f f i c e s t o apply P r o p o s i t i o n 2 . 5 .

Like-

(SEC. 3 )

wise, i n o r d e r t o prove t h a t b o r i z e t h e d e r i v a t i v e s of cy

axl 0

151

ELLIPTIC SYMBOLS

b =

a:,(:)

E

S-m(r) it i s s u f f i c i e n t t o m a j -

on A .

b

On A we have

and we have a l r e a d y observed i n Chapter I

( 6 . 8 . 2 ) t h a t t h i s i s a f i n i t e l i n e a r combination of terms of t h e

form

a$1

a

.... aB .

J.a

a

lBJ +

with

j.1-1

....+ le,I

bl

=

;

a

we t h e r e f o r e immediately deduce t h e d e s i r e d upper bound o f 3 . b x,e i n any domain ( K C ) € , w i t h K a compact s u b s e t o f

Conversely, i f w e t a k e a compact s u b s e t K of since

Ir[x,

, there

r E S-I(r)

Q)\ 5

Ib(x, 0 ) \

$

5 1(1

c la.bl 2 1 i n (Kc)' 2

for

exists

+ lQ\)-"

for (x,e)

and f i n a l l y

a

i n ( K C ) € which proves t h a t

DEFINITION 3. 3:

l(1 E

and p u t r = ab

We say t h a t a symbol a

E

S

1,

and

+

Consequently we have

KC.

E

18/)'m

1a(x,Q)1

2

21

s a t i s f i e s (3.2.1).

Suppose M i s a conic manifold, and m +m

-

> 0 such t h a t

E

E (KC)e

(x, 0)

r

r0 .

(M) i s e l l i p t i c of degree

m

E

IR

at a

point of M i f it i s an e l l i p t i c symbol of degree m in an open conic neighbourhood of t h i s p o i n t .

of p o i n t s of M a t which

a

We denote by E l l ( a ) the s e t

i s elliptic;

t h i s i s an open conic

s e t whose complement i s a closed conic s e t called the characteri s t i c s e t of

a

and denoted by c h a r ( a )

.

.

(CHAP. 3)

SYI’BOLS AND OSCILLATORY INTEGRALS

I n t h e c a s e of a symbol

a

#)

Sm(X x

f

, we

define t h e e l l i p t -

i c i t y , E l l ( a ) and c h a r ( a ) by c o n s i d e r i n g a

EXAMPLE 3.4: part a

4.

0’

i s a c l a s s i c a l symbol w i t h p r i n c i p a l

a

If

homogeneous o f degree

, we

m

-1

have c h a r ( a ) = a.

(0).

ASYMPTOTIC EXPANSIONS OF SYMBOLS

We can meaningfully c o n s i d e r a formal s e r i e s o f symbols ( a . ) J with degrees m

j

tending t o

-

,

m

when M i s a c o n i c manifold.

We s t a t e :

DEFINITION

4.1:

Let ( a . ) J

rn.

symbols

a:

be a c o w t a b l e family of j

,

~

E S J [ ~ ) ; we assume t h a t for any r e a l r , there

J

i s only a f i n i t e number of i n d i c e s say t h a t a f u n c t i o n a

E

Cm(M)

j such t h a t m .

satisfies a

-

J

aj

2

r.

We

i f there

J

e x i s t s a sequence of r e a l numbers (JJ) tending t o k f o r a l l k:

such t h a t

For example, w i t h t h e n o t a t i o n o f D e f i n i t i o n 2.12, we have

& - r _ 7 a j . j

2 0

THEOREM 4 . 2 :

Let

(aj)j

be a f a m i Z y of symbols as in

(SEC.

4)

D e f i n i t i o n 4.1.

Then there e x i s t s a

s-"(M), such t h a t a PROOF:

15 3

ASYMPTOTIC EXPANSIONS

-

The uniqueness o f

a

the existence of

, we

, with

aj

a

Sm(M) unique modulo

E

m = max m

j

j.

modulo SVm i s immediate.

To prove

w i l l not r e s t r i c t t h e g e n e r a l i t y , bear-

i n g i n mind P r o p o s i t i o n 2.10 ( T i ) , by assuming t h a t M = open c o n i c s u b s e t o f X x 1R".

r

i s an

We b e g i n w i t h t h e p a r t i c u l a r c a s e

where J = W and where t h e sequence (m.) i s s t r i c t l y d e c r e a s i n g J (and tends t o

-

x(0) = 0 if

m).

S

Suppose we have

X E C=(R~]

1/2, and x(0) = 1 i f 101 2 1.

e x h a u s t i v e sequence o f compact s u b s e t s o f

r.

such t h a t Let (K.) be an J

We can f i n d an

i n c r e a s i n g sequence ( t. ) such t h a t ; J

(we can e a s i l y c o n s t r u c t t h e t that for

j

,

j f i x e d , t h e r e . i s o n l y a f i n i t e number of upper bounds

( 4 . 2 . 1 ) t o b e r e a l i s e d , and t h a t

if 18 I

i n s t e p w i s e f a s h i o n by o b s e r v i n g

)

+

.

(

l i8

i)mj-l rn - 0

We t h e n p u t

+ m

X ( e ) aj(Xt 8 ) which i s c l e a r l y i n C m ( r ) s i n c e j=o j t h e sum i s l o c a l l y f i n i t e ; t h e convergence of t h e s e r i e s B(x, 95

2-j k- 1

a

=

a l l o w s us t o deduce from ( 4 . 2 . 1 ) t h a t

- j=o aj

rnk

E S

(r) for any

i n t e g e r k t 1.

1 54

We now pass on t o t h e e x i s t e n c e o f

p o s s i b l e values of m

bk =

-

rn

j

k bk

i n the general case;

j

j ranges over J .

when

n k E S (r) , and we c o n s i d e r a

aj

=n

a

we

t h e s t r i c t l y d e c r e a s i n g sequence of t h e

denote by ( n k l k

a

(CHAP. 3)

SD3OLS AND OSCILLATORY INTEGRALS

.

Then f o r a l l R

E

We p u t m

E

C

(r)

such t h a t

IN w e have:

k E M

mined by

n

-1

2

r

n

t h e requirements s i n c e

REMARK 4.3:

.

> nr+l

F i n a l l y , t h e symbol

a -+

r+l

'

m

We can confirm d i r e c t l y from t h e d e f i n i t i o n s Let ( J h ) h for h

be an a r b i t r a r y p a r t i t i o n of t h e s e t o f i n d i c e s J ;

%

(note t h a t

-

%

.

a j E J h E

S %(M),

a

Then we have

j mh=

where

m

max j E H

REMARK 4 . 4 :

J

-

]

E

H,

ah h E H

.

I n t h e case where J = TN and where t h e sequence

a

( m . ) i s d e c r e a s i n g , we have J any i n t e g e r k, we have

c a s e where M =

meets

.

- 0

t h a t we have t h e f o l l o w i n g a s s o c i a t i v i t y p r o p e r t y .

suppose

a

r,

[a

-

a

LV

k

J Z O

a.)

J

E

5

i f and o n l y i f f o r

Srnktl(M)

.

In the

j=o t h i s condition i s equivalent t o t h e following:

for any i n t e g e r k , and f o r any compact s u b s e t K o f r , t h e r e e x i s t s k "k+l- 1 c such t h a t la: aB, (a-( )1a j I ( x , I ~ ( +1 \ e l ) j-o

1 55

TOPOLOGY

1.1

f o r (x, 9 ) E KC and

+

k

I n f a c t , i n o r d e r t o majorise t h e d e r i v a t i v e s of o r d e r k' > k , it suffices t o write

k

k'

k'

and t o n o t e t h a t

k'

m

j=k+l

5.

TOPOLOGY ON THE SYMBOL SPACES

Let

r

be an open conic s u b s e t o f X x IR

subset K c

r

N

.

With any compact

j we a s s o c i a t e a semi-norm on

and any i n t e g e r

Sm(r ) by p u t t i n g

m( r ) a metrizable topological vector

t h e s e semi-norms d e f i n e on S

space s t r u c t u r e , because it s u f f i c e s t o consider an exhaustive

(5).

sequence of compact s u b s e t s

We have t h e approximation r e s u l t :

PROPOSITION 5.1:

If a

E

Sm(r), we take

p E

t o 1 i n the neighbourhood of 0 rmd we put p,[ Then

%=

pk.a

E S-(r)

and f o r a l l rn'

>

9)

Cz(lRN -i

p

identical

(j).

m, the sequence %

156

(CHAP. 3 )

SYMBOLS AND OSCILLATORY INTEGiiALS

converges t o

PROOF:

f o r the topology of

a

If CK

i s a semi-norm on

,j,m'

sm'(r). sm' ( r ),

since a

E

sm(r)

t h e r e e x i s t s C such t h a t

For'cc' = 0 , t h e v a l u e of t h e l e f t - h a n d s i d e i s O(km-m') because 18

I

2

k for 8

m-m'

a g a i n O(k

e

E

supp. ( 1

- pk)

) because w e have

-

E supp $(I pk)

.

;

i f a ' # 0 the left-hand side i s

k 5

\el

5 2k for

Consequently, by u t i l i s i n g L e i b n i z ' s

formula, we deduce t h a t

which proves t h e p r o p o s i t i o n . We can t h u s immediately deduce t h e f o l l o w i n g r e s u l t concerning e x t e n s i o n by c o n t i n u i t y :

PROPOSITION 5 . 2 :

space of

Let R be a linear mapping defined on the sub-

s-"(r) , formed

by the functions which vanish f o r

Ie I

large, and with values i n a Frechdt space F. We assume t h a t f o r all m

E

m,

2

i s continuous f o r the topoZogy induced by

Then there e x i s t s a unique extension of

sm(r).

l, : S+"(r] -. F which i s

(SEC. 6 )

conkinuous on each

6.

sm(r ) .

VARIOUS GENERALISATIONS

(6.1.) Up in C

157

GENERALISATIONS

.

t o now, we have considered symbols with v a l u e s

We a r e now going t o d e f i n e t h e symbol-sections

of a

Complex v e c t o r bundle above a conic manifold, under s u i t a b l e assumptions.

Let

M be a conic manifold and l e t m

on M; we s p e c i f y a C

a c t i o n ofIR

+

F be a v e c t o r bundle

on F which commutes, v i a t h e

p r o j e c t i o n of F onto M y with t h a t ofIR

+

in M

We can show t h a t t h e r e e x i s t s , above a conic neighbourhood of any p o i n t of M y a t r i v i a l i s a t i o n of F commuting w i t h t h e

+

allowing a r e d u c t i o n t o M = T ( a n open conic

actions o f B s u b s e t of Rn

N

x

Do) , i . e . F

I

r

d

x C ; we p u t i n t h i s case

Sm(M, F ) = ( S m ( r ) ) d , hence t h e g e n e r a l d e f i n i t i o n of Sm(M, F ) by r e s t r i c t i o n and t r a n s p o r t . If F

a

z(F,,

F),

where F

F

1’ 2

t h e preceding t y p e , we d e f i n e Sm(Y;

a r e two bundles on M of

z(F,,

F2)] accordingly.

158


We say t h a t

E Sm(M;

a

(CHAP. 3 )

i s r i g h t e l l i p t i c of

s(F1, F,))

degree m ( r e s p : l e f t e l l i p t i c ) i f t h e r e e x i s t s b

E S-”(M; s(FZ,

(resp : ba

-

IF

F,))

such t h a t ab

E

g(F1, F,))

S-’(M;

a

t h e c a s e of a c l a s s i c a l symbol part a

0

- 1F2 E S”(U:

.

e(F2, F,))

We can show t h a t , i n

w i t h homogeneous p r i n c i p a l

of degree m y t h e r i g h t e l l i p t i c i t y ( r e s p : l e f t ) of

a

is e q u i v a l e n t t o t h e r i g h t i n v e r t i b i l i t y ( r e s p : l e f t ) o f ao.

(6.2.)

I n f a c t , t h e r e i s not j u s t one s i n g l e t y p e of

symbols b u t r a t h e r we can s a y t h a t t h e r e a r e almost as many o f them as t h e r e a r e t y p e s of problems, and we need t o be a b l e t o a d a p t one t o t h e o t h e r .

For example, i f P(D) i s a h y p o e l l i p t i c

operator then i n order t o include functions o f t h e type

48

t h e upper bounds ( 6 . 9 . 2 ) i n Chapter I l e a d t o t h e

m

.

( r) With 0 < p < 1 m we d e f i n e t h e space S ( r ) o f symbols o f degree m and of t y p e p P f o l l o w i n g g e n e r a l i s a t i o n of t h e s p a c e s S

by r e p l a c i n g t h e upper bounds ( 2 . 3 . 1 ) by

\a, B a,W

a(x,

where 6 = 1 - p .

ell

I

c

(1

+

\ e l lm-P I 4 + s l e l

We can prove t h a t a l l t h e p r e c e d i n g r e s u l t s

i n t h i s c h a p t e r g e n e r a l i s e , w i t h o n l y minor m o d i f i c a t i o n s , t o symbols of t y p e p ( s e e HORMANDER C31, C51 f o r symbols which a r e

s t i l l more g e n e r a l ) .

(SEC.

7.

7)

OSCILLATORY INTEGRALS

159


7.1.

INTRODUCTION:

When

2 (5)

i s defined e x p l i c i t l y ,

e x p r e s s i o n (1.1)i s w r i t t e n f o r m a l l y as

however t h i s i n t e g r a l i s n o t a b s o l u t e l y convergent w i t h r e s p e c t to

5.

The o b j e c t i v e of t h i s s e c t i o n i s t o prove t h a t we can i n f a c t work w i t h ” i n t e g r a l s ” of t h e t y p e (7.1.1)and w i t h i n t e g r a l s which are even more g e n e r a l , as i f t h e y were a b s o l u t e l y convergent.

We s h a l l e x p l a i n t h e i d e a u n d e r l y i n g t h e method

i n t h e p a r t i c u l a r c a s e where P ( x , D ) i s t h e i d e n t i t y o p e r a t o r i n one dimension.

I n o r d e r t o g i v e a meaning t o t h e i n t e g r a l

we can c l e a r l y w r i t e it as an i t e r a t e d i n t e g r a l

however t h i s method i s n o t v e r y e a s y t o h a n d l e .

Since t h e

d i f f i c u l t y e x i s t s f o r 151 l a r g e , it i s s u f f i c i e n t , a f t e r truncation with respect t o i n t e g r a t e for 151 t 1.

6, t o o n l y c o n s i d e r t h e case where we We n o t e t h a t :

160

SYMBOLS

AND OSCILLATORY INTEGRALS

(CHAP. 3 )

which l e a d s u s , a f t e r formal i n t e g r a t i o n by p a r t s w i t h r e s p e c t to

y , t o put

Y € B

and t o d e f i n e t h e l e f t - h a n d s i d e by t h e a b s o l u t e l y convergent i n t e g r a l appearing i n t h e right-hand s i d e , t h i s being l e g i t i m a t e because it can e a s i l y be shown t h a t t h e r i g h t - h a n d s i d e o f

(7.1.2.)c o i n c i d e s w i t h t h e i t e r a t e d i n t e g r a l

T h i s t y p e of g e n e r a l i s e d i n t e g r a l h a s been s t u d i e d systema t i c a l l y by H@"DER

C51

under t h e name o f o s c i l l a t o r y

integral.

7.2.

DEFINITION OF OSCILLATORY INTEGRALS:

subset. of.IRn, ( n

2

0), l e t

cp E C"(x

x

#\

0)

Let X be an open be a r e a l - v a l u e d

7)

(SEC.

161


f u n c t i o n which i s homogeneous o f degree one w i t h r e s p e c t t o

e E R N \ o

and l e t

a E Sm(X x # ) be a symbol; we

consider t h e i n t e g r a l

It i s c l e a r t h a t t h i s i n t e g r a l i s a b s o l u t e l y convergent when

m

-

N , and i n t h i s case it depends continuously on a

E

Sm ;.

w e s h a l l prove t h a t we c a n , u s i n g i n t e g r a t i o n s by p a r t s , make it meaningful f o r a r b i t r a r y

m, i f

Q

We observe t h a t t h e mapping

(7.2

2.)

a E

s-"(x

x RN)

has no c r i t i c a l p o i n t s .

9, :

c

=

a(a)

1~(8u) E

c

i s a l i n e a r form; i n o r d e r t o demonstrate t h e c o n t i n u i t y of R i n t h e sense o f P r o p o s i t i o n 5 . 2 , we e s t a b l i s h t h e f o l l o w i n g :

PROPOSITION 7.3 :

f o r (x, 8) E X x

F?

If we assume t h a t

d x,

e

cp(x, 0)

f

\ 0 , then there e x i s t s a first-order

d i f f e r e n t i a I operator

PROOF:

For an o p e r a t o r o f t h e form (7.3.1), we have

We make t h e right-hand s i d e i d e n t i c a l t o 1 by p u t t i n g

o

p

where

Since

Q

- p)

(1

(CHAP. 3 )


162

E

i s i d e n t i c a l t o 1 f o r 10 I 5 1 and where

CIIRN)

h a s no c r i t i c a l p o i n t s , it i s c l e a r t h a t

-2

D E S

(X x

N

R )

and c o n s e q u e n t l y t h e c o e f f i c i e n t s

a j y bj, c are i n t h e s p a c e s i n d i c a t e d .

It i s u s e f u l f o r l a t e r work t o n o t e t h a t t h e t r a n s p o s e tL

b J l c ' i n t h e same symbol s p a c e s .

o f L h a s c o e f f i c i e n t s a:,

This lemma l e a d s us t o s t a t e :

DEFINITION 7 . 4 :

x

x(#

\ 0)

r is an open conic subset of

If

rp E

we say t h a t a function

c"(r)

is a

phase function if it is homogeneous of degree 1 with respect t o with r e a l values and such t h a t d

8,

xl

PROPOSITION 7 . 5 : X x[RN

\ 0)

,

If

Q

e

Q(X,

9)

f

0 on

r.

is a phase function on

then:

i ) the linear form 11 in ( 7 . 2 . 2 ) extends uniquely i n t o a

linear form 8 on -s+"(X

x

#) , which is continuous

on any Sm.

This extension is termed an o s c i l l a t o r y integraZ and i s denoted by (7.5.1

.)

A(,)

=

i(f'cx'

" a(x,

0) u(x) dx d 6

.

(SEC.

7)

For a

E

163

OSCILLATORY INTEGFULLS

m

, we

S

have the expression

when k > m + N, where L i s taken as i n 7 . 3 . ii) I f

pk

i s defined as i n Proposition 5.1, then f r

iii) For

a IJ

-

f i x e d i n Sm, the mapping

1 p 1 E

E cp1

C

i s a d i s t r i b u t i o n of order k (for k > m+N) which we denote by

(7.5.4

e

i

5

eiQ(x$ ,.(a)'

0) dQ

and which we caZZ the i n t e g r a l d i s t r i b u t i o n associated with the phase

tp

cmd the amplitude

PROOF OF i ) :

7.3, r e p l a c e e*

If

a

a

E

by Lk(e*)

S-m, we c a n , by v i r t u e of Lemma

i n (7.2.1), and by i n t e g r a t i o n by

p a r t s we o b t a i n

ela(x)ldx

de.

Furthermore, t h e p r o p e r t i e s o f t h e c o e f f i c i e n t s of tL imply t h a t

t~ maps SP c o n t i n u o u s l y i n t o sP-' Consequently, i f t h e i n t e g r a l which

a

E

f o r any r e a l v a l u e p .

Sm and i f we choose k such t h a t k > m+N,

OCCUTS

depends c o n t i n u o u s l y on

on t h e r i g h t - h a n d s i d e of

a

E

(7.5.5)

Sm; t h u s , it s u f f i c e s t o a p p l y

164

(CHAP. 3)


Proposition 5.2.

I n o r d e r t o prove i i ) , we u t i l i s e t h e c o n t i n u i t y of R on

m'

any subspace S

and we conclude by u s i n g P r o p o s i t i o n 5.1.

To prove i i i ) , we n o t e t h a t t h e r e p r e s e n t a t i o n

7.5.2)

allows us t o m a j o r i s e I ( a u ) by 0

when k > m+N.

O s c i l l a t o r y i n t e g r a l s which depend on one parameter behave l i k e o r d i n a r y i n t e g r a l s , i n t h e s e n s e t h a t we c a n , under c e r t a i n assumptions, d i f f e r e n t i a t e or i n t e g r a t e under t h e i n t e g r a l s i g n

ii* PROPOSITION 7 . 6 : Q

Let Y be an open subset of BP and Zet

be a phase f u n c t i o n on

dx,e Q

(x, y ,

u E C ~ ( Xx Y )

o

0)

,

X x Y x

.

we p u t f o r

If

a y

E

(

E

\~0) ~ such t h a t

S ~ ( Xx Y x

R ~ ) and

Y

cc

Then i ) We have

F E f(y)

and we can d i f f e r e n t i a t e m d e r

(SEC. 7 )

165

OSCILLATORY INTEGRALS .c

A s i n L e m a 7.3 we can again undertake w i t h t h e

PROOF OF ti): parameter

t h e c o n s t r u c t i o n of an o p e r a t o r

y

L[x, y, e, a, a,

J

E

ax)

So[X x Y x

with c o e f f i c i e n t s

#I,b,,J

c

E

S-'(X

x Y x

#)

.

If k > m+N,

then ( 7 . 5 . 2 ) shows t h a t w e have

=

F(Yl

/,/e%

(tlkl[au 1 dx d 0

and s i n c e t h i s i n t e g r a l i s

a b s o l u t e l y convergent , t h e u s u a l Fubini theorem allows us t o d r a w t h e r e q u i r e d conclusion.

PROOF OF i ) : symbol ition

a

By t r u n c a t i o n , we can reduce t o t h e case where t h e

i s zero f o r 18 I 2 1.

If we have p k ( 8 ) as i n Propos-

5 . 1 , we denote by F k ( y ) t h e f u n c t i o n o b t a i n e d when a

replaced by

%

is

= a.pk.

The o p e r a t o r L above allows us t o adapt t h e proof of P r o p o s i t i o n

7.5

i ) and i i ) t o t h e case where we have a parameter, i n o r d e r =

lim Fk(y) uniformly w i t h r e s p e c t t o y. k'w Fk E C y ( Y ] ; if a i s a multiFurthermore, it i s c l e a r t h a t

t o prove t h a t

F[y]

index, we immediately o b t a i n t h e expression

166


(CHAP.

3)

with

Consequently, t h e argument used f o r Fk can s t i l l be used t o prove that

uniformly w i t h r e s p e c t t o y .

-

Henceforth, we s h a l l omit t h e s i g n

on an o s c i l l a t o r y i n t e g r a l

s i n c e P r o p o s i t i o n 7.6 allows us t o manipulate it l i k e an o r d i n a r y integral.

