Quasihomogeneous Distributions (North-Holland Mathematics Studies)

QUASIHOMOGENEOUS DISTRIBUTIONS

NORTH-HOLLAND MATHEMATICS STUDIES 165 (Continuation of the Notas de Matematica)

Editor: Leopoldo NACHBIN Centro Brasileiro de Pesquisas Fisicas Rio de Janeiro, Brazil and University of Rochester New York, U.S.A.

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD

TOKYO

QUASIHOMOGENEOUS DlSTRIBUTlONS

Olaf von GRUDZINSKI Mathematisches Seminar der Universitat Kiel Kiel, Federal Republic of Germany

1991

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD

TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands Distributors for the U.S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas New York, N.Y. 10010, U.S.A.

L i b r a r y o f Congress Cataloglng-in-Publication

Data

G r u d z i n s k i . O l a f v o n . 1947OuasihornogeneOuS d i s t r i b u t i o n s / O l a f v o n G r u d z i n s k i . p. cm. -- ( N o r t h - H o l l a n d m a t h e m a t i c s s t u d i e s ; 165) Includes bibliographical references and index. I S B N 0-444-88670-2 1. Theory o f distributions (Functional analysis) I. T i t l e . 11. S e r i e s . O A 3 2 4 . G78 199 1 5!5'.782--dC20 90-23029

CIP

ISBN: 0 444 88670 2

0 ELSEVIER SCIENCE PUBLISHERS B.V., 1991 All rights reserved. No part of this publication may be reproduced, stored i n a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V. / Physical Sciences and Engineering Division, P.O. Box 103, 1000 AC Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Printed i n The Netherlands

V

Contents . . . . . . . . . . . . . . . ix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv List o f S y m b o l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi Introduction . . . . . . . . . . . . . . . . . . . . . .

Notation

C h a p t e r I . ( A l m o s t ) Q u a s i h o m o g e n e o u s F u n c t i o n s . Definitions a n d Basic P r o p e r t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. .

i

R e p r e s e n t a t i o n s o f t h e Multiplicative G r o u p 10,+a[ . . . . . . . . .

2

.

13

Quasihomogeneous Functions . .

.

. .

.

. . . .

,

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

( A l m o s t ) Q u a s i h o m o g e n e o u s Polynomial F u n c t i o n s

1’)

Non-trivial E x a m p l e s o f Q u a s i h o m o g e n e o u s Polynomial F u n c t i o n s

. . . . .

32

The Hypersurfaces Sx . . . . . . . . . . . . . . . . . . . . . . . . . .

41,

in c a s e M is N o t S e m i - s i m p l e

.

. . .

Almost Quasihomogeneous Functions D e t e r m i n i n g t h e Set !2( M 1

.

.

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

,

. . . . . . .

.

. . . . .

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

Quasihomogeneous Polar Coordinates

. . .

.

. . .

.

. .

30

. . . . . .

SO

.

h0

.

.

.

.

.

C h a p t e r 11. ( A l m o s t ) Q u a s i h o m o g e n e o u s D i s t r i b u t i o n s . Definitions a n d Basic P r o p e r t i e s . . . . . . . . . . . . . . . . . . . .

. . .

.

75

. . . . . . . . .

76

( b ) T h e F o u r i e r T r a n s f o r m of Q u a s i h o m o g e n e o u s D i s t r i b u t i o n s . . . . .

84

( c ) Meromorphic Functions of Quasihomogeneous Distributions, . . . .

88

( a ) Quasihomogeneous Distributions

,

,

. .

.

( d ) Almost Quasihomogeneous Distributions

.

,

. . .

. . .

.

.

. .

.

. .

.

. . . . . . . . .

93

( e ) M e r o m o r p h i c F u n c t i o n s o f A l m o s t Q u a s i h o m o g e n e o u s D i s t r i b u t i o n s 105 ( f ) Appendix: ( G , a )- i n v a r i a n t D i s t r i b u t i o n s . . . . .

,

,

,

.

.

.

.

. . . .

Ill

C h a p t e r 111. C o n s t r u c t i n g ( A l m o s t ) Q u a s i h o m o g e n e o u s F u n c t i o n s by T a k i n g Q u a s i h o m o g e n e o u s A v e r a g e s of F u n c t i o n s w i t h M - b o u n d e d S u p p o r t . . . 117 118

( a ) I n t r o d u c i n g t h e Q u a s i h o m o g e n e o u s A v e r a g e s fm,,,, . . . . ( b ) ( M , I ) - b o u n d e d S u b s e t s of X

.

.

.

.

. .

.

. . . . . . . .

.

. . . . . 125

vi

Contents

( c ) When is Every Compact Subset of X M-bounded?

.........

135

( d ) Describing Quasihomogeneous Functions on X a s Quasihomogeneous Averages

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

148

Chapter 1V . Constructing (Almost) Quasihomogeneous Distributions by Taking Quasihomogeneous Averages . The Case: X is Locally M-bounded

. . . .

153

. . . . . . . . . 154

( a ) Introducing the Quasihomogeneous Averages urn.,

( b ) Describing Quasihomogeneous Distributions in Terms of Quasihomogeneous Averages

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

160

( c ) Solving the Equation ( d M - m ) S = T . . . . . . . . . . . . . . . . . . 168 ( d ) Singular Support and Wave Front Sets o f the Distributions u,.., and

( e l The Quasihomogeneous Continuations ,v the Distributions xmp:(v).

. 170

u E a ) ' ( S x ) . . . . . . . . . . . . . . . . . 174

Chapter V . Constructing (Almost) Quasihomogeneous Functions by Taking Quasihomogeneous Averages o f Functions Not Necessarily Having M-bounded Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( a ) Defining f.,.,

by (3.1)' when Suppf is Not Necessarily M-bounded 182

( b ) Meromorphic Extensions o f t h e Maps f ( c ) Computing the Residues o f 331~., ( d ) A Formula for fm., ( e ) Introducing f.,

181

H ,.,.f

. . . . . . . . . . . 191

. . . . . . . . . . . . . . . . . . . 200

if Rem 2 0 . . . . . . . . . . . . . . . . . . . .

205

for Arbitrary mEC and fEY'(V) . . . . . . . . . . . 215

. . . . . . . . . . . . . . . . . . 220 ( g ) The Locally Convex Spaces ' W g . k ( E @ ) . . . . . . . . . . . . . . . . . 226 ( f ) The Locally Convex Spaces Q,(E,

)

Chapter V I . Constructing (Almost) Quasihomogeneous Distributions by Taking Quasihomogeneous Averages . The Case: (1.14) holds . . . . . . . . . . . 233 ( a ) Weakly ( M . 1 ).bounded

Subsets of X . . . . . . . . . . . . . . . . . 234

( b ) The Distributions urn.,

and Their Basic Properties

. . . . . . . . . 242

( c ) Describing the A l m o s t Quasihomogeneous Distributions on X with Support Contained in X \ X +

. . . . . . . . . . . . . . . . . . . 248

( d ) Characterizing (Almost 1 Quasihomogeneous Distributions o n X

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

260

Xg.k ( E @ ) . . . . . .

280

in Terms of Quasihomogeneous Averages ( e ) Solving the Equation ( a M - m ) S = T (f)

Duality Brackets for the Spaces Q r n ( E B ) and

276

vii

Contents

Chapter VII . Solvability of Quasihomogeneous Multiplication Equations and Partial Differential Equations

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

( a ) Quasihomogeneous Multiplication Operators ( b ) Reformulating (7.18); the Test Space

3,.

283

. . . . . . . . . . . . . 284

k ( q ) . . . . . . . . . . . . 300

( c ) Solvability of (7.1) for Individual T . . . . . . . . . . . . . . . . . . 308 ( d ) Quasihomogeneous Linear Partial Differential Equations w i t h Constant Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . .

316

( e ) Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

325

( f ) The Invariant Fundamental Solutions of the Heat and of the Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . . . .

338

Chapter VlIl . Extending (Almost) Quasihomogeneous Distributions on X. to the Whole of X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

353

( a ) Pulling Back Distributions on S x to (Almost) Quasihomogeneous Distributions o n X . . . . . . . . . . . . . . . . . . . . . . . . . . . .

354

( b ) Extending (Almost) Quasihomogeneous Distributions o n X , to the Whole of X ( c ) Extending Tf....

cJk

. . . . . . . . . . . . . . . . . . . . . . . . . . . to the Whole of X

( d ) The Fourier Transform of

Gn,.(..

364

. . . . . . . . . . . . . . . . 372

i f y Belongs to Y ( V ) . . . . . . . 370

Chapter I X . Quasihomogeneous Wave Front Sets

. . . . . . . . . . . . . . . . 383

( a ) The Wave Front Set W F M ( T ) . . . . . . . . . . . . . . . . . . . . . .

384

( b ) The Wave Front Set with Respect to C M * L. . . . . . . . . . . . . . 395 ( c ) Wave Front Sets of Almost Quasihomogeneous Distributions . . . . 407 ( d ) Quasihomogeneous Wave Front Sets of the Standard Fundamental Solutions of the Heat and of the Schrodinger Equation . . . . . . . 417 (d.1) Proof of Lemma 9.36

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

424

( d . 2 ) Proof of Theorem ').37. Part 1 : Establishing the Microlocal Decomposition of E

. . . . . . . . . . . . . . . . . . 429

( d . 3 ) Proof of Theorem '3.37. Part 2 : Estimating the Derivatives of f from Below . . . . . . . . . . . . . . . . . . . .

434

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

445

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

447

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ix

Introduction

A (generalized) function ( o r distribution) f : X + @

o n an open s u b s e t X of IR"

is called quasihomogeneous of degree m e C a n d of tSpe p e R "

f(M,x) = tm f ( x ) ,

\ 101 if U € X ,

t€lO,+coC,

where (*)

M,x : = (t P 1x , . . . . , t P n x , )

.

Moreover, a partial differential o p e r a t o r P ( 3 ) with c o n s t a n t coefficients is called quasihomogeneous of t j p e p if i t s defining polynomial P is quasihomogeneous of type P '

If p j = l f o r every j c i l , . . . , n ) t h e n , of c o u r s e , f ( r e s p . P ( 3 ) ) is called homogeneous.

Many classical differential o p e r a t o r s a r e homogeneous o r quasihomogeneous. For example, t h e Laplacian A o r t h e wave o p e r a t o r heat o p e r a t o r 3, - A ,

a: - A,

a r e homogeneous while t h e

o r t h e Schrodinger o p e r a t o r id, - A,

a r e quasihomogeneous

of type ( 2 , 1 , . . . , I ) .

Problem: Let P ( 3 ) be a quasihomogeneous partial differential o p e r a t o r with cons t a n t coefficients; given a quasihomogeneous distribution TE%'(IR"), d o e s t h e r e e x i s t a quasihomogeneous solution S E % ' ( R " ) of t h e equation

In particular, when is t h e answer in t h e affirmative f o r every quasihomogeneous

T ? ( i n t h i s c a s e - f o r t h e moment - we say t h a t P ( 3 ) h a s t h e quasihomogeneous solvability propertj.).

Choosing T to b e t h e Dirac distribution S (which is quasihomogeneous f o r every

p ) o n e s e e s t h a t t h e problem includes the question f o r t h e existence of quasihomogeneous fundamental s o l u t i o n s . As is well-known, f o r many of t h e classical

X

Introduction

d i f f e r e n t i a l operators t h e a n s w e r to t h i s m o r e special q u e s t i o n is in t h e a f f i r mative. For example, t h e wave operator has homogeneous fundamental solutions, a n d t h e Laplacian does so if n 2 3 . M o r e o v e r , t h e h e a t o p e r a t o r a n d t h e S c h r o d i n g e r o p e r a t o r have q u a s i h o m o g e n e o u s f u n d a m e n t a l s o l u t i o n s . H o w e v e r , a s is e q u a l l y w e l l - k n o w n , s o m e t i m e s t h e a n s w e r is in t h e n e g a t i v e . F o r e x a m p l e , in

case n = 2 the Laplacian does n o t have h o m o g e n e o u s f u n d a m e n t a l s o l u t i o n s , a n d t h e s a m e i s valid f o r t h e iterated Laplacian A k if n is e v e n a n d n o t l a r g e r t h a n 2k.

A m a j o r object of t h e t e x t is to p r e s e n t a s o l u t i o n to t h e q u a s i h o m o g e n e o u s sol-

vability p r o b l e m p o s e d a b o v e to which t h e a n s w e r w a s n o t previously k n o w n e v e n

for h o m o g e n e o u s d i s t r i b u t i o n s . In f a c t , u n d e r t h e s p e c i a l a s s u m p t i o n “ p E 1 0 , c c ~ C ”

”

n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n s f o r solvability are given, involving t h e d e f i n i n g polynomial P o n l y . Using t h e s e c o n d i t i o n s o n e c a n easily d e d u c e t h a t s p e c i a l P ( d ) h a p p e n to have t h e q u a s i h o m o g e n e o u s solvability p r o p e r t y while o t h e r s - e v e n i f t h e y a d m i t q u a s i h o m o g e n e o u s f u n d a m e n t a l s o l u t i o n s - do n o t . F o r e x a m p l e , it will be s h o w n t h a t t h e S c h r o d i n g e r o p e r a t o r h a s t h e q u a s i h o m o g e n e o u s s o l v a b i l i t y property while t h e heat operator d o e s n o t .

In o r d e r to p r o v i d e t r a n s p a r e n t p r o o f s a n d to o b t a i n better i n s i g h t i n t o t h e n a t u r e

of t h e p r o b l e m I m a d e t h e solvability t h e o r e m s p a r t of a s y s t e m a t i c e x p o s i t i o n of t h e b a s i c t h e o r y o f q u a s i h o m o g e n e o u s f u n c t i o n s a n d d i s t r i b u t i o n s o n X . In p a r t i c u l a r , t h e p r o b l e m s of e x i s t e n c e a n d r e g u l a r i t y of q u a s i h o m o g e n e o u s f u n c t i o n s a n d d i s t r i b u t i o n s are s t u d i e d a n d p r e s e n t a t i o n f o r m u l a s a r e g i v e n . A l s o i n c l u d e d is t h e solvability t h e o r y o f e q u a t i o n s of t h e f o r m

f o r q u a s i h o m o g e n e o u s d i s t r i b u t i o n s S a n d T o n X w h e r e q is a n y q u a s i h o m o g e n e o u s r e a l a n a l y t i c f u n c t i o n o n X . From t h i s t h e s o l u t i o n of t h e s o l v a b i l i t y p r o b l e m for q u a s i h o m o g e n e o u s 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 w i t h c o n s t a n t c o e f f i c i e n t s is o b t a i n e d via t h e Fourier t r a n s f o r m .

I n o w give a s h o r t o u t l i n e o f t h e c o n t e n t s o f t h e individual c h a p t e r s . T h e b a s i c m a t e r i a l o n q u a s i h o m o g e n e o u s f u n c t i o n s is c o l l e c t e d in Chapter I . In p a r t i c u l a r , q u a s i h o m o g e n e o u s polynomial f u n c t i o n s a s we1 I a s p o s i t i v e q u a s i h o m o g e n e o u s

xi

Introduction

differentiable functions are studied. Likewise, Chapter 2 is devoted to the basic definitions and properties of quasihomogeneous distributions. In particular, it is concerned with meromorphic f u n c t i o n s with values i n t h e set of quasihomogeneous distributions. The central tool for m o s t parts of the rest of the text is the so-called m e t h o d of taking quasihomogeneous averages which for homogeneous distributions was introduced by Girding C61 and the elements of which -again for the homogeneous c a s e - are also found in Volume 1 of Hormander's monograph Clll on linear partial differential equations. The method works w e l l i f X is locally p - b o u n d e d , i.e. if there exist p o s i t i v e continuous (resp. C")

functions on X which are quasihomogeneous of degree 1 .

The corresponding theory is expounded i n Chapters 3 (for functions) and 4 (for distributions). In particular, if X is locally p-bounded then the equation

( ***)

turns o u t to be always solvable in the set of quasihomogeneous distributions. However, the situation is not so nice i n case X is n o t locally p-bounded

(for

example if X = R").Then even for functions it is difficult to develop a sufficiently general theory of quasihomogeneous averages. A convenient way to do t h i s relies

o n meromorphic extension techniques. They, however, require the assumption " P E CO,+aC"

'*.

The corresponding theory is elaborated in Chapter 5. I t turns o u t

that for certain exceptional numbers m the averages obtained are not always quasihomogeneous but o n l y almost quasihomogeneous i n a sense made precise in Chapters 1 and 2 . For distributions on open subsets X of

R"

which are not locally p-bounded the

quasihomogeneous averages are introduced i n Chapter 6 . Since they depend on the quasihomogeneous averages of test functions one has to adhere to the assumption

" P E CO,+coC" ". Moreover, the flaws already found for functions in Chapter 5 entail that for certain values of m sometimes only so-called almost quasihomogeneous

distributions (introduced i n Chapter 2 ) are obtained as the result of t h e averaging procedure.

In C h a p t e r 7 t h e results of the preceding chapters are applied to t h e theory of quasihomogeneous multiplication equations on

IR" and - via the Fourier transform -

t o the theory o f quasihomogeneous linear partial differential equations on IR".

xii

Introduction

As a consequence o f t h e r e s u l t s of C h ap t er 6 , there d o not always exist quasihomo-

geneous, but in general only a l m o s t quasihomogeneous solutions. In particular, in general only a l m o s t quasihomogeneous fundamental solutions e xist. Necessary and sufficient conditions o n q and T a r e given for t h e existence of quasihomogeneous s o l u t i o n s S of t h e equation

( ***).

Several examples a r e tre a te d in detail.

In particular, f o r t h e heat and f o r t h e Schrodinger ope ra tor all t h e quasihomogeneous fundamental s o l u t i o n s which a r e invariant under t h e action of t h e orthogona l g r o u p on t h e space variables ar e determined. C h a p te r 8 d e a l s

- still

under t h e assumption " p E C O , + a C "

"

- with a nothe r a spe c t

of t h e method of taking quasihomogeneous averages, namely t h a t i t a llow s to e x te n d quasihomogeneous distributions o n I R " \ ( O )

to t h e whole of IR".

Finally, in Chapter- 9 - under t h e assumption " p E IO.+mC"

'*

- the singularities of

quasihomogeneous distributions o n IRn a r e studied. The m ost suita ble t o o l s for describing them a r e t he quasihomogeneous wave f r ont sets introduced by R. Lascar and Rodino. In C h a p t e r 9 t h e necessary p a r t s from t h e theory of quasihomogeneous wave f r o n t s e t s a r e presented in a way t h a t keeps close to t h e presentation of (homogeneous) wave f r o n t sets in Hormander's monograph. In a similar spirit quasihomogeneous wave f r o n t sets with respect to Gevrey c la sse s a r e introduced and employed f o r t h e description of singularities of quasihomogeneous distributions. A s an example, f o r t h e h eat and f o r t h e Schrodinger ope ra tor quasihomogeneous wave fr o n t s e t s of the invariant fundamental solutions determined in Cha pte r 7 are computed.

One observes t h a t t h e map I O , + a C -

GL(n ; IR) , t H M, , is a continuous ( o r ra the r

real analytic ) representation of multiplicative g r o u ps. Among all such representations t h e o n e s defined by

(*)

ar e distinguished in t h a t they a r e in real diagonal

f o r m . I t t u r n s o u t t h a t nearly everything of t h e theory sketched above can b e extended to cover generalized functions which ar e quasihomogeneous with respect to an arbitrary continuous representations of 10,+mC in GL ( n ; IR). In particular,

representations which a r e in complex diagonal f o r m a re a dm itte d. The only drawback is t h a t t h e formulations of s o m e of t h e results and many a proof become much more complicated than in t h e real diagonal c a se ( t h i s is particularly so if

M is not se m i - si m p l e) . So, from t h e outset we a r e going

to build t h e theory f o r

xiii

Introduction

an arbitrary continuous representation t

H M,

of I 0 , + 0 3 C in G L ( n ; l R ) , denoting

by M t h e infinitesimal generator of t h e representation and speaking of "quasihomo-

geneity of type M ".

For more detailed expository information see t h e introductions to each of t h e c h a p t e r s and sections below.

The main body of t h e t e x t c o n s i s t s of material not previously published. It requires o n l y a basic knowledge of distribution theory a s expounded, f o r e x a m p l e ,

in t h e basic c h a p t e r s of Volume1 of Hormander's monograph. Of c o u r s e , it is presumed t h a t t h e reader has a basic knowledge of linear algebra, of holoniorphic functions of o n e complex variable, of real variable theory, of Lebesgue integration theory, a s well a s of t h e rudiments of locally convex vector spaces. Otherwise t h e t e x t is self-contained. Complete proofs a r e given. In this way it becomes accessible to g r a d u a t e s o r even t o advanced undergraduates.

Acknowledgements. I thank my colleagues and friends Dieter Blessenohl, Rudolf Schnabel, and foremostly Volker Wrobel for helpful discussions and f o r their encouragement.

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xv

Notation Usually, I s t i c k to t h e s t a n d a r d notation of H o r m a n d e r C I 1 1 . In p a r t i c u l a r , I u s e t h e following conventions: No:=Nu(0);

/N,:={nEIN; n < k } , kEIN.

The m e m b e r s o f t h e s t a n d a r d basis o f IR" a r e d e n o t e d by ej, i.e. ( e i ) k: = h j k , j,kElN,,

w h e r e J j , d e n o t e s t h e Kronecker s y m b o l .

Moreover, if V is a finite dimensional normed real vector s p a c e t h e n

K ( s ,r ) (resp.

k(s,r ) )

:=

open ( r e s p . c l o s e d ) ball in V of radius r c e n t e r e d a t x

Sv : = unit s p h e r e in V ( a l s o d e n o t e d by S"-'

;

V=IR");

in c a s e

If X is any s u b s e t of V t h e n ,yx : = characteristic function of X

and X : = X \CO,.

I f W is a n o t h e r finite dimensional IR-vector s p a c e W t h e n L(V, W ) : = t h e R - v e c t o r s p a c e of all linear m a p s A :V+

W:

GL(V.W) : = set of all invertible e l e m e n t s of L ( V , W ) ; LIV):= L(V.V)

and

GLIV):=GL(V.V).

For A E L ( V ) a n d for any A-invariant s u b s p a c e U of V t h e e n d o m o r p h i s m of U induced by A is d e n o t e d by A,.

v * .. --

dual s p a c e L(V,IR) of V ;

The dual ( o r transposed) of a map A E L ( V , W ) is d e n o t e d by A ' € L ( W ' , V * ) .

If V a n d W a r e @-vector s p a c e s t h e n for I K E { ! R , @ } t h e set of IK-linear m a p s T : V d W is d e n o t e d by L , K ( V , W ) . Finally, for x E V and u E V * v ( x ) is a l s o w r i t t e n a s < u , x > . Likewise, t h e duality b r a c k e t b e t w e e n d i s t r i b u t i o n s T a n d test functions

'p

i s a l s o d e n o t e d by (7, (p >.

As for t h e special notation introduced in t h e t e x t , a fairly comprehensive list of

symbols is included b e l o w .

xv i

L i s t of Symbols

List of Symbols

2 . 19

X ( M ) . . . . . . . . . . . . . . . 2.5

pM(t) . . . . . . . . . . . . . . . 2

h(P) . . . . . . . . . . . . . . . . 2.5 p '. ( V ) . . . . . . 29.317. 320. 328

< x . y > .

. . . . . . . . .

M,. Mo . . . . . . . . . . . . . 3. 6 p . . . . . . . . . . . . . . . . 4 . 70

v,. v, . . . . . . . . . . . . . 4 . s . . . . . . . . . 4. S

G M ( X ) .E M ( h )

o , ( M ) . o ~ ( M )C. I = O ( M ) . K, N

MA

xx

. . . .

4

. . . . . . . . . . . . . . . . . 4

. . . . . . . . . . . . . . . . . 4 . . . . . . . . . . . . . . . . . s

Pa.€," . . . . . . . . . . . . . . . 31 O r d M ( q o ) . . . . . . . . . . . . . 39

. . . . . . . . . . . . . . . . 42

YM

. . . . . . . . . 4 4 . 188. 384 . . . . . . . . . . . 298. 384

Xmin

,,,A

. . . . . . . . . . . . . . 44

X..J.

X" . . . . . . . . . . . . . . 4 4 . 234

dim'V . . . . . . . . . . . . . . . 5

sy . . . . . . . . . . . . . . . . 47

G M ( T ) . E M ( T .) . . . . . . . . . . 6

S(M) . . . . . . . . . . . . . .

SO

Oo

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

P , . P - . P"

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

6

px.

an . . . . . . . . . . . . .

54

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

8

3 . . . . . . . . . . . . . . . .

62

. . . . . . .

02

O + . O - .

i,

RM(x)

6

. . . . . . . . . . . . . 8 . 18

XXMlO.+m[

ts. 0% . . b* . . . . . p ( x .a ) . . . CrPs(X ) . . Df . . . . . 3, . . . .

. . . . . .

. . . . . . . . . . .

62

. . . . . .

T(p) . . . . . . . . . . . . . . .

62

. . . . . .

. . . . . .

. . . . . . . . 14 . . . . . . 14. 30 . . . . . . . . is . . . . . . . . 16 . . . . . . 18. 21 . . . . . . . . 18

. . . . . . . . . . . . . . 19. 25 . . . . . . . . . . . . . . . . 20

Q, P

a U . a z j a. Lj . aj 1

N,

Pu.

1

. . . . . . . . . . . . . . . . 17

UM

. . . . . . . . . . . . . 6 7 +175 . . . . . . . . . . . . . . . . 67

-I

Vol

Vol,

PP.k

. Pi.

k

.

Vol

. . . . . . . . . .

6 7 . 76

. . . . . . . . . . . . . . 69

. . . . . . . . . . . . . . . . 72 (*)x . . . . . . . . . . . . . . . 76 . . . . . . . . . . . . . . . .

76

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

76

JM(K;X) . . . . . . . . . . . .

76

7.

I

.

4k

21

23. 26 29/30. 97

0 0

. . . . . . . .

. . . . . . . . . . . . . . . . 23

. . . . . . . . . 63. 141

x

T,

. . . . . . . . . . . . . . . 23

aM

-"

21

. . . . . . . . . . . . 23

O(X)

EM.EM(X)

. .

diag( . . . )

dz . i z ( x ) . xz

3,(X).

xa. q . . . . . . . . . . . . . . 19. 20

c (a)

A,.

. . . . . . . . . . . 13

U

'u

x,. x- . . . . . . . . . . . . . 5 3 . 54

T O M , . . . . . . . . . . . . . . 77 S o . S,

. . . . . . . . .

77. 260. 300

t ( d M - m ) . . . . . . . . . . . . 80

xvii

List of Symbols

. . . . . rM(x) . . T*Y. ~ : p . p*( r ) . . . (T. R ) O . . m*

. . . . . . . . .

80. 154

1

r

. . . . .

160. 260

. . . . . . . . .

168. 277

U r n . uk

82

T T .,.,

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

82 82 83

al,(x) . . . . . . . . . . . . . a'(X;Z ) . . . . . . . . . . . . < v qJ > . . . . . . . . . . . . .

83

v,

84

d ( T ) . . . . . . . . . . . . . 177. 358

84

aa,',.k ( X )

. . . . . . . . . . . . . . Y ( V ) .Y"(V) . . . . . . . . . . A

'p.T . . . . . . . . . . . . . . .

7 v . 7 . . . . . . . . . . . . 8 4 . 85

v** .

1:v-

h

. . . . . . . . . . .

p,* ( T )

h

Urn.11 U r n *

. . . . . . . . . . 84

k

.

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

171 171 175

175. 355

. . . . . . . . . 178. 179

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

179

C ' ( W ) . C r S s ( W ) . . . . . . 182. 183

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

85

I1 f

ai(zo;

h) . . . . . . . . . . . . .

88

@(cd . )

ord(z,

h) . . . . . . . . . . . .

88

w,

l)ol( h ) . . . . . . . . . . . . . .

88

0, . . . . . . . . . . . . . . . . 188

02

B(w)

D

.

V

E,.

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

ordM(T)

.

. . . . . . . . . . . . 03

a,. cw

L,

. . . . . . . . . . . . . . . 111

T .,f,

. . . . . . . . . . . 112. 114

G * . o * . @ ' . 0 ' . . . . . . . . 112 . 114 p,

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

,f

112

. . . . . . . . . . . . 117. 181. 218

.

.

. . . 117 110. 182 197. 215 . 219 fm. f,. wk . . . . . . . . . 124.216. 372 cfcx, . . . . . . . . . . . . . .

C;;,(X,

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

119 119

SJf.

. . . . . . . . . . . . . . 110

YM.I

. . . . . . . . . . . . . . 122

. . . . . . . . . . . 'ua;.k ( X ) . V=W* . . . . . 1/I

X,

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. 123. 154 . . . 140 . . . 151 . 154. 243

a . ( X ) . 3 b ( X ) . . . . . . . 155. 242

$ju. . . . . . . . . . . . . . . 155

urn.

. . . . . . . . . . . . 155. 243

. . . . . . . . . . . . . 183 9

9

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

.U+;UO. . a r . s

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

182

WN WN. rl w(b). 186. 188.190

9

G 0.63.S, . . . . . . . . . . 111. 283 111

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

IIW

C(O

. c.

.x,

191

. . . . . . . . 192

. . . . . . . . . . . . . 192 . . . . . . . . . . . . 193. 194 . . . . . . . . . . . . . 193

w o . wk

'U, ( M ) . . . . . . . . . . . . . . 1Y4

. . . . . . . . . . 194. 216. 219

3TEf.

x; Q ,

. . . . . . . . . . . . . . . 201 f . . . . . . . . . . . .

201. 204

Y G ( V ) . Y E ( V ) . . . . . . . . . 214

. . E M .2, . Q, ( E ) . . E, . . . . N(m) . . . O m( V ) N

m

Q;(K;O)

Q,(C;(X), Q,(Y(V),

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

215. 219. 284

. . . . . 219 . . . . . 220 . . 220. 2 8 0 . . . . . 221

. . . . . . . . . . . . ) . . . . . . . . . ) . . . . . . . . . .

.U m . k ( E ) . U Z ( E ) m

x z .k ( K ; @ )

urn.k ( $f ( v)@)

225

. . . . . . . . 226

. . . . . . . . . . 227

'uz.k ( c r ( x ) @ ). . . . . . . . 03

223 224

229

. . . . . . . . . 230

xviii

List OF Symbols

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

XI *

234

X

. . . . . . . . . . . . . . . . 234

Xo

. . . . . . . . . . . . . . . . 234 . . . . . . . . . . . . 238/239

x.i

mu., . . . . . . . . . . .

243. 247

Y(Y).Y ( F . Y ) . . . . . . . . .

245

Y P ; ( V ) . Y L ( V ) . . . . . . . . . 247 Q

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

248

. . . . . . . . . . . . . . . . 24')

u

"

u.

~

r-.

Y

. . . . . . . . . . . . . . 253

d. u

x + . xx . . . . . . . . . Q;(E)

a' :.

(E)

265.273. 358

B,.

A t.,

.,.,f

,f

. . . . . . . . . . . . 376 . . . . . . . . . . . . 384

(T)

WF,

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

h

V ( C . 1.)

(S. u ) " . . . . . . . . . . . . . . 385 . . . . . . . . . . . . . 38.5

H,.H, &M

. . . . . . . . . . . . . . . 385

. . . . . . . . . . . . . . . 390 Q M .X M . . . . . . . . . . . . . 393 P,

z,. M L.

. . . . . . . . . . . . . . . 3Y3

LN.

. . . . . . . . .

L,

A ( t ) . . . . . . . . . . . . . . . 396

. . . . . . 281.362. 363 . . . . . . . . . . 290

C .,

. . . . . . . . . 204

t B m . tC,

CMSL(X)

. . . . . . . . . . . . 3Y7

WFM.L(T)

. . . . . . . . . . . . 400

YM.-

. . . . . . . . . . . . . . 407

WF,.

. . . . . . . . . . . . . . 417

CpVL.C p . * ( f I )

% ( f ) . . . . . . . . . . . . . . . 300

rP(n) .

.k ( q )

. . . . . . . . . . . . . 300

.

Xm( 4 ; EcgI ) X., X ,

. . . . . . . . 305

.i ( 4 ; Ecq*) . X, .i

% m .k ( q ; E )

. . . . . . . 305

. . . . . . . . . . .

308

. . . . . . . . . . . . . . 308 aY;(v) . . . . . . . . . . . . . 310 e9 ( f )

2[yA. k ( v ) . . . . . . . . . . . .317

. . . . . . . . . . . . . 319 B3), ( q ) . . . . . . . . . . . 319. 321

3 , .

k(q)

0 .E

. E' . . . . . . . . . .

339. 418

. . . . . . . . . . . . . . 354 . . . . . . . . . . . . . . . . 354 M, . . . . . . . . . . . . . . . 355 QLv Ts

v

K & . j . wi. . . . . . . . . . . . . . 360

. . . . . . . . . . . . . . . . . . . . . . . . . . 364.365.

urn. E

361

f

371

Qk(T,,

305. 3Yh

C h ' L( X ) . . . . . . . . . . . . . 3Y6

xDA.k ( E ) . . . . . . . . . . . . 2YO

3.,

385

. . . . . . . . . . . . . 281

. . . . . . . . . . . . . . . . 284

.A,

. . . . . . . . . . . . . . . 373

. . . . . . . . . . . . . 280

< . . . .> M E

3.

ox)

. . . . . . . . . . 372

9W.P

. . . . . . . . . 417

. . . . . . . . . . . . . 417

. . . . . . . . . . . . . . 417

T o . p ( r . 5 ) . . . . . . . . . . . . 429

1

Chapter I

(Almost) Quasihomogeneous Functions. Definitions and Basic Properties

In section ( a ) of the present chapter we collect some basic facts on continuous representations of the multiplicative group 10,+aC in a finite dimensional vector space V . In particular, for the rest of the whole text w e fix such a representation, denoting its infinitesimal generator by M . The basic material on quasihomogeneous functions (with respect to this representation) is introduced in section ( b ) of this chapter. Section ( c ) is devoted to the treatment of quasihomogeneous polynomial functions on V . I f M is semi-simple they are easily described by means of mixed real-complex coordinates which are basic for much of the other material, as well.

If M is not semi-simple then the natural candidates for quasihomogeneous polynomial functions fail to be so but possess a more general type of invariance property, called almost quasihomogeneit,,. Section ( d ) contains some special examples

of quasihomogeneous polynomial functions i n case M is not semi-simple. In section ( e ) general almost quasihomogeneous functions are introduced, and their basic properties are exhibited. The f u l l motivation for the notion of almost quasihomogeneity w i l l come up in Chapter2. Section ( f ) is concerned with smooth positive functions x which are quasihomogeneous of degree 1 and w i t h the hypersurfaces S X : = x - ' ( 1 )associated w i t h them. The set Ltc(M1 of points where functions x a s above exist locally is described in section ( g ) . Finally, in section ( h ) , by means of s u c h functions x so-called quasihomogeneous polar coordinates are introduced and a few applications are given. There are several results of a more technical character which are required in later chapters. They can be skipped o n a first reading.

2

I.

( A l m o s t ) Quasihomogeneous Functions

O n c e a n d for a l l w e f i x n C N a n d a n n - d i m e n s i o n a l IR-vector s p a c e V , its n o r m u s u a l l y b e i n g d e n o t e d by

/ a / .

Of c o u r s e , s o m e t i m e s w e identify V w i t h IR". Eventu-

ally w e work with a scalar product
o n V w h i c h also c o m e s i n t o t h e p i c t u r e

implicitly w h e n w e identify V w i t h IR" a n d a u t o m a t i c a l l y a s s o c i a t e t h e scalar p r o d u c t d e f i n e d w i t h r e s p e c t to t h e s t a n d a r d b a s i s ( e l , . . . ,e,,) of

IR"

by

n

tx,y> :=

2 xjyj . j=1

N o t e t h a t s i n c e V is f i n i t e - d i m e n s i o n a l w e c a n w o r k w i t h any n o r m w h e n d e a l i n g with convergence, continuity and differentiability.

( a ) 1lcpres:enlations 01' Chc MulCLplicaCivc G r o u p IO.+-C

In t h i s s e c t i o n w e collect basic i n f o r m a t i o n o n c o n t i n u o u s r e p r e s e n t a t i o n s of IO,+coC in V , i.e. h o m o m o r p h i s m s o f t h e m u l t i p l i c a t i v e g r o u p I O , + a C i n t o G L ( V ) .

Example 1.1. I f M E L ( V ) then by p M ( t ) : = e s p ( ( l o g t )M )

a real analytic representation p M : 10, +at+ GL( V ) i s well-defined.

I

An i m p o r t a n t s p e c i a l c a s e is w h e n t h e m a t r i x (Mjkbj,keNnof M w i t h r e s p e c t to s o m e basis o f V h a s d i a g o n a l f o r m , i.e. w h e n w e c a n identify V w i t h IR" in such a way t h a t f o r s o m e pEIR"\(O)

Propoaitfon 1.2. I f p : I O , + ~ t + G L ( V ) is a continuous group representation then there e x i s t s a (unique) MELl V ) such that p = p M . In particular, p i s real anal-vtic, and M = p ' l l ) .

proOf. T h i s is, o f c o u r s e , a s p e c i a l c a s e of a g e n e r a l r e s u l t f r o m t h e t h e o r y o f Lie g r o u p s . A direct p r o o f is as f o l l o w s . T h e f i r s t s t e p is to s h o w t h a t

Q

is d i f f e -

3

1.a R e p r e s e n t a t i o n s of l O , + m C

rentiable. Since G L ( V ) is an open neighbourhood of p ( 1 ) = Id,

and since p is con-

tinuous we can find a > l such t h a t a

A:= sp(s) ds 1

is invertible. Making use of t h e homomorphism property of p and s u b s t i t u t i n g

s ' = t s we obtain f o r every t E l O , + a C t h a t a

a

at

p ( t ) = J p ( t ) p ( s ) d s A-' = f p ( t s ) d s A-' = 1

1

1'p(s')ds'A-'. t

Since p is continuous t h e right-hand side is differentiable a s a function of t , indeed. Defining a differentiable representation

of t h e additive g r o u p

differentiating t h e equation ? ( s + t ) = $ ( s ) ; ( t )

R in V by

with respect to s , and evaluating

t h e r e s u l t a t s = O one obtains

Since

gM

is a solution of t h e s a m e differential equation satisfying t h e s a m e initial

condition it f o l l o w s t h a t

Corollary 1.3. If

6=?M.

,y : 10. +CUT&

h. is

a continuous homomorphism of multiplicative

groups then y , is real analytic, and with m := ~ ' 1 1 1we have t €10,+a[.

xlt) =tm,

prooE.

We apply Proposition 1.2 t o p

: = @ o x

where CI,: C - G L ( 2 ; R )

is t h e real

analytic homomorphism of multiplicative g r o u p s defined by (1.3)

O(a+ib):=(:-:),

a,bER.

For t h e whole t e x t we now fix a non-trivial map M E L ( V ) and an n - t u p l e p E R n \ ( 0 ) and work with t h e representation p M s o m e t i m e s discussing t h e c a s e where M is

of t h e f o r m ( 1 . l . a ) .

Notation1.4. Usually instead of p M ( t ) we shall write M , .

4

I.

(Almost) Q u a s l h o m o g e n e o u s F u n c t i o n s

The following p r o p e r t i e s of M, are i m m e d i a t e l y verified

Remark 1.S. For every t €10,+ a 1 the following formulas hold:

In order to a n a l y z e p M m o r e t h o r o u g h l y w e e m p l o y d e c o m p o s i t i o n s of V i n t o M - i n v a r i a n t s u b s p a c e s of V . F i r s t o f all w e d e c o m p o s e V a c c o r d i n g to

v

(1.8)

= V,@V,

where (1.8.a)

@

V,:=

X€dR( M

GM(X) )

d e n o t e s t h e direct s u m of t h e generalized eigenspaces G M ( AI : =

u ker

(

M - X Idv)'

jeN

of M a s s o c i a t e d w i t h t h e e i g e n v a l u e s 0,

:=

X in t h e real spectrum

o , ( M ) : = { X € R ; M-Aid,

is n o t i n j e c t i v e )

a n d w h e r e V , is t h e u n i q u e M-invariant c o m p l e m e n t o f V L R . S i n c e t h e r e a l s p e c t r u m of t h e l i n e a r e n d o m o r p h i s m Mv,

of V ,

i n d u c e d by M is e m p t y o n e c a n

provide Vc w i t h t h e s t r u c t u r e o f a @ - v e c t o r s p a c e in s u c h a w a y t h a t Mv,

be-

c o m e s @ - l i n e a r . T h i s c o m p l e x s t r u c t u r e is unique if w e r e q u i r e , in a d d i t i o n , t h a t t h e e n d o m o r p h i s m o f V, d e f i n e d by x H i x is a polynomial in Mv, f o r every e i g e n v a l u e s u b s p a c e of V,

V,

a n d t h a t ImX > 0

X of M v C . T h i s i m p l i e s , in p a r t i c u l a r , t h a t every M - i n v a r i a n t

b e c o m e s a c o m p l e x s u b s p a c e of V,.

From now on w e consider

a s e q u i p p e d w i t h t h i s @ - s t r u c t u r e . S i n c e w e have to w o r k w i t h t h e m i x e d

r e a l - c o m p l e x s t r u c t u r e o n V w e i n t r o d u c e a unifying n o t a t i o n :

Notntlon1.6.A. ( i ) For every A€@ w e set K,:=IR ( r e s p . C ) if hElR ( r e s p . @ \ R ) r%

and M A := MvKx-

IdvKX' ?u

(ii)

By o = a ( M ) w e d e n o t e t h e s e t o f a l l

N o t e t h a t o is t h e d i s j o i n t union o f o,(M)

A € @ s u c h t h a t M x is n o t injective. and a , ( M ) : = o ( M v , ) .

S

1.a R e p r e s e n t a t i o n s of I O . + m C

( i i i ) For every eigenvalue X E O o f M we d e n o t e by E M ( A ) : = kerMix t h e so-called

eigenspace of M with r e s p e c t to A . ( i v ) For every X E o we d e n o t e by

t h e s o - c a l l e d generalized eigenspace of M with respect to A . In particular, we have (1.8.b)

@

V,=

GM(A).

h€~q.(M)

For arbitrarq xEV and h6.i we d e n o t e by \A t h e spectral projection of

(v)

o n t o G,(X), (1.0)

\

i.e. each X G V is (uniquely) decomposed according t o x =

1

where x A 6 G M ( X )

a€ o

( v i ) dim'V:= dimRVR+dimcVc

We fix X E o and observe t h a t

where f o r s o m e m 5 dim' G X (M ) . Each of t h e generalized eigenspaces G,(X)

is t h e direct s u m of M-cyclic sub-

spaces. I f U is such a n M-cyclic subspace of G M ( X ) of KA-dimension dCN we let

b =( b , , . . . , b,)

be a KA-basis of U such t h a t ,"

(1.11)

Nb,=O

and

N b j + , = b j f o r jEN,+,

This means t h a t t h e matrix of N with respect to 1

where

8

N:=(MX),I.

is of t h e form

if k = j + l

j,kEN,.

(1.11)' Njk=[

0 otherwise

I t f o l l o w s t h a t t h e matrix of P X ( S ) with ~ respect t o 23 is of t h e f o r m

Note t h a t one obtains t h e matrix of ( ( Q M o e x p ) ( s )u by multiplying

1.IO.b)' by

6

I.

e x p ( s A 1 . Recall t h a t in case A E o \ I R

to t h e real basis

( A l m o s t ) Quasihomogeneous F u n c t i o n s

one o b t a i n s t h e real matrix with r e s p e c t

!& : = ( b , , i b, , . . ., b d , i b d )

s t i t u t i n g t h e 2 x 2 matrix (",-,")

out of t h e complex matrix by sub-

f o r every complex number a + i b .

Lemma1.7. Let A E o , and let U be an M-invariant subspace o f G M ( A )

of KA-di-

mension d 6 N . Then there is a constant C > O such that , 1- ( l + l l o g t / )

(1.121

-'*'

t

Is I 5 / M t s / 5 C ( 1 + /log t 1) d - l t R e A Is I

for arbitrary x € U and t E I O , + O J C . proOf. I t suffices to prove t h e assertion when U is M-cyclic.

In t h i s c a s e t h e

discussion following (1.IO.b)' s h o w s t h a t there is a c o n s t a n t C s u c h t h a t t h e inequality o n t h e right-hand side of (1.12) is valid. A n application of this e s t i m a t e to Ml,,x

instead of x leads to t h e inequality o n t h e l e f t - h a n d side of (1.12). rn

A simple calculation in t h e c a s e d = 2 s h o w s t h a t t h e e s t i m a t e s (1.12) a r e s h a r p . We now make use of (1.12) t o study t h e behaviour of M,x if t approaches t h e boundary of IO,+coC.Here it will be useful to work with

Notatlon1.6.B. ( i ) For every s u b s e t

'c

of

( a ) o + _: = { X E o ; + R e X > O ) ;

(ii)

( i i i ) For every v € ( + , - , O )

0

we define

and

we d e n o t e by P,:V+V

(b t h e s p e c t r a l projection o n t o

GM(6,).

( i v ) The s p e c t r a l projection o n t o G M ( 0 ) is denoted by M,:V-V.

Remark1.8. Let s € i : = V \ 1 0 1 and s € / O , + w l .Set a : = { - + iiff ss==+O0 3 . Then ( i ) lirn M,s esists in V i f and onlv i f s E G m ( o a ) + ker M : i f this is the case t-*s

then lim M t s = Mox

(1.13)

;

t+s

more precisely, i f N, denotes the endomorphism induced by M t on G M (0, then lim /INt -No I/ = 0 . t +s

)+

ker M

7

1.a R e p r e s e n t a t i o n s of 1 0 . + o ~ C

(ii) I f x € G ~ ( o , ) + E ~ ( 6 0 )then lim d i s t ( M t x , E M ( b o ) )= 0 , the limit t+s

uniform i f x stays in any compact subset o f GM(a,)

being

+ EM(ao) .

(iii) The following conditions are equivalent: ( a ) lim sup l M t x / = + a ; t +s

the limit in ( b ) is uniform i f s varies in any compact subset o f the complement o f GMM(o,uaO)in V .

Corollary 1.9. The map (1.14)

to, + d x V + V ,

V = G M ( a +I + k e r M .

( t , x ) H M t s ,is continuous provided that

I

Proof of Remark 1.8. I t suffices to give t h e proof for t h e case cation of this case to -M instead of M s e t t l e s t h e case s =

s = 0 ,f o r

an appli-

+a.

W e f i x X E o a n d d e f i n e Q x ( s ) : = P x ( s ) x x , S E R , w h e r e P x isgiven by (1.10.b). Then, since each component ( w i t h respect to a fixed basis of G M ( X ) ) of Q x i s a polynomial with respect to s of degree smaller than m , Q x does not vanish identically if and only if x x # O . Moreover, Q x is constant if and only if xx belongs to t h e

eigenspace E M ( X ) . I t follows t h a t in case R e X > O ( s e e a l s o (1.12)) we have lim llexp(Xs) Px(s)ll = 0

s+

-m

and in case R e A < O and x x f O lim l e x p ( X s ) Q x ( s ) l =

s+

+m

--or)

where in view of (1.12) t h e limit is uniform if xx stays in any compact subset of G M ( X ) \ ( 0 ) . Hence we suppose t h a t X E i R . If xxfZEx(M) then d e g Q x is posit i v e so t h a t

lim l e x p ( A s ) Q x ( s ) l = lim IQx(s)l = + m .

s+

-03

s+

-03

Here, again t h e limits a r e uniform if xx stays in such a compact s u b s e t of G M ( X ) t h a t t h e degree of Qx does n o t change and t h e leading coefficient s t a y s away from zero.

8

I.

( A l m o s t ) Quasihomogeneous

Functions

If x x ~ E x ( M )then Q x ( s ) = x A . If, in addition, X f O then one observes that e x p ( s X ) does not converge a s s + t a . From this all the assertions follow.

Next we describe the topological structure of the so-called "quasihomogeneous r a y s " , the orbits under the action of IO,+aC induced by pM on V .

Proporltioni.iO. Let x E V . Then by i , : l O , + m C + V ,

t H M t s , a real analytic

Function i s defined, and

( i ) the Following conditions are equivalent: (a)

i, is immersive a t some (resp. ever),) point O F 1 0 , + w C ;

( b ) i, is non-constant; (c)

xQkerM.

( i i ) The f o l l o w i n g conditions are equivalent: (d)

i , induces a homeomorphism o n t o i t s image R M (Y ) := i , ( 1 0 , +a[) ;

I d ) ' there i s a non-compact closed subinterval I induces a homeomorphism (e)

\

OF IO,+wC such that i,

OF I o n t o i , ( 1 ) :

does not belong t o the set E M u ( o 0 ) :

(f) R M ( \ ) i s unbounded. ( i i i ) The Following conditions are equivalent:

Ig)

i , is not injective;

Ig)' there esists r > l such that For every k 6 Z we have i x ( r k t ) = i , ( t ) , t E I O , + m C , and, i n particular, i , ( t r k , r k + ' C ) = RMM(x): (h)

x € E M ( o , ) , and p : = { A E o , ;

(j)

R M ( s ) is compact.

,vA #0} C i w Z

( i v . A ) RM(.vJ is a real anal-vtic submanifold

For some W E I R ;

OF V OF dimension I if and on1-1, i f

i t i s non-trivial and compact or unbounded. ( i v . B ) I f R M ( , v ) i s bounded but non-compact then the dimension d o f the Q-vector space generated bq I I E o : s A # O I analytic subrnaniFold

is not smaller than -3. and RMM(s) i s a real-

OF V OF dimension d and i s equal to the closure OF i , ( I )

where I is an)' non-compact closed subinterval O F I O , + m C .

1.a

9

Representations of I O , + m C

The following lemma contains a part of t h e proof.

Lemma 1.11. Let r be a finite subset o f

h

and denote bj. d the dimension o f the

Q-vector space generated bj, r . Then the closure o f the image o f the map

r:w-

11.15)

c',

t

H

(exp(ivt)),,,

,

is a real analytic (compact) submanifold o f C r o f dimension d ; in fact, it is diffeomorphic t o a quotient o f the d-dimensional torus b-b a finite subgroup.

r is closed i f and only i f d = l . Finall). i f J is an unbounded r ( J )is dense in T(Ui): more generally. f o r every non-emptj

Moreover, the image of

subinterval o f lR then open subset U o f T(IR) there is a finite subset R o f J such that f o r ever) w ET(1R) there i s r € R such that ( w , e ~ p ( i v r ) ) , ~lies , in U . proOf. F r o m

'I

w e select a b a s i s ( v l , . . .,u,,) o f

zuEr Qv.

T h e r e m a i n i n g v ~ r .a r e

d e n o t e d by ud+, , . . . , vc- w h e r e c : = l ' ~ l T. h e n w e c a n fix n u m b e r s q k , i € Z a n d P j E Z

D e n o t i n g t h e d-dimensional torus by T " : = ( S ' ) d w e d e f i n e f : T d - C C

by

T h e n f is a real a n a l y t i c i m m e r s i o n o n t o a c o m p a c t s u b s e t K of Cc. M o r e o v e r , it is a h o m o m o r p h i s m of m u l t i p l i c a t i v e g r o u p s w h e r e , of c o u r s e , t h e m u l t i p l i c a t i o n

is c o m p o n e n t - w i s e . Its k e r n e l i s c o n t a i n e d in t h e p r o d u c t of t h e s u b g r o u p s c o n -

s i s t i n g of t h e roots o f unity of o r d e r P i , j € l N c , . I t f o l l o w s t h a t K is a real a n a l y t i c s u b m a n i f o l d of CC a n d t h a t f is a local d i f f e o m o r p h i s m o n t o K . Now l e t J be a n u n b o u n d e d s u b i n t e r v a l of

IR. S i n c e

((11,.

.., p d )

is linearly i n d e p e n -

d e n t over Q , a s w e l l , t h e i m a g e of J u n d e r t h e m a p y:!R-Td

d e f i n e d by

y ( t ) := ( e x p ( i p j t ) ) i E N d d

is d e n s e in T . M o r e p r e c i s e l y , it i s n o t d i f f i c u l t to p r o v e t h a t R

w h e r e cfz d e n o t e s t h e n o r m a l i z e d H a a r m e a s u r e o n T d ; in f a c t , if f is t h e restrict i o n to T

d

o f a polynomial f u n c t i o n t h e n t h e c o n d i t i o n (1.17) is verified by d i r e c t

10

I.

( A l m o s t ) Quasihomogeneous Functions

computation, and since by the Stone- WeierstraR theorem these functions are O

d

uniformly dense i n C ( T , @ ) the general case follows from the special one by approximation (see Arnold C 11 ,

5 11 . C , D , E ) .

Now w e fix a non-empty open subset W of Td. We first suppose that J contains Cb.+aC for some b > O . Let vETd. Denoting by v - '

the inverse of v with respect

to the multiplication of Td and applying (1.17) to non-negative continuous funct i o n s f w i t h support contained i n v

-I

W one finds a number t,ECb,+aC such that

y ( t v ) E V - ' ~ This . implies, in particular, that the set y ( C b . + a C ) is dense in T d , indeed. More generally, since the functions Tc'-Tcl,

z H z y ( t , ) . are continuous

one finds an open neighbourhood Z, of v in Td such that z y ( t , ) E W for every

z E Z , . Since Td is compact one can select a finite subset F of Td such that d

the sets Z, , v E F , cover the whole of T . Setting Rw:={ t,; v E F } we conclude that

( z y ( R w ) ) n W f Q ) for every zETc'.

In case J is bounded from above we apply the case treated above to -pi instead of pi and obtain a finite subset R,

of J having the properties above.

I t follows that the image of J under f o y is dense in K . Note that by (1.16) we have cl

n

j=l

d

c

' '

( e x p ( i p j t ) l q k *=j e x p ( ii = l q k , ' p ' t ) = e x p ( i v c l + k t ),

k€lN,-,

This implies that ( f o y ) ( t ) = ( e x p ( i v i t ) ) i f o \ r , . Now, i f U is a non-empty

.

open

subset of K then W : = f - ' ( U ) is a non-empty open subset of T d , and the s e t

R : = R,

found above has the property

( f ( z ) r ( R ) ) n U = f ( ( z y ( R ) ) n W ) # Q )for every Z E Td . Since f ( T C 1=) K =T(IR) the last part of the assertion is proved. For the proof of the last but one part we observe that t h e image of f o y is equal to K if and only if d = 1 .

Proof of Proposition 1.10. By the chain rule and by (1.5) and ( 1 . 4 ) we have (1.18)

i k ( t ) = t1 M t M x = tI M t i ; ( l ) .

This shows that i,

is immersive a t some point if and only if it is immersive a t

every point of 10,+ a t , Moreover, the equivalence ( a )

(6) follows.

In addition, we see that i, is immersive a t 1 if and only if M x # 0 , i.e. the equi-

1.a

11

R e p r e s e n t a t i o n s of l O . + m C

v a l e n c e (a).

(c) is valid.

( e ) B ( f ) :T h e c o n d i t i o n

m e a n s t h a t t h e r e is s E { O , + a ) s u c h t h a t

(f)

lim s u p I i,( t ) I =

+m.

t+S

By R e m a r k 1 . 8 . ( i i i ) t h i s is e q u i v a l e n t to ( e )

( i v . B ) and ( j ) * ( h ) : "(e)*(f)"

S u p p o s e t h a t R M ( x ) is b o u n d e d .

Then t h e

implication

a l r e a d y p r o v e n a b o v e s h o w s t h a t x E E M ( o 0 ) . W e may a s s u m e t h a t

x C k e r M so t h a t T : = ( - i p ) \ ( O ) is a n o n - e m p t y s u b s e t of IR. W e d e f i n e a n injective l i n e a r m a p cr:@'-EM(bnilR)

by a ( z ): = ~ v E r ~ , x iifu . T : I R d @ '

denotes

t h e m a p o f L e m m a 1.11 t h e n i,= x o + ~ o r o l o g , a n d by Lemma 1.11 t h e c l o s u r e N of R M ( x ) is a c o m p a c t d - d i m e n s i o n a l real a n a l y t i c s u b m a n i f o l d of V w h e r e d is

t h e dimension of t h e Q-vector

space

Z u E 1 Q w .M o r e o v e r ,

by Lemma 1.11 R M ( x )

is c o m p a c t if a n d o n l y if d = l , i.e. t h e r e is W ~ E sTu c h t h a t every W E X is o f t h e

f o r m q wo f o r s o m e q E Q , i.e. f o r s o m e k E Z t h e n u m b e r w : = wo/k

has t h e pro-

perty p C i w Z .

( h ) * ( g ) : if p a n d w a r e a s in c o n d i t i o n ( h ) w e may s u p p o s e t h a t w f O . H e n c e , EIR w e have X s E 2 r i Z f o r every X E p so t h a t setting s = 2 ~ / w

Z e x p ( X s )xX = x = e x p ( 0 M )x ,

exp(sM)x=

XECJ

.

. is n o t injective

1.e. I,

( g ) + ( g ' ) : If i, i , ( t l ) = i,(t,),

is n o t injective w e find t , , t , E l O , + a C

s u c h t h a t tl < t2 a n d

i.e. i,(r) = x w h e r e r : = t,/tl > I . I t f o l l o w s t h a t

i,(rkt) = i , ( t )

f o r a r b i t r a r y t ~ l 0+a[ , and ~ E Z . ( g ' J * ( j ) ; as t h e i m a g e of t h e c o m p a c t interval C l , r l u n d e r t h e c o n t i n u o u s m a p i,

t h e set R M ( x ) is c o m p a c t , i t s e l f . H e n c e t h e p r o o f o f (iij)is c o m p l e t e .

( d ) ' * ( e ) ; w e prove t h e contraposition. Hence w e a s s u m e t h a t x E E M ( o , ) t h a t by t h e implication " ( f ) + ( e ) "R,(x) t h e implication

"

(j )

+( g)'

so

is b o u n d e d . If R M ( x ) is c o m p a c t t h e n

a l r e a d y p r o v e d a b o v e s h o w s t h a t t h e r e s t r i c t i o n o f i,

to any n o n - c o m p a c t closed s u b i n t e r v a l o f I O , + a [ is n o t e v e n injective so t h a t

( d ) ' is v i o l a t e d f o r trivial r e a s o n s in t h i s c a s e . H e n c e w e may a s s u m e t h a t R M ( x ) is n o n - c o m p a c t . T h e n by t h e a s s e r t i o n ( i v . R ) f o r a r b i t r a r y tElR a n d s € { O , + a )

w e f i n d a s e q u e n c e (t,,),,EN in IO,+wC t e n d i n g to s a s n+m

and satisfying

i , ( t ) = lim i x ( t n ) . Since ( t n ) d o e s n o t c o n v e r g e to t t h i s m e a n s t h a t t h e c o n n+a>

d i t i o n ( d ) ' is v i o l a t e d .

12

I.

( A l m o s t ) Quasihomogeneous F u n c t i o n s

( e ) * ( d ) : T h e c o n d i t i o n ( e ) m e a n s t h a t t h e union of t h e sets K + _ : = { X E ~ , ; x x # O } a n d r o : = {XEO,;

xx4EM(X)}

is n o n - e m p t y .

W e f i x s E ( O , + a ) a n d d e f i n e a as in R e m a r k 1.8. If r o # @ or K - , # @

then we

deduce f r o m Remark 1.8.(iii) t h a t lim l M t x l = + a . t+s

O n t h e o t h e r h a n d , if TO = @ =

T

-

~t h e n

it f o l l o w s by Remark 1 . 8 . ( i i ) t h a t

limdist(M,x,EMM(~O)) 0 .

t+s

Putting everything together w e conclude t h a t ( e ) implies t h a t t h e map

is p r o p e r . S i n c e t h e c o n t r a p o s i t i o n of t h e implication " ( g ) = l ( h ) '* already p r o v e n

a b o v e tells u s t h a t i is injective if ( e ) is valid t h e c o n d i t i o n ( d ) f o l l o w s . S i n c e t h e implication " ( d ) * ( d ) ' "

is trivial t h e p r o o f of

lii) is c o m p l e t e .

( i v . A l : I f x d k e r M a n d if R M ( x ) is c o m p a c t t h e n in t h e p r o o f of ( i v . B ) w e s a w t h a t R M ( x ) is a o n e - d i m e n s i o n a l r e a l - a n a l y t i c s u b m a n i f o l d of V . S u p p o s e n o w t h a t R M ( x ) is n o n - c o m p a c t . T h e n by ( i i i ) a n d ( i ) i, is injective a n d i m m e r s i v e . C o n s e q u e n t l y , i,

is a h o m e o m o r p h i s m o n t o its i m a g e R M ( x ) if a n d o n l y if i,

a r e a l - a n a l y t i c e m b e d d i n g . S i n c e t h e l a t t e r is t h e case if a n d o n l y if R , ( x )

is

is a

r e a l - a n a l y t i c s u b m a n i f o l d of V t h e a s s e r t i o n f o l l o w s in view of t h e e q u i v a l e n c e "(d)-(f)".

m

Corollary 1.12. ( i ) ~ ~ 1 1+0 mC .) is bounded i f and onl@ i f o C i R and E M ( o ) = V . ( i i ) p M ( 3 0 . +wCI is compact i f and only i f E M l o ) = V and o C i w E for some w 6 R .

I f this is the case then there esists a real analytic representation p^M : S ' -

GL( V )

o f multiplicative groups and a positive number r such that (1.19)

P~M=$,,,,O+~

Note that + r : 10, +wC-

where

+,.(t):=tir-.

S' is a real analytic homomorphism of multiplicative

groups and a local diffeomorphism. 8

13

1.b Q u a s i h o m o g e n e o u s F u n c t i o n s

b) Quasihomogeneous Functions

The fact that p M : I0,+00l-

GL(V), t

H M,,

is a h o m o m o r p h i s m o f m u l t i p l i -

c a t i v e g r o u p s i m p l i e s t h a t if f : X + @

a n d ~ : 1 0 , + 0 3 C + @ are f u n c t i o n s s u c h

x

is a h o m o m o r p h i s m o f m u l t i p l i c a t i v e g r o u p s

t h a t f OM, = X ( t ) f , t E 10,+03l, t h e n

p r o v i d e d t h a t f does n o t vanish identically. I f , in a d d i t i o n , f is c o n t i n u o u s t h e n

x

is c o n t i n u o u s , as w e l l , a n d h e n c e by C o r o l l a r y 1.3 o f t h e f o r m X ( t ) = t m f o r

s o m e m E @ . So b e s i d e s a f u n c t i o n f : X t C w e fix a c o m p l e x n u m b e r m c C and define

Definition 1.13. f is s a i d to b e quasihomogeneous o f degree m land o f type M ) if a n d o n l y if

f ( M t x ) = t m f ( x ) f o r a r b i t r a r y ( x , t ) b e l o n g i n g to t h e set

If M is o f t h e f o r m ( 1 . l . a ) t h e n h e r e a n d in a l l t h e f o l l o w i n g s i m i l a r d e f i n i t i o n s w e say " o f tjpe p

"

instead of " o f tjpe M

" ,

F i r s t e x a m p l e s o f q u a s i h o m o g e n e o u s f u n c t i o n s a r e provided by p o l y n o m i a l s ( s e e s e c t i o n ( c ) ) . W h e t h e r or n o t p o s i t i v e f u n c t i o n s o n X e x i s t w h i c h a r e q u a s i h o m o g e n e o u s of d e g r e e 1 is a f u n d a m e n t a l q u e s t i o n t h a t will be d e a l t w i t h in s e c t i o n s ( f ) a n d ( g ) a n d a g a i n in s e c t i o n 3 . ( c ) b e l o w . For t h e p r e s e n t w e a r e g o i n g to give s o m e b a s i c r e s u l t s s h o w i n g how t o o b t a i n n e w q u a s i h o m o g e n e o u s f u n c t i o n s o u t o f given o n e s .

Remark 1.14. Let W be an M-invariant subspace o f V . Then the following assertions hold: ( i ) I f f is quasihomogeneous o f degree m and o f t j p e M then

geneous o f degree m and o f t j p e M w where b.), definition M,

fl,

is quasihomo-

is the element o f

L ( W ) induced bj M . /ii) Suppose that f is o f the form F = F o x W where F is a function from V / W

into a? and where ? r w : V +

V / W is the canonical projection. Then f is quasi-

homogeneous o f degree m and o f type M if and only i f F is quasihomogeneous

14

I.

of degree m and of type

( A l m o s t ) Quasihomogeneous F u n c t i o n s

M W where M W is

by definition the element of L ( V / W )

induced by M .

proOf. (i): t h i s f o l l o w s from

fii):

fl,

o ( M w ) t = fl,

o(M,),

= foMtJw.

this is a consequence of f o M t = F o ( r c w o M , ) = F ~ ( M W ) t ~ ~ w .

The following assertion is a direct consequence of Definition 1.13

Remark 1.1s. IF f : X -

.

C is quasihomogeneous of degree m and q : X +

C is quasi-

C q f is quasihomogeneous of degree m + P . homogeneous of degree ~ E then

I

Next we deal with linear differential operators preserving quasihomogeneity. I t is convenient to describe them independently of t h e choice of special coordinates o n V . To d o t h i s we have t o look upon X as a "linear manifold" and upon X x V *

as its cotangent bundle. Namely, fixing any basis b = ( b , , . . . , b,) of V we define n

linear charts t d : I R n + V

where by 8):=

by y H , z yi bj and CP,: )=I

( p i , . . . , p")

I R n x l R n ~ V x V *by t s x t % * , i.e.

we denote t h e basis of V* dual to 23. Note that i f

6 is another basis of V then (1.21)

Y:=

(PE'o(P,

equals B - ' x B *

where B : = t & ' o t c r

and where here B* denotes the adjoint of B w i t h respect to the canonical scalar product on IR". To introduce the data defining differential operators we fix r € N 0 and define

Deflnition 1.16.A. A function P : X xV*-

C is said to be a copolynomial function

on X of degree 5 r if and o n l y i f it is a polynomial of degree 5 r with respect to

the second variable, i.e. it is a Cm function with respect to t h e second variable

s u c h that D;"

P

E

0.

Of course, in local coordinates this more explicitly means that PB

function R:YxIRn-C (1.22)

R =

of the form

R,@C, la1 5s-

:=Po@,

is a

15

1.b Q u a s i h o m o g e n e o u s F u n c t i o n s

w h e r e Ga(q):=qa, w h e r e Y = X = : = ( F = ) - ' ( X ) a n d w h e r e R , : Y d @ is given by Zi 1 R,(y) = P , ( y ) : = g ( 3 , " P = ) ( y , O ) . If (3 is a n o t h e r b a s i s of V t h e n in view of (1.21) it f o l l o w s by t h e c h a i n r u l e t h a t PE is a l i n e a r c o m b i n a t i o n of t h e f u n c t i o n s P Z o B .

T h i s s h o w s t h a t t h e f o l l o w i n g d e f i n i t i o n is i n d e p e n d e n t of t h e c h o i c e of B .

Deflnltlon 1.16.B. Let 4 E N I -, u ( 0 )) . P is c a l l e d a CP-copolynomial function on X of degree 5 r if a n d o n l y if it is a copolynornial f u n c t i o n o n X of degree 5 r s u c h t h a t its c o e f f i c i e n t s a r e C

e f u n c t i o n s , i . e . : t h e f u n c t i o n s R , = P F in ( 1 . 2 2 )

are C' f u n c t i o n s .

Now w e f i x a copolynornial f u n c t i o n P o n X of d e g r e e 5 r a n d f o r any f E C " ( X ) d e f i n e P ( x , 3 ) f by

Remark 1.17. Under the preceding assumptions. bj (1.1731 a linear differential operator P ( u , 3 ) is well-defined independently o f the choice o f the basis $3. For every C'-function f : X + C (1.24)

prooE.

and ever-) A € G L ( V ) we have

P ( \ , d ) ( f o A ) = [ ( P o ( A - ' x A ')) ( \ , d ) f ] o A .

F i r s t of a l l w e a r e g o i n g to s h o w t h a t f o r any o p e n s u b s e t Y of

Cr f u n c t i o n g : Y -@

R", a n y

, a n d any polynomial Q of d e g r e e 2 r w e have

w h e r e h e r e a s in (1.21) we d e n o t e by B*EL(IR") t h e a d j o i n t of B w i t h r e s p e c t to t h e c a n o n i c a l s c a l a r p r o d u c t o n IR". I n d e e d , if Q a n d R a r e p o l y n o m i a l s of deg r e e 5 r s a t i s f y i n g ( 1 . 2 4 ) ' t h e n t h e p o l y n o m i a l s Q + R a n d - if d e g Q R 5 r - Q R s a t i s f y ( 1 . 2 4 ) ' . a s w e l l . H e n c e , w e may s u p p o s e t h a t Q = X i f o r s o m e jEINn. B u t in t h i s case ( 1 . 2 4 ) ' is e v i d e n t f r o m t h e c h a i n r u l e . N e x t , f o r every f u n c t i o n R : Y x R n d @ of t h e f o r m (1.22) w e d e d u c e f r o m ( 1 . 2 4 ) '

16

I.

(Almost) Quasihomogeneous Functions

N o w , if CS is another basis of V then applying (1.24)" to g=foP,

the equalities (1.21) and PwoY = , P P ~ ( za,) , ( )f, 0! f

and taking

into account we deduce that

= [ P,(y,d,

) (f

0

PB) I

0

B,

i.e. the definition of P ( x , d ) f does not depend o n the special choice of 6 . Finally, applying (1.24)" to equalities U $ ' o ( A - l x A * )

B-'xB*

oQ = ,

B:=(b'R3)-'oAof,

and

g : = foP,

and taking the

into account we conclude that

Note that the defining copolynomial function P can be reconstructed from the differential operator P( x , 3 ) via (1.25)

p(.

