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 <
a
,
*
> 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,.. . ,<,)
ces. In t h i s case we denote by k , M,
(resp. M , ) ,
(resp. k c ) the number of Jordan blocks in
and for every e 6 1 R : = N k l R (resp. L E I C : = l N k , )
(resp. cL) the dimension of the tth (resp. Lth) Jordan block of M, This means that for every t € I I R (resp. LEI,)
denote by d , (resp. Mc).
the sequence ( a D e - l + l , . . . , a D e )
(resp. ( b c P - l + l , . . . , b , - e ) ) is an M-cyclic IR-basis (resp. @-basis) of its span over IR (resp. 43) where L
P
D e . -- L d i
(resp. C l : = C c , ) .
j=1
Remark 1.30. fi1 For ever)'
J=l
c u = ( p , y , S ) ~ X there are a number N a 6 N o and n o n -
trivial poljmomial functions P a , k , I s k i N , ,
o f degree I / a /satisfying
such that t €10,+a[. k=l
24
I.
where the Functions c ~ k : l O , + w [ - I R ,
k E N , , are deFined by
I k!
(1.37)
W,(t):=-(logt)
k
(Almost) Quasihomogeneous Functions
.
(ii) Suppose that the assumption o f Convention 1.24.B. (ii) is satisfied. Then the
number N , is equal t o
where the data k R , d, , D p , and k , , c L , CL are defined in Convention I . 2 4 . B . ( i i ) . In particular, f o r ever) j E N , \
1a
is quasihomogeneous of degree
(Y-
i f and on/), i f p, = 0
I D P ; e 6 1 m } and Y J = O = & J For ever) J E N , \ { C L : L E I , / .
Moreover. f o r ever, [ E l R
(resp. LEI^) the degree of
the variables y D p - l + j , l _ < j > d p .(resp.
with respect t o
L C ~ - , + J , resp. Z c L - , + J
)
is not larger
than '-L
CL
dP
m F . Obviously, in view of o I . ~ k = i (+1 k )
(1.38)
j.k€N,,
Wj+ky
it suffices to prove the assertion for the case lrrl = I , only. In t h i s
C ~ Wthe
Lsser-
tion ( is an i J immediate consequence of (1.10).For the proof of lii)we may assume that k,+
k, = I . In this case we denote by
X the single eigenvalue of M. If k,= 1
then X is real, and the matrix of t-'(M,),
is of the form (l.10.b)' where s : = l o g t .
Hence we deduce that
Since on the right-hand side the leading term (with respect to the powers of s ) is equal to
the assertion follows. If k, = 1 then in a similar way one obtains c
-x I 71-7 I S 1 XaOMt=
TT [ (
J=1
C-J
i
C-J
i
S - 'J C $ z J + i ) " ( 2 F z J + i )] i=O i=O
Since here o n the right-hand side the leading term is equal t o
1.c
25
( A l m o s t ) Quasihomogeneous Polynomials
t h e a s s e r t i o n f o l l o w s i n t h i s case, as w e l l . m
Comllary 1.31. Suppose that M is semi-simple, i.e. f o r the matrices M R and
M, introduced in Convention 1.Pic.B above we have Mm = diag(p,, . . . , p d ) and M, = diag(5;,. . . , c,).
Then for ever), a E X the monomial function x u is quasihomo-
geneous o f type M and o f degree a,,,,.
I
I f M is n o t s e m i - s i m p l e t h e n R e m a r k 1.30 s h o w s t h a t xu is n o t a l w a y s q u a s i h o m o g e n e o u s b u t o n l y almost quasihomogeneous o f order 5 N , in t h e s e n s e o f
Deflnitlon 1.32. L e t N € N o . A f u n c t i o n q o : X-@ neous of degree m
,
of tjpe M . and of order , keN,,
o f f u n c t i o n s q k : X-QI
5
is c a l l e d almost quasihomoge-
N if a n d o n l y if t h e r e is a n N - t u p l e
such that
N
(1.39)
t-mqo(Mtx) =
wk(t) qk(x),
( x , t ) € X x ~ l O , + ~- C see (1.20).
k=O
Now w e fix a polynomial function P:V-@.
In o r d e r to g r o u p t h e t e r m s in
(1.33) a c c o r d i n g to 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 o f M, w e i n t r o d u c e
Notation 1.33. U n d e r t h e C o n v e n t i o n 1.24 w e set (i)
X(M):= { a M ; a: E[)
(ii)
I Q , , P ) ( , ~: ~ = a
;
;1! ( a a p ) ( o ) x U . ~
=
x€V;
m
( i i i ) A ( P I : = { t E @ ;Q p P $ O } .
For r e a s o n s to b e c o m e c l e a r i n a m o m e n t Q m P is c a l l e d the (almost) quasihomogeneous part Q,, P o f P of degree m (and of type M). O f c o u r s e , Q m P = 0 if r n C U ( M ) , i.e. A ( P ) C U ( M ) .