We can g i v e an i n c l u s i o n f o r t h e s i n g u l a r support of a d i s t r i b u t -

i o n of t h e t y p e ( 7 . 5 . 4 ) .

PROPOSITION 7.7:

We p u t

')a(x,

all

8

Suppose we have

E RoN

;

.

\ 0

X x RN

0

then the d i s t r i b u t i o n

c {x E X I there exists

PROOF:

J?

8)dB s a t i s f i e s

s i n g SUPP A

(7.7.1.)

=

If cp i s a phase f m c t i o n on

and if we have a E sm(x x RN) A = /eiv(xs

R",

x

0

E

X

8f 0

such t h a t

,

Q ~ ( x ,B)

e) f

Q ~ ( X ~ ,

t h e homogeneity and t h e c o n t i n u i t y of

t h a t t h e r e e x i s t s an open neighbourhood V o f xo such t h a t

o Q

I

03 for show

(SEC. 8)

9)

cpL(X,

167

INTEGRAL OPERATORS

-

f 0

f o r (x,

a) E

x

V

N

Ro

.

Furthermore, P r o p o s i t -

7.6 shows t h a t t h e mapping

ion

V 3 x

=

A(x)

eiY(”,

‘)a[x,

9)d8

E C

i s Cm and s i n c e we have

we t h e r e f o r e deduce t h a t inclusion

8.

IV

E

, which

C”[V]

demonstrates t h e

(7.7.1).

INTEGRAL OPERATORS ASSOCIATED WITH A PHASE AND AN AMPLITUDE

CONSTRUCTION 8.1: Q

a

A

E

Let X , Y b e open s u b s e t s o f lRn

be’ a phase f u n c t i o n i n Sm[X x Y x

.K E J1(X

x Y)

N

R )

.

X x ’f x P N

\

0)

Bp ;

let

and l e t

We c o n s i d e r t h e i n t e g r a l d i s t r i b u t i o n

a s s o c i a t e d w i t h t h e phase rg and t h e amplitude

a

i.e. l8.1.1.)

Let

< K, w > =

A : f(Y)

For w E f l y )

icp(x’yse)

J’[X)

, Av

&(x, y,

0 ) w[x, y) dx dy d9,

be t h e o p e r a t o r w i t h k e r n e l K .

i s t h u s t h e d i s t r i b u t i o n i n X d e f i n e d by

168

(CHAP. 3)


(8.1.2.)

< Av, u z

e+(x,y,el

a(x, y,

e)

u(x) ~ ( y )dx dy d o ,

and A i s termed t h e i n t e g r a l o p e r a t o r a s s o c i a t e d w i t h t h e phase and t h e amplitude

Q

a

i ) If K i s the kernel of A, we have

PROPOSITION 8 . 2 : t8.2.1.)

s i n g supp K

e c

iii) If

d .

.

there e x i s t s

c {(x, y) E X x Y I

~t with

cp(x, y , 0)

Ti(.,

f

0

y,

el

for

=

.

o 3

N

,

(x, Y, 8 ) E X x Y x 3,

x, e then A extends i n t o an operator which i s continuous from E ~ ( Y )into

P(x]

.

PROOF: i) f o l l o w s by a p p l i c a t i o n of ( 7 . 7 . 1 ) t o K . For i i ) , P r o p o s i t i o n 7.6 and ( 8 . 1 . 2 ) show t h a t i f m

t h e n Av i s t h e C

(63.2.)

,

v E f(Y)

f u n c t i o n i n X d e f i n e d by

(Av)(x)

ei~(x9y,91 , . ( a

=

y,

e)

v(y) dy d e

.

The c o n t i n u i t y of A i s o b t a i n e d by m a j o r i s i n g a semi-norm

'K, j

(A) a f t e r having r e p l a c e d ( 8 . 2 . 2 ) by an i n t e g r a l o f t h e t y p e

(7.5.2) with

k

> m+N+j

.

(SEC. 8 )

1.69

INTEGRAL OPERATORS

For i i i ) , we n o t e t h a t t h e o p e r a t o r tA i s o b t a i n e d by permuting t h e r o l e s of x , y i n (8.1.2),b u t i ) shows t h a t tA i s continuous from

c:(x]

into

c"(Y)

and consequently P r o p o s i t i o n 4 . 5 of

Chapter I shows t h a t A e x t e n d s i n t o an o p e r a t o r which i s c o n t i n -

&'(Y)

uous from

into

J~(x)

.

This formula is deduced from t h e F o u r i e r i n v e r s i o n formula by u t i l i s i n g t h e passage t o t h e l i m i t ( 7 . 5 . 3 )

EXAMPLE 8 . 4 :

m

Let P(x, D ) b e a d i f f e r e n t i a l o p e r a t o r w i t h C

c o e f f i c i e n t s on X .

By d i f f e r e n t i a t i n g i n t h e o s c i l l a t i n g i n t e g r a l

(8.3.1 we o b t a i n

(8.4.1

E W L E

8.5:

Composition w i t h a C

We d e f i n e t h e o p e r a t o r

f* : C"(Y]

, (8.3.1)

shows t h a t

For

x E X

m

mapping

+

c"(X)

f

from X i n t o Y .

by f*v = v

o

f.

170


EXAMPLE 8.6:

Trace Operator.

Suppose we have

and suppose Y i s an open s u b s e t o f T i r . of TRn

(CHAP. 3)

'A

=

R"

X

n'PR

L e t X be t h e open s u b s e t

d e f i n e d by

-

Let y be t h e t r a c e o p e r a t o r on X:

c;(Y)

3

v

yv

E Cp)

,

t h e n we have y = f * w i t h f ( x ) = (x, 0 ) .

4%01

IW1IXl =

i

Therefore Example 8.5

shows t h a t

E W L E 8.7:

Pseudo-differential operators.

These are t h e

o p e r a t o r s P d e f i n e d by ( 8 . 4 . 1 ) when we r e p l a c e t h e polynomial with respect t o p(xl y ,

5)

5, P(x, 5),

E

Sm(X X

by an a r b i t r a r y amplitude

x

X

R")

.

These w i l l be s t u d i e d i n d e t a i l

i n Chapter I V .

8.8

Cp

GENERALISATION TO THE CASE WHERE THE PHASE FUNCTION

I S DEFINED IN AN OPEN C O N I C SET.

set of

X

xPN\

0)

Let

r

be an dpen c o n i c sub-

and l e t Q be a phase f u n c t i o n i n

r.

The

methods of S e c t i o n 7 a l l o w u s , w i t h minor m o d i f i c a t i o n s , t o d e f i n e and study t h e o s c i l l a t i n g i n t e g r a l

ii

eirp(x'

when

a E Sm(X x RN)

a[.,

and when

0 ) u ( x ) dx d e a

satisfies

,

u

E cz(X]

171

STATIONARY PHASE

i s a phase f u n c t i o n i n an open conic s e t

If cp

r

cx

X

Y x(R

N

conic supp a

\

0 ) and a E Sm(X x Y x RN) i s such t h a t

c r u (x

+

o p e r a t o r A : C:(Y) t h e amplitude

y x

{o})

B'(X)

, then,

we define t h e i n t e g r a l

q,

a s s o c i a t e d w i t h t h e phase

and

as i n S e c t i o n 8.1, and we o b t a i n analogous

a

properties.

GENERALISATION TO THE CASE OF SYMBOLS OF TYPE

8.9

(see Section

6).

p >

0

All t h e p r e c e d i n g c o n s t r u c t i o n s and p r o p e r t i e s

remain v a l i d , w i t h minor m o d i f i c a t i o n s , when t h e amplitudes

a

a r e symbols from S:.

9.

STATIONARY PHASE THEOREM

9.1

CASE WHERE THE PHASE IS NEVER STATIONARY. Rp, Rq

be open s u b s e t s o f t h e behaviour f o r

I(xy

(9.1.1.1

where

T

cp E

c"(x

X

Y)

+

7)

+

m

;

Let X, Y

we o f t e n have occasion t o s t u d y

of i n t e g r a l s of t h e t y p e

.iT(x'

a(x, y, 71 dy

i s r e a l - v a l u e d and where a E Sm(X x Y x

I n o r d e r t o avoid any convergence problems, we assume t h a t zero f o r

y

o u t s i d e a compact s e t K c Y.

a

R +). is

172

SYMBOLS AND OSCILLATORY INTEGRAL3

I f the function

cp

(CHAP. 3 )

h a s no c r i t i c a l p o i n t s w i t h r e s p e c t t o t h e i n t -

e g r a t i o n v a r i a b l e y , then I i s of r a p i d decrease with r e s p e c t t o T;

more p r e c i s e l y , we have:

PROPOSITION 9 . 2 :

We asswne t h a t

(p(x, y)

Y

+ , where

x R )

E S-”(X

and i n t h i s case I(x,

d

T

f 0 on X x Y , is t h e frequency

variab t e .

PROOF:

The f i r s t - o r d e r o p e r a t o r

S u b s t i t u t i n g back i n t o ( 9 . 1 . 1 ) we o b t a i n , a f t e r i n t e g r a t i o n by parts ,

I(x,

7)

Since any

k

i.i.p

=

(?Ik

El

E

.

( t L ) k a dy

Swk(X

x Y x

and f o r any compact s e t

R+) H

, we cx

x

t h e n deduce t h a t f o r

R+ w e

have t h e upper

bound

The d e r i v a t i v e s of I are d e f i n e d by i n t e g r a l s of t h e same t y p e as I ;

t h u s we m a j o r i s e them i n t h e same manner, and t h i s proves

the proposition.

9)

(SEC.

9.3

STATIONARY PHASE

CASE WHERE

173

HAS A QUADRATIC CRITICAL POINT.

The

above p r o p o s i t i o n l e a d s us t o s t u d y t h e behaviour o f I when possesses a c r i t i c a l p o i n t . t h a t i n which

We b e g i n w i t h t h e s i m p l e s t c a s e , i . e .

i s a q u a d r a t i c form which is non-degenerate

g,

More p r e c i s e l y , w e assume t h a t

with r e s p e c t t o y .

Y1 where

=

1/2

< Q(x)Y,

Y t

Q ( x ) i s an i n v e r t i b l e symmetric m a t r i x which h a s

ence on x

E

X , and we t a k e Y = Bq

.

.)

I ( x , 71

E swq/2 (x

a

x

R+)

and t h i s symboZ admits an asymptotic expansion

(93.4.) where R(xl

aY )

=

-2i

< Q-’[x)

C

m

depend-

I n t h i s c a s e , we have:

satisfies (9.3.2

cf~

ay, ay >

and s i g n Q ( x ) = ( n m b e r of p o s i t i v e eigenvaZues of

Q(X

(number o f negative eigenvazues of Q ( x We begin w i t h a F o u r i e r t r a n s f o r m c a l c u l a t i o n .

174


LEMMA 9.4:

(CHAP. 3)

Let Q be an invertibZe symmetric matrix in IRq i

Then the Fourier transform of the f u n c t i o n exp (5 < Qy, y

.

is

given by the f u n c t i o n

PROOF OF THE L E W :

Let M b e an o r t h o g o n a l m a t r i x such t h a t

tM Q M i s d i a g o n a l ;

t h e change o f v a r i a b l e y = Mx i n t h e c a l c u l -

a t i o n o f t h e F o u r i e r t r a n s f o r m t a k e s us t o t-h e c a s e where Q i s 9 2 diagonal. I f we t h u s p u t < Q Y , Y > = h j YJ , we have

i

~ X P ( Z< Q Y , Y

>I

=

We a r e t h u s l e d t o c a l c u l a t e

fi

1

2

exp($ h j y j )

j - 1 S(e

i/2

A j !)

and we have seen

i n E x e r c i s e 10.3 of Chapter I t h a t w e o b t a i n t h e f u n c t i o n

We hence o b t a i n t h e lemma, by o b s e r v i n g t h a t

We now r e t u r n t o t h e proof o f t h e theorem.

We a p p l y P a r s e v a l ' s

formula i n t h e i n t e g r a l (9.3.1),and we o b t a i n

with

(SEC. 9 )

175

STATIONARY PHASE

(where * i n d i c a t e s t h a t we have t a k e n t h e F o u r i e r t r a n s f o r m w i t h respect t o the variable y ) .

The advantage of e x p r e s s i o n

(9.4.2)

l i e s i n t h e f a c t t h a t T appears i n t h e e x p o n e n t i a l v i a t h e f u n c t ion

71.

;

t h u s , we can o b t a i n t h e asymptotic expansion of J by

u t i l i s i n g t h e Taylor expansion o f t h e e x p o n e n t i a l f u n c t i o n .

LEMMA 9.5:

For any i n t e g e r N

2

1, we have

.)

(9.5.1

where the remainder rN s a t i s f i e s the upper bounds

\(z) d j

(9.5.2.)

PROOF OF THE LEMMA:

For j = 0 , t h e upper bound ( 9 . 5 . 2 ) i s i n

f a c t j u s t t h e u s u a l upper bound f o r t h e remainder i n t h e Taylor formula.

We o b t a i n t h e upper bounds ( 9 . 5 . 2 ) f o r N

2

j > 0 from

t h e c a s e j = 0 by d i f f e r e n t i a t i n g t h e e q u a l i t y ( 9 . 5 . 1 )

9.6

CONCLUSION OF THE PROOF OF THE THEOREM:

t h i s l e m a t o the exponential i n the integral s

by--

where

27

< Q-’(x)q,

>

.

We o b t a i n ,

j times.

We a p p l y

(9.4.2),r e p l a c i n g

(CHAP. 3)


176

and

I a(x,

?,,TI TI

9

; t h u s , i n view of

it s u f f i c e s t o show t h a t t h e remainder RN s a t i s f i e s

t h e upper bounds

uniformly f o r

1.1

+

w i t h i n a compact s u b s e t K c X and f o r

x

5 N

We b e g i n by p r o v i n g t h i s w i t h ci = B = 0.

,

U t i l i s i n g ( 9 . 5 . 2 ) , we o b t a i n :

(9.6.4.)

IRN(x,

Furthermore, s i n c e

a

from ( 9 . 6 . 4 )

E

K.

C T

i s a symbol o f degree

support with respect t o

uniformly f o r x

I

711

y

, we

m

w i t h compact

have f o r any i n t e g e r LI

2

0

By t a k i n g II s u f f i c i e n t l y l a r g e , we deduce

, and ( 9 . 6 . 5 )

the inequality

(SEC. 9 )

STATIONARY PHASE

177

I n t h e g e n e r a l c a s e , L e i b n i z ' s formula shows us t h a t

a:

3;

i s a f i n i t e sum o f terms o f t h e form

R"XI

r

with (y

= a'

+

(y"

1

fi = fi' i-

We can show by r e c u r r e n c e on

$1'

Iff' 1 +

,

\fi'

1

aa' ax8 ' rN( ...)

that

is

a f i n i t e sum of terms of t h e form

By combining Lemma 9.5 and t h e immediate Gpper bound

we e a s i l y o b t a i n t h e upper bound ( 9 . 6 . 3 ) .

EXAMPLE 9 . 7 :

Suppose we have X

n

c

I

Y =

R

n

X

W

n

;

a p o i n t of Y i s denoted by y = ( z , < ) and we d e f i n e Q(x) = Q by

< Qyp y > = < z I 5 > R(x1

i

a,)

e i z~. 5

.

We have d e t Q = 1, s i g n Q = 0 and

n

=

a2 3] azj a6j 1

I n t h i s c a s e Theorem 9 . 3 g i v e s

n

a(x, z ,

=

D

(10.3.3)

w[y) =

-in D rr-2 b ( y )

b)

i f n is even,

Pv;

i f n is odd.

From (10.3.1), (10.3.3) and from (10.9.1)of Chapter I ,

deduce t h a t when

{x E R"

1

1x1

n

i s odd, we have E

< l},

Chapter I ) t h a t

1

= 0 in

and t h e r e f o r e ( s e e s e c t i o n 10. 8 of

supp E l

P

{x E R"

t h i s ( s e e (10.8.1),Chapter I )

I

1x1 = I]

.

I n f e r from

t h a t t h e s u p p o r t of t h e funda-

mental s o l u t i o n E of t h e wave o p e r a t o r i s e q u a l , when odd and 2 3, t o t h e cone

c)

Suppose

for x

E

E?

:

{ ( t , x ) E R*'

1t

f : R-C

i s continuous.

= 1x13

n

is

. Prove t h a t we have,

186

where u

S =

o f En-', i s t h e area of t h e u n i t sphere S n-2

n-2

u0 = 2

d)

(CHAP. 3 )


sn-l

[Cut

with

up i n t o zones x.o = c o n s t a n t , and p u t

X ,.,I.

W ' e s h a l l h e n c e f o r t h assume t h a t

t o s t u d y t h e r e s t r i c t i o n of E

1

i s even, and we propose

n

.

c 11

t o (1x1

Deduce from (10.3.1), ( 1 0 . 3 . 2 ) and ( 1 0 . 3 . 4 ) t h a t t h i s r e striction is the C

m

function

where, for 0 5 r < 1, we have p u t

n-3

cn

J'

- (2in)-"

-1 (&)I

F""1 and Cn =

=

(1

+

s/r)l-"(l-s2)

on-2

ds

I

.

Prove, by i n t e g r a t i o n by p a r t s , t h a t n


a2k-2

t Re c a l l t h a t

'2k4 Calculate d i r e c t l y t h e i n t e g r a l

j

IR2 , 1x1 < 1.

@GI

when x

E

X.D+

1

dm


F2(r) =

1

(1

- r)-3

and t h u s , by u t i l i s i n g (10.3.5), t h a t t h e r e s t r i c t i o n of El t o (x

E R"

11x1

+ r

, formula

(1.1.4) of Chapter I shows t h a t

e-i$

a$8

Q ( ~ , D ~ Ie (i 9~

=

> - a,:*' a!

Q ( x , d X f ( x , B ) ) D ~ ( ae * )

CY

and s u b s t i t u t i n g i n t o (2.1.12),we o b t a i n

(2.1.13)

b =

cm 1

CYlB

( D ~as p ) ( x , x , d x t ( x , 8 ) ) . D ~ ( a eir) lY=x'

(SEC. 2 )

203

CHARACTEEiISATION

By i d e n t i f i c a t i o n w i t h (2.1.11)ywe deduce from t h i s t h e which proves t h e formula ( 2 . 1 . 4 ) .

e x p r e s s i o n f o r t h e terms d aa

A s an a p p l i c a t i o n of Theorem 2 . 1 , we give below a c h a r a c t e r i s a t i o n of p s e u d o - d i f f e r e n t i a l o p e r a t o r s on an open s u b s e t

x

c ?Rn.

PROPOSITION 2.2:

Let P be a oontinuous lin e a r operator

from C I ( X ) i n t o C m ( X ) . Then, P e-ixs P(a

(2.2.11

eixsl

E

E

S ~ ( Xx

Lm(x)'if and onZy if we have

R") for a l l

a

E

c:(x).

If we f u r t h e r assume t h a t P is proper, i t s u f f i c e s t o have (2.2.1) with a = 1.

PROOF:

I t remains t o prove t h a t t h e c o n d i t i o n i s s u f f i c i e n t - .

Let ( p . ) be a p a r t i t i o n of u n i t y on X w i t h f u n c t i o n s p J j and l e t

a J.

on supp p j ;

For u

m

be f u n c t i o n s i n C 0 ( X ) , w i t h

w e assume t h a t t h e supp

Rj j

E

Cz(X)

m

E

Co(X) identical t o 1

are also locally f i n i t e .

W

E

C (X),by F o u r i e r i n v e r s i o n we have: 0

J

J

I n t e r p r e t i n g t h i s l a s t i n t e g r a l a s an i n t e g r a l w i t h r e s p e c t t o ,€ with v a l u e s i n C w ( X ) , t h e c o n t i n u i t y o f P allows us t o w r i t e 0

(CHAP.

PSEUDO-DIFFERENTIAL OPERATORS

204

Since

u =

, we

pju

4)

t h e r e f o r e deduce t h a t

J

where t h e amplitude P(X,Y,e;I

-c

J i s c l e a r l y an element of

Pj(xlh)

Pj(Y1

Sm(X x X x

l o c a l l y f i n i t e with respect t o

R")

s i n c e t h i s sum i s

.

y

Since P i s p r o p e r , it d e f i n e s a continuous o p e r a t o r P

Ca(Rn)

* C"(X)

C"(X)

where t h e f i r s t arrow denotes t h e r e s t r i c t i o n t o X.

If we apply

t h i s operator t o the integral

m n c o n s i d e r e d a s an i n t e g r a l w i t h v a l u e s i n C (IR ) , we o b t a i n

r

with

(2.2.4)

p(x1sl =

=-ix[:

,,(,ixl:

) E Sm(X x R")

.

SYMBOL OF A P.D.O.

205

This l e a d s us t o :

I f we Let P be a proper operator taken

PROPOSITION 2 . 3 : from L m ( X ) ,

then there e x i s t s a unique

p E Sm(X x

R")

such

Moreover, i f p ( x , y , S ) is an amplitude f o r

that we have ( 2 . 2 . 3 ) .

P, we have (2.2.5) @

We say t h a t p ( x , S ) i s the complete symbol of P and we w rite P = p(x,D).

PROOF:

We have j u s t proved t h e e.xistence o f

ness a r i s e s o u t o f t h e d e n s i t y o f

&(I?"]

p

.

;

t h e unique-

The expansion

( 2 . 2 . 5 ) f o l l o w s from a p p l i c a t i o n of ( 2 . 1 . 4 ) t o t h e p a r t i c u l a r case ( 2 . 2 . 4 ) .

3.

SYMBOL OF A PSEUDO-DIFFERENTIAL OPERATOR

3.1

COMPLETE SYMBOL:

We s h a l l now e x t e n d t h e n o t i o n o f

a complete symbol t o an a r b i t r a r y o p e r a t o r P i n g P r o p o s i t i o n 1.5, w e can w r i t e P = P'

E

Lm(X).

By u t i l i s -

+ R' and we denote by

p ' ( x , S ) t h e complete symbol of t h e proper o p e r a t o r PI. consider a d i f f e r e n t decomposition o f t h i s t y p e , P = P" the difference p' symbols i s i n

-

S-o(X

If we

+ R",

p" between t h e corresponding complete x

R")

s i n c e t h e o p e r a t o r P'

- P

= R"

- R'

(CHAP. 4)


206

If we denote by o(P)(x,c) the

is both proper and in L--(X).

equivalence class of p' in Sm / S--, we have thus defined a linear mapping

Lm[X) 3 P ->

o(P) E Sm(X x R")

This is clearly surjective, and Proposition

/

S-"(X

x

.