J )

= e-'E"'

p(u,a)ecc."

FEVC.

,

In Chapter S we shall also have t o deal with separate differentiability properties w i t h respect to a prescribed partition o f the argument variables. In order to pre-

pare this we f i x two real subspaces V l and V, such that ( 1.26)

v = v,tB v, .

Note that if for every j E 1 1 , 2 ) w e denote by

7ri:

V-V,

the canonical projection

defined by (1.26) and identify Vf with the subspace { w o x , ; w E V 4 I ( 1.20)*

then

v * = v;@ vz' .

Moreover, if A E L ( V , V ) is such that the subspaces V, and V, are invariant under A then A commutes with

Finally, we fix r , s E N o u

and

7tl

7t2,

and V:

and V:

are invariant under A * .

and define

( ~ 0 )

Deflnitlon 1.18. ( i ) By C ' " s ( X I we denote the space of all functions f : X-@ such that DiD-$f exists and is continuous for arbitrary j , k E ! N O such that j'r

k5

s

and

(here the partial derivatives are taken with respect to the deomposition ( 1 . 2 b ) ) .

( i i ) A polynomial function P : V * +

if the derivatives D;*'P and D;"P

C is said to be OF degree 5 ( r , s ) if and only

vanish identically (here the partial derivatives

are taken with respect to the decomposition ( 1 . 2 6 ) * ) .

1.b Quasihornogeneous

17

Functions

( i i i ) A copolynornial f u n c t i o n P o n X is s a i d to be of degree 5 ( r . s ) if a n d o n l y

if P(x,

)

is o f d e g r e e 5 ( r , s ) f o r every X E X .

So a f u n c t i o n f : X - @

b e l o n g s to C'"s(X)

if a n d o n l y if P ( & ) f is w e l l - d e f i n e d

a n d c o n t i n u o u s f o r every polynomial f u n c t i o n P : V * d C o f d e g r e e 5 ( r , s ) . Of c o u r s e , t h i s c a n be e x p r e s s e d in c o o r d i n a t e s a s f o l l o w s : if 2 3 = ( b , , . . . , br,) is a n y basis of V s u c h t h a t (1.27)

( b , , . . . , b , l ) is a b a s i s o f V, w h e r e d : = d i m V , , a n d (b,,,,, , _ . b, t , ) is a b a s i s o f V,

t h e n F b e l o n g s to C r ' s ( X ) if a n d only if &"f is w e l l - d e f i n e d a n d c o n t i n u o u s f o r every a b e l o n g i n g to t h e set

w h e r e w e s e t a':=( a , , . . . , a , ~ )arid a',:= ( a d + l , . . . an). M o r e o v e r , a c o p o l y n o n i i a l f u n c t i o n P o n X is o f d e g r e e 5 ( r . s ) if a n d o n l y i f t h e f u n c t i o n s R, = P z vanish identically if a does n o t b e l o n g to t h e s e t ( 1 . 2 8 ) .

Remark 1.17'. Let P b e a r o p o l j - n o m i a l f u n c t i o n on

X of degree i (r.s). Then for

a n y F€C"*(X) by (1.133) a f u n c t i o n P ( \ , d ) f on X i s w e l l - d e f i n e d if 2 3 = ( b , , . , . , b,,) is c h o s e n to b e an). b a s i s of V s a t i s f j i n g (1.27). Moreover, if A € G L ( V , V ) is s u c h t h a t V, and V, are A - i n v a r i a n t t h e n P o ( A - ' x A ' )

is a copol.vnoniial f u n c t i o n on X of degree

5

(r.s). a s well, a n d t h e e q u a t i o n

( t . 2 4 ) is valid.

m f . In view o f w h a t w a s s a i d i n t h e t e x t s u b s e q u e n t to Definition 1.18 o n e o b t a i n s t h e a s s e r t i o n s e i t h e r by f o l l o w i n g t h e p r o o f o f Remark 1.17 or by k e e p i n g t h e v a r i a b l e s in V, ( r e s p . V,)

f i x e d a n d a p p l y i n g Remark 1.17 w i t h ( V . A ) r e p l a c e d

by ( V , , A V B ) ( r e s p . ( V , , A v l ) ) .

H

We n o w d e s c r i b e t h e behaviour of q u a s i h o m o g e n e o u s f u n c t i o n s u n d e r d i f f e r e n t i a t i o n . For t h i s w e o n c e a n d f o r a l l a s s u m e t h a t t h e s u b s p a c e s V, a n d V,

(1.26) are invariant u n d e r M a n d h e n c e u n d e r M, f o r every t € l O , + a C .

in

18

I.

( A l m o s t ) Quasihomogeneous Functions

Ropoaltlon 1.19. Let rENo and P E C , and let P : X x V x + C

be a copolynomial

function on X OF degree 5 r which is quasihomogeneous of degree 't and o f type

M x ( - M * ) . Then for every C" Function F : X +

C which is quasihomogeneous of

degree m the function P ( s , d l f is quasihomogeneous OF degree m + P . The assertion remains valid i f P is a copol-vnomial function o f degree S ( r , s ) with respect t o the decomposition 11.261 and i f f belongs t o C r ' s ( X I . proOf. Employing (1.24) and making use of the assumptions about f and P w e deduce ( P ( x , d ) F ) o M , = (P(x,a)(foM,oM,,,))oM,

=

= [Po(M,xM:,,)](x,3)(foMt) = t P + m P ( x , c 3 ) f . m Note that Remark 1.15 is a special case of Proposition 1 . 1 9 , Another special case is

Corollary 1.20. Let P be a poljnomial function on V ' which is quasihomogeneous of degree P6C and o f t j p e M

. and suppose that F is a

C"Eunction where

I' :=

deg P .

I f f is quasihomogeneous o f degree m then P ( d ) F is quasihomogeneous o f degree m-P.

8

Next we introduce an important special operator, namely the directional derivative into the direction of the quasihoniogeneous rajs R M ( , \ ) : = { M,x; t € l O , + m l } ,

X E V : the so-called Euler operator with respect to M . We denote the total derivative of a differentiable function f by DF.

Example 1.21. The differential operator 3 , deFined bj (1.29)

( 3 , f ) ( x ) := D f f s l - M s.

f€C'fXI.S€X.

a ) where the polwomial function PM : V x V ' 4 R is ( s , t )H < f ,M s > . Note that PM is quasihomogeneous OF degree 0 and

is equal t o P , ( s ,

defined

bj,

o f type

M x ( - M ' I . In coordinates the defining equation (1.29) reads as ri

n

1

Proposition 1.19 shows that for every quasihomogeneous C function f the function

3,f

is quasihomogeneous of the same degree. This, however, is also a trivial

1.c

19

(Almost) Q u a s i h o m o g e n e o u s P o l y n o m i a l s

c o n s e q u e n c e of t h e f o l l o w i n g p r o p o s i t i o n which e s t a b l i s h e s Euler's equation f o r quasihomogeneous functions.

Proposition 1.22. Suppose that f : X+ (I)

C is differentiable.

I f f is quasihomogeneous o f degree m then 3 , f = rn f

(ii) The converse implication is valid provided that f o r everj. S C X the set ( t 6 3 0 , + a C ;M t x E X } is an interval. proOf. S e t t i n g g ( t , x ) : = t - m f ( M t x ) w e o b t a i n by ( I S ) t h a t d,g(t,x) = -mt-m-'

f(M,x) + t - m D f ( M t x ) . ( p h ( t ) x )= t-"-'(d,f-mf)(M,x).

F r o m t h i s t h e a s s e r t i o n s are i m m e d i a t e l y d e d u c e d .

W e s h a l l see in P r o p o s i t i o n 1.58 b e l o w t h a t u n d e r s p e c i a l a s s u m p t i o n s o n X a n d M t h e only differentiable quasihomogeneous functions on X a r e polynomial functions. T h e s e a r e s t u d i e d in t h e f o l l o w i n g s e c t i o n .

(c1 (

A I mos1B Qua s1homog n (?ous 1% I y n o m i a I F u n c 1i o n s

In t h i s s e c t i o n w e d e s c r i b e t h e b e h a v i o u r of a r b i t r a r y p o l y n o m i a l f u n c t i o n s o n V u n d e r c o m p o s i t i o n w i t h M , . In p a r t i c u l a r . o u r a i m is to d e t e r m i n e t h e q u a s i h o m o g e n e o u s p o l y n o m i a l f u n c t i o n s . If V = R " a n d M i s of t h e f o r m ( 1 . l . a ) t h i s is very e a s y as w e s h a l l f i r s t see.

Eixample 1.23. Let a EN:.

Then the monomial function R"3x

t+

v m is quasihomo-

geneous o f type p and o f degree I1

( a , p > :=

1aI. pI. '

j = I

If P:IR"+@

is a polynomial f u n c t i o n w e d e f i n e by XE

< u , p >=I11

R".

20

I.

(Almost) Quasihomogeneous Functions

t h e so-called quasihomogeneous part Q,,P

OF P OF degree m (and OF type p ) .

Of course, QmP = O i f m does not belong to the set Z ( p ) := { < a , p > ;a E Z n } for Z = N o . Taylor's formula shows that P is decomposed into its quasihomogeneous

parts according to (1.30)

2

P =

QeP.

eEN,,(p)

Moreover, it follows that P is quasihomogeneous of degree m if and o n l y i f P equals Q,P

.

If M has complex eigenvalues the above can be carried over i f all the eigenvalues are simple. i.e. of algebraic multiplicity I . For t h i s it is appropriate to work w i t h complex coordinates o n V,

as fixed i n

Conventlon 1.24.A. Set d : = dim[RV[R and c : = dimc

vc .

and let A = ( a , , . . . , a d ) be

an R-basis of V I R and B = ( b , ,.... b,) be a @-basis of Vc-

w i t h IR" and V,

can identify V,

. So

via these bases one

with CC. The ( d + c ) - t u p l e ( a , , . . . , a d b, , . . , , b,) ~

is called a real-comple.\ basis OF V with respect to

M . By

denote the ( d + c )-tuple of coefficient functionals associated Note that q, , . . . , qc are real-valued and q C l + , , . . . , q c l + ,

( q, . . . . , q d + c )

we

w i t h t h i s basis.

are complex-valued.

If

y = ( q , ( x ) , . . . , q c l ( x ) ) and z = ( q d + , ( x ) , . . . , ~ l ~ l + ~ (we x ) also ) say that ( y , z ) are the real-comples coordinates OF

Y.

Note that i f we write z = u + i v then ( y , , . . . , y d , u1 , . . . , u, , v l , . . . , v,)

are the real

coordinates of x with respect to the 18-basis ( a , , . . . , a d , b , , . . . , b,, ib, , . . . , ib,) of V. Using real -complex coordinates we introduce special polynomial functions:

Notatlon1.2S.A. ( i ) X:=N,dxIN:xIN:;

the a € X are written as a = ( D , y , S ) where

BEIN," and y,SEN:. ( i i ) s c I : = y p z y Z s for every a = ( b . y , S ) E ' U and every x e V with real-complex coor-

dinates ( y , z ) , i.e. more explicitly, x a = qa(x) where

mostly w e shall denote the function qa by x " . (iii) If X E V has real-complex coordinates ( y , z ) w e write x l : = y j for j E N d , x ~ + ~z j: =

-

and x d + c + j: = z j for j E I N c , thereby looking upon ( x l ,. . . , x n ) a s some sort of pseudo-real coordinates of x

.

1.c

21

( A l m o s t ) Quaslhornogeneous Polynornlals

U s i n g t h i s n o t a t i o n o n e c a n c o n v e n i e n t l y w r i t e d o w n t h e a c t i o n of r e a l l i n e a r m a p s on V w i t h t h e h e l p of r e a l - c o m p l e x c o o r d i n a t e s : Let H be a c o m p l e x Banach

be a n IR-linear m a p , a n d l e t c E V ; w r i t i n g

s p a c e , l e t T:V-H

€,=x d

C

C

< j a j + z R e ( c i + d ) b i +x l m ( c j + d )( i b j )

j= I

j=i

j=i

and noting that

N o w , if g : X-H

is a d i f f e r e n t i a b l e f u n c t i o n o n e c a n apply (1.31) to T = D g ( x )

and obtains n

D~(.\).[ =Jtr,

11.32)

( a l f ) ( \ )=

)=I

where t h e derivatives

Notation 1.2S.B. ( i ) d,

:= I

(3,.+i3,.) I

a,

and

yiaaf)(\).

U€X.

(EV.

aE71 /a/=/

a"

a r e d e f i n e d a c c o r d i n g to

aa:=3e3l.3; , ( h e r e in aa w e

d : = f (13 -i3 ) a n d Li uj "i p r i n t t h e s y m b o l I3 in b o l d f a c e in o r d e r to a=(B,y.S)eX ,where

J

d i s t i n g u i s h 3" f r o m t h e real derivative 13" f o r a E I N t

of Remark 1.43 b e l o w ) ; i n s t e a d o f a"f w e a l s o w r i t e ( i i ) if j€N, w e w r i t e

a i : =13,,j;

i f jelN, w e w r i t e

-

c o m p a r e also t h e p r o o f

f(a).

I3

=i

a n d S c , + , + j .. -- d-

=i

W i t h t h e h e l p of t h e n o t a t i o n i n t r o d u c e d a b o v e T a y l o r ' s f o r m u l a c a n be w r i t t e n in t h e u s u a l f o r m :

Lemma 1.26. Let

n

b e an open subset o f V which is s t a r - l i k e with respect t o

the origin. let r e N , arid let f 6 C ' ( f ? ) . Then for ever) xt-f? we have 1

Proof. In view of (1.32) by t h e rnain t h e o r e m of c a l c u l u s w e have

22

I.

f ( x )- f ( O ) =

c

( A l m o s t ) Quasihomogeneous F u n c t i o n s

1

x = J ( a ~ f ) ( s xds ) .

lal=l

0

This is the assertion for r = l . Following the proof of t h e standard formula o n e can now deduce the general case by induction on r

:

In order to derive the formula

for r + l from the one for r one fixes a € U satisfying I a I = r , applies the case r = l to ( a " f ) ( s x ) instead of f ( x ) , and substituting t for s t , changing the order of integration and taking

?

J r(l-s)r-ids = (l-t)r

t€CO,11,

t

Putting everything together one completes the induction step.

Obviously, the product rule remains valid for the operators a , . Consequently, one deduces the following version of the Leibniz formula by following the lines of the proof of the standard version.

Lemma1.27. For arbitrary r € N , f , g E C ' ( X ) , and

a a ( f g )=

);(

cr62C

satisfying I a l S r w e have

apfaaP-ag.

p EU, Sa

The Leibniz rule can be used to carry out the easy proof of

h p 0 8 k i O n 1.28. ( i ) For arbitrary cr,p 6 2C we have

I 0

otherwise

( i i ) Consequently, the restrictions o f the functions sa, a € X , t o any non-empty

open subset o f V are linearly independent.

I

1.c

23

( A l m o s t ) Quasihomogeneous Polynomials

For the study of t h e behaviour of a polynomial function P: V-

C under t h e action

of Mt it is most convenient to express P as a polynomial in the variables y , z , and Z where ( y , z ) be real-complex coordinates. Here Lemma 1.26 yields

(1.33)

P ( ~ =)

c

UCU

1,(a'p)(o)

X € V .

Xu,

a.

In order to guarantee that the functions xu, a € % , behave nicely under the action of

M, we postulate additional properties of the bases introduced in Convention 1.24.A. Conventlon1.24.B. ( i ) From now on w e suppose that the bases A of V,

and B

of Vc are chosen in such a way that they contain bases of the generalized eigenX E ~ We . define p € I R d and C6CC by the conditions

spaces G,(X),

( i i ) If necessary we even require that the real matrix M1R of M,, to A and the complex matrix M C of M,,

with respect

with respect to B are i n Jordan cano-

nical form. This means, in particular, that the matrices MIR-diag(gl,. . . , p d ) and are nilpotent where by d i a g f . . . I we denote diagonal matri-

M, - diag(C1,.. . ,

, 4 , Q ' E E s u c h t h a t PfP', is a s u b s e t o f C o f m e a s u r e

0 . H e n c e w e c a n c h o o s e zEC s u c h t h a t

(1.45)

ID,I = ID1 a n d IE,I

= IEl

whereC,:={<j,z>; j 6 C ) . Wechoosek,t€DandK,LEEsuch that < k , z > = m i n D , , t K , z > = m a x D, , < Q , z >= m i n E, , a n d < L , z >= m a x E,

< k ' + t ' , z >= t k ' , z )

+

.

Making u s e of t h e e q u a t i o n

O s u c h t h a t t h e m a p N x I I - E , I + E C - Z , ( y , t ) H M t y , is a r e a l a n a l y t i c d i f f e o m o r p h i s m . I t f o l l o w s t h a t U : =N,

is a n

o p e n subset of V . In view of t h e c l a i m proven b e f o r e a n d in view of (1.18) a n d

(1.73) a l l t h e q u a s i h o m o g e n e o u s r a y s t h r o u g h t h e p o i n t s of N i n t e r s e c t N t r a n s v e r s a l l y . H e n c e w e may a p p l y P r o p o s i t i o n 1.63 to U i n s t e a d of X a n d o b t a i n t h e

desired x .

so

I.

( A l m o s t ) Quasihomogeneous Functions

Notatlon 1.65. By Y ( M I w e d e n o t e t h e s e t o f a l l p o i n t s

x E V satisfying o n e (and

h e n c e e a c h 1 of t h e c o n d i t i o n s o f Lemma 1 . 6 4 .

In f a c t , t h e p o i n t s in V ( M ) s a t i s f y t h e c o n d i t i o n ( c ) in a m u c h s t r o n g e r f o r m :

Lemma1.66. The s e t 9 Z : = { ( . \ , M t ~ ) :x € L t ( M ) , t E l O , + w C } is a real-analytic submanifold of V x V , and the map 3 ? - - + 1 0 . + d ,

( x , M , . \ ) H t , i s well-defined and

real -analytic. proOf. W e set Y : = Q ( M ) . By Y : Y x l O , + ~ C - Y x Y

w e d e n o t e t h e real a n a l y t i c

i m m e r s i o n d e f i n e d by ( x , t ) H ( x , M , x ) . Let x E Y , a n d c h o o s e U a n d x a s in c o n d i t i o n ( a ) of Lemma 1 . 6 4 . T h e n by ( y , z ) H x ( z ) / x ( y ) a m a p c I , : U x U ~ l O , + c c is ~l w e l l - d e f i n e d which is r e a l - a n a l y t i c . S i n c e x is q u a s i h o m o g e n e o u s of d e g r e e 1 it f o l l o w s f r o m E u l e r ' s e q u a t i o n t h a t Dx h a s n o z e r o s . H e n c e 0 is a s u b m e r s i o n . D e n o t i n g by FI,: U x U + U

t h e projection o n t o t h e first f a c t o r , w e conclude t h a t

From this t h e assertions follow.

Via t h e c o n d i t i o n ( b ) o f Lemma 1.04 w e d e d u c e f r o m Remark 1.8 t h e f i r s t i n c l u s i o n

o f t h e f o l l o w i n g r e m a r k ( e m p l o y i n g N o t a t i o n l.SO.(i) f o r X = V ) , t h e s e c o n d inclus i o n b e i n g valid in view of Lemma l.G4.(c) :

Remark 1.67. V , u V - c Y ( M ) C V \ E M ( o O ) .

I

N o t e t h a t t h e c o m p l e m e n t o f V + u V _ in V e q u a l s G M ( a o ) . A t f i r s t o n e m i g h t s u s p e c t t h a t f o r t h e i n c l u s i o n o n t h e r i g h t - h a n d side a c t u a l l y e q u a l i t y h o l d s . H o w e v e r , in case t h e r e e x i s t s a n e i g e n v a l u e X EoO\(0) 2 3 t h i s is n o t

of a l g e b r a i c m u l t i p l i c i t y

so. T h i s c a n b e s e e n f r o m t h e e x p l i c i t d e s c r i p t i o n o f t h e set

G ( M ) w h i c h is p o s t p o n e d to s e c t i o n ( g ) b e l o w . Now w e are g o i n g to c o n s t r u c t g l o b a l l y d e f i n e d positive q u a s i h o m o g e n e o u s f u n c t i o n x o n V +

,

In c a s e M is o f

t h e f o r m ( 1 . l . a ) t h i s is very e a s y :

Example1.68. Suppose that M is o f the form ( 1 . l . a ) . (i)

If p€l O.+a7C1' then ( X . N ) =

td". S f ' - ' )

satisfies the condition Ib) of Propo-

51

1.f The Hypersurfaces S"

sition 1.63; the corresponding x that satisfies ( a ) is denoted by x p .

(ii) In general, i f one sets

s,p : = { ' V E X :

ciEI,.Vi--3=1}

then the condition ( 6 ) o f Proposition 1.63 is satisfied f o r ( X , N ) = ( X , , S y ) ; the corresponding real analytic quasihomogeneous function x is denoted 6-v

Proof. ( i ) : Since the assumption o n p means that J , = N,

xi.

and S f = S"-' t h i s is

a special case of ( i i ) .

( i i ) : Let

5 E V = R"

x E S)l. We define xj

by

if jcj?

0 otherwise

Then

5

is a normal unit vector t o S)l at x . Since

t < f , i : ( ~ )( t> p j= ) xx 2i >

o

i EJ,

it follows that i i ( l ) , the generator of T, R M , ( x ) , does not belong to T,SZ.

Since

the map IO,+00[3 t H

2 j

EJ+

I

is strictly increasing (resp. decreasing), converging to

and to 0 (resp.

+ a )as

t + O the assertion follows.

+m

( r e s p . 0 ) as t + + m

H

For general M a similar construction works. For the proof we require

Lemma 1.69. For ever)'

E

> 0 there esists a scalar product

the following properties - here

(i)

> on V having

11 . I[ denotes the norm defined 6-1 >

:

are pairwise orthogonal:

(ii) f o r every A E o we have

11.731

'

(ReA - E ) //s// 5 > 5 I Re A + E ) //.Y//

(iii) f o r ever-)

.Y

V we have

'

,

s E G,

(A):

52

I.

(Almost) Quasihomogeneous

proOf. Let h ~ a let , U be an M-cyclic subspace of G,(X),

Functions

and let ( bl , . . . , b d )

be a Kx-basis of U w h i c h brings the matrix of Mx into Jordan canonical form, i.e. (1.11) holds. Setting ci : = Ei-'bi we obtain another basis of U such that Ncl = O and Nci+' =

E C ~ j, € l N d - l .

We now define

d d d : = R e c ziWi ' j=1

J

j=1

j=1

and observe that

so that by the Cauchy-Schwarz inequality we obtain

Writing V as a direct sun1 of such spaces U and defining
i n such a

way that these spaces are orthogonal to each other w e arrive a t the assertions

(il and

a).

For the proof of (iii) let x € G , ( h ) \ ( O ) ,

define y:IR+V

by y ( s ) : = e x p ( s M ) x ,

observe that y ' ( s ) = M y ( s ) , and compute

~dl l y ( s ) I =l ~ ~ Y ( s ) ~ L - r and m i n ( D - r , n + l + D - r - ( L - r ) ) = n - and obtain t h a t

I.

60

( A l m o s t ) Quasihomogeneous Functions

Moreover, by (1.88) again and in view of n = d - L = D-1 we see that (B1-'qlIL= ( - l ) n ( l i n ) / ( k ) = ( - l ) D - l DL '

Since L = D+1 it follows that

Inserting t h i s into (1.91) and dividing by

D

we conclude that a D = ( - l ) D c D , a s

desired.

(ii).

Let n be t h e largest natural number strictly smaller than D. N o t e that i n

case d is odd (resp. even) we have n = D - 1 / 2 , i.e. d - n = D + l / Z (resp. n = D - 1 , i.e. d - n = D t l ) . For every j € N 0 such that j 5 D we set a i : = P ( ' ) ( O ) / j ! . We are going to construct P, in the following form d

P, : =

ajTi +

OsjsD

aj(s)Ti. j=cl-n

where the coefficients a i ( s ) , d-n 5 j < d , are to be defined in such a way that for every i € { O ) u N , in (1.89), equality holds for L = i + l (and a n , j = a i ( s n ) )even before the limit is taken. To carry t h i s out we denote by B ( s ) the matrix w i t h entries si-i bii ,

O < i < n , d-nSj

w h e r e s , : = log t,

.

By c h o o s i n g s u b s e q u e n c e s w e achieve t h a t

w x : = lim e x p ( - X s , ) n+m

e x i s t s f o r every X E ~ , .

In view o f w h a t w a s s a i d a b o v e w e c o n c l u d e f o r every iclNdz w e have

64

I.

( A l m o s t ) Quasihomogeneous F u n c t i o n s

w h e r e u n d e n o t e s t h e p r o j e c t i o n o f y n o n t o Z . In view of (1.99) L e m m a 1.7S.(i) implies t h a t l o i z ( v ) - l 5 ( d z - l ) / Z f o r v € ( x , y ) , i.e. b o t h x and y satisfy (1.9S), a n d 2 0 if dz is odd t h e n ( y Z l i = ( - l ) i - l W x z ( x z ) i f o r j = ( d z + 1 ) / 2 . T h i s y i e l d s t h e equation 3,(y)

=,30(x)

a n d t h e c o n d i t i o n (1.98). Since in case S o ( x ) f @ t h e

a ( x ) - t u p l e w : = ( w x ) x E o ( x ) b e l o n g s to T ( a ( s ) ) w e see t h a t t h e a s s e r t i o n s (i), " C and (iil,

"+

"

"

are proved.

C o n v e r s e l y , w e s u p p o s e t h a t x a n d y s a t i s f y (1.9s) a n d t h a t in c a s e

, 3 0 ( ~ ) fw@ e

c a n c h o o s e w E T ( o ( x ) ) s a t i s f y i n g ( l . ( J 8 ) . By Lemma 1.11 w e f i n d a s e q u e n c e ( s , ) " , ~ ~ in 10,iaC c o n v e r g i n g to +as u c h t h a t w x = l i m e s p ( - X s i ) f o r every X E o ( x ) . By j +

c h o o s i n g a s u b s e q u e n c e w e achieve t h a t w x : = lim e x p ( - X s i ) j+m

e x i s t s f o r every X E c r o \ a ( x ) , a s w e l l . In view o f ( l . ( J 8 ) t h e a s s u m p t i o n o n i z ( x ) a n d i z ( y ) a l l o w s u s For every ZE,?, to a p p l y Lemma 1 . 7 5 . ( i i ) (its a s s e r t i o n b e i n g trivial f o r d = O ) to t h e d a t a d : = d z - l ,

C i : = ~ ~ ~ ( x ~ ) ~ - ~ / a( njd - Pl =) P! z,, y z .

In t h i s way w e o b t a i n a real a n a l y t i c f u n c t i o n T z : t h e a b b r e v i a t i o n yz Kxz-isomorphism lim Isl+-

:=

PZ

0

Tz

( h e r e E Z : (K,,

k-(K,z)dZ

) d Z -Z

such t h a t with

d e n o t e s t h e canonical

a s s o c i a t e d w i t h t h e b a s i s 2\z) w e have )(i-l)

rz(s) -

lim ( P z , ~ ~ ( ~ ( ) 0 )= ( Y Z ) i

-Isl+m

and lim

(Pz,yz(s) ,(i-l)

( s )=

Wxz

1 0

so s m a l l t h a t t h e c o m p a c t n e i g h b o u r h o o d K : = { y € V ; IP+yl S E , I M o ( y - x ) l < E

o f x is c o n t a i n e d in X . T h e n

C o n s e q u e n t l y , by Lemma l . W . ( i i ) t h e set L : = K M n S X is a c o m p a c t s u b s e t of X ,

1. h

73

Q u a s i h o m o g e n e o u s Polar Coordinates

S i n c e K is e q u a l to K \ k e r M a n d s i n c e K is c o m p a c t o n e f i n d s R > 0 s u c h t h a t

K is c o n t a i n e d in t h e set J : = { M , x ; x € L , t E I O , R l } . M o r e o v e r , by (1.107) a n d by c o m p a c t n e s s , a g a i n , t h e r e is S E l 0 , R l s u c h t h a t J \ K is c o n t a i n e d in t h e c o m p a c t s u b s e t { M,x : x E L , t E C6,RI } of X, . H e n c e f o r every k E ( 0 )N~ N the assumption o n qk implies t h a t

&IK

is i n t e g r a b l e if a n d o n l y if

&IJ

is i n t e g r a b l e , a n d w e

have to p r o v e t h a t t h e l a t t e r is t h e c a s e for every k E ( O ) u INN if a n d o n l y if Rem > - p .

( a ) * ( b l : w e may a s s u m e t h a t P = m a x { k E ( 0 ) u l N N ; x E s u p p & } a n d t h a t so s m a l l t h a t K n s u p p q k = @ f o r every k€lN,

E

is

such t h a t k > 0 . I t then follows

by t h e e q u a t i o n ( 1 . 3 0 ) ; - s e e t h e p r o o f of L e m m a 1 . 0 0 - t h a t (1.108)

qp(M,y) = t m i g ( y )

for arbitrary y E K a n d t ~ l O . + ~ [ .

Note t h a t by L e m m a 1.90 q P I L is i n t e g r a b l e w i t h r e s p e c t to x . If

1'1 q p ( 9 ) 1d G ( 9 ) L

w e r e e q u a l to z e r o t h e n in view of (1.108) a n d P r o p o s i t i o n 1.86 q p w o u l d v a n i s h 0

a l m o s t e v e r y w h e r e o n t h e set LM ( w h i c h e q u a l s K M \ k e r M ) so t h a t s u p p i p n K w o u l d be e m p t y in c o n t r a d i c t i o n to t h e c h o i c e of x a n d 1 . C o n s e q u e n t l y , s i n c e by ( a )

q p is

locally i n t e g r a b l e so t h a t

iplJis

i n t e g r a b l e w e d e d u c e f r o m Propo-

s i t i o n 1.86 a n d (1.108) t h a t t h e f u n c t i o n t H t ' n * p - l is i n t e g r a b l e o n 1 0 , R l . S i n c e t h e l a t t e r is t h e c a s e if a n d o n l y if Re m + p > O t h e p r o o f o f t h e i m p l i c a t i o n "(a)*(b)"

is c o m p l e t e .

( b ) + ( a ) ' : I t f o l l o w s by P r o p o s i t i o n 1 . 8 6 , by Fubini's t h e o r e m , a n d by ( 1 . 3 0 ) t h a t R

J'I{,(y)ldy K

5

R

N

5

.I'

5 j'Iio(y)Idy = , \ ' I q 0 ( M , 8 ) l d ~ ( B )t'? J 0 L

.[Iqk(a)ldG(9) k=O L

J'

Ilogtl

k

t

R e m +!'-I

dt

0

S i n c e by Lemma 1.90 t h e q k a r e G - i n t e g r a b l e o n t h e c o m p a c t s u b s e t L o f X, n S x t h e a s s u m p t i o n o n m i m p l i e s t h a t t h e r i g h t - h a n d side o f t h e p r e c e d i n g e s t i m a t e

is finite. C o n s e q u e n t l y , qo is locally i n t e g r a b l e o n X .

We close t h i s s e c t i o n b y a n o t h e r a p p l i c a t i o n of P r o p o s i t i o n 1.86:

Lemm 1.92. Suppose that u = o + . Let q o : V+

C' be continuous and almost quasi-

74

I.

( A l m o s t ) Quasihomogeneous F u n c t i o n s

homogeneous o f degree m . Suppose that qo does not vanish identically. Then the restriction o f qo t o

m.First of

V\ K(O,lI

is integrable i f and only i f Rern < - p .

all w e observe that by Proposition 1.51.(ii) the kth order deficiency

qk of qo is continuous, a s well, for every kEN,

where N : = ordMl(qo). Let x

be t h e function x, of Proposition 1.70. Then Sx = S"-' is compact. I t follows from Proposition 1.86 that qo is integrable o n V \ K ( 0 . 1 ) if and only if the function

g : C 1 , + ~ C x S x ~ Q (t,3) I ,

t"-'qo(M,3),

isintegrableon C t , + a l x S " with respect

to d t @ x " .

2". Since qo

i s locally integrable it follows from Remark 1.50 that the restriction

to V \ K ( O , l ) of every deficiency of qo is integrable. Hence, in view of Lemma 1.48 we may assume that qo is quasihomogeneous of degree m . B u t in t h i s case we have g ( t , 8 ) = t m + " - 'qo(3) so that i n view of the compactness of S" and since J'sx I q o ( 8 ) I d9 # 0 the Fubini theorem shows that the function t H tm+'-'

I' S

inte-

grable o n C l , + a C , i.e. R e m + p < O .

x'. By the equation

(1.39) we have

( t . 4 ) E 10,+03[ X S " . Since the assumption on m implies that the functions t

-'w , ( t ) ,

I+ t m + p

O = t m , and T is quasihomogeneous of degree m i f and

o n l y if (2.3)'

t € IO,+mI.

TOM,= t " ' T ,

Example 2.2. The Dirac distribution

So ( a t 01 is quasihomogeneous o f degree - p

.

I

The analogue of Proposition 1.I'J is valid:

Propoeltlon 2.3. Let T € B ' ( X ) be quasihomogeneous o f degree m . Let P E @ , and let P : X x V * -

C be a C"~-copol~rnomia/ function (in the sense o f Definition t.161

which is quasihomogeneous o f degree 4 dnd o f t j p e M x ( - M I

'.

Then P(.\,dI T is

quasihomogeneous o f degree m + 4 .

Corollary 2.4. Let TED'IXI be quasihomogeneous o f degree (;I

m

.

Then

P l d l T is quasihomogeneous o f degree m -4 f o r ever) polynomial function

P on V

'

which is quasihomogeneous o f degree 4 6 C and o f tqpe M

*:

78

(iil

11.

( A l m o s t ) Quasihomogeneous Distributions

q T is quasihomogeneous of degree rn+! for every q E Ca'(X) which is quasi-

homogeneous of degree !€ C . I

T h e r e s t r i c t i o n s of q u a s i h o m o g e n e o u s d i s t r i b u t i o n s o n X to o p e n s u b s e t s of X

are, of c o u r s e , q u a s i h o m o g e n e o u s , a s w e l l . O n t h e o t h e r h a n d we h a v e

Pmposltion 2 . 5 . ( i ) I f T C D ' ( X ) is quasihomogeneous o f degree rn then there exists a unique extension TM 6 D ' ( X M ) of

7 which is quasihomogeneous of degree m ,

as w e l l . l i i l The map T H TM is linear and ( w e a k l y ) continuous

proOf. Let K b e a c o m p a c t s u b s e t of X M . S i n c e t h e sets M , X ,

o p e n o n e f i n d s a f i n i t e s u b s e t R of 1 O , + a l s u c h t h a t K

C

r E l O , + a C , are

U r E R M r . X . Let ( Y , . ) , . ~ ~

b e a p a r t i t i o n of u n i t y o n K s u b o r d i n a t e d to t h e c o v e r i n g ( M , . X ) , . G R . F o r e v e r y ' ~ E C ~ ( K w e) set

< T M , 'p > : =

(2.4)

r'"

+'I

< T, ( cpr 'p )

0

M,.

>.

reR

Now w e f i x t ~ l 0 , t a Ca n d a f i n i t e s u b s e t S of

c h o o s e a p a r t i t i o n of unity

UscsMsX,

10.+aC s u c h

( $ s ) s e ~o n

M,K

t h a t M,K

C

s u b o r d i n a t e d to t h e

If

covering

(MsXISGs,

Ml/sK,.,s

t h e n M s x E suppcp,.oMl/,=Mtsuppcp,. so t h a t M s / t x E suppcp,.C M , X .

and

T h i s s h o w s t h a t t,.,s : =

set

K,.,,:= s ~ p p n+s u~p p ( ' p , . o M , , , ) n M , K .

E J M ( M l/sKr,s

;

xE

X ) . N o t e t h a t t h e last set is c o n t a i n e d

in J M ( s ~ p p ( $ , ( ' p , c p ) ~ M 1 ~ , ) ~XM ) , b; e c a u s e s u p p ( + s ( ' p r q J ) O M l / t ) is a s u b s e t

of K r , s . I t f o l l o w s t h a t sm+' < T , ( $ , ' p o M l / , ) o M S

t-v

>

=

sss

=

tm

c c

pl+v

nl+v

tr,s

(T,(J1,(cp,cp)oM,/t)oM,>

=

s€S r€R

= tm

C

rm+v

= tm

C

C



reR

w h e r e t h e l a s t e q u a l i t y comes about s i n c e b o u r h o o d of s u p p 'p .

cscs$ , O M ,

is e q u a l to 1 o n a neigh-

2.a

79

Quasihomogeneous Distributions

N o w , t h e case " t = 1

"

s h o w s t h a t t h e r i g h t - h a n d side of ( 2 . 4 ) does n o t depend

o n t h e c h o i c e o f R a n d ( v , . ) , . ~ ~i.e. , T M is w e l l - d e f i n e d . M o r e o v e r , t h e g e n e r a l case s h o w s t h a t ( 2 . 3 ) is valid f o r T M i n s t e a d o f T. F r o m ( 2 . 4 ) it is o b v i o u s t h a t

t h e r e s t r i c t i o n o f T M to C Z ( K ) is l i n e a r a n d c o n t i n u o u s . If K i s c o n t a i n e d in X we m a y t a k e R = ( 1 ) so t h a t ( 2 . 4 ) s h o w s t h a t TMI,=T.

T h e assertion ( i i ) a l s o

follows from (2.4). w

Support a n d singular support of quasihomogeneous distributions are quasihomogeneous:

Ropositlon 2 . 6 . Let T E D ' ( X ) be quasihomogeneous. Then (i)

supp T = Isupp T ) , n X :

( i i ) sing supp T = (sing supp T ) , n X

.

P r o o f . (i). Let Y b e t h e o p e n s u b s e t X \ s u p p T of X . T h e n f o r every x E YM n X o n e f i n d s a n o p e n n e i g h b o u r h o o d U of x in Y M n X a n d a n u m b e r t E l O , + a l s u c h t h a t M1,,U

C Y . H e n c e f o r every y € C F ( U ) w e d e d u c e f r o m (2.1) t h a

s u p p ( c p o M , ) C Y a n d t E J M ( s u p p ( c p o M , ) ; X ) so t h a t by a p p l y i n g ( 2 . 3 ) to y o M i n s t e a d of y w e o b t a i n : O = t m " ' < T , c p o M t ) = < T , y p ) . (ii).T h i s t i m e w e set Y : = X \ s i n g s u p p T a n d let f E C m ( Y ) b e s u c h t h a t TI,=T, S i n c e by t h e a r g u m e n t p r e c e d i n g Definition 2.1 f is q u a s i h o m o g e n e o u s it e x t e n d s to a q u a s i h o m o g e n e o u s cmf u n c t i o n f M : Y M

+

by P r o p o s i t i o n 1.57. C o n d i t i o n ( i )

i m p l i e s t h a t T , f M , a n d T c o i n c i d e o n Y M n X . H e n c e Y M n X C Y , i.e. Y M n X = Y . a n d t h e assertion follows. w

Corollary 2 . 7 . Suppose that (1.14) holds. Let T E D ' (V , ) b e quasihomogeneous. I f s i n g s u p p T n U is e m p t y for s o m e neighbourhood U o f k e r M then T is induced by a Ccufunction.

Proof. By R e m a r k 1 . 8 . ( i ) t h e a s s u m p t i o n (1.14) i m p l i e s t h a t lim M t x = M o x t+o

f o r every x E V +

.

Hence ( U n V , ) ,

E

U

= V + , a n d t h e a s s e r t i o n f o l l o w s by Proposi-

tion 2.6.(ii). w

We n o w c o m e to Euler's equation f o r q u a s i h o m o g e n e o u s d i s t r i b u t i o n s . L e t TE B'(X)

80

11. ( A l m o s t ) Q u a s i h o m o g e n e o u s D i s t r i b u t i o n s

be f i x e d . W e f i r s t have to d e t e r m i n e t h e t r a n s p o s e ( w i t h r e s p e c t to t h e d i s t r i b u t i o n a l d u a l i t y b r a c k e t ) of t h e E u l e r o p e r a t o r 3,

d e f i n e d by (1.29).

Remark 2.8. The transposed operator ' f 8 M ) o f d M with respect t o the duality bracket between 3 ' f X ) and C g f X ) is equal to - 8 M - p . Hence

proOf. Let T E B ' ( X ) a n d ' p € C g ( X ) . Recall t h a t t h e t r a n s p o s e d o p e r a t o r in q u e s t i o n is d e f i n e d by t h e e q u a t i o n

< d M T , ' p> = < T , t ( d M ) v ,>.

E x p r e s s i n g d M in coordi-

n a t e s via t h e f o r m u l a (1.20)' w e see t h a t n

( a M ) 'p

=c

n

( - dj ) (

j=1

I1

Mj k

X k 'p)

k=1

=

II

n

j=1

k=l

-cMi c ( c Mi j cp

i=l

-

k Xk

) dj 'p = - p 'p - d M ' P -

Lemma 2.9. The following conditions are equivalent: ( a ) T satisfies 12.3) f o r arbitrar) p€C;fX)

and t€lO .+ m C such that Cl, t l

is contained in J M ( s u p p p ;X) :

f b ) f o r ever) p E C F I X ) there t € l t- E , I

IS

s > 0 such that T satisfies 19.3) f o r ever)

+EL,

fc)

( 3 , - m ) T = 0.

T h i s is a n i m m e d i a t e c o n s e q u e n c e of

Lemma2.10. Let p6C;fX). g :J M Isupp p ;X I

+C

g ' ( t )= t-nl

Then bj g f t ) : =t-'"-I-l

a C' function

i s well -defined sa tis[\,ing -P- 1

( ( 8 M - m ) T, p ~ M l > / .~

proOf. T h a t g is a C1 ( i n f a c t , a C")

f u n c t i o n is c l e a r . Applying t h e c h a i n r u l e

t w i c e a n d m a k i n g u s e of (1.5) a n d (1.4) w e o b t a i n d t ( ' p 0 M 1 / , ) ( x ) = D q ( M l / , x ) * ( ( - l / t 7- ) f i1 M M l / , x )

=

= - t I D ( ' ~ o M i / ~ ) ( x ) * M -x -I=d M ( ~ o M l / t ) ( ~ ) . t f o r a r b i t r a r y x € X a n d t € J M ( S U p p V ; X ) . I t f o l l o w s t h a t g ' ( t ) is e q u a l to

81

2.a Quasihomogeneous Distributions

In view o f ( 2 . 5 ) this e q u a l s t h e right-hand s i d e of t h e desired equation.

H

An obvious consequence of Lemma 2.9 is

Proporltlon 2.11. (i) If T is quasihomogeneous of degree m then d M T = m T ; (ii) the converse is valid provided that JMM(K;X1 is an interval for evey)' compact

subset K OF X .

I

Of c o u r s e , t h e l a s t condition is satisfied if X is quasihomogeneous.

In passing we a r e going to recall t h e explicit description of all homogeneous distributions o n IR. To t h i s purpose we require

Lemma2.12. Let T ~ a ' ( l O , + a Cbe l homogeneous of degree m . Then there is c 6 C

M . Since

t h e distribution S : = x - " ' T is homogeneous of degree 0 it f o l l o w s

by Euler's equation t h a t 0 = 0 ' S= x S ' , i.e. S ' = O . Consequently, S is induced by a c o n s t a n t f u n c t i o n , and since T = x"'S t h e assertion f o l l o w s .

Roposltlon 2.13. The space S?h(R ) of distributions on R which are homogeneous of degree m is a two-dimensional

-m@".

and bj

(ii)

l I n .

So(-m-''

vector space spanned bj

if ' -m€N

lil

\tn. ,\rif

(here we adopt the notation of

Hormander C l l l . pp. 6 8 and 72). proOf. I f m @ -N then

xy

and xl" a r e homogeneous by ( 3 . 2 . 7 ) o n p. 71 in Horman-

d e r [ I l l . I f m E -N then it follows by Example 2.2 and Corollary 2 . 4 . ( i ) t h a t S d - m - l ) " is homogeneous of degree m . Moreover, since x"! = ( x r ) , m E C \ ( - N ) . we conclude f r o m t h e f o r m u l a s (3.2.10)' and ( 3 . 2 . 8 ) in C111 t h a t

xm is

homogeneous of

degree m in c a s e " m E -N ". On t h e o t h e r hand, fixing any non-trivial TE.!ijA(IR) we conclude from Lemma 2.12 t h a t there are constants c +E C such that

a2

11.

( A l m o s t ) Quasihomogeneous Distributions

S i n c e x y v a n i s h e s o n ~10,+00C this means t h a t t h e restrictions of t h e distribut i o n s T a n d c, x y + c- x!?

to I R \ ( O ) c o i n c i d e . T h i s i m p l i e s t h a t f o r s o m e c o m p l e x

polynomial P of o n e v a r i a b l e w e have

T = c,x~+ccx!?+PP(a)&,. If m d-IN t h e n P v a n i s h e s identically s i n c e S:J) jENo.

is h o m o g e n e o u s o f degree - 1 - j ,

M o r e o v e r , if - m € N t h e n we c o n c l u d e t h a t P = C X - ~ - 'f o r s o m e CELT ,

a n d t h e l e f t - h a n d side of t h e e q u a t i o n

T - ( - l ) - m c - ~ m- c S d - r n - l ) -

- (c,

- ( -1) -'"c-) x:"

( w h i c h is valid by t h e f o r m u l a (3.2.10)' in H o r m a n d e r C111) is h o m o g e n e o u s o f

degree n i . Since in view o f t h e f o r m u l a ( 3 . 2 . 8 ) in C111 t h e d i s t r i b u t i o n x y ' is n o t h o m o g e n e o u s t h e r i g h t - h a n d s i d e o f t h e p r e c e d i n g e q u a t i o n m u s t vanish so t h a t

T is a linear c o m b i n a t i o n o f

xm a n d

So( - m - ' ) , a s w a s to be s h o w n .

m

N e x t w e n o t e a n i m p o r t a n t c o n s e q u e n c e o f Euler's e q u a t i o n :

Ropositlon 2.14. I f T E ~ I ) ' ( Xi s) q u a s i h o m o g e n e o u s t h e n i t s a n a l y t i c wave f r o n t set WFA(T) i s c o n t a i n e d in t h e s e t

(2.6)

r,

( X I :=

i(,\, t i E X X i. *

:

c t.M \ > = o }

proOf. S i n c e r M ( X ) is t h e c h a r a c t e r i s t i c set of t h e d i f f e r e n t i a l o p e r a t o r d M - m t h e a s s e r t i o n f o l l o w s f r o m P r o p o s i t i o n 2.11.(i) a n d f r o m T h e o r e m 8.0.1 in H o r m a n d e r 1111. m

F o r t h e rest of t h e p r e s e n t s e c t i o n w e s u p p o s e t h a t w e c a n f i x a Cm f u n c t i o n

x:X-+lO,+wC

which is q u a s i h o m o g e n e o u s of d e g r e e 1 . By P r o p o s i t i o n 1.63 t h e

set S " = x - ' ( l ) is t h e n a Cm h y p e r s u r f a c e in X , a n d by P r o p o s i t i o n 1.72 t h e m a p p,

(see N o t a t i o n 1.71) is a Cm s u b m e r s i o n o f X o n t o S x . T h i s l e a d s to t h e f o l l o w -

ing d e s c r i p t i o n o f

r M ( x ) . Recall

manifold Y then for any s u b s e t

t h a t if Q : X + Y

r

is a

cm m a p

o f X i n t o a C'"

of t h e c o t a n g e n t b u n d l e T * Y o f Y t h e set

{ ( ~ , T : p q ) ; X E X , ~ E T ~ ( ~ ( )p Y ( x ,) , q ) € T } is d e n o t e d by p ' ( T ) .

Lemma 2.15. fi) r M ( X ) e q u a l s of t h e zero s e c t i o n in T * S x .

p," ( f x S x ) w h e r e ?*S"

d e n o t e s the complement

83

2.a Quasihomogeneous D i s t r i b u t i o n s

'*

(ii) The intersection of p:(T S " ) and the conormal bundle of S" in X is empty.

mf. (i)Let : x € X . Since px is a

submersion t h e set R : = p ; ' ( p , ( x ) )

is a sub-

manifold of X s u c h t h a t its tangent space T,R is equal to kerT,p,

and s u c h t h a t

t h e annihilator (T,R)O of T,R i n T:X

On t h e o t h e r

equals t h e image of T:p,.

hand, R is t h e image of t h e Cm map i,:

lO,+~C-

X

~

t

HMtx,

which in view

of Lemma 1.64.(c) and Proposition 1.10 induces a diffeomorphism of I O , + m C o n t o

R. Consequently, t h e t a n g e n t space T,R is generated by i i ( 1 ) = M x .

(ii):

By J : S " + X

T;p;j

we d e n o t e t h e inclusion map. Let x € S X a n d q€T:SX. Then

belongs t o (T,SX)O i f and only if 0 = < T ~ p x . q , T , J - v =tq,T,(p,oJ)*v> >

i.e. if and only if q = 0 .

= ,

vET,SX.

H

Combining Proposition 2.14 and Lemma 2.15 with Corollary 8.2.7 in Hormander C l l l o n e obtains

Theorem 2.16. I f T is quasihomogeneous then the restriction TIsx t o S x i s welldefined a s the pullback bj the inclusion map J : S " + X .

Conversely, since p x is a C-

H

submersion it follows t h a t every distribution on S x

can be pulled back by p x t o a distribution o n X ( s e e e . g . C h a p t e r VI in Hormand e r C111). From t h e r e s u l t s of C h a p t e r V l l l i n 111 1 one easily deduces

Theorem 2.17. The map C o ( S x )-+ COOi'), J , H + o p , , extends t o a continuous linear map p:

T : =p:(v)

--+

:d)'(Sx)

B ' ( X ) such that f o r every u ~ B ' t S " ) the distribution

has the following properties:

( i ) T is quasihomogeneous o f degree 0 . ( i i ) W F ( T )= p: W F ( u ) C

rMI X ) ;

(iiil i f x is real analJ,tic then ( i i ) remains valid with WF replaced bJ WF, ( s e e Definition 8 . 4 . 3 in Hormander C l l I ) :

( i v ) the restriction o f T t o S x - which is well-defined b-b, Theorem 2.16 equal t o v .

#

-

is

84

11.

( A l m o s t ) Quaslhomogeneous Distributions

Later we shall see that every T E D ' ( X ) which is quasihomogeneous of degree m is of the form

(2.7)

T = xmpz(v)

for some v E a ' ( S " ) ; i n fact, one can take v = T I S x (see Theorem 4.25 below).

t bb

Ihe Four 1e r 'I' r a nsI'orm o I' Q ua s 1homogencous I)1sC r i b ut 1ons

We have to fix a few notational conventions by way of which the standard IR"theory of the Fourier transform is reformulated in a coordinate-free manner. First of all, by .YpIV) we denote the FrCchet space of all rapidly decreasing complex-

valued Cm functions on V. By selecting any basis of V it can be defined as the space Y(IR").For any ' p E Y ( V ) the Fourier transform

$ : V*-@

$(el :=.~'exp(-i)'pd ( xx),

where < * ,

-- >

is defined by

5EV*,

V

denotes the canonical duality bracket between V* and V and where

- a s before- the integral is taken with respect to any orthonormal basis of V. The Fourier transform for functions 'pEY(V*) is defined in the same way once it is settled in what sense the integration on V* is to be understood. I n accordance

with the convention for V it suffices to fix a scalar product o n V*. To t h i s end the given scalar product < isomorphism V-V*, finition of

by

*, *. )

X H
, an almost quasihomogeneous A

C" function. Then sing supp T C (01.

proOf. Let g E C m ( V ) be s u c h t h a t TI;=T,,

a n d let m e @ be s u c h t h a t g is a l m o s t

quasihomogeneous of degree m . We fix mE4L and choose P E U ( M ) such t h a t R e 4 > R e ( m + r r M )+ p . Applying Remark 1.43 to V' i n s t e a d o f V a n d identifying V h * w i t h V , by Pa w e d e n o t e t h e p o l y n o m a l f u n c t i o n o n V s a t i s f y i n g P a ( i d ) =a" w h e r e here

a'

is a c t i n g o n d i s t r i b u t i o n s d e f i n e d o n V * . N o t e t h a t by C o r o l l a r y 1.36 a n d

R e m a r k s 1.43 a n d 2.18 Pa is q u a s i h o m o g e n e o u s of d e g r e e a M . Let Q : V * + @

be

any polynomial f u n c t i o n which is q u a s i h o m o g e n e o u s of d e g r e e P a n d of t y p e M'. N o t e t h a t by R e m a r k 2.18 a n d P r o p o s i t i o n 1.55 Q ( D ) ( P , I ; g )

is a l m o s t q u a s i h o m o -

g e n e o u s of d e g r e e m + a M - P t h e real p a r t of which is s t r i c t l y s m a l l e r t h a n - p . H e n c e , by L e m m a 1.92 t h e r e s t r i c t i o n of Q ( D ) ( P , I ; g )

to V \ K ( O , l ) is i n t e g r a b l e .

C h o o s i n g ' p E C F ( X ) e q u a l to 1 o n K ( O , I ) o n e c o n c l u d e s t h a t ( I - c p ) Q ( D ) ( P , T ) b e l o n g s to Z ' ( V ) . H e n c e its Fourier t r a n s f o r m is c o n t i n u o u s . Since t h e s u p p o r t A

of

'p

Q ( D ) ( P , T ) is c o m p a c t its Fourier t r a n s f o r m is a n a l y t i c . H e n c e Q aaT =

F ( Q ( D ) ( P , T ) ) is c o n t i n u o u s . S i n c e M is s e m i - s i m p l e , f o r every €,EV*\(O)

the

polynomial f u n c t i o n Q c a n be c h o s e n s u c h t h a t Q ( C ) # O . H e n c e , i t f o l l o w s t h a t A

a a T is c o n t i n u o u s o n V * \ ( O ) . T h i s i m p l i e s t h e a s s e r t i o n . rn

A c t u a l l y , t h e a s s u m p t i o n t h a t M b e s e m i - s i m p l e is s u p e r f l u o u s . In f a c t , a m o r e g e n e r a l a n d m o r e p r e c i s e v e r s i o n of P r o p o s i t i o n 2.23 will be p r o v e d in C h a p t e r 0 (see T h e o r e m 0 . 3 4 ) s h o w i n g , in p a r t i c u l a r , t h a t t h e c o n v e r s e of P r o p o s i t i o n 2.23

is valid, a s w e l l .

88

(Almost) Quasihomogeneous Distributions

11.

C c B M eromorp h1c Func 1I ons o I' Q uas 1homageneo us I)1s1r I b u1Lo ns

L e t fl be a c o n n e c t e d o p e n s u b s e t of C, a n d l e t h : fl-

function, i.e. t h e r e is a discrete s u b s e t

D

a ' ( X ) be a meromorphic

of fl s u c h t h a t for e v e r y q ~ e C g ( X by )

f l \ D 3 z H < h ( z ) , ' p > a m e r o m o r p h i c f u n c t i o n h,:Q-@ t h e p o l e order of h,

is d e f i n e d , f o r e v e r y

ZED

a t z b e i n g b o u n d e d by a c o n s t a n t i n d e p e n d e n t f r o m q . W e

fix z,EQ. A p p r o x i m a t i n g t h e i n t e g r a l by R e m a n n s u m s a n d e m p l o y i n g t h e B a n a c h S t e i n h a u s t h e o r e m ( n o t e t h a t t h e s p a c e CgCX) is b a r r e l l e d ) o n e d e d u c e s t h a t for every jEk a d i s t r i b u t i o n a j ( z o ; h )E 3 ' ( X ) is d e f i n e d by (2.10)

'p E

w h e r e y,:CO,2rl-C so s m a l l t h a t t h e

D

C,-C

x)

9

is d e f i n e d by y , ( t ) : = E e i t a n d w h e r e E E I O . + ~ Ch a s to be

closed disc

K(z,,,E)

w i t h t h e p o s s i b l e e x c e p t i o n of z,.

is c o n t a i n e d in

n

a n d c o n t a i n s n o p o i n t of

For o b v i o u s r e a s o n s o n e c a l l s a j ( z o ; h ) the

j t h Laurent coefficient o f h at z o . S i m i l a r l y , t h e n u m b e r

o r d ( z o ; h ): = inf { j 6 Z ; a i ( z o ; h ) f 0 ) ( w h i c h is f i n i t e by t h e d e f i n i t i o n of m e r o m o r p h y ) is c a l l e d t h e order o f h at z , . T h e set $ 1 ~ 1( h ) of p o l e s of h is, of c o u r s e , by d e f i n i t i o n e q u a l to { ~ € 0o r ;d ( z ; h ) < 0 } .

N o t e t h a t by t h e B a n a c h - S t e i n h a u s t h e o r e m t h e L a u r e n t s e r i e s rn

c o n v e r g e s to h ( z ) u n i f o r m l y o n every b o u n d e d s u b s e t of CTCX) a n d u n i f o r m l y f o r z in any c o m p a c t subset of K ( z o , r o ) \ v o l ( h ) w h e r e r,:=dist(z,,'F)ol(h)\lz,I u C \ Q ) . M o r e o v e r , let g : n +

C be a h o l o m o r p h i c f u n c t i o n s u c h t h a t g ( z o ) = m . In t h e

p r e s e n t s e c t i o n w e d e a l w i t h t h e s i t u a t i o n t h a t h ( z ) is q u a s i h o m o g e n e o u s of deg r e e g ( z ) . In view of P r o p o s i t i o n 2.5 it is n o loss of g e n e r a l i t y t h a t f r o m n o w o n w e a s s u m e t h a t X is q u a s i h o m o g e n e o u s . T h e p r i n c i p l e of a n a l y t i c c o n t i n u a t i o n s h o w s :

Remark 2.24. I f there is a non-empty open subset Z

o f O\ ~ u fL h ) such that f o r

every Z C Z the distribution h ( z ) is quasihomogeneous of degree g ( z ) then the same

is valid for every z €O\ pol Ih) . 8

2.c

M e r o m o r p h i c F u n c t i o n s of Q u a s i h o m o g e n e o u s D i s t r i b u t i o n s

89

How t h e quasihomogeneity of h ( z ) is reflected in the Laurent coefficients of h

is shown by

Propodtion 2.25. The following conditions are equivalent: ( a ) h ( z ) is quasihomogeneous o f degree g ( z ) f o r every

Z E ~ (lz,Iu~oZloC(h)); \

( b ) f o r every j E Z we have - w ) .

mf. By d M h we denote

the meromorphic function on fl mapping z E f l \ "pol h )

t o a M ( h ( z ) ) .From (2.10) onededuces foreveryjEZ that a i ( z o ; d M h ) = d M a i ( z O h) and

As two meromorphic functions o n the connected open set fl coincide if and only

if their Laurent expansions at z, do so the condition ( b ) is equivalent to the equality a M h = g h which in view of Proposition 2.11 is equivalent t o ( a ) . rn

If g = m then t h e condition ( b ) means that a j ( z o ;h ) is quasihomogeneous of degree m for every j E Z . However, in general the latter is valid for j = ord (z,; h ) , only. For example, if g ( z ) = z , z c n , then ( b ) reads as

so that aj(z,; h ) is definitely not quasihomogeneous of degree m if j > ord(z,; h ) and if h does not vanish identically.

In order t o obtain information on the behaviour of a j ( z o ; h ) O M , for j > o r d ( z o ; h ) one could compute aj(z,; h o M , ) in terms of the Laurent coefficients of t g h . However, for fixed j only finitely many of them are involved. Therefore we prefer to examine the condition ( b ) for arbitrary finite sequences of distributions not regarding whether or not they appear as coefficients in a Laurent series.

So we fix N E N , distributions To,...,TN o n X , and a sequence of complex numbers c k , k E INN. The analogue of condition ( b ) becomes j-1

(2.12)

( a M - m ) T~=

C

k=O

c ~ - ~ T ~

90

11.

( A l m o s t ) Quasihomogeneous Distributions

f o r every 0 5 j 5 N . To f o r m u l a t e t h e e q u a t i o n for T j o M , w h i c h is implied by (2.12) w e set N

:=

C(Z)

1 ce ( z - m ) e ,

ZEC,

e =I

a n d f o r a r b i t r a r y t € I O , + a C a n d k E N 0 d e n o t e by b , l t ) t h e k t h T a y l o r c o e f f i c i e n t

of t h e f u n c t i o n z

c

t C ( L )a t z = m , i . e .

t-+

m

(2.13)

( z - ~ T I )= ~t C ( = ) ,

b,(t)

ZEC.

k=O

Applying t h e binomial f o r m u l a o n e o b t a i n s m o r e e x p l i c i t l y t h a t k

(2.13)'

b k i (log t ) i

bk(t) = i=O

w h e r e boo : = 1 a n d N

cp""/4!

bki := aEA(k.j) 4=1

with

Proposition 2.26. Under the preceding h-bpotheses the relations i-I

tElO,+rnC,

hold f o r evegv j E N N u /01 if and on]-v if (2.12) is valid f o r every j 6 N N u 101.

A-oof. Let q € c g ( x ) F.o r

every j C N N u ( 0 ) w e d e f i n e f u n t i o n s g i : l O , + m [ l C

a n d hi : I O , + a I +QI by i

By Lemma 2.10 g i is d i f f e r e n t i a b l e s a t i s f y i n g

To c o m p u t e hi w e f i r s t d i f f e r e n t i a t e b o t h sides of (2.13) w i t h r e s p e c t to t a n d obtain

c

k=O

b k ( t )( z - m l k =

a

c(z)tC(L),

I n s e r t i n g ( 2 . 1 3 ) i n t o t h e r i g h t - h a n d s i d e , s u b s t i t u t i n g t h e d e f i n i t i o n of c ( z ) a n d comparing coefficients w e obtain t h a t

2.c

91

Merornorphic F u n c t i o n s of Q u a s i h o m o g e n e o u s D i s t r i b u t i o n s

t ~ l O , + m C ,k c N,

u{ O ) .

I n s e r t i n g t h i s w i t h k r e p l a c e d by j - k i n t o t h e e q u a t i o n o b t a i n e d by d i f f e r e n t i a t i n g t h e d e f i n i n g e q u a l i t y f o r h i , w r i t i n g L = k + t , a n d c h a n g i n g t h e o r d e r of s u m m a t i o n

N o w w e s u p p o s e t h a t ( 2 . 1 4 ) h o l d s . S i n c e t h i s m e a n s t h a t g j = hi it f o l l o w s in view

Of

bL-k(I) = SLk

that j-1

< ( d M - m ) TI . , c p > =g j ( l ) =h J ( 1 ) =

1cj-~

L=0

so t h a t ( 2 . 1 2 ) is valid. C o n v e r s e l y , w e s u p p o s e t h a t ( 2 . 1 2 ) h o l d s a n d t h a t ( 2 . 1 4 ) is a l r e a d y proved f o r every

LEN^-^

u ( 0 ) i n s t e a d of j . I n s e r t i n g t h i s i n t o (2.15) w e see t h a t j-i

h j ( t ) = t-"'-'

1c j - r < T L o M , , ~ > ,

t EIO,+~C.

L: 0

By (2.12) t h i s i m p l i e s t h a t hj = g i . S i n c e in view of b k ( 1 ) = S k ,

w e have g j ( l )=

< T i , ' > = h j ( l ) i t f o l l o w s t h a t h j = g j , i.e. ( 2 . 1 4 ) h o l d s . S i n c e by P r o p o s i t i o n 7.11 t h e c o n d i t i o n ( 2 . 1 4 ) f o r j = O is e q u i v a l e n t to (2.12) f o r j = O t h e a s s e r t i o n f o l l o w s by i n d u c t i o n . rn

In t h e s p e c i a l c a s e c k = c S l k w e have b k ( t ) =( c l o g t ) k / k !

so t h a t in view of

(1.37) t h e c o n d i t i o n ( 2 . 1 4 ) r e a d s as t E 10, +cot.

In t h i s case t h e f o l l o w i n g s l i g h t l y m o r e p r e c i s e version of P r o p o s i t i o n 2.26 h o l d s .

Proposltlon 2.26'. For ever) c E?!l (a)

the following conditions are equivalent:

(2.14)' is valid for j = N :

( b ) ( 2 . 1 4 ) ' i s valid f o r every j € N N ~ 1 0 1 : (cl

To is quasihomogeneous o f degree m , and

13, - m l T i = c q - , for

every j E N N .

proof. In

view of P r o p o s i t i o n 2.26 it s u f f i c e s to p r o v e t h e i m p l i c a t i o n " ( a ) J ( b )".

92

11.

( A l m o s t ) Quasihomogeneous

Distributions

Using t h e n o t a t i o n o f t h e p r o o f o f P r o p o s i t i o n 2.26 w e f i r s t o b s e r v e f r o m (2.15) that

N o w s u p p o s e t h a t (2.14)’ h o l d s f o r a f i x e d j € ! N N . T h i s i m p l i e s (see t h e p r o o f

of P r o p o s i t i o n 2.26) t h a t ( a M - m ) T j = c T j - ] so t h a t gj(t) = C t - m - l

< T j - , OM,,‘p > =

$ g j - l ( t ).

S i n c e (2.14)’ m e a n s t h a t gj = h j a n d h e n c e gi = hl it f o l l o w s t h a t g j - ] = h i - 1 so t h a t (2.14)’ i s valid w i t h j r e p l a c e d by j - I , a n d ( b ) f o l l o w s f r o m ( a ) by i n d u c t i o n .

To p r o v i d e a non-trivial e x a m p l e f o r a m e r o m o r p h i c f u n c t i o n of q u a s i h o m o g e n e o u s d i s t r i b u t i o n s w e f i x a polynomial f u n c t i o n P : V * - - + @ a n d r e c a l l t h e m e t h o d f o r o b t a i n i n g Bernstein’s fundamental solution E p of t h e d i f f e r e n t i a l p o l y n o m i a l P ( D ) . S u p p o s e t h a t P 2 0 . T h e n by

: = (2rc)-”.f

P(5)’

$(c)

d< ,

T€Y(V),

V*

a h o l o m o r p h i c f u n c t i o n v : { z < ( r : Rez > O } +

Y ’ ( V ) is d e f i n e d . As w a s s h o w n

by B e r n s t e i n in C21 (see a l s o Bjork 131 ) , it c a n be e x t e n d e d to a m e r o m o r p h i c

f u n c t i o n o n t h e w h o l e of C w i t h v a l u e s in Y ’ ( V ) . T h i s e x t e n s i o n is d e n o t e d by

p,

a s w e l l . O n e easily sees t h a t E p : = a o ( - l ; $$) is a f u n d a m e n t a l s o l u t i o n of P ( D ) ,

called Bernstein ‘s fundamental solution.

Example 2.27. Suppose that P is quasihomogeneous o f degree k‘6C and o f type M * . Then (il

p ( z ) is quasihomogeneous o f degree -lz - p for every

(ii)

setting N := - ord ( - 1 ;

$1)

we have N

in particular, i f e = O then E, is quasihomogeneous o f degree - p while in the case e t . 0 E p is quasihomogeneous ( o f degree l - p ) i f and onlv i f

is holomorphic

at z = - 1 . proOf. (il:If Rez > O t h e n 7 ( p ( z ) ) = Pz is q u a s i h o m o g e n e o u s of degree Oz a n d

of t y p e M * so t h a t by P r o p o s i t i o n 2.19 p ( z ) is q u a s i h o m o g e n e o u s o f degree

2.d

Almost Quasihomogeneous

93

Distributions

g ( z ) :=

- tz-p .

(ii): In

view of P r o p o s i t i o n 2.25 ( a p p l i e d to g ( z ) :=

H e n c e t h e a s s e r t i o n f o l l o w s by Remark 2.24.

- ez - p )

o n e h a s to a p p l y pro-

position2.26'to T i : = a i - N ( - l ; P ) , m : = P - p , and c : = - 0 .

F u r t h e r e x a m p l e s of m e r o m o r p h i c f u n c t i o n s of q u a s i h o m o g e n e o u s d i s t r i b u t i o n s will c o m e u p in C h a p t e r s 4 a n d 6 . In f a c t , u s i n g t h e s e o n e c a n s h o w t h a t f o r every T E D ' ( X ) w h i c h is q u a s i h o m o g e n e o u s of degree m t h e r e is a n e n t i r e f u n c t i o n h:@-+%'(X)

s u c h t h a t h ( m ) = T a n d s u c h t h a t h ( z ) is q u a s i h o m o g e n e o u s of

degree z f o r every z E @ .

tdB

A I m o s1 Quas i homogthn thou s I)1s1I- 1 b u 1ionh

In l a t e r c h a p t e r s a c e n t r a l r o l e is played by d i s t r i b u t i o n s ?' w h i c h a p p e a r a s z e r o o r d e r L a u r e n t c o e f f i c i e n t s of c e r t a i n m e r o m o r p h i c f u n c t i o n s h of q u a s i h o m o g e n e o u s d i s t r i b u t i o n s w h e r e t h e f u n c t i o n g in P r o p o s i t i o n 2 . 2 S . ( a ) is t h e i d e n t i t y m a p o n 43. P r o p o s i t i o n 2.20' s h o w s t h a t T t h e n s a t i s f i e s t h e e q u a t i o n ( i 3 M - m ) N T = 0 f o r

s o m e N E W . We c a l l such d i s t r i b u t i o n s "almost quasihomogeneous of degree m " . T h e i r b e h a v i o u r u n d e r t h e a c t i o n of M , is d e s c r i b e d by ( 2 . 1 4 ) ' f o r c = 1 . I t is t h i s p r o p e r t y - w r i t t e n in a s l i g h t l y d i f f e r e n t way

-

t h a t s e r v e s as t h e basis f o r t h e

f o l l o w i n g d e f i n i t i o n of a l m o s t q u a s i h o m o g e n e i t y . T h r o u g h o u t t h i s s e c t i o n w e f i x a number

NEW.