Proposltlon 1.34. For ever)' m E#(M) Q m P : V+C
is a poljnomial function being
independent of the choice o f coordinates in Convention 1.24 and having the f o l lowing properties:
26
( i ) Q,P
I.
( A l m o s t ) Quasihomogeneous Functions
is almost quasihomogeneous o f degree m ; in case M is semi-simple
QmP is even quasihomogeneous o f degree m . (ii.al
P=
A
QpP;
P€A(P)
is a family o f non-
lii.6) conversely, i f 2 is a finite subset o f C and (P,),,,
trivial polynomial functions on V such that for every ( € 2 P p is almost quasihomogeneous o f degree 4' and such that (1.40)
P = Z P p P € t
then d e = A ( P ) , and P p = Q p Pfor every ( € 2 . (iii) P is almost quasihomogeneous o f degree m i f and onlj, i f P = Q, P proOf. (il:This is a consequence of Remark 1.30.
.
u) This : is (1.33). For
later
purposes it is suitable to formulate a part of t h e proof of (ii.b) separately a s
Remark 1.35. For everj
k'E#IMI
the equations
t17
(1.411
IQpPIoM,=tPSwkft)Pp,k,
t € 1 0 , +a[.
k=O
define a sequence o f polynomial functions P p , k , k E N o
,
onlj finitely many o f
them not vanishing identically, having the following properties: ( i ) Pp,o = QpP ;
(iil PP,k = Q I ( P P , k;)
(iii) deg PP,k 5 deg P p . 0 5 degP; and i f QpP is o f degree
5
( r , s l with respect t o
an M-invariant decomposition (1.26) then the P p , k are so, as well. (ivl I f M is semi-simple then Pp,k = 0 f o r every k E N .
mf. The existence of
the polynomial functions
Pp,k
satisfying (1.41), ( i ) , ( i i ) ,
( i i i ) , and ( i v ) is an immediate consequence of Remark 1.30. Their uniqueness is
due to t h e fact that for every non-empty open subinterval I of I O , + c o C
(1.42)
the functions I h C , t H t Z u j ( t ) ,z € C and j E N o , are linearly independent.
rn
End o f the proof o f Proposition 1.34.
m:Let
( P P ) e E 2 be
as in assertion ( i i . b ) ,
and for every 4 6 % let ( R e , k ) k c N , be a sequence of functions such that (1.41) is valid with ( Q o P , Pp,k) replaced by ( P e , R p , k ) . Inserting these equations into
1.c
27
( A l m o s t ) Quasihomogeneous Polynomials
t h e e q u a t i o n s for PoM, obtained from ( i ) and (1.40). likewise combining ( i i . a ) a n d (1.41), and making use of
pe,k
+ 0 for
Z={PeU(M);
( 1 . 4 2 ) , again, we deduce t h a t
s o m e j € N o } and t h a t P e , k = R g , , f o r arbitrary P E Z and k c N o . Since
P e , o = Q e P and Rp,O=Pe and since by Remark 1.35 we have P e , k f O f o r s o m e k c N o
if and only if Pp,O+O t h e assertion is proved.
(iiil.
"j" is t h e special c a s e Z = ( m ) o f ( i i . b ) .
Finally, t h e fact t h a t Q,P
follows f r o m ( i ) .
'"'*
d o e s not depend o n t h e choice of coordinates is an-
o t h e r consequence of ( i i . b ) . rn
As an immediate consequence of Corollary 1.31 and Proposition 1.34.(iii) o n e obtains
Corollary 1.36. If M i s semi-simple then ever-) pol-vnomial Function which is almost quasihomogeneous is, in Fact, quasihomogeneous.
I
Concerning p r o d u c t s of polynomial functions o n e has
Corollary 1.37. For any other polynomial Function R on V we have (i )
Q,,,(PR)
=
.Y
QpP Qnl-pR
P€?C(M)
and (ii I
ProoF. -(il. Since t h e right-hand side of t h e equation in ( i ) is a l m o s t quasihomogeneous of degree P o n e obtains t h e assertion by computing PR with t h e help of Proposition 1.34.(ii.a) - applied t o P and R - and by alluding to t h e uniqueness
property ( i i . b ) of Proposition 1.34.
(ii). -
O n e has to c o m p o s e b o t h sides of formula ( i ) with M,, and o n t h e r i g h t -
hand side one h a s to insert t h e formula (1.41) for P and t h e o n e f o r ( R , m - O instead of ( P , t ) . Then t h e desired formula is obtained by t h e uniqueness of t h e functions ( P R ) m , k when one is taking (1.38) i n t o account.
rn
For polynomial functions t h e converse of Remark 1.15 is valid:
Roporltfon 1.38. Suppose that R is another poljwomial Function
on V such that
28
I.