R")

1.4 shows that its

kernel is L-m( X); therefore we have an isomorphism (3.1.1)

Lm /

1-OD

->

0.

Sm

/

S-"

called the isomorphism of the complete symbol.

REMARK 3.1.2.

:

The above shows that any operator be-

m longing to L (x) can be written, to within a regularising operator, in the form p(x,D) with

3.2. PRINCIPAL SYMBOL.

e Sm(x x

p

If P

E

R")

.

Lm(X), it is clear that we

have the equivalence p

E LW1(X)

>-(

a(P] E

s-'

/ s-=

.

Consequently, we deduce from (3.1.1),by passage to the quotient, a simpler isomorphism:

(3.2.1)

Lm

/

Om

LW1

sm / Sm-l

->

s

called the isomorphism of the principal symbol of degree m.

If P = P(x,D) is a differential operator, then om(P) admits a canonical representative in Sm( X part P,(x,E)

x

Bn ) , namely the principal

of the polynomial P ( x , S ) .

(SEC.

4)

ALGEBRA

& SYMBOLIC CALCULUS

207

Lm(X), we s h a l l o f t e n commit t h e n o t a t i o n a l m abuse o f i d e n t i f y i n g u (P) w i t h a r e p r e s e n t a t i v e i n S , and w e m Given P

E

s h a l l do t h e same for c r ( P ) .

4.

ALGEBRA AND SYMBOLIC CALCULUS OF P.D.O. ' s I n Chapter I we encountered t h e d e f i n i t i o n of t h e t r a n s -

pose tP and t h a t of t h e a d j o i n t P* o f an o p e r a t o r P on an open s u b s e t of lRn.

For p.d.o.'s we have:

THEOREM

4.1

:

If P

E

Lm(X) , then

t

P and P* are i n Lm(X)

and the complete symbols admit the following expansions: (4.1.1)

(4.1.2)

a PROOF: m

with C

Since t h e t r a n s p o s e or t h e a d j o i n t of an o p e r a t o r

k e r n e l i s an o p e r a t o r w i t h C

m

k e r n e l , we w i l l n o t

r e s t r i c t t h e g e n e r a l i t y by assuming t h a t P i s p r o p e r .

If we p u t p ( x , h )

t

o(P)[x,S)

, then

P c a n be w r i t t e n

208

(CHAP. 4)

PSEUDO-DIFFERZNTIAL OPEFATORS

By n o t i n g t h a t we have

,

<W,tP">

Q

v

E

,

CIfx)

we o b t a i n

We t h e r e f o r e deduce t h a t tP E Lm(X), and t h e expansion (4.1.1) t h e n f o l l o w s from ( 2 . 2 . 5 )

t

The P

c a s e can be deduced from t h i s s i n c e P

*

=

t-

P.

F o r composition, we have

THEOREM 4 . 2 :

Suppose we have P

E

Lm(X) and Q

E

Lm' (X);

we assume t h a t a t l e a s t one of t h e s e is a proper operator. Then, (4.2.1 )

R

,

P

E LWm' (X)

and for t h e complete symbol

PROOF:

We s h a l l n o t r e s t r i c t t h e g e n e r a l i t y by assuming

t h a t both of these operators a r e proper.

(4.2.3)

e-ixt(Q

p)(eiRs)

e-iXt

Q

rei"!

= eixzQ(u(P)eixz)

We have

e-ixI ~ ( ~ ~ ~ f ) ]

.

Consequently ( 4 . 2 . 1 ) can be deduced from ( 2 . 1 . 3 ) and from Proposition 2.2.

The expansion ( 4 . 2 . 2 ) f o l l o w s from

4)

(SEC.

ALGEBRA & SYMBOLIC CALCULUS

209

a p p l i c a t i o n of ( 2 . 1 . 4 ) t o ( 4 . 2 . 3 ) .

REMARK 4.3 :

The composition o f p r o p e r p . d . 0 . ' ~ i n d u c e s ,

v i a t h e isomorphism of t h e complete symbol, an a s s o c i a t i v e

l a w of composition on

/ s-".

S+w

This l a w i s denoted as

=#

and can be d e f i n e d d i r e c t l y by

(4.3.1)

-

(qSfd(x,g)

1 7 :a

a!

q(x,g)

.

:D P(X,E)

F i n a l l y , we have i n v a r i a n c e o f t h e p.d.0.'~ under diffeomorphism.

THEOREM

4.4

subsets X , Y o f B n .

be a diffeomorphism between two open

Suppose P

operator transported by (4.4.1

x

Let

:

I

x

x*P

E

L ~ ( x )and l e t X*P denote the Then

onto Y.

E

Lrn(Y)

and f o r the complete symbol we have

-

~(GPI(Y,TIcy

(4.4.21

PROOF:

1

7 a;

t

~ ( P H X , x'(X).T)DF(eir)

( Z P X

We w i l l n o t reduce t h e g e n e r a l i t y by assuming t h a t

P i s p r o p e r ; i n t h i s case

x*P i s a l s o proper and we u t i l i s e t h e

c h a r a c t e r i s a t i o n i n Proposition 2.2.

We have

consequently, ( 2 . 1 . 3 ) of Theorem 2 . 1 shows t h a t t h e l e f t - h a n d s i d e of ( 4 . 4 . 3 ) i s i n Sm(Y x B n ) . equal t o taking

g ( ~ P ) ( y , ~ ); in (2.1.4)

:

This l e f t - h a n d s i d e i s a l s o

hence t h e expansion

(4.4.2)by


210

a = 1

4.5

,

,

= u(P)(x,Z>

P(X,Y,$)

(CHAP.

4)

= X(X).T.

*(x,Tl)

For p r i n c i p a l symbols, it

REDUCED SYMBOLIC CALCULUS,

f o l l o w s from t h e e x p r e s s i o n g i v i n g t h e f i r s t t e r m of t h e expan-

(4.1.1.), (4.1.2),(4.2.2),(4.4.2),

sions

t

a,(

(4.5.11

(4.5.31

o,~Pl~x,- 51

=

p)rx,c)

OrrHm~( Q o

(4.2.2),we

A s an a p p l i c a t i o n of

(Q)(x,!I

0 ',

om(%Pl ( Y , 7) = O,(PI

(4.5.4)

t h a t we have

(x,

a,(P)(x,!)

t

X' (XI .Ill w i t h y = X(X]

.

deduce (under t h e c o n d i t i o n s of

Theorem 3 . 2 ) t h a t t h e commutator

(4.5.5)

[P

, Q]

Q

= P

-Q

where t h e Poisson b r a c k e t {

is in

P

L~

ml-I

(XI

and

} i s d e f i n e d by

n

REMARK 4.6

:

Case of vector-valued p . d . 0 . ' ~ .

L e t E , F be f i n i t e dimensional v e c t o r spaces on C

.

We

denote by Lm(X; E , F) t h e space of p . d . 0 . ' ~ of t h e form (1.2.1) when t h e amplitude

p

E Sm(X x R" ; L(E, F)) ca

.

Then, such

m

o p e r a t o r s a c t c o n t i n u o u s l y from C (X; E) i n t o C (X; F ) and 0

p o s s e s s a l l t h e p r o p e r t i e s of s c a l a r - v a l u e d p . d . o . ' s e x c e p t i o n of

(4.5.5) and (4.5.6). For

with t h e

example, Theorem 2.1

f o l l o w s from Remark 9.11 o f Chapter 111; it i s of c o u r s e

P.D.O.'s

a

necessary t o t a k e

< SW(X

x

RN ; F),

P E L ~ ( X ;

E

, F)

b

E Sm(X x

a,(P)

E Sq(X

211

ON MANIFOLDS

x

RN ; E)

and we t h e n f i n d

The p r i n c i p a l symbol of

i s c l e a r l y an element

R" ; f(E, F)) / Sm-'.

For t h e composition,we assume

Q , P E Lmm' (X ; E

Q E L m' ( X ; F, G ) and we f i n d t h a t

, G)

.

F i n a l l y , i f we t a k e i n n e r p r o d u c t s ( r e s p . Hermitian produ c t s ) on E and F, t h e t r a n s p o s e ( r e s p . t h e a d j o i n t ) of

P E Lm(X ; E

, F)

i s an o p e r a t o r tP ( r e s p . P*) belonging t o

L m ( X ; F, E ) .

5.

P.D.O.'s

ON MANIFOLDS

I n t h i s s e c t i o n X denotes a manifold of dimension

n.

Following t h e d e f i n i t i o n of d i f f e r e n t i a l o p e r a t o r s ( D e f i n i t i o n

7.6 o f Chapter I ) and from t h e c h a r a c t e r i s a t i o n o f p . d . o . ' s d e s c r i b e d i n S e c t i o n 2 , it i s n a t u r a l t o s t a t e :

DEFINITION 5 . 1 :

Suppose

m

ER U

;

{-o]

P : C:(X)

L m ( X ) the s e t of continuous operators

-

we denote by C"(X)

which s a t i s f y the following condition: f o r any real-valued f u n c t i o n

*(x,8)

E

C"(X

x

:R 1,

homogeneous of degree 1 with respect t o 6, and f o r any amplitude a

E

Sq(X x

dx$(x,8)

RN )

with compact support i n x such t h a t

f 0 f o r (x,6) E

conic supp a , we have

212

(CHAP. 4)


F i r s t l y , Theorem 2 . 1 and P r o p o s i t i o n 2.2 show t h a t t h i s d e f i n i t i o n c o i n c i d e s w i t h t h e former d e f i n i t i o n when X i s an If X i s a manifold and we have P

open s u b s e t o f IRn,

E

Lm(X),

and i f we l e t U be an open s u b s e t of X , we d e f i n e t h e o p e r a t o r PIu as b e i n g t h e compose

c"(x) 0 ->

->

CpI)

c"(x) ->

c"(u)

where t h e f i r s t arrow i s t h e n a t u r a l i n j e c t i o n and l a s t arrow denotes t h e r e s t r i c t i o n t o U ; D e f i n i t i o n 5 . 1 shows t h a t PIu

E

Lm(u).

D e f i n i t i o n 5.1 i s obviously i n v a r i a n t under diffeomorphism; consequently i f

->

-

%(PI,,)

EL

x :X

transported operator

U

3

s t r i c t l y positive density. P

E

Rn

m N

(u)

.

i s a c h a r t of X , t h e

We assume t h a t X i s equipped w ith a

PROPOSITION 5 . 2 :

of a p . d . 0 .

U c

Then t h e kernel d i s t r i b u t i o n

Lm(X) is Cm outside of Diag ( X

x

X).

= L-"(x)

=

Further-

more, we have the following e q u a l i t i e s : (5.2.1.)

Operators w i t h kerne l s E

PROOF:

Suppose (x

0

neighbourhoods o f x

0'

yo

c"[x

, yo) 8

x

x)]

Diag(X

n P(x) . m

x

X ) and l e t U , V be open

chosen s u f f i c i e n t l y small f o r them t o

be a l s o d i s j o i n t c o o r d i n a t e p a t c h e s .

Consequently i n t h e open

(SEC.

5)

P.D.O.'s

set W = U u V

213

ON MANIFOLDS

c R"

X :W

we can d e f i n e a c h a r t

.

Furthermore, we know ( s e e ( 8 . 2 . 1 ) , C h a p t e r I ) t h a t t h e r e s t r i c t ion o f P t o W

-

W i s t r a n s p o r t e d by

x

(x,x)

into a distribution

K such t h a t N

(5.2-2)

K=KNg

P where

g

is a C

, Since K

m

N

f u n c t i o n and P = x*P.

i s Cm o u t s i d e of t h e d i a g o n a l o f

? X F,

we deduce

P The same method allows

t h a t t h e k e r n e l of P is Cm i n U x V .

us t o prove t h e i n c l u s i o n

n Lrn(X)

c { o p e r a t o r s w i t h k e r n e l i n C"(X x X ) l

rn

.

A proof analogous t o t h a t o f P r o p o s i t i o n 9 . 2 of Chapter I11 m

proves t h e i n c l u s i o n : { o p e r a t o r s w i t h C F i n a l l y , we obviously have

L-=(X)

c

k e r n e l s ) c L-m(X).

n Lrn(X) ,

rn

and t h i s

concludes t h e proof of ( 5 . 2 . 1 ) .

The q u o t i e n t L" / L-m i s a s h e a f

PROPOSITION 5.3 :

s e t s (U ); ci

p

= lUiY

Let there be a covering of X by open

we s p e c i f y operators

Then, there e x i s t s P

pw

E

on X ; t h i s i s

Pcu E

L ~ ( U @ ) such t h a t

L m ( X ) unique modulo L W m ( X ) such t h a t

( t h e sign E s i g n i f i e s moduZo

.

214


PROOF:

(CHAP.

4)

Taking a f i n e r covering where a p p r o p r i a t e , we

w i l l n o t r e s t r i c t t h e g e n e r a l i t y by assuming t h a t t h e Ua are r e l a t i v e l y compact and l o c a l l y f i n i t e .

Suppose ( p a ) i s a

s u b o r d i n a t e d p a r t i t i o n o f u n i t y with f u n c t i o n s p

CY

E

and suppose we have f u n c t i o n s pa

W

Co(Ua)with

f CI(ue)

identical t o 1

c1

on SUPP P,.

We d e f i n e P by

(5.3.1

1

p u

b,

=

pa P&

cz It i s c l e a r t h a t P i s a continuous o p e r a t o r from C"(X)

into

0

s i n c e t h e sum ( 5 . 3 . 1 ) i s f i n i t e f o r a l l u

C"(X) Taking

a

E

C"(X). 0

and $ a s i n D e f i n i t i o n 5 . 1 , we have

(5.3.2) N t h e right-hand s i d e i s a f i n i t e sum o f symbols i n Sm+q(X x iE ) ;

consequently we c l e a r l y have P

We now show t h a t

p

rPP

Lm(X).

E

Since supp p

'

cy

rl

Up

c U, n U

8'

we have (5.3.31

R

where

us

'a p ~ IU l B

x u

=

(pa

- 1)

PB p,

=

a ' (

p

Pp

'8

p~)lUg p,

+ R

aB

i s an o p e r a t o r w i t h C

B r n

S u b s t i t u t i n g (5.3.3) i n t o (5,3.1),

we o b t a i n

m

kernel i n

215

P.D.O.'s ON MANIFOLDS

P

REMARK 5 . 4 :

P*

The above p r o p o s i t i o n and i t s proof remain

v a l i d i f we everywhere r e p l a c e t h e e q u a l i t i e s modulo L

m'

e q u a l i t i e s modulo L

This sheaf

-m

by

w i t h m' < m .

p r o p e r t y a l l o w s us t o g i v e an e q u i v a l e n t form,

which i s p o s s i b l y more n a t u r a l , f o r t h e d e f i n i t i o n of p . d . o . ' s on X.

PROPOSITION 5.5 :

C:(X)

4

x :u

C"( X)

. Then

H

4

u c R"

PROOF: necessary.

, the

Let P be a continuous Linear operator

P E Lm(X)

operator X * ( P

i f and onZy i f f o r any chart mc

IU

i s i n L (u)

We saT7 f o l l o x i n g D e f i n i t i o n

.

5 . 1 t h a t t h i s was

I n o r d e r t o show t h a t it i s a l s o s u f f i c i e n t , it

s u f f i c e s t o apply P r o p o s i t i o n 5 . 3 t a k i n g a c o v e r i n g of X by open c o o r d i n a t e p a t c h e s .

F i n a l l y , we n o t e t h a t P r o p o s i t i o n 1 . 5 and i t s proof remain v a l i d when X i s a m a n i f o l d .

216

(CHAP.


6.

4)

SYMBOLIC CALCULUS OF P.D.O,’s ON A MANIFOLD

6.1.

DEFINITION OF THE PRINCIPAL SYMBOL OF AN OPERATOR P E Lm(X).

x

With each c h a r t

lu)

-

sm(r x R”) / SW1. -1 allows diffeomorphism x

U,,,(&P

the

:U

/

U

cR”

us t o l i f t it t o an element denoted

S”l(T*U

\ 0) independently

x

choice of t h e c h a r t diffeomorphism same r e a s o n we have

we a s s o c i a t e t h e symbol

Formula ( 4 . 5 . 4 ) shows t h a t

E

E Sm(T*U\ 0)

by q,(P)

N

uu(p) = av(P)

of the

d e f i n e d on U. in

T*(U

For t h e

n V) \

0

I

thus

t h e combination p r o p e r t y of t h e symbols ( s e e P r o p o s i t i o n 2 . 1 0 o f Chapter 111) proves t h e e x i s t e n c e of a unique element

cr € Sm(T*X \ 0)

/ Sm-’

rn

for any open sub-

This symbol u i s c a l l e d t h e p r i n c i p a l symbol

set of c h a r t s . of degree

alu = uu

such t h a t

of P and i s denoted by crm(P).

The l i n e a r mapping p E Lm(X)

_I_*

/

E S”(T*X\O)

o,(P)

S-’

is s u r j e c t i v e ; Remark 5 . 4 allows us t o reduce t o c o n s i d r i n open s u b s e t s of IRn i n o r d e r t o prove t h i s . P r o p o s i t i o n 5 . 5 shows t h a t cr,(P) P

E

m- 1 L (X).

sm-l

i f and only i f

Hence w e have t h e isomorphism o f the p r i n c i p a l

symbol of degree

(6. 1. 1)

E sm-I /

Furthermore,

m

Lrn(X)

/ Lnc1[X)

For composition we have:

Om

*

Sm(T*x \O)

/

s”’(T*X

\O)

(SEC. 6 )

SYMBOLIC CALCULUS ON A MANIFOLD

P E Lm(X)

Suppose t h a t

THEOREM 6 . 2 :

217

Q E Lm’ (X)

where one a t l e a s t i s a proper operator; then Q

PROOF:

,P

E LWm1(X)

From D e f i n i t i o n 5 . 1 , we can immediately show t h a t Q

P E Lmcm’(X)

.

The d e f i n i t i o n of t h e p r i n c i p a l symbol shows t h a t i f it is s u f f i c i e n t t o v e r i f y ( 6 . 2 . 1 ) i n an open c o o r d i n a t e patch,U. Suppose we have

cp

bourhood of supp $.

$ Q P

,0

E Cz(U)

w i t h q = 1 i n t h e neigh-

We have

3

$ Q q P

e

6 Q cp P

+

+Q(I-(P)P

since

$ Q(I

- cp)

E L-”(x)

.

m

which concludes t h e proof s i n c e $ i s a r b i t r a r y i n C (U). 0

THEOREM 6 . 3 :

Assume t h a t X i s endowed with a s t r i c t z y

218

(CHAP. 4 )


p o s i t i v e density.

If P

E

(6.3.1)

tP

(6.3.2)

P+ E P ( X )

PROOF:

E

P(X)

L ~ ( x ,) we then have I

q t P 1 (x, 51 =

;

O,(P*I(X,51

=

+I

(X,-C)

m m

I

L e t t i n g U be an open s u b s e t of X , we c l e a r l y have

c o n s e q u e n t l y ( 6 . 3 . 1 ) and ( 6 . 3 . 2 ) f o l l o w from meorem 4 . 1 and from P r o p o s i t i o n 5 . 5 .

PROPOSITION 6 . 4 :

Any p . d . 0 . P

E

L m ( X ) extends contin-

uousLy i n t o an operator from &'(x) i n t o .I?'(x).

PROOF:

We have P = t ( t P ) ; now Theorem 6 . 3 i m p l i e s t h a t

t~ i s continuous from

m

c o (XI i n t o c

m

( X I ; consequently P r o p o s i t i o n

6 . 4 f o l l o w s from Remark 8.4 of Chapter I .

DEFINITION 6.5 :

i f f o r any chart

the operator h ( P

U

lu

A p.d.0.

P

c x &cc

E

R"

L m ( X ) i s called c l a s s i c a l ,

,

the complete symbol of

u ) E L m (U) is c l a s s i c a l .

We denote by

Lm(X) the subspace of c l a s s i c a l p . d . 0 . ' s of degree m. C

Formulas ( 4 . 4 . 2 ) , ( 4 . 1 . 1 ) ,

( 4 . 1 . 2 ) and ( 4 . 2 . 2 ) show t h a t t h i s

c l a s s i s s t a b l e under diffeomorphism, under t r a n s p o s i t i o n , under passage t o t h e a d j o i n t , under r e s t r i c t i o n and under composition, by u s i n g t h e proof of Theorem 6.2.

6)

(SEC.

SYMBOLIC CALCULUS ON A MANIFOLD

219

The p r i n c i p a l symbol o f an o p e r a t o r P E Lm admits a

c

CI(T*X

c a n o n i c a l r e p r e s e n t a t i v e u ( P ) which i s a

rn

f u n c t i o n and i s homogeneous of degree

PROPOSITION

E

(x0,&,)

6.6

T*X \

Suppose P

:

01 ,

with respect t o 5

m

s i n c e t h i s i s t r u e i n any l o c a l c h a r t .

\o)

Moreover, we have

.

E Lz(X)

Then f o r

we have

U

7-

where:

of

a E C:(X)

xo

$ E CO1(X)

PROOF: x

and has the value 1 i n the neighbourhood

,

dJl(xo) =

to

f

d$

on SUPP a.

0

Let U be an open c o o r d i n a t e p a t c h

containing

OD

0,

and suppose we have a , @E Co(U) w i t h a = 1 i n t h e

neighbourhood of supp B and B(x ) = 1. 0

A t the point x = x

0,

we have

e-iT* p(a

eiT9‘)

e-i~* p plCra .i~$>

+

e - i ~ tB P(I-U] iaei T * ) . -m

The second t e r m on t h e r i g h t - h a n d s i d e i s i n S c a l c u l a t e t h e f i r s t t e r m we can r e p l a c e P by P

(B+)and t o

IU

and conse-

q u e n t l y reduce t o t h e c a s e where X’is an open s u b s e t o f IR”. I n t h i s c a s e we can r e p l a c e P by u p ( x , D ) and a p p l y Theorem 2 . 1 ; we o b t a i n

e-iT*p(aeiT’)(xo)

= a,(P)(x,

, T d)(xo))a(xO)

md S”’.(R+)

2 20

(6.6.1).

which proves

REMARK 6.7: bundles.

Case of p . d . o . ' s

f ,9

Let

o p e r a t i n g on s e c t i o n s of

as,$$) t h e vector

denote by

C ~ ( X; 9)

.

9

3; i n t o

C"(X

4

F on C.

and

X

bundle above

rn

f

and

g

9

8 and

of

P :

, $1

if

above a l o c a l

x -

U of X we have ( w i t h t h e n o t a t i o n of ( 7 . 8 . 2 ) o f

-+

E

Lm(i;

E

, F)

We denote by vX t h e c a n o n i c a l p r o j e c t i o n denote by

* nx f

* %g 3

i n v e r s e images o f

modulo Sm-',

(3

U

E

t h e b u n d l e s on and

9

under nX.