Definition 2.28. A d i s t r i b u t i o n TE B ' ( X ) is called almost quasihomogeneous of degree m land o f type M ) and of order i N if a n d o n l y i f t h e r e e x i s t d i s t r i b u t i o n s

d, , . . . , d N E B ' ( X ) s a t i s f y i n g N

(2.16)

tCmToMt=T+zmk(t)dk k=l

In view of (1.42), f o r every kEN t h e d i s t r i b u t i o n d, a p p e a r i n g in (2.16) is u n i q u e ; it is called the k t h order deficiency o f T. I f k = l t h e n w e a l s o s p e a k of the deficiency of T . S e t t i n g d o : = T w e call t h e n u m b e r ordMMO : = m i n t k € N , ;

t h e (quasihomogeneity) order of T ( w i t h respect t o

MI.

dk#O}

94

11.

(Almost) Quaslhomogeneous Distributions

If X i s n o t s u p p o s e d to be q u a s i h o m o g e n e o u s t h e n in t h e d e f i n i t i o n o f a l m o s t q u a s i h o m o g e n e i t y o n e h a s to p o s t u l a t e t h a t t - m - p k we have k-/

(t)M-ZP)kPZ = I

17 ( z - i )

=o

[ ( d M - e ) P ] " pL-",

and i f ( 3 M - P ) P does not vanish identicallj then there is a countable subset D of C such that P ( z ) is not almost quasihomogeneous f o r every z € C \ D .

H e n c e t h e d e s i r e d e q u a t i o n f o l l o w s by i n d u c t i o n on k

.

If (aM - P ) P $ 0 t h e n t h i s

e q u a t i o n t e l l s u s t h a t f o r a r b i t r a r y kElN a n d z E C s a t i s f y i n g R e z > k + l t h e f u n c t i o n

Pz is n o t a l m o s t q u a s i h o m o g e n e o u s of d e g r e e z Q of order< k . H e n c e t h e s e c o n d p a r t o f t h e a s s e r t i o n f o l l o w s f r o m Remark 2.24' a n d P r o p o s i t i o n 2 . 4 0 . ( i i ) .

m

111

2.f ( G , o )- i n v a r i a n t D i s t r i b u t i o n s

6f'B Appendlx: 6Q . e ) - l n v a r l a n t D l s t r l b u t l o n s

L e t G be a c o m p a c t s u b g r o u p of G L ( V , V ) s u c h t h a t A ( X ) = X f o r e v e r y A E G . M o r e o v e r , let 0 : G - C

be a c o n t i n u o u s h o m o m o r p h i s m of G i n t o t h e m u l t i p l i -

cative g r o u p @ . N o t e t h a t s i n c e G is c o m p a c t t h e i m a g e o f 0 is c o n t a i n e d in t h e u n i t c i r c l e S'. In p a r t i c u l a r ,

IdetA 1 =

1 f o r every A E G . W e set 0 : = ( G , o ) .

Deflnltlon 2.56. A d i s t r i b u t i o n T E B ' ( X ) is c a l l e d @-invariant if a n d o n l y if T o A = n ( A )T ,

(2.32) If

0

AEG.

= I w e a l s o s a y t h a t T is G-invariant.

Recall t h a t t h e c o m p a c t n e s s o f G i m p l i e s t h a t G is u n i n i o d u l a r so t h a t t h e ( n o r m a l i z e d ) left-invariant Haar measure pc on G is r i g h t - i n v a r i a n t , a s w e l l . In t h e p r e s e n t s e c t i o n w e c o l l e c t t h e basic m a t e r i a l o n h o w to c o n s t r u c t (9-invar i a n t d i s t r i b u t i o n s by t a k i n g t h e a v e r a g e w i t h r e s p e c t to p G ,

Notation 2.57. F o r a n y s u b s e t L o f X w e set L ,

:=

u A(L)

AEG

Lemma 2.58. f i l ( i l , n X = Lxn X ; in particular, i f L is a closed subset of X so i s L, liil

.

I f L is compact so is L , .

Proof. - f i l : T h e i n c l u s i o n x i s valid by c o n t i n u i t y . To p r o v e

'2w e

fix x c T G n X

a n d c h o o s e a s e q u e n c e (xk)keO\r in L G c o n v e r g i n g to x . T h e n f o r every k e N w e f i x A k € G a n d t k E L s u c h t h a t x k = A k ( t k ) . By c h o o s i n g s u b s e q u e n c e s if n e c e s s a r y w e achieve t h a t (A,)

c o n v e r g e s to s o m e A E G a s k + a . By c o n t i n u i t y it f o l l o w s

-

t h a t e k = A i ' ( x k ) c o n v e r g e s to e : = A - ' ( x ) . H e n c e [ E L a n d x = A ( t ) E ( i ) , .

(iil:

L G is t h e i m a g e of t h e c o m p a c t set G x L u n d e r t h e c o n t i n u o u s f u n c t i o n

GxV+V,

(A,x) H A ( x ) .

h p O d u O n 2 . 5 9 . Let rEDVoulw1, and let f e C ' i X l . Then Q,

112

11.

( A l m o s t ) Quasihomogeneous Distributions

f @ Is) : = l a ( A - ' ) f ( A s ) d p c ( A ) ,

S€X,

C

a @-invariant C ' function

f w :X +

C i s well-defined having t h e following pro-

perties :

f@ c ( s u p p f ) ,

(i)

supp

fii)

fw = f

;

i f and only i f f is (9 -invariant;

( i i i ) the map C ' ( X )

---j C r ( X ) ,

mf. By d i f f e r e n t i a t i n g u n d e r n e s s of G o n e sees t h a t f,

f

H

f m , i s linear and continuous.

t h e i n t e g r a l s i g n a n d m a k i n g u s e of t h e c o n i p a c t -

b e l o n g s to C ' ( X )

a n d t h a t t h e r e is a c o n s t a n t B,.

only depending o n r and G such t h a t

S i n c e pc i s invariant a n d s i n c e o is a h o m o m o r p h i s m it follows t h a t f m , ( B x )= . I ' o ( B ( A B ) - ' ) f ( A B x ) d V , ( A ) = o ( B ) f C e ( x ) ,

x€X.BEG.

G

( i ) : If ~ d ( s u p p f t h) e~n f o r every A E G w e have A x d s u p p f a n d h e n c e f ( A x ) = O so t h a t f a ( x ) = O . Since by Lemma 2 . 5 8 . ( i ) t h e set ( s u p p f ) G is closed in X t h e a s s e r t i o n is p r o v e d . (iil:S i n c e f,

is @ - i n v a r i a n t t h e implication

v i o u s in view of Iic(G) = 1 .

is c l e a r . T h e c o n v e r s e is ob-

(iiil f o l l o w s f r o m ( 2 . 3 3 ) .

To f o r m u l a t e a s s e r t i o n s a b o u t t h e derivatives a n d t h e Fourier t r a n s f o r m of f, we introduce

Notation2.60. ( i ) By G * : = { A * ; A E G } w e d e n o t e t h e s u b g r o u p of G L ( V * , V * ) c o n s i s t i n g of t h e t r a n s p o s e s of t h e e l e m e n t s of G ; (ii) we define a continuous homomorphism o * : G * + & (iii)

w e set

w':=

by o * ( A I : = o ( ( A * ) - ' ) ;

(G*.G+).

Observe t h a t G* is a c o m p a c t g r o u p , a s w e l l , a n d t h a t its H a a r m e a s u r e pG* is d e s c r i b e d by j ' f ( A ) dyG+(A) = ('f(A*) d p G ( A ) , G'

G

fEC'(G*).

113

2.f ( G , a ) - i n v a r i a n t D i s t r i b u t i o n s

Roporltlon 2.61. Let

5:

G -+ 6 be a n o t h e r c o n t i n u o u s h o m o m o r p h i s m . Then. wri-

ting $ : = ( G , r ) , For arbitrary rEOVoulal a n d f ' E C r ( X I we have:

P ( d ) F ~ = I P ( 3 ) F ) ( C , o rFor ) every $*-invariant polynomial Function P : V X + 6

(i)

of degree n o t larger t h a n r : (ii) q F B = ( q f ) ( G , o r )for ever) $-invariant

Proof. -l i ) :

c o n t i n u o u s f u n c t i o n q : X -6.

The assertion follows from

P ( d ) ( f o A ) = ( ( P o A * ) ( a ) f ) oA = ( T ' ( A * ) P ( d ) f ) oA = r(A-') ( P ( a ) f

fii):

t h i s is a c o n s e q u e n c e of

)

0

A,

AEG.

q ( x ) = r ( A - ' ) q ( A x ) . A E G . rn

Concerning t h e Fourier t r a n s f o r m o n e o b t a i n s

Ropoeltion 2.62. S u p p o s e t h a t X = V a n d f EY'(V ) . Then fcs b e l o n g s to Y(V ) , a s well, a n d A

Stf,,, = (f I @ * .

(2.34)

+P(V ) , f

Moreover, t h e m a p P(VI

rj

fe

, is

linear a n d c o n t i n u o u s .

m F . Since G is c o m p a c t t h e r e is a c o n s t a n t C s u c h t h a t l + l x l C C ( l + I A ( x ) l ) f o r arbitrary x E V a n d A E G . C o n s e q u e n t l y , t h e f i r s t a n d t h e third p a r t of t h e assertion f o l l o w f r o m (2.33). For t h e proof of t h e second p a r t o n e observes f r o m ( 2 . 8 ) h

t h a t 9 ( f.A) = f

0

( A *) - ' .

By Fubini's theoreni a n d by t h e invariance of p G u n d e r

t h e t r a n s f o r m a t i o n A H A - ' o n e then o b t a i n s f o r every f € V *

9(f,)

j'O(A-') F ( f o A ) ( f )d p c ( A ) =

(f) =

G

= J ' o * ( ( A * ) - ' ) ? ( A * € , )d p G ( A ) = J'O*(B-') ? ( B € , )d p G + ( B ) . G G*

rn

By Fubini's t h e o r e m , by a t r a n s f o r m a t i o n of variables, a n d by t h e invariance prop e r t i e s o f pG w e d e d u c e Jf,(x)cp(x)dx X

G

X

f ( x ) cp(A-'x)dx d p G ( A ) =

= JA(A) G

= ~ b ( A - ' ) J f ( A x ) c p ( x ) d x d y C ( A )=

X

J f ( x ) J'A(A-' X

G

)

cp(Ax) d y G ( A ) dx =

114

11.

(Almost) Quasihomogeneous Distributions

= J ' f ( x )' p ( G , l / o ) ( x )d x X

for arbitrary fECo(X) and y E C T ( X ) . This motivates t h e following

Definition 2.63.

Let T E B ' ( X ) . W e s e t

w h e r e @ ' : = ( G ,l/cs). A n a l t e r n a t i v e way to w r i t e t h i s is (2.35)

< T a , r p > =J ' o ( A - ' ) < T o A , c p ) d p G ( A ) ,

'p E

c;c

X).

G

T h e r e s u l t of t h e c o m p u t a t i o n p r e c e d i n g Definition 2.03 c a n b e r e w r i t t e n as

Propoeition 2.64. For everj TEB'IXI Tc+,is a well-deFined @-invariant distribution on X having t he fol l owing properties:

(i)

supp T, C ( s u p p TIG :

(ii)

suppose that X = V and that T is temperate; then T , is temperate, as well.

t h e defining equation and 12.351 remain valid f o r arbitrarl p < E ( V V ) . and A

S(T& I = (TIr.

I

In general, let x € X be fixed, and let K be a compact neighbourhood of x in X . Then by the definition of C;(X) we find a compact subinterval J of IO,+coC such t h a t { t € I ; M , ( K ) n s u p p f # @ } is contained in J . This means, i n particular, t h a t

f,,,,

e q u a l s f,,,,x

J

o n K where

xJ

denotes t h e characteristic function of J . Con-

sequently, t h e proof is reduced to t h e c a s e " I is c o m p a c t " already d e a l t with above. Note t h a t ( 3 . 3 ) remains valid.

(ii):

For arbitrary t e l O , + m C and z € @ we have

t h e s e r i e s converging uniformly if t s t a y s in a compact s u b s e t of I O , + m l and z in a c o m p a c t s u b s e t of C . Let K be a compact s u b s e t of X , and let J b e a s

in t h e proof of ( i ) . In view of ( 3 . 3 ) it then follows t h a t f o r every a€N," s a t i s fying l a l < _ r t h e series m

converges to Of

@.

a"f,,,

uniformly o n K and uniformly f o r L in any c o m p a c t s u b s e t

120

111. Q u a s l h o m o g e n e o u s A v e r a g e s of F u n c t i o n s . Part 1

By e x p l o i t i n g ( 3 . 3 ) o n e c a n o b t a i n a c o n v e n i e n t f o r m u l a f o r t h e d e r i v a t i v e s of fm,w.

To t h i s e n d w e fix t € @ ,N € ! N o , a n d a c o n t i n u o u s c o p o l y n o m i a l f u n c t i o n

Po:XxV*-@

on X of degree 5 r ( i n t h e s e n s e of Definition 1.16) w h i c h is a l m o s t

q u a s i h o m o g e n e o u s of degree t , of t y p e M x ( - M

)*,

a n d o f order 5 N

.

Applying

Remark 1.50 to ( P , M x ( - M ) * ) i n s t e a d o f ( q o , M ) w e c o n c l u d e t h a t f o r e v e r y k € N N t h e k t h order deficiency Pk of P o is a c o n t i n u o u s copolynornial f u n c t i o n o n X , a s w e l l . C o n s e q u e n t l y , f o r every k € (0) uN,

P k ( x , 3 ) is a w e l l - d e f i n e d 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 o n t i n u o u s c o e f f i c i e n t s , a n d (1. 24) - a p p l i e d to A = M, - a n d (1.65 give N

(3.4)

P o ( x . d ) ( f o M t ) = t-'

( - l ) i q ( t )(Pi(x,d)F)oM,,

t €I O , + ~ C

i=O

S i m i l a r l y , by a p p l y i n g (1. 24) to ( f o M , , Ml,t)

one obtains that

N

(3.4)'

( P O ( x , d ) f ) o M ,= t e

q ( t )P i ( x , 3 ) (FOM,),

t€lO,+~C.

i=O

Propoeitlon 3.4. Under the preceding assumptions on Po. k', and N w e have (3.5)

P,(s,

a ) f,,,

N

,

= I

.y l - 1 ) ' =o

and

.

(Pi(s.a) f I,,

+

p . wr,,;

N

In particular, i f Po is quasihomogeneous o f degree P and of t y p e M x ( - M *) then

proof. Let

x e X . S i n c e ( 3 . 3 ) l e a d s to ( P o ( x , a )f , , , , , , ) ( x ) =

dt .f t - m P o ( x , d ) ( f o M , ) ( x ) w ( t ) T I

o n e o b t a i n s ( 3 . 5 ) by i n s e r t i n g ( 3 . 4 ) . For t h e p r o o f o f ( 3 . 5 ) ' o n e s i m i l a r l y d e d u c e s from (3.4)' that N

.

w 0 ( m f j m + e , w ( x )= 2 j

dt t-m p i ( x , a )( f o M , ) ( x ) w ( t ) w i ( t ) T

i=O I

a n d by ( 3 . 3 ) , a g a i n , t h e c o n d i t i o n ( 3 . 5 ) ' f o l l o w s .

In view of E x a m p l e 1.21 a special c a s e of ( 3 . 6 ) is

H

,

121

3.a I n t r o d u c i n g Q u a s i h o m o g e n e o u s A v e r a g e s

Another special case of Proposition 3.4 worth to be formulated separately is

Corollary 3.S. Let PEC a n d N E N o , a n d l e t q E C o ( X ) be a l m o s t quasihomogeneous of degree P a n d o f order 5 N . Then - for every i € N N denoting by qi t h e i t h order

deficiency of q (which is continuous, a s well, by Proposition 1.51)- we have

+z N

(3.8)

qfrn,w = ( q f ) r n + p , w

. ( - 1 ) ' (qif)m+P,w,.,i

i=l

and N

13.8)'

( q f ) r n + P , w = q f r n , w + s qi

Frn,wwi.

i=l

In particular. if q is quasihomogeneous of degree C then

We now come to t h e invariance of f m , w under linear changes of variables. Here we require

Lemma 3.6. Let L be a n IM,Il-bounded s u b s e t of X . (i)

Then A-'(L)

is an (M.1)-bounded s u b s e t of

A-'(Xl for every' AELIV,VI

commuting with M . (iil If G satisfies t h e assumptions of Remark 2.67. (ii) then LG (see Notation 2.57) i s an IM,I)-bounded s u b s e t of X .

mf.(i): Let K be a compact subset of A - ' ( X ) . Then H : = A ( K ) is a compact s u b s e t of X s u c h t h a t { t e l ; M , ( K ) n A - ' ( L ) # @ } i s e q u a l to { t C I ; M , ( H ) n L # @ } , and t h e assertion follows.

(ii): Let

K b e a compact s u b s e t of X , and let t e l be such t h a t M , ( K ) n L , = @ .

Then

M , ( K ) n A ( L ) # @ f o r s o m e AEG so t h a t M , ( A - ' ( K ) ) n L # @ , i.e. M t ( K G ) n L # @ . Since by Lemma 2.58.(ii) K,

is a compact s u b s e t of X t h e assertion follows.

Roporition 3.7. (i) If A6L(V. V) commutes with M then f o A (fOA),,,

= E,,,,

6

C;(A-l(X))

oA .

(ii) If ($3 satisfies the assumptions of Remark 2.67. (ii) then fa (fa)rn,w

and

= (frn,w)a

E

C;lX)

and

122

111. Q u a s i h o m o g e n e o u s A v e r a g e s of F u n c t i o n s . Part 1

proof. Ci.,: S i n c e s u p p f o A = A - ' ( s u p p f

)

the first part of the assertion follows

f r o m L e m m a 3 . 6 . ( i ) . T h e s e c o n d p a r t is a n i m m e d i a t e c o n s e q u e n c e o f t h e a s s u m p tion o n A and (3.1)'.

(ii):

S i n c e by P r o p o s i t i o n Z . S Y . ( i ) s u p p f g is c o n t a i n e d in ( s u p p f )c t h e f i r s t p a r t

f o l l o w s f r o m L e m m a 3 . 6 . ( i i ) . To p r o v e t h e s e c o n d p a r t w e f i x x € X . S i n c e in view o f L e m m a 3 . 6 . ( i i ) t h e set { t E I ; M t ( x I G n s u p p f f !ij = ( t € l ; M t x E ( s u p p f bC

1

is c o n t a i n e d in a c o m p a c t s u b i n t e r v a l J of I O , + a C w e o b t a i n by a p p l y i n g Fubini's

theorem t h a t ( f " , , , , ) @ ( x ) = J'o(A-') G

J

dt = ( f < g ) n , , w ( x ) . (fOA)(M,x) d F c ( A ) w ( t )T

= J't-"'J'o(A-') J

f t - " f ( M , A x ) w ( t )dtT d p G ( A ) =

G

Notatlon3.8. F o r any s u b s e t Y of V w e set

YM,l :=

u M1,,(Y)

tEI

I t is o b v i o u s t h a t f,,,,

v a n i s h e s o u t s i d e t h e set ( f - ' ( & ) ) M , I . To d e t e r m i n e its

closure w e require

Lemma 3.9. Let L be a n (M,II-bounded subset o f X . Then L X n X = ( c n X ) M , I . In p a r t i c u l a r , i f L i s closed in X so i s L M . 1 .

Proof. T h e i n c l u s i o n '2.'is o b v i o u s by t h e c o n t i n u i t y of M I / , . To p r o v e

-

w e fix X E L M , , n X , c h o o s e a s e q u e n c e

'z"

in L M , i c o n v e r g i n g to x as j + a ,

a n d l e t ( t j )be a s e q u e n c e in I s u c h t h a t M t j x i E L f o r every j C N . S i n c e t h e set

K : = {x} u { x i ; j € N) is a c o m p a c t s u b s e t o f X Definition 3.1 s h o w s t h a t t h e r e i s a c o m p a c t s u b s e t J of I

s u c h t h a t t j E J f o r every j € N . H e n c e , by c h o o s i n g

s u b s e q u e n c e s w e achieve t h a t ( t i ) c o n v e r g e s to a n u m b e r t e l . By c o n t i n u i t y it follows t h a t lim M t j x i = M t x .

j-3 rn

T h i s m e a n s t h a t M , x € L n X , i.e. x € ( i n X ) ~ , ~ .

Applying Lemma 3.9 to L = f - ' ( 6 ) o n e o b t a i n s in view of

Ln X

= supp f that

123

3.a I n t r o d u c i n g Q u a s i h o m o g e n e o u s A v e r a g e s

Ropodtion 3.10. The support o f f,,

The following lemma

- which

i s contained in l s ~ p p f ) ~m, ~ .

relies on Proposition 3.10 - prepares f o r t h e defini-

tion of quasihomogeneous averages of distributions o n X in C h a p t e r 4 below.

Lemma3.11. Let f € C ? ( X ) and g E C P / I ( X I where 1 1 1 : = { l / t : t c l } . I f the s e t s ( s u p p f )M, I n s u p p g and supp f n ( supp g I M ,

,, I

j ‘ f m,, , ( X I g ( u I d x = . / ‘ f ( x lg - , , , - c I , X

are compact then

( X I ds

x

where v ( t ) := w ( l / t ) (here the integrals are well-defined since the support o f each integrand is compact I . Proof. We s e t F : = s u p p f , G : = s u p p g , K : = FM.1 n G , and L : = F n GM,,/1

.

Since F

is an ( M , I ) - b o u n d e d and G an ( M , I / I ) - b o u n d e d s u b s e t of X t h e s e t

is a c o m p a c t s u b s e t of I . Note t h a t by Proposition 3.10 we have: s u p p f m , w g C K

and s ~ p p f g - , - , , ~ C L . Applying Fubini’s theorem, s u b s t i t u t i n g f i r s t x = M l / , y and then t = l / s , taking t h e inclusions s u p p ( f g o M , ) C F n M l , , ( K ) into account and applying Fubini’s theorem again one verifies t h a t

Next we deal with special choices of w .

C L, sEI/I,

124

111. Q u a s i h o m o g e n e o u s A v e r a g e s of

Functions.

Part 1

Lemma 3.12. Suppose that I is a closed subinterval of 3 0 , +a[, and let a (resp. b ) be its left (resp. right) endpoint. Then for arbitrary fEC,'(X) and jcNo we have

where w - I

= 0 , wj := uj X I , and

:

B .:= C.J

I"

i f CE{O,+WI

c - m wj(c)foMc

i f c ~ l O , + a C'

In particular, i f I = 10,ll then (d,,,, - m ) j + ' f r r r S w i = ( - 1 ) J f . Proof. The first equality is a special case o f ( 3 . 7 ) . To prove t h e second equality we first observe (see the proof of Proposition 1.22 for m = O ) that (3.10)

t1 ( d M f ) ( M t x )= 3 t f ( M t x ) ,

Moreover, since w j ( t ) = w i - l ( t ) / t

t€IO,+coC, X € X .

we have

a t ( t - r n U i ( t ) ) = t1( - m t - r n w j ( t ) + t - r n w i - l ( t ) ) . Consequently, for arbitrary a < c < d < b partial integration yields: d ( 3,

f ) rn ,c.,j

x

( X I = ~ ' t - " ' 3 t f ( M t x ) ( , ) i ( t ) d=t

,d

C

d

d

= B d , i ( ~-)B c , j ( x )

+

d t - J' t-"' f ( M , x ) w j - l ( t ) t dt . mJ't-'" f ( M t x )w i ( t ) i C

C

Letting ( c , d ) tend to ( a , b ) we derive the first assertion. To prove the second one we observe that i n case 1=10,11 the first assertion tells us that

( 3 M - m ) f r n , w o = f , and

(3,-m)f,,,,.=

Hence the last assertion follows by induction.

1

-fm,wi-,,

jCN.

H

Now we come to the special case w = wi .

Propodtion 3.13. Suppose that f E C G I X I . Then for any j E N o the function

fm,a. I

is

almost quasihomogeneous o f degree m and o f order 5 j ; more precisely, we have tElO,+wC.

In particular, by 13.1) a continuous function f m : X

+C' is

well-defined (coin-

ciding, o f course. with f m , , , I which is quasihomogeneous o f degree r n .

125

3.b ( M , I ) - b o u n d e d Subsets of X

proOf. S u b s t i t u t i n g u = t s a n d m a k i n g use o f t h e binomial f o r m u l a w e o b t a i n +OD

( t s ) - mf ( M S t x ) ( l o g t s - l o g t ) J

%=

0

By t h e d e f i n i t i o n o f wi t h i s i m p l i e s (3.11) A l t e r n a t i v e l y , we d e d u c e f r o m Lemma 3.12 t h a t ( a M- m ) f m , w o

-m)'f,,w.

I

= ( - 1 ) ' f,,wi-i

0 so t h a t t h e a s s e r t i o n f o l l o w s f r o m P r o p o s i t i o n 2.31.

Is every C ' f u n c t i o n form q =f,

(a,

q:X+@

and

m

which is q u a s i h o m o g e n e o u s o f d e g r e e m o f t h e

for s o m e f E C & ( X ) ? Of course, o n e cannot take f = q since the

s u p p o r t o f q is n o t a n M - b o u n d e d s u b s e t of X u n l e s s q - 0 .

So t h e idea is to

m u l t i p l y q by a c u t - o f f f u n c t i o n + E C h ( X ) w h o s e q u a s i h o m o g e n e o u s a v e r a g e o f d e g r e e 0 is i d e n t i c a l l y e q u a l to I . In t h i s way t h e q u e s t i o n is r e d u c e d to t h e case m = 0 and q

I

1

.

In a s l i g h t l y m o r e g e n e r a l f o r m u l a t i o n t h i s is t h e c o n t e n t of

Propoeitlon 3.14. Suppose that f o r ever)' compact subset K o f X there is given a function

+K

6 C ; ( X ) such that ( + K ) ( , is equal t o 1 on K ,

.

Let q € C o ( X ) .

Then q i s quasihomogeneous o f degree m i f and on/> i f q = (GK q),,, on K ,

for

ever) compact subset K o f X . Proof.

3': by C o r o l l a r y

"+";by

3.5 w e have ( + K q ) " , =

q (L)K)o

P r o p o s i t i o n 3.13 ( + K q ) , mis q u a s i h o m o g e n e o u s o f degree m .

In s e c t i o n ( c ) w e s h a l l d e t e r m i n e u n d e r which c o n d i t i o n s t h e a s s u m p t i o n s o f P r o p o s i t i o n 3.14 c a n be s a t i s f i e d . In o r d e r to p r e p a r e t h i s w e are g o i n g to s t u d y ( M , I ) - b o u n d e d s u b s e t s o f X in t h e f o l l o w i n g s e c t i o n .

(b) (M,t)-boundcd S u b s e t s of' X

If I is c o m p a c t t h e n , of c o u r s e , every s u b s e t o f X is ( M , I ) - b o u n d e d . If J is a n o t h e r closed subset of I O , + a E t h e n every s u b s e t L o f X w h i c h is ( M , I ) - a s

126

111. Q u a s i h o m o g e n e o u s A v e r a g e s o f F u n c t i o n s . Part 1

well as (M,J)-bounded is also ( M , l u J ) - b o u n d e d . Consequently, if I is an interval there are essentially three non-trivial cases to be distinguished: I = 10,+00[, 1=10,11, and I = C l , + m C . M o s t l y , we shall deal w i t h these cases, but the case

that I is not an interval might be of interest, as well, as the following example shows.

Example 3.15.

Let ( s k ) k Ebe~ a monotone sequence i n 1 0 , + m C such that

For every 46N l e t I p be a compact subset o f I O , + w C \

u

I:=

u C I / P , PI

sk . We set

I, E N

Ip

P € N

( n o t e that in view o f (3.12) one can easi?, achieve that

I is

unbounded

that 0 6 7 ) . Finall). l e t \ E X be such that { x l is an M-bounded subset

-resp.

of X . Then

is an I M ,10,I I u I ) - (resp. I M, C I. +atu I ) - ) bounded but n o t an ( M , C I, + a C ) - (resp.

( M , I O . l I ) - ) bounded subset of X .

proof. Since

{ski

L E N } c t t ~ I O , + a [M; , { x ) n L # @ ) it f o l l o w s from (3.12) that

L is not an ( M , C l , + a E ) - (resp. ( M . l O , I l ) - ) bounded subset of X . L e t K be a compact subset of X . Since ( x ) is an M-bounded subset of X the set J : = ( t E l O , + ~ sl E; M , ( K ) }

is compact, hence contained in C I / N , N I

N E N . N o w , let tElO,+mC be such that M , ( K ) n L # @ .

for some

Then one can choose

k E N such that X € M , , ~ ~ ( K )i.e. , t / s k E J so that

tEskJ

C

u [l/t,tlsk

if O N .

k€N

This implies, in particular, that t 2 s l / N

(resp. t 5 N

s,

1 . Moreover, i f , in addi-

t i o n , t belongs to 1 then it m u s t lie even in the compact set I , u . . . u

We are first going to deal w i t h the question when a given point x 6 X is ( M , I ) bounded i n

X . i.e. ( x ) is an (M,I)-bounded subset of X .

~ O p O l l l ~ O3.16. n Suppose that

i n g conditions are equivalent:

I is non-compact. Then f o r ever-)' s 6 X the follow-

127

3.b ( M , I ) - b o u n d e d S u b s e t s of X

( a ) x is (M,I)-bounded in X ;

( b ) the map I + X ,

t H M l / t ~ induces ~, a homeomorphism onto its image, and

{ M t s ; t E l / I } = ( s J M , I is a closed subset of X ;

(3.131

( c ) X ~ X \ E M ( O ~and ) , 13.13) holds. Proof. ( a ) * ( b ) : S i n c e f o r K ' : = i , ( I O , + c o C )

w e have { t c l ; M , ( K ' ) n ( x ) # @ } = I

a n d s i n c e ( x ) is a n ( M , I ) - b o u n d e d s u b s e t of X t h e set K' c a n n o t be c o m p a c t . H e n c e P r o p o s i t i o n l.lO.(iii) i m p l i e s t h a t i, m u s t be injective. Now let ( t j ) i c Nbe a s e q u e n c e in I s u c h t h a t ( M l / t j x ) i e N c o n v e r g e s to s o m e Y E X a s j+m.

Let K be a c o m p a c t n e i g h b o u r h o o d o f y in X . T h e n f o r s u f f i c i e n t l y

l a r g e j w e have M t i ( K ) n ( x ) #

#.

H e n c e a s u b s e q u e n c e of

( t i ) c o n v e r g e s to

s o m e t E I so t h a t by t h e c o n t i n u i t y of i x w e have y = Ml,,x E ( x IM , I . M o r e o v e r , t h e r e is a s u b s e q u e n c e c o n v e r g i n g to s o m e f o r a n y o t h e r s u b s e q u e n c e of ( t i ) i e N

S E I . S i n c e , a g a i n , w e have M l / , x = y

t h e injectivity of i, i m p l i e s t h a t t = s . O f

c o u r s e , t h i s m e a n s t h a t ( t j ) j c Ni t s e l f c o n v e r g e s to t .

( b ) + ( a ) : Let K be a c o m p a c t s u b s e t o f X , a n d let a n d ( k e ) p c N be a s e q u e n c e i n K s u c h t h a t M , , k p =

Y

(tp)peN be

a s e q u e n c e in I

f o r every t C N . By c h o o s i n g

s u b s e q u e n c e s w e achieve t h a t ( k e ) c o n v e r g e s to s o m e wCK as t + a . C o n s e q u e n t l y ,

By (3.13) o n e f i n d s t E l s u c h t h a t w = M , / , x .

T h e f i r s t p a r t of t h e c o n d i t i o n ( b )

t h e n i m p l i e s t h a t t h e s e q u e n c e ( t , ) c o n v e r g e s to t .

( b ) > ( c ) : It s u f f i c e s to s h o w t h a t (3.14)

c

x

I n d e e d , a s s u m i n g t h a t x E E M ( o O ) w e a r e g o i n g t o derive t h a t I is c o m p a c t in c o n t r a d i c t i o n to t h e a s s u m p t i o n o n I . In f a c t , let (t,),,,EN

-

b e any s e q u e n c e in I . S i n c e

t h e p r e s e n t a s s u m p t i o n o n x i m p l i e s t h a t ( x ) is~ c o m p a c t , by c h o o s i n g a s u b s e q u e n c e w e achieve t h a t (Ml/t,,x),,eN

c o n v e r g e s to s o m e p o i n t Y E ( X ) ~ By , ~ .

(3.13) a n d ( 3 . 1 4 ) t h e l a s t set is e q u a l to ( x l M , l so t h a t o n e f i n d s t C l s u c h t h a t y = Ml/,x.

H e n c e , by t h e f i r s t p a r t of ( b ) t h e s e q u e n c e (t,),EN

c o n v e r g e s to

t , i.e. I is c o m p a c t , as was to be s h o w n . F o r t h e p r o o f of ( 3 . 1 4 ) w e fix a p o i n t y b e l o n g i n g to t h e l e f t - h a n d side of ( 3 . 1 4 )

128

111.

Quasihomogeneous

A v e r a g e s of F u n c t i o n s . P a r t 1

a n d c h o o s e a s e q u e n c e ( t m ) m E N in 1 0 . + ~ s1u c h t h a t t h e s e q u e n c e o f p o i n t s y,

. _ Ml,t,x .-

, m E N , c o n v e r g e s to y a s m + a . S i n c e t h e e n d o m o r p h i s m s Mtm

a c t o n E M ( o ~ )as i s o m e t r i e s (see (1.79) a n d (1.10)) w e observe t h a t I I X - M , ~ Y I I =I ~ Y ~ - Y ~ , so t h a t lim Mtmy = x n+m

large m

.

. Since

mEN,

X is o p e n t h i s m e a n s t h a t M t m y E X f o r s u f f i c i e n t l y

S i n c e X is q u a s i h o m o g e n e o u s t h i s i m p l i e s y E X , as d e s i r e d .

f c ) + f b ) : If x d o e s n o t b e l o n g to E ~ ( a 0 t)h e n by P r o p o s i t i o n l . l O . ( i i ) i, i n d u c e s a h o m e o m o r p h i s m o n t o its i m a g e a n d so does its r e s t r i c t i o n to 1 .

In p a r t i c u l a r . t h e p o i n t s of k e r M a r e never ( M , l ) - b o u n d e d in X if I is n o n - c o m p a c t . S o m e t i m e s t h e a s s u m p t i o n "06 X " even i m p l i e s t h a t t h e r e a r e n o n o n - t r i v i a l (M.1)-bounded s u b s e t s of X a t all:

Remark 3.17. S u p p o s e t h a t 11.141 h o l d s , and t h a t I is u n b o u n d e d . Then no p o i n t o f X n M G ' ( X ) is ( M , I I - b o u n d e d i n X . In p a r t i c u l a r , i f X C M,'(XI

t h e n no

n o n - e m p t j ' s u b s e t o f X is ( M , I ) - b o u n d e d .

mf. By R e m a r k 1.8 t h e a s s u m p t i o n s o n M a n d 1 imply t h a t Mox E

(x )M,I\

{ x) M.I

f o r e v e r y x E V \ k e r M . I f M o x E X i t f o l l o w s by P r o p o s i t i o n 3.10 t h a t x is n o t ( M , l ) b o u n d e d in X .

N o t e t h a t by Remark 1.8 o n e h a s

(3.15)

(x)M,I\

( x ) M , ~C E ~ ( o 0 ) .

H e n c e , a s a n o t h e r c o n s e q u e n c e of P r o p o s i t i o n 3.16 o n e o b t a i n s

Propodtlon 3.18. S u p p o s e t h a t I is n o n - c o m p a c t . Then e v e r y p o i n t o f X is ( M , I ) b o u n d e d i f a n d only i f (3.16)

XnEMM(aoJ=@.

In p a r t i c u l a r , e v e r y p o i n t o f

is ( M , I I - b o u n d e d in V if and onlj, i f

do

= 0 ,I

W e now c o m e to t h e d e s c r i p t i o n o f g e n e r a l ( M , I ) - b o u n d e d s u b s e t s of X .

3.b

129

(M.I)-bounded S u b s e t s of X

Remark 3.19. I f L is an ( M , I ) - b o u n d e d subse t o f X so is L n X . proOf. L e t

K be a

c o m p a c t s u b s e t of X , a n d l e t U be a c o m p a c t n e i g h b o u r h o o d

of K in X . T h e n t h e c o n t i n u i t y of MI/, i m p l i e s t h a t { t c l ; M t ( K ) n < # @ } is c o n t a i n e d in { t c l ; M , ( U ) n L f Q ) } ; a n d t h e a s s e r t i o n f o l l o w s .

Corollary 3.20. Suppose that I is non-c om pac t. I f L is an ( M , I ) - b o u n d e d s u b s e t o f X then

proof. By

R e m a r k 3.19

Ln X

is a n ( M , I ) - b o u n d e d s u b s e t of X , a s w e l l , so t h a t

every p o i n t i n < n X is ( M . 1 ) - b o u n d e d in X . H e n c e , t h e c o n d i t i o n ( 3 . 1 7 . a ) is a c o n -

-

s e q u e n c e of P r o p o s i t i o n 3.16. M o r e o v e r , by Lemma 3.9 t h e set LM,I n X c o i n c i d e s w i t h ( L n X ) M , I . C o n s e q u e n t l y , s i n c e E M ( a o ) is M,-invariant f o r every t E 1 / 1 t h e c o n d i t i o n (3.17.b) follows f r o m ( 3 . 1 7 . a ) .

P r e p a r i n g f o r a c h a r a c t e r i z a t i o n of ( M, I ) - b o u n d e d s u b s e t s w e n o t e t h a t ( M I ) b o u n d e d n e s s i m p l i e s s o m e sort of local p r o p e r t y .

Remark3.21. Suppose that L is an ( M . I ) - b o u n d e d subset o f X and that (3.17.a) holds. Then

is a clos ed s ubs et o f X x X . and

(3.18.6)

t he map L I + l O , + w C .

( M , , , v , s ) H t . is continuous.

Proof. F i r s t of a l l w e o b s e r v e t h a t by (3.17.a) a n d P r o p o s i t i o n l . l O . ( i i ) t h e m a p in (3.18.b) is w e l l - d e f i n e d , i n d e e d . M o r e o v e r , by Remark 3.19 w e may a s s u m e t h a t L is closed in X . N o w , let ( x , ) , , , ~ , ~ ( r e s p . ( t m ) m e Nbe ) a s e q u e n c e in L (resp.

I ) , and let ( x , y ) ~ X x X be s u c h t h a t lim ( M l / t , x m , x m )

= (y,x).

m-*m

T h i s i m p l i e s t h a t t h e set K : = ( y ) u { Ml/tmx,; a n d t h a t t h e n u m b e r s t,

m r l N } is a c o m p a c t s u b s e t of X

b e l o n g to t h e set ( 3 . 2 ) . C o n s e q u e n t l y , w e find t E l s u c h

130

111. Q u a s i h o m o g e n e o u s

Averages of F u n c t i o n s . Part 1

t h a t a s u i t a b l e s u b s e q u e n c e of ( t m I m E N c o n v e r g e s to t . S i n c e a n y s u b s e q u e n c e of ( x ~ ) ~ ~t e nDd s\ to I x , a s w e l l , a n d s i n c e t h e m a p ( r , y ) H M r y is c o n t i n u o u s it f o l l o w s t h a t y = M l / , x .

S i n c e L is c l o s e d in X t h i s s h o w s t h a t ( y , x ) € L I , i . e .

(3.18.a) is p r o v e d . M o r e o v e r , b y t h e s a m e a r g u m e n t a s a b o v e w e see t h a t any s u b s e q u e n c e of (t,),,EN h a s a s u b s e q u e n c e c o n v e r g i n g to s o m e n u m b e r S EwI h i c h by t h e c o n t i n u i t y of t h e m a p ( r , y ) H M , y , a g a i n , s a t i s f i e s t h e e q u a t i o n Ml/,x = M l / , x . S i n c e by (3.17.a) and Proposition l.lO.(ii) this implies s = t w e conclude t h a t t h e whole sequence c o n v e r g e s to t as m + a . H e n c e ( 3 . 1 8 . b ) is p r o v e d , as w e l l .

(t,)

T h e f o l l o w i n g c h a r a c t e r i z a t i o n of ( M , I ) - b o u n d e d s u b s e t s of X i s e s s e n t i a l f o r Chapters 4 and 0 .

Propositlon 3 . 2 2 . Suppose that I is non-compact. Let L be a subset o f X satisQ i n g 13.17.a) and ( 3 . 1 8 . 6 ) . Then the following conditions are equivalent: ( a ) L i s an ( M , I ) - b o u n d e d subset of X ;

( b ) ever, s u bse t H o f the s e t LI defined in 13.1H.a) such that x1(H)is a relative/). compact subset of X is relativelv compact in L , (here r l :X x X - X d en o t e s t h e projection o n t o the first f a c t o r ) ; (c) L n K M M . l ,isI compact f o r every' compact subset K o f X

N o t e t h a t f o r t h e c o n v e r s e implication it does n o t s u f f i c e to p o s t u l a t e t h a t

- -

L n K,,,,,

n X is c o m p a c t (see E x a m p l e 3.24 b e l o w ) .

Proof. ( a ) = + ( b ) :Let (Om,),,,,

in I s u c h t h a t k,:=M1/,,Om

K : = ( X I u {k,;

be a s e q u e n c e in L n X a n d (t,)

any s e q u e n c e

c o n v e r g e s to s o m e x E X a s m + m . T h e n t h e s u b s e t

m c W ] of X is c o m p a c t . S i n c e t h e n u m b e r s t,,

lie in t h e set ( 3 . 2 )

w i t h L r e p l a c e d by L n X a n d s i n c e by R e m a r k 3 . 1 9 i n X is ( M , I ) - b o u n d e d in X ,

a s w e l l , w e achieve by c h o o s i n g s u b s e q u e n c e s t h a t (t,) By c o n t i n u i t y w e d e d u c e t h a t t , = M t m k m H e n c e t h e s e q u e n c e of p a i r s ( M l / t , t , , t m )

:=

M tm k,

b e l o n g s to

L

-

c o n v e r g e s to M t x = : 4 E L n X a s m + m . c o n v e r g e s to ( M l , , 4 , t ) E L , .

( b ) + ( c ) : Let ( t m ) m E N be a s e q u e n c e in I a n d (k,)

4,

c o n v e r g e s to s o m e t e l .

be a s e q u e n c e in K s u c h t h a t

f o r every m E N . C h o o s i n g s u b s e q u e n c e s w e a c h i e v e t h a t

3.b

131

( M . 1 ) - b o u n d e d S u b s e t s of X

(k,) c o n v e r g e s to s o m e x E K a s m + m . Applying ( b ) to t h e set H = { ( k , , 4 , ) ; w e f i n d 4 E L n X a n d t E I s u c h t h a t (k,,k'",)

-

c o n v e r g e s to (M1,,4,4)

mEN}

so t h a t 4,

c o n v e r g e s to 4 = M t x E L n KM,1/1 . ( c ) + ( a ) : Let K be a c o m p a c t s u b s e t o f X . A n d let (t ,)m EN be a s e q u e n c e in

t h e set ( 3 . 2 ) . I t s u f f i c e s to s h o w t h a t (t,)

h a s a s u b s e q u e n c e c o n v e r g i n g to s u c h t h a t 4,

s o m e t E l O , + m C . F o r e v e r y m E N w e f i x k,EK

to L . By c h o o s i n g s u b s e q u e n c e s w e achieve t h a t (k,) m + m . Hence, H : = ( x ) u { k , ;

: = Mt,k,

belongs

c o n v e r g e s to s o m e x E K as

m E N } is a c o m p a c t s u b s e t of X . S i n c e 4 ,

lies in

L n H M , , , l f o r every m E N t h e c o n d i t i o n ( c ) s h o w s t h a t by c h o o s i n g s u b s e q u e n c e s ,

-

t e n d s to s o m e 4 E L n H M , l / I a s m + m . T h e n w e fix

a g a i n , w e a c h i e v e t h a t 4,

h c H a n d t E l s u c h t h a t P = M , h . If h = x t h e n it f o l l o w s by ( 3 . 1 8 . b ) t h a t ( t m ) , , E N c o n v e r g e s to t a s m + m , a s d e s i r e d . If h # x t h e n f o r every N E I N r e p e a t i n g t h e p r e c e d i n g a r g u m e n t w i t h H r e p l a c e d by { x ) u { k,, s e q u e n c e of (krn),,,€N

;

m 2 N } we deduce that a sub-

b e l o n g s to ( t ) M , ISO t h a t x E { ~ ) ~ , ,In. view o f ( 3 . 1 5 )

t h e a s s u m p t i o n ( 3 . 1 7 . a ) s h o w s t h a t x E (4) M , I , i . e . 4 = M t x f o r s o m e t E I . A s above w e d e d u c e t h a t ( t m ) m E N c o n v e r g e s to t .

By a p p l y i n g P r o p o s i t i o n 3.22 o n e c a n easily d e t e r m i n e t h e ( M , I ) - b o u n d e d s u b s e t s o f V in c a s e

rs

=

6, :

Remark 3.23. Suppose that

v = G M f o + ) + E M / o ~and )

that X n E M ( o o ) = 0 . Let L

be a s u bs et of X . (i)

I f L is bounded then L is an ( M , C l . + m C ) - b o u n d e d subset of X ;

(ii) i f dist(L,EMM(oo))'0 then L i s an f M . l O . l l ) - b o u n d e d subset of X ; (iii) i f L is a relative?, compact subset of V \ E M ( o 0 ) then L i s an M-bounded

s u b s et of X ; ( i v ) i f o = o + and if X = V then the converse implications are valid.

h f . By Remark 1 . 8 t h e a s s u m p t i o n s o n M a n d X imply t h a t limt+m l M , x l a n d lim,,odist(

= +a

M , x , E M ( o 0 ) ) = 0 uniformly if x s t a y s in any c o m p a c t s u b s e t o f

V \ E M ( a 0 ) . F r o m t h i s t h e a s s e r t i o n s fi) - liii) a r e easily d e d u c e d . M o r e o v e r , if t h e a s s u m p t i o n s of (iv) are s a t i s f i e d t h e n P r o p o s i t i o n 1.70 s h o w s t h a t w e c a n choose t h e norm

I * I on V

in s u c h a way t h a t f o r every x E V t h e f u n c t i o n t

H1 M,x

I

132

I l l . Q u a s i h o m o g e n e o u s A v e r a g e s of F u n c t i o n s . Part 1

is strictly increasing. Hence, i t follows t h a t t h e compact s u b s e t K : = { X C V ; 1x1 = l }

of

satisfies t h e equations

where B : = { x C V ; 1x1 S l } . Hence t h e assertion ( i v ) is obtained by an application of Proposition 3 . 2 2 .

If t h e assumption o n V and M is dropped then t h e implications ( i ) - ( i i i ) are false in general; in fact, not every compact subset of V is M-bounded a s is illustrated by t h e following

Example 3.24. Suppose that n = 2 and p E l 0 . + ~ ~ C x l - ~ . OLet C. L : = C O , E ~ ~ ’ I X I I and I

E

> O . and set

K : = l l l x C O . ~ - .~ ~ l

Then ( t ~ l O , + m CM; , ( K ) n L f o } = 1 0 , ~ land , L n K M M , I , 3 0 , r 3 = L \ { ( O , l ) } . Inparticular, L is not a ( p , I O , l I ~ - b o u n d e dand K not a (p,Cl,+wC)-bounded subset of R 2 \ 101; and consequently no neighbourhood o f (0.1) (resp. 1 1 , O ) ) is a ( p , l O , l 3 ) -

(resp. ( p , C l , + m C ) - )bounded subset o f R2\101.

I

Since by Proposition 3.18 t h e assumptions of Example 3.24 imply t h a t t h e sets ((0,1)} and {(1,0)) are p-bounded

s u b s e t s of lR2\(0) t h e Example 3.24 s h o w s ,

in particular, t h a t ( p , l ) - b o u n d e d s u b s e t s of X d o not necessarily have ( p , I ) bounded neig hbourhoods. The following lemma is relevant f o r Lemma 4.5 below

Lemma 3.25. Suppose that I is a closed subinterval o f 10,+03C.Let A be an ( M , I ) bounded and B be an ( M , l / I ) - b o u n d e d subset of X . ( i ) If I is non-compact

then both A , , , n B

and A n B , , , , ,

are M-bounded

subsets of X ;

(ii) if I is compact then AM,, n B is an M-bounded subset o f X i f and only if A n BM,,,,

i s one.

For t h e proof of this and other assertions the following lemma is useful.

3 . b (M.1)-bounded

133

S u b s e t s of X

Lemma 3.26. Suppose that I is a proper closed subinterval o f 1 0 , +at. I f L is an IM,I)-bounded subset o f X so is L M , I .

mf. Let K be a c o m p a c t s u b s e t o f

X . Since t h e a s s u m p t i o n s o n L a n d I i m p l y

t h a t L is a n ( M , J ) - b o u n d e d s u b s e t of X w h e r e J : = { s t ; s , t € l } it f o l l o w s t h a t t h e set ( U C J ; M u ( K ) n L

#@I

is c o n t a i n e d in a c o m p a c t s u b i n t e r v a l Cc,dl o f IO,+ooC.

N o w , l e t t € l be s u c h t h a t M , ( K ) n L M , I # @ , a n d c h o o s e k € K , t € L a n d S C I s u c h that Mtk=Ml,,t,

i.e. M , , k = t .

H e n c e c < s t < _ d . If b € l O , + a C is s u c h t h a t I

e q u a l s 1 0 , b l ( r e s p . Cb,+aC 1 t h e n t h i s i m p l i e s t h a t t < l c / b , b l ( r e s p . C b , d / b l ) .

Proof o f Lemma 3.25. g ; I f I = I O , + a C then A and B are M-bounded subsets of X , a n d t h e r e is n o t h i n g to b e p r o v e d . So w e s u p p o s e t h a t I # I O , + a l . S i n c e I is n o n - c o m p a c t t h e c l o s u r e J o f I O , + m l \ I d i f f e r s f r o m 1 1 1 by a relatively c o m p a c t s u b s e t o f 10,+03C so t h a t B is a n ( M , J ) - b o u n d e d s u b s e t of X , as w e l l . S i n c e by L e m m a 3 . 2 6 A M , l is a n ( M , I ) - b o u n d e d a n d BM,l,I

an (M,I/I)-bounded and

h e n c e ( M , J ) - b o u n d e d s u b s e t of X t h e a s s e r t i o n f o l l o w s . ( i i ) . "j" Let:K b e a c o m p a c t s u b s e t o f X , a n d c h o o s e t € l O , + m [ a n d k < K s u c h

that a:=M,kEAnB,,,,,

. Then we find s€I/I such t h a t b : = M , a E B . This means

t h a t M , ( M , k ) = M , a 6 A M , l n B . Since by t h e a s s u m p t i o n o n I t h e set { M , k ; k C K , SE

l / l } i s a compact subset of X t h e assertion follows.

"e": o n e h a s to i n t e r c h a n g e t h e r o l e s of implication

"*".

( A , I ) and ( B , l / I ) and apply t h e

=

T h e f o l l o w i n g l e m m a is r e q u i r e d f o r t h e p r o o f of T h e o r e m 4.8 b e l o w

Lemma 3.27

. Let

I and J be closed subintervals o f 1 0 , +a[such that I nJ is com-

pact. Let L be an ( M , I ) - b o u n d e d subset o f X . Then for ever). compact subset K o f X the set L M . 1 n K M , j is a compact subset o f X .

Proof. If I = l O , + ~ Ct h e n t h e a s s u m p t i o n o n J i m p l i e s t h a t J is c o m p a c t so t h a t

K,,,

is a c o m p a c t s u b s e t o f X , a n d t h e a s s e r t i o n f o l l o w s by L e m m a 3 . 9 . H e n c e

we suppose t h a t I # l O , + a l . Then t h e assumption o n J and 1 means t h a t I and l/J d i f f e r by a relatively c o m p a c t s u b s e t o f I O , + a 3 C . S i n c e by R e m a r k 3 . 1 9 , Lemm a 3 . 9 , a n d L e m m a 3.26

rM,I n X is a n ( M , I ) - b o u n d e d s u b s e t of X i t is ( M , l / J ) -

134

111.

Quasihomogeneous

A v e r a g e s of

F u n c t i o n s . Part

b o u n d e d , as w e l l . S i n c e by Lemma 3.9 L M , I n X is a closed s u b s e t o f

1

X a n appli-

c a t i o n o f C o r o l l a r y 3.20 a n d P r o p o s i t i o n 3.22 l e a d s to t h e d e s i r e d a s s e r t i o n . rn

In s o m e s e n s e ( M , I ) - b o u n d e d n e s s is a l o c a l p r o p e r t y :

Remark 3.28. Let 14 be a locallq finite covering o f X consisting o f quasihomogeneous open subsets of X . Let L be a subset o f X . fi)

I f there exists a family (Lll)uE,l o f (M,I)-bounded subsets o f X such that

L , C U for every U E l l and L =

u

LLl

then L is an (M,I)-bounded subset o f X .

UEll

lii) L is an ( M . I) - bounded subset o f X i f and only i f L n U is an ( M ,I ) - bounded subset o f U f o r ever) U E l l . N o t e t h a t in ( i ) it is n o t s u f f i c i e n t to a s s u m e t h a t e a c h Lu is a n M - b o u n d e d subset of U

:

For e x a m p l e , if

do

#

t h e n in view of P r o p o s i t i o n 3 . 3 4 . A b e l o w every

c o m p a c t s u b s e t of X+_ i s a n M - b o u n d e d s u b s e t o f X+_, b u t in case X t # X + u X _ n o t every c o m p a c t s u b s e t of X + U X - is a n M - b o u n d e d s u b s e t o f X + u X _ .

Proof. Let K b e a c o m p a c t s u b s e t of X . Since every U E U is q u a s i h o m o g e n e o u s w e have ( Kn U ) M = K M n U . S i n c e

1I

is locally f i n i t e w e c a n f i n d a f i n i t e s u b s e t

23 of U s u c h t h a t KM n U = @ f o r every L l € U \ % . H e n c e , u n d e r t h e a s s u m p t i o n s of (il,f o r every t e l w e have M , ( K ) n L = U U , , z x ( M , ( K ) nLLI) f r o m which t h e c o n c l u s i o n of ( i ) f o l l o w s .

To p r o v e (ii)w e c h o o s e a family ( K " ) " , s such that

uUEsK u

of compact s u b s e t s K U of K n U

= K and observe t h a t then

Mt(K)nL = UUEgsMe(KLI)n(LnU).

We close t h i s section w i t h a n e l e m e n t a r y but i m p o r t a n t c r i t e r i o n f o r ( M , I ) - b o u n -

dedness. I t requires t h e existence of positive quasihomogeneous functions on X .

Lemma3.29. Let x : X + I O , + . i o l

be a continuous function which is quasihomo-

geneous o f degree I . Then a subset L o f X is ( M ,I) - bounded i f and only i f f o r ever)' compact subset K o f X the following condition holds:

135

3.c W h e n i s Every C o m p a c t S u b s e t of X M - b o u n d e d ?

I n J x ( L n K M ) i s a relatively compact subset o f I

(3.19)

for every compact subset J o f IO,+mC. Note that i f I i s an interval then 13.19) i s equivalent t o (3.19)'

Proof.

I n x ( L nK,)

"e": If K

i s a relatively compact subset o f I .

is a c o m p a c t s u b s e t o f X t h e n J : = l / x ( K ) is a c o m p a c t s u b s e t

of IO,+col. Since f o r arbitrary t E l O , + a [ and kEK t h e condition " [ : = M , k E L " i m p l i e s " t = x ( e ) / x ( k ) € J x ( L n K M ) " t h e set ( 3 . 2 ) is c o n t a i n e d in I n J x ( L n K , ) a n d h e n c e is relatively c o m p a c t in I by ( 3 . 1 0 ) .

"*":

- W e fix a c o m p a c t s u b s e t K o f X . a n d let J be a c o m p a c t s u b s e t of l O , + a C .

N o t e t h a t by t h e i m p l i c a t i o n

"+",a l r e a d y

p r o v e d a b o v e , t h e set x - ' ( l / J )

is a n

M-bounded s u b s e t of X . Moreover, note t h a t t h e existence of x implies (3.16) a n d t h e c o n t i n u i t y o f t h e m a p { ( x , M , x ) ; x C X . t C l O , + ~ C } .( x , M , x ) H t . H e n c e , it f o l l o w s by P r o p o s i t i o n 3.22 t h a t H : = x - ' ( l / J ) n K ,

is a c o m p a c t s u b s e t of X .

Now w e l e t j € J a n d ( E L n K M . set t : = j x ( P ) , a n d c h o o s e s E I O , + ~ [ a n d k C K such that 4=M,k. 4=M,(Ms,,k)

Then x ( M , , , k ) = x ( P ) / t = l / j E l / J ,

i.e. M , / , ~ E H so t h a t

E M , ( H ) . H e n c e w e c o n c l u d e t h a t I n J x ( L r l K M ) is c o n t a i n e d in

t h e set { t E 1 : M , ( H ) n L

#@ which I

is relatively c o m p a c t in I by t h e a s s u m p t i o n

o n L . rn

N o t e t h a t in view o f P r o p o s i t i o n 1.70 t h e i m p l i c a t i o n s ( i ) - ( i i i ) o f R e m a r k 3 . 2 3

are s p e c i a l c a s e s o f L e m m a 3 . 2 0 .

(c) W h e n is e v e r y (:ompael Subscl o f ' X M-bounded
O so s m a l l t h a t K , : = { x E V ; P , ( x ) = P ,-( y +- ) . a n d I P r ( x ) I C ~ }is c o n t a i n e d in U, . S e t t i n g s, : = 0 a n d s- : = +a w e o b s e r v e t h a t by R e m a r k l.&i.(i) w e have lim,,,+M,y,= -

f o r every t E I ,

0 . Hence w e can choose a E 11,+a[ such t h a t

I Mty,(

5 E

w h e r e I - : = C a , + a C a n d 1, : = 1 1 1 - = l O , l / a l . I t f o l l o w s t h a t f o r

every t E 1 - by P + ( x ): = M l / t y + a u n i q u e e l e m e n t x of K- is w e l l - d e f i n e d s u c h t h a t M,x E K +

.

This s h o w s t h a t

M , ( K - ) n K + f @ f o r every t E L . So t h e f i r s t p a r t o f t h e a s s e r t i o n is p r o v e d . F r o m t h i s it f o l l o w s t h a t M , ( L ) n L f Q ) f o r every t E l O , + a C \ I ' w h e r e L : = K + u K a n d l ' : = C l / a , a l . S i n c e M , ( M a , , ( L ) ) n L = M a ( L ) n L # 9 ) w e see t h a t M , ( K ) n L # @

for every t e l l w h e r e K is t h e c o m p a c t s u b s e t o f X d e f i n e d by

K:=

u Ma / t ( L )

te1'

S i n c e K c o n t a i n s L w e c o n c l u d e t h a t t h e set ( 3 . 2 ) is e q u a l to I O , + a C . In view of K C L,

C UM t h e p r o o f is c o m p l e t e .

Lemma3.33. Let

. z 6 GMM(oo) be such that z is M-connected to y in the sense

137

3.c W h e n is E v e r y C o m p a c t S u b s e t of X M - b o u n d e d ?

of Definition 1.81. Then for every neighbourhood U of z and for every neighbourh o o d W of y there is a finite s u b s e t R of I I , + 4 a n d a c o m p a c t s u b i n t e r v a l J

of I O , + w l such that M,(

M,I W ) ) n U f @ for every t E 10,+.coy 1J .

proOf. W e set r : = o o . By Lemma 1.84 w e find a point w E T ( r ) C C'

( s e e Defini-

tion 1.79.(iv)) a n d a c o n t i n u o u s f u n c t i o n S : l O , + a r \ { l ) - G M ( ~ ~ )s u c h t h a t t h e condition (1.101) is valid for every s E { O , + w ) with x replaced by z . Observing t h a t H:(x)

is c o n t i n u o u s a s a function of ( v , x ) E C ' x G M ( r ) w e find o p e n

:=

neighbourhoods U, of w in C' a n d Z of H,(z) H:(x)EU

:= (

z

~

/

w in~ G )M (~r ) ~ s u c~h t h a t

for every ( v , x ) E U , x Z . Then by Lemma1.11 w e find a finite s u b s e t R

of l l , + m E s u c h t h a t

(3.20)

x

f o r every v E T ( r ) t h e r e is r E R satisfying ( r v

~

)

~

~

~

E

Now, by (1.101) w e can c h o o s e a c o m p a c t s u b s e t J of IO,+mC s u c h t h a t S ( r t ) E W a n d N r t S ( r t ) E Z for arbitrary t € l : = l O , + ~ C \ J a n d r € R w h e r e N,

is defined by

(1.102). Let t e l . Since by t h e definition of T ( r ) t h e p o i n t y ( t ) : = ( t ' ) , , ,

belongs

to T ( r ) , by (3.20) w e find a n u m b e r r E R s u c h t h a t y ( r t ) E U , . I t f o l l o w s t h a t

H;(rt,(N,,S(rt))€U.

Since by (1.102) w e have H ; ( r t ) ~ N , . t = M r t = M t o M r

the

a s s e r t i o n is proved.

Proof of Proposition 3.31. ( b ) * ( a ) :

We o b s e r v e t h a t f o r arbitrary c o m p a c t sub-

sets K a n d L of X and f o r every t E l w e have: M , ( K ) n L = M , ( K n M , , , ( L ) ) # @

if and only i f K n M,,,( L) # @ . C o n s e q u e n t l y , t h e condition ( b ) remains valid if I is replaced by 1 1 1 a n d hence by J : = I u 1 1 1 . I f I is a n o n - c o m p a c t interval t h e n J

a n d 10,+00Cd i f f e r by a c o m p a c t subinterval of IO,+mC so t h a t ( a ) follows in t h i s c a s e . Moreover, f o r t h e proof of t h e general case w e may s u p p o s e t h a t I is unbounded. The proof of t h e general case is d o n e by contraposition. So w e a s s u m e t h a t t h e r e

are c o m p a c t subsets K a n d L of X s u c h t h a t J : = { t € l O , + m C ; M , ( K ) n L # @ } is n o t c o m p a c t . Hence, in view of t h e observation a t t h e beginning of t h e p r o o f , by interchanging t h e roles of K a n d L if necessary w e achieve t h a t J is u n b o u n d e d . C o n s e q u e n t l y , w e can c h o o s e a s e q u e n c e (t,),,N a n d a s e q u e n c e ( k m b m G N in K s u c h t h a t

converging to + a a s m + m

em:= Mtmkm

b e l o n g s to L f o r every

U

~

138

111.

Q u a s i h o m o g e n e o u s Averages of F u n c t i o n s . Part 1

m e N , By c h o o s i n g s u b s e q u e n c e s if n e c e s s a r y w e achieve t h a t (k,)

( r e s p . (g,,,))

c o n v e r g e s to s o m e k e K ( r e s p . @ E L ) a s m + w . In view of R e m a r k 1 . 8 . ( i ) w e c o n clude that

(3.21.b)

P + ( k ) = lirn P + ( k , ) m+m

= lim Ml,,mP+(Pm)

=0 .

m+m

Setting y + : = P ' ( P ) and y - : = P ' ( k ) (where P ' : = P + + P - ) we can choose compact n e i g h b o u r h o o d s U: of y+ in V ' : = G M ( o + U O - ) a n d U o f z : = P o ( 4 ) a n d W of y : = P o ( k ) in G M ( 6 0 ) s u c h t h a t U:+U

a n d U ; + W are c o n t a i n e d in X . N o t e t h a t in view

of (3.21) t h e p o i n t s y + a n d y - s a t i s f y t h e a s s u m p t i o n s of Lemma 3.32 if ( V , M ) is r e p l a c e d by ( V ' , M ' ) w h e r e M I : = M v o . Let

d

be a positive number such t h a t

t h e c o n c l u s i o n of Lemma 3.32 h o l d s . M o r e o v e r , n o t i n g t h a t z is M - c o n n e c t e d to y w e fix J a n d R as in t h e c o n c l u s i o n o f L e m m a 3 . 3 3 . Finally, w e fix b c I O , + m l s u c h t h a t l b , + m C n J = @ a n d r b ~ a f o r e v e r y r ~CRo n. s e q u e n t l y , f o r e v e r y t E l b , + m C w e find r E R , w E W , a n d u - EUY s u c h t h a t M,,u_ E U:

a n d M,M,.w E U , i . e .

M , ( M , . ( u - + w ) ) E L : = U : + U . S i n c e K ' : = U r G R M r ( U L + W ) is a c o m p a c t s u b s e t o f

X a n d s i n c e I is a s s u m e d to b e u n b o u n d e d t h i s s h o w s t h a t L is n o t ( M . I ) - b o u n d e d in X . S i n c e L is c o m p a c t t h e c o n d i t i o n ( b ) is v i o l a t e d .

( b l 2 (el: T h e f i r s t p a r t of ( c ) i s a c o n s e q u e n c e o f P r o p o s i t i o n 3.18. To d e d u c e t h e o t h e r p a r t s w e fix s e q u e n c e s ( x , ) , , , ~ ~ in X a n d (t,,,)meN in I s u c h t h a t c o n v e r g e s to s o m e ( x , y ) E X x X as m + m .

(x,.M,,x,)

L:=(y)u(M,,x,;mEN)

T h e n , in p a r t i c u l a r ,

is a c o m p a c t a n d h e n c e - b y ( b ) - a n ( M . I ) - b o u n d e d

s u b s e t of X . C o n s e q u e n t l y , Remark 3.21 i m p l i e s l o t h a t y = M , x i.e. ( x , y ) E X , , a n d 2 0 t h a t (t,)

for some t E 1 ,

c o n v e r g e s to t a s m + w . T h i s p r o v e s t h e s e c o n d

a n d t h e t h i r d p a r t of t h e c o n d i t i o n ( c ) .

( c l + ( b ) : Let K a n d L be c o m p a c t s u b s e t s of X , a n d let ( t m ) m e N be a s e q u e n c e in t h e set ( 3 . 2 ) . For every m E N w e fix k,,, E K a n d 4,,, E L s u c h t h a t MtrnkI,,= P r n . By c h o o s i n g s u b s e q u e n c e s if n e c e s s a r y w e achieve t h a t (k,,,) kEK and

(em)

c o n v e r g e s to s o m e

to s o m e P E L a s m + m . By t h e s e c o n d p a r t of t h e c o n d i t i o n ( c )

w e deduce t h a t 4 = M , k f o r s o m e t E l . A n d by t h e l a s t p a r t of ( c ) w e c o n c l u d e t h a t t,+t

as m * a ,

a s desired.

( a l + ( e l : W e c h o o s e a s e q u e n c e of c o m p a c t s u b s e t s X,,

m E N O , of X s u c h t h a t

139

3.c When is E v e r y Compact S u b s e t of X M - b o u n d e d ?

(3.22)

(a) @ = X g = X I C . . . C X m C i m + l , mEN; and

(b)

u X,=X.

mrN

S i n c e t h e q u a s i h o m o g e n e o u s h u l l o f a n o p e n s e t is a u n i o n of o p e n sets t h e sets

(3.23)

are c o m p a c t (3.24)

To s h o w t h a t L is a n M - b o u n d e d s u b s e t of X w e fix a c o m p a c t s u b s e t K of X , c h o o s e mEIN s u c h t h a t K C X,

a n d o b s e r v e t h a t for every t < lo,+-[

w e have

m

M,(K)nL C M,(X,)n(Xm)MnL=

M,(X,)n

u Lk. k=1

rn

S i n c e U k - l L k is c o m p a c t it f o l l o w s f r o m ( a ) t h a t t h e set (3.2) is c o n t a i n e d in a c o m p a c t s u b i n t e r v a l of I O , + a [ .

( d ) * ( a ) : L e t Z a n d K be c o m p a c t s u b s e t s of X . C h o o s e a n M - b o u n d e d s u b s e t L o f X s u c h t h a t K C L M . F o r e v e r y Y E 1 Z . K ) set J , : = { s E I O . + ~ C ; M , ( Y ) n L # @ ) . Let t E l O . + m [ satisfy M , ( K ) n Z f @ , a n d c h o o s e k E K such t h a t z : = M , k E Z . S i n c e K,

C

L M w e f i n d s E l O , + ~ [s u c h t h a t M , z E L . T h e n l.'sEJ,

a n d 20 M,,kE

L,

i.e. s t E J K . C o n s e q u e n t l y . t E J : = { 1.1s; r E J K , s E J z } . S i n c e J K a n d Jz a r e r e l a t i v e l y c o m p a c t s u b s e t s of I O . + a l so is J . H e n c e Z is a n M - b o u n d e d s u b s e t of X .

*

S i n c e t h e i m p l i c a t i o n s " ( e ) f d ) '' a n d " f a ) + ( b ) a r e trivial t h e p r o o f is com"

p l e t e . rn

T h e f o l l o w i n g p r o p o s i t i o n gives a fairly e x p l i c i t d e s c r i p t i o n o f locally M - b o u n d e d o p e n s e t s . W e divide it i n t o p a r t s A a n d B .

Ropodtlon 3.34.A. Suppose that

00 = 0

. Then t h e following conditions are

equi-

valent: (a)

X is local1.y M - b o u n d e d :

( b ) there e x i s t s a continuous ( r e s p . real analj,tic) function x: X - - + 1 0 , which i s quasihomogeneous o f degree 1

+mf

:

( c ) X = X + or X = X Proof. ( a ) + ( c ) : If ( c ) does n o t h o l d t h e n w e f i n d y - E X \ X + C G M ( o - )

and

140

111. Q u a s i h o r n o g e n e o u s

A v e r a g e s of F u n c t i o n s . Part t

y + E X \ X - C G M ( a + 1 . S i n c e X is o p e n w e c a n c h o o s e a c o m p a c t n e i g h b o u r h o o d

U o f ( y + , y - ) which is c o n t a i n e d i n X . By Lemma 3.32 w e f i n d a c o m p a c t s u b s e t K o f U M C X s u c h t h a t M , ( K ) n U f Q ) f o r every t E I O , + r u C . T h i s s h o w s t h a t U is n o t a n M - b o u n d e d s u b s e t o f X .