( A l m o s t ) Quasihomogeneous Functions
P R Z 0 . If P R i s a l m o s t quasihomogeneous so are P and R ; in particular, iF P R i s quasihomogeneous then so are P and R .
proOf. We set D : = A ( P ) a n d E : = A ( R ) - s e e N o t a t i o n 1 . 3 3 . ( i i i ) . In view of t h e a s s e r t i o n s ( i ) a n d ( i i . a ) of P r o p o s i t i o n 1.34 it s u f f i c e s to s h o w t h a t (1.43)
ID1 = 1 = I E l .
N o t e t h a t D a n d E are n o n - e m p t y f i n i t e s u b s e t s of @ . M o r e o v e r , if w e f i x m e @ such that PR=Q,(PR)
t h e n in view of C o r o l l a r y 1 . 3 7 . ( i ) f o r every W E C t h e fol-
lowing implication holds: (1.44)
if t h e set 3 , + , : = { ( a , b ) E D x E ; a + b = w } c o n s i s t s o f precisely one element then w = m .
In order to d e d u c e ( 1 . 4 3 ) f r o m t h i s p r o p e r t y w e d e n o t e by <
.;
> t h e real s c a l a r
p r o d u c t o n 63 a n d o b s e r v e t h a t t h e union of t h e sets k e r < k - k ' , * > , k , k ' E D s u c h that k#k', and ker<4-0';>
, 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
<4',z> w e d e r i v e f r o m ( 1 . 4 5 ) t h a t
By ( 1 . 4 4 ) it f o l l o w s t h a t k + P = m = K + L ,i.e.
J k + @ = . C ( m = J K + L so
t h a t by ( 1 . 4 6 ) ,
I = 1 = IE, I
so t h a t ( 1 . 4 3 )
a g a i n , w e c o n c l u d e t h a t k = K a n d 4 = L . T h i s i m p l i e s ID, follows from (1.45). m
Corollary 1.39. Let R b e another pol-vnomial function such that P R f 0 . IF P R i s a monomial Function then so are P and R .
m F . I t f o l l o w s f r o m E x a m p l e 1.23 t h a t a s a m o n o m i a l f u n c t i o n PR is q u a s i h o m o g e n e o u s of t y p e p f o r every P E N " . By P r o p o s i t i o n 1.38 t h e s a m e r e m a i n s valid f o r P a n d R . T h i s , in t u r n , i m p l i e s t h a t P a n d R are m o n o m i a l f u n c t i o n s , as w e l l . I n d e e d , let a , p EN," b e s u c h t h a t P ' a ' ( 0 ) $ 0 $ P'"(0).
T h e n < a , p >= < f i , p ) f o r
1.c
2Y
( A l m o s t ) Q u a s i h o m o g e n e o u s Polynomials
k=
2 '.* I . ( p i - a i )
every P E N " , in p a r t i c u l a r ,
(uj-pj)
i . e . uj= p j f o r e v e r y j E N,,
as c l a i m e d . rn
for arbitrary j€Nn and k C N ,
In view o f P r o p o s i t i o n s 1 . 3 4 . ( i i i ) a n d 1 . 2 8 . ( i i ) t h e d i m e n s i o n of t h e s p a c e $ l i r I ( V )
of p o l y n o m i a l f u n c t i o n s o n V w h i c h a r e 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 is e q u a l to t h e n u m b e r of e l e m e n t s o f t h e set
T h i s set is easily d e t e r m i n e d in c a s e V is M - i n d e c o m p o s a b l e : if V = V, is obviously e q u a l to {PEN:';
a n d if V = V,
IPI = m/X} where
then it
X is t h e only e i g e n v a l u e of M :
it is d e s c r i b e d in
Remark 1.40. Let h E C \ R . and s u p p o s e that V = C n is M-c-iclic and that h i s the o n l y point in 01M). (i)
I f R e l = O then t h e s e t 11.47) is equal t o { ( y . S ) ; y . h 6 N g . l y l - l ~ S l = m / h } ,
( i i ) I f R e l f O then 11.47) equals { ( y . h ) ; y , S E N t . l y l = m R , lSI=m,} m R ,.= 1
in particular.
mf. Let
R e irr JReA
I m in + K) and
ml :=
in case n = I there i s precisel-, o n e
where
J1 m - - l : CY E
21 such that a M = m . -
a=(y,S)~#=lN:xN,".
Then a , = l y l X + I S I X ,
a n d t h e a s s e r t i o n u is
c l e a r w h i l e (ii)is a n i m m e d i a t e c o n s e q u e n c e of t h e f o l l o w i n g l e m m a . rn
Lemma 1.41. I f A i s a s in Remark 1.40. ( i i ) then for everj z E C the equation 11.481
rl +sl =z
W e close t h i s s e c t i o n by c o n s i d e r i n g polynomial f u n c t i o n s P o n t h e d u a l s p a c e V' o f V w h i c h a r e ( a l m o s t ) q u a s i h o m o g e n e o u s of t y p e
M ' .
In order to e m p h a s i z e
t h i s w e are g o i n g to d e n o t e t h e f u n c t i o n s P g , k associated to s u c h a p o l y n o m i a l f u n c t i o n P in Remark 1.35 by
Pi,k.