. T*X

Sm(T*IJ\o

of t h e choice of

combine i n t o an element o f

\0

* T X\

and

X

which a r e

0

g

.

Sm(T*X \ 0 ; L(n>

independent, These elements

, r;

g e n e r a l i s a t i o n of t h e r e s u l t s of S e c t i o n s 5 and

Definition

,g)

6

5

U

9 ) ) / Sm-'

c a l l e d t h e p r i n c i p a l symbol of P and denoted by u,(P).

Lm(X ; 9

and We

which can be l i f t e d

* , '9))

; $(nu% f

4

The p r i n c i p a l symbol of

S"(c x R" ; L(F,G))

i s an element of

i n t o an element

type

We

of t h e morphisms

i s an element of L (X ; 9

; 9)

Chapter I ) :

Pf,g

E

, having

We s a y t h a t a continuous o p e r a t o r

f o r any t r i v i a l i s a t i o n s chart U

X

be v e c t o r bundles above

as f i b r e f i n i t e - d i m e n s i o n a l v e c t o r s p a c e s

of

(CHAP. 4)


The

t o p . d . 0 . ' ~ of

p r e s e n t s no new d i f f i c u l t i e s .

Likewise,

6 . 5 and t h e p r o p e r t i e s of c l a s s i c a l p . d . o . ' s g e n e r a l i s e

t o t h e spaces

L:(X

;9

, Y)

.

7)

(SEC.

7.

ELLIPTIC P.D.O.'s

221

ELLIPTIC PSEUDO-DIFFERENTIAL OPERATORS

I n t h i s s e c t i o n , X d e n o t e s a manifold. To t h e concept of an e l l i p t i c symbol t h e r e corresponds t h e concept o f an e l l i p t i c o p e r a t o r .

DEFINITION

of degree

7.1:

If P

E

L m ( X ) , we say t h a t P i s e l l i p t i c

i f i t s principaZ symboZ o,(P)

m

symboZ of degree m i n

i s an e l z i p t i c

.

Sm(T*X\ 0)

If P i s a d i f f e r e n t i a l o p e r a t o r o r more

EXAMPLE 7 . 2 :

generally a c l a s s i c a l p.d.0.

of degree

m

, t h e n it i s

i f and o n l y i f t h e homogeneous f u n c t i o n a,(P)

elliptic

does n o t v a n i s h on

T* X \ 0.

I n f a c t , e l l i p t i c o p e r a t o r s a r e t h o s e which a r e i n v e r t i b l e i n t h e s e n s e of t h e f o l l o w i n g theorem:

The operator P

THEOREM 7 . 2 :

m

E

Lm(X) i s e l l i p t i c of degree

i f and only i f there e x i s t s a proper operator Q

that P.Q.

1

Identity.

Q.P. E I d e n t i t y ;

E

L-,(X)

such

Such an operator Q then s a t i s f i e s

it i s unique modulo

and i s termed a para-

metrix of P. PROOF:

The e x i s t e n c e o f Q c l e a r l y i m p l i e s t h e e l l i p t i c i t y of

om(P) s i n c e we have

222

(CHAP. 4)


. g,(Q)

o,(P)

-

1 .E

*

-1

S

(T X l 0)

I n o r d e r t o prove t h e e x i s t e n c e o f Q, w e make use o f an approximation

procedure which i s important i n i t s own r i g h t .

By h y p o t h e s i s , we know t h a t t h e r e e x i s t s q

0

E S'"(T*X

\ 0)

such t h a t

(7.2.1) Let

orn(P).qo

Qo

E L'm(X)

-1

-1

E S

*

.

(T X \ O )

be proper such t h a t

,

orn(QO)= q,

then

( 7.2.1) implies

(7.2.2)

P.

- I = A,

0,

.

E L-'(X)

We now prove t h e f o l l o w i n g :

m LEMMA 7.3 :

such t h a t rnk

We take a sequence of p.d.o.'s -0.

k=

i ) There e x i s t s

S

.

Then:

f

L "(X)

Rk E L k ( X )

rn

proper, such t h a t for any

integer h

(s

(7.3.1)

-

h

E

Rk)

Lmh+'(x)

Such an operator S i s m i q u e modulo L--(X)

ii) If

R E L-'(X)

.

and we w r i t e

i s proper, and if

T E Lo(X)

is proper such t h a t (7.3.2)

- &

(-llkRk

then we have

(7.3.3)

(1+ R).T

P

T.(I

+ R)

I

Id

.

7)

(SEC.

ELLIPTIC P.D.O.'s

PROOF:

u

If

x u

223

i s a l o c a l c h a r t of X , we can

-D

rn define

Su E L '(U)

by c o n s i d e r i n g t h e asymptotic sum of t h e CI

complete symbols i n U o f t h e o p e r a t o r s X ( R

*

k1U)

( t h i s being Then, we

p o s s i b l e i n view of Theorem 4.2 of Chapter 111). d e f i n e S s a t i s f y i n g ( 7 . 3 . 1 ) by combining t h e S

U by v i r t u e of

--m

The uniqueness modulo L

Proposition 5.3.

(7.3.1).

i s obvious from

The v e r i f i c a t i o n of ( 7 . 3 . 3 ) i s immediate from t h e -

d e f i n i t i o n of T.

We r e t u r n now t o t h e proof of t h e theorem.

We t a k e

Q = Q0 .T where T is d e f i n e d by ( 7 . 3 . 2 ) and R i s a proper op-0

erator equal t o R

0

modulo L P . Q

(7.3.4)

I

; then Q

Id

E

L-m(X)

i s proper and

.

The same method w i l l a l l o w us t o c o n s t r u c t a proper such t h a t

Q'

(7.3.51

.

P a I d

GI' E )X("'L

.

M u l t i p l y i n g ( 7 . 3 . 5 ) on t h e r i g h t by Q, ( 7 . 3 . 4 ) i m p l i e s t h a t Q e Q '

which proves t h a t Q i s a p a r a m e t r i x o f P and t h a t a p a r a m e t r i x

i s unique modulo L-O0'

REMARK 7 . 4 : bundles

Case of p . d . o . ' s

o p e r a t i n g on s e c t i o n s of

.

We once more adopt t h e n o t a t i o n o f Remark

P

E Lm(X ; 3

,3)

e x i s t s a symbol

;

6.7.

Suppose

we say t h a t P i s r i g h t e l l i p t i c i f t h e r e


224

q,

E

S-m(T*X \ 0

.

o,(P)

q,

- Id

IT?))

; L(?T:$,

E

(CHAP.

such t h a t

*

*

; L(nxY, nx9))

S-'(T*X\O

4)

.

Theproof of Theorem 7.2 shows t h a t P i s r i g h t e l l i p t i c i f and only i f t h e r e e x i s t s

Q

E

; 3, 9 ) such t h a t P . Q E I ;

L-"(X

we say t h a t Q i s a r i g h t parametrix of P.

We have t h e analogue of t h i s "on t h e l e f t " , and we have t h e concept of a two-sided

parametrix when 3 and

9

a r e of t h e

same rank.

For o p e r a t o r s of t h e c l a s s i c a l t y p e , we have a simple c r i t e r i o n of e l l i p t i c i t y ; t h i s i s

PROPOSITION 7.5

P E L:(X

Supposing

:

;9

,'3) ,

then

P is l e f t e l l i p t i c ( r i g h t e l l i p t i c ) if and only if for any

E

(x,5)

T*X \ O

the linear mapping

i s injective (surjective). and

9

a,,,(~)(x,5l

E 4sx

Moreover, if the bundles

9,) 3

are of the same rmk, any Zeft ( r i g h t ) parametrix of P

i s a two-sided parametrix.

To prove t h a t P admits a l e f t p a r a m e t r i x , we s h a l l

PROOF:

t

now d e f i n e t h e a d j o i n t P

of P.

To do t h i s , we endow X w i t h a

s t r i c t l y p o s i t i v e d e n s i t y and we d e f i n e a Hermitian s t r u c t u r e on

3; - a n d on

7

(we can do t h i s l o c a l l y by means of a

(SEC. 7 )

ELLIPTIC P.D.O.'s

225

t r i v i a l i s a t i o n and we re-combine by p a r t i t i o n o f u n i t y ) .

and C i ( X ; 3) a r e each equipped w i t h an

Ci(X,S)

t h e spaces

.

, lg (p u

9

L:(X

=

v$

(u

; 3 , 9)

*

p

9

c;(x,sJ ,

u E

Vl9

whose p r i n c i p a l symbol s a t i s f i e s

operator equivalent t o P

*

and

X

,

.

3 X

i s a proper

*

-*

The p r i n c i p a l symbol P P i s for a l l

*

rn

gEx ;

d*

If P

it t h u s has t h e same p r i n c i p a l symbol

t h e i n j e c t i v e l i n e a r mapping (0 ( P ) ( x , C ) )

Fx i n t o

v € CO"(X,Y)

6 . 7 ) t h a t t h i s i s an o p e r a t o r belonging

f o r t h e Hermitian s t r u c t u r e s on 9

a s P*.

and

The a d j o i n t P* o f P i s d e f i n e d by t h e e q u a l i t y

and we know ( s e e Remark to

, l9

(

i n n e r product denoted r e s p e c t i v e l y as

(

Thus

it i s t h e r e f o r e i n v e r t i b l e .

-*

.

(x,c) E T X \ O

a,(P)(x,S)

Consequently t h e Let Q E L4m(X

o p e r a t o r P P i s e l l i p t i c of degree 2m.

from

;9

, 9)

be a p a r a m e t r i x of t h i s o p e r a t o r ; we have 4

Q. P

(3.5.1) The e q u a l i t y

.P

=

I

+

A

with

R €L--(X;9,9)

(7.5.1) shows t h a t t h e o p e r a t o r Q?* E L-,(X

;9 ,9)

i s a l e f t p a r a m e t r i x of P.

The o t h e r a s s e r t i o n s of t h e p r o p o s i t i o n are immediately obvious,

.

226


7.6

(CHAP.

4)

CASE OF SYSTEMS WHICH ARE ELLIPTIC I N THE SENSE OF DOUGLIS-NIRENBERG

. P =

We s a y t h a t a system

...

where

(‘jk) j=l, J k=Ii*.o9K

rn pJk

E

i s of t h e t y p e ( s , t )

Lcjk (X)

E

IRJ x IRK i f

f o r any ( j , k ) E J x K . We d e s i g n a t e a s t h e mjk = tk - j p r i n c i p a l symbol of P of t y p e ( s , t ) t h e f u n c t i o n from Cm(T*X

\0

;

46

cJ))

d e f i n e d by t h e m a t r i x

A system P of t y p e ( s , t ) i s s a i d t o be l e f t ( r i g h t ) e l l i p t i c i n

t h e sense of Douglis-Nirenberg, if f o r any t h e m a t r i x CT

s, t

(P) ( x ,

THEOREM 7.7 :

5) is

(xi{)

E

injective (surjective).

T*X\ 0

We have :

A system P of type ( s , t ) i s l e f t ( r i g h t )

e l l i p t i c i f and only i f there e x i s t s a system Q of type ( t , s ) which i s proper and such t h a t

Q

.

P t

I

(P

.

Q E

.

I)

Moreover, when K = J , one-sided e l l i p t i c i t y i s equivalent t o the existence of a two-sided parametrix 9.

PROOF:

We equip X w i t h a Riemannian m e t r i c s o a s t o be

a b l e t o d e f i n e t h e l e n g t h 151 of a cotangent v e c t o r .

r

E

With

IR, we denote by A r a proper o p e r a t o r from Lz(X) having 151

f o r i t s p r i n c i p a l symbol.

we denote by A

The o p e r a t o r A

-1 a p a r a n e t r i x of A r r

.

r

r

i s t h u s e l l i p t i c and

With t h e v e c t o r s

E

J B ,

(SEC. 8 )

P.D.O.'s

&

SOBOLEV SPACES

we a s s o c i a t e t h e d i a g o n a l system of t y p e (s,O) matrix A

t

E

S

(ASJjk = As

with

if

227

d e f i n e d by t h e

j = k , and 0 o t h e r w i s e .

For

J

K -1 IR we s i m i l a r l y d e f i n e a d i a g o n a l system A t o f t y p e ( O , t ) .

Then , t h e formula

p l a c e s systems P of t y p e ( s , t ) i n b i j e c t i o n w i t h t h e o r d i n a r y

8 , GJ)

E Lo(X ;

systems

.

Using t h e symbols it can be

immediately shown t h a t P i s l e f t ( r i g h t ) e l l i p t i c i n t h e s e n s e c

( s , t ) i f and o n l y i f P i s l e f t ( r i g h t ) e l l i p t i c .

Consequently,

t o prove Theorem 7.7 it s u f f i c e s t o use P r o p o s i t i o n 7.5 and t o c

CI

observe t h a t i f Q i s a l e f t ( r i g h t ) p a r a m e t r i x o f P , t h e n

-1

Q = At

. -Q .

As

REMARK 7.8 :

i s a l e f t ( r i g h t ) p a r a m e t r i x of P.

Tne d e f i n i t i o n s and r e s u l t s concerning

systems of t y p e ( s , t ) g e n e r a l i s e immediately t o t h e c a s e where

d(

and

GJ

9 and 9 above X which

a r e r e p l a c e d by bundles

decompose i n t o d i r e c t sums o f b u n d l e s :

J

K

9 = 0 Sk k= 1

8.

P.D.O.'s

59.; 0 j=1

AND SOBOLEV SPACES

Let X be an open s u b s e t of IRn. the p.d.o.'s,

gj

To prove t h e c o n t i n u i t y o f

we b e g i n by showing t h a t a p o s i t i v e e l l i p t i c

228


(CHAP.

4)

o p e r a t o r admits a k i n d o f approximate square r o o t .

Let C

LEMMA 8.1 :

E

L o ( X ) be such t h a t

i t s principal

symbol u 0 ( C ) is r e a l and s a t i s f i e s for any compact subset K

Then, there e x i s t s a proper (8.1.2)

C

PROOF:

=

B*B

B

+

R

E

c

X

L o ( X ) such t h a t

with

R E L-"(X)

.

The assumption (8.1.1)means t h a t f o r any compact

s u b s e t K c X , t h e r e e x i s t s c > 0 such t h a t

Thus d ( C ) i s an e l l i p t i c symbol and we can c o n s t r u c t a f u n c t i o n 0

p

E

c"(X

x

R")

of t h e same t y p e a s t h a t u t i l i s e d during t h e

proof o f P r o p o s i t i o n 3 . 2 i n Chapter 111.

bo = p j4 -

E So(X x

R")

Then

and bo i s e l l i p t i c .

i s a proper o p e r a t o r w i t h p r i n c i p a l symbol b

it s a t i s f i e s

c

- Bo* . Bo = R-,

ELqX)

0'

If Bo

Lo(X)

E

by c o n s t r u c t i o n

.

We can t h e n c o n s t r u c t by r e c u r r e n c e proper o p e r a t o r s B

j

E

L-J(X)

such t h a t f o r any i n t e g e r j 2 0 , we have

Knowing t h e B for k < j, t h e c o n d i t i o n (8.1.3) means t h a t B k j must s a t i s f y

(SEC. 8 )

P .D.O.' s & SOBOLEV SPACES

229

(8.1.4)w i l l be

Noting t h a t u- ( R - j ) is r e a l , t h e c o n d i t i o n

j

s a t i s f i e d i f t h e p r i n c i p a l symbol b 2 bo

t h i s i s possible since b

0

of B

u-j(R_J3

=

bj

j

j

satisfies

i

is elliptic.

i n order t o

It now only remains t o t a k e

obtain ( 8 . 1 . 2 ) .

REMARK 8.2 :

The same proof a l l o w s us t o c o n s t r u c t an

approximate square r o o t of C , i . e . a proper B

L o ( X ) such t h a t

E

2

C - B .

REMARK 8 . 3 :

case where C

lgl-m

E

This lemma g e n e r a l i s e s immediately t o t h e

Lm(X).

We need of course t o r e p l a c e u 0 ( C ) by

i n (8.1,1),and we f i n d B i n Lm'2(X).

u,(c)(X,5)

As an a p p l i c a t i o n of t h i s lemma, we have

THEOREM

8.4

:

Let P

E

Lo(X) be proper, such t h a t there

e x i s t s M such t h a t f o r my compact subset K

c

Then, there e x i s t s a proper and seLf-adjoint.R

X

E

L-O0(X) such

that (8.402)

lpu 1,2

sM

IIUIC

+

(R u

,u)

for

u E c:(x).

2 30


I n particular, any p.d.0. into

'Lcow (XI

PROOF: C =

8-

(CHAP.

4)

beZonging t o L o ( X ) i s continuous from

2

Lloc(x1

a

The assumption (8.1.1) i m p l i e s t h a t t h e o p e r a t o r

*

P P s a t i s f i e s t h e assumptions of Lemma 8.'l.

sequently, t h e r e e x i s t s a proper B

E

0

L (X) and R

Con-m

E

L

(X) such

that

( 8.4.3

)

P*P

+

B*B

=

h?

+R

a

The e q u a l i t y ( 8 . 4 . 3 ) shows t h a t R i s s e l f - a d j o i n t and a l l o w s us

where (

,

) denotes t h e i n n e r product i n LL(X); t h i s proves

(8.4.2).

THEOREM 8.5 :

If P

E

L m ( X ) , then f o r any s E D , P i s

I f i n addition P i s continuous from H'c omp (XI i n t o H:,:(x). S proper, it i s continuous from H c omp (X) i n t o HE;Zp(X) and from HSl o c(XI i n t o H;,?X).

PROOF:

Since t h e theorem i s t r i v i a l f o r a r e g u l a r i s i n g

o p e r a t o r , we w i l l n o t r e s t r i c t t h e g e n e r a l i t y by assuming t h a t P i s proper.

We r e c a l l t h a t t h e norm HS of u

E

C:?Bn)

is

d e f i n e d by

1M1,

=

IIAsulb

o p e r a t o r whose complete symbol i s

where

(1

As

h e r e denotes t h e

+ 1512 ) 4

2

L e t A:

(SEC. 8 )

P.D.O.’s

& SOBOLEV SPACES

be a proper o p e r a t o r e q u i v a l e n t t o A

: A’-:

A:

metrix of

IIPu/Is-m = ll~s,mPuli,

but

E

A:-’

P,A ,s

A:

= I

5

IlA&

Lo(X)

S

+ Rs.

and l e t A;

For u

”A:

A u;

E

I, p Rs

I

231

-1

C”(X), 0

+

be a parawe o b t a i n

I l ~ ~ m p ~ s u /I( o

E L - W

.

Consequently Theorem 8.5 follows from Theorem 8 . 4 .

We now g i v e a r e s u l t which i s known by t h e name of

Garding’s i n e q u a l i t y .

THEOREM 8 . 6 :

If P

E

Lm(X) , we assume t h a t there e x i s t s

a > 0 such t h a t f o r any compact subset K c X

Then, f o r any s

m

5,

E

>

0 and any compact subset K, and for aZZ

there e x i s t s c such t h a t

PROOF:

We d e f i n e t h e o p e r a t o r

t h e assumption ( 8 . 6 . 1 ) allows us t o a p p l y Lemma 8 . 1 (or more p r e c i s e l y Remark 8 . 3 ) t o t h i s o p e r a t o r ; t h u s . t h e r e e x i s t s a proper B

E

Lm’*(X)

and R

E

L-“(X)

such t h a t


232

1

* )U

~ ( +p P

= (a

- 2) Am A,, E

*

s z

u

+ 6*B u + R u

(CHAP.

with

4)

u E CI(X)

Taking t h e i n n e r product w i t h u , we o b t a i n

F i n a l l y , f o r any K , t h e r e e x i s t s C2 such t h a t for

u

E B,

,

by s u b s t i t u t i n g t h e hence we deduce ( 8 . 6 . 2 ) w i t h C = C C 12' upper bound f o r

1 (R

REMARK 8.7 :

u, u) I i n t o ( 8 . 6 . 3 ) .

G e n e r a l i s a t i o n s t o t h e c a s e of systems.

Let E,F be Hermitian v e c t o r spaces of f i n i t e dimension. We now g i v e a b r i e f i n d i c a t i o n o f how t h e r e s u l t s of s e c t i o n

8 g e n e r a l i s e t o t h e c a s e where P i s a p . d . 0 . b e l o n g i n g t o Lm(X ; E , F ) .

By s e l e c t i n g b a s e s of E and F , we can reduce t o

t h e s c a l a r c a s e i n o r d e r t o prove t h e c o n t i n u i t y of P from HS (X c omp

j

E ) i n t o HS-m(X loc

j

F).

For Ggrding's i n e q u a l i t y , we

(SEC. 8)

ON SOBOLEV SPACES

P.D.O.'s

assume t h a t

;E

P E L:(X

, E) and we

233

replace the inequality

( 8.6.1) by t h e i n e q u a l i t y

where t h e s i g n 2 i s t a k e n i n t h e s e n s e of t h e o r d e r r e s p e c t i n g t h e Hermitian o p e r a t o r s . Next we n o t e t h a t w e do n o t need t h e f u l l p r e c i s i o n of Lemma I

it s u f f i c e s t o prove t h e

8 . 1 t o prove Ggrding's i n e q u a l i t y ; e x i s t e n c e of a proper

*

C

8 . 8

+

B E Lmj2(X ; C

,

R,

with

, E)

E

RIEL

such t h a t m-I

.

(X;E,E)

To do t h i s , we t a k e for B a p r o p e r o p e r a t o r w i t h homogeneous

Jo,(C)(x,~f

p r i n c i p a l symbol e q u a l t o

(here

J

denotes t h e

p o s i t i v e square r o o t i n t h e p o s i t i v e Hermitian o p e r a t o r s ) .

The

remainder of t h e proof o f ( 8 . 6 . 2 ) i s p r a c t i c a l l y unchanged.

The c o n t i n u i t y o f t h e p . d . o . ' s

allows us t o g i v e a new def-

i n i t i o n of Sobolev spaces :

PROPOSITION 8.8:

u E lfoc(X)@

(8.8.1)

If s

I

E

u E ,f)"X), proper,

Moreover, the semi-norms proper P

E

IR, we have the equivalence P

and for a l l we have

1 1 P~ ullo ,

2

E LIDc(xl

p u

for

E L"(X)

cp

E

Ls ( X ) , also define the topology of H S o c ( X ) .

Cz(X) and

.