( c )+ ( b ) ; see E x a m p l e 1 . 6 8 . ( i i ) a n d P r o p o s i t i o n 1.70. fb)+fa)

:

Let K be a c o m p a c t s u b s e t of X . If x is c o n t i n u o u s t h e n x ( K ) is a

c o m p a c t s u b s e t o f I O , + ~ C .H e n c e it f o l l o w s by L e m m a 2.20 a p p l i e d to K i n s t e a d of L t h a t K is a n M - b o u n d e d s u b s e t o f X . rn

In order to h a n d l e t h e c a s e

"

G # ~0 "

Notation3.35. F o r every yEG,(o,)

Propodtion 3.34.B. S u p p o s e t h a t i f for arbitrary

J

, z 6 G,,,,(a,)

we introduce

w e set X , : = ( x € X ; P o ( x ) = y }

00 f

8.Then X is locall-v M - b o u n d e d i f and on/-\

s u c h that z is M - c o n n e c t e d t o y (see Definition 1.81)

we have

(3.25)

la) X,,uX, C V , or

M .F i r s t (3.2s)'

:

or

Ib) X , . u X , C V - :

or

(c) X-,,c V , n V - ;

fd) X , C V , n V - .

o f all w e c o n v i n c e o u r s e l v e s t h a t ( 3 . 2 5 ) c a n be r e p l a c e d by x,cV+,

or X , c V -

In f a c t , ( 3 . 2 5 ) m e a n s t h a t ( 3 . 2 5 ) ' is valid f o r b o t h ( y , z ) a n d ( z , y ) so t h a t o u r c l a i m f o l l o w s in view of C o r o l l a r y 1 . 8 3 . ( i ) . "j; Let y , z be p o i n t s in G M ( o o ) s u c h t h a t z is M - c o n n e c t e d to y a n d s u c h

t h a t (3.25)' is violated. T h e n w e f i n d p o i n t s x - E X , \ V + a n d x , E X , \ V -

.

W e set

y, : = P ' ( x , ) a n d M ' : = M v * w h e r e P ' : = P + + P - a n d V ' : = G M ( o +U O - ) . By t h e c h o i c e o f x , w e have ~,EGMM'(o;). W e c h o o s e a c o m p a c t n e i g h b o u r h o o d Ug o f y,

in

V' a n d f o r every w € ( y , z ) a c o m p a c t n e i g b o u r h o o d U& o f w in G M ( d o ) s u c h t h a t

K : = U'_+U: a n d L : = U : + U E are c o n t a i n e d in X . Let (t,) a n d (y,)

be a s in Defi-

nition 1.81.(i) s u c h t h a t (1.97) is valid w i t h x replaced by z . T h e n by L e m m a 3.32 t h e r e is N E N s u c h t h a t f o r every m 2 N w e c a n f i n d k&,€UI s a t i s f y i n g M,,k; By m a k i n g N l a r g e r if n e c e s s a r y w e a c h i e v e t h a t y,

E Ui

a n d Mt,ym

E

E

Ug if m

U: . 2N

.

3.c When Is E v e r y Compact

S u b s e t of X M - b o u n d e d ?

141

T h i s m e a n s t h a t Mtm( k h + y , )

E L f o r m ? N , i . e . t h e s e q u e n c e (t,),2N

is c o n t a i n e d

in t h e set (3.2) so t h a t L is n o t a n ( M , C l , + m C ) - b o u n d e d subset of X , a n d X

is n o t locally M - b o u n d e d by P r o p o s i t i o n 3.31.

"+":S u p p o s e o t h e r w i s e . T h e n

w e f i n d c o m p a c t s u b s e t s K a n d L o f X a n d a se-

q u e n c e ( t m ) m E N in I O , + a C c o n v e r g i n g to +aas m + w s u c h t h a t M , , ( K ) n L f # . F o r e v e r y mE N w e fix k,E

K such

ern:=

M,,k,

E L . Since K a n d L a r e c o m p a c t ,

by c h o o s i n g s u b s e q u e n c e s if n e c e s s a r y w e achieve t h a t (k,) k E K a n d (4,)

c o n v e r g e s to s o m e

to s o m e [ E L as m + w . In p a r t i c u l a r , t h e p o i n t z : = P o ( @ )is M - c o n n e c -

ted to y : = P o ( k ) . M o r e o v e r , in view of Remark l . 8 . ( i ) w e c o n c l u d e t h a t t h e e q u a t i o n s ( 3 . 2 1 ) a r e valid. C o n s e q u e n t l y , k b e l o n g s to X,\V+ ( 3 . 2 5 ) ' is v i o l a t e d .

I f V - = # or V +

e

and

to X , \ V -

,

i.e.

H

= # t h e assertion

Corollary 3.36. Suppose that

a-

of P r o p o s i t i o n 3 . 3 4 . B b e c o m e s m u c h s i m p l e r :

= p or

u + = @ . Then X is locallj~ M-bounded i f

and on/), i f X d o e s not intersect the s e t 13.26)

@ ; , ( X ) : = { z E P o f X I :z is M-connected t o s o m e

-\

EPofX)}.

I

Special a s s u m p t i o n s o n G ~ ( 6 o lead ) to o t h e r s i m p l i f i c a t i o n s o f t h e c o n d i t i o n o f P r o p o s i t i o n 3 . 3 4 . B . For e x a m p l e . if cS,(X)

C EM(a,)

o r , m o r e g e n e r a l l y , if t h e

a s u m p t i o n of C o r o l l a r y 1.83. ( i i i ) holds t h e n t h e f o l l o w i n g c o n s e q u e n c e o f P r o p o sition 3.34.B applies.

Corollary 3.37. Suppose that

00 f

@ and that M-connectedness defines an equi-

valence relation on the s e t ( f ~ f X defined ) in ( 3 . 2 6 ) . Then X i s locallj~M-bound ed i f and on!, i f f o r ever) M-connectedness equivalence class R C F M f X ) the open quasihomogeneous subse t X i ' = { P ' ( \ I .

V ' : = GMlu,

UG-)

IS

\ E X ,

Po(\) E R } o f the vector space

locall) M'-bounded where P ' : = P , +P- and M I : = M v ..

Pro o f . Let R be s u c h a n e q u i v a l e n c e c l a s s . First of all w e observe d i r e c t l y f r o m Definition 1.81.( i ) t h a t R is q u a s i h o m o g e n e o u s so t h a t X k is q u a s i h o m o g e n e o u s ( o f t y p e M a n d h e n c e o f t y p e M'),i n d e e d . By P r o p o s i t i o n 3 . 3 4 . A t h e s e t X i is locally M I - b o u n d e d if a n d only i f

142

111. Quasihoniogeneous Averages o f F u n c t i o n s . Part 1

x ~ c V , or X ~ C V -

(3.27)

Now, obviously the condition ( 3 . 2 7 ) implies that ( 3 . 25)' is valid for arbitrary y , z E R . Conversely, suppose that the latter is the case. Then if X k Q V, then we choose

y e R such that X, Q V , so that ( 3 . 2 5 ) ' implies that X,

C

V - for every z E R , i.e.

xk=u P'(X,) c v- . Z€R

I t f o l l o w s that X k is locally MI-bounded if and only if ( 3 . 2 5 ) is satisfied for arbitrary y , z E R . Since X , = @ for every y € . C s M \ c S ~ ( x ) the assertion follows by Proposition 3 . 3 4 . B .

E x a m p l e 3.38. Let p 6 R 3 such that p, X:=

u

>

0 , p2

0. and p3 = 0 . We deFine

R , ( Q ) x l a l C IR3

a€&

where Q := 10. +a[' and where R, E GLII?:R ) denotes the rotation by the angle a . Then X is open, connected. quasihomogeneous OF t j pe p , and 1ocall.v p-bounded, but X # X-+ . I

Now we come to the main result on locally M-bounded open s e t s . I n particular, it includes the analogue of the condition ( b ) in Proposition 3.34.A for the case "do #

@ . "

Theorem 3.39. The Following conditions are equivalent: l a ) X is locallj M-bounded: ( b ) there is a continuous (resp. C'") Function x : X - - + l O . + ~ C which is quasihomogeneous OF degree I

;

( c ) there exists a (non-negative) Function + E C G ; ( X ) such that + o = l

;

( c l ' f o r every family 1 I O F open subsets o f X such that ( U M ) u c l l is a locally Finite covering OF X there e i i s t s a Fami?, $,ECZZ;X), U E U , such that

OF non-negative Functions

3.c

143

W h e n i s E v e r y Compact S u b s e t of X M - b o u n d e d ?

( c )" f o r every compact subset K o f X there esists

GK 6 C

z ( X ) such that

(q5K)Of0 on K .

T h e first s t e p in t h e p r o o f is t h e c o n s t r u c t i o n o f a s u i t a b l e c o v e r i n g o f X . S i n c e i t i s u s e d in s e c t i o n ( d ) b e l o w , as w e l l , w e f o r m u l a t e it as a s e p a r a t e l e m m a .

Lemma3.40. Suppose that X is locally M-bounded. Let 11 be a family o f open subsets o f X such that I l l , ) , , , ,

covers X . Then there eAist sequences ( K n , ) , r , c N

o f compact subsets o f X , (V,),,, and (Urn)I,,

o f relativelj compact open subsets o f X ,

in 12 having the following properties:

cN

-

K,

(3.30)

U

(3.311

C V, C V , C U,,

(K,)M

mgN;

,

=X:

m €N

and (3.357)

f o r every m E N onlj, finite1.k man] o f the

(

5 ) M ,j , is

holomorphic. The formula a b o u t t h e derivatives of

Q u follows from < $ , ,(,k( m ) ),y>

= ( - 1 ) k ;k,)(m*) > ,

k€",

from the corresponding formula i n Proposition 3 . 3 . ( i i ) , and from (1.65) .

(ii) ;

We set S : = supp u and

Proposition 3.22 t h e s e t

Q, : =

Q,M,,/I

s u p p y . Then by t h e assumption on 0 and by

n S = S n ( Q , n S M , I ) M , l , I is a compact s u b s e t

of X . Since by Proposition 3.10 t h e support of rprn*,"

is contained in

the

first part of t h e assertion is proved. T o prove t h e o t h e r p a r t s of t h e assertion we are first going to find an open neighbourhood U of S such t h a t U is an ( M , I ) - b o u n d e d s u b s e t of X and such t h a t K : = Q n U M , , is a compact s u b s e t of X . Indeed, since it was proved above t h a t L : = QM,l,l n S is a compact s u b s e t of X we can choose

E

0 such

t h a t L + K(O,E ) C X . Moreover, by Proposition 3.42 we find an open neighbourhood W of S which is an ( M , I ) - b o u n d e d s u b s e t of X . Then by Lemma 3.cl

U : = ( W \ Q , M , , / l ) u ( L + K ( O , f ) )is an open ( M , I ) - b o u n d e d s u b s e t of X containing S such t h a t L ' : = U n Q M , l / I (which is equal t o ( L + K ( O , f ) )nQ,,,

)

is a com-

pact s u b s e t of X . Since by Proposition 3.22 t h e s e t Q n L'M,] is a compact s u b s e t of X containing t h e s e t Q, n ( U n O M , l / l ) M , , which coincides with Q n U M , ] we see that U has t h e desired properties, indeed.

We now choose x € C g ( X ) equal to 1 near K . I f x € U then f o r every t € J , : =

4.a

157

The Quasihomogeneous Averages

( t E I / I ; M , x E O } w e have M , x € O n l l M , ~so t h a t X ( M , x ) = l a n d (x) = Vm*,v

f

tCm*(xrp)(Mtx)v ( t ) T dt =

(X'~),,,*,~(X),

JX

i . e. rpmX,v a n d ( ~ r p ) ~ * , ,c o i n c i d e o n U . A n d s i n c e s u p p

is c o n t a i n e d in U M , ,

i t f o l l o w s in view of R e m a r k 4.1 t h a t ~u,cpm*,,>===I'

dt tm* ( X ' p ) ( M l / , x ) w ( t ) T = ( ~ ' p ) , * , ~ ( x ) ,

XEU,

I

so t h a t < u O T , J,

> = < u , ( x ' P ) ~ *> ,. ~In

view of t h e p r e c e d i n g e q u a l i t i e s t h e p r o o f

of ( 4 . 4 ) is c o m p l e t e .

N e x t w e are g o i n g to p r o v e t h e a n a l o g u e s of P r o p o s i t i o n s 3.4 a n d 3.7 a n d of Corollary 3.5 f o r d i s t r i b u t i o n s .

158

I V . Quasihomogeneous Averages of D i s t r i b u t i o n s . Part 1

Proporitlon 4.4. ( i ) Let N E N o and d € C , and let P o : X x V * + @

be a C a c o p o l ~ , -

nomial function which is almost quasihomogeneous of degree t,of type M % ( - M I *, and of order 5 N . Then we have N

(4.6)

Po Is,3 ) u ,

,

s

=

(-1)

(Pi Is,3 ) u ) ,

+

p,

wi

i=0

and N

(4.6)'

IP, ( x , 3 ) u Jrn

+p

,

=

sPi

(s,3 ) u,, ,wwi

i= 0

where

(1.7)

p . := ( 3M

X ( - M)'

ieN.

-t)I p ,

(iii) ( U O A ) , ~ ~=, u,,,,, ,. oA for ever>' A E G L I V , V ) commuting with M ( i v ) ( u ~ ~ ) ,=~( ,u ,, ~, , , ~ ) i~f ~@ satisfies the assumptions of Remark 2'.67.(ii) proOf. Note first that P i ( x , d ) u , q u , and - b y Lemma 3 . 0 . ( i i ) -

a;(X) and

U~

belong to

that by Lemma 3 . 6 . ( i ) u o A belongs to D ' , ( A - ' ( X ) ) , indeed.

l i ) : Let c p € C g ( X ) . Then

by Proposition 3.4 and by ( 1 . 6 5 ) w e have

and

Iii): This

is a special c a s e o f ( i ) . It a l s o follows from ( 3 . 8 ) and ( 3 . 8 ) ' .

159

4.a T h e Quasihomogeneous A v e r a g e s

liv):

From Proposition 3 . 7 . ( i i ) we deduce

Finally, (iii) f o l l o w s f r o m Proposition 3 . 7 . ( i ) in a similar way.

By making use of ( 4 . 4 ) o n e can prove t h e a s s e r t i o n s in a perhaps more d i r e c t way w i t h o u t recourse to t h e defining equations ( 4 . 3 ) .

In view of Example 1.21 we obtain a s a special case of ( 4 . 6 ) :

T h e following lemma is required f o r t h e proof of Proposition 4.9 below.

( X;) . and Lemma 4.S. Suppose that I i s an interval. Let f E C c I ( X I and ~ € 3 assume that s u p p f n ( s u p p u ) ~ i,s ~an M-bounded subset o f X . Then f o r every

P E @ the distributions f u , , ,

and

fp-,rl,v

u belong t o D h , l X ) s a t i s o i n g

I;

(f~m,w)F,"'k

=

s (fP-rrl, "< ,uJP.,,I;-i ,i

7

kENJ,.

i=O

N o t e that 6). Lemma 3.25. (il t h e assumption on the supports o f f and u i s automatically s at i s fi ed i f I i s non-c om pac t.

hf. The f i r s t

p a r t of t h e assertion follows by Propositions 4 . 3 . ( i ) and 3.10 and

by Lemma 3.25. Let q E C r ( X ) . Then by Propositions 4 . 3 . ( i ) and 3.10 we have S U P P U ~ . , . , n s u ~ p ( f c p , * , ~ ,c, ~() S U P P U ) ~ ,nI s u p p f n ( S U P P C P ) ~

and s u p p u n s u p P ( f t - r n , v m i Y J ~ * , ) ~C ~S -U P~ P U n ( s u p p f ) M , , , , n ( ~ ~ p p c p ) M . By Proposition 3.22 and Lemma 3.25 t h e s e sets a r e c o m p a c t . Hence, taking ( l . b 5 ) and Remark 4.1 i n t o account and applying Corollary 3.5 to t h e function q = q e x , , k ( n o t e t h a t by Proposition 3.13 i t s i t h o r d e r deficiency q i is equal to ( - l ) i q g * , , , , k -) i we deduce t h a t

160

(b)

I V . Q u a s i h o m o g e n e o u s A v e r a g e s of D i s t r i b u t i o n s . P a r t 1

Descrlblng Quasihomogc?ncousDlstribulions in ‘I’erms of‘

Qunslhomogeneous Avc?r:igc?s

For t h e whole section w e fix a number k € N o

by u r n , , ; if I = l O , + ~ tlh e n w e

Notatlon4.6. F o r a n y u E % ; ( X ) w e d e n o t e a l s o w r i t e u , , i n s t e a d of un1,1.

Propodtlon 4.7. Let u E a ) k f X ) . Then

u , , , , ~ is almost quasihomogeneous o f de-

gree m and o f order 5 k . In particular. u,,, is quasihomogeneous o f degree m . Moreover. for every i E N k we have

Proof. Let r p € C F ( X ) . Applying (3.11) to ( q , m * ) i n s t e a d of ( f , m ) , m a k i n g use

of P r o p o s i t i o n 3 . 7 . ( i ) a n d t a k i n g (1.0S) i n t o a c c o u n t w e c o n c l u d e t h a t f o r every

t E I O , + w [ w e have

< u ~ . , ~ , ~ o M ~= ,t -(l ’P< U>m

,,.,k , ( ~ O M l / t > =

= ( - l ) k t - ” < ~ , ( c p o M , / ~ ) ~ , , * , ‘ ~( ~- l>) k t-I’ < u , v , , , , * . ~ ~ o M ~= / ~ > k

= t-1’ ( l / t ) ” ’ *

c

Ui(l/t) (-l)k-i

< u , ‘p,,,*,wk-i>

=

i=O

H e n c e , t h e a s s e r t i o n f o l l o w s by P r o p o s i t i o n 2.31. An a l t e r n a t i v e p r o o f c a n be b a s e d o n ( 4 . 4 ) . F o r still a n o t h e r a l t e r n a t i v e p r o o f o n e verifies ( 3 , - m ) i e N , a n d (3, - m ) u,

= -

= 0 by m a k i n g u s e of ( 2 . 5 ) a n d L e m m a 3.12.

W e c a n n o w f o r m u l a t e t h e main t h e o r e m of t h i s s e c t i o n

Theorem 4.8.

Let T E D ’ ( X ) . Then the Following conditions are equivalent:

(a)

T is almost quasihomogeneous OF degree m and OF order 5 k :

(b)

( T , r p > = O for every p E C ; ; ’ I X ) such that p r n , , w k s O ;

fc)

T = u , , , , , , ~ For some u € D L , ( X ) .

,

4 . b Ouasihomoaeneous Distributions are Quasihomogeneous Averages

h o o f . (a)*fb):

that

'p,,,~,,,~

Let ' ~ E c ~ ( x )a n, d set V k : = 6 & ~ , 0 , ~ a, n d

= 0 .This

means that @ =

- ' p m + , w k X C , , + o a C .In

161

~ : = ' p ~ * Suppose , ~ ~ .

view of Proposition 3.10

t h i s implies t h a t SUPP @

( s ~ P P ~ ) M , l O . l (l sn u P P ~ ) M , C l , + ~ C

Since by Lemma 3.27 t h e r i g h t - h a n d side is a c o m p a c t s u b s e t of X t h i s m e a n s t h a t @EC:(X).

N o t e t h a t by Lemma 3.12 a n d (2.5) w e have

'p

= - t(dM-m)k+lQ .

H e n c e by ( a ) a n d Proposition 2.31.(i) w e d e d u c e t h a t

< T , ' p > = - < T , t ( d ~ - mk)+ l @ ) = - < ( a M - m ) k + l T , @ > = O .

Applying ( 2 . 5 ) a n d ( 3 . 7 ) w e o b t a i n

( b ) + ( a l : Let (r,€C:(X).

(t(dM-m)iq)m*.Wk=

(dM-m*)'

'pm*,uk

i€iN,,.

9

Hence, setting : = tm*(r,oMl,t -

k 2 q ( t )9 a M - m ) i y i=O

w e o b t a i n by Proposition 3 . 7 . ( i ) a n d in view of (1.65) t h a t k

C ( > + ( 1 / t(a, )

Xmr ,idk = ( l / t ) - m * ( r , m , , ~ , k o ~ l , t

-m*)'

(r,m*,L,k

i=O

Since by Proposition 3.13 qm*,cdkis a l m o s t quasihomogeneous of d e g r e e m * a n d of o r d e r 5 k t h e r i g h t - h a n d s i d e of t h e preceding equation vanishes, a n d t h e con-

dition ( b ) - applied to

x-

implies t h a t k

O = ,

i=O

i.e. ( a ) h o l d s .

Ic)+la):

t h i s follows f r o m Proposition 4.7

For t h e proof o f t h e implication " ( a l = ? I c l '* we require t h e following partial generalization of Corollary 3.5 to d i s t r i b u t i o n s . T h e proof m a k e s use o f t h e implication "(a)*(b)"

of T h e o r e m 4 . 8 .

Propoaltlon 4.9. L e t l ' € C . N E N O . a n d let T E D ' ( X ) b e of degree

eCC'

almost q u a s i h o m o g e n e o u s

a n d of order 5 N . Then s e t t i n g T j : = ( 3 M - r n ) i T w e have for

162

IV. Q u a s i h o m o g e n e o u s Averages of D i s t r i b u t i o n s . Part I

m f . L e t v € C g ( X ) . F o r t h e p r o o f of (4 . 1 2 ) w e f i r s t observe t h a t

=

where

+:=f,*,cp

a n d - in view of (1.65) N

< ( f T ) , , , + e , w+ C ( - l )( f'T i ) , m + r , w , .,, i q > = < T , x > i=l

where N

x

t(aM -[)'

:=

( f 'p -",p-I-',v,,,i) .

i=O

H e n c e , by t h e e q u i v a l e n c e " ( a )M ( b ) '' of T h e o r e m 4 . 8 i t s u f f i c e s f o r t h e p r o o f of ( 4 . 1 2 ) to s h o w t h a t ( + - x ) ~ * , ~ ~ ~ To = O t .h i s e n d w e d e d u c e by m a k i n g u s e of

( 2 . 5 ) a n d ( 3 . 7 ) a n d of P r o p o s i t i o n 3.13 t h a t

a s desired (4.12)': N o t e t h a t

where N

x

:=

2 t(a,-4)'(f,,,"icp)

.

i=O

H e n c e , by T h e o r e m 4 . 8 , " ( a ) H ( b ) " , a g a i n , it s u f f i c e s to s h o w t h a t

($-x)e*,WN

4.b

163

Q u a s i h o m o g e n e o u s D i s t r i b u t i o n s are Q u a s i h o m o g e n e o u s A v e r a g e s

v a n i s h e s i d e n t i c a l l y . To t h i s e n d w e m a k e u s e of ( 2 . 5 ) a n d (3.7) a n d of P r o p o s i t i o n 3.13 to o b t a i n

Applying L e m m a 4 . 5 to ( t * - m , t * , T , , v , N )

i n s t e a d of ( m , t , u , w , k ) a n d t a k i n g

(4.2)' into account we conclude t h a t N

=,x

+e*.wN

I=O

(fm.wq

'p)e*,C.iN-i= x t * . f d N

.

Corollary 4.10. Let T E Z J ' l X ) be almost quasihomogeneous o f degree order 5 k

m and of

. Then k

T = X ( - l ) ' ($(dM-m)iT)m,t,,,

(1.13)

i=o

End o f the proof o f Theorem4.8: ' ' f a ) * f c ) " . S e t t i n g u i : = c l , ( a M - m ) ' T w e de-

d u c e f r o m (4.11)a n d (4.10)t h a t

H e n c e in view of (4.13)t h e e q u a t i o n T = u,,,,~

is valid f o r

i=O

As a c o n s e q u e n c e of C o r o l l a r y 4.10 w e n o t e in p a s s i n g

Corollmy 4.11. Let T E D ' ( X ) be almost quasihomogeneous o f degree order 5 k

.

m

and o f

Then there exists a sequence f q jl j C N in C"'(X) converging weakly t o

T as j + a such that f o r every j 6 N q j is almost quasihomogeneous of degree m and o f order 5 k , the support o f qi being contained in K ,

f o r some compact

subset K o f X .

m f . Let ( f i )jcN

be a s e q u e n c e in C g C X ) c o n v e r g i n g weakly to T as j + a , l e t

+ E C G ( X ) be s u c h t h a t J l o = l , a n d set g.1 . 1. : = ( q ~ ( 3 ~ - r n ) ' f ~ ) ~ , , ~ .

T h e n in view of ( 4 . 2 ) ' f o r every iEiNku(0) t h e s e q u e n c e of d i s t r i b u t i o n s

164

I V . Q u a s i h o m o g e n e o u s A v e r a g e s of Distributions. P a r t 1

Tgi,i= ( 4 ( a M - m ) i Tfi)m,wi converges weakly to (J, ( a M -m)iT)m,l,,i a s j + a . The assertion follow s from Corollary 4.10 a n d f r o m Proposition 3.10.

Our next aim is to find mo r e general distributions u € 3 b ( X ) than those given by

( 4 . 1 4 ) which satisfy t h e equation T = u m , + . First we a r e going to rewrite ( 4 . 1 4 ) . Applying Lemma 2.34 and s u b s t i t u t i n g I = k - s we obtain

so t h a t ( 4 . 1 4 ) becomes k

(4.14)'

=

2 C k , I ( - a ~ ) ~ - (' d$M - m ) ' T I=O

where the coefficients

ck,i

ar e defined by c k , O : = l and by

In Proposition 4.13 below t h e coefficients

ck,i

appear via a property described in

assertion ( i ) of t h e following lemma. The assertion ( i i ) of this lemma s h o w s t h a t t h e functions

appearing i n ( 4 . 1 4 ) ' satisfy t h e assumption of Proposition 4.13 below.

Lemma 4.12. ( i ) If we s e t

:=I

then t h e unique solution of t h e s y s t e m of

equations

i s given by ( 4 . l S ) ; moreover, t h e numbers

ck,i

satisf),

(ii) If $ E C G ( X ) satisfies cOOzl then bj. (4.161 a sequence i s defined s u c h t h a t

( $ i ) o _ c i _ c kin

CG(X)

4.b

165

Q u a s i h o m o g e n e o u s D i s t r l b u t i o n s are Q u a s i h o m o g e n e o u s A v e r a g e s

is valid For

.J=NO and

such that

Proof. (i): W e f i x j c N a n d set m : = m i n ( j , k ) . I n s e r t i n g ( 4 . 1 5 ) , c h a n g i n g t h e order of s u m m a t i o n a n d s u b s t i t u t i n g i : = i - l w e o b t a i n

C o m p u t i n g t h e s u m in s q u a r e b r a c k e t s by a p p l y i n g Lemma 1.76 to t = k - I w e c o n c l u d e t h a t t h i s is e q u a l to rn I=O

S i n c e in case " j 5 k " by t h e binomial f o r m u l a t h e l a s t s u m is e q u a l to ( 1 - l ) j = 0 w e see t h a t t h e k - t u p l e

( c k , ~ ,. . , c k , k ) is a s o l u t i o n of ( 4 . 1 7 ) . S i n c e in t h e

e q u a t i o n ( 4 . 1 7 ) t h e c o e f f i c i e n t of

ck,,

is e q u a l to 1

t h e s y s t e m of e q u a t i o n s

( 4 . 1 7 ) c a n be r e w r i t t e n in s u c h a way t h a t it gives a r e c u r s i v e d e f i n i t i o n of t h e 'k,i.

T h i s s h o w s t h a t t h e s o l u t i o n is u n i q u e

Finally, f o r t h e d e r i v a t i o n of ( 4 . 1 7 ) ' m a k i n g u s e of

(1)

=

1/11,+ ( ; ; I )

IEN,-, ,

1

w e conclude t h a t

a n d ( 4 . 1 7 ) ' f o l l o w s by t h e c o m p u t a t i o n s a b o v e .

( i i ) : Since

( - d ~ ) ~ $ = ci k , i 4 0

(ck.i QO)O.wk+,

=

it f o l l o w s by ( 3 . 7 ) a n d by P r o p o s i t i o n 3.13 t h a t (-aM)' (h)O.uk,,

=

('h)O.wk+i-i

9

jcNo,

i.e. ( 4 . 1 8 ) is valid. M o r e o v e r , t h e c a s e i = k a n d j = O s h o w s t h a t ($o)o,wk= ( $ ) o

so t h a t (4 .1 9 ) is a c o n s e q u e n c e of t h e a s s u m p t i o n o n $ . The following proposition provides more general distributions u satisfying t h e r e b y giving a n o t h e r p r o o f of t h e implication " ( a ) * ( c ) "

T=urn,,k,

of T h e o r e m 4 . 8 .

166

IV. Q u a s i h o m o g e n e o u s

Ropodtion 4.13. Let NEN,such

Averages of

D i s t r i b u t i o n s . Part

1

that N 2 k . and let T E a , ' ( X )be almost quasihomo-

geneous o f degree m and o f order 5 N . Then For any sequence f$i)05i

sk

in C G l X )

satisfying (4.18) f o r , J = l O l u N N and (4.191 we have

Jli are

N o t e t h a t in case N = k a n d if t h e

c h o s e n as in Lemma 4 . 1 2 . ( i i ) t h e n one

o b t a i n s ( 4 . 2 1 ) by i n s e r t i n g (4.14)' i n t o t h e e q u a t i o n T = u , , , ~ .

p r o O F . FOI- every i E N k u l O ) w e a p p l y ( 4 . 1 2 ) ' to ( ( a M - m ) i T , m , O , + i , w k ) i n s t e a d

of ( T , t , m , f , w ) a n d o b t a i n N-i (+i ( a M

- m ) ' ~ ) , , , , u=~

G

(+)O,cakml ( a M

-

m)i+l

T.

1=O

C o m b i n i n g t h i s w i t h t h e e q u a t i o n ( 1 . 3 8 ) a n d t a k i n g ( 4 . 1 8 ) i n t o a c c o u n t w e see t h a t t h e f i r s t s u m o n t h e r i g h t - h a n d side o f ( 4 . 2 1 ) is e q u a l to

w h i c h via t h e s u b s t i t u t i o n j = i + l t u r n s o u t to b e e q u a l to N

c

d k , j ( $ O ) O , C ~ , +(~a

M-m)iT

j=O

where

S i n c e by L e m m a 4 . l 2 . ( i ) w e have d k , O = l . d k , j = O for J c l N k , a n d d k , j = ( - 1 ) k ( j -k1 ) f o r j > k t h e e q u a l i t y (4.21) f o l l o w s in view of (4.19).

P r o p o s i t i o n 4.13 e n a b l e s us to p r o v e t h e f o l l o w i n g g e n e r a l i z a t i o n of T h e o r e m 3 . 4 8 to d i s t r i b u t i o n s .

Theorem 4.14. Let A be a finite subset of C , let f j n , ) r r a c A be a family in N o , and f o r every m € A let T,,, €.?)'(X) be almost quasihomogeneous o f degree m and OF order

5

j r n . Then For ever) open subset Y

OF X satiscving YM = X there exists

u ~ a h ( X with ) support contained in Y such that Jrrl

m F . Let x : X + I O , + a C

= T,

f o r every m € A .

b e a C m f u n c t i o n which is q u a s i h o m o g e n e o u s o f de-

gree I . W e fix z € A , set k : = j , ,

a n d fix i E N k u ( 0 ) , a n d set J : = { ( m , e ) E A x N O ;

167

4 . b Q u a s i h o m o g e n e o u s Distributions are Q u a s i h o m o g e n e o u s Averages

t<j,+k-i}.

F o r every ( m , P ) E J w e d e f i n e

(4.22.a)

qm,e : =

w h e r e qj : =

(-1)j

~I ) , ~ ~ - ~x" +

c

~

in case m = z a n d

t 2 k-i

otherwise

o j o x . T h e n by L e m m a 2.34 a n d by P r o p o s i t i o n 2.31 a n d R e m a r k 1.74

w e have ( a M - m ) J q m , e = ( - l ) J q m , e - , f o r a r b i t r a r y ( m , t ) E J . In view of P r o p o s i t i o n 2.31 t h i s m e a n s t h a t t h e c o n d i t i o n ( b ) of T h e o r e m 3 . 4 8 is satisfied. H e n c e , there exists (4.22.b)

$,,i

E CGCX)

w i t h s u p p o r t c o n t a i n e d in Y s u c h t h a t ( m , t ) EJ .

( $ z , i ) m , c . 1 ~ =qm,e

O n c e in t h i s way f o r a r b i t r a r y z E A a n d i E N j , u ( 0 ) t h e f u n c t i o n s $ z . i are c o n s t r u c -

ted t h e d i s t r i b u t i o n U

:=

C

j

3

Crn

rn€A

$m,i

(aM-m)'Trn

i=O

is w e l l - d e f i n e d , its s u p p o r t lying in Y . In p a r t i c u l a r , u b e l o n g s to 9 h ( X ) . In

order to verify t h e o t h e r desired p r o p e r t i e s of u w e fix z € A , a g a i n , a n d u s e t h e a b b r e v i a t i o n s i n t r o d u c e d above. N o w , f o r a r b i t r a r y mEA a n d i E N j m u ( 0 ) w e a p p l y

(4.12)' to ( ( 3 ,

-m)'T,,

j m - i , m , ~ - m , x - ~ $ , , ,w~k ) i n s t e a d of ( T , N , P , m , f , w )

and obtain i,-l

( x - $m,i ~ ( a M - m ) i T m ) z , W k=

(X-rn

I=O

$m,i)z

-",.

o,k'JI

(a~-m)'+'T, .

Applying ( 3 . 9 ) to t h e f u n c t i o n q = x - r n ( w h i c h is q u a s i h o m o g e n e o u s of d e g r e e - m ) a n d t a k i n g (1.38) i n t o a c c o u n t w e d e d u c e t h a t

N o t e t h a t by ( 4 . 2 2 ) t h e r i g h t - h a n d side v a n i s h e s in case m f z . M o r e p r e c i s e l y , t h e c o n d i t i o n ( 4 . 2 2 ) leads to

S u m m i n g over mCA a n d i E N j m u ( 0 ) w e o b t a i n T, by a s i m i l a r a r g u m e n t as in t h e p r o o f of P r o p o s i t i o n 4.13 or by a p p l y i n g P r o p o s i t i o n 4.13 to T, i n s t e a d of T a n d to t h e f u n c t i o n s +i : = x - , $ , , ~ . T h i s s h o w s t h a t U , , ~ . , ~ = Trn~ .

T h e f o l l o w i n g division t h e o r e m is o b t a i n e d by a n a p p l i c a t i o n of T h e o r e m 4 . 8

Theorem 4.15. Let PEC, and let q C"IX) be quasihomogeneous of degree P .

168

IV. Q u a s i h o m o g e n e o u s Averages of D i s t r i b u t i o n s . Part 1

Suppose that f o r every T € a ' ( X ) the equation

(4.23)

qS=T

has a solution S E 2 J ' t X ) . Then f o r ever). T E Z J ' f X ) which is almost quasihomogeneous o f degree rn and o f order 5 k the equation (4.23) has a solution S E B ' ( X ) which is almost quasihomogeneous o f degree m-P and o f order i k .

N o t e t h a t by t h e Lojasiewicz division t h e o r e m (see C141) t h e a s s u m p t i o n is s a t i s fied if q is real a n a l y t i c .

proOf. L e t T E % ' ( X ) be as in t h e t h e o r e m . T h e n by T h e o r e m 4 . 8 w e f i n d u € % D ' , ( X )

such that U ~ , ' . , ~ = T By. t h e a s s u m p t i o n w e c h o o s e a s o l u t i o n V E % ' ( X ) o f t h e e q u a t i o n q v = u . In view o f P r o p o s i t i o n 3 . 4 2 w e c a n m u l t i p l y v by a s u i t a b l e c u t - o f f f u n c t i o n w i t h M - b o u n d e d s u p p o r t a n d still o b t a i n a s o l u t i o n of q v = u . H e n c e w e may a s s u m e t h a t v b e l o n g s to % D ' , ( X ) . I t t h e n f o l l o w s by P r o p o s i t i o n s 4 . 4 . ( i i ) a n d 4.7 t h a t S : = v m - o , c d k is t h e desired s o l u t i o n of ( 4 . 2 3 ) .

tc)

Solving l h e Eyualion t a M - m ) S ='I'

Let x : X + l O . + ~ C gree 1 , a n d set

be a c o n t i n u o u s f u n c t i o n which is q u a s i h o m o g e n e o u s of de-

U o : =x-'(lO,2C) a n d

ll,,,:=x - ' ( l l , + a - . C ) . T h e n

~ E C - ( X ) s u c h t h a t s u p p x C Uo a n d s u p p ( 1 - x ) C U,. s u p p x is a n

( M,

[I,

+mC 1 -

we

choose

H e n c e , by L e m m a 3.20

bounded and s u p p ( 1 - x ) an ( M ,10,11)-bounded s u b s e t

of X . Finally, w e fix T E % ' ( X ) . T h e n ( l - x ) T b e l o n g s to % \ , , , , ( X )

a n d xT to

% k , , + m L ( X )so t h a t it m a k e s s e n s e to d e f i n e (see N o t a t i o n 4 . 0 )

n-

Theorem 4.16. TI,, is a well-defined distribution on X having the following properties: (i)

( 3 , - m ) TI,, = T ;

4.c

S o l v i n g t h e Equation

fmc (supp T ) ,

169

( 3 ~m-) S = T

(ii)

supp

(iii)

let r E N o u l a ) ; i f T is induced by a C' function so is

(iv)

i f q 6 C L v ( X )is quasihomogeneous of degree P E @ then q ? m = ( q T ) , + p .

;

f,,, ; w

mf.(il: Let

cpECT(X). W e observe t h a t by Lemma 3.12 w e have

( a M ~ ) m * , l O . l l = m '*P m * . I O . l l + ' P

and

( d M v ) m * ,C 1 , +

mC

= m x 'Pm+, C1, + m C - 'P

Hence, using (2.5) w e o b t a i n

m:

is a c o n s e q u e n c e of Proposition 4 . 3 . ( i ) .

(iii): f o l l o w s f r o m Proposition 3 . 3 . ( i ) a n d ( 4 . 2 ) ' .

(iv):

f o l l o w s f r o m Proposition 4 . 4 . ( i i ) .

Corollary 4.17. For ever!. d E ' a ' f X ) which is almost quasihomogeneous of degree m there exists u C D ' f X ) which is almost quasihomogeneous o f degree m with deficiencj d .

I

A s a n o t h e r c o n s e q u e n c e of Theorem 4.10 o n e o b t a i n s t h e following generalization

of t h e Division T h e o r e m 4.15.

Theorem 4.18. Let q and P satisfj, the assumptions o f Theorem 4 . 1 5 , and let T , c € B ' ( X ) . I f q c = ( d M - r n ) T then there ezrists a solution S E B ' I X ) o f the equation ( 4 . 2 3 ) such that

(aM - (m -P)) S

=c .

In particular, i f TE B ' (XI is almost quasihomogeneous o f degree m with deficiency d and i f c € B ' ( X ) is a solution o f the equation q c = d then there exists a solution S E B ' ( X ) o f (4.23) which is almost quasihomogeneous o f degree m-P with deficiency c .

170

-.

IV. Q u a s i h o m o g e n e o u s A v e r a g e s OF D i s t r i b u t i o n s . Part 1

By Lemma2.34 and by Theorem 4.16.(i) (applied to ( c , m - P ) instead of

( T , m ) ) w e have (aM-m)(qc",-o)=q(aM-(m-P))z , , - p = q c = ( a M - m ) T , i.e.

T- q

cm- e

is quasihomogeneous of degree m . By Theorem 4.15 w e find a distribu-

tion R E % ' ( X ) which is quasihomogeneous of degree m-P suc h t h a t q R = T - q z m - p . Hence in view of t h e preceding application of Theorem 4.16.(i) t h e distribution

."

S : = R + c,,,-@ is t h e desired solution.

(d) Singular Support and W a v e Fro111Scls 01' l h e Uistributions u,,.,

Propoeltton 4.19

. Sing sirpp u,,,,

C (sing s u p p

ii ) M . I

For ever,' u €3;( X I .

m F . L e t K be a compact s u b s e t of X \ ( s i n g s u p p u ) M , I . Then K,,,,,

n s u p p u is a c om pa c t

intersect sing s u p p u . Since by Proposition 3.22 L : = K M , s u b s e t of X we can fix a function x € C T ( X ) s u ch tha t of L and s u c h t h a t s u p p

d o e s not

x=1

on a neighbourhood

x is contained i n t h e open set Y : = X \ sing s u p p u . We

choose f € C m ( Y ) s u c h t h a t T , = u l y . Then x f exte nds to a function gEC;;'(X) satisfying x u = T g . Let v € C T ( K ) . Then by Proposition 3.10 L contains t h e s e t so t h a t

s u p p u n s u p p rpr,,*,

< u,,,,,

'p

> = < x u , Y ~ , , * , , > = = < ( T g ) m , w, V J > .

Since by ( 4 . 2 ) ' we have (T,),,,,=T,m,w

and since by Proposition 3 . 3 . ( i ) gm,,

is a C m function it fo l l o ws t h a t o n K t h e distribution urn,,

is induced by a C m

function. Since by Proposition3.10 ( s i n g s u p p u ) ~ , is I a closed subse t of X t h e a s s e r ti o n follows.

H

For t h e wave f r o n t s e t s of urn,,

a similar r es u l t is valid:

Theorem 4.20. For ever) u 6 3 i f X ) the set W F ( U ) M , I:=Ute,M , ' W F ( u ) - where M ~ W F ( u ) : = { ( M * , , , . , M ~ l l ) ;( ) , , q ) € W F ( u ) )- is a closed subset OF X x \ ; * containing W F ( u , , . , ) .

7he assertion remains valid i f WF is replaced bj

WF,.

4.d

Singular

Support

171

a n d Wave F r o n t Sets of

As for t he proof. The a s s e r t i o n s a r e special c a s e s of more general r e s u l t s t h a t

will b e proved in C h a p t e r 9 ( s e e Remark 9.10 and Theorems 9.11 and 9.28 applied

to ( M , l d v * ) instead of ( N , M ) ) .

Suppose t h a t w

1 . Then t h e assertion of Theorem 4.20 neglects t h e s m o o t h i n g

e f f e c t which c o m e s a b o u t by t h e integration along t h e quasihomogeneous rays

{ M t x ; t E l O , + c o C } , x € X . In f a c t , since urn is quasihomogeneous Proposition 2.14 For WF instead of WF,

s a y s t h a t W F A ( u r n ) is contained in T,(X).

we a r e going

to prove this inclusion more directly, thereby, in addition, obtaining continuity properties which a r e required f o r t h e proof of Theorem 4.25 below. Recall t h a t f o r any closed conic s u b s e t

r of

X x +* t h e space a;(X)

a s t h e s e t of all distributions T E a ' ( X ) such t h a t W F ( T ) C

is defined

T. The topology of

a ' , ( X ) is defined by t h e s e m i - n o r m s of t h e weak topology of a ' ( X ) and t h e s e m i n o r m s of t h e f o r m (4.25)

T H s u p ( r N [ F ( c p T ) ( - c < ) lr;: E I O , + a C , < E H }

where N E N and cp€C;'(X)

and where H is any compact s u b s e t of

(suppc~,)xH n T = 0 ( s e e Definition 8.2.2 in Hormander C I I I

\i*

such that

).

Moreover, f o r any closed s u b s e t Z of X we d e n o t e by a ' ( X ; Z ) t h e s p a c e of all distributions o n X with s u p p o r t contained in Z . Equipped with t h e topology of uniform convergence on t h e bounded s u b s e t s of CFCX) it is a closed s u b s p a c e of a ' c x ) .

Theorem 4.21. Suppose that

c('

is a C'" function. Let L be a closed M-bounded

s u b s et o f X . Then the map

a'tx;L ) --+a;-,,,,

(X), u

H

u,n,w

,

is w el l-defi n ed and continuous.

Note t h a t t h e assumption t h a t w be a Cm function cannot be o m i t t e d . For e x a m p l e , t h e conclusion is false if w equals w l : =

xlo,,

o r w 2 : = x ~ , , . ~Indeed, ~ . since

t h e conclusion is valid f o r w 5 1 and since urn = u , , , ~ ~ +

it suffices to see

t h a t it c a n n o t be valid f o r both w1 and w 2 . But t h i s , i n t u r n . is clear since t h e r e e x i s t distributions U E ~ ' ( X ; Ls u) c h t h a t W F ( u ) is not contained i n T,(X)

and

172

IV.

Q u a s i h o m o g e n e o u s A v e r a g e s of D i s t r i b u t i o n s .

Part 1

s i n c e by t h e a s s e r t i o n ( i ) of Theorem 4.16 o n e h a s

Proof of Theorem 4 . 2 1 . Let K ( r e s p . H ) be a c o m p a c t s u b s e t of X ( r e s p . such that K x H n

rM(x)= @.

+*)

Since K x H is c o m p a c t , as w e l l , o n e f i n d s a c o m p a c t

neighbourhood K' of K in X s u c h t h a t

By t h e a s s u m p t i o n o n L a n d by Proposition 3.22 t h e set LnK,,,,,

is a c o m p a c t

s u b s e t of X . Let W b e a c o m p a c t neighbourhood of LnK,.,,,

in X , a n d let

. Then by Proposition 3.10 a n d Remark 4.1

X E C ~ C Wb)e equal to 1 near LnK,,,,, w e have f o r every u € a ) ' ( X ; L ) (4.27)

9(cyu,,,,)(

)m4,v

,

L E U,,

,K

H I w e fix

> O s u c h t h a t W , : = W + K ( O , E ) is contained in X . Since t h e m a p ( t , x ) H M , x is

c o n t i n u o u s o n e f i n d s for every Y E W c o n s t a n t s 6 , , € 1 0 , ~ C a n d y y > 0 s u c h t h a t { M , x ; t E I y + C - 2 y , , 2 y y l . xEK(y,G,)} C K' w h e r e I y : = { t ~ l O , + m lM; , y E K } . Since t h e sets I,,

x E W , , a r e all c o n t a i n e d in

t h e c o m p a c t subset ( t E l O . + ~ [ ;M , ( W , ) n K # Q ) } of l O , + m C w e may a s s u m e (after having m a d e S,

s m a l l e r i f necessary) t h a t I, C l y + l - y y ,yyC for arbitrary

Y E W a n d x E K(y.S,). Since W is c o m p a c t it is covered by finitely many of t h e

balls K ( y , 8,).

C o n s e q u e n t l y , o n e o b t a i n s a finite o p e n covering U o f W o f o p e n

s u b s e t s of W, a n d a family ( I L I ) U ~ L I c o n s i s t i n g of finite unions ILI of c o m p a c t s u b i n t e r v a l s of IO,+o3C s u c h t h a t for every U E U (4.28)

( a ) { M t x ; t E I L I , x E U } C K'

;

(b)

u I,

X€U

C

i)L1.

Now, in view o f (4.28.b) it f o l l o w s f o r arbitrary y E C T ( K ) , €,EH,

UEU, and (4.29)

K E E ~ , + ~ ,

x E U that ((P

e - i < = e**>

),,,*,v(x) = J t - m * y ( M , x )

e x p ( - i < r < , M t x ) ) v ( t )d Tt .

'LI

By combining ( 4 . 2 8 . a ) a n d ( 4 . 2 6 ) o n e o b s e r v e s t h a t o n a neighbourhood of t h e set D : = U , , , I U x U x H

by

173

4.d S i n g u l a r S u p p o r t a n d Wave F r o n t Sets of

g ( t , x , < ): = a C-function

-it

) = 1- r g ( t , x , ) ,

( t , x , c ) E D , r >0 .

W e i n s e r t t h i s e q u a t i o n i n t o t h e r i g h t - h a n d side of ( 4 . 2 9 ) a n d do p a r t i a l i n t e g r a t i o n ; h e r e t h e b o u n d a r y t e r m s vanish s i n c e by ( 4 . 2 8 . b ) f o r a r b i t r a r y X E U a n d t E 3 I U t h e p o i n t M t x does n o t b e l o n g to K . R e p e a t i n g t h i s p r o c e d u r e w e see

t h a t f o r a r b i t r a r y U E U , x E U , a n d N E N t h e r i g h t - h a n d side o f ( 4 . 2 9 ) is e q u a l to r - N J' Q N ( t - r n * - l

v ( t ) q ~ ( M , x ) )e x p ( - i < r < , M,x

>)

dt

'U

where Q =Q(t,x,c.3,)

d e n o t e s t h e differential operator f Hd,(gf).

Using t h e

Leibniz r u l e , m a k i n g u s e of (3.5),a n d a p p l y i n g t h e p r e c e d i n g c o n s i d e r a t i o n s to t h e derivatives of q , as w e l l , w e c o n c l u d e t h a t f o r a r b i t r a r y q E C F ( K ) a n d N E N t h e set ~N:={rNX(qe-i

; rE

Cl,+al, . i=O

By T h e o r e m 4 . 2 5 . ( i ) t h e l e f t - h a n d s i d e of t h i s e q u a t i o n is e q u a l t o k

=

(P,,,*..,~I~~>.

( 3 . t ) ' :

is continuous;

(iii) f o r every k € M the conditions ( i ) and ( i i ) remain valid i f w is replaced by

W ( J ; ~

( i v ) f o r ever) fEE the map D

+F ,

m H f,.,

, is holomorphic, f o r ever)'

k E N i t s derivative o f order k being given b), m e k ! ( - 1 )

k

In order to i n t r o d u c e t h e r e l e v a n t s u b s p a c e s of C o ( X ) w e fix a c o n t i n u o u s w e i g h t function W :X

C O , + ~ [.

Not.tlonS.1. ( i ) By C o ( W ) w e d e n o t e t h e Banach s p a c e of c o n t i n u o u s f u n c t i o n s f:X+@ that

such t h a t / / f / l W : = s u p { I f ( x ) l / W ( x ) ; x € X \ W - ' ( O ) } i s finite and such

1 f 15 11 f I I w W . N o t e t h a t t h i s d e f i n i t i o n m a k e s s e n s e f o r any w e i g h t f u n c t i o n

W:X+CO,+al,

b u t t h e n C o ( W ) need n o t be a Banach s p a c e .

( i i ) Let r , s E N o u ( a ) ; r e f e r r i n g to Definition 1.18.(i) ( w h e r e t h e s p a c e C'"(X)

w a s d e f i n e d w i t h r e s p e c t to s o m e p r e - f i x e d M - i n v a r i a n t d e c o m p o s i t i o n of V of t h e f o r m (1.26)) w e d e n o t e by C r ' s ( W ) t h e s p a c e of all f u n c t i o n s f E C r o S ( X )

183

5.a Defining f m , w by ( 3 . 1 ) '

s u c h t h a t f o r every polynomial function P : V * d @ of degree 5 ( r , s ) t h e function

P ( a ) f b e l o n g s t o C o ( W ) . Of course, if V, = ( O ) w e write C r ( W ) instead of C r S s ( W ) . Note t h a t equipped with its natural n o r m s C r P s ( W ) is a Banach s p a c e in c a s e r , s < + aand a FrCchet s p a c e otherwise. Now we fix a n o t h e r continuous weight function U : X

+CO,+mC

and c o n s t a n t s

c , d E IR s u c h t h a t c < d . We s u p p o s e t h a t there e x i s t s Y E % ' ( I O . + ~ C d: t / t ) such that

Lemma 5 . 2 . Under the preceding assumptions the assertion ( 5 . 1 ) is valid f o r E = C O ( W ) . F = C O ( U ) . and D = a : ( c , d ) : = { z ~ C :c c R e z \ d } .

mf.(i):Let r n E @ ( c , d ) .Then I t - m I = t - R e m 5 m a x { t - C , t - d } = l / m i n { t C , t d } . Hence ( 5 . 2 ) implies t h a t

fm,,,

Consequently, f o r every f € C o ( W ) one deduces t h a t

is a well-defined

function o n X satisfying +m

(5.4)

I1 f m , w IIL,

5

c II FII,

where

C :=

'

t < y ( t )d T

+a.

0

In o r d e r to s h o w t h a t fm,w is continuous we fix x E X and a c o m p a c t neighbourhood K of x in X and deduce f r o m ( 5 . 3 ) t h a t

where C ' : = I( f

IIw

s u p { U ( y ) ; y E K } is finite. Hence Lebesgue's Dominated Conver-

gence Theorem s h o w s t h a t f m , w ( y ) t e n d s to f,,,(x)

( i i i ) : First

of all we fix

E

as y + x .

~ 1 0 , y and C observe t h a t

1 w k ( t ) 1 m i n t t C , td 1 5 c

~mint, t C~+ ' , td-'

,

t E IO,+mC, k E N ,

where C , , k : = i n f ( t - E ~ k ( t t)E; C l + a C } . Consequently, (5.6)

(5.2) remains valid if ( w , y , c , d ) is replaced by ( w ' d k , C E . k y , c + ~ , d - ~ )

so t h a t everything we did in t h e proof of ( i ) remains valid f o r wwk instead of w provided t h a t m belongs t o @ ( C + E , d - E ) .

184

V. Q u a s i h o m o g e n e o u s A v e r a g e s of Functions. Part 2

I t follows t h a t t h e a s s e r t i o n ( 5 . l . i ) remains valid f o r D = @ ( c + E , d - E )with w replaced by w u k . Since

E

can be made arbitrarily small t h i s is a l s o t r u e f o r D = @ ( c , d ) .

For t h e assertion (5.2.ii) t h e s a m e a r g u m e n t applies o n c e it is proved in its original f o r m .

lii): We fix m E @ ( c , d ) , c h o o s e

E

> O so small t h a t Rem+C-ZE,ZEI C I c , d l , a n d

let ~ E @ ( - E , E ) Note . t h a t then m + h E @ ( c + E , d - E ) .By t h e main theorem of calculus we obtain f o r every f 6 C o ( W ) t h a t + nr,

is well-defined and finite f o r v : = I w log I by t h e f i r s t part of condition i i i ) already

completely proved above. Applying t h e inequality ( 5 . 4 ) to s t e a d of ( f , m , w ) we deduce t h a t N ( f ) 5 C C , , , Ilf

(

I f 1 , Re(m s h ) , v ) in-

I l w . Combining t h e preceding

e s t i m a t e s with

11 fn, + ti . w - gm,w 11 LI

5

11 f i n +

11 ,

w

-

fm,w

11 u 11 ( f - g ) r n , w 11 u +

9

g € C O (w ) ,

and with ( 5 . 4 ) (applied t o f - g instead of f ) we derive t h e continuity of t h e map under consideration a t t h e point ( m , g ) E @ ( c , d ) x C o ( W ) . (iv):Let m , E , and h be a s in t h e proof of ( i i ) . Since by ( 5 . 6 ) t h e e s t i m a t e (5.5)

is valid with ( w , C ' ) replaced by ( w l o g , C ' C , , l ) we can apply Fubini's theorem to t h e right-hand side of ( 5 . 7 ) to obtain 1

hI ( f m + t i , ~ - ~ m , w ) (=X-)J ' f m + s h , w l o g ( X ) d s

*

X€X.

0

Since by ( i i i ) t h e function

@(-E,E)

+Co(U),

h H f m + h , w l o g , is continuous

o n e deduces t h a t t h e map Q : @ ( c , d ) d C o ( U ) ,z H f , , , , , point z = m with derivative - f m , w l o g .

is holomorphic a t t h e

In view of ( 5 . 6 ) , f o r every k E N we c a n

apply t h i s to w u k instead of w and in this way obtain t h e formula f o r t h e k t h o r d e r derivative of Q by induction.

Next we a r e going t o have a look a t t h e s t a n d a r d properties of f m , w . A s f o r t h e

185

S.a Defining f m , w by (3.1)'

multiplication by quasihomogeneous functions, t h e following a s s e r t i o n is easily verified directly f r o m t h e definitions.

Remark 5.3. Let P E C a n d N E N , a n d let q : X +

C b e a l m o s t quasihomogeneous

of degree P a n d of order I N . For every j E N N l e t qi be i t s j t h order deficiency. Then for arbitrary f 6 C " ( X ) a n d m 6 C s u c h t h a t (q f )m + 8,

, (qj F ) ,

+

e, , a n d

fm,ware well-defined by (3.1)' t h e equations ( 3 . 8 ) a n d ( 3 . 8 ) ' hold. In p a r t i c u l a r ,

iF

q is quasihomogeneous of degree P then ( 3 . 9 ) i s valid.

I

We now f o r m u l a t e a n assertion about derivatives. For simplicity we r e s t r i c t ourselves t o f i r s t o r d e r derivatives. Of course, one can deduce a corresponding a s s e r tion f o r higher o r d e r derivatives by induction if one carefully f o r m u l a t e s s u i t a b l e conditions on t h e set of admissable m .

Lemma 5.4. Let

f 6 C 1 ' O ( W ) (see Notation 5.l.ii). a n d l e t P: V' &IR

Then For evegk rn 6

function s u c h t h a t k e r P 3 V:.

b e a linear

nrEn @(c+ R eP, d

+Ref)

the

function P ( d ) F m , w is well-defined a n d equal t o

1(Pi,, (a) f ) n , - p . w c . , . I

P€A(P) j€N0

where t h e polj,nomials P ,:

a r e defined in t h e t e s t preceding Remark 1.42. More-

over, we have + il',

( P ( 3 )F m , w )

(5.81

(A)

=

J' tC"'P131

dt ( F o M , ) Is) w l t ) t ,

A

EX.

0

Proof. First of a l l we observe t h a t t h e assumption o n P means t h a t P is a polynomial function o f degree 5 (1.0) such P ( 3 ) is t h e directional derivative o p e r a t o r with r e s p e c t t o a unique vector y E V, . In particular, t h e polynomial functions P z , j a r e of degree 5 ( l , O ) , too, so t h a t t h e derivatives P : , i ( 3 ) f ,

e E h ( P ) and jEIN,,

a r e well-defined and belong to C o ( W ) . We fix x E X . Then by t h e main theorem of calculus, by ( 1 . 4 0 ) , and by Fubini's theorem - w h i c h can be applied in view of (5.5) with ( f , w ) replaced by ( a i f , w w j )

*

o r ( P p , j ( d ) f , w w j-) we deduce f o r every sufficiently small hE!R

(5.9)

~1 ; ( f ~ , , ( x + h y-) frn,,(x))

=

186

V. Q u a s i h o m o g e n e o u s A v e r a g e s of F u n c t i o n s . Part 2

S i n c e by P r o p o s i t i o n 5.2 t h e f u n c t i o n s (P:*j(a) f ) m - e , w w j are c o n t i n u o u s t h e f i r s t part of t h e assertion follows. F r o m t h i s o n e o b t a i n s t h e c o n d i t i o n (5.8) by a n o t h e r a p p l i c a t i o n o f (1.49). m

T h e p r o o f o f t h e f o l l o w i n g a s s e r t i o n is trivial

Remark 5.5. Let f E C o ( W ) . I f A E L I V , V ) commutes with M then f o A belongs t o C O I W o A ) , and ( f o A ) , , , = f , ~ , o A

f o r every m E C ( c , d ) .

I

Now let G a n d (9 s a t i s f y t h e a s s u m p t i o n s o f R e m a r k 2 . 6 7 . ( i i ) . F i r s t w e observe t h a t if f E C o ( W ) t h e n f,

b e l o n g s to Co(WG) w h e r e W G : X - - 9 E 0 , + m C is t h e

c o n t i n u o u s w e i g h t f u n c t i o n d e f i n e d by

F u r t h e r m o r e , w e n o t e t h a t (5.2) r e m a i n s valid if t h e pair ( W , U ) is r e p l a c e d by ( WG

, U G ) . By a p p l y i n g Fubini's t h e o r e m ( c o m p a r e t h e p r o o f of P r o p o s i t i o n 3.7. ( i i )

w e derive

Remark 5.6. I f f 6 C o ( W ) then (fn,,,)@ (f@)",.W.

belongs to C o ( U G ) and is equal to

=

As f o r t h e s u p p o r t o f f,,,,

o n e immediately obtains

Remark 5.7. Let fECo(X) be such that by (3.1)' a function f , , , : X + C well-defined. Then supp f , ,

,C

(supp f ) M ,I

where I : = supp w .

is

187

5.a D e f i n i n g f m . w by ( 3 . 1 ) '

N o t e t h a t t h e set ( s u p p f ) M , I n e e d n o t be c l o s e d e v e n i f s u p p f is compact:

HxampleS.8. ( i ) Suppose that

(1.14) holds. I f

XEV,

then O E i \ L

where

L := c , ~ ) M , [ l , t < m c (iil Suppose that p E l O , + ~ t ' . Let P > p 2 / p r , and set

K : = { (x,y) E t - l , l J x C 0 , l I : / ~ l ' S y } . Then

L := K p , , o , , , = l R x 1 O , + ~ t u { ( O , O ~and ~, i\L=kxlO1.

proOf. (i): T h e a s s u m p t i o n (1.14) i m p l i e s t h a t Ml,,x

( i i ) : Let

t e n d s to 0 a s t + + a .

( u , v ) E I R x l O , + ~ C T. h e n w e c a n c h o o s e t E l O . 1 1 so s m a l l t h a t x : = t P 1 u

E C - 1 . 1 1 , y : = t P 2 v ~ 1 0 , 1 1 ,a n d t P 2 - p p i

_>

Iu~'/v,

i.e. y

> I x 1'.

rn

-

I n g e n e r a l , f o r K M , , \ K M , i t h e f o l l o w i n g i n c l u s i o n is valid:

LemmaS.9. Suppose that ( 1 . 1 4 ) holds. Let I be a closed subset o f I O , + l f ~ t .and let K be a compact subset o f V . Then K M . l \ K M , l C

KO

:={'

KO

u K,,, where

i f 1 / 1 i s bounded Mi'(KnM,(K))

i f 1/1 i s unbounded

and

ld

i f I i s bounded

Ka' ' = { M O ( K )

i f I i s unbounded

Proof. Let x E K M , l . T h e n w e find s e q u e n c e s ( t j ) j e N in 1/1 a n d ( k j ) j e N in K s u c h t h a t M t i k i c o n v e r g e s to x as j + m .

By c h o o s i n g s u b s e q u e n c e s if n e c e s s a r y

w e a c h i e v e t h a t ( k j ) c o n v e r g e s to s o m e k E K a n d ( t i ) to s o m e s E ( O . + ~ ) ~ l / l . If s
k + l

w h e r e P 2 , i d e n o t e s t h e j t h o r d e r deficiency of t h e polynomial P,

s a t i s f y i n g (1.53).

T h i s i m p l i e s t h e a s s e r t i o n ( i ) a s w e l l as t h e e q u a l i t y in ( i i ) f o r i = O .

G O , i . e . f'"(0) = 0 f o r every @EX:.

N o w , s u p p o s e t h a t Q,f

S i n c e P 2 , i is a l m o s t

q u a s i h o m o g e n e o u s of d e g r e e CLM so t h a t by Remark 1.42 P , * , i ( d ) is a l i n e a r c o m -

a',

b i n a t i o n of t h e o p e r a t o r s bitrary

@EX;,

t h i s i m p l i e s t h a t [ P z , j ( a ) f ] ( O ) = O f o r ar-

CLE'U~Aa n d j € ! N o . C o n s e q u e n t l y , t h e p r e c e d i n g f o r m u l a s s h o w t h a t t h e

a s s e r t i o n ( i i ) t u r n s o u t to be valid f o r every i E N o in c a s e Q,f

=O.

In t h e g e n e r a l c a s e o n e o b t a i n s t h e a s s e r t i o n ( i i ) by applying t h e s p e c i a l c a s e to g : =f

- Q,f

( n o t e t h a t Q,g

= 0 ) a n d t a k i n g E x a m p l e 5.22 w i t h f r e p l a c e d by

Q,

f

i n t o a c c o u n t . N o t e t h a t in c a s e k = O in view o f P r o p o s i t i o n 2.31 t h e a s s e r t i o n ( i i ) is a l s o a c o n s e q u e n c e of t h e c o n d i t i o n s (5.30) a n d ( 5 . 4 0 ) The case

Setting

"E

#

0 ":By R : IR +C w e d e n o t e t h e C" f u n c t i o n d e f i n e d by

V : = X ] ~ , ~ ,a n d

u : = v R o n e d e d u c e s f o r every z ~ @ ( - a , O )t h a t

S i n c e by P r o p o s i t i o n 5.13 t h e f u n c t i o n z

H

f z - r - l , u k is h o l o m o r p h i c o n @ ( - m , r + l )

t h e p r i n c i p l e of a n a l y t i c c o n t i n u a t i o n s h o w s t h a t I'

1

% t f , w - jToI ! is h o l o m o r p h i c o n

@(

(-E)J

gltf,vk(

(

*

)

-j)

- 0 3 . r ~ ~ C) o. n s e q u e n t l y , r

a - i ( m : m f , w )=

i=O

LI ! ( - s ) ' a - l ( m - j ; % t f , v k ) ,

iEN.

H e n c e . t h e a s s e r t i o n ( i ) is a s i m p l e c o n s e q u e n c e of t h e c o r e s p o n d i n g a s s e r t i o n for E = O . S i n c e by t h e a s s e r t i o n ( i i ) for t h e case E = O a - k - I - i ( m - j ; 3 1 3 f , v k ) is e q u a l

to - ( k ; i ) ( , M - ( m - j ) ) L ~ n , ~ j ft h e a s s e r t i o n ( i i ) Follows.

Applying P r o p o s i t i o n 5.25. ( i i ) to

E

= 0 and i = 0 o n e obtains

204

V . Q u a s i h o m o g e n e o u s A v e r a g e s of F u n c t i o n s . Part 2

Corollary 5.26. For every r n E C ( - a , r A m i n ) the C'T''s function Q,f pend on the choice of coordinates.

does not de-

8

I t i s n o w e a s y to c o m p u t e ( 3 M - m ) f m , w :

Propodtion 5.27. For every rn (d,

- rn) f,

=E

f, -

,,

E CI - 03,r cE) we have

- f,

$

-,- Sk,

,,,k

(- E )

Q, - i f + b -

w ( b )f 0 M b .

jGNo

Proof. W e n o t e first t h a t by E x a m p l e 1.21 a n d by P r o p o s i t i o n s 5.10 a n d 5.19 w e have

(5.41)

aM frn,w

=

)m,w

'

M o r e o v e r , w e o b s e r v e t h a t ( 5 . 2 2 . a ) c a n be w r i t t e n a s

H e n c e , if R e m 10 t h e n in view of ( 5 . 2 0 ) t h e e q u a t i o n (5.25) b e c o m e s

By t h e principle of a n a l y t i c c o n t i n u a t i o n t h e e q u a t i o n ( 5 . 2 5 ) ' r e m a i n s valid f o r every m E @ ( - m , r c , ) \ ' U , ( M ) , a n d in view of (5.41) t h e desired e q u a t i o n is p r o v e d for these m .

Now let m E U,(M ) . T h e n by t h e f i r s t p a r t of t h e p r o o f a n d by ( 5 . 4 1 ) o n e o b t a i n s f o r every z E @(-a, rc,) \*a,( M ) t h a t

H e n c e , u s i n g (5.41) a g a i n w e c o n c l u d e t h a t

In view of P r o p o s i t i o n 5.25 t h i s is t h e desired e q u a t i o n . w

5.d

205

A Formula f o r f m . w if R e m 2 0

(dB A Formula I'or

II' H c m 2 0

First we are going to derive a n explicit formula for f m , w in c a s e m d o e s n o t belong to ' U , ( M ) . A few abbreviations a r e required. (see Notation 5.15) we set A a : = c r M

We set A o : = l ,

and f o r every c r E X ' " \ ( O )

and p a : = ReX,.

We fix a number !€lo. r c , l and define

j=1 ? . ,

Note t h a t I B I ' r

PN z O

imply

I @ 1-1

-

if ( N , p ) E l p u J p (For I P - P N ~ c , < ppl+ . . . + p < r , and p p l + . . . + pBN< 4 similarly implies

EN-1

IpI

< @ s r c , and

5 r ).

Finally, we define

and

Ropoaltion 5 . 2 8 . There i s a (unique) farnil) of polynomial functions R p , o: V -

( p , o )€8,u@". of degree

5

C,

( r . 0 ) . on/> finitel., inan) of t h e m not vanishing iden-

tical/-\. having the following properties: (i)

for arbitrar) E E C and r n € C ( - w , r c : ) \ 2 1 : ( M ) the function

f m , w is

equal t o

206

V . Q u a s i h o m o g e n e o u s Averages of F u n c t i o n s . Part 2

( i i ) For every

I p , a) E B P u (YP the function RpVo is almost quasihomogeneous o f

N

degree

(p)M.

(iii) Suppose that M is semi-simple, and let ( P . o ) E 3 p Iresp. F P ) : then R g , o vanishes identicallq i f and onl-v i f o does not belong to the set

S k ( N ) := { O € N o X N N ;

N su=o O,=k+N}

(resp. S k ( N + t ) ) where N is defined by the condition (N,PIE I p u J p ; moreover, i f Ro,o f 0 then R p , o I ~ \ ) = A F s E .

proOf. By t h e principle of a n a l y t i c c o n t i n u a t i o n a n d in view of Remark 5.17 it suff i c e s to prove t h e e q u a l i t y in sequence ( t i ) ,

54

(5.42)


0 .We set L : = { ( N , f i ); N E N , B E ( U 1 ' o ) N , Apt+ . . . + X

PN

(see p . 2 0 5 ) . Note first that for arbitrary N E N and P E ( U 1 ' o ) N w e have N

(5.56)

(P)M+

-.,

N - IP 1 =Agl+

For arbitrary j E N o and aEU'

...+

X PN

*

we set

K j , a : = { ( N , P ) ;N = j + l a I , P E ( U

Hence it f o l l o w s from ( S . 5 6 ) that

1.0

)

N

, P=a).