As a c o n s e q u e n c e o f Remark 1.17 ( r e s p . l . 1 7 ' ) ,
P r o p o s i t i o n 1 . 3 4 , a n d o f (1.41) w e n o t e t h e f o l l o w i n g f o r m o f t h e c h a i n r u l e .
30
I.
Finally, we show that the derivatives
a"
(Almost) Quasihomogeneous Functions
introduced i n Notation 1.25.B are induced
by almost quasihomogeneous polynomial functions on V * . To t h i s end we have to introduce real-complex coordinates on V* related to those for V introduced in Convention 1 . 2 4 . A . As i n Convention 1 . 2 4 . A . by ( q l ,.. ..qd.qd+l , . . .
we
denote the ( d + c )-tuple of coefficient functionals on V associated with the R-basis A of V,
and the @-basis B of V,
23': =
(ql
as introduced in Convention 1.24. Then
, . . . , q d , Re q d + , . . . , Re '
is the IR-basis of V* dual to the IR-basis
~ d Im + ~rid+ 1 , . . . I
I
Im '
~ d + ~ )
b : =( a l , . . . , a d , b l ,..., bc,ibl , . . . , ib,) of
V . It follows that ( q l , . . . , q d ) is an IR-basis of ( V * b I R and that (Reqd+l, . . . , Reqd+c.
I m q d + l , . . . , I m q d + = ) is an R-basis of ( V * ) c . Moreover, i f J:V,-V,.
veiv,
denotes the complex structure introduced i n the text preceding Notation I.6.A then
J* is the corresponding complex structure on ( V * ) , . (1.50)
I t is easily seen that
Imqj= -J*(Reqj)
so that ( R e q l , .. . , Req,) is a @-basis of V * . In order to identify t h i s as the @-basis (ql, . . . , q c ) of ( V ' ) ,
(1.51)
which is dual to the @-basis B of V,
3:L,( V,, IR) -+ Lc( V,,
@) ,
x
Hx
we note that the map
- i J'x ,
is a @-isomorphism, its inverse being given by v H R e v . We have c
(1.SI)'
c
3( ~ ( ~ i R e q j + u j I m q i ) ) = ~ ( ~ i - i u i ) q j , j=l
<,UEIRC.
j=l
Conventlon 1.24*. Let ( q l , . . . , qd+,=) be as above. We consider V* as equipped w i t h real-complex coordinates defined by the IR-basis ( q i , . . . , q d ) of ( V * ) m and
t h e @-basis (2qd+1,...,2qd+c) of ( V * ) , where the latter space is identified w i t h L,(V,,@)
via t h e map 3 defined i n ( 1 . S l ) . This defines a real-complex basis
L'* ' .' -
(q1 ,...,qd,2q,+,,...,2qd+c) of
v*.
1.c
31
( A l m o s t ) Q u a s i h o m o g e n e o u s Polynomials
Note that under the identification of (V*),
and L,(V,,@)
via 3 the IR-basis
%* of V* becomes t h e R-basis (4,,. . . , f l d r q d + l , . . . , ' I d + c , i ' l d + l l ..., i r l d + c ) . over, note that by Convention 1.24.B.(i) the real-complex basis cS*
More-
contains a
basis of G M * ( X ) for every X € a ( M ) = o ( M * ) ,and that (1.34) transforms into
Finally, we observe that (1.34) and (1.34)* imply that
The reason w h y the factor 2 h a s been introduced into the complex part of V * becomes clear in the proof of
Remark 1.43. ( i ) For every
P, : V *
--f
C Y E there ~
is a (unique) polynomial function
C such that
P, is almost quasihomogeneous o f degree a,,,, and of tjpe M'. More precisel!.
Pa is the monomial function basis @'
which is associated with the real-comple\
o f V * (introduced in Convention 1.24 * ) when in Notation 1.2S.A V is
replaced bj V ' . In other words, for ever! ( E V ' we have
lii) For ever-v polynomial function Q : V x+C o f degree r E N the Leibniz rule can be written as
where the derivatives
proOf.
u: By
(u,c,iJ)
Q(a)
o f order a are formed with respect to the basis Q * .
we denote the real coordinates o n V* with respect t o the
IR-basis 8* introduced in the text preceding Convention 1.24*. For every k € N d (resp. jEN,)
( resp.
we define Pk:V*-C
(resp. Qj:Vx-C)
( S j - i u i ) ) . Then
( Q j 0 ts*1 ( v , [ , i ~ ) : =
-
d Y k = P k ( 3 ) , 3 = Q j ( 3 ) ,and dLi = Q j ( 3 ) . 'j
It follows that (1.53) is valid for
by ( P k o t B * ) ( v , < , u ): = uk
32
I.