234


PROOF:

If

u

E

S

Theorem 8.5 i m p l i e s t h a t P u

Hloc(X),

cp

and t h a t f o r e v e r y

E

C"(X)

t h e r e e x i s t JI f o r every u

Conversely, suppose P

let Q

E

L-'(X)

Thus i f

+

R u

,

and t h i s a l s o shows t h a t f o r

H

S

loc ( X )

L:oc(X)

and C such

C:(X)

For u

E

R E L-"(X)

where

, we

P u E LFo,(X)

E

E

E

4)

.

L s ( X ) i s e l l i p t i c and p r o p e r , and

E

be a p a r a m e t r i x o f Q.

u = QPU

(CHAP.

have u

E

cp E Co(X)

H

S

loc

8 ( X ) , we i s proper

have:

.

( X ) from Theorem 8 . 5 ,

, l l c p ~ I ( ~i s

bounded

I\$,P u \ b + \I$, R u \Io , w i t h

above by a q u a n t i t y of t h e form

From t h i s w e deduce t h e i n v a r i a n c e of Sobolev spaces under d i f f eomorphism.

COROLLARY 8.9 :

Let

x

be a diffeomorphism between two open

subsets X , Y of B n . Suppose u

E

a' ( x ) ;

then

(8.9.1)

moreover,

x

*

defines a homeomo2yphism between these two SoboZev

spaces.

The proof i s immediate i n view of P r o p o s i t i o n 8.8 and t h e i n v a r i a n c e under diffeomorphism o f spaces of proper pseuao-diffe r e n t i a l o p e r a t o r s of g i v e n degree. A s r e g a r d s m a n i f o l d s , w e have

(SEC. 8 )

P.D.O.'s

If X is

THEOREM 8.10:

& SOBOLEV SPACES

a

235

,

of dimension n

Cm manifoZd

then Theorem 8 . 5 generalises word f o r word t o t h i s case.

Let K be a compact s u b s e t o f X and l e t

PROOF:

( Uj)

jtl,.

..,

be a f i n i t e c o v e r i n g o f K by open c o o r d i n a t e p a t c h e s U. of c h a r t s J N xj : u 4 U j of X . We p u t u o = x \ uj. I n view o f N

u

J

i

Theorem 2.19 then Pu

E

o f Chapter 11, we have t o show t h a t i f u

HS-m(U ) and depends c o n t i n u o u s l y on u ( 0 S j loc j

Suppose w e have

E

'pj

C,"(Uj)

the neighbourhood o f supp

rpjPu

=

Since +,u

cpj

pj

'pj

=

o

+

P 4 . u

J

and

R~

P jl. u f o r 1 S j J

Xj

*

(p

lu -1.) -

.

and

Q

j '

R.u

J

S

N.

Co(Uj)

I

N)

.

equal t o 1 i n

We have where

-a,

EL

' m

qj E

H:(X)

E

, we .(XI

Rjt=qjP(1-.qj)

.

are l e d t o study

We have

Consequently Theorem 8.5 t o g e t h e r w i t h

D e f i n i t i o n 2.18 o f Chapter 11, show t h a t

rp P q j u E H T i 3 X )

5

.

The c o n t i n u i t y i s e s t a b l i s h e d i n t h e same manner.

E l l i p t i c p.d.o.'s

THEOREM 8.11:

manifoZd and l e t P

are hypoelliptic;

more p r e c i s e l y we have

(On e l l i p t i c r e g u l a r i t y ) . E

L m ( X ) be a p.d.0.

which i s e Z l i p t i c and

proper. i)

If u

E

Let X be a Cm

&'(X) , we have the equivalences:

236

(CHAP. 4)


more generally

s i n g sup2

I f we have s, t

ii)

E

=

PU

.

s i n g supp u

IR , then for any compact subset K

c

x,

there e x i s t s c such t h a t

PROOF:

Q E Lem(X)

Letting

be a p a r a m e t r i x o f P , t h e proof i s

based on t h e e q u a l i t y

(8.11.4)

Thus

u = Q

.PU +

Ru

,

where

u Eb(X)

and R E L-OJ(X).

(8.11.1.)~ ( 8 . 1 1 . 2 ) and (8.11.3)f o l l o w from t h e c o n t i n u i t y

of t h e p . d . 0 . ' ~ . (8.Il.S)

We always have t h e i n c l u s i o n ( s e e ( 1 . 3 . 2 ) )

P u c s i n g supp u

s i n g supp

and t h e i n v e r s e i n c l u s i o n f o l l o w s from inclusion

(8.11.4)and from t h e

(8.11.5)for Q.

REMABK 8.12:

Let

F

of f i n i t e dimension over X.

and

9

b e complex v e c t o r bundles

Taking

p

E

L ~ ( X;

F

, 3) , we

l e a v e t o t h e r e a d e r t h e t a s k of showing t h a t a proof s i m i l a r t o t h a t of Theorem 8.10 allows us to prove t h a t P i s continuous from

(SEC. 8 )

I-?

a P

P.D.O.'s

( X ; 3) i n t o HSm '

loc

237

& SOBOLEV SPACES

( X ; 3) ,

Furthermore, if P i s proper

and l e f t e l l i p t i c , Theorem 8.11 and i t s proof g e n e r a l i s e immediately.

I n t h e c a s e o f c l a s s i c a l p . d . 0 . ' s , we have a

converse t o t h e i n e q u a l i t y (8.11,3),as f o l l o w s :

PROPOSITION 8.13:

If P E LE(X ; 9

a s s me t h a t f o r any compact s e t K

c

, 9) , and t

< m,

we

X, t h e re e z i s t s C such t h a t

Then, P i s l e f t e l l i p t i c . From P r o p o s i t i o n 7 . 5 , it s u f f i c e s t o prove t h a t for a l l

PROOF: (xo

, so)

from

E T*X \ 0

sx i n t o

, the

gx.

of p.d.0.'~ i n I R n .

,5 )

l i n e a r mapping cr,(P)(X

We b e g i n by reducing t o t h e c a s e of a system Let U b e a neighbourhood o f x

0

which i s an

open c o o r d i n a t e p a t c h o f X o v e r which t h e bundles M N a r e isomorphic t o U x 6 and U x C , Suppose we have i d e n t i c a l t o 1 i n t h e neighbourhood of x t o 1 i n t h e neighbourhood o f supp and i f u

is i n j e c t i v e

9

0

,

and $

E

3; and

cp

9

m

E

Co(U)

Cz(U) i d e n t i c a l

If we f i x K = supp Q ,

m

E

C K ( X ) we have

Pu = P cpu = +$P ( p u Since ( i - $ ) P q

+

(1-JI)Pcpu

,

i s r e g u l a r i s i n g , we can r e p l a c e P by $ P cp

in

t h e i n e q u a l i t y (8.13.1)and t h i s allows us t o work w i t h i n t h e open s e t U.

X=R",

Consequently, w e w i l l n o t reduce t h e g e n e r a l i t y by t a k i n g

3 = X x $ ,

% = X X $

.

238


Suppose we have

E C;(R"

v

; C)'

(CHAP. 4)

and let u (x) = v ( x ) e i7X.go 7

f o r T > 0. We study the behaviour, as T

-+

+

m,

of both sides of

(8.13.1) when we take u = uT.

If we let Am be the p.d.0. with complete symbol (1

, we

+ 1012)rn/2

11

'7

have

Ilm =

11

'7

\b

consequently we obtain

Likewise, we obtain

112

By combining (8.13.1) and these two latter inequalities, we obtain

and as v

is arbitrary, this implies that

I+) 1 and consequently

c

l~,(P)(xtr,~

u (P)(x rn

,

5),

.

V(X)

I

is injective.

We now assume that X is a compact manifold endowed with a

(SEC. 8 )

P.D.O.'s

239

& SOBOLEV SPACES

s t r i c t l y p o s i t i v e d e n s i t y and we l e t 3 be a Hermitian v e c t o r bundle of f i n i t e rank over X. d e f i n e s a continuous o p e r a t o r

]p

in

P from

C"(X

, 9)

it

; 9) i n t o

we can a l s o c o n s i d e r it a s an unbounded o p e r a t o r

; 9) ;

c"(X

P E L"'(x ; 9

If

L2(X ; 9)

of dense domain

D

P

.

Hm(X ; 9)

Under t h e s e c o n d i t i o n s we have

Suppose t h a t P is e l l i p t i c .

THEOREM 8.14:

Then,

i ) The a d j o i n t of t h e unbowzded operator (ByD) i s the unbowzded

operator (P*,D*) associated w i t h t h e e l l i p t i c p . d . 0 .

P*E Lm(X;S)

with

i i ) The operator

dim Ker P

0

.

If P i s an o p e r a t o r w i t h c o n s t a n t c o e f f i c i e n t s t h e

commutator [P, p,*]

i s i d e n t i c a l l y z e r o , and we can reduce t o

c o n s i d e r i n g t h e case where t h e symbol of K .

E H'~

p

is zero for x o u t s i d e

Using F o u r i e r t r a n s f o r m a t i o n , we can w r i t e

hence, for u

E 8 , we

obtain

I n similar f a s h i o n , we o b t a i n

We u t i l i s e t h e Taylor formula up t o o r d e r N t o expand w i t h r e s p e c t to

E

t h e difference


252

( C W .

4)

By s u b s t i t u t i n g (10.1.3) and ( l 0 , l . b ) i n t o t h e l e f t - h a n d s i d e of

(10.1.1)and t a k i n g account of (10.1.51,we o b t a i n

The i n t e g r a l e x p r e s s i o n for t h e remainder rN shows immediately t h a t , for any k

0, t h e r e e x i s t s C such t h a t

2

(10.1.6)

I n a d d i t i o n , t h e assumption concerning t h e s u p p o r t of p ( x , E ) i m p l i e s t h a t for a l l h

2

0 , t h e r e e x i s t s C such t h a t

A (10.1.7)

I P ( M

8

511

To m a j o r i s e t h e norm

c

(1

(IR (e)ull

N

+ 151," (1 +

s+a

, we

l%51)-h

specify

v E 8 and we

c o n s i d e r t h e i n n e r product

Taking account of (10.1.7)and of P e e t r e ' s i n e q u a l i t y , t h e r e

follows

FRIEDRICH'S LEMMA

(SEC. 1 0 )

253

t h e right-hand s i d e of (10.1.8) i s bounded above by

Consequently, by t a k i n g

h

s u f f i c i e n t l y l a r g e , we t h u s deduce

which p r o v e s (10.1.2).

A s a p a r t i c u l a r c a s e , we have F r i e d r i c h ' s Lemma

COROLLARY 1 0 . 2 :

We adopt the same asswnptions as i n the

preceding theorem, together with the f u r t h e r condition P(0) = 1. Then for aZZ s (10.2.1

1

E

B, there e x i s t s C such t h a t

IICP,

P,*3lIS

for

Moreover, PROOF:

[P

, p,*3

5

cIlulls+ncl

u EH~++"

0

and e

in HS when u

>o E

H

s+m-1

E - 0

The i n e q u a l i t y (10.2.1) i s o b t a i n e d by s e t t i n g

254


N = u = 1 and by r e p l a c i n g

If u

E

S

s

i s dense i n H

s +m- 1

-

4)

by s - 1 i n t h e i n e q u a l i t y ( 1 0 . 1 . 2 ) .

u

p,*u

we know t h a t

(CHAP.

in

s

and s i n c e g

O'E

t h e i n e q u a l i t y ( 1 0 . 2 . 1 ) a l l o w s us t o e x t e n d

t h e conclusion t o a l l u

E

H

s +m- 1

We now g i v e t h e proof o f Theorem 6 . 5 o f Chapter I1 w i t h i n a more t h i s i s as f o l l o w s :

general s e t t i n g ;

Let P be a p . d . 0 . a s i n Theorem 10.1; l e t

COROLLARY 1 0 . 3 :

there e x i s t s

E

IR

and

p

s a t i s f y i n g the supplementary condition

( 6 . 3 . 3 ) of Chapter 11, w i t h h > s + l .

Then, there e x i s t s C such

that

PROOF:

We r e p l a c e

[P

we m a j o r i s e each term.

6.4 of Chapter

, p,*Ju For

by i t s e x p r e s s i o n (10.1.1)and

0

0

7 ,

( i n what f o l l o w s , such a l o c a l axn

c h a r t w i l l be c a l l e d a l o c a l c h a r t of

y)

)

, P i s written

and we have t h e jump formula ( c f (l.b.l), Chapter 1):

A l l t h e above c l e a r l y h o l d s f o r an a r b i t r a r y r e g u l a r open

s e t fi and f o r an a r b i t r a r y d i f f e r e n t i a l o p e r a t o r P.

Carrying o u t a l e f t convolution of b o t h s i d e s o f (1.3) with a fundamental s o l u t i o n E of P = P ( D ) we o b t a i n , p u t t i n g P u = f : (1.6.)

u

=

(E * f o )

In

+

(E * F ~ U ) , ~

This formula e x p r e s s e s u as t h e sm of a volume p o t e n t i a l and a s u r f a c e p o t e n t i a l ; we s h a l l show t h a t t h e s e two p o t e n t i a l s a r e in C m ( E ) , so that taking traces i n ( 1 . 6 ) :

274

(CHAP.

ELLIPTIC BOUNDARY PROBLEMS

5)

(1.7.)

In p a r t i c u l a r , i f

u

C : C"(aL2,

o b t a i n yu = C ( y u ) , where

i s d e f i n e d by

c

v

y

=

E c"(E)

[(E

*

and u

c"(an, c")

C")

Cm(ahl,

J

].

~ v ) , ~

C")

It can e a s i l y be

From (1.7),i f u

shown t h a t C i s a p r o j e c t o r .

(l.l), then yu E

s a t i s f i e s PU = 0 , we

E

Cm(?2)

satisfies

satisfies

i s given by ( 1 . 6 ) .

Conversely, i f

and i f we p u t

u E Cm(z)

v E

u = (E

*

cm(an, c"')

follhl

satisfies

+ (E * pt) In '

then

and s a t i s f i e s (1.1).

We t h u s see t h a t (1.1)reduces t o i n v e s t i g a t i n g t h e system

I n what f o l l o w s , we show t h a t t h e p o t e n t i a l s

(SEC. 2 )

275

REGULARITY OF BOUNDARY POTENTIAL

have t h e s t a t e d r e g u l a r i t y a t t h e boundary, and t h e n go on t o i n v e s t i g a t e t h e o p e r a t o r C ( c a l l e d t h e CalderBn p r o j e c t o r ) . F i n a l l y , we apply t h e r e s u l t s o b t a i n e d t o t h e i n v e s t i g a t i o n o f

(1.1)under c e r t a i n e l l i l s t i c i t y assumptions.

2.

REGULARITY OF THE POTENTIALS AT THE BOUNDARY We use t h e n o t a t i o n d e f i n e d a t t h e s t a r t of t h e I n t r o -

duction.

u E J'(L-2).

Let

t r a c e s up t o o r d e r k ( k

x

:

u

v

cnn

of

u has s e c t i o n a l

We say t h a t E

IN) on 30. i f , f o r any l o c a l c h a r t

(fi,

V ) and f o r any cp

t r a n s p o r t e d i n t o a f u n c t i o n o f c l a s s Ck i n x s u f f i c i e n t l y s m a l l ) with values i n J ' [ R X ~ ' ) naturally define the traces

y J. u = y 0

[(l

">j u ]

If u

Ik

E

ulan

('i), then

on ail up t o o r d e r k f o r k < s

for 0 S j

u =

-

a,

(for x

.

We t h e n

u has s e c t i o n a l t r a c e s

and

yj u E Hs- j-$? loc

(an)

has a s e c t i o n a l t r a c e of o r d e r 0 on a R , we

j

a.

Let

(1,v ) , we

IN ; by means of l o c a l c h a r t s of

E

define t h e l a y e r v

8 Dj 6

s e c t i o n a l t r a c e s up t o o r d e r

( 1 . 3 ) , (1.5).

t 0

( s e e Chapter 1 1 ) .

&'(a)

&'(an) and.

n

is

1

n a t u r a l l y d e f i n e i t s e x t e n s i o n u o by 0 o u t s i d e v

TU

as d i s t r i b u t i o n s on as2.

(0 5 j 5 k )

u E Hsoc

For example, i f

yo

n

E c;(u),

E 5'an(R")

.

If u

E

B'(C2)has

m y we a g a i n have t h e jump-formula

276


A

Now l e t operator.

< L:(Rn)

(CHAP. 5 )

be a c l a s s i c a l p s e u d o - d i f f e r e n t i a l

We i n t r o d u c e t h e c o n d i t i o n s A i s proper, and each term of the complete symbol

(2.1.)

1 aj(xI j s

of A i s a r a t i o n a l f r a c t i o n i n 5.

5;)

Using t h e formulas of t h e symbolic c a l c u l u s for p . d . o . ' s ,

it can be shown t h a t t h e c o n d i t i o n s ( 2 . 1 ) a r e p r e s e r v e d under diffeomorphism, t r a n s p o s i t i o n , composition, and passage t o t h e parametrix i n t h e e l l i p t i c c a s e . t h u s s a t i s f i e d when

A

In p a r t i c u l a r , they are

i s t h e proper parametrix of an e l l i p t i c

d i f f e r e n t i a l operator.

Suppose we have

THEOREM 2 . 2 : (2.1) and

u

E

B'(Rn]

zero i n a.

A E LE(Rn)

Then (Au)

In

satisfying

has sectional

traces of any order on an.

PROOF:

By p a r t i t i o n of u n i t y and l o c a l c h a r t s we can

n reduce t o t h e c a s e 0 =IR+.

We can assume

u t o have compact

s u p p o r t ; we t h e n know t h a t t h e r e e x i s t c and R such t h a t

1;(5)\ that

5 c[l A'u

A' E LP'(R")

+

151) a for 5

k n C [R )

.

E

IRn, and it can e a s i l y be shown

when k

- -n-1

=

0 ( f o r x describing

supp 0 )

I t m a y be n o t e d , i n view of t h e homogeneity of can choose

rs

contour o f

{ Im

for

&! E r

5'

I

'

=

IS'\r

for

> 0 1.

5' E R"

1

~

~

2

11 ,

We t h e n have

.

\ $1

where

a , t h a t we

r

is a f i x e d

5 c(l

+ 15'1 )


(SEC. 2 )

We t a k e i n p a r t i c u l a r SUpp $

C

{ xn > 0 1

.

@ (x)

3

cp(xl) $(xn)

279 with

From ( 2 . 2 . 3 ) , ( 2 . 2 . 2 ) , we o b t a i n

It can be shown t h a t t h e o r d e r o f i n t e g r a t i o n can be changed, s o t h a t we can w r i t e :

where

( B y i n t e g r a t i n g by p a r t s w i t h r e s p e c t t o x ' , we s e e t h a t G(xn,

6') i s o f r a p i d d e c r e a s e w i t h r e s p e c t t o C ' , and hence The

t h a t t h e right-hand s i d e of ( 2 . 2 . 4 ) i s m e a n i n g f u l ) . e q u a l i t y (2.2.4)i s c l e a r l y t r u e f o r deduced f o r imating

u E &'(R")

1

and i s

by proceeding t o t h e l i m i t , approx-

u by i t s r e g u l a r i s e d forms pk

support i n { x s 0 n

,

u E C ;(R')

*

u

, with

p

having

( s e e (l.l.l),Chapter I).

F i n a l l y , ( 2 . 2 . 4 ) and ( 2 . 2 . 5 ) show t h a t , i n

Cx,'OI,

(CHAP. 5 )


280

ca

Bu c o i n c i d e s w i t h t h e f u n c t i o n from C

xn C->

(2.2.6.)

y,(Au)

into

&'(an).

P ( R-1 ) )

d e f i n e d by

Tne above proof shows t h a t t h e mapping

i s continuous from ~u

In

I

(cp

REMARK 2.3 : u -->

-+ (R

E

D'(B") I

u =

o

QI

in

THEOREM 2 . 4 : ( R e g u l a r i t y of t h e s u r f a c e p o t e n t i a l s a t t h e boundary )

Suppose we have

A

i ) The operator

E L:(R")

satisfying ( 2 . 1 ) . A(v,@ 6 )

K : v >-t

3,

Then

i s contin-

uous from c"(~Q) i n t o crn(E).

If v E

p(an)

Kv has sectional. traces of a n y order on

I

i i ) The operator

v -->

A f t e r transport b y a local chart of symbol. i s a

0

(XI'

5 ' ) ->

\r

i s the principal. s y m b o l of A, and

the poZes En of

ao(x', 0 ;

iii) For all. s

E

5'' %)

(n,

0 ;

' . ( , a

r

LF+'(an)

i s in

yO(Kv)

an

.

i t s principal.

y),

s', $1 8% ,

where

i s a contour enclosing such t h a t I m 5, > 0 .

IR, K is continuous from H '

loc

(an) i n t o

2 81


(SEC. 2 )

For ( i ) ,we can r e p l a c e A by t h e o p e r a t o r B = A - R

PROOF:

used i n t h e previous proof;

u = v(x')

i c u l a r case

taking (2.2.4), (2.2.5) i n the part-

,v

@ b[xn)

€ C:[R*'),

we s e e t h a t f o r

xn > 0 :

(2.4.1.)'

=

(Bu)(x)

k(x,

5') ; ( I 1 )$5'

where

so t h a t ( i ) t h e n f o l l o w s as t h e right-hand s i d e o f ( 2 . 4 . 1 ) i s Cw with r e s p e c t t o x E 7 R n .

For ( i i ) ,w e reduce t o ~ ' i 8 v ti) 1.1'

>-

v

&'mI ([ A ( v @ 6lll"Io

i s the operator

T'

:

Cmm

1

i

> c"(an1

f

t

7

Y , ( [ ~ A ( ~ ~ ~ I , ~ ,I )

288


We need t o prove t h a t < f , Kv > = < T'f,

PROOF: f

Cim)

E

.

= Bn \

where

,v

E C;(an)

n

,

We reduce t o t h e c a s e R = IR,

t h e e q u a l i t y i s immediate when f

E

C"(E), 0

-EI

f E

.

C:(n)

;

When

we c o n s i d e r a r e g u l a r i s i n g sequence X

pe(xn) = ] , (p

n

with

Supp p

lim

To c o n c l u d e , we show t h a t i n C"(Q).

v > for

c

xn > 0 ]

yo t Atp,

*

, We have

f o )= y o ( [ t A ( f o ) ] , n r )

e -0 This i s obvious i f A i s a p s e u d o - d i f f e r e n t i a l

o p e r a t o r o f degree l e s s t h a n -n,

so t h a t the required equality

f o l l o w s by a s t r a i g h t f o r w a r d passage t o t h e l i m i t i n (2.5.1).

lFnder the conditions of Proposition 2.7,

COROLLARY 2.8:

the operator T' extends continuouszy from

d (5)t o

ti'(an).