= z } C I,

5 . d A Formula for f,

if R e m

213

2 0

where all t h e unions are disjoint. Our aim is to l e t m tend to z f r o m t h e left in every t e r m in t h e formula of Proposition S.28.(i). We d o this f o r each summand separately.

If ( N , P ) E L and

O = ( O ,-,,...,ON

Iim

) E N o x N N then

N

II ( m - X p l - . . . - X g , - l

=

)--by

m+z

and - by Propositions 5.25 and 5.30 -

0

If ( N , B ) E 1,\ L then h g l + . ..+ADN f z so t h a t - using Corollary 5.29 - w e conclude f o r every

0E

No x N N t h a t

Finally, if ( N , ( 3 ) C J p we have p

PI+ " ' +

fiN


k , as w e l l , a n d f o r p = k is e q u a l to

fi$

(

*

)a.

rn

C o m b i n i n g P r o p o s i t i o n 5.32 w i t h P r o p o s i t i o n s 5.13 a n d 5.16 a n d R e m a r k 5.17 o n e obtains

Corollary 5.33. For every m E C ( - w , r c , ) n I , ( M ) the map C r ' s ~ X ~ - + C r ' s ~ X ) . f

H fm,,,,

i s well -defined and continuous.

More precise!, , let W : X

+CO, +wC

the map C ' " s I W ) - - + C ' " s ( U ) ,

be any continuous weight function: then

f H f r n , + ,,

is well-defined and continuous where

U : = Cl+lP+(.)l)"W(b) with

and with W ( b ) being defined by (5.18).

8

Notatlon 5.34. F o r every N E C O , + a C w e d e n o t e by Y:(V) a l l Cm f u n c t i o n s g : V+ SUP {

C such that

1 g'"'(X)

is finite f o r arbitrary

t h e FrCchet s p a c e of

UE%

I ( 1 + I P + ( x ) ()-N ( 1 + I M o ( x ) 1 )L ; X E V } a n d L E N . W e set

Y E ( V ): = i n d Y E ( V ) . N++m

5.e Introducing f,

215

f o r Arbitrary m a @ and f E Y P ( V )

Recall that the space 6 , ( V )

of all multiplication operators o n the space Y ( V )

consists of all Cm functions x : V d @ such that for every ~ € 2 1there is a constant N E N such that I d " f l ( l + l * l ) - N is bounded.

Remark S . S . Let x E 6 m ( V ) be with support contained in fi, (see Notation 5.10) f o r s o m e boundedsubinterval J OF CO,+mC. Then the map Y G ( V ) + Y ( V ) ,

g-xg,

is well-defined and continuous.

m.By

the assumption on J there is a constant CJ such that for every

X E " ~

the estimate ( l + l x l ) 5 CJ ( 1 + ( M o ( x ) I ) is valid. Hence the assertion follows by the Leibniz rule.

Propodtlon 5.36. For every m E C' the Function 6 , well-deFined, linear and continuous: For ever,

Y';(V)

(V)

6,

(V),

F H F,

w ,

is

N E C O , + w C it maps the space

continuously into Y , & + N ( V ) where v is defined bj, (5.53). resp. ( 5 . 5 9 ) , i f

r E N is chosen such that Rem < r c , . Moreover, For every F E ~ D ~ ( we V ) have lim f,,

c+o

mf. We apply

=

F",x,o. b 3 wk

in the topologj o f 6,(V).

Proposition 5.12 to u = ( I

+

I * I )N 0 M o ( see (5.10)) and take Pro-

positions 5.16, 5.28 and 5.32 i n t o account.

tc) Introducing I*,

for Arbitrary m € C and r ' E p t V )

If no restricting condition on the support of w is assumed we decompose w according to w =

w +

x,,,+-~ w .

If f m , w is well-defined by (3.1)' then this

decomposition immediately leads to

In general, f m , w can be defined by (5.60) if f m , " and f m , " are well-defined. Conditions ensuring t h i s are obtained by combining t h e results of sections ( a ) and ( b ) . So we assume i n this section that (5.17) holds and that for a fixed i n t e ger k c N o we have

216

V . Quasihomogeneous A v e r a g e s of F u n c t i o n s . Part 2

We are going to deal with the cases

The case

"E

"E

= 0 " and

"E

> 0" separately.

= 0".I n t h i s case the condition (5.12) w i t h a = 1 i s satisfied for L < 0

o n l y so that the domain of definition of f m , " cannot be the whole of X , but is

X, o n l y . In order that f m , u be defined for every m E C f has to be a Cm function w i t h respect to the variables in G M ( o + ) . So we fix s € l N o u ( m ) and f E C ~ ' ~ ( X ) . Finally, we require that the growth of f i s restricted as follows: s u p ( I f ' a ' ( x ) l (l+JP+(x)l)N: xEMo'(K)} < +a

(5.02 )

for arbitrary c i ~ Z l and ~ ' ~N E N , and for every compact subset K of X . We can now state and prove the main result on the quasihomogeneous averages

Theorem 5.37. Under the preceding assumptions for every m E C b.), 15.60) (restricted t o X,)

a fuiiction f m , w € C " " S( X +I is well-defined having the following

properties : lil

I f m € C ( - m , O ) or if ( * 5 . 1 0 ) holds then f,,,,,

(ii) s u p p f m s , liii) f,.

is given by ( 3 . 1 ) ' .

c (suppf)M.

is almost quasihomogeneous o f degree m : more precise?,

.

we have

here the term in the second line on the right-hand side o f (5.631 vanishes in case m @ X ( M ) ,i.e. in this case (3.11) is valid. ( i v ) By

D?f,,(m) := f m , w , m E C \ XC(MI. a meromorphic function 2Vf,w on C

with values in C " . s ( X ) is defined, its poles lying in X ( M ) ; moreover, if m c X ( M ) then ao(rn;2Vf,,) =En,,,

and

( v ) Let P c C and " E N . and let P o : X x V * - C

be a continuous copolynomial

function o f degree i (~0,s)which is almost quasihomogeneous of degree!, o f

I n t r o d u c i n g f,

5.c

for Arbitrary rn E C a n d f

C

217

W (V)

order I N ,and o f type M x ( - M ) * . For every j E N N let Pi:XxV*-C

be i t s

Jth

order deficiency which is a continuous copolynomial function o f degree 5 ( r , s ) , as well. Moreover, suppose that for every j E ( O I u N N the function P I ( x , 3 ) f be-

longs to C"'s(X) and satisfies the assumptions o f Theorem5.37. as well. Then on X , the formulas (5.341 and ( 5 . 3 4 ) ' are valid. ( v i ) Let PEC and N E N , and let qo:X+C

be almost quasihomogeneous o f de-

gree P and o f order 5 N such that for every j E ' l O l u N , the Function q j f belongs t o C m ' s I X ) and satisfies the assumptions o f Theorem 5.37, as well, where in

case j ? l qj:X+C

be the j e h order deficiency o f qo. Then on X , the formulas

(5.35) and (5.35)' hold. (vii) f m , w o A = ( f o A ) , , ,

for every A E L ( V , V ) commuting with M .

(viii) ( f m v w ) O= ( f w ) m . w i f @ satisfies the assumptions o f Remark 2.67.(iil; in particular, i f f is @-invariant so is f,,. b

(is) The formulas o f Propositions 5.28 and 5.32 remain valid i f J, ... is replaced by

J',''I'

... and i f all the other terms containing b and

E

e\plicitlq are deleted

m f . The assumptions o n t h e growth of f mean t h a t f o r arbitrary

N E INo t h e r e is a c o n t i n u o u s f u n c t i o n u, SUP {

, N :Mo( X )

I f ' " ' ( X ) I ( 1 + I P + ( X )I ) N / U , ,

N ( MoX) ;

( T E ( U ~ a' n~d

lo,+a C such that X

Ex }

is f i n i t e . T h e n o n e o b t a i n s t h e f i r s t p a r t o f t h e a s s e r t i o n by c o m b i n i n g P r o p o s i t i o n s

5.11.(i) a n d 5.12 a n d Lemma 5 . 4 w i t h P r o p o s i t i o n 5.16 a n d Corollaries 5.29 a n d 5 . 3 3 .

( i ) :t h i s (ii):

is c l e a r f r o m P r o p o s i t i o n s S . I l . ( i ) a n d 5.12 a n d f r o m R e m a r k 5.14

f o l l o w s by R e m a r k 5.7 a n d P r o p o s i t i o n S . l b . ( i i ) .

( i v ) : since

by P r o p o s i t i o n 5.11.(i) t h e f u n c t i o n C 3 m H f , , ,

(where

V

:

=

~

~

~

,

is h o l o m o r p h i c t h e a s s e r t i o n f o l l o w s f r o m P r o p o s i t i o n s 5.16 a n d 5 . 2 5 .

(iii): U n d e r

t h e a s s u m p t i o n s of p r o p e r t y ( i ) t h e c o n d i t i o n (3.11) is d e r i v e d as in

t h e p r o o f of P r o p o s i t i o n 3.13. By t h e principle o f a n a l y t i c c o n t i n u a t i o n i t r e m a i n s valid f o r e v e r y m E C \ U ( M ) . H e n c e h : = nZ,

satisfies t h e condition ( a ) of Propo-

s i t i o n 2.51 f o r ( a , b ) = ( 1 , O ) . By P r o p o s i t i o n 2.31 t h e c o n d i t i o n (3.11) m e a n s t h a t (dM-m)'f,,,

= (-~)'f,,~~-~,

i€Nk,

+

~

~

W

218

V . Q u a s i h o m o g e n e o u s A v e r a g e s o f F u n c t i o n s . Part 2

holds for every m € @ \ X ( M ) . Since by ( i v ) the assumptions of Remark 2.54 are satisfied for N = k and j = 0 it follows that the preceding equations hold for arbitrary m E X ( M ) , as well. Moreover, since by property ( i v ) , again, the assumptions of Proposition 2.53 are satisfied for N = k and j = 0 , i.e. k? = - k - 1 , w e conclude that

and

Hence the condition (S.63) follows by property ( i v )

( v ) : by

Proposition 5.19 the assertion is valid for 9Rf,". To prove it for 9Rf,, w e

observe that by Lemma 5.4 i n the defining integral for f,,,,

we may interchange

differentiation and integration. Hence performing the computations done in the proof of Proposition 3.4 and taking (1.38) into account we arrive a t the desired formulas.

(vi):

t h i s is a special case of ( v ) or can similarly be proved directly.

(viil and Iviii): see Remarks 5.5 and 5.6 and Proposition 5.21. (is): One easily verifies that under the assumptions of Theorem 5.37 the results of sections ( b ) , ( c ) ,and ( d ) remain valid if in (5.28) the term if J'," is replaced by

omitted provided that

x

~

is deleted, ~ ,

, j ' i m and if all the other terms containing b explicitly are E

t 0 . Alternatively. one could argue that f m , w . , , O , b , con-

verges pointwise t o f m , w as b + w .

NotationS.38. If k = O we write fm instead of f m . w . Note that under the assumptions of Theorem 5.37.(i) f,

is given by (3.1). More-

over, the assertion of Theorem 5,37.(iii) means that f,

is quasihomogeneous of

degree m in case mi?X(M) but may not be quasihomogeneous in case m € . U ( M ) .

Estimates for f m , w can be obtained from the results i n sections ( a ) - ( d ) . For the purposes i n the following chapters, it suffices to have estimates if f belongs

to

Y(V).Observe

that in this case the growth conditions (5.62) are satisfied.

~

5.e

Introducing f,

219

f o r Arbitrary m a C and f € s P ( V )

Notation 5.39. F o r every N E C O , + a C w e d e n o t e by K G t h e F r e c h e t space of a l l C- f u n c t i o n s g : V++@

such that

s u p { 1 g ' " ' ( x ) I ( 1 + 1 P + ( x ) ~ )( -1 +~ 1 Moxl)'

;

x E f 2 c q , + 0 31c

i s f i n i t e for a r b i t r a r y a € % , L E N , a n d q > 0. W e d e f i n e

Similarly as R e m a r k 5.35 o n e p r o v e s

Remark 5.40. Let

y, E ~ , ( V )

be such that i t s support is contained in 0, ( s e e Nota-

tion 5.10) f o r some compact subinterval J of 10,+ m y . Then the map g

HX g ,

is well-defined and continuous.

,FL+Y(V ) ,

I

Applying P r o p o s i t i o n s 5.11.(i) a n d 5.12 t o u = ( l + I

- I)-'oMo

a n d t a k i n g Proposi-

t i o n s 5 . 2 8 . 5.32 a n d 5.30 i n t o a c c o u n t o n e o b t a i n s

Proporltion 5.41. Let f € Y ( V ) . Then f o r ever) m6C' the function

fm,w

belongs t o

2;. More precisel),, let rEN and choose v according t o 15.53). resp. ( 5 . 5 9 ) : then (il

the map 3 ? ~, defined , ~ in Theorem 5.37.(iv), maps the set C I - w . r c , ) mero-

morphicallj into Z&

;

( i i ) by f H f,,,,

a continuous linear map from Y ( V ) into ,F& i s defined provided

that R e m c r c , .

I

The case

"E

nition of f,,w

0

".

H e r e (5.12) is valid for every L E R so t h a t t h e d o m a i n of defi-

will be t h e w h o l e of X . As for t h e a s s u m p t i o n s o n f , w e c o n t e n t

o u r s e l v e s w i t h s t a t i n g t h e r e s u l t s for f E 6 ~ ~ ) t ( V ) .

Theorem 5.42. Suppose that the function f,,,,,

0,IV).

(where

E

tv

> 0.Let

f ~ 6 ! , ,V~) .( Then f o r ever)' m E C by (5.60)

is given b.,

( 5 . 6 1 ) ) is well-defined and belongs t o

If f belongs t o Y G ( V ) ( s e e Notation 5.341 so does B-v J l f , ( r n ) := f,.

with values in 6,(V)

fm.w

.

, rn € C \ XSI M),a meromorphic function Jl?f, ,on C

i s defined, i t s poles lying in 2 1 , ( M ) ; i f m E X s ( M ) then

220

V. Q u a s i h o m o g e n e o u s A v e r a g e s of F u n c t i o n s . P a r t 2

ao(m;!Wf,,l

, and f o r every iEN a _ , (m;!Wf,,) is given by the formulas

=fn1,,

o f Proposition 5.25. The assertions ( i ) , (iil, (vii), and (viii) o f Theorem 5.37 remain valid. The assertions ( v ) and (vil carry over i f in their assumptions one postulates that the

, ( functions P I ( x , d ) fand q j f belong to 6 tions 5.27, 5.28, and 5.32 remain valid i f

V ) . Moreover, the formulas o f Proposi-

so... is replaced by f,“?.. and i f all b

the other terms containing b exp1icitl.v are deleted. Finally, by f

H

f,.

,continuous

linear maps

6,

(V

)

+6,

(V

) and

YG(V ) +YG(V ) are defined. Proof: is a n a l o g o u s to t h e p r o o f s of T h e o r e m 5.37 a n d P r o p o s i t i o n 5.41.

H

W e c l o s e t h i s s e c t i o n w i t h a n a p p r o x i m a t i o n r e s u l t r e q u i r e d in C h a p t e r 8 .

Proporition 5.43. For arbitrar-b, f E Y ( V ) and m64‘ F t , l , w / v + in the topology of

mf. Let

2;

converges t o

f,r,,wk

as E -s’ 0 ,

v ~ { q i E; N o } , a n d set ~ , ( t ) : = e - ~ ~ v t (Etl )O,, + a C . T h e n f r o m Pro-

p o s i t i o n s 5.11 a n d 5.36 w e c o n c l u d e t h a t f o r every m E ( C ( - c o , O ) f m , Y c c o n v e r g e s to f m , v in t h e t o p o l o g y of Z z as E + O . T h i s i m p l i e s t h e a s s e r t i o n in view of

Theorem 5.37.(ix) and Theorem 5.42.

H

L e t @ s a t i s f y t h e a s s u m p t i o n s of Remark 2 . 6 7 . ( i i ) . In t h i s s e c t i o n w e are g o i n g to describe t h e spaces Q,CCF(X),)

and Q , ( Y ( V ) , )

which a r e defined accor-

d i n g to

Notatlon5.44. F o r any s u b s p a c e E of C m ’ ” ( X ) w e w r i t e Q,(E) : = { Q m f ; f E E } and E * : = { fB; f E E } .

In p a r t i c u l a r , for t h e s e s p a c e s w e s h a l l i n t r o d u c e n a t u r a l locally c o n v e x t o p o l o g i e s

221

5.f T h e L o c a l l y Convex Spaces Q , ( E a )

required in Chapter 7 . We begin by noting a few generalities o n (almost) quasihomogeneous Co3 functions on X . Recall that we are still assuming (5.17).

Roposltion 5.45. Let R 6 C 'a lil

X I . Then

the Following conditions are equivalent: (a)

R is almost quasihomogeneous o f degree m ;

(bl

R=Q,R;

(cl

R=Q,f

f o r some f E C i l ' , s ( X ) :

lii) i f one (and hence each) o f the conditions o f ( i ) is valid then R is almost quasihomogeneous OF order not larger than 15.65)

N ( m ) :=

l o

i f rng2lllM)

1 ma\{N,.

if m<X(M)

a6?/Ii}

where N , is defined bj the condition that

\n

is almost quasihomogeneous o f or-

der N , (see Remark 1.301: in fact. R vanishes i f ni 621(Ml. proOf. The implication " f b l * f c ) " is trivial. Since by Remark 1.30 the functions xu, cx€'UA, are almost quasihomogeneous of degree

cation " f r ) J ( a l definition of Q,f.

"

m and of order N, the impli-

and the assertion(iil are an immediate consequence of the

By freezing the variables in ker M one reduces the proof of

" f a l + ( b l " to the case " a = a + " . Since i n this case (5.17) means that X = V the assertion then follows from Propositions 1.58 and 1.34.(iii). w

To formulate an assertion about the support of Q,f

we require

Lemma 5.46. IF L is a closed subset o f X then the sets L \ X, ( = L n M o ( L l ) and M o - ' f L \ X + ) are closed subsets o f X . proOf. Since X + is open L \ X + is closed i n X . I f ( x k ) k e N is a sequence i n M o l ( L \ X + ) converging t o some point x € X as k + a then the sequence ( M o x k ) converges to M o x as k + a . Since by (5.17) M o x belongs to X and since M o x k E L it follows that

M o x €L \ X +

, i.e. x E M , ' ( L \ X + ) .

Note that one could d o without

assuming (5.17) i f L \ X + were closed i n M o ( V ) . w

222

V . Q u a s i h o m o g e n e o u s A v e r a g e s of F u n c t i o n s . Part 2

Cor~ll~ 5 4y7 . Supp Q,F

C MG'((suppE)

mf. If x @ M o ' ( ( s u p p f ) \ X + ) every aCU'.

\ X,) For every F 6 C m * * f X )

t h e n M o x @ s u p p f so t h a t f ' a ' ( M o x ) = O

for

C o n s e q u e n t l y , t h e a s s e r t i o n f o l l o w s f r o m Lemma 5.46.

Differentiation and multiplication by ( a l m o s t ) q u a s i h o m o g e n e o u s f u n c t i o n s as well as taking @-averages c o m m u t e s with t h e operation Q,

Propodtion S.48. Let f (i)

d'(Q,

E

C n''s (X). Then

F ) = QrII-,,fdaf)

(ii) q Q, f = Q,

+

:

f o r every a € 2 L m ' s :

( q F ) For ever-) P E C and f o r ever) function q : X

+C

which

is almost quasihomogeneous o f degree P and which is a C"'s function on a neigh-

bourhood OF Mo-'(supp F ) : (iii) Po(,\,S)[Q,,, F ] = Q , J I + p ( P o ( \ , 3 ) f )f o r ever) P E C and ever) C'm'scopoIJno-

mial function Po: X x V *-

C of degree

5 ( as) ,

which is almost quasihomogeneous

OF degree C and o f type M x ( -MI * ; ( i v ) ( Q , , , f ) o A = Q,,,(foA) For ever) A E L I V , V ) commuting with M ,

(v)

(Qrnf ) ,

proOF.

= Q r J I ( f ( $ )in : particular, i f f is @-invariant so is Q m f

L): this

is a special c a s e of ( i i i ) ; for a direct proof w e w r i t e a = p + y

w h e r e p : = a + a n d y : = a o (see Notation 5.15) and obtain by Proposition 1 . 2 8 . ( i )

Since y + q = ( q - p ) + a a n d { q - p ; n € X , A , q ? B } =211:1-aM

(ii):

t h e assertion follows.

again, t h i s is a special c a s e of ( i i i ) ; f o r a direct proof w e derive f r o m t h e

Leibniz f o r m u l a t h a t

223

S . f T h e Locally Convex Spaces Q,(Ew)

fiii): Instead

of performing a direct computation based o n ( i ) and ( i i ) we employ

Proposition 5.27 in o r d e r to reduce everything to t h e r u l e s of computation already established. Indeed, supposing t h a t w = xlo,ll and E = 0 we obtain f r o m Proposition 5.27 and (5.41) t h a t

Setting Pj : = (3,

-,)*

Q,f

=f

- t ) i Po

- ((3,

-m) f

I,,,

so t h a t

w e deduce f r o m Proposition 5.19 t h a t rn

By Lemma 2.34 we have

Finally, applying Proposition 5.27 obtain in view of (5.41) [ ( 3 , - m - t )( P i( x , a

Since t h e right-hand side is equal to S j 0 [ - Q , ( P o ( x , a )

f ) + P o ( x . d ) f ] o n e arrives

a t t h e desired formula by putting everything t o g e t h e r .

( i v ) : Here,

again, instead of performing a direct computation we apply Proposi-

tion 5.27 ( t o w = x , o , l l , i.e. k = O , b = 1 , and

E

= O ) twice and t a k e Remark 2 . 6 7 . ( i )

and Proposition 5 . 2 l . ( i ) i n t o account to obtain

(v):

this follows from ( i v ) and t h e definition of f,

and 5.21.(ii) and Remark 2.67.(ii) ) .

( o r from Propositions 5.27

H

Now we a r e going t o deal with t h e case E = CgCX) f i r s t .

Notation 5.49. L e t K be a G-invariant compact s u b s e t of V . By Q ; : ( K : @ ) we den o t e t h e s p a c e of all @-invariant functions R e C m ( V ) with s u p p o r t contained in

M,'(Kn

M o ( K ) ) which a r e a l m o s t quasihomogeneous of degree m .

224

V . Q u a s i h o m o g e n e o u s Averages of F u n c t i o n s . Part 2

Lemma S.50. (i) Equipped with the semi-norms R H ~ ~ ( R ) : = ~ u ~ { I R ( ~ + ~~ E' ( Vsa,)EIZ;l ; J ,

BE2K0,

Q Z ( K ; @ ) is a Frbchet space.

( i i ) Let

U be any compact neighbourhood of K n M o ( K I

in

V, and

set

L : = U n M i l ( K n M o ( K ) ) ; then Q , , , ( C ; ~ K ) @c) Q ; ( K : @ )

cQ~,(c;(L)~~),

proOf. li): We first observe that n g ( R ) which in view of d"R = ( d " R ) O M , is equal to sup{ I R ( " + " ( x ) I ;

xEKnM,(K), a€'U;}

is finite for every fi€'uo. I f ( R i ) j E N is a Cauchy sequence i n Qz(K;(31) then for

every =EX;

the sequence (d'Ri)iEN converges to a C

support contained in M,'(K

m

function R , : V + Q I

nM,(K)) such that H a o M O = R,.

with

Consequently, by

t h e desired limit R of ( R j ) is defined. (iil:In view of Proposition 5 . 4 8 . ( v ) and Corollary 5.47 the inclusion on the left-

hand side is obvious. To prove the other inclusion we choose XEC:(U) to 1 near K n M , ( K ) .

equal

Then for every R E Q Z ( K ; @ ) the Leibniz formula and Pro-

position l.ZH.(i) show that R = Q , ( x R ) . Again by Proposition 5.48.(v) we conclude that R = Q , [ ( X R ) ~ ~ I .

For the following proposition we drop the assumption (5.17)

Proporltion S.S1. Via the identit-)

the space Q , ( C , " C X ) m )

is equipped with the structure of a strict (LFI-space

( s e e , for example. $23.5.3.4 in Floret-Wloka C 5 1 ) ; a s a quotient of CGv(X) i t

-

i s a nuclear ( S ) -space. proOf. The first assertion is an immediate consequence of Lemma 5.50. To see

that F:=Q,(CT(X),)

is a quotient of CgCX) we define P : C g ( X ) + F

by

225

5.f T h e Locally Convex Spaces Q m ( E a )

'p I+

Q m v a and observe t h a t P

is linear, continuous and surjective. Since C g ( X )

is a strict ( L F ) - s p a c e , a s well, t h e open mapping theorem (see e . g . 5 2 4 . 4 . 1 in Floret-Wloka CSI) s h o w s t h a t P is a homomorphism. Since F is c om ple te (see 524.3.1 in C51) t h e l a s t assertion follows f r o m 527.2.4 and 527.1.13 in C 5 1 .

rn

Now we come to t h e cas e E = Y ( V ) .

Pmpodtion5.52. ( i ) Q,(Y'OIV),) tions R : V - + C

is equal to the space o f @-invariant C'I'func-

which are almost quasihomogeneous o f degree m such that

ITp(R) := s u p { / R ( B ' a ' ( v ) /( l + / M o s / ) e ;x E V , LYE~I:,, p 6 X o , /PI'

P)

is finite f o r every [ENo . (ii) The functions ITp defined above are semi-norms providing Q , , , , Y ( V ) , I with the structure o f a nuclear FrPchet -Schwartz space.

For t h e proof and f o r o t h e r purposes t h e following lemma is required.

Lemma 5.53. ( i ) I f F and H are closed subsets o f V such that (5.66)

dist ( F . H )

->

0

then there esists a function , y E C " " ( V ) equal to 1 near F with support contained in V \ H such that 0 5 x5 1 and all derivatives o f y, are bounded.

(ii) I f J and K are disjoint closed subintervals of 1 0 . + w l then ( 5 . 6 6 ) i s satisfied f o r the choices ( a ) F=OJ and H = n K (see Notation 5.10) and ( b ) F = x -'(J) and

H=

- 1 ( K ) where x := x

proof.

(ii):

+

( s e e Proposition 1.70 ) .

u:see f o r example Corollary 1.4.11

in Hormander C 1 1 1 .

Of c o u r se , we may as s u me t h at J and K ar e non-empty and t h a t J or K .

say K , is c o m p a c t . Then P + ( n K ) is co mp act , a s well. Since i t is disjoint from t h e closed s u b s e t P + ( n J ) of V it follows t h at d i s t ( P + ( n K ) , P + ( n J ) )is positive. Since

I x - y l 2 I P , ( x ) - P + ( y ) l t h e assertion is proved for t h e choice

w . For

the

proof of (b) we f i r st deduce f r o m (1.70) and the defining equality in t h e proof

of Proposition 1.63, " ( b ) * ( a ) " ,

t h a t x o P + = x so tha t by continuity we have

k o P , = k . Hence in view of (1.77) it follows t h a t P + ( i - ' ( K ) ) is c om pa c t and

226

V . Q u a s i h o m o g e n e o u s Averages OF F u n c t i o n s . Part 2

P + ( i - ' ( J ) ) is closed, and a similar argument as above leads to t h e desired result for the choice ( b ) .

Proof of Proposition 5.52. (i):If R = Q,

'p

for some

'p E

Y( V )

then in view of Pro-

positions 5.45.(i) and 1,28.(i) it is obvious that I14(R) is finite for every 4 Conversely, let R E C"(V)

€

No.

be almost quasihomogeneous of degree m such that

l l e ( R ) is finite for every 4 E N o . By Lemma 5.53 we choose a function x E Ca(V) equal to 1 near fl,,-,,=Mo(V)

w i t h support contained in

n c o , l csuch

that all

the derivatives of x are bounded. Then X R belongs to Y ( V ) , and we have Q,(xR)

= R . Since by Proposition S.48.(v) we have Q,[

R E Q m ( Y(V ) @ )

(ii): If

(xR),]

= R it follows that

.

( R j ) j e N is a Cauchy sequence in F : = Q m ( Y ( V ) , ) then it converges (with

respect to the semi-norms I I p ) to a @-invariant Cm function R : V + @ Q,R=R,

such that

i.e. R is almost quasihomogeneous of degree m and hence, by assertion

( i ) , belongs to F. Note that the map P : Y ( V ) - + F , v H Q n , y @ , is linear, conti-

nuous and surjective, hence a homomorphism by Banach's open mapping theorem. Consequently, F is a quotient of the nuclear FrPchet-Schwartz space Y ( V ) , hence a nuclear FrPchet-Schwartz space itself by

5 27.2.4

and

5 27.1.13

in Floret-Wloka

C51.

In t h i s section w e drop the assumption (5.17), f i x k E N 0 , and introduce natural locally convex topologies (required in Chapter 7 ) for t h e spaces

xz,k(cg(x)@,)

and % z , k ( Y ( v ) @ which ) are defined according t o

To deal with the case E = C Z ( X ) first we fix a G-invariant compact subset K of

V and introduce

s.g

227

The Locally C o n v e x Spaces X z , k ( E m )

az,k( K ; 8 ) w e d e n o t e t h e s p a c e o f a l l a l m o s t q u a s i h o m o g e n e o u s Cm f u n c t i o n s f : V + + C w i t h s u p p o r t c o n t a i n e d in KM s u c h t h a t

Notation 5.55. By @-invariant (3M-m)k''f

e x t e n d s to a C c O f u n c t i o n R : V + C

b e l o n g i n g to Q z ( K ; @ ) (see

Notation 5.49). Note t h a t the pairs ( f ,R) satisfy k

To h a n d l e t h e s u p p o r t o f t h e s e f u n c t i o n s f w e r e q u i r e

Remark S S . There is a compact subset L of V , such that

-

K , n V+ = L ,

(5.68)

proOf. T h e s u b s e t L : =

.

Knn,,

o f V, i s c l o s e d in V a n d in view o f R e m a r k 1.8

s a t i s f i e s ( 5 . 6 8 ) . I t is b o u n d e d s i n c e by L e m m a 5.9 it is c o n t a i n e d in t h e set

Lemma 5.57. ( i ) Suppose that

u

i s chosen a s in Proposition 5 . 3 1 . Then, equipped

with the s e m i - n o r m s defined bj

fH

~ p ( f ) : = s U ~ { I f ( a ' ) ( X () lI +

az,k ( K ; @ )

l P + h ) l ) - v : Y E O C 1 / p , + < , , C r / a / 0 such t h a t t h e closed polydisc

-

P ( x , E:)=

{ y € V ; I P + y l 5 ~ I, M o ( Y - x ) I ~ E i) s c o n t a i n e d i n X . T h e n K : = ( y c P ( x , ~ )l ;P + y l = ~ } is a c o m p a c t subset o f

X, . In view of t h e l a s t a s s e r t i o n in P r o p o s i t i o n 1.70 a n d

228

V.

Quasihomogeneous Averages of Functions. Part 2

-

in view of Remark 1,8.(i)w e have K M , l / ~ o , l l = P ( x , E ) n X + . I f L is an ( M , l O , l l ) bounded s u b s e t of X,

it follows by Proposition 3.22 t h a t i n ( P ( x , E )n X , )

compact. Since L C X, this implies t h a t x does not belong to

'L.

is

H

h o o f of Lemma 5.57.(ii) : The inclusion on the left-hand side is obvious in view of Proposition 2.59.(i) and of t h e assertions ( i i ) , (iii), and ( v i i i ) of Theorem 5.37.

To verify t h e o t h e r inclusion we set

E

: = d i s t ( 3 U , K n M o ( K ) ) / 2 and choose a

finite s u b s e t N of K n M o ( K ) such that t h e set K n M o ( K ) is covered by t h e polydiscs P ( x , E ): = { y E V ; ( M o ( y - x ) l , ) P + ( y ) 0 and

[,EN s a t i s f y i n g I I p ( R ) < Cp K % ( f ) ,

fElIz,k(K;@),R=(3M-m)k+'f,

N o w , l e t ( f i ) j E N be a C a u c h y s e q u e n c e in ( W z , k ( K ; ( $ ) , a n d d e n o t e by Rj t h e e x t e n s i o n o f ( a M - m ) k + i f j to V. T h e n ( R j ) is a C a u c h y s e q u e n c e in t h e s p a c e Q , " ( K ; @ ? i ) . By L e m r n a S . S O . ( i ) it c o n v e r g e s to s o m e R E Q z ( K ; ( g ) . Let f be t h e l i m i t of t h e s e q u e n c e ( f j ) in t h e s p a c e C m ( V + ) . I t t h e n f o l l o w s t h a t s u p p f i s c o n t a i n e d in K M . M o r e o v e r , ( 5 . 6 7 ) is s a t i s f i e d s i n c e it is valid w i t h ( f , R ) r e p l a c e d by ( f i , R j ) . In p a r t i c u l a r . ( 3 , - m ) " + ' f

=

R),+.

W e c o n c l u d e t h a t f is t h e desired

l i m i t o f ( f i ) in t h e t o p o l o g y o f X Z , k ( K ; @ ) . rn

Proposition 5.59. Via t h e identitr.

Xz,ktC,^'(X),)

= ind KCX

2(z.k ( K : @)

X z , k t C A v ( X ) c e ) carries t h e s t r u c t u r e of a s t r i c t ILF) - s p a c e ( s e e e . g . $.?3.5.3.4 in Floret- Wloka t 5 1 ) ; a s a quotient OF C ; ( X )

h-oof. T h e

it is a nuclear (?)-space.

f i r s t a s s e r t i o n is a n i m m e d i a t e c o n s e q u e n c e o f L e m m a 5 . 5 7 . To see

t h a t F : = x Z , , ( C T ( X ) @ ) is a q u o t i e n t of c r ( X ) w e d e f i n e P : C r ( X ) + F 'p

('p@)n,,wk

by

a n d o b s e r v e t h a t P is l i n e a r , c o n t i n u o u s a n d s u r j e c t i v e , h e n c e o p e n

by t h e o p e n m a p p i n g t h e o r e m (see e . g .

5 24.4.1 in

f o l l o w s as in t h e p r o o f of P r o p o s i t i o n 5.51.

I S ] ) . T h e rest o f t h e a s s e r t i o n

rn

S i n c e w e d r o p p e d t h e a s s u m p t i o n (5.17) a r e m a r k is in order c o n c e r n i n g t h e s u p p o r t

of t h e f u n c t i o n s qrn a n d of

'pm

( r e s p . Q,?)

is

Q,'p

if ' p E C g ( X ) . Recall t h a t t h e d o m a i n of d e f i n i t i o n

V+ (resp. V ) .

Remark 5.60. For ever!' p EC;=(XI w e have ( i ) supp pm,wk C X , ,

and

(iil s u p p Q , p C M , ' ( X l

CX

230

V . Q u a s i h o m o g e n e o u s A v e r a g e s of F u n c t i o n s . Part 2

a f . We f i r s t observe t h at (5.70)

M,'(X) C X .

Indeed, let xEV b e such t h a t y : = M o x E X . Since M t x converges to y a s t + O and since X is open t her e is t ~ l O , + m Cs u ch t h a t M , x E X , and since X is quasihomogeneous t h i s implies t h at x E X .

( i ) :By

Theorem 5.37.(ii) t h e s u p p o r t of

p ',

lies in t h e s e t

( s u p p ' p ) n V,

which

is contained in X by Lemma 5.9 and ( 5 . 7 0 ) .

Iiil:

By Corollary 5.47 t h e s u p p o r t of Q m y lies in M,'((suppcp)\

X , ) which is

contained in X by ( 5 . 7 0 ) . rn

Finally, we come to t h e case E = Y ' ( V )

Propodtion 5.61. (i) 2L,'z,k(P( V ) , ) is equal to the space of all almost quasihomogeneous @-invariant C ^ functions f : V+-+

a function R 6 Q, I Y'( V ) , )

C such that I d M - m j k + 'f e\tends to

and such that

is finite For ever)' ( € N o where v is an, constant chosen as in Proposition 5.41, lii)

The functions ~p defined above are semi-norms providing 2LA.,kIP( V ) , )

with

the structure of a nuclear Fr&het -Schwartz space.

For t h e proof t h e following supplement to Theorem 3.48 is required.

.

let A and ( j m ) r I I E , , be as in Theorem 3 . 4 8 ,

ENjmU

101 let qm,l E C " ' ( X ) be given such that the

LcmmaS.62. Suppose that X = V , and for arbitrary m E A and j

condition Ib) OF Theorem 3 . 4 8 is valid and such that

f o r arbitrary q E D:= { qrIl,j; m [ A , j E N j i m u{ O l } , a € # ,and P E N . Then f o r anj.nonemptj' open subinterval I of 10,twC there exists a function f E P ( V ) satisfying the condition l a ) of Theorem 3 . 4 8 for Y = x - * ( l ) where x is the function x + defined in Proposition 1.70.

5.8

231

The Locally C o n v e x S p a c e s U Z , k ( E @ )

Proof. W i t h o u t loss of g e n e r a l i t y w e may a s s u m e t h a t I is relatively c o m p a c t in I O , + c o C . I t t h e n f o l l o w s f r o m (1.77) t h a t x - ' ( l ) C

nJ for

s o m e compact subinter-

val J o f I O , + a C . N o w , s e t t i n g Y = x - ' ( I ) w e p r o c e e d a s in S t e p 1 of t h e p r o o f o f T h e o r e m 3 . 4 8 . In p a r t i c u l a r , w e o b t a i n f by ( 3 . 4 3 ) . In order to see t h a t f bel o n g s to Y ( V ) w e o b s e r v e t h a t in view of (1.76) a n d t h e d e f i n i t i o n of x + w e have

S i n c e P + ( n J ) is a c o m p a c t s u b s e t of V ,

x and

p x are b o u n d e d o n

t h i s i m p l i e s t h a t all t h e d e r i v a t i v e s of

Y . H e n c e it f o l l o w s f r o m ( 3 . 4 3 ) t h a t f o r e v e r y a ~ 4 L

t h e r e is a c o n s t a n t C, s u c h t h a t If'"'(x)l s c a m a x { Iq("(p,(x))I:

I P I < I ~ I q, E D } ,

kEY.

C o m b i n i n g t h i s w i t h t h e a s s u m p t i o n o n ,XI w e o b t a i n f o r a r b i t r a r y a E X a n d P E N a c o n s t a n t C,,t

such that

lf(a)(x)l

0 such t h a t t h e closed polydisc

238

VI. Q u a s i h o m o g e n e o u s A v e r a g e s of D i s t r i b u t i o n s . Part 2

em)

for otherwise the desired subsequence of

is found by the compactness of

M,(%x,E)) C X . h

Next we convince ourselves that it sufficc and sequences ( s m ): , Om=

(6.7)

N

if m

?

In fact, t h i s implies that s , E { t E l ;

to find a compact subset K of X, A

in I and ( y m )

M,,,ym

i

N

in K such that

N

M , ( k ) n i n X + f Q ) )so that by the assumption

on L and by Remark 3.19 a subsequence of

(st,)

converges to some s E l a s

m+a;

A

and since K is compact, by choosing subsequences for a second time we achieve A

that (y,)

converges to some y E K so that by continuity and in view of ( 6 . 7 ) a

(em)

subsequence of A

To find K , s,,

A

converges t o M s y

E

K,,,,,

C X , as m + a .

and y m w e first observe that since ( l / t m ) is a bounded sequence

it follows by continuity (see Corollary 1.9) that M,/',(tm-

M t 111k,)

converges

to 0 as m + a so that

lim MlIt,tm

= x

m-m

.

If X E X , then for sufficiently large N the requirements above are satisfied by s m : = t m y, m : = M l I t n , t m ,

h

and K : = { y , ; m t N } u { x ) .

In case x @ X + then - taking Mo(Ml/,,t,,) can choose N so large that

I MO(MI/,t,,-x)I

= Mo(M1,,tm)
Cthen XI = X, , and one obtains the assertion by combining Lemma 3.20 (applied to X,

instead of X )

and Proposition 6 . 3 . 8 . Suppose now that I is bounded, i.e. X I = X . Since for every x E X \ X , ( x ) ~ = { x and ) c ( x ) = O € l the condition ( 6 . 8 ) for

tained in X,

. Since

we have

K = ( x ) shows that L is con-

by Proposition 6.3.A.(ii) the same is satisfied i f L is a weakly

(M,I)-bounded subset of X we may assume t h i s to be valid. B u t then (6.8)becomes (3.19)' so that bq Lemma 3.20 the condition ( 6 . 8 ) holds for every compact subset K of X, if and only if L i s an (M,I)-bounded subset of X, . By Proposition6.3.A.(i) t h i s means that L is a weakly (M,I)-bounded subset of X . Hence, to complete the proof w e have t o show that if ( 6 . 8 ) is valid for every compact subset K of X, it is so for everj compact subset K of X , as well.

To prove this we choose such a compact subset K of X and by Remark 5.56 find n V, = J M . By Lemma 5.9 and by (5.70) J is a subset of X , . Since we assumed that L C X , the set c ( L n K M )n I is cona compact subset J of V,

such that

tained in x ( L n J M )n l . From this the desired conclusion follows. Finally, if I = I O , + a C then one obtains the assertion by combining the cases already dealt w i t h above.

w

As a corollary of Lemma 6.7 the following variant of Remark 5.56 is obtained.

Rcmark 6 . 8 . For every compact subse t K of X the s e t L := K,nSX s u b s et of X , satisfying (6.9)

-

KMnX+=LM.

is a compact

240

VI. Q u a s i h o m o g e n e o u s Averages of D i s t r i b u t i o n s . Part 2

proOf. L is c o m p a c t since by Lemma 6.7 S y is a weakly M - b o u n d e d s u b s e t of X . T h e condition ( 6 . 9 ) is satisfied s i n c e ( S x ) =~ X + .

T h e following l e m m a is required f o r t h e proof of t h e a s s e r t i o n o n s u p p

in

Proposition 6.15 b e l o w .

Lemma6.9. Let L b e a weakly ( M , I ) - b o u n d e d s u b s e t of X . Then we h a v e (i) L;

(ii)

n X = ( i n x ) M , Iu

if I is b o u n d e d

{ @X n M o ( L ) if

I is u n b o u n d e d

if I is u n b o u n d e d then X n M o ( L ) is a closed s u b s e t of X a n d e q u a l to

X nMo(i nX): liii)

if K is a c o m p a c t s u b s e t of X s u c h t h a t K n L , , ,

is emptj' t h e n so is

i n x n K M , ,,I . proOf. &J: Let ( t i ) i E Nb e a s e q u e n c e in I a n d ( t i ) b e o n e in L s u c h t h a t y j : = MlItjtj converges to s o m e point x E X . W e set K : = ( y i ; j C N } u { x ) . T h e n A : = L n X nK,,,/I is a c o m p a c t s u b s e t of XI,, . Since t h e t i belong to A , by choosing s u b s e q u e n c e s i f necessary

z ~ i n X , , , a n d ( t i ) to s o m e (0.10)

S E T .By

w e achieve t h a t

continuity w e o b t a i n

if s
0 , and i f s < b e l o n g s to ( i n X ) , , , .

+ a t h e n (6.10)

implies t h a t x = M l / s z

Finally, if s = + m then I is u n b o u n d e d , a n d it f o l l o w s by

continuity t h a t x = lim y i = j

i

Moz

E

Mc,( L n X,, I )

Hence t h e proof of t h e inclusion (6.11)

-LnMX, I

.

( 1 ,

"L" is c o m p l e t e .

Note t h a t w e have even s h o w n

c (LnX)M,Iu(XnM,(LnX)).

-

To prove t h e inclusion '2"w e f i r s t o b s e r v e t h a t by continuity LM,I n X c o n t a i n s ( L n X ) , , , . Hence

w e may s u p p o s e t h a t I is u n b o u n d e d . Let x E X n M o ( L ) . W e in L converging to y a s choose y E L such that x = M o y and a sequence ( t i ) j E l N j+m.

Then ( M , e i ) converges to x . Since M , / , t i t e n d s to M o t i as t + + m a n d

6.a Weakly ( M , I ) - b o u n d e d Subsets of

241

X

s i n c e I is u n b o u n d e d w e f i n d a s e q u e n c e ( t j ) in I s u c h t h a t x = lim M i , t i t j .

-

This

j + co

m e a n s t h a t x b e l o n g s to L M , ~ .

lii): W e n o t e f i r s t t h a t ( i n X ) M , l \ X + = ( L n X ) \ X ,

C X n M o ( L n X ) . Conse-

q u e n t l y , s i n c e X \ X, is a c l o s e d s u b s e t of X o n e o b t a i n s t h e a s s e r t i o n by i n t e r s e c t i n g b o t h sides of ( i ) , r e s p . ( 6 . 1 1 ) w i t h X \ X , .

fiii):

I t s u f f i c e s to deal w i t h t h e c a s e s I.'

I = l O , b l for some bcIO,+mC and

2.' I = [ a , + m C f o r s o m e ~ E I O , + ~ C .

If c a s e 1 h o l d s t h e n by ( i ) w e h a v e T , n X = ( L n X ) , , ,

so t h a t t h e a s s u m p -

t i o n o n K i m p l i e s t h a t L n X n K M , i / I = @ . In view of ( 6 . 5 ) a n d P r o p o s i t i o n 6 . 3 . A . ( i i ) the assertion follows -

If C a s e 2 h o l d s w e fix P E L n X n K M , l / I a n d c h o o s e s e q u e n c e s ( t j ) j E Nin I a n d (ki)iEN in K s u c h t h a t M t j k i c o n v e r g e s to P a s

j+m.

By c h o o s i n g s u b s e q u e n c e s

w e a c h i e v e t h a t ( I / t i ) c o n v e r g e s to s o m e s E C O , l / a l a n d ( k i ) to s o m e x E K as j+m.

By c o n t i n u i t y it f o l l o w s t h a t x = lim Ml,,iM,jkj j+m

=

M,t E

(

LnX),,,

-

u Mo(LnX).

By t h e a s s e r t i o n ( i 1 t h i s i m p l i e s t h a t x E K n L M . l . H e n c e t h e c o n t r a p o s i t i o n o f

t h e desired implication is p r o v e d . rn

T h e n e x t a s s e r t i o n is t h e a n a l o g u e of L e m m a 3 . 2 6

Lemma6.10. Suppose that I f l O . +a[.If L is a weak?,. (M.1)-bounded subset of

x so is

LM.1.

Proof. I f I = 1 0 , b l f o r s o m e b E I O , + m C t h e n by a p p l y i n g P r o p o s i t i o n h.3.A t w i c e w e d e d u c e t h e a s s e r t i o n f r o m Lemma 3 . 2 6 . If I = l a , + a C f o r s o m e a E l O , + a l t h e n by P r o p o s i t i o n 6 . 3 . 8 a n d L e m m a 3.26 t h e set L M , , n X + - b e i n g e q u a l to ( L ~ X + ) M , Iis- a n ( M , I ) - b o u n d e d s u b s e t of X , so t h a t t h e a s s e r t i o n f o l l o w s by a n o t h e r a p p l i c a t i o n of P r o p o s i t i o n 6 . 3 . B .

rn

T h e f o l l o w i n g a n a l o g u e of Lemma 3.27 is r e q u i r e d f o r t h e p r o o f o f T h e o r e m 6.37 below.

Lemma 6.11. Suppose that I is non-compact. Let J be a closed subinterval of 10.+a[

242

VI.

Quasihotnogeneous Averages of Distributions. Part 2

such that InJ i s compact. IF L is a weaklj (M,I)-bounded subset OF X then For

- -

everq compact subset K OF X the set L M , , n K M , , n X

is compact. Note that i F - L is compact or i F J is bounded then this set is equal t o LM,, n K M , , .

Proof. If

J is c o m p a c t t h e n K M , j is a c o m p a c t s u b s e t o f X . H e n c e w e s u p p o s e

t h a t J is n o n - c o m p a c t . T h e n t h e a s s u m p t i o n s o n I a n d J m e a n t h a t I a n d 1/J d i f f e r by a relatively c o m p a c t s u b s e t of 1 0 , + 0 0 C . C o n s e q u e n t l y , s i n c e by L e m m a 6.10 LM , I is a weakly ( M , I ) - b o u n d e d s u b s e t o f X it f o l l o w s t h a t LM,I is a w e a k l y

( M , l / J ) - b o u n d e d s u b s e t o f X . Replacing I by l / J in Definition 0 . 1 w e o b t a i n t h e first assertion. F o r t h e p r o o f o f t h e s e c o n d a s s e r t i o n w e first o b s e r v e t h a t in c a s e J is b o u n d e d L e m m a 5.9 a n d ( 5 . 7 0 ) imply t h a t

C K M , j u M,'(K)

C X . On the other hand,

if J is u n b o u n d e d t h e n I is b o u n d e d , so t h a t i f , in a d d i t i o n , L is c o m p a c t t h e s a m e a r g u m e n t s h o w s t h a t L M , l is c o n t a i n e d in X .

Finally, w e n o t e t h e f o l l o w i n g c o n s e q u e n c e o f P r o p o s i t i o n 0 . 3 . A

Remark 6.12. Every G6C{o.ll I X , ) uniquely estends t o a C'Function (I on X with weak/) (M.I)-bounded support.

e F . Let u € X \ X , . S i n c e L : = s u p p J , is a n ( M , 1 0 , 1 1 ) - b o u n d e d s u b s e t o f X , P r o p o s i t i o n 6 . 3 . A s h o w s t h a t J, v a n i s h e s o n K ( x , E )n X , H e n c e by J , J x , x + : - O

t h e desired e x t e n s i o n

4 of

f o r s o m e E E IO,+mnC.

J, is d e f i n e d .

Below w e s h a l l identify t h e f u n c t i o n s J , € C;o.l,(X+)

w i t h t h e i r e x t e n s i o n s to X

,

Notation 6.13. By 3;( X ) w e d e n o t e t h e s p a c e of d i s t r i b u t i o n s u E D'CX) s u c h t h a t s u p p u is a weakly ( M , I ) - b o u n d e d s u b s e t of X . If I = I O , + ~ Cw e a l s o w r i t e D k ( X ) .

W e f i r s t observe t h a t t h e a s s e r t i o n o f Remark 6.12 c a r r i e s o v e r to d i s t r i b u t i o n s :

243

6 . b The Distributions

Remark 6.14. Suppose that 1/1 is unbounded. Then every distribution u E a ; ( X + I E D ; ( X ) which vanishes on a neighbour-

(uniquely) estends t o a distribution

hood of X \ X + . In this was.

a; ( X I

is canonically identified with a ; ( X +I .

m f . If u E a ; ( X , ) t h e n by P r o p o s i t i o n

0.3.A t h e d i s t a n c e of every p o i n t of X \ X ,

to s u p p u is p o s i t i v e . C o n s e q u e n t l y , t h e d e s i r e d e x t e n s i o n ti e x i s t s , a n d its s u p p o r t is e q u a l to s u p p u . H e n c e P r o p o s i t i o n 6.3.A i m p l i e s t h a t

W e n o w f i x k E N , a n d s u p p o s e t h a t w:IO,+mC-lR

b e l o n g s to ' 3 ; ( X ) .

is given by

T h e n by (1.65) w e have

Ropositlon 6.15. Let u € 2 l i ( X ) . Then bj, (4.31 a distribution u , , , , E 2 l ' ( X I is welldefined. Its support is contained in (supp U Moreover. b-t 9)?,,,,(m) : = u , , ~ , ,. ni

3lU,,:@+3'(X)

€@

)

~

(compare Lemma 6 . 0 ) . , ~

1 ( - 2 C ( M ) - p ) , a meromorphic function

is defined, i t s poles Ijing in the set ( - 2 l ( M ) - p ) : f o r everj

m E C we have a o ( m : I U ? , , , , ) = u , , , , , . I f I is bounded then 51?un, is holomorphic.

proOf. W e set L : = s u p p u a n d fix ~ , E C T ( X S) i. n c e by R e m a r k 5 . 7 , P r o p o s i t i o n 5 , 1 6 . ( i i ) . a n d T h e o r e m S . 3 7 . ( i i ) w e have (6.13)

SuPPvrn*,v C ( S U P P ' P ) M , I / I " X I / ,

a n d s i n c e by D e f i n i t i o n s 0.1 a n d 0.13 t h e i n t e r s e c t i o n of the r i g h t - h a n d side w i t h

L is a c o m p a c t s u b s e t o f XI,, s p a c e D,,(X,/l),

w e see t h a t t h e f u n c t i o n Y,,,*,~

i . e . R e m a r k 4.1 ( a p p l i e d to Y = X I , , )

b e l o n g s to t h e

shows that the right-hand

side of t h e e q u a t i o n in ( 4 . 3 ) is w e l l - d e f i n e d .

To p r o v e t h a t t h e linear f u n c t i o n a l urn,,

d e f i n e d by ( 4 . 3 ) is c o n t i n u o u s w e fix

a c o m p a c t s u b s e t K o f X . T h e n by Definitions 6.1 a n d 6.13 t h e set H : = L n KM,1/1

is a c o m p a c t s u b s e t of X 1 / l . H e n c e w e c a n fix a c o m p a c t n e i g h b o u r h o o d W of

H in X l / I a n d c h o o s e X E C Z C W ) e q u a l to 1 n e a r H . C o n s e q u e n t l y , in view of (6.13) o n e c o n c l u d e s by Remark 4.1 t h a t f o r q E C ; ( K )

t h e d e f i n i t i o n o f urn,,

a m o u n t s to (4.5).S i n c e by P r o p o s i t i o n s 5.11, 5.36, a n d S . 4 l . ( i i ) , by ( 3 . 5 ) , a n d

VI. Q u a s i h o m o g e n e o u s A v e r a g e s of Distributions. Part 2

244

by Remark 5 . 4 0 t h e map C g ( K ) - C g ( W ) ,

y H ~ y ~ , +is , well-defined, ~ linear

is linear and continuous o n C g ( K ) , indeed.

and continuous it follows t h a t

The assertion a b o u t ~ u p p u ~ , is, an immediate consequence of Lemma 6.9.(iii) and Remark 4.1 . The assertions a b o u t

mu,,

follow by t h e continuity of t h e restriction of u to

C g ( W ) from the corresponding assertions on

m,

(see Propositions 5.11 and

5.16 and Theorem 5.37.(iv) ) and because (6.141

< a i ( m ; ~ u , , ) , y >= ( - I ) ' < u , a i ( m * ; 9 1 , , , ) > ,

Roporltion 6.16. For every u Ea;fX)

j€z.

the assertions OF Proposition 4 . 4 a s well

a s the condition ( 4 . 1 0 ) remain valid.

mf. We

first observe t h a t a a u , a € U , and q i u belong t o D ; ( X ) , indeed. Let

A e G L ( V , V ) commute with M . To verify t h a t u o A belongs t o 9 ; ( A - ' ( X ) ) we set L : = s u p p u and observe that supp u o A = A - ' ( L )

and t h a t for every compact

s u b s e t H of A - ' ( X ) t h e s e t K : = A ( H ) is a compact s u b s e t of X satisfying

Finally, to see t h a t u~ belongs to 3;CX) we observe - u s i n g Lemma 2.58.(i) t h a t f o r every compact s u b s e t K of X we have

-

LGnXnKM,I/I c ( L n X n ( K G ) M , I / ~ ) G

and t h a t t h e right-hand side is compact by Lemma 2.58.(ii) A s for t h e formulas, i n view of Lemma 5 . 4 , Propositions 5.16. 5.19, and 5.21,

Corollary 5 . 2 0 , Remarks 5.3. 5 . 5 , and 5 . 6 , and Theorem 5.37 t h e proofs of Proposition 4 . 4 carry over. w

The following lemma shows to what extent the equality ( 4 . 2 ) ' remains valid in t h e present context.

Lemma 6.17. Let u 6 3;(X).and l e t r E No u l a I and f Then f , , + ,

is a well-defined C' Function on Um,w/

XI

= Tf,,,,

.

6 C'tX,

XI satisfying

I such that u I

XI

= Tf

.

245

6.b T h e Distributions

m. By P r o p o s i t i o n 6 . 3 t h e set

L:= s u p p f is a n ( M , I ) - b o u n d e d s u b s e t of XI.

H e n c e t h e f i r s t p a r t of t h e a s s e r t i o n f o l l o w s f r o m P r o p o s i t i o n 3.3. If I is c o m p a c t

or if X , = X + t h e n t h e s e c o n d p a r t of t h e a s s e r t i o n is valid by ( 4 . 2 ) ’ . H e n c e w e may s u p p o s e t h a t I = I O , b l f o r s o m e b r l O , + m C ; in p a r t i c u l a r , w e h a v e X I = X . By P r o p o s i t i o n 6.3.A

L is t h e n c o n t a i n e d in X + ; by L e m m a 6 . 0 it f o l l o w s t h a t

LM,I i s a c l o s e d s u b s e t of X w h i c h is c o n t a i n e d in X + . H e n c e w e c a n f i x a f u n c e q u a l to 1 o n LM.1 w i t h s u p p o r t c o n t a i n e d in X + . C o n s e q u e n t l y , if

tion xEC-(X)

q E C z ( X ) t h e n x q E C z ( X + ) so t h a t by P r o p o s i t i o n s 3.10 a n d 3 . 2 2 a n d L e m m a 3 . 1 1

it f o l l o w s t h a t J’f,,,(x)

(6.15)

( x p ) ( x ) d x = J ’ f ( x ) (xp),,,*,.(x)

x+

dx

I

X+

S i n c e by P r o p o s i t i o n 3.10 w e have s u p p f m , w C L M , , t h e l e f t - h a n d side of ( 6 . 1 5 ) coincides with Jxf,n,,(x)

cpP(x) d x . O n t h e o t h e r h a n d , s i n c e

x = 1 o n LM,I a n d

s i n c e by P r o p o s i t i o n 5.11 yrn*,. is d e f i n e d by t h e i n t e g r a l f o r m u l a (3.1)’ i t f o l l o w s that

(x’p),+,,(x)

=~J,,,*,~(X)

f o r m u l a ( 3 . 1 ) ‘ it f o l l o w s t h a t

f o r every x E L . I n s e r t i n g t h i s i n t o t h e r i g h t - h a n d ( x c p ) m * , v ( x )=p,,,,(x)

f o r every x E L . I n s e r t i n g

t h i s i n t o t h e r i g h t - h a n d side of ( 6 . 1 S ) c o m p l e t e s t h e p r o o f .

W e close t h i s s e c t i o n by f o r m u l a t i n g c o n d i t i o n s u n d e r which

is t e m p e r a t e .

T h e f i r s t s t e p is to c a r r y over Remark 4.1 to t e m p e r a t e d i s t r i b u t i o n s .

Notatlon6.18. Let F be any s u b s e t of V, a n d let Y be a n o p e n s u b s e t of V. (i)

For any E > O w e set F , : = { x e V ; d i s t ( x , F ) < E } ;

(ii)

by Y ( Y I w e d e n o t e t h e s p a c e of all cp€Y’P(V)

some

E

s u c h t h a t ( s ~ p p q C) ~Y f o r

> 0;

( i i i ) by Y ’ ( F ; Y I w e d e n o t e t h e s p a c e of all Cm f u n c t i o n s v : Y - + C

for some (6.16.a)

E

such that

> 0 w e have

(Fnsuppcp), C Y

O n e observes t h a t by q H q l y a linear i s o m o r p h i s m of Y ( Y ) o n t o Y P ( V ; Y ) is de-

246

V1. Q u a s i h o m o g e n e o u s A v e r a g e s OF D i s t r i b u t i o n s . Part 2

fined. In t h i s way Y ( Y ) will be identified w i t h the subspace Y ( V ; Y ) of Y ( F ; Y ) .

Lemma 6.19. Let S E Y ' ( V ) . Then For any open subset Y OF X there exists a unique linear functional

2:Y(supp S ; Y) --?, C

estending the restriction OF S to Y ( Y ) *

and vanishing on the subspace { p E C " ( Y ) ; supp Q nsuppS = # } . S has the Following properties: (i)

t (ass)

> = ti',( - a ) u Q> ,

Below w e shall omit the superscript

l*wl*

qPE50(suppS; Y), LYEN,";

and any reference t o Y , considering S

as a functional on the union of all the spaces Y ( s u p p S ; Y ) .

m F . Let c p E Y ( s u p p S ; Y ) . We choose

E

0 such that (6.16) is satisfied for

F := supp S . We write L : = F n supp 'p . To prove the uniqueness part w e choose q ~ 1 0 , ~and C - b y Lemma S.S3.(i) - find a function ~ E C - ( V ) equal to I o n L,

w i t h support contained in L, such that all the derivatives of

x

are bounded. Then

by the Leibniz rule we derive from (6.16.b) that x ' p E Y ( Y ) . Since s u p p ( 1 - x ) is contained in V \ L , (6.17)

so that s u p p ( l - ~ ) ' p n s u p p S = Qit) f o l l o w s that

< S , q> =

< s , x'p > .

To prove the existence w e define S by (0.17) and have t o verify that the definition does not depend on the choice of t h e n supp ( x - x ' ) is contained i n L,\ so that by (h.16.b) the function

E ,q

and

x.

If

E', q'

and x' are other choices

0

L,

where y := maxf

E , E'

1 and 6 := min ( q , q'

)

( x - x ' ) ' p belongs to 9 " V ) . Since s u p p S does

not intersect s u p p ( x - x ' ) ' p it f o l l o w s that the right-hand side of (6.17) is equal to < S , x " p > , indeed. N

The assertion on 3" S follows by the Leibniz rule since L, n supp 3'x = 9, if 0 The last assertion is obvious.

#

0.

H

As we shall show below, if u is a distribution belonging to Y ' ( V ) n a ; ( V ) a s u f -

ficient condition for urn,,, to be temperate is that its support belong t o a special class of weakly (M,I)-bounded subsets of V. These are described by t h e following remark which is a simple consequence of the estimates (1.77).

247

6 . b The Distributlono urn,,

Remark 6.20. Let L be a subset o f V. Then the following conditions are equivalent: ( a ) L COJ (see Notation 5.10) for some closed subinterval J of IO.+col such that J n i i s a compact subset o f I O , + m T ; Ib) X l L )

nr

is a relatively compact subset of I O , + m C i f x = x + .

I

Deflnitlon6.Zl. ( i ) A s u b s e t L of V is called (M,l)-temperate if o n e ( a n d h e n c e e a c h ) of t h e c o n d i t i o n s of R e m a r k 6 . 2 0 is s a t i s f i e d ; if I = l O , + ~ Cw e a l s o s a y

M-temperate i n s t e a d of ( M , I ) - t e m p e r a t e . ( i i ) By Y'iIVI

w e d e n o t e t h e s p a c e of all t e m p e r a t e d i s t r i b u t i o n s o n V w i t h

( M , I ) - t e m p e r a t e s u p p o r t ; if 1 = 1 0 , + ~ 0tC hen we also write Y ' k f V ) ,

N o t e t h a t it f o l l o w s f r o m Lemma 6 . 7 t h a t ( M , I ) - t e m p e r a t e s u b s e t s of V a r e weakly ( M , I ) - b o u n d e d in V . In p a r t i c u l a r , Y i ( V ) is c o n t a i n e d in Y ' ( V ) n D ; ( V ) .

belongs Proporition 6.22. ( i ) I f L is an (M.1)-temperate subset of V then p O m * , " t o the space F(L;Vl,I) for ever)' rpEY'(V). V ) . Then u,,,,,, is temperate: more precisel).. for ever). F E Y ' ( V )

( i i ) Let u

the right-hand side o f the equation in (4.31 is well-defined, defining a continuous linear functional on Y'( V ) . denoted bj. u,,,,

. as

well. Moreover. W .,

(see Pro-

position 6.15) is a meromorphic function with values in F'(V ) .

-Proof, Case I :

I f I is c o m p a c t t h e n V is ( M , I ) - t e m p e r a t e , i t s e l f , a n d

'p H

T,,,*,,,~

defines a continuous linear map from Y ( V ) into itself, a n d t h e assertions follow.

CasesSand3: S u p p o s e t h a t 1 = 1 0 , b l ( r e s p . Cb,+mC) f o r s o m e c E l O , + a C . For t h e p r o o f of ( i ) o n e d e d u c e s f r o m R e m a r k 0 . 2 0 t h a t L is c o n t a i n e d in t h e s e t

nCc,+mE ( r e s p . n,,,,,)

f o r s o m e c E I O , + c o C . C o n s e q u e n t l y , m a k i n g u s e of Lem-

m a S . S 3 . ( i i ) o n e f i n d s E , d € l O , + a C s u c h t h a t L, is c o n t a i n e d in

nro,d7 ) , Hence,

f o r F = L a n d Y = V,,,

nCd,+-[

(resp.

t h e c o n d i t i o n (b.1b.a) is trivially valid f o r

a r b i t r a r y ~ J E C ~ ( X , / a~n d) , by P r o p o s i t i o n S . l l , ( i ) ( r e s p . P r o p o s i t i o n 5 . 3 6 ) t h e condition (6.lb.b) holds with

'p

r e p l a c e d by

'p,,,x,v

f o r every ' p E Y ( V ) .

For t h e p r o o f of ( i i ) w e observe t h a t by t h e a s s u m p t i o n o n s u p p u L e m m a 5.53

p r o v i d e s u s w i t h a f u n c t i o n x 6 C m ( V ) e q u a l to 1 o n ( s u p p u ) , f o r s o m e

E

>0

248

V I . Q u a s i h o m o g e n e o u s A v e r a g e s of D i s t r i b u t i o n s . Part 2

with support contained in the set f l ) ~ d , +(resp. ~[ f l c o , d 1 ) for some d ~ l O , + c o l such that the derivatives of

x are bounded. Consequently, by Proposition 5.11.(i)

(resp. Remark 5.35 and Proposition 5.36 ) the map defined by

'p H X ( P ~ * is , a~

continuous linear operator of Y ( V ) into itself. In view of (6.17) the assertions follow. C a s e 3 : If I = I O , + ~ [ the proof is analogous or is obtained as a combination of the cases 2 and 3 . m

(c) Describing Lhc Almosl yuasihomo~:.cllc.ous1)istribuliona on \ w i l h S u p p o r l Conlainccl in X \ k +

To introduce a method for constructing distributions having the properties spelt out i n the title of the present section w e require

Lemma 6.23. IF K is a compact subset o f X then M i ' ( K n M , ( K ) ) is a closed subset OF X which is contained in L M , 3 0 , 1 3 For some compact subset L OF X o . m F . Since K n M,(K) is compact we find

E

> 0 and a finite subset F of K n M,,(K)

such that -

K n M o ( K ) C L : = U x E F P ( x , ~ C) X where t h e closed polydiscs

~ ( X , E )are

defined as i n the text preceding ( 6 . 6 ) .

Since in view of the last assertion i n Proposition 1.70 we have P( x , f ) M , 1 0 , 1 3 = { y E v ; I M,-,(y - x ) I 5 E } = M,' (Po(, E ) ) we conclude that MG'(KnM,(K))

MG'(KnM,(K)) C L M , ] , , , ]

By

Lemma 5.46 the set

is even closed in V .

Proporltlon 6.24. Let u (6.111)

C X .

( X ) . Then b y

( Q ; , u , P > : = < u , Q,,*P>,

pE C,-(X),

a distribution QA u E 3 ' f X ) i s well-defined having the following properties:

2 49

6.c A l m o s t Q u a s i h o m o g e n e o u s D i s t r i b u t i o n s w i t h S u p p o r t in X \ X +

(i)

Qh u

is almost quasihomogeneous of degree m and of order

_
aM = rn*

(-a)a,yo

where S , E ~ ' ( V ) is t h e Dirac distribution a t 0 .

proOf. Let K be a c o mp act s u b s e t of X . Then by Lemma 6.23 and Definition 6.1 t h e set Z : = M i ' ( K n M o ( K ) ) n s u p p u is a compact s u b s e t of X . We fix a function equal t o 1 near Z . Clearly, t h e map CTCX) + C F ( s u p p x ) .

xEC:(X)

is well-defined and continuous. By Corollary 5.47 for every 'p€C:(K)

of Q,*

'p

is contained in M G 1 ( K n Mo( K ) ) so t h a t Q,+

'p

'pHxQ,n%cp, the support

is equal to x Q m + ' p

near s u p p u . In view of Remark 4.1 it follows t h a t Q k u is a well-defined distribution o n X satisfying

(i): 'p

< QL,

u. 'p > = < u , x Q,*

'p

>

for every

'p E

CFC K ).

Let P : = N ( m * ) . Then by Propositions5.4X.(iv) and 5.45 we deduce f o r every

E CTC X ) t h a t

= t-'I

< u . (Q,*

(p) O M l / ,

>=

e = t-' ( l / t ) m *

wi(l/t) - < (3,

-m)"'d,

41 > .

The condition ( a ) means t h a t ( a M - m ) j + ' d = O f o r sufficiently large j . Hence t h e condition ( b ) follows.

( b l q f c ) :see Proposition 6 . 2 4 . ( i ) , ( i v ) . ( c ) + ( a l : this is obvious in view of Remark 1.43.(i) and Proposition 2.35.

f i i ) : In

a" xp = p! sap

view of Proposition 1.28. ( i ) we have

for arbitrary a ,

E~LA*.

Hence, f o r every ~ E C T ( X we ) conclude that n

< x P d , v >= < d , x P y o M o > =

1

~=

cr€'u;*

5 = < d g , v o M o > = < d p ,'p> .

ae'U+,* 7.