(Almost) Quasihomogeneous
Functions
In view of ( 1 . S l ) ' P, t u r n s o u t to b e n o t h i n g else t h a n t h e m o n o m i a l f u n c t i o n x e x , which c o m e s a b o u t if in N o t a t i o n 1.2S.A V is r e p l a c e d by V' a n d if V' is e q u i p p e d w i t h t h e r e a l - c o m p l e x basis cS* i n t r o d u c e d in C o n v e n t i o n 1.24'. H e n c e R e m a r k 1.30 - a p p l i e d to ( V * , M * ) i n s t e a d o f ( V , M ) - s h o w s t h a t P, 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 d e g r e e orM* w h i c h e q u a l s a M by ( 1 . 5 2 ) . Finally, (1.53.a)' is a c o n s e q u e n c e o f (1.25) a n d (1.53); a n d in view o f t h e chain r u l e (1.53.6)'f o l l o w s f r o m ( 1 . 5 3 . a ) ' . (ii).Taking ( i ) i n t o a c c o u n t a n d applying P r o p o s i t i o n l . 2 8 . ( i ) t o ( V * . M * ) i n s t e a d
of ( V , M ) o n e o b t a i n s f o r a r b i t r a r y a,PE'U t h a t
H e n c e , c h o o s i n g c, E C such t h a t Q =
ca P,
which is p o s s i b l e in view o f
( i ) a n d (1.33) - o n e d e d u c e s t h a t
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 by Lemma 1.27
t d B Non- lr iv ia I
Exa mp Ic!s o I' Q u a si homagenc w u s 1'0 I y nom1a I
F u n r * l i o n s in (*;iscB M i s No1 Scnii-siniplcb
If M is s e m i - s i m p l e t h e n a c c o r d i n g t o C o r o l l a r y 1.30 every a l m o s t 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 Function o n V is q u a s i h o m o g e n e o u s . So in t h i s c a s e P r o p o s i t i o n 1.34.(iii) describes a l l 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 o n V . In t h e p r e s e n t s e c t i o n w e s h o w h o w to o b t a i n 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 in t h e p r e s e n c e o f m u l t i p l e e i g e n v a l u e s . F i r s t of all w e give a c o o r d i n a t e - f r e e r e f o r m u l a t i o n of t h e c o n t e n t o f t h e l a s t b u t o n e p a r t o f Remark 1 . 3 0 . ( i i ) , d e n o t i n g t h e c a n o n i c a l p r o j e c t i o n o n t o t h e real vector s p a c e
by d : V -
V': i f P : V'+
C i s a polJ.nornial Function which is quasihomogeneous
33
1.d E x a m p l e s of Q u e s i h o m o g e n e o u s Polynomials
of degree m and o f type M ' where M ' € L ( V ' I is induced by M then P o x ' is a polynomial function on V which is quasihomogeneous o f degree m and o f type M . Note t h a t M' h a s t h e s a m e eigenvalues a s M b u t of algebraic multiplicity 1 . Are t h e r e any o t h e r quasihomogeneous polynomial functions o n V ? Under special a s s u m p t i o n s t h e a n s w e r is "no". In a n y c a s e , t h e n u m b e r o f 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 is r a t h e r r e s t r i c t e d u n d e r t h e p r e s e n c e o f m u l t i p l e e i g e n v a l u e s . To see
t h i s m o r e c l e a r l y w e a s s u m e f o r t h e rest o f t h e p r e s e n t s e c t i o n t h a t V be
X t h e o n l y e i g e n v a l u e o f M w e let b = ( b , , . , . , b,) b e a
M-cyclic. D e n o t i n g by
K X - b a s i s o f V s a t i s f y i n g (1.11) so t h a t t h e m a t r i x of N w i t h r e s p e c t t o B is o f t h e f o r m (1.11)'. I t f o l l o w s t h a t M, is given by n-j
(1.55)
( M , X ) ~= t X i=O
$ si
w h e r e s : = log t ,
w h e r e t h e p o i n t s x o f V a r e d e n o t e d by t h e c o o r d i n a t e s
j€Nn, (x
l , . . . ,)in) E ( K ,
)"
with
r e s p e c t to t h e basis B . We s u p p o s e t h a t n 2 2 . T h e f o l l o w i n g r e s u l t s t a t e s o n l y a very w e a k restrictive c o n d i t i o n s h a r e d by a l l 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 tion o n V.
Propodtion 1.44.
Under the preceding assumptions everr pol-vnomial function
P: V - C
-
which is quasihomogeneous o f degree m
f
0 vanishes on the l-dimen-
sional subspace ( M A) ''-I I V ) . T h e p r o o f is a b y - p r o d u c t o f m o r e p r e c i s e c o n s i d e r a t i o n s . We begin w i t h
The case " I ED? ". H e r e it f o l l o w s f r o m E x a m p l e 1.21 t h a t n
dM= ~
(
~
x
~
x~ n +) l : =~ ~ )~. + ( h~e r e ~w e set +
j=l
We fix a polynomial f u n c t i o n P : V - @
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
degree m , i . e .