Xhe same is true for the t r a c e operator

T :

f

->

Yo-([

A(fo]]

) when the degree u of A i s l e s s than

In

- 1/2. PROOF: i n t o -C:tc)

Since

, TI

TI

=

+,

K and s i n c e K i s c o n t i n u o u s from C:(an)

e x t e n d s c o n t i n u o u s l y from

8 (z)t o a' (an).

Since T'f

289


(SEC. 2)

,

2 (f) Lloc

f O

= Tf

Now t h e o p e r a t o r from

Y'

0

with

RICE) t o B'(an, C") Q i s of degree

Y[Q(f0)],

extends continuously

i n view of C o r o l l a r y 2.8, s i n c e

- 1 a t most f o r 0 5

,

j 5 m-1

f o r e i t s l e f t composition with r B i s continuous from into

C"(an,

Ch)

and i t s l e f t composition w i t h R P A B to

has i t s k e r n e l i n Coo(;

X

cmmn). E)

The o p e r a t o r f

tQ

+

tQ

,

with

( f , g ) i s i n t h e orthogonal of

(f, g)

t

- > [ ~ ( f o ) ] , ~

R

i s therefore

The orthogonal of I m 6 c o i n c i d e s w i t h Ker

tp = I

- tf?(f, g)

E

is

and t h e o t h e r o p e r a t o r s which

appear i n ( 4 . 1 . 2 ) a r e c l e a r l y r e g u l a r i s i n g .

now

#(E)

w

continuous from

regularising.

There-

C"m)

t 6 ;

t

f ? r e g u l a r i s i n g ; hence i f

Im p , we have

x C"(a0,

Cu).

F i n a l l y , i f 52 i s

(SEC. 4)

APPLICATION

bounded, we know t h a t

~"(5) x c"(an,

R is

.

CP)

%us

297

a compact o p e r a t o r from

Im(1

+ e) i s

c l o s e d and of f i n i t e

codimension, and t h e same a p p l i e s f o r Irn p which c o n t a i n s

+ e)

Irn (I

.

formula f o r t~ :

tP(f")

=

( t P f ) O

+ ;;(Yf)

and we t h e r e f o r e deduce t h a t

i s i n Ker

t 33 i f and o n l y i f

We now proceed t o t h e c a s e of Sobolev s p a c e s .

We denote

by m ' t h e s m a l l e s t i n t e g e r 5 m such t h a t B j = 0,

..., p-1.

= 0 f o r k 2 m', j ,k Then B y u i s w e l l d e f i n e d when u has s e c t -

i o n a l t r a c e s on a R up t o o r d e r m'

-

1; from Theorem 2.9, t h i s

i s t h e c a s e , i n p a r t i c u l a r , when u i s an e x t e n d a b l e d i s t r i b u t i o n

Pu

E

Hs(,E)

Let s , u

E

IR w i t h

such t h a t

with

s > rn'

s > rn'

- rn -

o p e r a t o r ( n o t n e c e s s a r i l y bounded)

- rn - 1/2 1/2

.

. We c o n s i d e r t h e

29 8

This domain is dense i n H

U

loc

(5)

; u s i n g Theorem 2 . 9

is c l o s e d .

and Remark 2 . 1 0 , it can be shown t h a t ps

IU

Suppose t h a t

THEOREM 4.3 : s,a

(CHm. 5 )


E R

0s,0

a r i g h t quasi-inverse

Q

=lo

,

s > rn'-m-1/2

with

'

:

Hloc

(z)

F is

surjective.

a 5 s+m.

Then Ps 1u a d n i t s

-

i n the sense t h a t Hr61/2 loc

(an)

is continuous, with P s , o ~ s , o = I + l?s,a~ from H ( ,n ;l

orthogonaZ of

r61/2[an)

into

x Hloc

Im ps

,a

(in

H Z

c"(;)

($1

n, cornp

i s the same as the orthogonal of Irn P bozmded, then I m ps (equal t o t h a t of I m

PROOF:

Let

c o n t i n u o u s from H

f S

loc

.

x

c"(ahl,

x

Hemp

HYoc

(-3

continuous

Cp)

.

The

(an)I

If, furthermore, R i s

i s cZosed and has f i n i t e codimension

SU

P

Let

).

7

be an e x t e n s i o n o p e r a t o r ,

( 5 ) i n t o 'H

loc

Dn)

; we d e f i n e

Q

=,a

( f , g]

(SEC. 4 )

APPLICATION

299

as b e i n g t h e r i g h t - h a n d s i d e of formula ( 4 . 1 . 1 ) d e f i n i n g Q ( f , g ) ,

a

c

b u t w i t h f o r e p l a c e d by f .

The c o n t i n u i t y of

SIU

f o l l o w s from P r o p o s i t i o n 3 . 2 and from t h e assumption

05

It can be shown t h a t

S+m

=

I

+

i s d e f i n e d by t h e r i g h t - h a n d s i d e of

fs,u(f, 9)

where

Q s,u N

,;'("C

viously has values i n

f[aQ

x

belongs t o t h e dornaizi

t

thRt

Ker Ker

(F, G) =

c C:fi) t p

SlU

BS,,

t

p

.

t p ( ~ G, )

x C:(aQ

= Ker

Of

Cb)

, which

i s bounded,

es,o i s

HS(E)x H**1/2(a*l

it follo*rs i n m e d i a t e l y

%if,

Since

, we

c B;,,

obtain

g i v e s t h e s t a t e d e q u a l i t y between

L i e o r t h o g o n a l of t h e images o f

R

If (F,G)

Cp)

t

; fs,u ob-

f

in t h i s formula r e p l a c e d by

( 4 . 1 . 2 ) w i t h f0

%U

,p

.

F i n a l l y , when

a compact o p e r a t o r from

,

and t h e r e f o r e I m ps

?U

i s closed,

with f i n i t e c o d h x n s i o n .

Suppose t h a t b

s , E~ R

with

,

u

E p (0)

be extendabZe t o IRn such t h a t

i i ) Suppose

u' m1-wI/2

i S

0 5 s+m

and l e t K be a compact subset of

; then there e x i s t s a constant C such t h a t

.

300

(CHAP. 5 )


f o r aZZ u with support i n K s a t i s f y i n g the assumptions of ( i ) . iii) We have iv)

K e r ps

If, furthermore,

ta 51

= Ker P

c

C-m]

is bomded, then

,

Ker bJ

st0

= K e r 63

i s of f i n i t e dimension.

For ( i ) ,we p u t Pu = f and

PROOF: we have Pv =

7 , with

N

cp = -(Rf

,

1,

,

v

I

u

- (Gjf),, H

;

I

I n view o f Theorem 2 . 9 ,

v h a s s e c t i o n a l t r a c e s of a l l o r d e r s on a R , and we have P ( v o ) = yo

+ g(yv),

Applying Q on t h e l e f t t o t h i s e q u a l i t y ,

we o b t a i n (4.4.2.

1

(I

')

YV

E c"(an,

C")

@

0 - 6 1/2 Hloc (an1

.

Now t h e i n j e c t i v i t y of t h e p r i n c i p a l symbol o f

i s equivalent t o t h a t of

left-elliptic that

F; t h e r e f o r e

i n t h e s e n s e of Agmon-Douglis-Nirenberg, so

(SEC. 4 )

APPLICATION

301

From ( 4 . 4 . 2 ) and from P r o p o s i t i o n 3 . 2 , we deduce t h a t

E

w

for

, which

HYocfi) u

and y

j

then gives t h e s t a t e d r e g u l a r i t y p r o p er ties

u

u since

-v

=

($9

E)if:H::

,

We prove ( i i ) i n t h e same way by u s i n g t h e e l l i p t i c i n e q u a l i t y

bounded,

(4.4.1)shows

by H'(E)

and H''(E)

4.5

t h a t t h e t o p o l o g i e s induced on Ker 63

a r e c o i n c i d e n t , which gives ( i v ) .

LOPATINSKI'S DETERMINANT AND CONDITION:

The boundary-

w

value problem (1.1)i s s a i d t o be e l l i p t i c i f b i s b i j e c t i v e . me number

u

of s c a l a r boundary c o n d i t i o n s must t h e n be e q u a l t o

m/2, and t h e preceding theorems show t h a t 63 o p e r a t o r s ( w i t h t h e same i n d e x ) when

n is

63s,a

I

bounded.

..

are indexed We conclude

t h i s s e c t i o n by g i v i n g i n v e r t i b i l i t y c r i t e r i a f o r b i n t h e case I.I

= m/2.

Following t r a n s p o r t by a l o c a l c h a r t of

note by p ( x , 5 1 , b j Y k ( x ' ,

(E,v), we

5') t h e p r i n c i p a l symbols of P, B

de-

j ,k

r e s p e c t i v e l y , and we p u t

p+-(x*;

+-

Irn

% ) a r e polynomials i n En whose ze,ros

5 1 ,

>

o

; pf i s o f degree m/2 i n L1.-

1

5')

6

5,.

5n a r e such t h a t

Let denote t h e remainder

(CHAP. 5 )


302

of t h e E u c l i d i a n d i v i s i o n of

$1

5''

bj(x';

by

pi(x';

$,I

9. .

E l ) i s i n v e r t i b l e i f and o n l y i f t h e

From P r o p o s i t i o n 3 . 3 ¶ b(x:

o n l y bounded s o l u t i o n U , for x 2 0 , o f n

i s U = 0. S i n c e t h e bounded s o l u t i o n s U , f o r x t 0 o f t h e e q u a t i o n n

5''

p(x1, 0;

tion

+

p

=

0

a r e t h e s o l u t i o n s U o f t h e equa-

U(xn)

=

0

b!(x';

s',

Dxn) U(xn)

[XI;

5''

)

Dx

, and

since

n

b

5',

.(XI;

3

Dxn)U

-p +( x * ;6')5'' i sDx b(x',

(b;, k

=

)U = 0

J

, we

Dx,]U

when

s e e t h a t t h e i n v e r t i b i l i t y of

n

e q u i v a l e n t t o t h a t of t h e m a t r i x

1

5'

1) jyk=o, ~, I p-

1

.

The d e t e r m i n a n t o f t h i s m a t r i x

i s c a l l e d t h e L o p a t i n s k i determinant o f (1.1)r e l a t i v e t o t h e l o c a l chart considered. Lopatinski condition i n

We s a y t h a t (1.1)s a t i s f i e s t h e

(x', 5')

-

i f t h i s d e t e r m i n a n t i s non-zero;

t h i s condition i s equivalent t o t h e i n v e r t i b i l i t y of b ( x ' ,

5.

6').

EXAMPLES The D i r i c h l e t problem i n

o p e r a t o r P of degree

m

is

for the elliptic differential

303

EXAMPLES

We t h u s have i n t h i s c a s e p = m/2

P

!?he corresponding operators 5

Bj,k

= gj,k

I

,

m' = m/2 ,

The D i r i c h l e t problem ( 5 . 1 ) i s e l l i p t i c .

THEOREM 5.2:

u

,

Ps,o

(with s

>

m+l

- -,2

s+m) are therefore indexed ( w i t h the same index) when Q i s

t h i s index is zero when P is formazly s e l f - a d j o i n t or

bounded;

strongZy e l l i p t i c .

PROOF:

Using t h e n o t a t i o n o f

b.(x', 5)

J

=

4

=

p = mj2); t h e r e f o r e

0,

j,k

.,., p-1

,

view of Remark 4 . 2 ,

(f,

g)

(5.2.1

E

c"fi]

x

bv.(xl,

J b1

51

(xl,

4 . 5 , we have for

I*) = BjIk

j,k and t h i s shows t h a t

Ker

c"(an,

t P

cb)

j = 0

,..., G-I( p u t t i n g for

(5.1) is elliptic.

i s t h e s e t of t h e such t h a t

.)

Now, we have l o c a l l y

that (5.2.1) is equivalent t o

In

304

(CHAP. 5 )


=

tPf

,

0

yj

=

f

0

- c(yf)

g . b e i n g e q u a l t o t h e c o e f f i c i e n t of D j 6 i n J t h e r e f o r e follows t h a t

cf

K e r tp

(5.2.2.)

E

, yj

o

=

{

E

f

=

Crn(qP*f

t

,

yj

=

f

if P = P

+

~ ( u ,u)

5

G

We t h e r e f o r e deduce t h a t K e r p' = { 0

Ker

t

p'

=

Let s >

{ 0 }

nH1 -2

+ 1.

,

}

E

hence t h e i n d e x o f P'

and u = s+m; t h e n

ps 9 0

REMARK 5 . 3 :

IR such t h a t

E

%(nl

0

6' i s assoc-

I n view o f ( 5 . 2 . 2 ) , we s i m i l a r l y have

, and

Index

Garding's

u

where

G's,o. 1

Index

i s t h e operator

. =

pi

Index p'

9U

I n t h e c a s e i n which P =

we have Green's formula

i s equal t o zero.

PS,,

-

(If, 0) A s t h i s o p e r a t o r i s compact, we have:

p =

63 t h e n

JIUII~

f->

Index

3

2

for all

i a t e d with P' = P

3

We now suppose

i n e q u a l i t y e n s u r e s t h e e x i s t e n c e o f c > 0 and A Re(Pu, u)

that

i s bounded.

t h a t P i s s t r o n g l y e l l i p t i c and t h a t

o

O 5 j 5 p-I

, so

h a s z e r o i n d e x i n t h e c a s e where $2 i s bounded.

(5.2.3.)

f =

for

0

*

Ker p

p

Ker

In particular

0

and it

0 5 j 5 1.1-1

for x

,

Osjsp-I

for

-

=

0

.

A and f2 i s bounded,

-

G&rding's i n e q u a l i t y ( 5 . 2 . 3 )

where N i s t h e inward u n i t normal. t h u s h o l d s f o r A = 0.

P

:

co.rnl

Therefore

cmfi] (-

and

H'(5)

c

AU,

yo u )

1----------b (-

but

Hr1/*(an)

x

Yo u5

w i t h continuous i n v e r s e ( f o r

is bijective

,

u(s+~!)

Now l e t P be e l l i p t i c of degree 2 ;

we c o n s i d e r t h e problem

Pu

=

f

in

0

y, u

=

g

in

an

(5.4.1

p = m/2 = 1

I n t h i s case of

i s an isomorphism,

Hsm>

u

~ > - 3 / 2

c"(an)

-

u *-I

: Bs,o

x

, ,

4.5 , b(x', 5 ) = 5,

m'

= m

b'(x',

.

and, u s i n g t h e n o t a t i o n

5')

=

i-

0 such t h a t 1P,(%rl

2

c

C: f R"

for x

151 = 1

where p

m

i s t h e homogeneous p r i n c i p a l symbol o f degree m o f P .

" Suppose t h a t b i s i n j e c t i v e , and t h a t we have such t h a t S

>

m'-m-&

P u

E H'(E)

,

,

0 5 s

+m

E H O I . (an) ~ ~

BY u

.

4.4.1) i s

fying t h e above assumptions.

5 . 5 remains t r u e f o r

6.6

with

Using Theorem 4 . 4 and a

parametrix of P i n BL-mfJRn), show t h a t u p r i o r i e s t i m a t e of t h e t y p e

u f) :(aH

E

Ha(;)

and t h a t an a

v a l i d for a l l

u

satis-

Deduce t h a t p a r t ( i ) o f Theorem

P and

CONVERSE OF THE REGULUITY THEOREM

Assume t h a t t h e boundary-value problem (1.1)s a t i s f i e s p a r t ( i i )

o f t h e s t a t e m e n t o f Theorem 4 . 4 ( a p r i o r i e s t i m a t e ) .

By

applying t h e i n e q u a l i t y ( 4 . 4 . 1 ) f o r K c Cl and by u s i n g Propo s i t i o n 8.13, Chapter I V Y show t h a t P i s e l l i p t i c i n

By a p p l y i n g t h e i n e q u a l i t y where

v

E &,,(an

, (3 m

E)

,

(4.4.1) f o r

show t h a t :

;

u = (QPv)

.,


312 for all

v E

JK,(an

,

.

E)

(CHAP.

Then deduce t h a t t h e o p e r a t o r

injective.

6.7

CASE OF SYSTEMS WHICH ARE ELLIPTIC I N THE SENSE OF DOUGLIS-NIRENBERG

Suppose

5

i s a manifold w i t h boundary, imbedded i n a man-

i f o l d M ( s e e Remark 9 . 6 , Chapter I ) .

J E

L:

0 Ej jt 1

I

,

F =

, where

Fi

@

Let

tne E

i= 1

j'

F. are vectorial 1

bundles above M.

Suppose we have an e l l i p t i c d i f f e r e n t i a l o p e r a t o r P:

, t) E I

(. i.e.

zJ

,'

( s e e S e c t i o n 7.6, Chapter I V ) an o p e r a t o r P o f t h e form

(pi j)i=1,. j-1,

.....,

P i,j

where

,I

Cw(M

Let

G

d 0 Gj

P

j=l

an.

Ej)

Y

* CaD(M, Fi)

J

i s a d i f f e r e n t i a l o p e r a t o r of degree t

above

x

Let

,

j

- si '

with

where G . a r e v e c t o r i a l bundles J

8 : Cm(M

e n t i a l o p e r a t o r of t y p e ( r

, E)

4

, t ) E IRd

Cm(M x

'

IRJ.

, G)

be a d i f f e r -

5)

(SEC.

6)

ADDITIONAL NOTES

313

Consider t h e boundary-value problem

I n o r d e r t o a l l o w t h i s problem t o be s t u d i e d by means of a formalism analogous t o t h a t i n Chapter V , it i s convenient t o introduce a r e a l f i e l d v t r a n s v e r s a l t o

an,

and t o p u t

, f E) ,

where

m

i s an i n t e g e r l a r g e enough t o e n s u r e t h a t t h e f o l l o w i n g two conditions a r e s a t i s f i e d :

(6.7.1.)

t h e boundary c o n d i t i o n

i n t h e form

B(w) = g W

, w i t h B = (Bo,B1

Bk : C"(m,E) * C (W,G) i s (r

...B,-i,

i s expressed where

a d i f f e r e n t i a l o p e r a t o r of t y p e

..., m-1

t-k) for k = 0,

(&)lang

(we p u t ( t - k )

j~

= t.-k for

j = ly...yJ).

(6.7.2.) y j t$

The jump formula for P does n o t i n v o l v e any terms

such t h a t j 2 m.

We can t h e r e f o r e w r i t e t h i s formula i n

the form

~(u") = (pu]

O

+

F(yu )

for

u E

cW(E , E)

,

#u

where P(yu) i s a d i s t r i b u t i o n c a r r i e d by

an

which depends o n l y

on P and YU.

It i s t h e n a simple m a t t e r t o a d a p t S e c t i o n s 3 and taking account of t h e f o l l o w i n g remarks:

4,

314


(CHAP.

5)

) i s of type We t a k e a parametrix Q of P such t h a t Q = (Q j ,k

a)

( t , s ) and each o p e r a t o r c o n s t i t u t i n g Q

satisfies (2.1).

j ,k

The e x i s t e n c e of such a parametrix i s immediate: we t a k e Qo of t y p e ( t , s ) having as p r i n c i p a l symbol t h e i n v e r s e matrix of 0

( P ) ; p u t t i n g , f o r example, PQo = I

s ,t

Q

degree -1, and we t a k e

b)

-

R , we have R of

.

Q,Rk

N

We d e f i n e t h e Calder6n p r o j e c t o r C = y Q

, Qm E) * C"(W

C : C"(S

Q

m

,

E)

C

p

p s e u d o - d i f f e r e n t i a l o p e r a t o r of t y p e ( t

o,[c)

Let

[uwSt-a

s

be t h e p r i n c i p a l symbol of

C

('k,

.

F.

(%,A)

-

9

k, t

A3'k1 Iro,

E, F

U

and t h a t t h e mapping onto t h e image of

0

0

,

+.

that

yU

; 5'

... I

m-1

(E,

V)

and

i s a projector,

+

, where S+ ( X I , 5')

(C)(x', 5 ' )

,0

s,t (p)(X'

- a).

i s an isomorphism of S (x', 5 ' )

n

U

, 5')

uo(X'

space o f s o l u t i o n s U , bounded f o r x system

a

Ck,j

It can be shown, following

t r a n s p o r t by a l o c a l c h a r t t o t h e boundary of t r i v i a l i s a t i o n s of

We have

,D

i s the

2 0, of the differential

) U(x,)

= 0,

We then deduce

x"

t h a t t h e image of uo(c)(x'

5')

does not depend on t h e choice

( t o w i t h i n an isomorphism) of t h e i n t e g e r m y and a l s o ( s e e f o r example I N C E [11 ) t h a t t h e rank of a o ( C ) ( x ' , 5 ' ) i s e q u a l t o one h a l f of t h e degree i n 5, i.e. t o

3

(g J

tj

-

dim Ej

of

- j'

det

0

s,t

(P)(x',

dim F . )

J

I

0 ; 5'

, 5") ,

with, i n t h e

c a s e n=2, a r e s t r i c t i o n analogous t o t h a t i n d i c a t e d p r i o r t o t h e statement o f P r o p o s i t i o n 3.3.

(SEC. 6 )

c)

ADDITIONAL NOTES

315

Consider t h e p r i n c i p a l symbol b of B:

b =

- k (Bk))k,o

I...I

and t h e r e s t r i c t i o n

I

i; of

b

"

t o Image

0

0

Thus b i s a morphism o f Image u ( C ) i n t o G.

(C).

0

N

I t can be shown t h a t t h e p r o p e r t y of b b e i n g i n j e c t i v e o r s u r j e c t i v e depends n e i t h e r on t h e i n t e g e r m n o r on t h e f i e l d v chosen,

d)

.. lan

For j = 1,. ,J l e t m! denote t h e s m a l l e s t i n t e g e r f o r

which

bu]

m'

can be e x p r e s s e d by means o f t h e t r a c e s

..

j = 1,. ,J and 0

y u . such t h a t

k J assumption

J

-4

+

X(x')

.