Applying t h e assertion ( i ) t o ( d p , - p ) instead of ( d , m ) we see t h a t

dp=da.

Moreover, since by Proposition 2.35 the distribution x p d is a l m o s t quasihomogen

neous of degree - p t h e same argument yields t h e equality ( x p d )

= xpd.

In view of Proposition 6 . 2 4 . ( i ) we deduce

Corollary 6.28. Ever,, distribution on X with support in X \ X + which i s almost quasihomogeneous of degree m is so o f order 5 N ( r n * ) ( s e e (5.651) and vanishes i f m d I -21(M)- p ) . I n particular, if M is se m i-sim ple then every such distribution i s quasihomogeneous.

#

I n order to complete t h e characterization of t h e distributions satisfying the con-

dition ( a ) of Proposition 6.27.(i) w e have to describe all distributions with s u p p o r t contained in X \ X + which a r e quasihomogeneous of degree - p . If k e r M = ( O ) , i.e. X \ X + = (01,then t h i s , of course, i s simple: the complex multiples of t h e Dirac distribution 8, are the only distributions of t h e desired type. To formulate t h e answer in case ker M # (0)w e write

and define t h e open s u b s e t X " of Vo by t h e condition (6.22.b)

X n M , ( X ) = (0)x X " , i.e.

X " = V, x X "

253

6.c A l m o s t Q u a s i h o m o g e n e o u s Distributions w i t h S u p p o r t in X \ X +

Fropodtion 6.29. Under the preceding conventions suppose that Vo is non-trivial. (i)

For every R E D ' ( X " I the distribution

(6.23) - where

Jv*@ R denotes the Dirac distribution on V 1 - estends t o a unique distribution

on X with support contained in X \ X + which is quasihomogeneous of degree - p . lii)

The restriction t o X o o f every distribution d 6 B ' I X ) with support contained

in X \ X + which is quasihomogeneous o f degree - p is of the f o r m (6.173) f o r some U

R E B ' ( X " I . This distribution R is uniquelj determined; it will be denoted bj d

;

and it i s given b) (6.24)

where 4, : V,

-

IR be the constant function

'H I .

Proof. (il:Denoting the distribution (6.23) by T we observe that the support of

T , being equal to ( 0 )x supp R , is a closed subset of X . Hence, Remark 4.1 shows that T extends to a distribution d € B ' ( X ) with the same support as T . I t is immediately seen that T is quasihomogeneous of degree -p so that ( d M + p ) T= O . Since d vanishes on X,

so does ( d M + ( l ) d . Hence, ( a M + I i ) d = O ,i.e. d is quasihomo-

geneous of degree - p , a s well.

(iil.

First of all, with the help of Remark 4.1 o n e deduces that the right-hand U

side of (6.24) defines a distribution d o n X " . To compute it we fix cp,EC;(V,) and c p 2 ~ C ; ; ) ( X "Then ). ( c p 1 @ c p 2 ) o M o = c p , ( O ) c 1 @ q 2 . Consequently, since by PropoA

sition 0.27 the distributions d and d coincide w e conclude that < d , cp,8qp,> = c p , ( O ) < d , e , @ q 2>

U

<Sv,,cpl

> < d .cp2

>.

u

This means that (6.23) is valid for R = d , indeed. rn U

r\

If d = u for some u ~ a ) ; , , + , ~ ( X ) then properties of u carry over to d

Remark 6.30. For any

u

:

+c,,c ( X I the distribution

(6.2551 (which is well-defined in view o f Proposition 6 . 2 4 ) has the following properties:

254

V I . Q u a s l h o r n o g e n e o u s A v e r a g e s of D i s t r i b u t i o n s . Part 2

if s ~ N ~ u ( and a ) if u is induced by a C O ' s f u n c t i o n then

(i)

u"

is induced

by a C s function;

if X = V and if U E ~ ' ; ~ , + ~ [then ( V )u" belongs t o 9 " f V o ) .

(ii)

mf. (i): w e

l e t f E C o S S ( X ) be s u c h t h a t u = T f a n d f i x a c o m p a c t subset K"

o f X". T h e n

K := { x'cV1

K,,30,11=Vl~K".

;

1 x'I

5 1 } x K"

is a c o m p a c t s u b s e t o f

X satisfying

S i n c e s u p p f is a w e a k l y ( M , C l , + a C ) - b o u n d e d s u b s e t o f X it

f o l l o w s t h a t t h e set { X ' E V ~ (; x ' , x " )

E

s u p p f f o r s o m e x " E K " ) is c o m p a c t . C o n -

s e q u e n t l y , by

I' f ( x ' , x " ) d x '

g ( x " ) :=

"1

For t h e proof o f

.

C i s d e f i n e d s a t i s f y i n g Tg=

a C Efunction g : X " +

w e observe t h a t t h e a s s u m p t i o n o n s u p p u m e a n s t h a t s u p p u

is c o n t a i n e d in K' x Vo

f o r s o m e c o m p a c t s u b s e t K' o f

V1

. W e c h o o s e x1 E CFC V l )

e q u a l to 1 o n a n e i g h b o u r h o o d o f K'. T h e n t h e m a p Y'(V1)--+Y'(V),

v,Hyl@~,,

is w e l l - d e f i n e d , l i n e a r , a n d c o n t i n u o u s so t h a t t h e a s s e r t i o n f o l l o w s f r o m t h e

equality

< u,yz>= < d , ' p 2

> = $ ( O ) <Sv,@

A

n

W

d

,(pI@'pz

> = < d , c p I @ ' p 2 >.

n

d = d t h i s m e a n s t h a t u = d . If t h e a s s u m p -

S i n c e by P r o p o s i t i o n 0 . 2 7 w e have

t i o n of (ii) is valid t h e n by Reniark 6 . 3 0 . ( i ) t h e r e is a f u n c t i o n gE C s ( V o ) s u c h

"

that d

T,

so t h a t u = T+,Sg. U n d e r t h e a s s u m p t i o n of (iiil w e k n o w f r o m Reu

m a r k 0 . 3 0 t h a t t h e d i s t r i b u t i o n d b e l o n g s to Y ' ( V , )

so t h a t u E Y ' ( V ) .

To p r o v e t h e g e n e r a l case w e s u p p o s e t h a t X f V a n d d e f i n e a f u n c t i o n S:kerM-

IO,+mC, x

I+

dist(p,(V+nMo'(x)).V\X).

S i n c e by ( 5 . 7 2 . b ) a n d (1.76) w e have (6.26)

s ( x ) = d i s t ( x + S , V \ X ) where S : = { v E G , ( o + ) ; I v l = l )

w e d e d u c e t h a t I S ( x ) - S ( x ' ) l 5 I x - x ' l f o r a r b i t r a r y x , x ' E k e r M , i.e. 6 is c o n t i n u o u s . N e x t w e are g o i n g to verify a p r o p e r t y of S which is crucial f o r t h e p r o o f , n a m e l y , t h a t f o r every x E k e r M w e have

*

I n d e e d , s u p p o s i n g f i r s t t h a t x@'X

o n e f i n d s a s e q u e n c e of p o i n t s x,

in t h e c o m -

p l e m e n t of X u k e r M c o n v e r g i n g to x as m + m . T h e n o n e c a n c h o o s e t,E s u c h t h a t y m : = Mt,xm

10,+00C

lies in S x . Since y m b e l o n g s to p x ( V + n M:'(Mox,,,))

as well a s to V \ X o n e c o n c l u d e s

256

VI. Q u a s i h o m o g e n e o u s A v e r a g e s of D i s t r i b u t i o n s . Part 2

=0

6(Moxm) 5 dist(y,,V\X)

so t h a t by t h e c o n t i n u i t y of M, a n d 8 o n e o b t a i n s 6 ( x ) = 0 . C o n v e r s e l y , s u p p o s i n g that

XE%

o n e f i n d s E > O s u c h t h a t t h e set

a n d h e n c e its q u a s i h o m o g e n e o u s h u l l { v E V + ; I M o v - x l < ~ } a r e c o n t a i n e d in X . S i n c e , in p a r t i c u l a r , t h i s is t r u e a b o u t its c o m p a c t s u b s e t x + S it f o l l o w s by ( 6 . 2 6 ) t h a t S ( x ) > 0 . T h i s c o m p l e t e s t h e p r o o f of ( 6 . 2 7 ) . N o w , s i n c e S is c o n t i n u o u s a n d s i n c e ( 6 . 2 7 ) t e l l s u s t h a t , in p a r t i c u l a r , S is p o s i t i v e o n X n k e r M w e c a n find s e q u e n c e s

(Zk)ke,,,,

in k e r M a n d ( q k ) in I O , + c o L s u c h

t h a t t h e b a l l s K ( z k , q k ) , k € I N , cover X n k e r M , s u c h t h a t t h e b a l l s K ( z k , 2 q k ) , k E N , are c o n t a i n e d in Xo f o r m i n g a locally f i n i t e f a m i l y , a n d s u c h t h a t

Moreover, w e choose a decreasing sequence

(tk)kalN

in 1 0 , I I in s u c h a way t h a t

t h e p o i n t s y k : = MtkyL , k E N , have t h e f o l l o w i n g p r o p e r t y : t h e s e q u e n c e of n u m -

bers l P + y k l is s t r i c t l y d e c r e a s i n g a n d c o n v e r g e s to z e r o a s k + a . T h e n w e c a n find a decreasing sequence

(Ek)

in 10,+mC c o n v e r g i n g to z e r o a n d s a t i s f y i n g t h e

e S t i m a t e S I P + Y k l - ? E k 2 I P + Y k + l l + 2 E k + l ,kEIN. W e Set

S i n c e p x is c o n t i n u o u s , by m a k i n g t h e Ip,(P+X)-p,(P+yk)l

Ek

s m a l l e r w e c a n achieve t h a t

s(zk),

XELk,

k€N.

- -

Referring to t h e n o t a t i o n i n t r o d u c e d in (6.22) w e c a n w r i t e L k = U k X W k w h e r e t h e U, are relatively c o m p a c t o p e n s u b s e t s of V, s u c h t h a t t h e r k , k E N , are p a i r w i s e d i s j o i n t s u b s e t s of V , \ ( o ) a n d w h e r e t h e wk f o r m a n o p e n c o v e r i n g 113 of X" s u c h t h a t t h e w k c o n s t i t u t e a locally f i n i t e f a m i l y of c o m p a c t subsets of

x".

A

F o r every kE:N w e fix a f u n c t i o n + k E C g ( U k ) s u c h t h a t $ k ( o ) = 1 . Finally,

w e c h o o s e a p a r t i t i o n of unity ( X k ) k c N o n X " s u b o r d i n a t e d to t h e c o v e r i n g IB and define v

Uk:=T+k@(Xkd).

257

6.c A l m o s t Q u a s i h o m o g e n e o u s D i s t r l b u t i o n s w l t h S u p p o r t i n X \ X +

Then u k € &'(V) with support contained i n

Lk

. Since there exists a locally finite

sequence of pairwise disjoint relatively compact open subsets Yk 3

Yk

of X such that

Lk for every k E N , we conclude that

is a well-defined distribution on X w i t h support contained in the closed subset

L:=

u

Lk

kEN

of X . In order t o see that u belongs t o 3 k l , + m r ( X ) we have to verify that L is a weakly ( M , C l , + a l ) - b o u n d e d subset of X . I n view o f Propositionh.3.B it

suffices to show that L is an (M,Cl,+aC)-bounded subset of X , . a compact subset of

X + . Then c : = inf x( K ) is positive. Since I P,

to 0 as k + a , since the restriction of i t o G,(a+)

So let K be

y k I + Ek converges

is continuous at 0 , and since

(5.72.a) holds we can choose N E N so large that sup x ( L k ) < c for every k > N . Consequently, for every t E C1 ,+a[ we have N

M,(K)nL= Mt(K)nH

H:=

where

u L,

k=l

Since H is a compact, hence an M-bounded subset o f X,

it follows that the

set { t E C l , + a l ;M , ( K J n L f @ } is a relatively compact subset o f I O , + a 3 C , indeed.

"

To see that G = d we fix ' p E C T ( X ) and - b y applying ( 0 . 2 4 ) first to u k instead of d and then to d itself

-

observe that

u

n

Consequently. making use of Remark 4.1 and of the fact that tik extends Sv,@uk we obtain +m

=

k=1

c

+m

W

< u k , ' p ( o ,' )

k=l

Since there are open neighbourhoods Y k of Lk i n that

(Yk)kEN

c

+m

>=

.

k=l

x

and

zk

o f wk in

x" s u c h

(resp. ( Z k ) k c N ) is locally finite in X (resp. X " ) , actually, the

s u m s are finite. And since +a

e,@xk= 1

on a neighbourhood of supp

'poMO

k=1

the right-hand side o f the preceding equation is equal to

< d ' , 'p >

which coincides

with < d , ' p > by Proposition 6.27.(i). For the proof of

li) we first observe that by Lemma6.h it suffices to verify the

258

VI. Q u a s i h o m o g e n e o u s

Averages

i n c l u s i o n in ( i ) f o r every c o m p a c t s u b s e t K o f X , ,

OF D i s t r i b u t i o n s . Part 2

o n l y . To t h i s e n d w e f i x

X E L ~ K ~a n, d~c h, o~o s e t E 1 , z E K , a n d kElN s u c h t h a t x = M , z a n d XELk g,(x) =Q,(z) Ep,(K) so t h a t by ( 5 . 7 2 . b ) . b y t h e e q u a l i t i e s

&(Yk)

.

Then

= y k a n d Mo(yk)

= z k , a n d by t h e e s t i m a t e s above w e o b t a i n

5 6 ( z k ) + q k + 2 6 ( z k )5 4 6 ( Z k ) 5 8 6 ( M o X )

w h e r e t h e l a s t inequality is valid s i n c e

M ~ ( X ) E K ( Z ~ , I BY J ~ )t h. e

continuity of

Mo a n d 6 t h i s i m p l i e s t h a t S ? r / 8 o n k e r M n L n K M , l / I . Since p , ( K )

is a c o m -

p a c t s u b s e t of X it f o l l o w s t h a t r > O . C o n s e q u e n t l y , it f o l l o w s f r o m (6.27) t h a t k e r M n L n K M , I , I is c o n t a i n e d in

2.

S i n c e i n ( X \ X + ) = @ t h e p r o o f o f ( i ) is

complete. U

If t h e a s s u m p t i o n o f lii) is valid t h e n Remark 6 . 3 0 . ( i ) i m p l i e s t h a t d is i n d u c e d by a C s f u n c t i o n so t h a t t h e u k a n d h e n c e u a r e i n d u c e d by C'n's f u n c t i o n s .

In view o f P r o p o s i t i o n 6 . 3 . A t h e c o n d i t i o n ( i ) in L e m m a 6.31 does n o t q u i t e m e a n t h a t t h e s u p p o r t o f u is a weakly ( M , I O , l l ) - b o u n d e d s u b s e t o f X . In f a c t , it is n o t a l w a y s p o s s i b l e to c h o o s e u w i t h weakly M - b o u n d e d s u p p o r t :

Corollary 6.32. Let d be as in Lemma 6.31. Then distributions u having the properties asserted in Lemma 6.31 can be chosen to belong to . 9 & ( X ) i f and onl-), i f d w

extends t o an almost quasihomogeneous distribution on X .

Proof.

'2": In view

of ( 6 . 2 . a ) Remark 6.14 s a y s t h a t u e x t e n d s to a d i s t r i b u t i o n

G E ~ D ( M ( % )w i t h t h e s a m e s u p p o r t as u . S i n c e QAK e x t e n d s

QAu

the assertion

f o l l o w s by P r o p o s i t i o n 6 . 2 4 . ( i ) .

"e": we

.r

may a s s u m e t h a t X = X . T h e n in view o f P r o p o s i t i o n 6 . 3 . A t h e c o n d i -

t i o n ( i ) o f Lemma 6 . 3 1 , a c t u a l l y , m e a n s t h a t s u p p u is a n ( M , I O , l l ) - b o u n d e d s u b set of X . H e n c e , L ~ m m a h . 3 1g i v e s , in f a c t , t h e desired d i s t r i b u t i o n U E ~ & ( X ) .

A s a f i r s t a p p l i c a t i o n of Lemma 6.31 w e o b t a i n

6.c A l m o s t Q u a s i h o m o g e n e o u s Distributions w i t h S u p p o r t in

259

X\X+

Ropodtlon 6.33. Let Pt? @, and let q 6 C C m ( X )be almost quasihomogeneous o f degree P such that q-jI0) c x \

(6.28)

x, .

Then fo r every distribution d € D ' ( X ) with support contained in X \ X ,

which is

almost quasihomogeneous o f degree m the equation q c = d has a solution c E a ' l X ) with support contained in X \ X , such that c is almost quasihomogeneous o f degree m - e .

I f in the

case

X = V q i s a polqnomial function and d is temperate then c can

be chosen t o be temperate, as well.

mf. By

Lemma 6.31 w e find u E 4 ; , , + , , ( X )

s u p p u is c o n t a i n e d in X ,

such t h a t Q L u = d and such t h a t

.

1

H e n c e , it f o l l o w s f r o m ( 6 . 2 8 ) t h a t v : = - u is a w e l l 4 d e f i n e d d i s t r i b u t i o n o n X b e l o n g i n g to % ; , , + , , ( X ) . Making u s e of P r o p o s i t i o n s 6 . 2 6 . ( i i ) a n d 6.24 w e d e d u c e t h a t c : =Q L n - @is v t h e desired s o l u t i o n . U n d e r t h e a s s u m p t i o n s of t h e s u p p l e m e n t a r y a s s e r t i o n w e f i r s t o b s e r v e t h a t by L e m m a 6 . 3 l . ( i i i ) w e may a s s u m e t h a t u is t e m p e r a t e a n d t h a t its s u p p o r t is cont a i n e d in f I [ , , 2 , .

In view of P r o p o s i t i o n 6 . 2 4 . ( i i i ) it s u f f i c e s to s h o w t h a t v is

t e m p e r a t e , By t h e Leibniz r u l e t h i s f o l l o w s f r o m a n e s t i m a t e of t h e f o r m

(6.29)

Iq(x)l 2

c (l+lxl)-N,

XEnC1/2,31

w h e r e C a n d N a r e p o s i t i v e c o n s t a n t s . By H o r m a n d e r 1 8 1 t h e r e a r e p o s i t i v e c o n s t a n t s C',

E ,

and N such that

I q ( x ) l 2 C ' ( l + l x l ) K Nd i s t ( x , q - l ( O ) ) ' ,

X€V.

S i n c e by ( 6 . 2 8 ) w e have d i s t ( x , q - ' ( O ) ) 2 I P + ( x ) l t h i s i m p l i e s ( 6 . 2 9 ) , i n d e e d .

260

VI. Quaslhomogeneous A v e r a g e s of Distributlons. P a r t 2

(d) C h a r a c l e r l a l n g (Almosl) Quaslhomogeneous D l s l r l b u t l o n s o n X In T e r m s o f Quaslhomogeneous A v e r a g e s

Notatdon6.34. For every u E D ; ( X )

w e denote

by u r n , , ; if I = I O , + ~ Cwe

a l s o write u,.

Propodtlon 6.35. Let u 6 ah (X). Then u , " . , ~ is almost quasihomogeneous o f degree m such that

In particular, urn is almost quasihomogeneous of degree m with deficiencj, QA u , and hence it is quasihomogeneous of degree m i f and onlv i f m 6 ? ( - X ( M ) - p ) or ( s a u )

proof. Let

n

= O f o r every a ~ Z , i *.

'p

X ) . The assertions

E C:

v i i ) and ( i i i ) of Theorem 5.37 and (1.65)

imply t h a t (6.31)

f o r every t € I O , + a C . Making use of (6.12)' and ( 2 . 5 ) repeatedly, one deduces t h e first part of t h e assertion. The second part is a consequence of Proposition 6.27.

Example 6.36. Let y € V + , and let S, be the Dirac distribution at

'

J

every polynomial function Q : V +C the distribution ( Q ( -a) S,, I,,,

. Then for is almost

quasihomogeneous of degree m with deficienq

I f Q is almost quasihomogeneous OF degree mt- and o f t-pe M * then this is equal t o Q(-d)'7M0,

.

proOf. For arbitrary a€%;* t h e form of (1.54) t h a t

and

'p

E C g ( V ) w e deduce by t h e Leibniz formula in

261

6 . d Characterizing ( A l m o s t ) Q u a s i h o m o g e n e o u s Distributions

If p t?Uo then

a'( ( aacp) oMo)

vanishes identically. Consequently, the summation

is over ?lo, and the first part of the assertion follows by Propositions 6.35 and

6.24.(iv). The second part is a consequence of Propositions 6.35 and O.Zh.(iii).

We n o w formulate the main theorem of t h i s section, the analogue of Theorem 4 . 8 .

Theorem 6.37. Let T E D ' ( X I and k (i)

E N ,

.

Then the Following conditions are equivalent:

( a ) T is almost quasihomogeneous o f degree m such that the support o f its ( k + l )t h order deficient) ( d M - r n J k + ' T is contained in X

( b ) < T . p > = O f o r ever-) y E C : ( X )

\X,

:

satisfqing p , , l * , c , , k = O ;

(c) there e.\ists a distribution u E D L ( X ) and a distribution d E D ' l X ) satis-

[)ling the condition ( a 1 o f Proposition 6.27. ( i ) such that

( i i ) If % i = X then the distribution

LI

in condition (c) o f (il can be chosen in

such a wa) that (6.331 is valid f o r d = O . (iii)

If X = V and i f T

is temperate and satisfies the conditions o f (i1 then the

condition I b ) is valid f o r ever) p E Y ( V J , and the distribution u in condition (c) of ( i ) can be chosen t o be temperate, as well, with M-temperate support, in such a was that (6.331 is valid f o r d = O . livl

The distributions u and d in condition ( c l o f ( i l can be chosen t o be @-in-

variant provided that T is @-invariant and that @ satisfies the assumptions of Remark -7.67. ( i i l .

The proof follows the lines of the proof of Theorem 4 . 8 except that in case mE ( - U ( M ) - p ) complications arise; in particular, the direct proof of the implication " ( b ) * ( a ) " does not seem to carry over. We begin w i t h the

262

VI. Q u a s i h o m o g e n e o u s Averages of D i s t r i b u t i o n s . Part 2

Proof of " f a ) J ( b ) " . Let ( P E ~ ( V satisfy ) q m c , w k = O . Then @ k : =( is equal to

- ( P m * . ~ C l , , m C W ~

V,

on

,

P

~

~

Applying Proposition 5.11 to t h e derivatives

of t h e l a t t e r function and taking L e m m a 5 . 4 into account we deduce t h a t

is valid f o r fl =

ncl,+mc.

Similarly, applying Proposition 5.13 and Corollaries 5.29

and 5.33 t o b = 1 and W ( x ) = ( l + I M o ( x ) I ) - N (l+(P,(x)l)-N w e conclude t h a t @ k belongs

to

= f l c o , l l , a s well. Consequently,

t h e condition ( 6 . 3 4 ) holds for

Y(V).Moreover, if

cp belongs to

c F ( x ) then

so does @ k . In f a c t ,

since by R e m a r k 3 . 2 5 . ( i ) s u p p ' p is a weakly ( M . L l , + a C ) - b o u n d e d s u b s e t of X it follows by Lemma 0 . 1 1 t h a t ( ~ ~ p p c p ) M n, ~( ~~ ~, p+ p~c ~p ) M , ]is~ ,a~ lc o m p a c t s u b s e t of X so t h a t in view of Remark 5.7 and Proposition 5.16.(ii) t h e s u p p o r t

of

@k

is a c o m p a c t s u b s e t of X

A s in t h e proof of Proposition 6.27 we now derive from Proposition 5.27 t h a t t h e equation (6.21) is valid f o r j = k . Since both

'p

and @ k and hence t ( d M - m ) k ' l @ k

belong to Y ( V ) this implies, in particular, t h a t

Q,,,*'p = O . Assuming t h a t

belongs t o CgCX) in c a s e T E % ' ( X ) we deduce t h a t

< T , (P) = - < d , @ k ) where d : = ( d M - m ) k ' l T . Since by t h e assumption on T Proposition 0.27 s h o w s t h a t d = Q k , ( d ) and since by Corollary 5.31 we have Q m h o k = 0 we conclude t h a t ( d , @ , ) = O and hence < T , c p ) = O , a s desired.

For t h e proof of t h e implication " ( b ) + ( c ) " we a r e going to derive a variant o f Proposition 4.13. To t h i s end we fix a sequence ( L ) i ) 0 5 i s k

(4.18) f o r ~ = ( O ) U I N ~ and + ~ +(4.19) ~ where t h e c o n s t a n t s

in

c; ( x ) satisfying a r e defined by

(4.15) and where (6.35)

N : = N(m')

- see (5.65)

That t h i s is possible - even f o r

3 =No-

follows by Lemma 4.12.(ii) and Remark

6.12. When working in t h e c o n t e x t of temperate distributions it is necessary to c h o o s e t h e functions

Jli

more carefully. This requires

,

~

6.d

263

characterizing ( A l m o s t ) Q u a s i h o m o g e n e o u s D i s t r i b u t i o n s

Lamma 6.38. Suppose that x =x, (see Proposition 1.70). Let I be a compact subset o f 1O,+moT, and let J be a relatively compact open neighbourhood of I in 1O,+wC. Then there exists a Cmfunction

#: V 4 W equal t o 1 on x - ' ( I ) satis-

fying (6.36)

** = I

and having the following properties: (il

supp# C x - ' ( J ) ;

( i i ) all the derivatives of (I, are bounded: (iii) for ever) j r N , all the derivatives of

are bounded on ever, M-tem-

i,!~~~,~,,.

I

perate subset of V : ( i v ) i f I C l r , e r t for some rElO.+-C then (I, 2 0

Proof. We choose a , b , c , d E l 0 , + m C s u c h t h a t b . d > c > a and C a . b l C J \ 1 . By Lemma 5.53 we find non-negative such t h a t all their derivatives a r e on x - ' ( [ c . d l )

( r e s p . x - ' ( l ) ) and

x , 'p E C m (V ) with values i n CO.ll bounded, such t h a t x ( r e s p . 'p) is equal t o I such t h a t t h e s u p p o r t of x ( r e s p . 'p) is confunctions

tained in x - ' ( l a , b l ) ( r e s p . x - ' ( J \ [ a , b l ) ) . Since t x ( x ) = x ( M , x ) E l c , d I i f and only if t E C c / x ( x ) , d / x ( x ) l it follows in view of T h e o r e m 5 . 3 7 . ( i ) t h a t

+

d/x(*)

(6.37)

xo(x) 2

J'

d =log, > 0 ,

X€V+.

c/x(x)

Consequently, by

+:='p+(l-'po)x/xo

a C m function o n V is well-defined. Ob-

viously, i t is equal t o 1 o n x-' ( I ) , and its s u p p o r t is contained in x-' ( J ) . Since l-cpo

and

xo a r e quasihomogeneous of degree

0 i t follows from Remark 5.3 t h a t

~ o = ' p o + ( l - ' p o ) - l . To p r o v e 0 we choose R > p > O s u c h t h a t J = l g , R C so t h a t

s u p p 'p C x-' ( I g , R C ) . Moreover, P:V*-C

we fix t E'U(M) and

a polynomial

function

which is a l m o s t quasihomogeneous o f degree 0 and of type M I . For

every i E [ N we s e t Pi : = (aM* - t ) i P . Applying Proposition 3.4 we deduce t h a t (6.38)

and

t(p(a)'po)(x)t 5

te (Pi(a)cp)(M,x)w i ( t )

T dt

I

5

264

VI.

I ( P ( a ) y o ) ( x ) l5

Quasihomogeneous A v e r a g e s of D i s t r i b u t i o n s . Part 2

2

b/a

J'

tRe@-l

Iwi(t)l dt llPi(a)xll~a

ieNo a / b

f o r arbitrary x e ~ - ~ ( C a , b lCombining ). t h e l a s t estimate with ( 6 . 3 7 ) s h o w s t h a t t h e derivatives of l / y o a r e bounded on s u p p y , and ( i i ) follows by ( 6 . 3 8 ) . The property (iii) is a consequence of t h e following estimate which - i n view of Proposition 3 . 4 and (1.38)- is valid f o r arbitrary

E

> 0 and x E x - l (

C E , J / E 1) :

Finally, if t h e assumption of (iv) holds then in t h e proof of ( i i ) above we can choose p and R such t h a t R = e p . Consequently, making use of ( 0 . 3 8 ) f o r P = l

we obtain

By Lemma 4.12.(ii) we obtain

Corollary 6.39. Under the assumptions o f Lemma 6.38 the Functions +i, 0 5 i 5 k can be chosen so a s t o have the properties ( i ) , ( i i ) . and (iii) of L e m m a 6 . 3 8 .

Besides t h e properties s p e l t o u t i n Lemma 4.12 t h e c o n s t a n t s c

~ p o, s s ~ ess

. I

other

important properties required for the proof of Theorem 6 . 3 7 .

Lemma6.40. For every j € N o we have

e F . Inserting t h e defining equations (4.151, interchanging t h e order of summation and substituting i = i - I we s e e that the left-hand side of ( 6 . 3 9 ) is equal to

26.5

6 . d Characterizing ( A l m o s t ) Quaslhomogeneous Distributions

Applying L e m m a 1 . 7 6 to ( j , j , k - l ) i n s t e a d o f ( j , k , e ) w e see t h a t t h e s u m in s q u a r e b r a c k e t s v a n i s h e s in case j < k - l a n d e q u a l s

(j+i-k)

in case j ? k - I .

Hence t h e

t h e l e f t - h a n d side of ( 6 . 3 9 ) is e q u a l to

Izk-j

If j < _ k t h i s e q u a l s ( - 1 ) k ( 1 - l ) J = ( - l ) k S ~ j ,a n d t h e a s s e r t i o n f o l l o w s in t h i s case. I f , o n t h e o t h e r h a n d j > k t h e n by ( 4 . 2 0 ) t h i s e q u a l s

( J k ' ) , as

desired.

N o w , l e t T be a d i s t r i b u t i o n s a t i s f y i n g t h e c o n d i t i o n ( b ) o f T h e o r e m 6 . 3 7 . ( i ) . T h e n a p p l y i n g P r o p o s i t i o n 4.13 to ( T ) , + , X , , k ) i n s t e a d o f ( T , X , N ) w e o b s e r v e t h a t t h e s u p p o r t of t h e d i s t r i b u t i o n k

is c o n t a i n e d in X \ X + . H o w e v e r , a t t h i s s t a g e it i s n o t c l e a r t h a t d is, in f a c t ,

a l m o s t q u a s i h o m o g e n e o u s o f d e g r e e m . In o r d e r to verify t h i s w e r e q u i r e a n o t h e r auxilliary f u n c t i o n : w e fix a C m f u n c t i o n

x+: X +43

s u c h t h a t s u p p q, is a weakly

( M , C l , + c o C ) - b o u n d e d s u b s e t of X a n d s u c h t h a t w e have (6.41)

h J I x 0) O ' b . ' k + j =

(J10)o.'J2k+1 + i

1

( XO),

jE3,

'

f o r 3=(01uNN w i t h N b e i n g d e f i n e d by ( 0 . 3 5 ) ( r e c a l l t h a t X o : =M G ' ( X ) ) . N o t e t h a t in view of P r o p o s i t i o n s 6 . 3 . 8 a n d 3.3 t h e a s s u m p t i o n o n s u p p x+ i m p l i e s t h a t (X+)O.WjX[l

. + m[

is w e l l - d e f i n e d o n X + by (3.1)' w h e r e a s f o r f : =

t i o n fo,ch.X is w e l l - d e f i n e d o n Xo by t h e r e s u l t s o f J 10.11

5 S.(b)

x+IxO

t h e func-

so t h a t fo,W.I is

w e l l - d e f i n e d o n ( X O ) + by ( 5 . 6 0 ) w i t h w = w i . In f a c t , f s a t i s f i e s t h e a s s u m p t i o n s

of T h e o r e m . 5 . 3 7 w i t h X r e p l a c e d by X". verify t h e e x i s t e n c e o f f u n c t i o n s

x+

By way o f L e m m a 6 . 4 7 b e l o w w e s h a l l

having t h e a b o v e p r o p e r t i e s a n d , at t h e s a m e

t i m e , s h e d s o m e l i g h t o n t h e meaning o f t h e c o n d i t i o n (b.41).A t p r e s e n t , in t h e f o l l o w i n g l e m m a w e n o t e a very e a s y way o f o b t a i n i n g t h e d e s i r e d d a t a by first prescribing

Lemma 6.41. Let

x+ a n d t h e n c o n s t r u c t i n g t h e f u n c t i o n s $i d e p e n d i n g o n

x: V +

a? be any C'?' function such that supp

x+.

is a weakly

(M,Cl,+wCI-bounded subset of V and such that x = 1 on a neighbourhood of k e r M . Then the functions

266

V I . Q u a s i h o m o g e n e o u s A v e r a g e s o f D i s t r i b u t i o n s . Part 2

belong to C z f V ) and s a t i s f y ( 4 . 1 8 ) For J = M o ,

x 9 = x , and

( 4 . 1 9 ) , and ( 6 . 4 1 ) For X = V .

,7=Mo.

proOf. Since x - 1 near V \ V + one deduces w i t h the help of Proposition 6.3.A.(ii) that s ~ p p $is~a weakly (M,IO,ll)-bounded subset of V . By the assumption o n suppx t h i s means that s ~ p p $is ~a weakly M-bounded subset of V , i.e. J l i ~ C E ( V ) .

I t follows from (3.7) and (5.63) that ( - a M ~ ) o = ( - a ~ ) ~ o = ~ ~ M Hence, ol"+~l. s assumptions of Lemma 4.12.(ii) so that the function Jl = - a ~ x ) ~ + s a t i s f i ethe

(4.18) for J = I N ,

and ( 4 . l Y ) hold. Moreover, the conditions (3.7) and (.5.63),

again, imply that

x defined on X satisfying ( 4 . 1 8 ) ,

Functions $i and

( 4 . 1 9 ) , and (6.41) are obtained

as restrictions to X of the corresponding functions in Lemma 0.41. This is obvious

in view of Corollary 0 . S . We now come to the description of the distribution d defined by ( 0 . 4 0 )

Theorem 6.42. Let T C B ' I X ) and

be such that the condition ( b ) OF Theo-

kElNO

rem 6.37.( i ) is satisfied. Then k

(6.43)

T=

z($i( d ~ - m ) ' T,,,.,,,)

kc-+

Q;,,(xq,T).

i=O

Note that in case the Functions I), are given as in Lemma 4.l.?.(ii) then

-F.

We define N by (0.35). Let y J E C g ( X ) . Then by applying the formula ( 3 . 8 ) '

to ( ' ~ ~ , , , ~ , x l ~ + . m * , Oinstead , o ~ ) of ( q , f , C , m , w )and making use of the equality

(xIx+)o,w=xO,w

we obtain i n view of (5.63) for arbitrary xEC;(X)

that k ( 6.45 )

S

.

( ~ ' ~ m * , m k ) r n +=, jw= ~ O ( - l ) ' x O , w q m j ' ~ m + . w ki- N

and qEN0

267

6.d Characterizing ( A l m o s t ) Q u a s i h o m o g e n e o u s D i s t r i b u t i o n s

Since for every i € { O ) u N k s u p p J l i is a weakly M-bounded s u b s e t of X it f o l l o w s

in view of T h e o r e m 5.37.(ii) t h a t Jli qm*,wk e x t e n d s to a f u n c t i o n in C g ( X + ) , again d e n o t e d by $ i p n , * , w k ,so t h a t by k

a f u n c t i o n in C T ( X + ) is well-defined. By ( 3 . 7 ) a n d (5.63) a n d by (0.45) it follows that k

where k

k

and k

N

By (1.38) a n d (4.18) a n d by changing t h e o r d e r of s u m m a t i o n a n d s u b s t i t u t i n g

J = k - j w e obtain

Since by ( 0 . 3 9 ) t h e sun1 in s q u a r e b r a c k e t s is equal to ( - l ) k S o k - J

i t f o l l o w s by

(4.19) t h a t A = p m + , , d k . Moreover. again by (1.38) a n d ( 4 . 1 8 ) a n d by ( 6 . 3 0 ) w e

concl ude t h a t

Since by Corollary 5.47 t h e s u p p o r t of Q,,,*q is a closed s u b s e t of V being contained in X" w e d e d u c e t h a t

xJ, Q m * q e x t e n d s

to a Cm function h : V-+@

( u n c h a n g e d ) s u p p o r t contained in XO. Since by Lemma 6.23 s u p p

Qm.v

,f o r s o m e c o m p a c t s u b s e t L of

xJ,

tained in L M , , o , ,

XO

a n d since s u p p

with is con-

is a weak-

ly ( M , [ l , + c o C ) - b o u n d e d s u b s e t of X it f o l l o w s t h a t s u p p ~ , , , n s u p p Q , ~ q c o m p a c t s u b s e t of X . i . e . h b e l o n g s to C;(X''). (applied to q = Q m * q

I xo

),

is a

Finally, w e d e d u c e f r o m ( 3 . 8 ) '

(1.38), and ( 6 . 4 1 ) t h a t t h e equality

268

VI. Q u a s i h o m o g e n e o u s A v e r a g e s OF D i s t r i b u t i o n s . Part 2

is valid o n ( X o ) + . Since by Remark 5.60 the support of hm*,wk is contained in

(Xo)+, a s well, equality holds on the whole of V + . Consequently, in view of what was proved above we see that the functions

@m*,wk

and rpm+,wk-

h m * , w k coincide

o n X + . Since by Remark S.60.(i) both functions vanish on V + \ X, that the function @ : =

'p

-@ -h

satisfies

@m*,cdk

we conclude

0 . Hence, by the condition ( b )

of Theorem6.37.(i) we conclude that < T , ( I ) > = O ,i.e.

< T , r p >= < T , @ > + JIo

1

o n L . C o n s e q u e n t l y , by i n d u c t i o n w e can find a s e q u e n c e of f u n c t i o n s x ~ E C ~ ( X ) ,

4 E No, s u c h t h a t J

(6.47)

Y

( - I + + ~ +o,cdJ-e+l~p

e=o

s ~ Jo n

L ,

JcNo.

Finally, w e observe t h a t by C o r o l l a r y 6.28 t h e n u m b e r N d e f i n e d by ( 6 . 3 s ) h a s t h e p r o p e r t y t h a t ( 3 M - m ) N " c = 0 f o r every d i s t r i b u t i o n c E a ' ( X ) w i t h s u p p o r t c o n t a i n e d in X \ X + which is a l m o s t q u a s i h o m o g e n e o u s of degree m . E m p l o y i n g R e m a r k 6 . 4 3 w i t h ( u , J l o , k ) r e p l a c e d by ( X e v , J l , O ) a n d a p p l y i n g ( 3 M - m )

e w e obtain

in view of ( 4 . 1 0 ) (see P r o p o s i t i o n 6.16 1 t h a t N-8

[(3M-m)e ( x p v - J l ( x p ~ ) , ) I , =

C (-l)'(3M-m)''eQ~(Jlo,~i+,~p~). j=O

S u m m i n g o v e r 4 , s u b s t i t u t i n g J = j + 4 , a n d c h a n g i n g t h e order of s u m m a t i o n we

see t h a t t h e d i s t r i b u t i o n

6.d c h a r a c t e r i z i n g

271

( A l m o s t ) Quasihomogeneous Distributions

By ( 6 . 4 7 ) this is equal t o Q k v , a s desired.

(ii).

We now s u p p o s e t h a t X = V , t h a t d is temperate, and t h a t x = x , . Then b\

Lemma6.31.(iii) t h e distribution v can b e chosen t o be t e m p e r a t e with s u p p o r t contained in

n c l . 2 1so t h a t

we may s u p p o s e t h a t L = 0 , 1 , 2 , 3 c .

By (1.77) we have

x < 3 d o n L f o r a suitable c o n s t a n t d > O . We set a : = e 3 d and Y : = x - ' ( l a , a + l C ) . Then { t E l O . + m C ; M t ( L ) n Y

#@I

C Ce,+aC. Hence we c h o o s e

f o r J = l a , a + l C .Then, a s b e f o r e , we deduce t h a t

+o,,a,l2

+ a s in

Lemma 0 . 3 8

1 o n L so t h a t t h e deri-

a r e bounded o n L . a s well. Consequently, t h e functions

vatives of l / + o , u l

xp

satisfying ( 0 . 4 7 ) can be c h o s e n in s u c h a way t h a t their derivatives a r e bounded on L . It follows by Proposition 6.22 t h a t t h e distribution u defined i n (6.48)is temperate. Since its s u p p o r t is contained in L u x - ' ( C a . a + l I ) w e s e e t h a t u has t h e desired properties.

End of the Proof of Theorem 6.37. ( b ) * ( c ) : By Theorem ( ~ . 4 2and Proposition 6 . 2 4 t h e distribution d defined by ( 6 . 4 0 ) is a l m o s t quasihomogeneous of degree m , its s u p p o r t being contained in X \ X + . Consequently. t h e equation ( 6 . 3 3 ) is valid f o r k

u=

(6.49)

+i

(3, - m ) ' T .

i=O

Since t h e implication "(c) * ( a ) " follows from Proposition 6.35 t h e proof of (il is c o m p l e t e .

(ii).

w

I f X = X t h e n by Lemma 6 . 4 4 we find v E D ~ ( X such ) t h a t d = v m . Since by

(4.10) and (6.30) we have v , = ( ( - a M + m )

k

v),,,~

t h e equation

T=

holds

for

(iii). I f T -

is t e m p e r a t e t h e n in view of Corollary 6.3') we can achieve t h a t t h e

distributions

Jli (3,

- m ) ' T . O S i < k , belong to Y'LCV). Hence, by Proposition 6.22

272

VI. Q u a s i h o m o g e n e o u s A v e r a g e s o f D i s t r i b u t i o n s . Part 2

t h e distribution d defined by ( 6 . 4 0 ) is t e m p e r a t e , a s well. Then by L e m m a 6 . 4 4 t h e distribution v can b e chosen to belong to V & ( V ) so t h a t t h e distribution u defined by ( 6 . 4 9 ) ' lies in V & ( V ) , a s well.

( i v ) : Suppose t h a t h.lfl.(v) ua

(6.33) is valid and t h a t T is @-invariant. Since by Propositions =

belongs to a L ( X ) and satisfies t h e equation

it follows by Proposition 2.64.(iii) t h a t T=T,= ( ~ m ) ~ , ( , , ~ + Here d @ . um is @ - i n variant by Proposition 2.64. Moreover, by t h e s a m e Proposition d m is @-invariant with s u p p o r t contained in X \ X + ( n o t e t h a t ker M is G - i n v a r i a n t ) , and by Remark 2.67.(ii) d,

is a l m o s t quasihomogeneous of degree m . Note t h a t d = O implies

dm=O. Finally, n o t e t h a t f o r arbitrary b t f > O there a r e b ' > is contained o n

f'>

0 such that ( f l c c , b , ) c

Hence, in view of Proposition 2 . 6 4 . ( i i ) if u belongs to

nCc.,bql.

VLCV) so does ~ ~ 3 , .

We end this section with a few supplements to Theorem 6 . 4 2 . For t h e f i r s t t w o supposing t h a t X = V we ask under which conditions t h e c o n s t i t u t i n g p a r t s of t h e formula ( 6 . 4 3 ) a r e temperate distributions. It t u r n s o u t t h a t under special hypotheses o n M every a l m o s t quasihomogeneous distribution o n V is t e m p e r a t e . This is t h e c o n t e n t s of t h e following consequence of Theorem 6.42 ( o r r a t h e r Proposition 4.13) and Corollary 0 . 3 0 .

Theorem 6.45. Suppose that ker M = 101. Then every distribution T €a'(V ) satisfying one (and hence e a c h ) o f the conditions of Theorem 6 . 3 7 . l i ) is t e m p e r a t e .

mf.In view of Remarks 0.14 and 3.23 t h e assumption o n M implies t h a t Dh(V)

=&'tc)= Y h ( V ) .

Hence Proposition 6.22 implies t h a t t h e distributions

C + i ( d M - m ) i T ) , , , , c s , ka r e temperate. Since, moreover, t h e s u p p o r t of t h e distribution d defined by (6.40)is contained in V \ V , = (01, i.e. d 6 & ' ( V ) it f o l l o w s f r o m ( 6 . 4 0 ) t h a t T is temperate, a s well, indeed. rn

However, i n c a s e X = V and k e r M f 10) t h e n , of c o u r s e , T is n o t necessarily t e m perate. If i t is so by a s s u m p t i o n , then each summand o n t h e right-hand side of ( 6 . 4 3 ) should belong to Y ' ( V ) , a s well. Establishing t h i s requires special choices

of

+ and xJ, which

a r e possible by Lemmata 5.53 and 6.41

:

273

6.d Characterizing ( A l m o s t ) Q ua siho mo g e ne o us Distributions

Remark 6.46. Suppose that X = V , that the functions +bi, O s i l k , have bounded derivatives and M-temperate support, and that

xJ.

has bounded derivatives and

(M,Cl,+wC)-temperatesupport. Let T be a distribution on V satiscving one (and hence each) o f the conditions o f Theorem 6 . 3 7 . ( i ) . I f T i s temperate then so are n

(Gi T ) m , c d k ,( x a x + TI

A

and ( x f f ~T i)

mf. This is a consequence

.

for O < i < k and m621:+

of Propositions 0.22 and 6.24.(iii) & ( i v ) . rn

To prepare for the final result of this section we come to the precise lemma about t h e existence and the properties of the functions

q,, alluded

to in the

t e x t preceding Lemma 0.41.

Lemma 6.47. Let

C be an) C'"function such that supp y, is a weak?,

,y : X +

( M , t l , + w t ) - b o u n d e d subset o f X . and let h € N , .

Then the following assertions

hold.

(i)

The following conditions are equivalent: l a ) there e\ists

a function

,Y+ 6

C " > ( X ) satiscling ( 6 . 4 1 ) f o r 3 = l O l u l N ,

such that the support o f X - X ~ is a weak?, ( M . Cl.+wt)-bounded subset o f X and. at the same time. an M-bounded subset of ( X o ) + :

lii)

The following assertions are equivalent:

( a ) there exists a function such that the support o f

x - xJ. is

( 6 ) for some (resp. ever) 1

x+

EC"'IX) satisfying ( 6 . 4 1 ) for .J=10) u N h

a weaklj. M - bounded subset o f X

-

i,h

6

C G ( X ) the function

x+

16.411 with .J= ( O j u N h f o r a suitable choice o f the functions

Note that the existence of functions

; w

:=

x - +b

satisfies

+bi;

x satisfying the condition

( c ) in

( i i ) is

established by t h e assertion ( i ) . Of course, if (5.17) holds, i.e. if X = Xo then ( c ) is trivially satisfied, and the formulation of Lemma 6.47 becomes much simpler.

proOf. Throughout the proof we set W : = ( X o ) + and f : =

XIXO.

274

VI. Q u a s i h o m o g e n e o u s A v e r a g e s of D i s t r i b u t i o n s . P a r t 2

(i) :

x'. From Theorem 5.37.(iii) we

(x+Ixo)o,wk

deduce that the function

is

almost quasihomogeneous of degree 0 such that ( - ~ M ) ~ " ( X + ~ ~ O ) O . =C Qo(x+lxo)lw J~

= X+"Mo(,

.

Since by Proposition 3.13 the function ( J , o ) o , , , ~ ~ + is almost quasihomogeneous of degree 0 satisfying (-aM)k+t( + O ) O , C J ~= ~ +(dr'O)O.wk ~

t h e condition ( b ) is a consequence of (6.41) for j = O and ( 4 . 1 0 ) .

E". We define

q : W - - - ; , @ by

q : = fo,"'k+~l-(+O)0,'J2k+l+hlW'

By (5.63) and (3.11) we have

(-i3M)k'h'1

q=xoMol,-

(+c))o,c,,kIw.By the condi-

t i o n s ( b ) and (4.19) we conclude that the latter function vanishes identically, i.e. q is almost quasihomogeneous of degree 0 and of order 5 j o : = k + h , I n other words, the condition ( b ) of Theorem3.48 is satisfied for A = ( O ) and q u , i = ( - d M M ) i o - i q with X replaced by W . Before going to apply Theorem 3.48 we have to construct a suitable subset Y of

X . To t h i s end we fix a € I O , + c o [ . Then

La : = ( x E X

;

( x ) 5 a min ( 1. dist( M o ( x ) , V \ X ) I )

subset of X

is a closed

contained in

X".

By

Lemma0.7

La

is a

weakl)

( M , Cl,+aC)-bounded subset of X . Moreover, La is a neighbourhood of M , ( X ) n X ; 0

more precisely, i f

E E

I0.aC then La is an open neighbourhood of L , .

Note that

0

(La\L,)M

= W . Consequently, we can apply Theorem 3.48 to the data described

above and to Y : =

La\

L, and obtain a function 'i"ECG( W ) with support contained

0

i n La\ L, such that

q n , j for 0 C j Cjo.

Since X n ( L a \ L,) C W the support of Y is a closed subset of X . Hence the -,

+

extension

of Y to X defined by

$Ix,,

: - 0 is a Cm function with the same

&,

support as '4'.

In particular, supp J, is a weakly ( M , C 1 , +aC)-bounded subset of r_

X and an M-bounded subset o f W. Consequently, t h e function x , , , : = x - $ has the desired properties.

m:f a ) +Ic). w

w

If

+:=x-x+

belongs to CGCX) then by Remark 0.12 and Proposi-

tion 3.13 +o is quasihomogeneous of degree 0 , and in view of (0.41) fo,cL,k+,l is

275

6.d Characterizing ( A l m o s t ) Quasihomogeneous Distributions

m

e x t e n d e d b y t h e f u n c t i o n g : = ( J ) ) o , ' , , ~ + +~ ( + o ) o , , , ~w~h i+ c h~by + ~(3.11) a n d ( 4 . 1 9 ) k+h+l

satisfies t h e e q u a t i o n ( - d ~ )

grl.

,

I c l j f b l . Let g E Cm( X + ) be a n e x t e n s i o n of fO,c,,k+ having t h e p r o p e r t i e s a s s e r *

I "

ted in ( c ) . T h e n f o r a n y + E C G ( X ) t h e f u n c t i o n q : = g - ( + ) o , w k + hb e l o n g s to C m ( X + ) a n d - b y t h e a s s u m p t i o n s o n g a n d by (3.11) - is a l m o s t q u a s i h o m o g e n e o u s of d e g r e e 0 a n d of order 5 k + h + l s a t i s f y i n g ( - d M ) k ' h ' l q - 1 .

Hence, by Theo-

r e m 3 . 4 8 a n d R e m a r k 6.12 o n e f i n d s a f u n c t i o n + E C G ( X ) s u c h t h a t In p a r t i c u l a r , by (5.63) w e have + 0 - 1 . Defining 4.12.(ii) t h a t ( 4 . 1 8 ) f o r

2=Noand

+i

+O,L,k+h+l= q

.

by ( 4 . 1 6 ) w e see f r o m L e m m a

(4.19) a r e v a l i d . M o r e o v e r , s i n c e J,o=( - d M ) k +

t h e c o n d i t i o n ( 5 . 0 3 ) s h o w s t h a t q = ( + o ) o , c , , 2 k + h + l . Finally, s e t t i n g

x+:= x - J , we

derive f r o m (5.63) t h a t

a n d s i n c e t h e t e r m in l a r g e b r a c k e t s is e q u a l to 41,

3 = ( 0 )u N,

by a n o t h e r a p p l i c a t i o n o f ( S . 0 3 ) .

-

r
1~+ ~h+is

t h e desired extension

fo~ldk+h.

fc) * ( a ) .

,

Let g E Cm( X + ) b e a n e x t e n s i o n o f fo.lA,k having t h e p r o p e r t i e s a s s e r -

ted in ( c ) . T h e n by t h e a s s u m p t i o n s o n g a n d by (5.63) a n d ( 4 . 1 0 ) t h e f u n c t i o n q : = g - ( J ) O ) 0 , c , , 2 k +is l t ahl m o s t q u a s i h o m o g e n e o u s of d e g r e e 0 a n d of o r d e r 5 k + h .

-

H e n c e , by T h e o r e m 3.48 a n d Remark 0.12 w e o b t a i n a f u n c t i o n $EC;(X) that

(+)O,wk+h=

q

Setting x+:=

x-

such

r>

J, w e see t h a t t h e t e r m in l a r g e b r a c k e t s in

e q u a t i o n ( 6 . 5 0 ) is e q u a l to ( J , ~ ) O , . , ~ ~ + , + ~ ,HI e~n.c e t h e c o n d i t i o n ( 6 . 4 1 ) f o r

3 = (O)uN,

f o l l o w s f r o m ( 0 . 5 0 ) by m e a n s of ( S . 0 3 ) .

H o w does t h e s u m o n t h e r i g h t - h a n d side o f ( 6 . 4 3 ) d e p e n d o n t h e c h o i c e of t h e functions

+i?

In view of ( 0 . 4 3 ) a n d ( b . 4 1 ) it o n l y d e p e n d s o n

Is it a l w a y s p o s s i b l e to c h o o s e

(+O)0,L,2k+N+1.

Gi in s u c h a way t h a t

k

Of c o u r s e , if T f O t h e n a n e c e s s a r y c o n d i t i o n is t h a t s u p p T is n o t c o n t a i n e d in

X \ X + f o r o t h e r w i s e w e have J,i ( d M - m ) ' T = 0 .

276

V1. Q u a s i h o m o g e n e o u s A v e r a g e s OF D i s t r i b u t i o n s . Part 2

Roporitlon 6.48. Suppose that

m*E

211M). Let TE B ' ( X ) satisfy one land hence

each) of the conditions o f Theorem 6.37.li). Then the equation (6.43)' is valid f o r some choice of the functions

GI i f and only i f f or some (resp, every)

y, E C?X)

satistving the assumption and the condition (c1 OF Lemma 6.47. lii) there exists

$E CElX I

such that Q;,,lxT) = QA,l$T).

16.51)

By L e m m a 6 . 4 7 . ( i i ) w e c a n c h o o s e xQ in s u c h a way t h a t

Proof.

" j " :

r.

JI:=x-x+

b e l o n g s to C E C X ) . If ( b . 4 3 ) ' is valid t h e n ( 6 . 4 3 ) s h o w s t h a t Q&(x,,T) = 0 , i.e. h o Ids.

( 6.51)

x: By L e m m a O . 4 7 . ( i i ) . ( b ) w e s a t i s f i e s (6.41) f o r 3 = (0) uN,.

can choose

JIi

*

in s u c h a way t h a t x + : = x - J '

Since (6.51) m e a n s t h a t Q;,,(x+T) = O t h e a s s e r -

t i o n f o l l o w s by ( 6 . 4 3 ) , a g a i n .

(e) Solving Che Eyualion ( 3 , - m ) S = I

Theorem 6.49. Suppose that (6.52)

(a) md(-X(M)-p),

or

l b ) x"=X.

Then For ever>' TE D' IX) there esists a solution S E B ' ( X )o f the equation (6.53)

(d,,,,-m)S=

T

having the following properties: li)

let r 6 N o u {a); i f T is induced by a C' function then so is S I X + :

( i i ) suppose that X = V ; i f T is temperate so is S; ( i i i ) let 8 satisfj, the assumptions o f Remark 2.67. l i i l ; i f T is @-invariant so is S.

The main s t e p in t h e p r o o f is s u m m a r i z e d in t h e f o l l o w i n g l e m m a . W e f i x a function X E Cm(X) such t h a t

(6.54)

(a) supp x

ci

- l [~ O , a [ ) ,

( b ) x = 1 on ; - ' ( C O , E I )

6.e

277

S o l v l n g t h e Equation ( 3 M - r n ) S = T

for s o m e a > € > O . Then t h e s u p p o r t of 1 - x is contained in x - ' ( C E , + ~ C ) , a n d in w

view o f Lemma 6.7 it makes s e n s e to define

T ,

by ( 4 . 2 4 ) . In C h a p t e r 7 w e re-

quire t h e more general distributions

&

Lemma 6.50. For every T E B ' ( X ) b y (6.551 a sequence o f distribution Tm,k~ . f i l ' ( X ) . k € N o , i s well-defined having the Following properties:

-

z

(i)

( 3 M - m ) Tm,k+l = Trn,k;

(ii)

(3M-m)

k+l

-

-

Tm,k = ( 3 M - m ) T m = T - Q ; , l ( x T ) ; w

(iiil let r € N o u ( w l ; if TIx

+

is induced bj a C' Function so is T I r l , k I X ; * -
= < d , >.

If th e restriction of d to Q,*

( E m # ) vanishes identically then

6.57) implies t h a t

d = O . On t h e o t h e r hand, s u p p o s e t h a t < * , Q , * q @ i > = O . Since for arbitrary

y e X \ X + and a € U A , t h e distribution d =a;(

(-a)"S,)@

belongs to Q;(E)@

( w h e r e 6 , d e n o t e s t h e Dirac distribution a t y ) it then follow s by (6.57) and by Propositions S . 4 8 . ( v ) and 5.45 t h a t

281

6 . f Duality Brackets

o = k o . M o r e o v e r , for N : = N ~ k e r C r , , ~ : = m a s { o r dC~€ fQc,~' ,;, - ~ ( E ) (q~c, = 0 } t h e Following assertions hold: (iii) N _ < N ( m ' + e ) : ( i v ) F ~ 2- [ci~:= fdM - i n + t ) ' ( k e r C , , , ) f o r e v e r y i C N N + , : Iv )

Wk = WN +

Proof. fiii): In

(fk + 1

view of kerC,,C

m:If cE ker C, so that

cEkerC,

for e v e r y k EN0 ; in particular, Wk = WN for ever). k 2 N .

Q A - e ( E ) t h i s is a consequence of Corollary 6 . 2 8 .

then by Corollary 2.36.(ii I we have q ( a M- m + e ) c =( a M- m ) ( q c )= 0

6,C 6 i - l f o r every i E N . On the other hand, we fix ielNN+' and choose s u c h that ( 3 M - m + e ) i - 1 c E C S i ,i.e. ( a M - m + e ) ' - ' c = ( d M - r n + e ) ' b for

some bEkerC,.

N+l-i

Applying ( d M - m + 4 ) N

to both sides of t h i s equation we

deduce that ( d M - m + o ) c = O . Hence, ordM(c)"-1.

In other words, if cEkerC,

is such that o r d M ( c ) = N then ( d M - m + 4 ) ' - ' c does not belong t o E i . Since by the definition of N there exists cEker C, ( i v ) is complete.

such that OrdMM(C)= N the proof of

7.a

291

Multiplication Equations

For t h e proof of t h e o t h e r a s s e r t i o n s w e fix k e N , a n d REWk and c h o o s e S E k e r B m , k s u c h t h a t R = ( d M - m + 4 ) k ' 1 S . Note t h a t S € x i n - e , k ( E ) e a n d q s = 0 .

(i):

Since T : = ( d M - m + @ s ) b e l o n g s to x L - g , k - l ( E ) ~a n d - b y Corollary 2.36.(ii) -

satisfies

q T = ( a M - m ) ( q s ) = o w e c o n c l u d e t h a t R = ( d M - m + t ) k T b e l o n g s to

W k - i , indeed. For t h e n e x t steps w e f i x , in addition, x and

x

a s in $ 6 . ( e ) s u p p o s i n g t h a t in

case E = Y ( V ) x be equal to x + . Moreover, w e c h o o s e u E { O , m a x ( N - k , O ) } . By Proposition 2.6S.(ii) a n d Lemma 6 . 5 0 . ( i v ) and by (2.39). Lemma 6 . 5 0 . ( i i ) a n d Pro?"

position 2.64.(iii) t h e distribution T : = ( S m - e , " ) m s a t i s f i e s t h e e q u a t i o n s ,."

( a ) q T = ( (qS)m,")s-= 0 ,

(7.13)

and

( b ) ( d M - m + 4 ) " + 1 T =S - c ,

w h e r e c : = Q L - g ( X S ) c y . I t f o l l o w s f r o m (7.13.b) a n d L e m m a 6 . 5 0 . ( v ) t h a t T lies in ~ ~ - e , k + , + l ( E ) BMoreover, . we n o t e t h a t by t h e a s s e r t i o n s ( v ) a n d ( i i ) of Proposition 6.26 w e have q c = Q k ( x q S ) Q = 0 .

0: Now,

s u p p o s e t h a t k 2 N . Then u = O a n d ( d M - m + 4 ) k + 1 c = 0so t h a t by (7.13.b) ( d M - m + 4 ) k ' 2 T = ( d M - m + 4 ) k + 1 S=

In view of (7.13.a) t h i s s h o w s t h a t

R.

R b e l o n g s to

wk+1.

H e n c e in view of ( i )

w e have proved t h a t W k t l = w k f o r every k t N . From t h i s it f o l l o w s by induction that

wk= WN

for every k 2 N .

O n t h e o t h e r h a n d , in case k < N we d e d u c e f r o m (7.13.b) f o r u = N - k and f r o m (7.13.a) t h a t

R = (aM-m+P)k+'S = (dM-m+P)"'T+

( 3 M - m + P ) k + 1 cE W N + C S k + l .

Hence, in t h i s case w e have proved t h a t wk C WN + 6 k + l . Since in view of t h e inclusions

"

k e r C , C k e r B,"

a n d ( i ) t h e inverse inclusion is obvious t h e proof

of ( v ) is c o m p l e t e . (iil: W e s u p p o s e t h a t k E N N a n d t h a t W k = w k - l . I t s u f f i c e s to s h o w t h a t wk+1= w k ,

i.e. w e have to s h o w t h a t t h e distribution REW, fixed above b e l o n g s

to W k + i . Now, s i n c e in case E = Y ( V ) by P r o p o s i t i o n 6 . 2 4 . ( i i i ) a n d by t h e choice

of

x t h e d i s t r i b u t i o n c defined

above b e l o n g s to Y ' ( V ) , Proposition 6.26.(iii) s h o w s

t h a t c E X ~ l + ~ , k - i (C~o n)s ~e q.u e n t l y , ( a M - m + e ) k c b e l o n g s to wk-1 a n d hence, by a s s u m p t i o n , to w k , i.e. w e find U € X ; - e , k ( E ) e

such that ( a M - m + k ? ) k c =

( a M - m + P ) k + l U a n d q U = O . I t f o l l o w s f r o m (7.13.b) for u = O t h a t

292

VII. Solvability of Quasihomogeneous Equations

= (dM-m+4)k'2T+ (3M-m+4)k+'c = (3M-m+4)k+2(T+U).

R = (dM-m+k')k''S

In view o f (7.13.a) t h i s s h o w s t h a t RE Wk+, , a s desired. Let u s n o t e a f e w c o n s e q u e n c e s o f Remark 7 . 8 . F i r s t o f a l l , Remark 7 . 8 l e a d s to a n alternative proof of t h e f a c t t h a t t h e conditions (7.10); are equivalent f o r a l l Moreover, i f in a c o n c r e t e s i t u a t i o n o n e sets o u t to verify t h e validity of

kEN,.

(7.10); o n e s h o u l d work w i t h k as s m a l l a s possible. On t h e o t h e r h a n d , i f t h e aim is to s h o w t h a t (7.10); is f a l s e it is m o s t promising to d e a l w i t h t h e case k > N(m*+ 4 ) . However, I c a n n o t produce e x a m p l e s having t h e p r o p e r t y t h a t t h e is positive or t h a t W , # W k - ,

c o n s t a n t N ( k e r C,)

for s o m e ( r e s p . e v e r y ) kEN,.

N e x t w e a r e going to see t h a t t h e conditions of Theorem 7.5 a r e n o t a l w a y s valid. T h e first s t e p is to see t h a t t h e condition (7.10);

b e c o m e s s i m p l e r if ( 6 . 2 8 ) h o l d s :

Remark 7.9. Suppose that q -'(O) n X + = @ . Then (i)

M is s emi -s i mple provided that t # O ;

f i i ) k er B,, = ker C,,

;

( i ii ) A, - e ( k e r B ,

i s trivial.

)

proof. m: Let Z be any M-irreducible ( a n d hence, in particular, M-cyclic) subs p a c e Z of G M ( o + ) . By t h e a s s u m p t i o n " X o # @ " o n e f i n d s x o E k e r M s u c h t h a t GM(rs+)+ x o is contained in X ( s e e also ( 6 . 2 2 ) ) . Then t h e polynomial f u n c t i o n qz:Z-

C , z H q ( z + x o ) , is quasihomogeneous of d e g r e e

P a n d of t y p e M,.

If

d i m ' Z > 1 t h e n Proposition 1.44 implies t h a t q z h a s a non-trivial z e r o . Since t h i s c o n t r a d i c t s ( 6 . 2 8 ) it f o l l o w s t h a t d i m ' Z = 1 . T h i s m e a n s t h a t M is s e m i - s i m p l e .

f i i ) . If

4 = 0 t h e n q has n o z e r o s a t all so t h a t k e r B,=

(0). If P # 0 t h e n by ( i ) w e

have N ( m * + P ) = 0 . And s i n c e t h e a s s u m p t i o n o n q implies t h a t s u p p S C X \ X + f o r every SE k e r B,

t h e a s s e r t i o n f o l l o w s f r o m Proposition 6 . 2 7 .

f i i i ) . In view of C o r o l l a r y h . 2 8 t h i s f o l l o w s f r o m ( i ) a n d ( i i ) .

As a first information a b o u t t h e l e f t - h a n d side of (7.10); w e n o t e

Remark 7.10. Suppose that G = { I d , I . Then dim kerC,, 2 l X A + + e l - lZ;,*l. ker M = /01 then equality holds.

If

293

7.a M u l t i p l i c a t i o n E q u a t i o n s

proOf. W e set A : =

U;*+@

if y e U ' \ U ;

the fact that q"'(y)=O of c o m p l e x n u m b e r s c,

a n d fix y e X

\ X , . From t h e Leibniz r u l e a n d f r o m

it f o l l o w s f o r every family c = ( c , ) , € *

t h a t t h e distribution

s,:= 1 c,(-a)as, aeA

s a t i s f i e s t h e e q u a t i o n q S c = 0 if a n d o n l y if c is a s o l u t i o n of t h e s y s t e m of linear equations

S i n c e t h e d i m e n s i o n of t h e s o l u t i o n s p a c e o f t h i s s y s t e m is n o t s m a l l e r t h a n IAl

- IBI

the first assertion follows.

If k e r M = ( 0 ) t h e n t h e s p a c e s Q f ( l P ( V ) ) are f i n i t e - d i m e n s i o n a l so t h a t P r o p o s i t i o n 7.2 implies t h a t dim kerC,=d,-p-d,

w h e r e f o r j € ( m - P , m ) di : = d i m Q I ( Y P ( V ) )

is e q u a l to IU,*,I.

Proposltlon 7.11. Suppose that q -'(O) n X , = @ and G = I I d ,

I.

Then each of the

conditions of Theorem 7.5 is violated if and on/-).if

proof. In view

of w h a t w a s said in t h e t e x t f o l l o w i n g t h e p r o o f of T h e o r e m 7.3

w e may a s s u m e t h a t ( 7 . 1 4 . a ) h o l d s . In view o f Remark 7 . 9 . ( i i i ) t h e c o n d i t i o n (7.10); is valid if a n d o n l y if k e r C , , = (0).I f e = O t h e n t h e a s s u m p t i o n ( 6 . 2 8 ) i m p l i e s

t h a t q - ' ( O ) = 8 so t h a t C,

is injective, i n d e e d . H e n c e w e s u p p o s e t h a t ( 7 . 1 4 . b )

is valid, as w e l l .

If ( 7 . 1 4 . ~ )h o l d s it s u f f i c e s to s h o w t h a t

I#i,*+pI

> lU;*I

for t h e n R e m a r k 7.10

y i e l d s t h a t k e r C, # ( 0 ) . I n d e e d , if m* @ U ( M ) t h e n t h e desired e s t i m a t e is trivial in view of ( 7 . 1 4 . a ) . And if d i m ' G M ( o + ) > 2 t h e n (6.28) a n d ( 7 . 1 4 . b ) i m p l y t h a t

1%;

I2

2 , i.e. w e c a n fix B , y € % ; s u c h t h a t P # y so t h a t in case m* b e l o n g s to

U ( M ) t h e injective m a p s U k *+% ,;

+ @

d e f i n e d by a H a + p a n d a H a + y , re-

s p e c t i v e l y , have d i s t i n c t i m a g e s , a n d t h e desired e s t i m a t e f o l l o w s a g a i n . Finally, w e s u p p o s e t h a t ( 7 . 1 4 . ~ is ) f a l s e . T h e n by Remark 1.40 IU;++oI

= I = lUk*l

so t h a t in case d i m ' V = 1 R e m a r k 7.10 d i r e c t l y i m p l i e s t h a t k e r C , = ( 0 ) . T h i s l e a v e s u s w i t h t h e c a s e d i m ' G M ( a + ) = l < dim'V. Writing V = I K x V o w h e r e I K x ( 0 ) = G M ( o + ) and (0) x V o = k e r M w e see t h a t Xo is of t h e f o r m X o = IK x X' w h e r e X' is a

294

VII. Solvability o f Quasihomogeneous Equations

non-empty open s u b s e t of Vo. By Proposition 5.45 we find g E C m ( X ' ) s u c h t h a t q ( x ) = x y g ( x ' ) , x = ( x , , x ' ) E X o , where v is t h e unique e l e m e n t of

Xi.

Now let

d c Q A - t ( 3 ( X ) ) . Then by Propositions 6 . 2 7 . ( i ) and 6.29 t h e r e e x i s t s a distribution

[(-a)""s]@u

U E D ' ( X ' ) s u c h t h a t dl,,=

where 6 d e n o t e s t h e Dirac distribution

on IK and where p is t h e unique element of 3 ; . If d E kerC,,, t h e n

Since by t h e Leibniz rule we have

.;

(-a)('+us

=

(-a)ps #

0

i t f o l l o w s t h a t g u = 0 . Since by ( 6 . 2 8 ) g has no zeros this implies t h a t u = O .