P=
c Pa xu meld
w h e r e d : = m / X , I d : = { u € N , " ; l u l = d } , a n d P u : = P ( u ) ( O ) / u ! .We a r e g o i n g to t r y to f i n d 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 on t h e c o e f f i c i e n t s Pa w h i c h e n s u r e t h a t P is q u a s i h o m o g e n e o u s . To t h i s e n d w e e m p l o y Euler's e q u a t i o n . S i n c e n
dMXu=
c
a,? 1
'
u-e.
(hxj+Xj+,)ujx
j=l
34
I.
( A l m o s t ) Quasihomogeneous
Funct.ions
we conclude that
where
I t follows that P is quasihomogeneous of degree m if and o n l y if (1.56)
L,(P) = 0
for every
Peld
Note that J p consists of one element only if and only i f
p = (d-P,+,,O ,..., O , p p + l ,,O. . . , 0 ) w i t h
p
is of the form
P P + l E N d for some @€lNn-i.
Hence, (1.56) implies that
r€Nd .
In case n = 2 the equations (1.57.a) mean that
Remark1.45. Suppose that under the preceding assumptions we have n = ? . Then P is quasihomogeneous variable x,.
of degree m i f and on/), i f
P d o e s not depend on the
I
In case n > 3 , however, there are choices of p such that the equation (1.56) relates a t least two of the coefficients of P to each other. We restrict ourselves t o t h e case n = 3 when besides the equations (1.57.a) and (1.57.b) there appear equations involving two of the coefficients of P , only: for the triples P E l d satisfying p2 L 1 and b3->1 the equation (1.56) leads to
1.d E x a m p l e s of Q u a s i h o m o g e n e o u s
35
Polynomials
Propodtion 1 . 4 6 . Suppose that under the preceding assumptions we have n = 3 . Then the polynomial functions (1.S8)
s E G V ~ ,SSd/.?,
w 3 3 ( x , y , z )H ( y 2 - 2 X Z ) S Z d - 2 s I
constitute a basis o f the space of polvnomial functions on W 3 which are quasihomogeneous of degree m = dA . proOf. S i n c e ( y + ~ z ) ~ - 2 ( x + s y2 z+/ 2s ) z = y 2 - 2 x z
t h e f u n c t i o n s are q u a s i h o m o g e -
n e o u s . T h e rest of t h e a s s e r t i o n i s p r o v e d by i n d u c t i o n o n d . If d = l t h e n t h e a s s e r t i o n is o b v i o u s . Now s u p p o s e t h a t d ? 2 , a n d let P be q u a s i h o m o g e n e o u s of
degree d X . S u p p o s e f i r s t t h a t z divides P , i . e . P ( x , y , z ) = z Q ( x , y , z ) . 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 1.38 t h a t Q is q u a s i h o m o g e n e o u s of degree m-X = ( d - I ) X so t h a t by a p p l y i n g t h e i n d u c t i o n h y p o t h e s i s to Q w e d e d u c e t h a t P 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 (1.58). H e n c e w e s u p p o s e t h a t z does n o t divide P, i.e. P ( r , d - r , O ) # O f o r s o m e r E { O ) u N , .
In view of (1.57.a) t h i s m e a n s t h a t
c : = P ( O , d , O ) # O . F r o m ( 1 . 5 7 . ~ )w e d e d u c e b y i n d u c t i o n t h a t
p( k.d - 2k . k ) =
(-,)k
d(d-Z)...(d-Zk+Z)
k EN. k
k!
If d w e r e odd, i.e. k : = ( d - l ) / Z E N t h e n , in p a r t i c u l a r , w e w o u l d have P ( k , l , k ) # O w h i c h c o n t r a d i c t s (1.57.b) for r = k + l . H e n c e j : = d / 2 b e l o n g s to !N, a n d , c o n s e 2
q u e n t l y , by R ( x , y , z ) : = P ( x , y , z ) - c ( y - 2 ~ 2 a) ~polynomial f u n c t i o n R is d e f i n e d w h i c h 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 dX a n d which is divisible by z . H e n c e t h e p r o o f of t h e i n d u c t i o n s t e p is c o m p l e t e d by a n a p p l i c a t i o n of t h e first p a r t of
w
t h e i n d u c t i o n s t e p a l r e a d y d e a l t w i t h above.
The case "A € @ \ I ? S i"n .c e h e r e
b
is a @ - b a s i s of V w e d e n o t e t h e c o o r d i n a t e s
of a p o i n t z in V w i t h r e s p e c t to b by ( z , , . . . , z n ) . I t f o l l o w s f r o m E x a m p l e 1.21 a n d f r o m (1.32) a n d ( 1 . 1 1 ) t h a t n
aM C [ ( x z ~+
-
-
~ ~ ++ ~(xyi ) +3% ~ )aj]
j=l
-
w h e r e h e r e w e set z , + ~: = 0 , 3i : = d2, a n d di := 31
2i
.