\

and

are

< X'(X') T[x')

Show t h a t t h e L o p a t i n s k i d e t e r m i n a n t of

(6.8.1) i n t h e chart

b)

+{

i a v

i s equal t o

Deduce from t h i s t h a t t h e boundary-value problem (6.8.1)

is a l v a y s e l l i p t i c when n = 2 , and t h a t for n

2

3 it i s

e l l i p t i c i f and o n l y i f t h e v e c t o r v ( x f ) i s nowhere t a n g e n t i a l to

c)

an.

i s bounded; l e t u = Kh denote t h e unique

Suppose t h a t

s o l u t i o n o f t h e D i r i c h l e t problem

Show t h a t

u

i s a s o l u t i o n of t h e oblique d e r i v a t i v e

318

(CHAP. 5 )


problem

i f and only i f

T

where

=

1i A av

O

K

i s a ( s c a l a r ) p . d . 0 . of degree 1 i n

an, of which t h e p r i n c i p a l symbol, following t r a n s p o r t by a l o c a l chart

an, i s

I n v e s t i g a t e t h e problem ( 6 . 8 . 1 ) i n t h e case n = 2 ,

d)

Q = ' ~ =z x p

of

xl,&_yv

E

+ iy E

\z\ =

zp f o r z

E

an, where

More p r e c i s e l y , c o n s i d e r t h e

d e f i n e d by

Suppose we have

u E

C"(3)

~ = b ; = ~ () 2. - i 3U

'g i f and only i f we have

with r e a l values.

Put

Show t h a t u i s i n t h e k e r n e l of

(SEC. 6 )

319

ADDITIONAL NOTES

u E cmc2 u

holonorphic i n

R e (zPU(z)) = 0


dim Ker p = 2

+ 2i

zP~(z) = a-P

0 ,

when p = 0 .

e n t i r e - s e r i e s e x m n s i o n of U.

121

= 1

d i m Ker p = 1

I n t h e c a s e p < 0 , l e t U( z ) =

for 0
0 ;

we d e f i n e t h e s e t s

~ ~ = { x ~ ~ ~ ~ I m fa i x } ; j < x = ~{ xl ~ ~ + i x ~ I r } ; j f o r r s u f f i c i e n t l y small, we have t h e i n c l u s i o n s

p;CB r C V . The Cauchy-Kovaleski Theorem 1 . 5 and Remark

1 . 4 show t h a t

it i s p o s s i b l e t o f i n d an r 0 s u f f i c i e n t l y small t h a t f o r any h

E ]

0, ro

[

Cauchy problem

and for any polynomial

q(xl,,.,,

x

n-1

1

the

(SEC. 1)

(1,10.4.)

THEOREMS

?(x,

DX)v

=

0

admits an a n a l y t i c s o l u t i o n

v

335

in

Br

i n Br.

X

T

L

we a p p l y Green's formula i n t h e compact domai

n

6;:

I

then t a k i n g account of (1.10.3)and (l.lO.b)y we o b t a i n

where C ( x ' ) i s a f u n c t i o n which depends o n l y on t h e c o e f f i c i e n t s of P and which does n o t v a n i s h .

Equations (1.10.3) and (1.lO.b) imply t h a t t h e l e f t - h a n d s i d e of (1.10.5)i s e q u a l t o z e r o ; for

lx']

4 r h

since

q

consequently u ( x ' , h ) = 0

i s an a r b i t r a r y polynomial;

this

, It i s t h e r e f o r e s u f f i c i e n t t o t a k e

shows t h a t u = 0 i n '.6 rO

W = B r . 0

REMARK 1.11:

The d e t e r m i n a t i o n of ro shows t h a t t h e open

s e t W depends o n l y on t h e open s e t Y , on t h e r a d i u s of converg-

336

(CHAP. 6 )

EVOLUTION EQUATIONS

ence of t h e Taylor s e r i e s expansions a t x

0

of t h e c o e f f i c i e n t s

of P and o f J, and on t h e modulus of t h e c o e f f i c i e n t s as i n Remark 1 . 4 .

REMARK 1 . 1 2 :

I n a l l t h e c a s e s f o r which t h e r e i s an e x i s t -

ence theorem f o r problem (1.10.4),t h e above proof shows t h a t t h e r e i s a l s o a uniqueness theorem (1.10.1).

I n t h e c a s e where P has c o n s t a n t c o e f f i c i e n t s and S i s a Theorem 1.10 and Remark 1 . 8 show t h a t i n

hyperplane,Holmgren's

o r d e r t o have l o c a l uniqueness, it i s n e c e s s a r y and s u f f i c i e n t t h a t S be n o n - c h a r a c t e r i s t i c .

We t h e n a l s o have g l o b a l unique-

ness :

Suppose P(D) is a d i f f e r e n t i a 2 operator

PROPOSITION 1.13:

with constant c o e f f i c i e n t s ;

is qon-characteristic for P. and supp u c

Pu = 0 i n

PROOF: I ) with

suppose t h a t the hype2rpZane xn = 0

Let

8" +

.

u

E .P(R"]

be such t h a t

Then u = 0.

Let p k be a r e g u l a r i s i n g sequence ( s e e (l.l.l), Chapter

p

,

E):R(:C -n

w i t h support i n IR,,

u

The f u n c t i o n

and s a t i s f i e s

P

\

k

= u

*

= 0 in

Pk

.

co

is C

,

A s P has

c o n s t a n t c o e f f i c i e n t s , Holmgren's Theorem 1.10 and Remark 1.11 imply t h a t

\

= 0 and consequently t h a t

u

=

lim k m

uk

=

0

,

337

NECESSARY C O N D I T I O N FOR WELL-POSEDNESS

(SEC, 2 )

NECESSARY C O N D I T I O N FOR THE CAUCHY PROBLEM TO BE WELL POSED

2.

The Cauchy-Kovaleski and Holmgren theorems a r e e s s e n t i a l l y l o c a l r e s u l t s ; i n t h i s s e c t i o n we give a necessary c o n d i t i o n f o r the Cauchy problem t o be w e l l posed g l o b a l l y .

x = [xo,

Let

XI

,...,

xn)

;

E R"+'

we p u t x = (xo, X I ) .

We denote by P ( x , D ) a d i f f e r e n t i a l o p e r a t o r of degree m w i t h X

coefficients i n C ~ ( I R ~ + ' ) .

DEFINITION 2 . 1 :

Let T E D ; we say t h a t th e Cauchy prob-

lem i s we l l posed for P i n t h e half-space xo

,

f E C"(R"+')

a wique u

.I

(2.1 .I

E

gj

E Cm[R")

j

P

0,

..,

m-I

2

T, i f f o r any

,

there e r k t s

Crn(Dn'l) such t h a t ~

u

=

f or

x

for

x

0

0

> T

,

= T

and

O;U=

g3

j = 0,

...,

m-I

.

we say t h a t t h e Cauchy probZern i s w e ll posed i n t h e d irection x

2

N = (1,O

..., 0)

if it i s we12 posed in each h a lf space

T.

It i s p o s s i b l e t o o b t a i n another formulation by r e p l a c i n g t h e i n i t i a l c o n d i t i o n s by a c o n d i t i o n on t h e s u p p o r t . s i m p l i f y t h e n o t a t i o n we t a k e T = 0 .

To

338

(CHAP. 6)

EVOLUTION EQUATIONS

Suppose t h a t the hyperplane x 0 = 0 i s

PROPOSITION 2.2 :

noncharacteristic for P.

Then the Cauchy probZem i s well posed

for P i n the half space x 0

2

f o E C”(Rw’)

with

e x i s t s a unique u

0

(2.2 2.)

supp f o c

E C”(R”+’)

P u*

PROOF:

0 i f and only i f for any

-’R+

= (x

1

x

0

2

o

}

there

c

y’.

such t h a t =

and

fo

supp uo

The proof of necessity is immediate, since all the

derivatives of the solution u of (2.1.1)with g = 0, j m-1 are equal to zero on the noncharacteristic j = 0,

...,

hyperplane x0 = 0.

Conversely, Borel’s Theorem allows us to construct a function v E C”(A”+’)

D’ v = 0

g

j

of which the traces on x0 = 0 satisfy

for j = O,..., m-1 and f = Pv 0

-

f flat on xo = 0 .

Then if uo is the solution of (2.2.2), the function uo + v is a solution of (2.1.1).

The problem of giving a general characterisation of operators for which the Cauchy problem is well posed remains open; nonetheless, a large number of partial results exist.

We give

below a necessary condition on the principal part of P.

THEOREM 2.3 : (Lax-Mizohata)

Suppose t h a t for any T

E

IR,


(SEC. 2)

339

the hyperplane xo = T i s noncharacteristic f o r P , and t h a t the

. . , 0).

Cauchy probZem i s well posed i n the d i r e c t i o n N = (1,0,. We t h e n have t h e i m p l i c a t i o n

We s t a r t by g i v i n g t h e f o l l o w i n g lemma:

n+l

Suppose K i s a compact subset of lR

LEMMA 2 . 4 :

; then

there e x i s t s a number C and an i n t e g e r k 2 0 such t h a t

.I

(2.4.1

lul,

with

=

‘v13

5

c IPul,

I

IDcy

v(x)I

sup

for a l l

u E

cK OD

la1 5 j x 5 0 0

m

PROOF:

L e t f E CK ; t h e S e e l e y e x t e n s i o n ( P r o p o s i t i o n

9.2, Chapter I ) of

C;(R”+’)

f- = flRn+,

-

which w e denote by Sf-.

t o B n + l i s a function i n

Suppose V i s a compact neigh-

bourhood of K ; we can f o r c e Sf- t o have s u p p o r t i n V f o r any f . The c o n t i n u i t y of t h e Seeley e x t e n s i o n shows t h a t f o r any int e g e r k t 0 t h e r e e x i s t C and q such t h a t

I~fl,

(2.4.2.)

For u

m

E

5

c 1f-I;

,

aD

f o r any

f

E cK

C K , we p u t f = Pu, g = Sf- and we d e f i n e , w-ith

.

EVOLUTION EQUATIONS

340 E

(CHAP.

6)

> 0, t h e function

fe

ee

=

f

+

-

(I

E

ee)g

c;

9

X

Oe(x) = e(f)

where x

0

9 E Cm(R)

and

t 1 and e q u a l t o u n i t y f o r xo

Let uE E Cm(TRn+’) =

P U

(where

,

f

E

E

inf

=

a

i s e q u a l t o zero f o r

0.

be t h e unique s o l u t i o n of

c

suppue

lxD.a}

(x

=

H

a

.

xg)

X E V

Since f

n+l = f onB, we have u

E

E

= u onIRy+l and

(2.4.3.)

By h y p o t h e s i s , P i s a continuous b i j e c t i o n of t h e Fr6chet space E

{u I

=

u E

c”(R”+’),

supp u

c H~

1

; consequently

t h e r e e x i s t C , an i n t e g e r k and a compact s e t K’ such t h a t

(2.4.4.)

CY

1u(x)1 S

sup

1.1

xEK

C

( D Pu(x)i

,

f o r any

u E E,

5 k x E K’

Applying ( 2 . 4 . 4 ) t o uE and t a k i n g account of ( 2 . 4 . 3 )

, we

obtain

]uI,

(2.4.5.1

I

c

Furthermore, we have function

(7

c

- g)

ee(f

-

m

Ifelk

,

Ifelk

-

for any u

Igik

E cK ,

e > 0.

since the e‘o g) has support i n t h e s t r i p

(SEC. 2 )

0 5

x0

341


i s f l a t f o r x0 = 0.

5 E and

We t h e r e f o r e deduce from

(2.4.5) that

14;

c

5

Id,

8

and t a k i n g account of ( 2 . 4 . 2 ) we f i n a l l y o b t a i n

I n o r d e r t o prove t h e theorem, we c o n s i d e r t h e p o i n t x ? 0 and we use a method adapted from I V R I I HORMANDER

-

PETKOV [ll

and

C71.

Let Co

E

C be such t h a t

If I m 5 ,

We denote i t s m u l t i p l i c i t y by r .

# 0 , we can

assume for example t h a t I m 5, < 0 , s o t h a t we n e c e s s a r i l y have

5 ' # 0.

The v e c t o r

plane x

0

(Re Go,

5')

= 0 ; hence we can apply a l i n e a r change of v a r i a b l e s

which p r e s e r v e s t h e v a r i a b l e x

0

into (0,

..., 0 ,

5 ' = en = ( 0 , (2.4.6.)

i s t r a n s v e r s a l t o t h e hyper-

1).

..., 0 , P,,,(o;

and which transforms (Re

We t h e r e b y reduce t o 5 , =

5 0' 5 ' )

- i,

1) and we have

D,,

0

,..., D,,)

where Q i s homogeneous of degree m

=

-

[ D + ~ i 0")'

Q ( D ~ , D,,]

r and Q ( - i , 1) # 0.

I n o r d e r t o l o c a l i s e t h e o p e r a t o r a t x = 0 and t o g e t r i d

o f terms o f lower o r d e r , I v r i i and Petkov i n t r o d u c e t h e change

(CHAP. 6)

EVOLUTION EQUATIONS

342

of v a r i a b l e s Yj

j = O

= x j P' j

,.,., n

,

with

p > O

,

sj>0

,

The o p e r a t o r P(x, D ) is transformed i n t o X

The i n e q u a l i t y ( 2 . 4 . 1 ) i m p l i e s t h a t f o r any compact s e t

K there exists

po

1

where S = max s

j

such t h a t

j'

The i d e a behind t h e proof c o n s i s t s of c o n s t r u c t i n g an asymptotic s o l u t i o n u

P

of

pP "p

=

o(p- ")

i n a neighbourhood

of 0 s o as t o c o n t r a d i c t t h e i n e q u a l i t y ( 2 . 4 . 7 ) .

s = 2r

3

if

j = I,..., n-I

and

so = sn = 4 r

then

We seek an asymptotic s o l u t i o n i n t h e form

w i t h v o ( 0 ) = 1.

We p u t

;

(SEC , 2 )

343


Expression ( 2 . 4 . 6 ) l e a d s u s t o t a k e f o r t h e phase f u n c t i o n cp(y) =

(o0 + ID,^

-

- iyo

+

, since

y,

O0cp, on v'l

0

Cp

this satisfies

f

=

; moreover

0

With t h i s choice o f c p t h e asymptotic expansion of P

-

s t a r t s w i t h a term of degree m

q(Do

l e a d s t o t h e equation sufficient t o take v

E

Cm

O

of 0.

+

the

P

r, and s e t t i n g t h i s t o zero

c On)

r

vo(y)

=

0

*It is

e q u a l t o u n i t y i n t h e neighbourhood

K1

The equations which f o l l o w a r e of t h e form

r

q ( D o + i On)

(2.4.8.) where F

j

E

=

Vj(Y)

K1 i s determined from t h e v o ,

Cm

Fj(Y)

. . , vj-1'

We s o l v e ( 2 . 4 . 8 ) i n t h e neighbourhood of 0 by convolution with t h e fundamental s o l u t i o n ( 6 . 1 . 3 ) of Chapter I of t h e Cauchy-Riemann o p e r a t o r

i(D 2

0

+ i On)

.

We can t h u s f i n d an

i n t e g e r J such t h a t t h e p a r t i a l sum

J

J

juo

satisfies

f o r p s u f f i c i e n t l y l a r g e ; t h i s c o n t r a d i c t s ( 2 . 4 . 7 ) and concludes t h e proof of t h e theorem.

REMARK 2.5

:

A completely analogous theorem e x i s t s f o r

344

EVOLUTION EQUATIONS

(CHAP. 6 )

d i f f e r e n t i a l systems, it b e i n g s u f f i c i e n t t o r e p l a c e P by t h e m determinant of t h e p r i n c i p a l p a r t i n c o n d i t i o n ( 2 . 3 . 1 ) ( s e e IVRII-PETKOV c11).

REMARK 2 . 6 :

Apart from t h e c a s e of o p e r a t o r s w i t h con-

s t a n t c o e f f i c i e n t s , a c h a r a c t e r i s a t i o n of o p e r a t o r s f o r which t h e Cauchy problem i s w e l l posed i s known o n l y i n a number of special cases.

For example, i f we assume t h a t t h e r o o t s a t

5, of ( 2 . 3 . 1 ) a r e of c o n s t a n t m u l t i p l i c i t y , we have a c o n d i t i o n which i s n e c e s s a r y (FLASCHKA-STRANG c11) and s u f f i c i e n t (CHAZARAIN [I]) and which i n v o l v e s lower-order terms of P , i . e .

t h e Levi c o n d i t i o n .

If we suppose t h a t t h e r o o t s a t 5

0

a r e of

m u l t i p l i c i t y l e s s t h a n o r e q u a l t o 2 , a g a i n we have a number of v e r y complete r e s u l t s ( s e e IVRII-PETKOV [ L ] , HORMANDER

[TI).

When t h e c o e f f i c i e n t s are a n a l y t i c , BONY and SCHAPIRA 111 have shown t h a t c o n d i t i o n

( 2 . 3 . 1 ) i s s u f f i c i e n t f o r problem ( 2 . 1 . 1 )

t o admit a unique a n a l y t i c s o l u t i o n when t h e d a t a a r e a n a l y t i c .

I n S e c t i o n 4 , we s h a l l show i n p a r t i c u l a r t h a t i f t h e r o o t s

a t 5, are simple, t h e n c o n d i t i o n ( 2 . 3 . 1 ) i s n e c e s s a r y and sufficient.

3.

HYPERBOLIC OPERATORS WITH CONSTANT COEFFICIENTS I n t h e c a s e where P h a s c o n s t a n t c o e f f i c i e n t s , Ggrding

(SEC. 3 )

HYPERBOLIC OPERATORS

345

has given an a l g e b r a i c c h a r a c t e r i s a t i o n of t h e o p e r a t o r s P(D)

for which t h e Cauchy problem i s w e l l posed.

The s u b j e c t i s

d i s c u s s e d i n depth i n HORMANDER [11; however, for completeness, we s h a l l reproduce a p a r t of t h i s d i s c u s s i o n h e r e , s i n c e c e r t a i n

4 i n connection with quest-

r e s u l t s w i l l be needed i n S e c t i o n

i o n s r e l a t i n g t o speed o f p r o p a g a t i o n .

Throughout t h i s s e c t i o n , we s h a l l be i n En.

Assume t h a t the Cauchy problem i s

PROPOSITION 3 . 1 :

well posed f o r the operator P(D) i n the d i r e c t i o n N Then there e x i s t s a r e a l number y

+

[3.1.1.) P ( S

7

PROOF:

#

0 for

En \ 0.

such t h a t

5 E R",

T

E C and I m

7

< Yo.

The g e n e r a l i t y w i l l n o t be r e s t r i c t e d i f we t a k e

.., 0 ,

N = (0,

N]

0

E

1).

The argument used i n proving (2.4-.4)

a p p l i e s p a r t i c u l a r l y i n t h i s c a s e ; consequently, for any compact s e t K t h e r e e x i s t C , an i n t e g e r k and a campact s e t K' such that (3.1.2.)

lU(xj1 I

sup

x

EK

c

ID~(P(D)

sup

I4 s x

E K'

and

Let

T E

put 5 = ( 5 '

,

C be such t h a t P(S', T)

E C~(R"I

~(X)II

k

7)= 10 with

supp u

5'

E

C

En-',

; we s h a l l show t h a t t h e r e e x i s t s yo

xy and

.

346

EVOLUTION EQUATIONS

independent o f 5 ' such t h a t suppose t h a t I m T 5 0.

Im

T

x

n

E

Cm(R)

We can t h e r e f o r e

R:,)

t o t h e function

etx,')

i s e q u a l t o u n i t y for

Since P(D) u(X) = 0 f o r

2 0.

E

e i(x-a).6

u(x)

0

6)

To show t h i s , we apply t h e i n e q u a l i t y

( 3 . 1 . 2 ) ( w i t h K reduced t o a p o i n t a

where

.

Yo

2

(CHAP.

x

n

x

,

an

and zero for

2~

2 -an

and u ( a ) = 1,

2

we o b t a i n

1

c

(1 +

1611k+m e& a,,.Im

T I

or, w i t h r e a l c o n s t a n t s b and c , (3,1,.3.)

b Log(1

+ /GI) + c

Im

T

,

when

P(5',

T)

= 0,

However, a r e s u l t due t o Seidenberg-Tarski ( s e e HORMANDER

[11, Appendix) shows t h a t t h e l o g a r i t h m i n ( 3 . 1 . 3 ) may be replaced by a c o n s t a n t , and t h i s proves t h e p r o p o s i t i o n .

This p r o p o s i t i o n l e a d s t o

DEFINITION 3.2 :

An operator with constant c o e f f i c i e n t s is

said t o be correct i n the sense o f Petrovski i n the direction N = (0,

i/

..., 1) i f

the c o e f f i c i e n t of the term of highest degree i n En i n

P ( 5 ) i s independent o f 5 '

ii/ condition (3.1.1) i s s a t i s f i e d for a certain y o .

(SEC. 3 )

347


EXAMPLES :

The h e a t o p e r a t o r

- Ax'

3%

The SchrBdinger o p e r a t o r

the direction N.

i s correct i n Q

Xn

-A

X'

i s c o r r e c t i n t h e d i r e c t i o n s N and -N.

Remark 1 . 8 shows t h a t i n g e n e r a l we cannot expect t o have uniqueness o f t h e Cauchy problem for a c o r r e c t o p e r a t o r ; nonet h e l e s s w e have

PROPOSITION 3.3 :

Suppose P is a correct operator in the

d i r e c t i o n N; then it adnits a fundamentaZ s o l u t i o n E with

-

support i n R:

.

PROOF:

By h y p o t h e s i s , t h e polynomial P ( E ' , T ) f a c t o r i s e s P Const. XI (7 Aj(I')) , consequently t h e r e e x i s t s

into

-

j=1

c > 0 such t h a t

(3.3.1.)1

~P(T,Y I I

2

c IIm

7

- Yojp

I

for and

7

5' E '-?I

-< yo

.

This i n e q u a l i t y allows us t o d e f i n e a d i s t r i b u t i o n F E

a'

348

EVOLUTION EQUATIONS

6)

.

1

if E = F

and t h e r e f o r e P(D) E =

(CHAP.

I n o r d e r t o show t h a t Cp E CR (:]: Cp

in (3.3.2).

supp

F

C

- ,

Rn

we t a k e

The Paley-Wiener theorem a p p l i e d t o

i n c o n j u n c t i o n with t h e Cauchy theorem, shows t h a t expression

( 3 . 3 . 2 ) i s independent of y f o r y < y

0

and t h a t t h e r e e x i s t s a

c o n s t a n t C such t h a t

(3.3.3.) where

.1

F, 'P =

5

inf

..

x t SUPP cp upper bound ( 3 . 3 . 3 ) shows t h a t

-

uently t h a t

supp E

REMARK 3.4 :

C

< F,

~p

> = 0

and conseq-

.

We say t h a t P(D) i s an e v o l u t i o n o p e r a t o r i n

t h e d i r e c t i o n N i f it admits a fundamental s o l u t i o n with n support inIR+.

An o p e r a t o r which i s c o r r e c t i n t h e sense o f

P e t r o v s k i i s an e v o l u t i o n o p e r a t o r ; t h e converse does not h o l d , however, and we r e f e r t o HORMANDER

C41

f o r a c h a r a c t e r i s a t i o n of

evolution operators.