Since by Proposition 6 . 2 4 . ( i i ) t h e s u p p o r t of d is contained in Xo

i.e. dl,,=O.

it f o l l o w s t h a t d = O . Hence kerC,=

(0).

In order to analyze t h e condition (7.10);

f u r t h e r we a r e now going t o employ

t h e duality brackets of section 6 . ( f ) and make use of t h e locally convex p r o p e r t i e s of

' U z , k ( E m ) . In particular, we a r e going to describe t h e polar s e t of A,,-@(kerB,,)

and of kerC,,, . To this end we first c o m p u t e t h e t r a n s p o s e s of t h e o p e r a t o r s A,,, ,Bin, and C,

(introduced i n Notation 7.7 ) with respect t o t h e duality b r a c k e t s

defined in section 6 . ( f ) .

A simple technical remark is i n order: i f v E C T ( X ) then by Remark S.60 we have s u p p Qc ,p

of

Q,,v

C

X and s u p p cpm,wk C X,

(resp.

(P,,,~.,~)

function o n X ( r e s p . X , ) Qc ,p

(resp.

vln,c,,k)and

to V by qIv,,

;

a l t h o u g h , actually, t h e domain of definition

is the whole of V ( r e s p . V,) we may consider it a s a ;

i n o t h e r words, we shall n o t always distinguish between

(Qmcp)IX

(resp.

~ P , , , , ~ +~) I. , Moreover,

if q is extended

: = 0t h e n t h e equations involving q appearing below can also be

read a s equations f o r functions o n V ( r e s p . V ,

).

This has t o be taken into a c c o u n t ,

i n particular, when Proposition 5 . 4 8 . ( i i ) o r Theorem S.37.(vi) a r e a p p l i e d .

295

7 . a Multiplication Equations

M.(i): We

let R€Q,+(E,*)

and fix T E E such t h a t R = Q , * ( q @ * ) . Then by

( 5 . 6 3 ) , (4.10), a n d (2.39) we have

(ii): For

arbitrary T E X A - t , k ( E ) < H and f p E E we deduce by ( 0 . 5 8 ) and by Proposi-

tions Z.bI.(ii) and Theorem 5.37.(vi) t h a t

(iii):For arbitrary d € Q L , - e ( E ) w and cpEEgi

position 5 . 4 8 . ( i i

)

we conclude by ( 0 . 5 7 ) and by Pro-

that

The relevant topological properties of t h e o p e r a t o r s introduced in Notation 7.7 a r e s t a t e d in

Proporltlon 7.13. The operators A,.

B,,, , and C,, are (surjective) weak homomor-

phisms, the images of their transposes being closed.

w. We d e n o t e by

u : U+

W any o n e of t h e o p e r a t o r s tA,,

tB,,

or

k,.

Since X , is d e n s e in X and in view of ( 7 . 6 ) u is injective. By T h e o r e m s 6.49 and 7.3 and Proposition 7 . 2 , respectively, its t r a n s p o s e 'u is surjective.

The case " E = Y ( V I " : Here by Propositions 5.52 and 5.61 t h e s p a c e s U and W a r e FrCchet s p a c e s ; and t h e assertion follows by Proposition V1.2.2 i n d e Wilde C41.

The case "E = d ) ( X ) " :Here by Propositions 5.51 and 5.59 U is a c o m p l e t e nuclear Schwartz space, and W is a countable strict inductive limit of FrCchet s p a c e s

296

VI1. S o l v a b i l i t y of Q u a s i h o m o g e n e o u s E q u a t i o n s

a n d h e n c e b a r r e l l e d . By c o m b i n i n g P r o p o s i t i o n s IV.3.3 a n d V1.3.8 in de W i l d e C41 w e d e d u c e t h a t u is relatively o p e n . C o n s e q u e n t l y , s i n c e U i s c o m p l e t e so is i m u . In p a r t i c u l a r , i m u is c l o s e d in W . T h e b i p o l a r t h e o r e m t h e n i m p l i e s t h a t i m u = ( k e r r u b o . Applying P r o p o s i t i o n V1.1.4 in de W i l d e C41 to 'u that

t~

i s a weak homomorphism.

we conclude

m

As a f i r s t c o n s e q u e n c e of P r o p o s i t i o n 7.13 w e n o t e t h a t

(7.16)

( a ) kerDm=(imtD,)O,

and ( b ) (kerDm)O=imtD,,

DE(A,B,C).

I n d e e d , t h e e q u a t i o n ( a ) is o b v i o u s , a n d t h e e q u a t i o n ( b ) f o l l o w s f r o m ( a ) by t h e b i p o l a r t h e o r e m s i n c e by P r o p o s i t i o n 7.13 i m t D m is closed in t h e r a n g e s p a c e of D . In p a r t i c u l a r , s i n c e ( A , - q ( k e r

B,))O

= tA,l-p((ker

B,,)O)

t h e condition

( 7 . 1 6 . b ) f o r D = B leads to (7.17)

(A,

~

e ( k e r B,))O

= tA ,

-q

(irn tBm )

F r o m t h e s e e q u a t i o n s w e o b t a i n t h e f o l l o w i n g d u a l d e s c r i p t i o n of ( 7 . 1 O l k .

Theorem 7.14. The conditions OF Theorem 7 . 5 are valid i f and o n / ) i f For s o m e ( r e s p . ever),) k €IN* the Following conditions hold: (7.18)

ever> F E X ~ * , , ( E . ~such , ~ ) that qIIx F extends t o an element +

OF Q n , c + p ( E c 9 0e s t e n d s t o a function in Q,,.

(E,$,#)-

and (7.19)

A,,,

-p

( k e r B,,,)is weak!, closed in QA,-l ( E l ,

e F . S i n c e k e r C , , is weakly closed in Q L - p ( E ) Bit f o l l o w s by t h e b i p o l a r t h e o r e m t h a t (7.10); is valid if a n d o n l y if (7.19) h o l d s a n d ( 7.18 ) "

(A,-p(kerBm))o

= (kerC,)O

By (7.17) a n d ( 7 . l O . b ) t h e l a s t c o n d i t i o n a m o u n t s to (7.18)'

t

-1 A,-,(irn

tB,)

-

= im k m .

S i n c e by ( 7 . 6 ) t h e m a p C m ( X + )

C C n ( X + ) ,f - q l

by R e m a r k 7.12 t h a t ( 7 . 1 8 ) is e q u i v a l e n t to ( 7 . 1 8 ) ' .

x +f

, is injective it f o l l o w s

297

7 . a Multiplication Equations

W e a r e n o w g o i n g to i l l u s t r a t e t h e u s e f u l n e s s o f c o n d i t i o n ( 7 . 1 8 ) . b e g i n n i n g w i t h a rather simple example.

Remark 7.15. IF rn*= - l then the condition (7.18) i s valid provided that k e r M

= (01

and q - ' ( O ) n + # @ .

p r o O F . T h e a s s u m p t i o n s o n rn* a n d M i m p l y t h a t Q , * + o ( E ) s t a n t f u n c t i o n s . C o n s e q u e n t l y , if q1

c o n s i s t s of a l l c o n -

f e x t e n d s to a n e l e m e n t of Q , , * + ( ( E )

X*

then

by t h e a s s u m p t i o n o n q it v a n i s h e s identically so t h a t by ( 7 . 6 ) f does so. a s w e l l . rn F o r d e a l i n g w i t h a n o t h e r s p e c i a l c a s e w h e r e t h e c o n d i t i o n (7.18) c a n easily be c h e c k e d w e have to s h o w t h a t t h e c o n d i t i o n ( 7 . 1 8 ) c a n be f o r m u l a t e d in a n a p parently slightly weaker form:

Remark 7.16. Let R E Q , r r + + P ( E c i t *and ) hEC"'(X+) Y:= I X o ) , . IF h extends t o a C"' Function

such that R / y = q / Y h where

on X* then, in Fact, it e.\tends

to a

Function PEQ,,,* (EeG*) satistbing R = q P .

m F . I t s u f f i c e s to f i n d P E Q , , * ( E b i ) s u c h t h a t R = q P ; f o r it t h e n f o l l o w s by c o n t i n u i t y f r o m ( 7 . 0 ) t h a t t h e f u n c t i o n s h a n d P coincide o n X ( i n case E = 3 ( X ) o n e h a s to t a k e i n t o a c c o u n t t h a t t h e s u p p o r t o f R a n d h e n c e t h a t o f P a n d h a r e c o n t a i n e d in X

n

).

If E = 3 ( X ) t h e n by C o r o l l a r y 5 . 4 7 w e c h o o s e a G - i n v a r i a n t c o m p a c t s u b s e t K of X n M o ( X ) s u c h t h a t s u p p R C M,'(K). tion x€C;(X)

M o r e o v e r , w e fix a G - i n v a r i a n t f u n c -

e q u a l to 1 o n a n e i g h b o u r h o o d U o f K . T h e n by t h e a s s u m p t i o n

o n h it is o b v i o u s t h a t Xh e x t e n d s to a f u n c t i o n in EQi

- again

d e n o t e d by Xh - ,

a n d by P r o p o s i t i o n s 5 . 4 8 . ( i i ) a n d S.45 w e c o n c l u d e t h a t

q Q,*(xh)

= Q,*+e(qXh)

= Q m * + e ( ~ R=) Q,*+e(R)

=R

0

If X = V ( h e n c e X = V ) a n d E = ( 'Y V ) t h e n by Lemma 5.53 w e c h o o s e a G - i n v a r i a n t

C" f u n c t i o n x : V + @

e q u a l to 1 n e a r V \ V + w i t h s u p p o r t c o n t a i n e d in

n,,,,,

f o r s o m e r > O s u c h t h a t a l l its d e r i v a t i v e s are b o u n d e d . S i n c e X R a n d h e n c e q ( X h ) b e l o n g to Y ( V ) it f o l l o w s by ( 7 . 5 ) t h a t Xh b e l o n g s to Y'(V), a s w e l l . A s a b o v e we conclude t h a t qQ,*(Xh)=R.

Since in view o f t h e a s s e r t i o n s ( i i ) a n d ( v ) o f

298

VII. S o l v a b i l i t y o f Q u a s i h o m o g e n e o u s E q u a t i o n s

P r o p o s i t i o n 5.48 w e may a s s u m e t h a t h is @ ' - i n v a r i a n t w e see t h a t P : = Q , * ( x h ) is t h e desired f u n c t i o n . w

T h e p r o o f of t h e f o l l o w i n g r e m a r k s h o w s t h a t in c a s e q is s u f f i c i e n t l y s i m p l e t h e c o n d i t i o n (7.18) c a n s o m e t i m e s be c h e c k e d d i r e c t l y f o r a r b i t r a r y m .

Remark 7.17. ( i l I f there e s i s t s v 6 G M ( O +l \ (01 such that the directional derivative 3,q of q vanishes identically on X o then the condition (7.18) o f Theorem 7.14 is always s at i s fi ed.

(iil

IF ReP

< A,,,,,:=mas{ReA: A E o , }

then the hjpothesis o f assertion ( i ) is

au torna tically s at i s fi ed.

w.(i): By

c h o o s i n g a s u i t a b l e b a s i s w e identify V w i t h IRxW in s u c h a w a y

t h a t I R x ( 0 ) is identified w i t h IRv a n d W w i t h C e k e r M w h e r e C is a c o m p l e m e n t o f IRv in G M ( o + ) . W e set X ' : = { x ' E W ; ( 0 , ~E ' X) o } . Then a look at (6.22) s h o w s t h a t IR x X'=Xo. N o w , let R E Q , * + ~ ( E c H * ) b e s u c h t h a t R I , + = q J , + h

Q n l , + e vwe

t h e d e f i n i t i o n of G M ( b + )*C,

f o r s o m e h E C m ( X + ) . By

f i n d d E N s u c h t h a t f o r every y c k e r M t h e f u n c t i o n

z H R ( z + y ) , is a polynomial f u n c t i o n o f d e g r e e 5 d . In p a r t i c u l a r ,

fixing x ' E X ' w e see t h a t R (

- , X I )

is a polynomial f u n c t i o n o f d e g r e e n o t l a r g e r

t h a n d . N o w , by t h e a s s u m p t i o n o n q w e have q ( * , x ' ) = q ( O , x ' ) .S i n c e in case q ( 0 , x ' ) f O t h e function h ( deduce t h a t h (

*

- , X I )

coincides with R ( * . x ' ) / q ( O . x ' ) o n l O , + m t w e

, x ' ) ) , ~ , +is~ a~ polynomial f u n c t i o n of d e g r e e n o t l a r g e r t h a n

d a n d h e n c e e q u a l to P ( * , X ' ) I , ~ , + , w~ h e r e P : X " + @

is t h e Cm f u n c t i o n

defined b y

.

I t f o l l o w s t h a t t h e polynomial f u n c t i o n s R( , x'

)

and

( qP)(

- ,x')

coincide o n

I O , + a 3 C a n d h e n c e o n t h e w h o l e o f IR. Since t h i s is trivially valid if q (

,x')=O

0

w e have proved t h a t R = q P o n X .

If X = V t h e n Xo = V , a n d t h e p r o o f is c o m p l e t e . I f X # V t h e n E = a ( X ) , a n d by LemrnaS.SO.(ii) w e see t h a t , in f a c t , R b e l o n g s to

Q m 4 + @3(( X o ) , e ) .

Applying

R e m a r k 7.16 to E = 9 ( X 0 ) w e d e d u c e t h a t P e x t e n d s to a f u n c t i o n H b e l o n g i n g to

a,*( a ( X o ) , i )

C Q,*(EQ*).

S i n c e t h e s u p p o r t s of R a n d H a r e c o n t a i n e d in

Xo it f o l l o w s t h a t R = q H o n X .

29Y

7.a M u l t i p l i c a t i o n E q u a t i o n s

lii): W e c h o o s e

aEX'

s u c h t h a t la1 = 1 a n d R e a M > R e 0 a n d o b s e r v e t h a t d"q

is a l m o s t q u a s i h o m o g e n e o u s o f degree t - a ~ .S i n c e R e ( d - a M ) < 0 L e m m a 1.60 i m p l i e s t h a t a"q v a n i s h e s o n Xo.

m

N o w w e a r e g o i n g to h a v e a look at t h e c o n d i t i o n ( 7 . N ) i n T h e o r e m 7.14. If q - ' ( O ) n X, = @ t h e n by R e m a r k 7.Y t h e c o n d i t i o n (7.19) is s a t i s f i e d f o r t r i v i a l

r e a s o n s . I t is a l s o valid if k e r M = { O ) f o r t h e n Q;-@(E)

a n d h e n c e all o f its

s u b s p a c e s a r e f i n i t e - d i m e n s i o n a l a n d h e n c e c l o s e d . W h e t h e r (7.19) is valid f o r g e n e r a l q a n d M I do n o t k n o w . T h e r e is, h o w e v e r , a n e l e m e n t a r y r e f o r m u l a t i o n of it f o r w h i c h w e m a k e u s e of

Notation 7.18. We set 21i7A,k(E) : = ? t i ( l k , k ( X ) n E ' (see N o t a t i o n 4 . 2 8 ) ; i . e . t h e space XDU:,,k(E) c o n s i s t s of a l l d i s t r i b u t i o n s TE E' w h i c h are a l m o s t q u a s i h o m o -

g e n e o u s of d e g r e e m a n d of o r d e r i k .

Lemma 7.19. The condition (7.19) holds if and on!, i f the space q 2[D,L-t.A ( E l , is weaklj cl os ed in X,i,,k(E),c,.

hoof.In

view o f P r o p o s i t i o n 7.13 t h i s is a s p e c i a l case of L e m m a 7.20 b e l o w . rn

Lemma 7.20. Let F . G . and H be locallj conve\ vector spaces, let A :F + a surjective continuous linedr hornomorphisni. dnd let B : F+

G be

H be a continuous

linear map. I f L3iker.A) is c lose d in H then A(X e rBI i s closed in G . proOf. L e t g E G be in t h e c l o s u r e of A ( k e r B ) . S i n c e A is s u r j e c t i v e w e f i n d f c F s u c h t h a t A ( f ) = g . I t s u f f i c e s to s h o w t h a t B ( f ) is in t h e c l o s u r e of B ( k e r A ) . F o r t h e n by t h e a s s u m p t i o n w e f i n d f ' E k e r A s u c h t h a t B ( f ) = B ( f ' ) so t h a t f - F ' E k e r B a n d g = A ( f ) = A ( f - f ' )E A ( h e r B ) .

So l e t W be a n o p e n n e i g h b o u r h o o d of B ( f ) in H . S i n c e B is c o n t i n u o u s t h e set U : = B - ' ( W ) is a n o p e n n e i g h b o u r h o o d of f in F. S i n c e A is o p e n t h e set A ( U ) is a n o p e n n e i g h b o u r h o o d of g i n G , h e n c e by t h e a s s u m p t i o n o n g c o n t a i n s a n

e l e m e n t of t h e f o r m A ( f " ) w h e r e f " E k e r B . C h o o s i n g f " ' E U s u c h t h a t A ( f " ' ) = A ( f " ) w e conclude that

f':=f"'-f"EkerA

a n d B ( f ' ) = B ( f " ' ) E B ( U ) C W . rn

300

VII. S o l v a b i l i t v o f O u a s i h o m o a e n e o u s E q u a t i o n s

In this section we a r e now going t o analyze t h e condition (7.18) f u r t h e r . To t h i s end we introduce a canonical subspace of ker B,,

consisting of a l m o s t quasihomo-

geneous distributions whose ( l + k ) t h o r d e r deficiencies a r e easily c o m p u t e d .

-

Notatlon7.21. ( i ) I f f E C m ( X ) w e define S(fI a s t h e set of all pairs sisting of a point y C X and a polynomial function Q : V * (7.20)

( y , Q ) con-

a3 satisfying

f Q(-a)S, = O

where 8, d e n o t e s t h e Dirac distribution a t y . Since by t h e Leibniz r u l e in t h e f o r m

of (1.54) we have

=

=

( R , (dM-m*-P)k+'cp> =

(

R , ( ( a ~ - m * - [ )k + l c p ) m * + e . W k > ~ .

S i n c e by ( 3 . 7 ) a n d ( 5 . 0 3 ) w e have

( ( dM

- m r -P )

+ I rp*,)

+

,',,k

=

(

aM - m * - P

)

k+l

*,pc

+

,',,k = (

-1)

k+l

e9 ( f )

1

+

it f o l l o w s t h a t < d , e q ( f ) > = < R , e q ( f ) l x + > MN.o w , c h o o s i n g @ E E Q - s u c h t h a t

f = @ m * , c . , kw e d e d u c e f r o m T h e o r e m S . 3 7 . ( v i ) t h a t e q ( f ) J x + = ( q @ ) , , * + t , w k so t h a t < R , e q ( f ) l x , > M = < R . q @ >= < q R , @ > = < q R , f > M . This implies ( 7 . 3 0 ) . (i): -

w e a p p l y ( 7 . 3 0 ) to ( S , c ) i n s t e a d o f ( R , d ) .

(ii): In a f i r s t s t e p w e p r o v e t h e a s s e r t i o n f o r the special case c = O . By T h e o r e m 7 . 3 w e c h o o s e a @ - i n v a r i a n t s o l u t i o n R E E ' of q R = T which is a l m o s t q u a s i h o m o g e n e o u s of d e g r e e ni-0 s u c h t h a t d : = ( d M - m + e ) k + 1 R E Q ~ - 4 ( EBy ) . Lemm a 7 . 6 . ( i i ) - a p p l i e d to t h e s i t u a t i o n d e s c r i b e d in t h e p r o o f of T h e o r e m 7 . 5 - it s u f f i c e s to s h o w t h a t d b e l o n g s to t h e s p a c e H : = A , - e ( k e r B , ) . a s s u m p t i o n of ( i i ) t h e s u b s p a c e H is weakly c l o s e d in Q A - t ( E ) o i

S i n c e by t h e t h e bipolar

t h e o r e m a n d R e m a r k 7 . 3 4 . ( v i ) imply t h a t d b e l o n g s to H if a n d o n l y if i t l i e s in t h e p o l a r set of e q ( z , * , k ( q ; E ) G * ) . In view of ( 7 . 3 0 ) t h i s m e a n s t h a t < T , f > M = O

for every f ~ z , . , ~ ( q ; E ) ~, , S i n c e by ( 7 . 2 9 ) a n d t h e a s s u m p t i o n c = O t h e l a t t e r c o n d i t i o n is valid t h e p r o o f of t h e f i r s t s t e p is c o m p l e t e .

To p r o v e the general case . by C o r o l l a r y 6.51 w e c h o o s e a d i s t r i b u t i o n RE E' s u c h that

( a M - m + 4 ) k ' 1 R = c . By P r o p o s i t i o n 2 . 6 4 . ( i i i ) a n d R e m a r k 2 . 6 7 . ( i i ) w e may

312

VII. Solvability of Q u a s i h o m o g e n e o u s Equations

a s s u m e t h a t R is @ - i n v a r i a n t . Applying ( 7 . 3 0 ) to d = c w e d e d u c e f r o m ( 7 . 2 9 ) that < T - q R , f > , = O

f o r every f E S , * , k ( q ; E ) G * .

H e n c e by t h e f i r s t s t e p of

t h e p r o o f w e f i n d a n a l m o s t q u a s i h o m o g e n e o u s @ - i n v a r i a n t s o l u t i o n S ' E E' of t h e equation q S ' = T - q R

s a t i s f y i n g ( a M - m + t ) k + ' S ' = O so t h a t S : = S ' + R is t h e

desired s o l u t i o n of ( 7 . 1 ) .

In c a s e M is s e m i - s i m p l e w h e n c h e c k i n g ( 7 . 2 9 ) o n e may t a k e a d v a n t a g e of

Remark 7.37. Suppose that M i s semi-simple. Then the l e f t - h a n d side o f ( 7 . 3 ) i s equal t o (i)

- in and

is any function s a t i s f j i n g + o = ~ and

< T , [ ( - ~ , ) ~ Q ] Ewhere > QEC;(XI case E = Y ' ( VI

-

having the properties (i)

-

( i i i ) of Lemma 6.361;

- i n case E = . B ( X ) - t o

,flSx

( i i ) < [ ( d ~ - m ) ~ T ] / ~ x > where x:X+--?,1O,+wCis any C"'function which is quasihomogeneous o f degree 1 .

If M is n o t s e m i - s i m p l e it is m o r e c o m p l i c a t e d to c o m p u t e t h e l e f t - h a n d

side

of ( 7 . 2 9 ) - s e e , f o r e x a m p l e , ( 8 . 1 2 ) b e l o w .

Proof. In view of t h e a s s u m p t i o n o n M Remark 7 . 3 4 . ( i ) s h o w s t h a t f is q u a s i h o m o -

g e n e o u s of d e g r e e m * . H e n c e , by T h e o r e m 3 . 4 8 a n d Lemma 5 . 6 2 w e f i n d q E Eg' a n d s u c h t h a t f = q P m , , w kIn .

s u c h t h a t s u p p q is a n M - b o u n d e d s u b s e t of X, view of (6.58) t h i s i m p l i e s t h a t M==. Consequently, defining functions

$i

by (4.16) w e d e d u c e f r o m P r o p o s i t i o n 4.13,

(2.5). and Corollary 2.36.(ii) t h a t k

=

1=0

k

=

c < T , [ ( - d ~ ) ~ $f j>]

=

Ck

f

i f r = p'-"'.

m f .

:

T h e c o n d i t i o n ( 7 . 4 4 ) " of S u p p l e m e n t 7 . 5 3 is violated if a n d o n l y if

m+eENo but m d N o .

(ii): If

m E -Ne t h e n t h e c o n d i t i o n ( 7 . 4 4 ) " of S u p p l e m e n t 7.53 a m o u n t s to

p,+p(IR)m= (0).S i n c e V,+e(!R)

= Cx

p m + ' = p a = p p e r , i . e . r=p-,+'

m

.

~ t h i+s m ~ eans t h a t g m + P f o , i.e.

N o t e t h a t u n d e r t h e a s s u m p t i o n s of E x a m p l e 7.63,

t h e Dirac d i s t r i b u t i o n So is

h o m o g e n e o u s of d e g r e e m = - 1 ; h e n c e it is t h e a s s e r t i o n ( i i ) ( a p p l i e d to T = 1 a n d CI

e

= p ) - a n d n o t ( i ) ! - t h a t i m p l i e s t h e e x i s t e n c e of a h o m o g e n e o u s f u n d a m e n t a l

s o l u t i o n ( w i t h p a r i t y ( - l ) e ) . Of c o u r s e , s u c h a f u n d a m e n t a l s o l u t i o n is e x p l i c i t l y 1

given by 7 ( E

+

(-1)

ev

E)

w h e r e E :=

x P - l H a n d H d e n o t e s t h e Heaviside f u n c -

tion. M o r e g e n e r a l l y , s u p p o s e t h a t u n d e r t h e a s s u m p t i o n s of E x a m p l e 7.63,

the number m

b e l o n g s to t h e e x c e p t i o n a l set (-INe). In order to describe t h e o p e r a t o r

m o r e p r e c i s e l y , w e f i r s t recall f r o m P r o p o s i t i o n 2.13 t h a t t h e a r g u m e n t a n d t h e t a r g e t s p a c e s a r e t w o - d i m e n s i o n a l . Since 4 + m + l t 1 t h e p r e c e d i n g a r g u m e n t s h o w s t h a t t h e r e is a h o m o g e n e o u s d i s t r i b u t i o n T s u c h t h a t T'e'm'l) b e l o n g s to X&(IRR,

we conclude t h a t Oe,,(T)

- S . Since

s,,(-"-"

. Since by E x a m p l e 7.63,

326

V I I . Solvability of Q u a s i h o m o g e n e o u s E q u a t i o n s

t h e operator O p is not surjective we conclude t h a t im Oe,m= @ I r ~ - m - l ). This, in t u r n , implies t h a t t h e kernel of Oe,m is 1-dimensional; indeed, it is spanned by t h e function x m + ' - note t h a t 0 5 m+4 5 4 - 1 . Of course, in view of t h e equation ( ~ - 1 ~ ( - m - =l )( - r n - l ) ! (

t h e fact t h a t t h e distribution

- ~ ) - ~ - ~ z ~

zm does

not belong to im O',,,, is a direct conse-

quence of t h e fact t h a t every solution u € a ' ( l R ) of t h e equation U ' = X - '

1s '

of

t h e form u = T c + l o g l . l f o r some constant c € C (see, for example, Hormander C 111, (3.2.13), p.73).

Example 7 . 6 3 ~ Suppose . that dima V = I . and let A

E Q' \ R

be such that M = A Id v .

Then there is a unique element ~ ' 6 sirch 3 that q ( v 1 = const v a ' , and the following assertions hold: (i1 I f G = l l d v l then the conditions o f Theorem 7.51 are violated i f and on!, i f m belongs t o IX(M1- P I \ X ( M ) . l i i l Suppose that G = I ? I d , I ; then the assumptions on @, u=p

I,and

q mean that

/a'/

r where p ( - + I d , ) : = - + :l consequentl~.i f m belongs t o the eAceptiona1 set

of assertion ( i ) then the conditions o f Theorem 7.51 hold i f and on1-p i f ( 7.4 71,

r=p

/a/ - /a'/ +1

where a is the unique element of X satiscving a M = n i + P ; note t h t (7.47), means that r = p in case l a l - l a ' l is even and r = l in case l a / - l a ' l is odd.

Of course, under t h e assumptions of Example 7.63, similar considerations are valid

if G is an arbitrary subgroup of S O ( V ) . proOf. The first assertion is a consequence of Remark 1.4O.(i). Hence, q satisfies

( 0 . 2 8 ) , and t h e solvability of (7.34) is handled by Supplement 7.53.

( i ) : In

f a c t , Remark 1.40 shows that d i m v , , , ( C ) = l for every m e X ( M ) . Conse-

quently, t h e condition ( 7 . 4 4 ) " is valid f o r G = { l d v ) if and only if m and m + P both belong to X ( M ) o r both d o not belong to 2 I ( M ) .

lii): Suppose t h a t m belongs to the exceptional s e t of assertion ( i ) . Then it

is

by Remark 1.40 t h a t we can fix a unique aC'11 such t h a t a M = m + 4 so t h a t

V m + t ( C ) = Cx". Moreover, t h e condition (7.44)" of Supplement 7.53 a m o u n t s

to

7.e

327

Examples

v,+o(C)s={Ol. Q C " ~ Q ' ~ ' ' T ,

Since x a o ( - l d c ) = ( - l ) ' O L ' x u this means t h a t p l O L l f a , i.e. p l a l =

i.e. ( 7 . 4 7 ) c h o l d s .

H

N o t e t h a t u n d e r t h e a s s u m p t i o n s o f E x a m p l e 7.63,

t h e Dirac d i s t r i b u t i o n 6, is q u a s i h o m o g e n e o u s of degree m = - p w h e r e h e r e 1-1= 2 Re X = X + X = ( 1.1) ; h e n c e , if a'? ( 1 , l ) t h e n - a s in t h e c a s e V = R

-

t h e a s s e r t i o n ( i ) does n o t i m p l y t h e e x i s -

t e n c e o f h o m o g e n e o u s f u n d a m e n t a l s o l u t i o n s . B u t in c o n t r a s t to t h e case V = R t h e s a m e is t r u e f o r t h e a s s e r t i o n ( i i ) . In f a c t , c h o o s i n g a = ( p , y ) a s in a s s e r t i o n ( i i ) a n d w r i t i n g a ' = (p',y') w e c o m p u t e -

BX + y X = a M = m + 4 = - ( I , I ) M + ( a ' ) M = ( p ' - t ) X

+

(y'-t)X

so t h a t by L e m m a 1.41 w e have p = B'-I a n d y = ~ ' - 1 . S i n c e t h i s i m p l i e s la1 - Ia'l = - 2 w e c o n c l u d e t h a t f o r ~ = t hl e c o n d i t i o n ( 7 . 4 7 ) , t h a t under t h e assumptions of Example7.63,

is a l w a y s v i o l a t e d , i n d e e d . N o t e

t h e o p e r a t o r q ( D ) in g e n e r a l d o e s

n o t have h o m o g e n e o u s f u n d a m e n t a l s o l u t i o n s , i n d e e d ; f o r e x a m p l e , if q = 4 2 2 t h e n q ( D ) is e q u a l to t h e Laplacian A a n d h a s t h e f u n d a m e n t a l s o l u t i o n & l o g 1 . 1 w h i c h is n o t h o m o g e n e o u s . T h e Laplacian in n d i m e n s i o n s is t h e object of

Hxample7.64. Suppose that V = R " and M=ld,.

Let h6lN and q l D I = A h w h e r e

by A we denote the Laplacian. Then P = 2 h . I f m EZ with m 2 - 9 h then the condition ( a ) ' o f Theorem 7.51 is valid For G = SO( V ) and a = r

?

1 i f and on/-),i f m ? 0

or iF m is odd.

In fact, it follows From Example 7.39 that i f n is even and not larger than 211 then A h has no homogeneous fundamental solution; indeed, a s is well-known. in this case A h has a Fundamental solution o f the Form c / * / ' " - " l o g 1.1 which is almost homogeneous but not homogeneous so that it also follows by Proposition 2.48 that in this case A h has no homogeneous fundamental solution.

p r o O F . I t is easily s e e n t h a t d i m ~ h ( R n ) S O ( n e) q u a l s 1 if h € 2 N 0 a n d 0 o t h e r w i s e . H e n c e o n e o b t a i n s t h e a s s e r t i o n via t h e c o n d i t i o n ( 7 . 4 4 ) " of S u p p l e m e n t 7 . 5 3 .

N e x t w e are g o i n g to have a look a t t h e h e a t o p e r a t o r .

328

VII. S o l v a b i l i t y of Q u a s i h o m o g e n e o u s E q u a t i o n s

Hxunple 7.65. We define P, , p , o and r as in Esample 7 . 4 0 . Let G be the group S O ( n - I ) operating on the space variables

F.

Then q = P , satisfies the conditions

o f Theorem 7.51 i f and only i f m f ?No - 2 . proOf. I t is easily verified (see Proposition 7.73 below) t h a t

Hence (7.44)" is valid if and only if m d 2 N o - 2 , and t h e assertion follows b y Supplement 7.53.

m

N o t e t h a t f o r every hElN t h e assertion of Example 7.65 implies t h a t the operator

P b ( D ) has a quasihomogeneous (of type p : = ( 2 . 1 , . . . , I ) ) fundamental solution provided t h a t n 2 2h o r n is even. In f a c t , unlike t h e situation in Example 7.64 t h e last assertion is valid without any restrictions on n o r h a s a look a t t h e explicit form of t h e standard fundamental solution of P:(D)

s h o w s (see t h e

references in O r t n e r C151 ) . A less explicit proof is provided by Example 7 . 4 0 . For two further special, b u t classical examples we a r e going to show t h a t t h e conditions of Theorem 7.51 are satisfied. This time we d o this by verifying t h e condition (7.45)'. Here t h e following lemma is required.

Lemma7.66. Suppose that kerM =101. Let m E # ( M ) , and let Y be an open subset o f V \ 101. Then there is a subset E o f 1' such that the polFnomial functions

constitute a basis o f the space 'Pm t V ) o f all po?,~~omial functions V+

Q' which

are almost quasihomogeneous o f degree m . I f M is semi-simple then one can choose E to be a subset o f Y,nSv.

w. We set A : = { a € ' U ; a M = m } . By Propositions1.34.(iii) and 1.28.(ii) t h e funca

tions 1 x u , a € A , constitute a @-basis of Cp,(V).

Consequently, f o r proving t h e

main part of t h e assertion it suffices to find a family t h e AxA-Matrix

( ( [ u ) a ' ) ( c r , a o ) E A x A with

([a)aeA

in T such t h a t

complex entries has a non-zero deter-

minant, i.e. is invertible. For then t h e functions

7.e

329

Examples

a'€ A , c o n s t i t u t e a @-basis of

p,(V),

as well.

N o w , f o r a r b i t r a r y a E A a n d j€Nn w e d e n o t e by Y,,j

a transzendental variable.

W e s u p p o s e t h a t a l l t h e s e v a r i a b l e s a r e i n d e p e n d e n t . W e set

Y,:= ( Y a J , . . . , Y , , n ) a n d d e f i n e primitive m o n o m i a l s d

TT

Y,$=

c

(Ya,j)pj

j=1

TT

( Y , , j + d Y j ( Y a , j + d + c )'j

,

j- 1

( c o m p a r e N o t a t i o n 1.25.A). By 37 w e d e n o t e t h e A x A - M a t r i x

T h e n P : = d e t X is a polynomial in t h e variables Y a , j w i t h c o e f f i c i e n t s in Z . Explicitly, w e have (7.48)

P(Y)

=

C

Y:'"'

E(G)

ocr(A)

a S A

w h e r e Z ( A ) d e n o t e s t h e set o f p e r m u t a t i o n s o f A a n d

E

t h e sign homomorphism

given by any o r d e r i n g of t h e e l e m e n t s o f A . W e observe t h a t a l l t h e p r i m i t i v e m o n o m i a l s a p p e a r i n g o n t h e r i g h t - h a n d side o f ( 7 . 4 8 ) are d i f f e r e n t , a n d h e n c e i n d e p e n d e n t . C o n s e q u e n t l y , P is n o t t h e z e r o p o l y n o m i a l . T h i s i m p l i e s t h a t P in*

duces a non-trivial function P : (Cn)A+

C . N o t e t h a t via t h e p s e u d o - r e a l coor-

d i n a t e s i n t r o d u c e d in N o t a t i o n 1.2S.A o n e o b t a i n s a f u n c t i o n o n V A w h i c h is den o t e d by ( < , ) n E A ~ P ( ( E , , ) ) . I t f o l l o w s t h a t o n e c a n f i n d a family

r

such that P (

( E m ) ) f 0 . Indeed,

d e n o t i n g by

?

( 0 . I f w1 , . . . , w, E C a r e s u c h t h a t P = a , n F = , ( T - w j ) w e c o n c l u d e t h a t q =nr=oqiw h e r e q i ( x , y , z ) : = ( y2 - 2 x z ) - w j z l ,

jEN.

In view of Remark 7.57 it suffices to prove t h e a s s e r t i o n for t h e q i . In view of w h a t w a s said a b o v e a b o u t q o t h i s leaves u s with t h e c a s e q ( x , y , z ) = y 2 - 2 x z - v z ' f o r s o m e w E C . In case w is real q h a s many real z e r o s so t h a t t h e m e t h o d of t h e proof of E x a m p l e 7 . 6 8 leads to a s i m p l e p r o o f :

The subcase v € R . W e f i r s t o b s e r v e t h a t t h e set Y : = 1 ( x , z ) E I R 2 ; 2 x z + w z 2 > O ) is n o n - e m p t y .

W e fix C = ( x , z ) E Y and - s e t t i n g y

:=JGobserve that

q , : = ( x , + y , z ) is a non-trivial real z e r o of q . For Q = I a n d q = q +- t h e polynomial function in ( 7 . 4 2 ) b e c o m e s

where

w i t h I,(p) being defined as in t h e proof of Example 7.67. If p = -2 t h e n

+

Ri = Sc,, = 1 ,

a n d s i n c e dim'Po(IR3) = 1 t h e condition (7.45)' of S u p p l e m e n t 7.54 is s a t i s f i e d . S u p p o s e now t h a t

Q

2 - 1 . Then s e t t i n g QC : =

3 (R;

2 R;)

we obtain

S e t t i n g B : = l z ( p + 2 ) a n d I ' : = l a ( p + l ) , by L e m m a 7 . 6 6 w e c a n c h o o s e families ( < a ) e E B and (

w

~

i n ) T s~u c h~ t h a~t t h e polynomials S , , , , ,

P E B , ( r e s p . Swy,,, Y E T , )

a r e linearly independent. I t f o l l o w s t h a t t h e polynomials Q:B,

P E B , and

QCY,

333

7 . e Examples

Y E T , are linearly i n d e p e n d e n t . S i n c e t h e y b e l o n g to D m + 2 ( q w ) e conclude that (7.52)is valid. In view o f (7.51) ( f o r n = 3 a n d m = p ) t h i s i m p l i e s t h a t t h e c o n d i t i o n ( 7 . 4 5 ) ' is satisfied.

The subcase u ~ RIn. t h i s c a s e t h e r e are n o t so many real z e r o s : q ( x , y , z ) = O if a n d o n l y if y = O = z . C o n s e q u e n t l y , a p r o o f via t h e t h e c o n d i t i o n ( 7 . 4 5 ) ' h a s to m a k e u s e of t h e e l e m e n t s ( q , Q ) in % ( q ) w h e r e Q is n o n - t r i v i a l . H o w e v e r , h e r e w e a r e g o i n g to p r o c e e d a l o n g d i f f e r e n t l i n e s in t h a t w e verify t h e c o n d i t i o n ( c ) o f P r o p o s i t i o n 7 . 4 0 . H e n c e w e f i x RECp,,,+Z(V*) s u c h t h a t R I , = q l , H

for s o m e

H e C m ( f l ) w h e r e n : = R 3 \ i O ) . Now w e fix r E I R \ ( O ) a n d o b s e r v e t h a t by q , ( x , y ) : = q ( x , y , r y )= ( i - v r 2 ) y 2 - 2 r x y a polynomial f u n c t i o n is d e f i n e d which is i n d u c e d by a p o l y n o m i a l o f d e g r e e 2 w i t h r e s p e c t to y w i t h c o e f f i c i e n t s in t h e polynomial r i n g @[XI

with invertible

l e a d i n g c o e f f i c i e n t . C o n s e q u e n t l y , s i n c e t h e f u n c t i o n R, c a n be c o n s i d e r e d a s a t h Ie Euclidean a l g o r i t h m p o l y n o m i a l w i t h r e s p e c t to y w i t h c o e f f i c i e n t s in @ [ X

provides u s w i t h f u n c t i o n s T , U

E C"(IR3)

which a r e p o l y n o m i a l s w i t h r e s p e c t to

t h e first t w o variables s u c h t h a t

a n d s u c h t h a t t h e d e g r e e of t h e polynomial f u n c t i o n U ( * ; - , r )

w i t h r e s p e c t to y

is n o t l a r g e r t h a n I . In view o f t h e a s s u m p t i o n o n H w e d e d u c e for every x E I R \ i O ) that limy,oU(x,y.r)/q,.(x,y)

e x i s t s . S i n c e degyq,.=2 t h i s i m p l i e s t h a t U ( x ; , r )

GO

f o r x ~ ! R \ ( 0 ) By . c o n t i n u i t y t h i s m e a n s t h a t U = 0 , i.e.

In order to c o m p l e t e t h e p r o o f w e have to s h o w t h a t (7.54)

3

by ( x , y , z ) H T ( x , y . z / y ) a polynomial f u n c t i o n P:IR -C is w e l l - d e f i n e d .

F o r t h e n it f o l l o w s f r o m ( 7 . 5 3 ) t h a t R = q P , a n d t h e c o n d i t i o n ( c ) of P r o p o s i t i o n 7.49 is verified. In order t o p r o v e ( 7 . 5 4 ) w e have to gain i n f o r m a t i o n a b o u t t h e coefficients

bj ,k(r ) : =

1

(

3; T ) ( 0 , O , r )

of t h e polynomial function T( . ; - , r ) . we c o m p u t e

Since t h i s is to b e o b t a i n e d o u t of ( 7 . 5 3 . a )

334

VII. S o l v a b i l i t y of Q u a s i h o m o g e n e o u s E q u a t i o n s

On the other hand, denoting the coefficient of x i y k z h i n R by aj,k,h w e obtain

where

Inserting the preceding equations into (7.53.a) and comparing coefficients w e arrive a t the equations

Observing that t h e elements of 9 , + 2 ( V * )

are precisely the polynomial functions

which are homogeneous of degree g+2 so that a j , k - h , h = 0 and hence A j , k = O i f j + k # g + Z we deduce that bi,k-O if j + k f p . I t follows that the first of the pre-

ceding equations gives (7.55)

a,*2,0.0 = 0

and the other ones lead to

We can now show by induction on k that, actually, the function Bk is a polynomial function of degree 5 k . Indeed, since the coefficients of a polynomial function on !R2are universal linear combinations of its derivatives a t an arbitrarily chosen point of ( l R \ ( O ) ) x l R , say ( l , l ) , we conclude in view of (7.S3.b) that the functions bj,k are linear combinations of t h e functions R-43,

r H r Y 3 H ( Y ) ( 1 , 1 , r )Y, E N , 3.

In particular, the functions Bk are C” o n the whole of R . Hence, i n view of (7.56.b) we deduce that the zero order coefficient in A , , a p + l , l , o , vanishes so

335

7 . e Examdes

t h a t B o ' - a p + l , ~ , l / Z .S i n c e A k + l is a polynomial f u n c t i o n o f degree l a l 2 j - 2 and r s j - 1 and t = O . Now, observing that

(3JtaGP

- . 'I ( Z , + Z , )( l '- s=- uA) !

if s + u < i

(z2+G)i-S-u

w e deduce that

i?s+u

If s + u = O then r = l a l ? j - 2 so that i n case r = j - l this equals

and in case r = j-2 is equal to

In case v : = s + u 2 I substituting I = i - v we obtain

If l a l = j - l then t h i s equals a j , v( j - v ) ! v! Z ,

-

+

a i , l + v( j - I - v ) ! ( v + l ) ! ( Z 2 + Z 2 )

and in case l a l = j - 2 this is equal to I

2

a j , v F( j - v ) ! v! ( Z I )

+

-

( v + l ) ! Z, ( Z 2 + Z 2 ) +

a j , v + (rj - 1 - v ) ! +

2 a j , v + 2(j-2-v)! i( v + 2 ) !(z2+Z,) . 2

Taking the equations

( a z 2 + a - )2 q = 2 u ,

( a , 2 + 3 T 2 ) q= ( z , - q ) + w ( ~ + z 2 ) ,

azi (az2+az2) q

1,

azlq = g

22

I

(azl)2

n o ,

into account w e deduce that [ ( Q j ) ( a + ) ( a ) q ] ( l , O ) =provided O that in case j ? 2 the following conditions are valid:

7.e

337

Examples

a i , v + l( j - l - v ) ! ( V + I ) ! + a i , v + 2( j - z - v ) !

I(v+z)! 2

zv = 0 ,

V€Ni-,

.

T h i s leads to t h e f o l l o w i n g r e c u r s i v e d e f i n i t i o n o f t h e a i , i: iENj-, .

a j , i : = - v i+fa. ,-I J .. I + l '

S i n c e t h e r e c u r s i o n s t a r t s w i t h a j , j = l w e see t h a t t h e a j , i a r e w e l l - d e f i n i e d n o n z e r o c o m p l e x n u m b e r s . C o n s e q u e n t l y , s e t t i n g Q o := 1 w e c o n c l u d e t h a t t h e p a i r s

( ( l , O ) , Q i ) , j E N o , b e l o n g to 2 3 ( q ) . N o w , f o r every j € N o w e d e n o t e t h e polynomial ( 7 . 4 2 ) f o r Q = Q i a n d q = ( 1 , O ) by

P i . O u r c l a i m is t h a t t h e p o l y n o m i a l s P i , O C j < b + c , a r e linearly i n d e p e n d e n t . In view o f S u p p l e m e n t 7 . 5 4 a n d ( 7 . 5 7 ) t h i s i m p l i e s t h e a s s e r t i o n .

C h o o s i n g u as above. fixing j E N b c r , a n d a p p l y i n g t h e binomial f o r m u l a w e c o m p u t e i

S

Q j ( a h m =i = l

1

r!

r - j + i -t Ji ql

aj,i

i2j-r

k=O

(i) k

S!

s-k

(s-k)!'12

U!

-u-i+k

(u-i+k)!q2

i-uck5s

S e t t i n g q 2 = 0 w e see t h a t t h e t e r m s in t h e s e c o n d s u m vanish e x c e p t w h e n s = k = i - u . T h i s m e a n s , in p a r t i c u l a r . t h a t i = v : = s + u . It f o l l o w s t h a t [Qj(a)q"](l,O) v a n i s h e s e x c e p t w h e n m a x ( 1 . j - r ) < s + u < j in which case it is e q u a l to r! aj,v ( r - j + v ) !

(:)

S !

u!

N o t e t h a t if a M = m + 4 t h e n b j Remark 1 . 4 O . ( i i ) w e have r = b - s a n d t = c - u . H e n c e

where

I j : = { ( s , u ) ~2N os ;< b , u < c . m a x ( l , j - b + s ) < s + u L j )= 2

= { ( s , u ) ~ N , ;s C b , j - b < u C c , s + u E N i )

Finally, it is i m m e d i a t e l y c l e a r f r o m ( 7 . 4 2 ) t h a t

p0 =

&!( i Z l ) b

(izllc.

Now s i n c e t h e c o e f f i c i e n t s a i , j are e q u a l to 1 w e o b s e r v e t h a t f o r e v e r y j E N b + = t h e monomials

-

(iZl)b+u-J(izz)J-u a p p e a r in Pi b u t n o t in Pi for 0 5 linearly i n d e p e n d e n t , as desired.

(iZ2)",

is j - I . w

Hence w e conclude t h a t

j-bsu 5 c ,

Po,...,Pb+care

338

VII. S o l v a b i l i t y of Q u a s i h o m o g e n e o u s E q u a t i o n s

Note that in view of Remark 7.55 the polynomial functions q appearing i n Proposition 7.60 satisfy the conditions OF Theorem 7.51 i f M is replaced by Id,;

in other

words, for these q the equation (7.34) has a homogeneous solution S for every homogeneous distribution T on V . Finally, I would like to point out that in case M is not semi-simple I cannot produce an example q such that the conditions of Theorem 7.51 are violated. On the other hand, the q’s appearing i n Proposition 7.60 seem to be too simple as to lend support to the conjecture that in case M is not semi-simple the conditions of Theorem 7.51 be always valid.

I n t h i s section w e suppose that V = I R x l R ” and that for some fixed w = v - i u ~ C \ ( O )

q is of the Form (7.58)

q ( T , < ) = iwr+I ~ I ’ ,

(r.,€,)EV*=IRxIR”.

Moreover. in this section we set

Obviously, q is quasihomogeneous (of degree 2 ) . It is also invariant under orthogonal transformations of the space variables. We are going to describe the fundamental solutions of q ( D) having the corresponding invariance properties. For the sake of brevity, in the present section we are going to use the following terminology:

Deflnltlon 7.70. Suppose that X is quasihomogeneous of type M and invariant under the action of t h e orthogonal group O f n ) on the space variables x . Then a distt-ibution T E % ’ ( X ) is called ( a l m o s t ) invariant of degree r n C C if and only if it is (almost) quasihomogeneous of degree m and invariant under the action of O ( n )

o n the x variables.

7.f

339

T h e He a t a n d t h e Schrodinger Equation

From t h e theory o f partial differential equations we recall t h e s t a n d a r d m e t h o d f o r c o n s t r u c t i n g fundamental s o l u t i o n s o f q ( D ) with s u p p o r t i n a half space. By a formal application o f t h e partial Fourier t r a n s f o r m with r e s p e c t to t h e s p a c e variables x - here d e n o t e d by

"-"

- t h e equation q ( D ) E = S o becomes

(7.59) d e n o t e s t h e Dirac distribution o n !R' a t t = 0 . Although in general t h e

where S,

restriction of distributions t o lines is n o t well-defined o n e l o o k s upon ( 7 . 5 0 ) a s a n ordinary differential equation depending o n

< a s a parameter.

As is immediately

checked, it has t w o distinguished s o l u t i o n s , namely ( 7.60 )

t

H

$ H ( o t ) exp( - 1 < 1 2 ) ,

0E( +l,-l),

d e n o t e s t h e Heaviside function which i s equal t o 1 on I O , + c o I t =Gtw 0 a n d vanishes o n I - a , O l . Since w , a s a function of ( t . 5 ) t h e expression where H:IR-IR

to t h e right of t h e arrow in ( 7 . 6 0 ) defines a t e m p e r a t e distribution if and only i f o v l w l - ' = R e , ?0O , i.e. b v _ > O .

Notation 7.71. We fix

1 , - 1 ) s u c h t h a t o = s i g n v if v f O , a n d d e n o t e by E t h e

t e m p e r a t e distribution o n R x IR" w h o s e partial Fourier t r a n s f o r m with respect t o x is induced by t h e function ( 7 . 0 0 ) . If v = O and O = 2 1 we a l s o w r i t e E'

in-

s t e a d of E .

I t is t h e n obvious t h a t E is a fundamental solution of q ( D ) . In o r d e r to e x p r e s s

E without r e c o u r s e t o t h e partial Fourier t r a n s f o r m one makes use of q e x p ( - a l . 1') where

(

*

)'/'denotes

= (rr/a)t1/2exp(-I * ?/Qa),

acC\(O), Ima20,

t h a t branch of t h e complex r o o t function defined o n C \ l - ~ , O l

which is positive o n I O . + ~ C .Introducing (7.61)

w

e ( t . x ) : = ( o w ) ' " - 2 ) / 2 H ( a t ) ( 4 ~ 6 t ) - " / ~ e x p- t( l x I

2

)

we arrive a t

O n e observes t h a t t h e s u p p o r t of E is contained in the half plane I S C O , + ~ ~ X I R " .

its analytic singular s u p p o r t being equal t o t h e hyperplane ( 0 ) x i R " . Note t h a t a

340

V I I . Solvability of Q u a s i h o m o g e n e o u s Equations

priori t h e order of integration cannot be changed. However, in case v f O t h e function e is Cm outside t h e origin, and t h e order of integration in ( 7 . 6 2 ) is arbitrary. In particular, q is hypoelliptic in this case. Moreover, o n e observes t h a t E is invariant of degree -n (in t h e s e n s e of Definition 7.70). The main object of the present section is to prove

Theorem 7.72. ( i ) I f v # O then E is the unique fundamental solution of q ( D I which i s almost invariant. (iil

If v = O then a distribution FEB'(IRx1R"I is an invariant fundamental solution

o f q ( D I if and on/), i f i t i s o f the f o r m

F=

(7.631

in particular, E'

f o r some Z E C ;

(I-ZIE++ZE-

is the unique invariant fundamental solution w i t h support con-

X . tained in the h a l f space -f LO, + ~ C IR"

proOf. First of all we note t h a t by Example 2.2 and Corollary 2.36.(i) every a l m o s t invariant fundamental solution of q ( D ) is so of degree - n . Since t h e difference

T of t w o fundamental solutions is a solution of t h e homogeneous equation (7.64)

q(D)T=O

we have to determine all solutions T E ~ ' ( I R X [ R "of ) ( 7 . 6 4 ) which are a l m o s t invariant of degree - n . For t h e case v f O this is done in Theorem 7.75 below. In case v = O o n e has t o employ Theorem 7.77'. I t shows, i n particular, t h a t t h e space of

invariant solutions of ( 7 . 6 4 ) is 1-dimensional so t h a t it is spanned by E ' - E - , indeed.

rn

First of all we determine all a l m o s t invariant polynomial functions. Note t h a t by Corollary 1.36 they are all invariant.

h p O d ~ 0 1 7.73. 1 Let

Q:R xlR"

-----i,

C be a non-constant polynomial function, and

let P E C . Then Q is invariant o f degree P i f and only i f P is an even natural number and there are comples numbers a o , . P/2

(7.65)

Q ( ~ ~ S a Il r j=/ l i~e - 2 j ,

j=o

. . , a p / Z such

that

341

7.f T h e H e a t and t h e Schrodinger Equation

h.oof. Let

el b e a unit v e c t o r in IR”. For any fixed X E ! R ” \ ( O ) w e c h o o s e S € O ( n )

such that Sx = sel where s := I x

I . I f Q is invariant of d e g r e e P t h e n

w h e r e P ( z ) : = Q ( z , e l ) , Z ~ C T.h i s m e a n s t h a t d

Q(t,x)

=c

a j r i Ixle-*j

j=O

w h e r e d : = d e g P a n d w h e r e a j : = P ( J ) ( O ) / j ! . Since

X H I X I ~ - ~ ~IS ‘

a polynomial

function if a n d only if P-2d is a non-negative integer t h e proof is c o m p l e t e .

H

N e x t w e a r e going to d e t e r m i n e t h e polynomial s o l u t i o n s of ( 7 . 6 4 )

Pmporltion 7.74. Let k € N O ,a n d l e t

where

Moreover, let Q : R x R ” - C

be a p o l j n o m i a l f u n c t i o n which i s invariant o f de-

gree -7k. Then q ( D ) Q = 0 i f and on1-v i f Q = z Qk for s o m e z 6 C . P r o o f . By Proposition 7 . 7 3 Q is of t h e f o r m (7.65) with P : = 2 k . We may a s s u m e t h a t Q + O so t h a t t h e l a r g e s t index d € N O satisfying a,,#O is well-defined. T h e n

k‘t 2 d . N o w , for any a = 2b E NO w e have

a n d h e n c e A,

151“ =

a (n+a-2)

C o n s e q u e n t l y , w e deduce

Since a d # O t h i s vanishes identically if a n d only i f

342

(7.67.A))

V l l . S o l v a b i l i t y of Q u a s i h o m o g e n e o u s E q u a t i o n s

(4-2d)(n+t-Zd-2) = 0

and (7.67.8)

w ( j + l ) a j + , - aj (P-Zj)(n+4-2j-2) = 0 ,

O < _ jC d - 1 .

Since 4 - 2 d is even a n d non-negative t h e equation n+Q-Zd-2 = 0 c a n only be valid

if 4 = 2 d a n d n = 2 . Hence (7.67.A) is equivalent to 4 = 2 d , i.e. d = k . T h i s , in particular, implies t h a t ( P - Z j ) ( n + t - Z j - Z ) t 2 n > 0 if O S j S d - 1 . H e n c e , if (7.67.A) is a s s u m e d to b e valid t h e n (7.67.B) a m o u n t s to

By induction t h i s is s e e n to be equivalent to O<j b e l o n g s to 6,(

V* x V )

so t h a t b y T h e o r e m 5 . 4 2 - a p p l i e d to ( V * x V , M * x M ) i n s t e a d o f ( V , M ) - f m , , j , E is a w e l l - d e f i n e d Cm f u n c t i o n b e l o n g i n g to Y';**M(

function v , , ~ : I O , + ~ [ - [ R

V'x V ) w h e r e t h e w e i g h t

is d e f i n e d by ~ ~ , :~= u( j (t 2)t ) e - 2 t t . N o t e t h a t by ( 1 . 6 4 )

v ~ is, a ~l i n e a r c o m b i n a t i o n of t h e f u n c t i o n s w i , Z E , O < i < j . If R e m < 0 w e d e d u c e

by s u b s t i t u t i n g t = 2 s t h a t +cn

( 8 . 9 ) K:,,([,x)

=J't-mexp(-i<M,",2)wj(t)

t =2-mf

m,v,

,r(t x ) . 9

0

By T h e o r e m 5 . 4 2 a n d by t h e principle of a n a l y t i c c o n t i n u a t i o n t h i s r e m a i n s valid f o r e v e r y m E @ \ ' U , ( M ) (see (S.31));in f a c t , t h i s e s t a b l i s h e s a n e q u a l i t y of m e r o -

8.a Pulling

361

Back D i s t r i b u t i o n s o n S x

m o r p h i c f u n c t i o n s o f m E C . C o m p a r i n g t h e Laurent coefficients a t t h e p o i n t s m E X , ( M ) w e obtain t h a t

( t ,. )

2 07

= 2-m

KA.j

7 1 (-1og2)J a - j ( m ; m f , , .

j = 0

1.

) ( M = O . M o r e o v e r , s i n c e by P r o p o s i t i o n 3.51 w e find + E C g ( X + ) s u c h t h a t P,(f) a n d s i n c e s u p p Q;(xxT)

C

=+m+,wk

X \ X + w e see t h a t

M = =O. C o n s e q u e n t l y , w e derive f r o m c o n d i t i o n ( d ) of T h e o r e m 8.8 t h a t (8.12)

M=

[

k

(-l)k-i -p t h e n t h e d i s t r i b u t i o n fm.wk is c o m p u t e d as in E x a m p l e 8.17:

Remark 8.29. I f Re m fm,wk

> -ji

defined by q / x , x

+

then fm.wk = Tq where q : X

+C

is the e.\tension

of

: = O . Note that q is locall>, integrable, indeed. and in

case R e rn > 0 even continuous (in Proposition 1.91 q was denoted b> fn,,LJk) .

H. In view

of T h e o r e m 5.37.(iii) w e d e d u c e f r o m Proposition 1.91 t h a t T,

a n e x t e n s i o n o f Tf

mscJk

which i s a l m o s t quasihomogeneous of d e g r e e m . Since

R e m > - p T h e o r e m 8.15' s h o w s t h a t T, e x t e n s i o n of T,

is

is t h e unique a l m o s t q u a s i h o m o g e n e o u s

. Hence by Theorem 8 . 2 8 it coincides with F m , w k .

m.Uk

m

W e close t h i s s e c t i o n by verifying t h a t t h e s t a n d a r d r u l e s of c o m p u t a t i o n which

are valid for f m , w k carry over to t h e i r e x t e n s i o n s f m , w k . N o t e t h a t in a s e n s e t h i s is in c o n t r a s t t o Proposition 8.19.

378

VIIl, Extending ( A l m o s t ) Quasihomogeneous Distributions

PropoaltJon8.30. ( i ) L e t N E N o a n d P E C , a n d l e t P o : X x V * + C b e a

Cmcopoly-

nomial function which i s almost quasihomogeneous of degree P , of type M x ( - M ) *, and of order 5 N . Then

and N

1)I;'( j=O

(8.30)'

P j ( s , d ) irnSwk+. J

(Po(x,d)f),+p,,,k =

where the copolynomial Functions Pi are defined bj (4.71 ( i i ) For ever,. PEC and ever,- C'?' Function q :X

-+

C which is almost quasihomo-

geneous of degree P we have N

(8.311

qirr,.uk =

s

)i''

(qj'f)rr,+t. 0 so s m a l l t h a t

Now s u p p o s e t h a t ( 8 . 3 2 ) h o l d s . By Remark 8 . 2 0 and by t h e definition of T,,,,',,~ w e then conclude that

< Gm,,+, x > = .I'$ J ~ , ~ , ,1 ~x ( xx ) d x = J v+

. +,m - m

\

t

cp(

M,x)

x(x )

d t dx t)T

v+ 0

for every x € C F ( V ) . Since (bm,l,,k

is t e m p e r a t e it f o l l o w s by Lemma 8 . 3 3 a n d by

t h e Fubini t h e o r e m t h a t frn

(8.33)

< +rn,'.,k

x > = J' I

0

t - m \ ' c p ( M , x ) x ( u ) dw ( , i k ( t )-d,t t

X€YP(V).

V

Now w e fix x E Y ( V * ) . Applying ( 8 . 3 3 ) to

^x

i n s t e a d of

x,

using Parseval's e q u a t i o n ,

taking t h e e q u a l i t j ?F(cpoM,) = t-" $OM:/, ( s e e ( 2 . 8 ) ' ) i n t o a c c o u n t a n d substituting s = l / t

we obtain +m

+m

T h e l a s t d o u b l e integral is equal t o

< ($)m*,c,,k,x )

to ( V * , M * , m * , $ ) i n s t e a d of ( V , M , m , c p ) .

I

as o n e sees by applying ( 8 . 3 3 )

382

VIII. Extending ( A l m o s t ) Quasihomogeneous Distributions

h o o f . Since (bo is t h e unique a l m o s t q u a s i h o m o g e n e o u s e x t e n s i o n of TVo w e 'po I 1 H&,=TI

conclude t h a t B(+o) = ($)-,,

.

7(&)= ( 2 ~ ) 6", .

By T h e o r e m 8.31 w e have

H e n c e t h e a s s e r t i o n follows.

C l o s i n g t h i s s e c t i o n w e f i x a Cm function + E O , ( V )

with M - t e m p e r a t e s u p p o r t .

A

Recall t h a t i t s Fourier t r a n s f o r m s J, lies in O k ( V * ) , t h e s p a c e of c o n v o l u t i o n operators o n

Y (V* ) .

" Theorem 8.35. Let X E Y ( V * ) . Then t h e convolution p r o d u c t J, *im,LJk is i n d u c e d A

Note t h a t t h i s is a f o r m u l a for t h e Fourier t r a n s f o r m of

in t e r m s of ^u

provided t h a t J, is equal to 1 o n t h e s u p p o r t of u . Recall t h a t by L e m m a 5 . 5 3 o n e c a n c h o o s e J, so a s to have t h i s property provided t h a t s u p p u is M - t e m p e r a t e . Proof of Theorem 8.35. By Remark 5.40 t h e function

J,2rn*,wkb e l o n g s

to Y ( V ) .

Hence it f o l l o w s f r o m Theorem 8.31 a n d f r o m t h e Fourier inversion f o r m u l a t h a t

v ( + 2 1 ~ * , ' , k ) = v ( J , ~ m =(2x)-" * , ' J k ) 5*( F ( ; m * , w k=) ) h

A

=

(2K)rn A

In particular,

A h

J,*j(m,cdk=

A

A

"

J,*j(In,',,k.

V

1

im,a,k b e l o n g s to Y ( V * ) .C o n s e q u e n t l y ,

Fourier inversion f o r m u l a t h a t

w e c o n c l u d e by t h e

383

Chapter IX

Quasihomogeneous Wave Front Sets

The present chapter contains the basic theory of quasihomogeneous wave front sets of type M where t h i s t i m e M is a linear endomorphism of V* s u c h that all

of its eigenvalues have positive real part. These types of wave front sets generalize the classical notion of (homogeneous) wave front set (see Hormander C111) which appears a s the special case M=ld,*.

As in t h e classical case, they lead

to a refined description of the singularities of distributions. In fact, the main result of the present chapter (Theorem 9.34 in section ( c ) below) shows that the singu-

larities of distributions on V which are quasihomogeneous of type M* are best described in terms of quasihomogeneous wave front s e t s of type M . With the quasihomogeneous wave front s e t s the basic idea is as follows: for every

v ~ 8 ' ( V )one keeps track of the behaviour of $ along the quasihomogeneous rays

{ M , < ; r E C l , + o o C } , < € S V * ,where by S p we denote the u n i t sphere of V* w i t h respect t o a scalar product satisfying (1.79) w i t h V replaced by V*. In case M is a real diagonal matrix quasihomogeneous wave front sets have been introduced

by R. Lascar in C121 in the context of quasihomogeneous pseudodifferential operators.

In section ( a ) w e treat the basic properties of the quasihomogeneous wave front s e t s keeping as close a s possible to Hormander's way of presenting the theory of (homogeneous) wave front s e t s in C111 (compare also

5 1.6 in

Hormander 1101).

In a similar spirit in section ( b ) we introduce Gevrey type versions of quasihomo-

geneous wave front sets which include those by Rodino C161 ( s e e also Liess-Rodin0 C131 and the literature cited there). Section ( c ) is devoted to the main theorem of t h i s chapter alluded to above. In addition, two further propositions on wave front set inclusions are given. Again, all these results generalize corresponding results in C 11 1 .

384

IX. Q u a s i h o m o g e n e o u s Wave Front Sets

C o n c r e t e e x a m p l e s are treated in s e c t i o n ( d ) . A s s u m i n g t h a t M is a real d i a g o n a l m a t r i x w i t h e n t r i e s of t h e f o r m p = ( r , I , .

.., I )

f o r s o m e r E C I , +a[,w e c o m p u t e

t h e q u a s i h o m o g e n e o u s w a v e f r o n t sets o f t h e invariant f u n d a m e n t a l s o l u t i o n s o f t h e h e a t a n d of t h e S c h r o d i n g e r o p e r a t o r s t u d i e d in s e c t i o n 7 . ( f ) .

As a l r e a d y i n d i c a t e d a b o v e , f o r t h e w h o l e c h a p t e r w e fix M E L ( V'

)

s u c h t h a t Re X > 0

f o r every X E a ( M ) . W e a s s u m e t h e C o n v e n t i o n s 1.24 a n d 1.24' to be valid w i t h M r e p l a c e d by M*, c o n s i d e r i n g M* as a n e l e m e n t o f L ( V ) via t h e c a n o n i c a l identific a t i o n of V*'

w i t h V . N o t e t h a t , in p a r t i c u l a r , N o t a t i o n s 1.25, 1.29, a n d 1.33 a s

w e l l as R e m a r k 1.43 a r e to be u n d e r s t o o d in t h i s s e n s e . In t h i s and t h e f o l l o w i n g s e c t i o n X is n o t r e q u i r e d to be q u a s i h o m o g e n e o u s o f a n y t y p e . Let T E % ' ( X ) , a n d let x O C X . T h e n by t h e Paley-Wiener

t h e o r e m xo

does n o t b e l o n g to s i n g s u p p T if a n d o n l y if t h e r e is a test f u n c t i o n q C C ; ( X ) s a t i s f y i n g cp(x0) # 0 s u c h t h a t (9.1)

s u p { I F ( ~ ~ T ) ( Is S) II ~ S; E V * \ K ( O , I ) J < +a f o r every N E I N .

S i n c e V * \ K ( O , I ) = U r E C , , + 0 3 C M T ( S ~w*h )e r e S v * d e n o t e s t h e u n i t s p h e r e in V* a n d s i n c e by Lemma 1.7 f o r every E > 0 t h e r e e x i s t c o n s t a n t s c, , d E> 0 s u c h t h a t

(9.2)

cE

T X m i n - ~151 5

w h e r e A,,,:=min{ReX;

l M , t I 5 d E T C X m a Y151, +C

T E C l , + c o C , <EV*,

X E a ) a n d A,n,,:=max{ReX;

X E ~ }t h e c o n d i t i o n (9.1)

is e q u i v a l e n t to

Defldtion 9.1. WF,,,,(T) is by d e f i n i t i o n t h e c o m p l e m e n t in X x S v * o f t h e set of all (x,-,, ;x E K } ;

f o r v E & ' ( V ) w e also w r i t e H , : = H s u p p u . Finally, in s e c t i o n ( b ) w e have to d i s t i n g u i s h t w o c a s e s , n a m e l y . w h e t h e r or n o t M is s e m i - s i m p l e . T h e r e a s o n a p p e a r s in t h e f o l l o w i n g s l i g h t l y m o r e p r e c i s e f o r m of ( 9 . 2 ) w e have to w o r k w i t h : in

order to be able to h a n d l e b o t h c a s e s s i m u l t a n e o u s l y w e i n t r o d u c e t h e n o t a t i o n

PM

(9.4)

if M is s e m i - s i m p l e

:=

if M is n o t s e m i - s i m p l e

A,)[

a n d n o t i c e t h a t by L e m m a 1.7 w e c a n c h o o s e a family of c o n s t a n t s c , , d, > O , f

EeM,

(9.2)'

in s u c h a way t h a t 2

cL.

y

T

2ReX-2c

15x12

5

XEd

f o r e v e r y E E ~ , w h e r e by

1 o n U t h e r e is xEC:(U)

such that

n o t depending o n

'p

iiaPxiiL,

x + = ' p . S i n c e by t h e Leibniz r u l e t h e r e

are c o n s t a n t s Ag

such that 5 Apsup(

t h e conclusion follows f o r

iiaav,iiL,;

CL

5

PI,

DEX.

( 2 . t )= ( U , K ( E , , E ) ) .

S t e p ? : W e fix x o € X . By S t e p 1 a n d by t h e c o m p a c t n e s s of F w e f i n d a f i n i t e

o p e n c o v e r i n g 93 of F a n d a family ( U w ) w E g s o f n e i g h b o u r h o o d s o f xo s u c h

389

9.a The Wave Front Set WFMVI(T)

that for every WET8 t h e conclusion holds w i t h

t h e n the conclusion is also valid for

(c,?)replaced by

( U w , W ) . But

(c,?)= ( ( - l w E m U w , u T 8 ) .