Now w e fix a polynomial
f u n c t i o n P o n V 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 d e g r e e m . T h i s m e a n s t h a t P=
c
P,,@Z a.--P 2
(a.P)a'Z,,,
36
I . ( A l m o s t ) Quasihomogeneous Functions
w h e r e U r n : = { ( a , @E)N , " x N , " ; I a l l . + l B I X = m }
(dZd$P)(O)/(a!p!).
and
O u r aim is to f i n d c o n d i t i o n s o n t h e c o e f f i c i e n t Pa,,,
w h i c h are n e c e s s a r y a n d
s u f f i c i e n t f o r P to be 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 . H e r e , a g a i n , w e m a k e u s e o f Euler's equations. Since n
dMzaZp =
n
(Xzj + z j + l ) aj Za-eJ j=1
?Ig +
(7, + G )@ j z'yg-ei
j=1 Pj"
uj 2 1
we conclude t h a t
(aM-m)P =
2
P,,@
(~l,p)c'u,
[
2
n
aiz a-ej+ej+l YP
j= I
j=1
Pi? 1
a i l1 I1- i
n-l
a-e'+e. =j =C C ajPu,pz 1 I+I,P+ C 1 (u,P)E'U, j=l "j 2 1 I1
~ ~ ~ ~ y p - ~ i + ~ i + 1 ]
+
u-P-ej+ej+r C PjPu,pz z (a.@)eXrn
Pj' 1
-1
where
w i t h J u : = { j € N l l - i ;m j + i 2 1 } . I t f o l l o w s t h a t P 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 if a n d o n l y if
Applying t h i s to a = ( k - l , l , O , . . . , 0 ) a n d p = ( 4 , 0 , . . . , O ) a n d to m = ( k , O ,..., 0 ) and p = ( P - I , l , O ,...,0 ) w e obtain
In p a s s i n g w e c a n n o w give t h e
Proof of Proposition f.44. In view of (1.11) w e have ( G X ) " - ' ( V ) = K , b , .
Conse-
q u e n t l y , if A E R ( r e s p . A 4 R ) t h e n t h e a s s e r t i o n f o l l o w s f r o m (1.57.a) For r = d (resp. from (1.60.a)).
37
1.d Examples of Quasihomogeneous Polynomials
N e x t w e o b s e r v e t h a t t h e a n a l o g u e o f R e m a r k 1.45 f o r A € o \ R is f a l s e : if n = 2
( Xo I
t h e n a s s u m i n g t h a t V = @ X @ a n d t h a t M i s given by t h e m a t r i x
M,z = t X(2,
t
w e see t h a t
Lz2, z 2 ) w h e r e L : = log t , a n d s i n c e
it f o l l o w s for a r b i t r a r y k , r , s E N O t h a t t h e f u n c t i o n d e f i n e d by (1.61)
zH
(ZIG- F Z 2 ) k 2 2r -2s2 -
i s q u a s i h o m o g e n e o u s o f d e g r e e k X + k X + r A + s X . In f a c t , in c a s e n = 2 e s s e n t i a l l y t h e s e a r e a l l 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 o n V :
Ropoaltlon 1.47. Suppose that under the assumptions s t a t e d a t the beginning of' th e present section we have n '13 and ImX # 0 . Then a basis o f the space o f pol.,nomial functions on V which are quasihomogeneous o f degree m is given b-i. the functions o f the f o r m (1.61) where Ik.r.s) runs through the elements OF the set
[ 3X:=/
3
i f ReA = O
{ ~ k , r . s ) € N:or - s = d : = m / A } i(k,r.s)cN:: k_iminImR.m,).r = m R - k .s=m,-k}
if ReA#O
the numbers m R and m , being defined in Remark t . - J O . ( i i ) . Pr o o f . W e fix a polynomial f u n c t i o n o n V 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 . Besides ( 1 . 6 0 . a ) o n e o b t a i n s f r o m ( l . S 9 ) t h e f o l l o w i n g c o n d i t i o n s :
N o w , t h e p r o o f t h a t P is o f t h e d e s i r e d f o r m is d o n e by i n d u c t i o n o n d e g P , t h e case d e g P = 1 b e i n g o b v i o u s . If P is divisible by z2 o r
then w e apply t h e
i n d u c t i o n h y p o t h e s i s to t h e q u o t i e n t ( w h i c h is q u a s i h o m o g e n e o u s , a s w e l l ) a n d o b t a i n t h e a s s e r t i o n f o r P. H e n c e w e s u p p o s e t h a t P is divided n e i t h e r by z2 n o r by
G . This
3':=((u,p)~2L,;
m e a n s t h a t b o t h t h e sets P2=O,
P,>,#O)
J : =( ( a , p )E X r n ; a 2 = 0 , Pa,,#O} a n d
are non-empty.
Note that
in view of
the
c o n d i t i o n s ( 1 . 6 0 . ~ ) a n d (1.6O.b) w e have 3 = { ( a , f i ) E % , ; a 2 = O = [ j l , P,,,#O} a n d 3 ' : = ((a , p )EX,;
p z = O = al ,
0 1 , In f a c t , w e a r e g o i n g to s h o w t h a t
38
1.