Remark 1 . 8 and P r o p o s i t i o n 3 . 1 show t h a t a n e c e s s a r y c o n d i t i o n f o r t h e Cauchy problem t o be w e l l posed f o r P(D) i n t h e d i r e c t i o n N i s t h a t P be h y p e r b o l i c i n t h e sense of t h e following d e f i n i t i o n :

(SEC. 3 )

349


DEFINITION 3.5

We say t h a t P(D) i s hyperbolic i n the

:

direction N i f

(i)

Pm(N) # 0

(ii)

The condition (3.1.1) i s s a t i s f i e d f o r some yo

.

The l a s t p a r t of t h i s s e c t i o n w i l l be devoted t o proving t h a t t h i s c o n d i t i o n i s s u f f i c i e n t ; t o t h i s end, we r e q u i r e c e r t a i n a l g e b r a i c r e s u l t s r e l a t i n g t o hyperbolic o p e r a t o r s .

F i r s t , we have:

PROPOSITION 3.6 :

Suppose P(D) i s a hyperbolic operator

i n the d i r e c t i o n N; then i t is hyperbolio i n the d i r e c t i o n

-

N

and its principal p a r t Pm (D) is hyperboZio i n the d i r e c t i o n N.

PROOF: P(5 +

Since Pm (N) # 0, t h e sum of t h e r o o t s i n

-T

of

N) i s an a f f i n e f u n c t i o n of 5 of which t h e imaginary

T

p a r t i s bounded above by m y

*

0,

it i s t h e r e f o r e c o n s t a n t .

Consequently, t h e r e e x i s t s some y such t h a t P(S+ T

N) f 0

for

(5,

7)

E R“

x C

and ]Im 71

Y

and t h i s proves t h e f i r s t a s s e r t i o n .

+x

-

PI(, + T N) ; h e + = now t h e r o o t s i n T of t h e l e f t - h a n d s i d e a r e l o c a t e d w i t h i n a

We have

strip

1Im TI

I””

3

T

N)

; therefore the roots i n T

of t h e l i m i t

(CHAP. 6)

EVOLUTION EQUATIONS

350

are real.

rn of P i s h y p e r b o l i c , t h i s does not

If the principal part P

i n g e n e r a l imply t h a t P i s i t s e l f h y p e r b o l i c ( s e e SVENSSON [11); however, we have

Suppose t h a t a12 the r o o t s of the

PROPOSITION 3.7 :

equation i n (3.7.1

.]

:

T

Pm(l+

T

N)

=

0

are r e a l and d i s t i n c t for 5 n o t proportiona2 t o N. P(D) is hyperbolic

Then

We say t h a t it i s

in the d i r e c t i o n N .

s t r i c t Zy hyperbo l i c .

a2 xn

For example, t h e wave o p e r a t o r hyperbolic i n t h e d i r e c t i o n ( 0 , proposition, we t a k e N = ( 0 ,

LEMMA 3.8 :

If

P

,

. ..,O,

PROOF:

.)I

2

c IIm

Let X h ( E ’ )

factorising P

m’

we o b t a i n

is strictly To prove t h e

1).

1);we s t a r t by f i r s t proving

is s t r i c t l y hyperbozic in the

d i r e c t i o n N , then there e x i s t s c (3.8.1.) I P ( S * ,

. .. , 0 ,

Ax,

TI

(

> 0

such t h a t

ISI +

be t h e r o o t s i n

171)”+’

T

of

m (s,,

P

7) =

0 ;


(SEC. 3 )

Gh(E1,

where

7)

n

=

(7

- Xj(c'))

351

a

and

i s a non-zero

j# h

constant. m

IGhCI1l

The f u n c t i o n

T)I2

does n o t vanish f o r

h=l

( I 1 ,7) E

(R"" x

c )\

0 and i s homogeneous of degree 2(m - 1);

t h e r e t h e r e f o r e e x i s t s c > 0 such t h a t

We now r e t u r n t o t h e proof of t h e p r o p o s i t i o n . P(g',

7) =

bound

pm(!',

lQ(sl,.)I

( 3 . 8 . 1 ) shows t h a t

+ Q(E', 7) Since we C [ltll + lT1)w' I the

7)

I

P[%',

7) f Owhen

1 I m 71

We put

have an upper inequality 2

y

for y

sufficiently large.

N a t u r a l l y , P r o p o s i t i o n 3 . 3 a p p l i e s f o r hyperbolic opera t o r s ; we s h a l l show f u r t h e r t h a t E has support i n a cone of

Q J E o 1

.

(CHAP. 6)

EVOLUTION EQUATIONS

352

Suppose P is a hyperboZic operator i n the

DEFINITION 3.9:

we denote by r(P, N) the connected component o f N

domain N ; the open s e t

c

5

1

5 E R"

and P,(S)

f

]

0

in

.

We s h a l l now show t h a t P i s a l s o h y p e r b o l i c i n each d i r e c t i o n of We s t a r t w i t h t h e following lemma:

r(P, N).

LEMMA 3.10:

r(P, N) i s equaZ t o the s e t o f the 6

such t h a t the poZynomiaZ Pm(5 + roots in

T.

PROOF:

Let

r'

and by d e f i n i t i o n

I n o r d e r t o show t h a t

r'

i s connected. T/ =

then

is+p

r'

C

t h i s s e t i s c l e a r l y open

5 E

for P

,

r1

and f u r t h e r , N

numbers which sum t o u n i t y .

E

and

A

p

5E

are positive real

By h y p o t h e s i s

we have t h e

factorisation

and consequently t h e p o l y n o d a l

P,(v

+

T

N)

= P,(ht

admits t h e r o o t s

h T

+PN +T J

-)z

r N)

< 0

.

= P,(N)

II (p + 1

T

-h

Tj]

F i n a l l y , i f 5 belongs t o t h e boundary of TI, t h e r o o t s of P (5 + m .

T

N) a r e n e g a t i v e or zero and t h e r e e x i s t s a t l e a s t one

zero r o o t s i n c e 5

4 rl.

.

r'

it remains for us t o prove t h a t

More p r e c i s e l y , we s h a l l show t h a t i f

N E ri where

En

N) has onZy s t r i c t l y negative

T

be t h i s l a t t e r s e t ;

Pm(c) 4 0

E

Hence P,(5)

P

0

which proves

re,

(SEC. 3 )


353

t h a t 5 a l s o belongs t o t h e boundary o f r ( P , N ) and consequently

rl

6

.

q p , N)

The cone r(P, N) depends only on P,;

however we have

Suppose P i s hyperboZic i n the direction N.

THEOREM 3.11:

~ e t 9 E q p , N) ; -then (3.11.1.)

P(5+ T

N

+ P?)

f 0

andImb S PROOF: i e n t of

is

0

7

< yo

.

, ZER"

= 0, t h i s i s condition (3.1.1).

For Im

urn

Im

for

f

P,(n)

0 ;

The c o e f f i c -

consequently t h e number of zeros

p such t h a t I m y < 0 i s c o n s t a n t when t h e parameter T remains i n

t h e h a l f plane I m T equation

P(5 +

(3.11 2.)

7

N

However, f o r

yo.

+ PI)

T - ~P[{

which converges t o P,(N

0

+ +

T(N

A?)'=

Lemma 3.10 shows t h a t

P(,N

'I

l a r g e and p = AT, t h e

i s equivalent t o

+ An))

= 0

0 vrhen Im

T +

+

0

I?)

f

-

m

.

f o r Re h > 0 ,

consequently, t h e same a p p l i e s f o r (3.11.2)when I m

'I

< Y o , and

t h i s proves t h e theorem. A s an a p p l i c a t i o n , we deduce

THEOREM 3 . 1 2 :

I f P i s hyperbolic i n the direction N, it

i s hyperbolic i n my direction of r(P, N) and, moreover, t h i s

(CHAP. 6 )

EVOLUTION EQUATIONS

354 cone is convex.

q E

Let

PROOF:

r ( P , N)

hyperbolic i n t h e d i r e c t i o n

;

we s t a r t by showing t h a t P i s N with

q -k E

E

In fact,

> 0.

Theorem 3.11 i m p l i e s

~ ( +5 ~ ( +7

#

e N))

o

and Lemma 3.10 shows t h a t ing

1 = (51

by t a k i n g

E

-e

+

N)

e

N

Irn T c inf(0, y o

-58

for P,(q

, we

+

e N)

#

0

for

t h e n deduce t h a t

> 0 small enough t o ensure t h a t r-

-

E

E

, 5 > 0.

E Writ-

‘Q € r ( p , N) N remains i n

t h e open s e t r(P, N ) . To prove t h a t t h e cone r(P, N ) i s convex we show t h a t t h e

segment j o i n i n g two of i t s p o i n t s qI and q2 remains i n i t s i n t erior.

I n f a c t we showed d u r i n g t h e proof of Lemma 3.10 t h a t

t h i s segment is i n s i d e t h e cone

r[p,

7,)

= r ( P , N)

An analogous r e s u l t holds for s t r i c t l y hyperbolic o p e r a t o r s ; this is: THEOREM 3.13: N , then

I f P is s t r i c t l y hyperbozic i n the direction

it is s t r i c t l y hyperbolic in any direotion o f the erne

r(P, N).

PROOF:

For 5

f a c t o r i s a t ion

(3.13.1 .)

E

IRn and not p r o p o r t i o n a l t o N , we have t h e

(SEC. 3 )

where t h e


a r e Cm f u n c t i o n s which a r e mutually d i s t i n c t f o r

T.(*)

g E R" \

355

J

(mN)

the single root

#

, Since P,"] T =

-

h when

, the

0

5 t e n d s towards

J

E r(p,

N)

, not

J

consequently,

h N;

.

t h e f u n c t i o n s T . (5) a r e continuous on lRn Suppose we have

r o o ts T . ( C ) tend t o

p r o p o r t i o n a l t o N , and i-1

E

IR ;

we deduce from ( 3 . 1 3 . 1 ) :

(3.13.2 .) We now show t h a t

The homogeneity and t h e c o n t i n u i t y of t h e f u n c t i o n s

T

j

show t h a t

which proves ( 3 . 1 3 . 3 ) s i n c e Lemma 3.10 t e l l s us t h a t t h e

T

. (TI)

J

are s t r i c t l y negative. The a s s e r t i o n ( 3 . 1 3 . 3 ) and t h e c o n t i n u i t y of t h e f o r any j t h e r e e x i s t s p

j

( 3 . 1 3 . 2 ) shows t h a t t h e p

j

a r e r o o t s of

now only remains t o show t h a t t h e y

5 i s not proportional t o p

j

= pk = y with

Tj(h

X

E

+ pll) IR

.

=

~

J

q.

j

P,(5

imply t h a t

j

+ pJ. 7)

T.(S

such t h a t

T

= 0

+ p7)

, and

= 0

.

It

a r e d i s t i n c t i n p a i r s when

In f a c t , i f there existed

j and k d i s t i n c t , we would have

~ + p7) ( 5= 0

Then t h e e q u a l i t y

shows t h a t A = 0 , 2 . e . t h a t

and consequently

P,[s

s + pl"rl

+ pl"rl) 0

=

'5 +

Pl"rl = h N w i t h

0 = Am p,")

, which

i s a contrad-

(CHAP. 6)

EVOLUTION EQUATIONS

356

iction. We a r e now i n a p o s i t i o n t o prove t h e p r i n c i p a l r e s u l t of t h i s this is

section;

THEOREM 3.14:

direction N ;

Suppose P i s a hyperbolic operator i n the

then P admits a unique fimdamentaZ s o l u t i o n w i t h

support i n the dual cone

r*p, N) PROOF:

=

c x E R" I

x.5

2

o for

rp,

~ Z Z5 E

N)

3

Since P i s , i n p a r t i c u l a r , c o r r e c t i n t h e sense of

P e t r o v s k i i n t h e d i r e c t i o n N , P r o p o s i t i o n 3.3 shows t h a t i t

-

admits a fundamental s o l u t i o n E w i t h support i n By and defined

bY

with

Re y

< y,

.

Theorem 3.11 i m p l i e s t h a t t h e f u n c t i o n

i s holomorphic i n t h e open s e t the fact6risation

shows t h a t

Rn

- i r(P,

N)

.

Furthermore,

(SEC. 3 )


357

(3.14.2.) when

Re y

.

< y,

The Paley-Wiener theorem a p p l i e d t o cp

implies t h e e x i s t e n c e of C such t h a t exp HJYN

IG(g + iy

(3.14.3.)

,

5 E R"

for

X

+ i ~ 7 1 15

N

and y r e a l .

c

+ 17)

(151 + I Y l +

I~lIW1

We deduce from ( 3 . 1 4 . 2 ) ,

( 3 . 1 4 . 3 ) t h a t we can apply a t r a n s l a t i o n of t h e contour of i n t e g r a t i o n i n ( 3 . 1 4 . 1 ) and o b t a i n

y v v ,] for

369

HYPERBOLIC CAUCHY PROBLEMS

lipj v f

o

>

J

S t r i c t l y hyperbolic s c a l a r operator of

o r d e r m.

Suppose we have

m-I

(4.8.1 .)

where

P ( t , x, Dt'

AWj

Dxl

= a w j (t, x, D,)

suppose t h a t a

=

- j=o

D;

with e

m-j

:D

€ss"s(R"l

x

R"]

0

5 E R" \ 0

and which

i s independent of ( t, x) for 1x1 s u f f i c i e n t l y l a r g e .

The Cauchy problem f o r P i s w r i t t e n f o r m a l l y as f o l l o w s :

(43.2.) PU = f

,

k Dt ulk0

=

k

gk

01.m.s

-1

We s h a l l now reduce t h i s t o a f i r s t - o r d e r system; t o do t h i s , we p u t u

= D5-1 A,,+~ u and j Then, ( 4 . 8 . 2 ) i s e q u i v a l e n t t o

u

;

a (t, x, 5) m-J

admits a p r i n c i p a l p a r t

m- j which i s homogeneous of degree m-j f o r

Am-j

=

(ul,

..., ),u

370 l4.8.3.

EVOLUTION EQUATIONS

)

DtU-AU

=

F

(CHAP.

1

r

U t=o

=

6)

G

where

with

A

B = j

F = G = Consequently B . i s a p . d . 0 . .3

a It, X I *j+l

o f degree 1 w i t h symbol

14(wj) and

11(1 + 151

A h a s as homogeneous

p r i n c i p a l symbol o f degree 1,

It can e a s i l y be shown t h a t p,(t,

x, T,

5)

where p

m

det(23

- Al(t,

X,

5))

=

is t h e p r i n c i p a l symbol of P.

The above a l l o w s us t o s t a t e t h e f o l l o w i n g :

THEOREM AND DEFINITION

4.9

:

The operator P defined i n

(4.8.1) i s said t o be s t r i c t l y hyperbolic i f for a l l (t, x, EJ E '"R

x

[R"

\

03

the zeros i n

of i t s principal

(SEC. 4)

symbol

371


x,

p,(t,

~j =

T,

-

7’

0

aWJ [t, x,

5)

are

7’

J=O

real and d i s t i n c t .

We assume i n addition t h a t p is m independent of ( t , x) f o r 1x1 s u f f i c i e n t l y large. Let

,

sE R

f E

Lz([ 0 T

1 ; H’),

j = 1,

g j E H’+*j

..., m.

Then there e x i s t s a unique m-I

n

E

k = o

ck([ o

T

3;

H

such t h a t Pu = f i n & ( ] 0 T [ x R”) Furthemnore, there e x i s t s C

~

,

1

D$’

~

~

~ ( 0 .) ,

=

gd

such t h a t

IT

for a l l

s a t i s f y i n g the above assumptions and

f,gj

.

t E [ O T ]

FinaZZy, i f

have

u E C”([

PROOF:

E C”([

f

0T

3;

Hs”)

0T

.

3;

Hs”)

and

g5

E H*

we

It i s s u f f i c i e n t t o a p p l y Theorem 4.5 t o t h e

system ( 4 . 8 . 3 ) .

The s t a n d a r d example o f a s t r i c t l y h y p e r b o l i c o p e r a t o r i s t h e wave o p e r a t o r

P

~i

2 Dt

-c

2

(X)

4(

where

c

is a strictly

m

positive C large).

f u n c t i o n (which i n t h i s c a s e i s c o n s t a n t f o r 1x1

P h y s i c a l l y , c ( x ) i s i n t e r p r e t e d as b e i n g t h e wave

propogation speed.

We s h a l l show more g e n e r a l l y t h a t we

.

~

372

EVOLUTION EQUATIONS

(CHAP.

6)

have t h e n o t i o n o f a propagation speed when P i s a s t r i c t l y hyperbolic d i f f e r e n t i a l o p e r a t o r .

Consider a d i f f e r e n t i a l o p e r a t o r of degree m, D ) , with C

P = P ( t , x, Dt, of IRn+l.

m

c o e f f i c i e n t s i n an open s u b s e t S2

X

Let p

m

be i t s p r i n c i p a l p a r t ; saying t h a t P i s

s t r i c t l y hyperbolic w i t h r e s p e c t t o s a y i n g t h a t f o r a l l ( t , x) P ~ , ~ ( T ,=~ p], ( t , t o N = (1; 0).

x, 7 ,

E

(see Definition 3.9).

i s equivalent t o

S2 t h e polynomial

5) is

We denote by

t

s t r i c t l y hyperbolic with respect

r(t,

x ) t h e cone T(Pt,xl

N]

We s t a r t by g i v i n g a d e f i n i t i o n of t h e

hypersurfaces which w i l l p l a y a r o l e analogous t o t h a t o f t h e hyperplane t = 0 .

DEFINITION 4.10

:

1 in IRn+l A hypersurface S of c l a s s C

i s said t o be spacelike f o r P a t a p o i n t ( t , x) covectors normal t o

s

a t ( t , x) are i n

r(t,

X)

E S

if the

u (- r(t,

x))

I t i s said t o be spacelike f o r P i f t h i s i s true a t a l l points.

,

(SEC. 4)


373

We n o t e t h a t w i t h t h i s d e f i n i t i o n , Theorem 3.13 shows that p

t YX

i s s t r i c t l y h y p e r b o l i c w i t h r e s p e c t t o any non-zero

conormal o f S a t ( t x),

This w i l l form t h e b a s i s of t h e

f o l l o w i n g l o c a l uniqueness r e s u l t .

DEFINITION 4.11 : of degree

m

Suppose P i s a d i f f e r e n t i a l Operator

with Cm c o e f f i c i e n t s which i s s t r i c t l y hyperbolic 0

in a neighbowhood V of a p o i n t ( t o y x 1. surface of '?.

L e t S be a Cm hyper-

Suppose t h a t s is spacelike for P a t ( t o y xo).

Then there e x i s t s a neighbowhood W of ( t o yxo) such t h a t i f u E we

P[v)s a t i s f i e s

have u = 0 i n

PROOF:

Let

bourhood of (to,

~u =

o

in

v

,

D~

uIs =

o

Icy]

Imi

w.

x X

0

be a diffeomorphism d e f i n e d i n t h e neigh-

1 which

transports S into

,

374

(CHAP. 6)

EVOLUTION EQUATIONS

L1

3 I r =0 1.

('E;

S = {

r )

and (to, xo) into 0.

Let P be the op-

erator obtained by transporting P. Now ( 7 . 3 . 5 ) , Chap. I, shows

(7, 3

that

(7, 5 ) ;

= Pt

0'0

0'

0

consequently

is strictly 0'0

~

-

hyperbolic with respect to N = (1; 0). ument shows immediately that

p-t,x

c

c

c

A transversality arg-

is still strictly hyperI)

bolic with respect to N' when (t, x) and N' are respectively .*

sufficiently near 0 and N.

The above allows us to prove the

existence of a neighbourhood W in which uniqueness applies, by proceeding as in the proof of Theorem 1.10 (see Remark 1 . 1 2 ) . We are in fact then led to using an existence theorem for a Cauchy problem of the type (1.10.4). For this, it is suffitcient to apply Theorem 4.9 to P after extending it to the whole of IRn+l.

This is possible by virtue of:

LEMMA 4.12 :

Let Q = Q(t, x, Dt, Dx) be a d i f f e r e n t i a ? ,

operator ( o r pseudo-differential operator w i t h respect t o x) of degree m, with Cm c o e f f i c i e n t s , which i s s t r i c t l y hyperbolic i n a b a l l Br of radius r a t 0.

Then there e x i s t s a s t r i c t l y

c

hyperboZic operator Q of degree m which coincides w i t h Q i n the ball B

r/2

and which has constant c o e f f i c i e n t s outside the b a l l

PROOF: z e r o for s

2

Let

@(s]

E C"(R)

be equal to unity for s

r , and lying between 0 and 1.

The mapping

5

r

-2'

(SEC. 4)

375


and i s z e r o o u t c o i n c i d e s w i t h t h e i d e n t i t y map i n t h e b a l l B r/2 s i d e t h e b a l l Br.

x),

Then t h e o p e r a t o r Q w i t h complete symbol Q ( p ( t ,

7,

g)

meets t h e r e q u i r e m e n t s .

This l o c a l uniqueness r e s u l t can be g l o b a l i s e d .

For ex-

t h e f o l l o w i n g theorem e s t a b l i s h e s t h e e x i s t e n c e of a

ample,

f i n i t e wave p r o p a g a t i o n speed for s t r i c t l y h y p e r b o l i c d i f f e r e n t i a l operators :

Suppose P i s a d i f f e r e n t i a l operator of

THEOREM 4.13 :

degree m, s t r i c t l y hyperbolic i n

[ 0, T ]

constant c o e f f i c i e n t s f o r 1x1 large.

u p

x

R"

and with

We put

SUP

t€[ O,T1,xERn

151 = where

T

j

let Let

DJ u t

c s0

E

(to,xo>

= { (t, x ]

=

c

u E C"([ jt=o

j

I

are the zeros with respect t o

For cone

1

=

t

0, T

E

I -

(0, x) ~x 0T

o

I

[

in

s0

1; b )

2

x

R"

O, to [ xo/

'I

of

p,.

we define the backwards

,

< v to

1x

- xoI < v ( t o - tl 3

3 denote

i t s trace on t = 0 .

be such -that Pu = 0 i n C and

f o r j = 0,

..., m-1.

;

Then u = 0 i n

c.

PROOF: For e

C

u E Cm[[

F i r s t we suppose t h a t

E ] 0,

, we

to

tE E

put

t

0

-

0

0

,

5X

Introduction to the Theory of Linear Partial Differential Equations (Studies in mathematics and its applications)

Introduction to the Theory of Linear Partial Differential Equations (Studies in mathematics and its applications)

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Linear partial differential equations and Fourier theory

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