Step 3 : By Step 2 and by the compactness of K we find a finite open covering

U of K and a family ( F u ) , E u

of open neighbourhoods of F in V*\(O) such that

for every U E U the conclusion holds for ( U , F u ) instead of *

N

F := n u E u F U

let

and fix a compact neighbourhood K of K i n be a Cm partition of u n i t y on

Applying the assertion to

2

(k,p). Now UU.

Moreover, we

subordinated to the covering U .

x u r p and observing that by the Leibniz rule we have

IdP ( x u ' p ) I 5 B p , s ~ p l ( ' p ( ~ ) (a~ w > p 2 + .. . + p

. RPk

R,;,

k

w e derive t h e f o l l o w i n g i d e n t i t y of o p e r a t o r s

2

1 =S,-

R,oS,+E,.

PEBm

C o m b i n i n g t h i s w i t h (9.17) - a p p l i e d to w =S,(x)

-

and taking (9.14)' into account

we conclude t h a t ,\,

A

u(M,l.

I

Notatlon9.13. F o r any N E N , w e w r i t e L N : = L ( N )

Definltlon 9.14. Let h:'U-IO,+coC

be a n a d d i t i v e f u n c t i o n . By

d e n o t e t h e s p a c e of Cm f u n c t i o n s f : X + C

C h V L ( X ) we

s u c h t h a t f o r every c o m p a c t s u b s e t

K o f X t h e r e is a c o n s t a n t C s u c h t h a t (9.33)

If'a'(x)l 5 C(CLJh'='

,

x E K , cre'u,

where L,:=(Loh)(a).

Since (9.34)

m i n { h ( i ) ; ~ € 3l t l,= I ) la15 h ( u ) 5 m a x { h ( t ) ; ~ € 3l t ,l = l } l a l ,

a€'&,

we observe

Remark 9 . E . I f s u p l L ( Z t ) / L ( t ) ;t E C O . + a C } is finite then in (9.33) the term L , can be replaced b-v L , , ,

.

8

Remark 9.16. C"'L(X)is a C-algebra which is closed under differentiation.

397

9.b T h e W a v e Front S e t with Respect to C M , L

mf. Let f , g E C h S L ( X ) ,let

K be a compact s u b s e t of X , and let C be a con-

s t a n t such t h a t (9.33) is satisfied for f and g . Since L is increasing we obtain by t h e Leibniz rule t h a t f o r every

C

c2(CL,)

h(a)

L

p

XE

K

( F ) = C22'O1' ( C L , ) h ( a )

La

. by ( 9 . 3 0 . b ) we In view of (9.34) t h i s implies t h a t f g belongs t o C h V L ( X ) Since have (La+, it f o l l o w s t h a t f o r every

LE(W

t h e function a L f belongs t o C h ' L ( X ) , as well.

rn

We now introduce t h e spaces C M S L ( X ) .As already indicated above, we have to distinguish t h e c a s e s where M is semi-simple and is n o t . In o r d e r t o be a b l e t o handle b o t h c a s e s simultaneously we introduce

Notatlon 9.17. From now o n C ~ + X ): =

we set

h ( a ) : = ReaM and define

n

Ch,Li+n(x) rl€VM

where (fM is defined in ( 9 . 4 ) .

We s u p p o s e from now on that

In view of Lemma 1.3.6 in Hormander C111 t h i s assumption is necessary f o r t h e following lemma to hold.

Lemma 9.18. Let K be a c o m p a c t s u b s e t of V . a n d l e t It be a f i n i t e o p e n covering o f K . Then t h e r e eAist c o n s t a n t s C and A g ,

p€z,and

functions

XN,Ll

ECF(U),

U ~ l and l N E N , w i t h values in CO.11 s u c h that 3

x

~ - 1 ,on K~

UELl

and s u c h t h a t for arbitrar-p p E 2 l

x

~

s a t i, s f i e s~ (9.5) ~ for ( A , B ) = ( A g , C N )

398

I X . Q u a s i h o m o g e n e o u s W a v e Front Sets

Bf. Since

(9.35) implies that

( 9.35 1'

IQI O s u c h t h a t u € C M B Lo n K ( x o , 3 ~ ) . By Lemma9.111 w e

choose a constant C > 1 and a sequence of functions e q u a l t o 1 o n K ( x ~ , E s)u c h t h a t ( a ' X N I < - C ( C N ) h ( a )

sup{ IIaBxNIIL,;

xN ECg( K ( x o , Z E ) ) ,

N EN,

if la1 < N + 1 a n d s u c h t h a t

N E N } is f i n i t e for e v e r y P E X . I t f o l l o w s f r o m t h e last c o n d i t i o n

t h a t t h e sequence of distributions

is b o u n d e d in

&'(X). M o r e o v e r , by

5 La+,. 5 L ( N + A,,,,,)

5 A ' + X m a x LN ( b y

UN:=XNU

t h e Leibniz r u l e w e o b t a i n

S i n c e N 5 LN ( b y ( 9 . 3 0 . a ) ) a n d La+,.-P I

I

(9.30.b)) this implies t h a t if R e a M 5 N a n d tEXl where D : = 2 C A

l+hrnax

. Fixing

a n d a p p l y i n g Lemma 9 . 2 ( t o k = 0 = P ) w e

q EWM

conclude t h a t I G ~ ( M , < ) 5I a C 2 (

D

S i n c e by (0.31) w e have LN 5 A"'

x: F i r s t of a l l w e f i x q E W M

L

~( b () D L~N ) ' ~+ ' / T~)

N

,~

< € S V * r, . E [ l , + c o [ .

t h e assertion follows.

a n d t h e n p E W M s u c h t h a t ( l + p ) 3 < - l + q . Let a E X .

W e fix k E N s u c h t h a t k 2 p + l a n d c h o o s e N E I N s u c h t h a t (9.37)

N-1 5 ( t + p ) h ( a ) < N

W e d e d u c e f r o m ( 0 . 2 ) ' t h a t t h e r e is a c o n s t a n t B,.O

n o t depending o n a such

that (9.38)

l(M,E)"I 5 B , T ( ' + ~ ) ~ ( ~ ) ,

TECl,+coC, < E S V * .

Combining this with (9.36) and (0.37) w e obtain for arbitrary rECl,+mC a n d <ESV*

E m p l o y i n g q u a s i h o m o g e n e o u s p o l a r c o o r d i n a t e s (see P r o p o s i t i o n 1 . 8 6 ) a n d t h e Fubini-Tonelli t h e o r e m w e c o n c l u d e t h a t t h e f u n c t i o n d e f i n e d by is i n t e g r a b l e o n t h e set Z : = V * \ K ( O , I ) s u c h t h a t


O , a n d a b o u n d e d s e q u e n c e

(UN)NEN

in & ' ( X ) s u c h t h a t

f o r every N c N uN is e q u a l to T o n U a n d (9.36) is s a t i s f i e d f o r c f k ( f o , 2 ~ ) . W e f i r s t p r o v e t h e a s s e r t i o n for ( U , K ( E , ~ , E )i )n s t e a d of

(2.:).

Since

(uN)

is

b o u n d e d in & ' ( X ) w e f i n d c o n s t a n t s W a n d s E C O , + a C s u c h t h a t u N s a t i s f i e s ( 9 . 8 ) f o r every N E N . W e set A : = s u p ( A p ; I P l < k + 4 + 1 } w h e r e k a n d 4 are c h o s e n s o a s to f u l f i l l t h e r e q u i r e m e n t s of Lemma 9 . 3 . In view of (9.3Y) a n a p p l i c a t i o n of L e m m a 9.3 yields c o n s t a n t s c , b , d s u c h t h a t

I7 ( X N T ) ( M , 5 ,

IJ)

I = I7 ( x NuN+,.) ( M, €, , I J ) I

5

5 e x p ( H X N ( i ~ )( )c A r - ' ( b ( A L ~ ) ' + " / r ) ~ d+A C , ( C , ,

( L N + , . ) ' + " / T ) ~ +) ~

f o r a r b i t r a r y €,EK(€,,,,E), I J E V * , a n d r E C l , + a C . S i n c e by ( 9 . 3 0 . b ) a n d (9.31) w e have LN+,. 5 A r + ' m i n { A N , LN} w e d e d u c e t h e d e s i r e d e s t i m a t e s . S t e p ? : By S t e p 1 a n d by t h e c o m p a c t n e s s of F w e f i n d a f i n i t e o p e n c o v e r i n g

X ' 3 of F a n d a f a m i l y (UW)WEm of o p e n n e i g h b o u r h o o d s of xo s u c h t h a t f o r every W ~ u t3h e c o n c l u s i o n h o l d s f o r ( U w , W ) i n s t e a d of _

N

t h a t t h e c o n c l u s i o n r e m a i n s valid f o r ( K , F ) = ( n w , , m U w ,

( i , : )I t. t h e n urn).

follows

S t e p 3 : By S t e p 2 a n d by t h e c o m p a c t n e s s of K w e f i n d a f i n i t e o p e n c o v e r i n g

U of K a n d a family ( F u ) u , u

of o p e n n e i g h b o u r h o o d s of F in V * \ ( O ) s u c h t h a t N

_

for every U E U t h e c o n c l u s i o n h o l d s f o r ( U , F u ) i n s t e a d of ( K , F ) . Now w e set N

F : = nLIE,FU

a n d fix a c o m p a c t n e i g h b o u r h o o d

i

of K in U U . M o r e o v e r , by

L e m m a 9 . 1 8 f o r every N E N w e c h o o s e a C m p a r t i t i o n of unity

N

(

x

~

,

~

o n) K~

s u b o r d i n a t e d to t h e c o v e r i n g U a n d s a t i s f y i n g t h e c o n d i t i o n s of L e m m a Y . 1 8 . S i n c e b y t h e a r g u m e n t a t t h e beginning of t h e proof f o r every U C U t h e f u n c t i o n s

x

~

x s a t~i s f y,

(9.39) ~ w i t h p o s s i b l y n e w c o n s t a n t s it f o l l o w s t h a t t h e desired w

e s t i m a t e s are valid f o r E, C F w i t h

xN r e p l a c e d

by XN X N , U . S i n c e

t h e a s s e r t i o n f o l l o w s by t h e t r i a n g l e i n e q u a l i t y .

C

u = xN

XNx N ,

UEU

C o m b i n i n g P r o p o s i t i o n 9.1'~w i t h Lemma 9.22 o n e o b t a i n s t h e f o l l o w i n g a n a l o g u e

of P r o p o s i t i o n 9 . 5 .

ROpo8ltlOn 9.23. Under the assumption of Proposition 9.19 the projection of the

,

~

403

9.b T h e Wave Front Set w i t h R e s p e c t to CMvL

set W F M , L ( T )on the First factor is equal to s i n g s u p p ~ , ~ ( Twhich ) i s by definition the smallest closed subset o f X outside which T i s induced by a C M P Lfunction.

I

F r o m t h e f o r m u l a s c o m p u t e d i n t h e p r o o f of Remark 9 . 6 w e i m m e d i a t e l y d e d u c e

Remark 9.24. The assertions of Remark WFM,. throughout.

9.6 remain valid i f WFM is replaced by

I

A s a n o t h e r c o n s e q u e n c e o f L e m m a 9.22 w e o b t a i n

Proof. Let -

( x . < )E X x S , * \ W F M , , ( T ) ,

and let

E >

0 be s u c h t h a t K ( x , 2 ~ C) X a n d

K(X,ZE)XK(S,E)~WF,,,(T)=(~. By L e m m a 9 . 1 8 w e f i n d c o n s t a n t s C , A p ? l a n d

f u n c t i o n s x N E C ; ( K ( x , 2 ~ ) ) e q u a l to 1 o n

K ( x , E ) with

for arbitrary N E N and @ E X t h e estimates

(9,s) a r e

v a l u e s in C0,ll s u c h t h a t

satisfied for

(xN , A p , C N )

instead of ( x , A , B ) . N o w , by R e m a r k 9.16 f o r every PEU a'f

b e l o n g s to C M ' , ( X ) . H e n c e , f o r a n y

qc(fM w e f i n d a c o n s t a n t A; s u c h t h a t l a a + P f ( x ) l 5 A; ( A ; ( L ~ ) ' + ~ ) ~ ( ~ ) ,x E K ( x , 2 ~ ) a, E X , h ( u ) 5 N .

Applying t h e a r g u m e n t a t t h e b e g i n n i n g of t h e p r o o f of L e m m a 9.22 to rpN = f w e see t h a t t h e f u n c t i o n s f x N s a t i s f y t h e e s t i m a t e s ( 9 . 3 9 ) w i t h p o s s i b l y n e w c o n s t a n t s A g . C o n s e q u e n t l y , L e m m a 9 . 2 2 i m p l i e s t h a t ( x . 0 does n o t b e l o n g to

WFM.L(T).

In view o f (9.11) a n o t h e r a p p l i c a t i o n o f Lemma 9.22 y i e l d s WFM,,(DaT)

C WFM,,(T).

Combining this with Proposition 9.25 w e obtain

Proporltlon 9.26. WFM,L(P(x.3) TI C WFM,L(T) for every linear differential ope-

rator P ( x , 3 ) on X with coefficients in C M ' L ( X ) .

I

404

IX. Quasihornogeneous

Wave F r o n t Sets

A s for the converse inclusion, the expected assertion holds:

Theorem 9.27. Let m € R e 2 l ( M ) , and let P be a differential operator a s in Theorem 9 . 8 . Suppose that i t s coefficients are real analytic functions a ,

C.

:X

Moreover, suppose that i t s quasihomogeneous principal part P,r, :X x V *+

C de-

fined in t h e t e s t preceding Theorem 9 . 8 i s quasihomogeneous o f degree m . Then

T€.D'(X).

W F M , , ( T ) C W F M , L ( P ( ~ . d l Tu IP r i ' ( 0 ) .

(9.40)

Proof. The proof of Theorem 8.6.1 in Hormander I111 is suitably modified. E X xSv4 be such that it does not belong to the right-hand

Let

side of

( 9 . 4 0 ) . Then we can choose compact neighbourhoods K of xg in X and W of E0 in V x \ { O ) , a family of constants C , , q E V M , and a bounded sequence

(vN)N~IN

i n & ' ( X ) such that for every N E I N (9.14) and hence (9.14)'are valid w i t h v re-

placed by

vN,

such that

I ~ ^ ~ c M , E5, )c I, , ( c , L ; * ' / T ) ~ ,

(9.41)

SEW, rE[l,+03[. T e V M ,

and such that (9.16) holds. Applying LemniaO.18 to M = l d v we find a sequence of functions x ~ E C ~ ( K ) equal to 1 on a fixed neighbourhood U of x g and constants Cp such that (0.42)

Then the distributions

ct,p

=

cN(M, < -

du()]OM,*,,> =


,

yEV.

I

t6JO.+~~l,

i=O

where T i : = ( d M * - m ) ' T . N o w we decompose each Ti according to Ti = q T i + ( I - ( p ) T i and first compute - making use of ( 3 . 8 ) ' (T

=

T ~ r, y (

^x

0

M

t ) ) = < Y T ,~t

-p

T ( e i ( l + r + R e m + y + k X m a , ) / X m i n

where r : = m a x { R e p M ; @ E x , I P l S k } we obtain a constant Iap($NoM:)(Z-y)l

c,,k

such t h a t

5 C m , k( l + I M : ( Z - y ) l ) - k t-CI-Rem-l (C,,kN1*,/t

)N

for arbitrary N E N . t E C l , + a C , ~ E K ( x , E )z ,E V \ K ( ~ , E )and , BE'u satisfying I P I < - k . In view of ( 0 . 5 6 ) and (0.58)and by t h e Leibniz formula one obtains new c o n s t a n t s cA,k

such t h a t s u p { ( l + l ~ Id'[(l-cpN) ()~ 5

r Y ( i N o M : ) ] ( z ) ( ; la15 k , z C V , ~ € K ( x , E ) i}

c-dk t - ~ - R em-1 (C,,kN"'/t

N

)N,

E N,

t E C 1 ,+a[.

Since Ti is temperate it follows t h a t for some k E I N there is a c o n s t a n t C;' only depending o n q such that Itrn+'+' < ( l - q ~ ) T Ti ,y

A

>I

( X ~ O M t )

C

c;

(C,,kN"q/t)N

. this with ( 9 . 6 0 ) and f o r arbitrary N E N . t € C l , + m C , and y € K ( x , ~ ) Combining A

(0.54) we conclude t h a t ( < , - x ) +! WFM*.,(T), a s desired. Proof of ( i i i ) . By t h e Fourier inversion formula, again, t h e assertion ( i i i . b ) follows from ( i i i . a ) . In order to prove ( i i i . a ) w e first suppose that for some k E N t h e support of t h e distribution Tk : = ( a M * - t n ) k T is equal to ( 0 ) .In t h i s case we have

(a,*

and since by Lemma 2.21 the distribution

n

h

- m * ) k T coincides with

(

- l ) , T,

and hence is induced by a non-trivial polynomial function o n V* t h e s u p p o r t of n

T equals t h e whole of V*, a s desired. Tk is induced by a polynomial function,

Next we suppose t h a t f o r some kEN, n

i.e. t h e restriction of T to V*\(O) is almost quasihomogeneous so t h a t by Pron

position 2.37 t h e s u p p o r t of T is quasihomogeneous which, in t u r n , implies t h a t h

n

(supp T )M,ao = S p n supp T

. n

Hence Proposition 9.31 s h o w s that W F M , L ( T ) C V x s u p p T . Consequently, to com-

9.c Wave F r o n t Sets of A l m o s t Q u a s i h o m o g e n e o u s

415

Distributions

A

p l e t e t h e p r o o f o f ( i i i . a ) w e have to verify t h a t t h e c o n d i t i o n " < ~ S , * n s u p p T " implies

"

( O , < ) E W F M ( T ) ".

To do t h i s , w e c h o o s e a polynomial f u n c t i o n s P o n V s u c h t h a t u : = T - P i s a l m o s t q u a s i h o m o g e n e o u s o f degree m a n d o f t y p e M * . In f a c t , by T h e o r e m 8.15 t h e r e A

is a n e x t e n s i o n S E Y ' ( V * ) o f T I P , ( o ) which is a l m o s t q u a s i h o m o g e n e o u s o f deA

gree m * ; a n d s i n c e s u p p (T-S) C ( 0 )t h e d i s t r i b u t i o n T - B - * S is i n d u c e d by a poly-

n o m i a l f u n c t i o n P , a n d u = 7 - ' S is a l m o s t q u a s i h o m o g e n e o u s o f t y p e M*. i n d e e d , by P r o p o s i t i o n 2 . 4 0 . ( i ) . A

N e x t we o b s e r v e t h a t W F M ( T ) = W F M ( U ) a n d T = G + ( 2 ~ ) P" ( -D)S, so t h a t A

(suppT)\(O) = (supp G)\{O). T h e r e f o r e it s u f f i c e s to p r o v e t h e a s s e r t i o n f o r u i n s t e a d o f T . S i n c e ^u, being e q u a l to S , is a l m o s t q u a s i h o m o g e n e o u s of degree m * a n d s i n c e by P r o p o s i t i o n 2 . 4 0 . ( i i ) w e have ( & M - m * ) i G = ( - l ) i G j w h e r e u j : = ( a M f - r n ) j u it follows that (9.61)

A

u =x

-m'

k

TEIO,+~C,

w j ( r ) GjoM, j=O

w h e r e k : = ordM( u ) = OrdM* ( u ) . A

N o w , let E , E S ~ * .a n d s u p p o s e t h a t (O,E,)

< WFM(u).

every j E N , , a n d w e c a n c h o o s e XEC:CV)

a n d ~ E l O , 1 1s u c h t h a t x = 1 o n a neigh-

b o u r h o o d of 0 a n d

Then (O,E,) d W F M ( u j ) f o r

41h

I X . Ouasihomotzeneous Wave Front Sets

w e are going to s h o w t h a t in t h e t o p o l o g y o f Y ( V * ) f o r every J , E Y ( V ' ) .

lim c p , * J , = J ,

(9.64)

,+m

I n view o f

weakly to on

< cp,*^u ^u

"

,J,

"

> = < ^u , y,* J, >

it f o l l o w s from ( 9 . 6 4 ) t h a t

'ps*

^u c o n v e r g e s

a s ~ + + m .By w h a t w a s p r o v e d a b o v e t h i s i m p l i e s t h a t

K ( < , E ) , i.e.

< does

n o t b e l o n g to s u p p

6,

^u

vanishes

a s w a s to be s h o w n .

N o w , f o r t h e p r o o f of ( 0 . 6 4 ) w e f i r s t observe t h a t in view o f

('pr*J,)(n)

=

~ J , * J , ( ~ ) ,

a e N , " . it s u f f i c e s to s h o w t h a t for a r b i t r a r y J I E Y ( V * ) a n d kElN w e have

f o r a r b i t r a r y < , < E V * ,s E C O , I l , a n d r E C l , + m E . P u t t i n g e v e r y t h i n g t o g e t h e r w e arrive a t ( I + l F , l ) k l('p,*iJ,-J,)( i L&,

icN.

,

C o m b i n i n g t h i s w i t h ( 0 . 7 3 ) w e see t h a t f does n o t b e l o n g to C P S La t ( 0 , ~ ~ H e)n.c e ( 9 . 7 2 ) i m p l i e s t h a t ( ( O , X ~ ) , ~ U , O ) ) ~ W ~ S, i~n c( eE by + ) (. 9 . 7 2 ) , as w e l l , W , , , , ( E + ) a n d W p , L ( E - ) do n o t i n t e r s e c t t h i s m e a n s t h a t ( ( O , x o ) , ( w , O ) ) b e l o n g s to W , , , L ( E ) . By t h e c h o i c e of u t h i s i m p l i e s : ( a ) if v f O t h e n r e p l a c i n g w by - v w e d e d u c e t h a t e q u a l i t y h o l d s in (9.71); a n d ( b ) if v = O t h e n W , , , ( E )

contains ((O,xo),(signu.O)).

S i n c e by a s s e r t i o n ( i ) t h e p o i n t ( ( O , x o ) , ( - s i g n u . O ) ) b e l o n g s to WF,,(E) C W P , L ( E ) e q u a l i t y h o l d s in ((1,711 in t h e c a s e " v = O " , a s w e l l . rn

Discussion. In case v = 0 t h e r e s u l t s of T h e o r e m s 9.35.A a n d 9 . 3 5 . 8 o n t h e w a v e

f r o n t sets WFp(E' ) , r 2 1 , may be i n t e r p r e t e d a s follows. T h e high f r e q u e n c i e s c a u s i n g t h e Cm s i n g u l a r i t i e s of E' concentrated near t h e direction

a t t h e p o i n t s ( 0 , ~ of~ {) O ) x ( k " \ ( O ) )

are

no:= ( - s i g n u , O ) b u t n e v e r t h e l e s s k e e p a l i t t l e

a w a y f r o m it: t h e y a r e c o n t a i n e d in t h e sets { ( - s " s i g n u , s t ) ; s Z 1 , < E K ( O , E ) } ,

421

9 . d T h e H e a t a n d t h e S c h r o d i n g e r Equation

E

> O , ( w h i c h are t h e s m a l l e r t h e larger r is) in case r < 2 b u t s t a y in t h e i r c o m -

p l e m e n t in case r 2 2 a n d

E

is s u f f i c i e n t l y s m a l l . M o r e o v e r , in c o n t r a s t to t h e case

r < 2 t h e c a s e r t 2 s h o w s t h a t it d e p e n d s o n t h e p o i n t xo h o w t h e €,-components o f t h e high f r e q u e n c i e s c a u s i n g s i n g u l a r i t i e s look like. As f o r t h e C p s Ls i n g u l a r i t i e s o f E t h e i n t e r p r e t a t i o n o f t h e r e s u l t s o f T h e o r e m s 9 . 3 5 . A a n d 9 . 3 5 . B is m o r e c o m p l i c a t e d s i n c e w h e n varying r o n e does n o t o n l y c h a n g e t h e s h a p e of t h e f r e q u e n c y d o m a i n s involved in t h e d e f i n i t i o n o f WF,,,,(E)

b u t a l s o t h e t y p e o f Gevrey regu-

l a r i t y described by W F , , L ( E ) .

Finally, in case v = O w e c o m p u t e t h e q u a s i h o m o g e n e o u s w a v e f r o n t sets o f t h e o t h e r a l m o s t invariant f u n d a m e n t a l s o l u t i o n s of q ( D ) .

Theorem 9.38. Suppose that v = 0 . Let F be an almost invariant Fundamental solution OF q ( D ) which is diFFerent f r o m E' (i)

IF r?-3 then WF,(F) = WF,,

(ii)

If r

c

-3 then

A(F)=

and E - (see Notation 7.71). W F P , * ( E + ) uW F , , , , ( E - ) .

WFp(FI = W F p ( E ' ) and WF,,,(F) = W F p , , ( E f ) .

For t h e p r o o f w e r e q u i r e i n f o r m a t i o n o n t h e q u a s i h o m o g e n e o u s w a v e f r o n t sets o f t h e a l m o s t invariant s o l u t i o n s o f ( 7 . 0 4 ) d e t e r m i n e d in s e c t i o n 7 . ( f ) :

Theorem 9.39. Suppose that v = O . Let T E Y ' I I R x R " ) be anj almost invariant solution of the equation ( 7 . 6 4 ) which is not induced bj a poljnomial Function. Then i t s wave fr on t s e t W F , ( T ) coincides with WF, , , (T) i((O,S[)*

and is equal t o

( 0 . E ) ) ;E E S r l - ' . S E R J

s f ) , ( r , f ) ) ;( r , E ) E S r ' n q - ' ( O 1 , S E R } 1/01x 1 ~ " )x { I -sign u , o)}

if r > - 3 iF r=-3 if

r-x-3.

ProoF. F i r s t of all w e verify t h a t in case r < 2 w e have (q-'(o))p,m= { (-signu,O)). I n d e e d , if < = ( r , c ) ~ S b"e l o n g s to t h e l e f t - h a n d side t h e n w e f i n d a s e q u e n c e o f points

j € N , in q - ' ( O )

as j+a s u c h t h a t

a n d a s e q u e n c e ( s j ) in 10,+00C c o n v e r g i n g to z e r o

422

I X . Q u a s l h o r n o g e n e o u s W a v e F r o n t Sets

Since t h e condition on

(Tj

,cj) amounts

to uTj =

- lcjl 2

this implies

As r < 2 t h i s is equal to 0 .In view of sign 'cj = -sign u we conclude t h a t (= ( -sign u , O ) ,

a s claimed above. h

Since s u p p T C q-'(O) we deduce from Proposition 9.31, Remark 9.30, and ( 9 . 6 6 ) t h a t

Next we observe t h a t it suffices to prove t h e equalities outside t h e origin of IRxIR" because since t h e wave f r o n t sets are closed it then follows t h a t they contain t h e set ( 0 ) XI,, so t h a t in view of ( 9 . 7 4 ) their intersection with ( 0 ) x S " coinx 2,. cides with ( 0 )

Since by Theorem 7.80 t h e analytic singular support of T is equal to ( 0 ) x I R " t h e assertion follows in t h e case r < 2 . For the proof of t h e o t h e r cases we observe that by t h e O(n)-invariance it follows by (7.74) and Theorem 9.27 that WF,,A(T) is contained in t h e s e t of all ( ( t , x ) , ( T , < ) ) in ( [ R x [ R " ) x s " such t h a t X j t ; k - X k < j = o f o r arbitrary j , k E N , , . Since in case

(#o

this means that x = s E f o r some S E I R we conclude that WF,,A(T)

C (

( ( t , S < ) , ( T , < ) )t ;, s E I R ,

(~,€,)€z,,}.

Since t h e distribution Sm*,k defined in (7.60) is real valued it follows t h a t t h e distribution T m , k defined in Theorem 7.77' satisfies t h e assumption of Remark 9.6. ( i i ) . Hence, if ( ( 0x,) , ( r , < ) )E W F P , A ( T m , k ) then ( ( 0-,x ) , ( T , < ) ) E WFp,*(Trn,k), and - s i n c e T is invariant under t h e map ( t , x ) H ( t , - x )- it follows by Remark

~ ) . by Theorem 7 . 8 0 the analytic 9.6.(iii) t h a t ( ( O , X ) , ( ~ : , - S ) ) E W F ~ , A ( T , ,Since singular support of T m , k is equal to ( 0 ) x I R " the assertion follows in case T=T,,,

for some k C N o . In order to remove this restriction on T we employ Theorem 7.77' to find N E I N O and c o n s t a n t s a o , . . . , a N E C such t h a t a N # 0 and

423

9.d T h e He a t e n d t h e Schrodinaer Eauation

N

T-

C ajTrn,j j=O

i s induced by a polynomial function Q . Since ( d , - m ) S m , j = - S m , j - , it f o l l o w s

by Proposition 2 . 4 0 . ( i i ) t h a t ( d , - m ) T m , j= T r n , j - l ,j E N . Moreover, s i n c e by Proposition 7.76 t h e s u p p o r t of ( a p - m ) S m , o is contained in { O ) , t h e d i s t r i b u t i o n (3,-m)

N

T m , N - L= ( d p - m ) Tm,o is induced by a ( q u a s i h o m o g e n e o u s ) polynomial

function R . Consequently, (

a,

-m

)NT

= a N Tm,o + a N - T,

+

(a, - m ) N TQ .

Since a N # O t h i s implies t h a t

so t h a t t h e a s s e r t i o n is valid f o r general T , a s well.

Proof o f Theorem 0 . 3 H . Since F i s d i f f e r e n t f r o m E'

t h e distribution T + : = F - E'

is a non-trivial s o l u t i o n of ( 7 . 0 4 ) which is a l m o s t invariant of d e g r e e - n . W e

set u : = sign u

(i).

.

H e n c e , if r 2 2 T h e o r e m s 0.35.A a n d 9.39 s h o w t h a t

From t h i s it f o l l o w s t h a t

{((O,TUSC),(T, O w i t h c e n t e r 0 . T h e n

a , : P ~ x C 0 , 1 l ~ Q 1 " (, t ; , t )H O t ( F , ) , is a n ( n + l ) - c h a i n , its b o u n d a r y da, c o n s i s t i n g

of t h e t w o n - c h a i n s yj:P:--+@".

€,Haj(€,), j€(O,l},

a n d t h e 2" n - c h a i n s

, o r , y k , . . . ,y n - l ) . By ( 9 . 7 7 ) ( G z ) * ( d L 1 A . . . ~ d < , , ) is o f ontinuous function w h e r e h ~ : P ~ - l ~ C O . is l l a~ c@

for s u i t a b l e c o n s t a n t s C, a n d

P which are i n d e p e n d e n t o f r .

N o w , let K be a c o m p a c t s u b s e t o f 0 . W e c h o o s e E > O s u c h t h a t K , : = K + K ( O . E ) i s c o n t a i n e d in

n . By

t h e Paley-Wiener T h e o r e m w e f i n d a c o n s t a n t C, s u c h t h a t

for every r p € C Z ( K ) t h e f o l l o w i n g e s t i m a t e h o l d s :

c o n v e r g e s to 0 as r - + + m . C o n s e q u e n t l y . s i n c e by S t o k e s ' t h e o r e m

5

J' F d < 1 A . . . ~ d < ,= O , aar t a k i n g t h e limit a s r + + a s h o w s t h a t c l o = p 1 . Proof o f Proposition Y . 4 0 . By t h e c h o i c e of a (see N o t a t i o n 7.71) w e have a v = I v l . Hence

( u r + v o s + I t; 12)'

+ (

- u a s + v r ) ' = ( u I + I ) is absolutely i n t e g r a b l e o n A2

g is well-defined, indeed. Since f o r any c o m p a c t s u b s e t K of fl t h e n u m b e r (9.90)

i n f { < [ o , x > ; ( t , x ) E K for s o m e t c R }

is positive Fubini's t h e o r e m s h o w s t h a t

i.e. E,=T,.

< E 2 , 9 >= J n g ( z ) r p ( z ) d z , cpECz(fl),

In order to verify t h a t g is differentiable w e fix a = ( j , p ) E N , x I N , " .

T h e n w e have

429

9.d.2 Proof of Theorem 9.37. Part 1

where

c , : = s u p t ~ ~ ~ ' ~ ' e x p ( - c ITcE IA ), ;I . Since for any a > O t h e derivative of t h e function CO,+mC3 t H t a e x p ( - c t ) vanishes a t t = a / c , o n l y , w e have

(9.92) Hence C, 5

s u p { t a e x p ( - c t ) ; t E C O , + ~ C=($)a, }

a>O.

la1 (z) . Since

Jexp(-cl.l) A2

c J' R X

e x p ( - c l * l ) = c-n-i

exp(-l*~) IR x R"

R"

w e c o n c l u d e t h a t g is a Cc" f u n c t i o n satisfying I g ' O L ' ( z ) l5

Blal*' < [ o , x >- 1 m l - n - i

lclllul a t

e

,

z = ( t , x )E n ,

w h e r e B is a c o n s t a n t n o t depending o n z and a . In particular, g is real analytic. C o n s e q u e n t l y , W F ( p , A ( E l n ) = WF,,,(E,),

and since A is t h e union of a l l t h e sets

A C , ~ t h e a s s e r t i o n of L e m m a 0 . 3 6 f o l l o w s f r o m ( 7 . 8 6 ) .

(d.2) Pr-ool' 01' 'i'heorem ! B . : B i .

Par-1 1: Eslablishing

lhcb Mic*rolocal I)ccomposilion el'

E

We f o l l o w t h e p a t t e r n of t h e proof OF L e m m a 0 . 3 6 . Again, t h e first s t e p is a d e f o r m a t i o n of t h e integration c o n t o u r which is b a s e d o n t h e following l e m m a . W e set

r0:={r~R\1-i,5C; signr= signuif Lemma 9.42. There are constants a . c,, (9.93)

v=O}xR"

and

plr,E):=( l ~ l + ~ E , ~ ~ ) ~ ' ~

so E 10. I C on]), depending on w such that

I q ( r - i o s , < + i v ) l >co

for arbitrary s ~ C O , s , l .( r . E ) E r O . and v 6 R " satisKving l u l S a p ( s , t ) . proOf. W e have z : = q ( r - i o s , [ + i u )

= U T + Ivls+

2

- l u 1 2 + i ( v r - u o s + 2 < E , , u > ) . We

s u p p o s e first t h a t v = O a n d c h o o s e a E l 0 , l C s u c h t h a t a'< I u I . Then t h e conditions

430

IX. Q u a s i h o m o g e n e o u s Wave F r o n t Sets

s i g n r = s i g n u and I T I? 1/2 imply t h a t Rez = I u l l r l + 1512 - I u I 2 t I r l ( l u l - a 2 )+1512(1-a2)t ( l u l - a 2 ) / 2 = : c , We suppose now t h a t v f O , fix b E l 4 l u l , + ~ C ,and choose a ~ 1 0 , l Cs u c h t h a t a 2 ( 1 + l / b ) < 1 / 2 and 2 a ( b 2 + b ) i / 2 < I v I . If

I < [ > (blrl)i’2 then l i ~ l < a ( l + l / b ) ~ / ~ 1 < 1

and Rez t -lullr1+1~12~1-a2~1+l/b)]?~ ~ ~ 2 ( - l u l / b + l t/ b2 /) 8 . On t h e o t h e r hand, if 1 F , l 5 ( b l ~ l ) ’ / then ~ ~ < < , u > ~ < ~ F , I I L 5J I a ( b 2 + b ) 1 / 2 1 r l so t h a t 2Ilmzl ? [ l ~ l - 2 a ( b ~ + b ) ” ~ ] /1111s. Choosing so sufficiently small one arrives a t t h e assertion.

B

Now we fix c o n s t a n t s a , c , s , having t h e properties in t h e assertion of Lemma 9 . 4 2 , with support contained in To such t h a t l q l < a @ ,

c h o o s e a Ci function q:RxRn-!Rn

and choose another C’ function x : I R x I R ” ~ Rwith values i n C O , s , l such that (9.04)

x = s o on I R x R ” \ T o .

Moreover, let U be an open s u b s e t of R” such t h a t

-

If t h e partial derivatives of 1 and x of order 1 are bounded by c o n s t ( 1 + I I b N f o r s o m e NEN then Proposition 9.40 and Lemma 9.41 (applied to F = l / q , g ( ( , t ) = ( -a ( 1 - t ) so - a t x ( 5 ) . t q( 5 ) ) and

n = IR x U )

yield

” h

Next we fix a Ci function

x : R XR

n d CO, 11 with support contained in To which

is quasihomogeneous of degree 0 and of type p on t h e complement of K ( 0 , l ) .

Since the partial derivatives of

x

are bounded all the preceding hypotheses a r e

satisfied if we s e t q:=axpt0,

x : = s , ( l - x ) , and U : = X where X is defined in ( 9 . 6 9 ) .

In order t o define the desired microlocal decomposition of E we choose a quasihomogeneous (of type p ) open s u b s e t constants E , C > O where

r of R x R ”

such that

rEC r C Tc

f o r suitable

431

9.d.2 Proof of Theorem 9.37. Part 1

M o r e o v e r , w e f i x R E C l , + a C a n d set

r,

:=r\(l-R,RCxR")

and

r-:=IRxIR"\r+.

In addition, w e s u p p o s e t h a t

x = 1 on T\C-l,lIxlR"

(9.98)

a n d d e f i n e E , ~ % ' ( f l ) by

"

h

<E,

>= (2~)-"-'

,'p

'p E

: ( O ( < ) ) J( = ( 25r ) - n - l

J(S) dT,

'p E

c; f(n),

I'+ nI',

s i n c e Tc is q u a s i h o m o g e n e o u s , a n d s i n c e in t h e b o u n d e d set I - R , R [ x K ( O . E R ' " ) W F p , A ( T f - h ) C nxr, . H e n c e W F , , * ( E , ly s m a l l

E

r c \ ( r + n r=cr,n ) I-R,RCxlR" )

is c o n t a i n e d

it f o l l o w s f r o m P r o p o s i t i o n 9.32 t h a t C nxr, . Since t h i s is valid for s u f f i c i e n t -

we conclude t h a t

(9.105)

WF,,,(E+)

c

nX{(U,O)).

In view of ( 9 . 0 9 ) t h i s i m p l i e s t h a t W F , , , , ( E + ) n W F , , , ( E - )

is e m p t y . H e n c e in

view o f ( 9 . 6 7 ) a n d ( 9 . 6 6 ) w e c o n c l u d e t h a t (Y

. I06 )

WF,,A(E?) = WF,,,A(E)n ~ x { ( + u , O ) ) .

Now w e s u p p o s e f o r a m o m e n t t h a t ( 9 . 7 3 ) is p r o v e d . T h i s i m p l i e s t h a t f does n o t b e l o n g to Tp a t t h e p o i n t s of ( O I X I O . + ~ [ <so~ t h a t in view of (9.1OS) a n d (9.106) w e have

( ( 0x) 1 0 , + ~ C ~ o ) x { ( ~ , OC )W} F , , , ( E + )

C WF,,,(E).

Since this

is valid f o r any < o € S " - l it f o l l o w s t h a t ( ( O ) x l R " ) x { ( v , O ) } C W F , . , ( E ) .

Using

(9.106) a n d ( 0 . 6 7 ) , a g a i n , w e c o n c l u d e t h a t

WF,,A(E+)= ( ( O ) X X ) X { ( U . O ) } . If v f O t h e n r e p l a c i n g u by - u w e d e d u c e t h a t ( ( O I x l R " ) x { ( - u , O ) } C W F , , , ( E ) . If v = O t h e n t h i s f o l l o w s f r o m t h e f a c t t h a t t h e s i n g u l a r s u p p o r t o f E is e q u a l to ( 0 ) x l R " : i n d e e d , s i n c e f € C m ( n ) w e have W F , ( E + ) = @ so t h a t by (9.106) w e

see t h a t W F , ( E ( , )

= WF,(E-) C n x { (

d e r i v e s f r o m (9.106) t h a t W F , , , ( E - ) is c o m p l e t e .

I t r e m a i n s to p r o v e ( 9 . 7 3 ) .

=(

C o n s e q u e n t l y , in b o t h c a s e s o n e

x ( ( - w , O ) } , and t h e proof of (9.72)

43 4

I X . Q u a s i h o m o g e n e o u s W a v e Front Sets

(d.3) Proor or Theorem 9.37. Par1 2 : Esllmallng the Derlvatlves

or

I' Prom Below

Here, again, it is necessary to d e f o r m t h e c o n t o u r of integration. However, t h i s t i m e t h e deformation is a c r o s s poles so t h a t additional t e r m s arise. This procedure is based o n t h e following lemma and t h e discussion following it.

Lemma 9.43. There i s a constant A E I O , + w C on/)* depending on w such that for arbitrar). (Y. 107)

tcIR,

b 6 1 0 , +wC,and A 6IR the following holds: I f

Q ( A ) : = A a + i w t + b= O

then (9.108)

/Re;\ I ( ? I W ~ I ) ' / ~

and -provided that u r l O i f v = O -

mf. We s e t c : = ReX ( 9.107 I '

and d : = l m X . Then t h e equation (9.107) a m o u n t s t o

(a) d2-c2=uT+b.

and

(b) 2cd=-vr.

If IcI 5 21dl t h e n by (9.107.b)': c2 5 21cdl = I v T I , i.e. IcI 5 1 ~ ~ 1 " ~I f . 1cI ? 21dl then by (9.107.a)': c 2 = d 2 - u r - b 5 l u r l + c 2 / 4 so t h a t IcI 5 ( $ I u ~ l ) ' / ~Hence . t h e proof of (9.108) is complete. If r = O t h e n by (9.107.a)' we have d 2 = l T l + c 2 + b , and (9.109) is trivially valid if A is chosen to be 5 1 . So w e may a s s u m e t h a t

T

f0.

We f i r s t s u p p o s e t h a t v = 0 . Since in this case by t h e assumption we have

UT

t0

it follows f r o m (9.107.a)' t h a t Id1 2 IcI so t h a t c = O by (9.107.b)'. Hence (9.107.a)'

yields: d 2 = I u r l + b 2 A2 ( I T I + c2 + b 1 i f A : = min{ 1 ,

}.

Next we s u p p o s e t h a t u = 0 . Then d 2 = c 2 + b , in particular: 1cI 5 Id1 . Hence by (9.107.b)' we have d2 t lcdl = lv11/2.

Consequently, d 2 2

$ ( c 2 +b + lvr1/2),

and

(9.109) is valid if A2 5 min{ 112, lv1/4}. Finally, we s u p p o s e t h a t v f O f u

and c h o o s e A > O such t h a t 8 1 w l ( l + S l w l ) A 2< v 2

and A 2 5 mint IuI , 1 / 3 } . If b t 3 1 u r l then by (9.107.a)' we have

435

9.d.3 Proof of Theorem 9.37. Part 2

d 2 = c 2 + u r + b > c 2 + I u r l + b > A 2 ( l r l + c2 + b ) . 3 -

If b 5 3 l u r l t h e n t h e a s s u m p t i o n t h a t (9.109) be f a l s e l e a d s in c o m b i n a t i o n w i t h (9.108) a n d ( 9 . 1 0 7 . b ) ' to

8 I w 1(1+5 I w I ) A 2 r 2 = 4 . 2 I W T I A'

(

I rI + 2 I w r I + 3 IwrI ) 2

2 4c2A2 (Irl+c'+b) L (2cd)'=v2r2 which in view o f " r f O " c o n t r a d i c t s t h e c h o i c e o f A . H e n c e (9,109) is valid.

W e n o w f o r m u l a t e a f e w a d d i t i o n a l t e c h n i c a l a s s u m p t i o n s . F i r s t o f all w e f i x a c o n s t a n t A > O s u c h t h a t t h e a s s e r t i o n of Lemma 9 . 4 3 h o l d s a n d a s s u m e t h a t a < A w h e r e a is t h e c o n s t a n t in L e m m a 9 . 4 1 a p p e a r i n g in t h e d e f i n i t i o n of q . M o r e o v e r , w e fix

E

> O . In order to s i m p l i f y c o m p u t a t i o n s w e a s s u m e t h a t R 2 ( m a x { 81wl , I }

r / (2-1.) / E ~ )

T h i s i m p l i e s t h a t w i t h t h e a b b r e v i a t i o n d : = l / r w e have ( 9.110)

(21wr1)'/2 5

$111 d ,

T E!R\I-R,RC.

Defining Q a s in (9.107) w e are g o i n g to derive (9.111)

sEIR, TER\I-R,RC.

i ~ 22 1 ~ 1 2 c ' ,

(Q(+EITld+is)I

In f a c t , if ho is a z e r o of Q w i t h n o n - n e g a t i v e real p a r t t h e n I ~ l ~ l ~ + i s2+E hI T~l dl+ R e X o ? ~

1d ~

1

a n d - by (9.108) a n d (O.Il0) IElrld+is-hol

>

EITld-ReX,,?

il~ld 2

so t h a t t h e e s t i m a t e (9.111) f o l l o w s in view o f Q ( X ) = ( X + X o ) ( X - X o ) . Now w e c h o o s e o r t h o n o r m a l c o o r d i n a t e s in

O . max{l
Deforming t h e contour o f integration i n 10. We fix r E C R , + m C a n d < ' E K ' ( O , E T ~ ) .

To identify l o ( r , < ' , x l ) as a p a t h i n t e g r a l w e d e f i n e a m e r o m o r p h i c f u n c t i o n H:C+C

by Xk eixiX

H(X):=

q w ( r , X ,t')

and define a path yo:C-~rc',~rdl--+@ ( 9.113 )

by s H s + i a p ( T . s , < ' ) . T h e n

l o ( r , < ' , x l ) = J ' H ( X ) dX . Yo d

To d e f o r m t h e c o n t o u r of i n t e g r a t i o n w e set b : = a p ( r . E r d e f i n e y + : Cb,cI-C

Z : = y o + y +- S - y -

,[I),

by s H + E r d + i s a n d 8 : [ - E T ~ , E T " I - C

a n d f o r any c > b by s H s + i c . T h e n

is a s i m p l e c y c l e . I f c is s u f f i c i e n t l y l a r g e t h e n in view of Lem-

m a O . 4 3 a n d (0.110) precisely o n e of t h e t w o p o l e s of H lies in t h e i n t e r i o r o f

Z , namely t h e o n e w i t h p o s i t i v e imaginary p a r t ; w e d e n o t e it by ih, w h e r e

here

d e n o t e s t h e h o l o m o r p h i c b r a n c h of t h e c o m p l e x s q u a r e root w i t h p o s i t i v e

real p a r t a n d w i t h @ \ l - a , O l a s its d o m a i n of d e f i n i t i o n ( n o t e t h a t i W r > O if v = O ) . S i n c e t h e o t h e r p o l e of H lies in t h e e x t e r i o r of Z a n a p p l i c a t i o n o f t h e r e s i d u e t h e o r e m a n d t a k i n g t h e l i m i t a s c + + a y i e l d s in view of (0.113):

where +m

437

9.d.3 Proof of Theorem 9.37. Part 2

In view of qw(f,X.S')= ( X - i X o ) ( X + i X o )

t h e residue of H a t i X o is equal to

By inserting (9.114) w e decompose (9.112) i n t o a s u m J - ( s -)J + ( x ) + r J , ( x ) and a r e going to e s t i m a t e each t e r m separately. The third o n e will give t h e main contribution. So in o r d e r to obtain an e s t i m a t e of ('9.112) from below we have to e s t i m a t e J , ( x ) f r o m above whereas J l ( x ) is to be estimated f r o m below.

Estimating (a-2+1)1'2s.

J _ + ( fsr )o m above. If

s2b

then

ET

d

5s/a

that

so

I f ~ s ~ + 5i s (

In view of (9.111) this gives (a-2+1)k'2T-2d sk e x p ( - x l s ) .

I H (+Esd+ i s ) /5 2 E - ' Hence IJ+_Cx)lis not larger t h a n

+m

+m

( a - 2 + ) k/2

21K'(0,1)(~'~l+"-~

1'

'

J

T j + ( n - 3 + I y I )d

R

sk e x p ( - x l s ) d s dr .

arrd

Substituting f i r s t xls = 3 t , i.e. d s = 3 d t / x l , and then

f

x1 T~ = a , i.e

one sees t h a t t h e d o u b l e integral above is equal t o +m

+m

orj+ n - 3 + I y I + r - 1

J' as x

Rd/ 3

J'

tk e-3tdt do.

n

Here, in t u r n , t h e double integral is rnajorized by

1' e - o do I'

+m

0

+m

t r j +n - 4 + l P I + r

-2t

dt

0

provided t h a t r j + l y l + n + r2 4 which is t h e c a s e if j t 3 - n . Combining (9.92) with r i m e - t d t = 1 o n e obtains t h a t +m

( 9 . I IS )

J'tee-2t d t 5 (P/e)'

P €10.+03C.

0

Consequently, t h e second integral above is not larger than rj+ I p l + n + r - 4

( r j +I b l + n + r - 4 )

Putting everything t o g e t h e r o n e finds a c o n s t a n t C, only depending on ~ , n and , a

such that

438

IX. Q u a s i h o m o g e n e o u s W a v e F r o n t

Estimating J l l x ) from below: the c a s e " n = l " . Substituting o =

A , i.e.

K =

Sets

o2 and

dr = 20do w e see t h a t +m

"ljl(,) = i k s

e x p ( - x l f i m ) dr

J+(k-1)/2

(m)k-i =

R +OD

= 2ik(m)k-1

exp(-xlom)do

From this o n e o b t a i n s

by estimating t h e integral above with t h e help of t h e following lemma t h e f u l l s t a t e m e n t of which is required f o r t h e c a s e " n ? ? " .

Lemma9.44. Let c€CO,+wC, k€GV,. and A6C such that ReA > O . Then

N o t e that c(~&)".

c k c l e - " i f cReA 2 k

proOf. Partial integration leads to +m

+m

J' t k + l

dt =

k+l x j' t k e - X t d t

0

0

so t h a t by induction one obtains t h a t +m

s

0

k! t k K X dt t =- X k + i '

Hence t h e t e r m to be estimated is equal to C

J' tk e - X t d t 0

which is not larger than C S where S : = s u p { g ( t ) ;t E C O , c l } with g ( t ) : = t k e - R e X t . Since g ' ( t ) = O if and o n l y i f t = t o : = - k

Re X

t h e function g has a unique maximum

a t t o . If t o 2 c then glc0,', is increasing and S = g ( c ) ; if t O < c t h e n S = g ( t o ) .

Estimating

Jl(s)

M(r,x):=

f r o m below: t h e case "n??". We set

1'

(- x i ( i W r + p

]dp.

0

Substituting p =

6, i.e.

p 2 = r s and p d p = $ d s we obtain

2 1/2

(iWr+p )

= fiX(s)

where X ( s ) : = ( i W + s ) l " ,

2 1/2 )

= -fix,

i p < 8 , x ' >- x l ( i W s + p

A(s. 1 . We a r e going to apply Lemma 0.44, making use of both of its c a s e s . To distinguish between them w e s e t h l ( s ) : = h ( s ) R e X ( s ) and fix increasing o n

Cz,+aCand such t h a t

now t h a t (9.119)

A

j+lal+n-1 ?xihi(s)

2tT

such t h a t h, is strictly

h , ( C O , z C ) n h , ( E $ , + a C ) is empty. We s u p p o s e

440

I X . Q u a s i h o m o g e n e o u s Wave F r o n t S e t s

T h e n w e c a n c h o o s e s,E

C;,+wC

s a t i s f y i n g j + l a l + n - 1 = x1 h l ( s , ) .

It follows that

By L e m m a 9 . 4 4 - a p p l i e d to c = h ( s ) a n d X = x l A ( s , < 3 , y > )- w e c o n c l u d e t h a t +m

(9.120)

IJ l ( x ) l=

IJ' R

7'

I I r 3'l

+m

M ( r , x ) dr =

sn-2

I

smX ( s ) k - l N ( s , x l , < 3 , y > )ds d 3 ?

o

and

with

Estimating K , ( . v , ) f r o m above. W i t h t h e a b b r e v i a t i o n B : = ( I w I + I ) ' / ~

(9.121)

w e have

ReX(s) 5 I X ( s ) l < B m a x ( & , l l ,

SE

IO,+CoC

By Lemma 9 . 4 3 w e f i n d a c o n s t a n t A > 0 s u c h t h a t

H e n c e , m a k i n g u s e of R e X ( s ) / l X ( s ) l 5 1 a n d o f (9.118) w e d e d u c e t h a t

where

x : = m t - 1( k - j - l a l - n ) + + 2 d-2

- ~(lyl+n-3tk-j-lal-n)+- 1 -2 4d-2

= -j+jo-l

with jo:=7(m-3)+1 1 1 = - -I-d r-1 2d-1 - 2 - r '

C o n s e q u e n t l y , t h e l a s t i n t e g r a l is f i n i t e a n d e q u a l s s,i*iO/(j-jo) d e f i n i t i o n o f s, a n d h , , by (9.118) a n d by (9.121) w e h a v e

if j > j o .

By t h e

9.d.3 Proof of Theorem 9 . 3 7 . Part 2

441

Putting everything t o g e t h e r we find a c o n s t a n t C, n o t depending o n a and x l such t h a t

cpI+I

xl-rj-IPI-n-r+2

K2(Xl)

(9.124)

lalrj+l@l

i>io.

Estimating K 3 ( x I ) from above. By ( 9 . 1 2 2 ) , by t h e definition of h and by (9.123) we have

1

('+(xl,s)ds 5 -B k ( fi ) j + in I + n A

1

.I'

exp(-Qxis

Srn

0

0

Here t h e integral o n t h e left-hand side is not larger than

I/(Z-r)

)ds.

J'd d s / 6 = 2 . Again

by (9.121), (9.122), and (9.118) we deduce t h a t

Here t h e integral o n t h e right-hand side i s majorized by

Finally, t h e conditions (9.121), (9.122), and (9.118) a l s o imply

T

where

0

=( 2tG 2 -~ r ) 4d-2 ( 2 - r ) t l - r d t , s h o w s t h a t t h e last integral is equal to

O:= m - ; i + - + m . I

and d s = (Qx1/2)'-'

2

Substituting

0

Since

Q x 1 s 1 / ( 2 - r ) = 2 t , i.e. s

442

IX.

Quasihomogeneous

W a v e F r o n t Sets

( 2 - r ) P + l - r = ~ ( l y l * n - 3 - l + k t 2- -I r- ( j + l a l + n ) + 2 ] - l =

= 1 p 1 + n - 3 + ~ ( j + 1 ~ 1 + n - 1 p 1 - =n +r j2 t) ~ p ~ t n t r - 3 it f o l l o w s b y (9.115) t h a t t h e i n t e g r a l a b o v e is n o t l a r g e r t h a n

[ ( r j +101t n + r - 3 ) / e ] r J +1 ’ 1 + n + r - 3 . P u t t i n g e v e r y t h i n g t o g e t h e r o n e f i n d s a c o n s t a n t C3 n o t d e p e n d i n g o n a a n d x 1 such that K3(X1)

( 9.125 1

~

cAaI+l [ X l - r j - l P l - n - r + 2

lalrj+lpl + I

I.

Estimating I K , (y)/ f r o m below. S i n c e

we obtain

@ ( s , b )= ( i W ) - i s - 2 G ( i W / s , b ) where

X E U : = C \ 1-a, -1 1 , b E IR . S u b s t i t u t i n g t = I w l / s , i.e. s = l w l / t a n d C 2 d s = d t / l w l

we obtain

+OD

+a2

j ‘ @ ( s , b )d s = ( i W ) - ’ - ’ J ‘ G ( t i Z , b ) i Z d t 0

0

w h e r e Z : = W / I w l . Note t h a t G ( * , b ) is h o l o m o r p h i c o n U . H e n c e w e may d e f o r m t h e c o n t o u r of i n t e g r a t i o n . To d o t h i s w e fix c > O a n d d e f i n e y o : C O , c l - + U y o ( t ) : = t i Z , yl:CO,cl-U

by

by y l ( t ) : = t , a n d S c : C O . ~ 1 ~byUS , ( t ) : = c e i t w h e r e

9~ 1 - 7 r , + ~ c Cis d e f i n e d by t h e c o n d i t i o n i Z = e i a . W e set

t:=

s i g n $ . Then yO-t6,-yl

is a s i m p l e c y c l e in U so t h a t by Cauchy’s t h e o r e m it f o l l o w s h a t

j ‘ G ( X , b ) dX = \ ’ G ( X , b )dX yo

y1

+

t . J ’ G ( X , b ) dX . SC

If c is s u f f i c i e n t l y l a r g e w e have l ( c e i t + l ) ” 2 - i b l ? 6 / 2 a n d I ( c e i t + l ) 1 ’ 2 1 5 2 6 , a n d hence IG(S,(t),b)

I5

2k+1+i+1a1+ncP

where

S i n c e IS:[

IC

this implies t h a t

443

9.d.3 Proof o f T h e o r e m 9.37. Part 2

lim J ' G ( X , b ) dX = 0 . c+m

8,

Consequently, +m

+m

J ' G ( t i Z , b ) i Z d t = J ' G ( t , b ) dt 0

0

S u b s t i t u t i n g t = s2- 1 , i.e. s =

6 - iand

d t = 2 s d s one obtains

Estimating t h e right-hand side o f (0.126) from below may be difficult in g e n e r a l , b u t it is easy when b = O : +m

+m

+OD

J'

j'G(t,O) d t 2 2 J ' ( 1 - s - 2 ) i s - 1 r l - n d s 2 2-j+'

n

0

- l Y 1 -n

=

A

Finally, w e have to deal with t h e integration over S n - 2 . Using polar c o o r d i n a t e s and s e t t i n g c : = -& weiobtain C ~ y ~ + n - ~

J'

9'dB

sn-2

lyl+n-l

=

J'

,I'

(t9)Ytn-2d8dt =

0 sn-2

J'

(x')' d x ' .

K'(0,c)

If y c ( 2 N 0 ) " - ' t h e n ( x ' ) ' is non-negative, and t h e integral o n t h e l e f t - h a n d side is not s m a l l e r than

J'

n-l

( x ' ) Y d x '=

c-1.11"-1

i=l

2 -

~ i + l

Putting everything t o g e t h e r one obtains a positive c o n s t a n t C, not depending o n a and x1 such t h a t (9.127)

I K,(O)(

2 CP"'

if Y E ( 2 ~ , ) " - ' .

Combining a l l t h e estimates. Suppose t h a t

X I =

0 , i.e. y = 0 . Note t h a t

(j+lal+n-l)! t (lal/e)J+lal. Combining t h i s with (9.117) and (9.116) if n = 1 and with (0.120), (9.127), (0.124), (9.125), and (9.116) i f n t 2 we find positive c o n s t a n t s C , D , and w n o t depending o n a and x1 s u c h t h a t with t h e abbreviation c ( s , a ): = we have

Dial ( l a l ( r - 2 ) j

(Z-r)(j+l)

+

lal-j-lal

j + l a l + n)

444

(9.128)

IX. Q u a s i h o m o g e n e o u s Wave F r o n t

~ f ( u ) ( ~ c, -~ l u)l -l l

( X l ) j + l u l + nI u l - j - l a l

Sets

2 3- c(xl,a)

provided t h a t j + l a l + n - 1 ? 2 ( ~ x l j, > m a x { Z - n , j o I a n d y € ( 2 [ N o ) " - ' . T h e r i g h t - h a n d

side of (9.128) is n o t s m a l l e r t h a n 1 if xi s t a y s in a fixed f i n i t e i n t e r v a l a n d j is s u f f i c i e n t l y l a r g e a n d if l a l / j does n o t g r o w too f a s t . M o r e p r e c i s e l y , s e t t i n g b : = 2 - r a n d fixing p t I w e have c ( s , a ) = ~ 1 " ' l U l - b i S b ( i + l )[ l + ( s / i u o r J + l f i l s n - b5] < 2 [~ -

l a l / bl u l - i s i + l

Ib

w h e r e S , ( a ) : = m i n { lorl/p ,p""(r'-b)

if

1.

0 5 s <S,(a)

N o t e t h a t the t e r m in s q u a r e b r a c k e t s ,

being s m a l l e r t h a n B'"'

(s/IuI)J+~

where

B:=eD'/b,

is n o t larger t h a n 1 if pi'' t B'"' a n d lul 1 p s . H e n c e , o b s e r v i n g t h a t S,(a) = l a l / p

if p 2

a n d s e t t i n g Bo : = m a x i

1

u} w e a r r i v e a t

Theorem 9.37'. There are positive constants C . B . B,,, and j l such that (9.129)

l f ( " ) ( O ,( s , ~ ) ) l

~ l n l ~ - j - l n l - r I la l i + / n l

for everj' s ~ l O , + wand t ever) a = ( j . k . y) E I N ~ , X I N ~ X ( _ ~ Isatis[j,ing N ~ ) ' ~ - ~the f o l lowing conditions: j _> j ,

and

l a l / s 2 ma,, { B O , B'"''}

.

I

N o t e t h a t t h e a s s e r t i o n o f T h e o r e m 9.37' is c o n s i s t e n t w i t h ( 0 , 1 0 4 ) , i n d e e d . S i n c e ( 0 . 7 3 ) i m m e d i a t e l y f o l l o w s t h e p r o o f o f Theorem 9.37 is c o m p l e t e .

445

References

References

C 11 A r n o l d , V.I.

:

Geometrical Methods in the Theocv o f Ordinary Differential

Equations. G r u n d l e h r e n der M a t h . W i s s . 2 5 0 . S p r i n g e r - V e r l a g , B e r l i n , H e i d e l b e r g , N e w Y o r k , T o k i o 1983. C21 B e r n s t e i n , 1.N. : The dnalytic Continuation o f Generalized Functions with Re-

spect t o a Parameter. F u n c t i o n a l A n a l . A p p l . 6 , 273 - 285 (1072). C31 Bjork, J.E.

Rings o f Differential Operators. N o r t h - H o l l a n d Publ. C o . M a t h .

:

Library Vol. 21, A m s t e r d a m , London 1979. [

4 1 De W i l d e , M . : Closed Graph Theorems and Webbed Spaces. P i t m a n R e s e a r c h Notes in M a t h . 19

L o n d o n , S a n F r a n c i s c o , M e l b o u r n e 1978.

I 5 1 F l o r e t , K. u n d J . W l o k a : Einfuhrung in die Theorie der lokalkonvexen Raume. S p r i n g e r L e c t u r e N o t e s in M a t h . 56, Berlin, H e i d e l b e r g , New Y o r k 1968. C 6 1 & - d i n g , L. : Transformation de Fourier des distributions homogenes. B u l l ,

SOC.m a t h . F r a n c e , 8 9 , 381 - 4 2 8 (1961).

C71 von G r u d z i n s k i , 0. : On the Standard Fundamental Solutions o f the Schro-

dinger and of the Heat Operator. P r e p r i n t Univ. Kiel 1986 C81 H o r m a n d e r , L . : On the Division o f Distributions b-v Polynomials. A r k . M a t . 3 ,

555 - 568

( 1058)

C 9 1 H o r m a n d e r , L.

:

.

An Introduction t o Complex Analysis in Several Variables.

Znd e d . , N o r t h - H o l l a n d Publ. C o . , A m s t e r d a m , London 1973. C l O l H o r m a n d e r , L.

:

On the Esistence and the Regularity o f Solutions of Linear

Pseudodifferential Equations. L' E n s . M a t h . 1 7 , 99 - 163 (1971) . C 11 1 H o r m a n d e r , L.

:

The Analysis o f Linear Partial Differential Operators. Vol. I .

G r u n d l e h r e n der M a t h . W i s s . 2 5 6 . S p r i n g e r - V e r l a g , B e r l i n , H e i d e l b e r g , New Y o r k , T o k i o 1983. C 121 L a s c a r , R.

:

Propagation des singularitds des solutions d ' dquations pseudo-

diffdrentielles quasi homogrhes. A n n . I n s t .

Fourier

(Grenoble) 27,

79 - 123 ( 1 9 7 7 ) . C131 L i e s s , L. a n d L. R o d i n o : lnhomogeneous Gevrey Classes and Related Pseudo-

differential Operators. Boll. Un. M a t . I t a l . C ( 6 ) 3 , n o . 1 , 2 3 3 - 3 2 3 ( 1 9 8 4 ) .

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References

C 14 1 Lojasiewicz, S. : Sur l e probldme d e division. Studia Math. 18, 87 - 136 (1959). C 153 Ortner, N . : Regularisierte Faltung von Distributionen. T e i l 2 : Eine Tabelle

von Fundamentallosungen. ZAMP 31, 155 - 173 (1980) .

I: 161 Rodino, L . : On the Cevrey Wave Front Set o f the Solutions of a Quasielliptic Degenerate Equation. Conference on linear partial and pseudodifferential operators (Torino 1982). Rend. Sem. Mat. Univ. Politec. Torino 1983, special issue, 221 - 234 (1984) . C 17 1 Tougeron, J . C . : Idhaus d e fonctions diffhrentiables. Ergebnisse der Math. 71.

Springer-Verlag, Berlin, Heidelberg, New York 1972.

Indcx

almost quasihomogeneous - function,

-

93

distribution,

- extension,

-

order,

14,82

cotangent bundle,

2S,30

degree,

30.93

13,25, 77 ,(I3

degree 5 r , 25

polynomial,

40. 93, 96

deficiency,

43

14, 15

degree 5 ( r , s ) , 16, 17 338

(almost) invariant,

339,348

analytic singular support, analytic wave front s e t ,

82, 100, 419

differential operator,

167, 169

division theorem, basis

14, 30

dual basis,

real -,

6.14

duality bracket,

179, 280, 282, 294,

20,30

real-complex - , dual - ,

15

77, 260, 300

Dirac distribution,

362 - 364

14,30

Bernstein's fundamental solution, 92,104 330

bipolar theorem,

296

eigenspace,

5

generalized

-,

4, 5

Euler operator w i t h respect to M , 18

complex structure on V ,

-

on V',

30 385

conormal bundle,

83

generalized -, extension,

Convention 1.24*,

19, 75, 79/80

75, 06 153,168, 276

43, 78,364,365,371, 376

30

Conventions 1.24.A & B ,

20,23

coordinates

Fourier - inversion formula,

quasihomogeneous polar pseudo-real - ,

- -

80

inhomogeneous - ,

67

real-complex

transposed - , Euler equation,

complexification, contraction,

4

-

- , 1 , 8,67

20

on V ,

- transform, 1,20

o n V * , 30

84

- transform of u,,,, Frkchet space,

353,382

8 4 , 183, 219,224,227

FrCchet-Schwartz space, copolynomial function,

14, 15

84/85

- representation formula, 409, 424

functional equation,

225, 230

181, 194

448

Index

fundamental s o l u t i o n ,

317,325,327,

3 2 8 , 3 3 0 , 3 3 8 , 3 4 0 , 417 - 421 Bernstein's -,

nuclear ( S ) - s p a c e ,

-

224,229

Frhchet-Schwartz s p a c e , 225,230

92,104,330 order of h ,

88

G-invariant,

111

order 5 N ,

25.93

@-invariant,

111, 112, 114

outward normal unit ve c tor, 4, 5

generalized eigenspace,

181, 216, 218

growth conditions,

Paley- Wiener the ore m ,

3 8 4 , 410, 425 partial Fourier t r a n s f o r m , 33'). 420

Haar m e a su r e ,

111, 112

polar s e t ,

heat o p e r a t o r ,

3271328,338, 417

poles of h ,

hypoelliptic,

294, 302 88

104, 3 4 0 quasihomogeneity o r d e r ,

infinitesimal g e n e r a t or ,

1

invariant of degree m ,

338

Jordan canonical f o r m ,

23

average, lI7/ll8,1S3/1S4, 181, 242 continuation,

40,93

exte nsion,

hull,

Laurent s e r i e s,

88

( LF) - s p a c e ,

77

13

42

h u l l (of type M ) a t infinity,

88

407

left-invariant Haar measure, Leibniz r u l e ,

174/175.355

78

func tion, 327

Laurent coefficient,

03

quasi h omog en eo u s

distribution,

k t h order deficiency, Laplacian,

69

111

part,

2 0 , 25, 101/102,201

22, 31

polar c oordina te s,

224,229

principal p a r t ,

linear manifold,

14

locally convex topologies,

179/180,

2 2 4 , 2 2 5 , 2 2 7 - 230,2Y4

h7

390, 404

r a y,

8 , 18, 171, 383

set,

42

wave front s e t ,

383/384, 400

locally M - b o u n d e d , 135,139 - 142,145

m a trix, M-bounded s u b s e t of X , weakly -, M-connected ,

118

234

0

spe c trum 4 1 , 12

r ep r ese nta tion,

62,136,140,141

( M , I ) - b o u n d e d in X ,

126

Riesz isomorphism, r o t at i on,

84

142

( M , I ) - b o u n d e d s u b s e t of X , 118, 125 weakly -, M- te m p e r a t e ,

scalar produc t o n V ,

234

( M , I ) - temperate,

-

247

234

meromorphic f u n c t i o n ,

on

v*,

84

Schrodinger o p e r a t o r ,

88,lOS

s emi - norm s,

2 3 3 0 , 3 3 8 , 417

171,224,225,227,230

Index

449

semi-simple,

25

semi-simple part,

307, 323

singular support, 7 9 , 9 7 , 9 9 , 170, 3 4 8 , 389,403 solvability condition,

5 . 6 , 385

spectral projection, support,

286, 287, 320

7 9 , 9 7 , 9 9 , 1 2 3 , 155, 171, 186, 194 + 2 1 6 , 2 2 2 , 2 2 9 , 2 4 3

supporting function, tangent s p a c e ,

385

47

Taylor's f o r m u l a ,

21

temperate distribution, topological dual, torus, type M , type p ,

84

180,280,282

9

1 3 , 2 5 , 42. 77,03 13

wave operator,

330/331

weak homomorphism, weak topology,

295

171

weakly ( M , I ) -bounded, weakly M-bounded, weight function,

234

234

118, 182. 183

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