( A l m o s t ) Quasihomogeneous F u n c t i o n s
W e c a n t h e n d e d u c e t h a t t h e polynomial f u n c t i o n R d e f i n e d by R ( 2 ) : = P ( Z )-
C P(,,o),(o,k) ( Z l G - 5 2 ~ ) ~ k€Q
is divisible by z2 ( a n d , in f a c t , by
2 ) .Applying
the first part of t h e induction
s t e p to R w e c o n c l u d e t h a t P is o f t h e desired f o r m which c o m p l e t e s t h e p r o o f of t h e i n d u c t i o n s t e p .
In order to p r o v e (1.02) w e fix ( ( k , O ) , ( O , s ) ) E Z I a n d ( ( O , P ' ) , ( r ' , O ) ) E , 3 ' a n d d e d u c e f r o m (1.hO.d) t h a t j!
(1.63)
P ( k , O ) , ( O . s ) = ( - ' ) j k ( k - l ) . ..(k-j+l)'(k-j,j),(j,s-j)
1
j<min{kls),
and ( ( 1.03 )'
- 1 ) J r' ("-1 ) . '
P ( o , e * ) , ( r * , o=)
('I-'
j!
+
1 ) P, - j ) , (,.o
~
,j )
, j 5 min {
(1,
r' ) .
In t h e case " R e A f 0 " w e d e d u c e f r o m Remark 1 . 4 0 . ( i i ) t h a t k = m R = 4 ' a n d s = m I = r ' . If m l < m R t h e n (1.63)s h o w s t h a t
P ( k - s , s ) , ( s , o ) # O w h i c h in view of (1.6O.b)
i m p l i e s t h a t k = s , i . e . m I = m R a n d t h e e q u a t i o n in (1.6'2.d) is valid f o r k = m R ; in p a r t i c u l a r , w e have U = ( m R ) . If m I 2 m R t h e n ( 1 . 0 3 ) ' s h o w s t h a t P ( e * , o ) , ( , . q - p * , e#' )O which in view o f ( 1 . 6 0 . ~ ) i m p l i e s t h a t 4 ' = r ' , i.e. m I = m R a n d t h e e q u a t i o n in (1.02.d) is valid f o r k = m R ; in p a r t i c u l a r , w e have 5i= ( m ~ ) a,g a i n . S i n c e in b o t h s u b c a s e s w e have s h o w n t h a t m I = m R t h e c o n c l u s i o n s of b o t h s u b c a s e s a r e valid, i.e. ( 1 . 6 2 ) is p r o v e d . Finally, w e c o m e to t h e case " R e A = O " . H e r e so t h a t k - s = d = P ' - r ' .
' U , = { ( a , P ) E N ~ x W ~ ;I a l - l p l = d }
If d ? O t h e n ( 1 . 6 3 ) s h o w s t h a t P ( k - s , s ) , ( s . o ) f O w h i c h
in view o f ( 1 . 6 0 . b ) i m p l i e s t h a t k = s , i.e. d = O , a n d t h e e q u a t i o n in ( 1 . 6 2 . d ) h o l d s .
*) in On t h e o t h e r h a n d , if d < O t h e n ( 1.63) ' s h o w s t h a t P ( ~ * , o ) , ( r ~ - e f*O, ~ which view of ( 1 . 6 0 . ~ )i m p l i e s t h a t r ' = P ' , i.e. d = O , a n d t h e e q u a t i o n in (1.62.d) h o l d s w i t h k r e p l a c e d by r'. S i n c e , a g a i n , in b o t h s u b c a s e s w e have s h o w n t h a t d = O t h e c o n c l u s i o n s of b o t h s u b c a s e s a r e valid, i.e. ( 1 . 6 2 ) i s p r o v e d .
1.e
A l m o s t Quasihomogeneous Functions
39
te) Almost Quaslhomogcneous Functlons
In t h e p r e s e n t s e c t i o n w e give a s h o r t d i s c u s s i o n o f g e n e r a l 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 a s d e f i n e d in Definition 1.32 a b o v e . In s e c t i o n ( c ) t h i s n o t i o n w a s m o t i v a t e d by t h e a p p e a r a n c e of a l m o s t 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 which are n o t q u a s i h o m o g e n e o u s . H o w e v e r , in view o f C o r o l l a r y 1.36 in case M i s s e m i - s i m p l e t h i s m o t i v a t i o n b e c o m e s p o i n t l e s s . In s e c t i o n s ( c ) a n d ( d ) of C h a p -
ter 2 b e l o w w e s h a l l m e e t 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 ( a n d d i s t r i b u t i o n s ) in a q u i t e d i f f e r e n t c o n t e x t , t h e r e b y o b t a i n i n g a m o r e s e r i o u s m o t i v a t i o n which is a l s o valid in case M is s e m i - s i m p l e . Below w e have to m a k e u s e of t w o s i m p l e p r o p e r t i e s of t h e f u n c t i o n s w i . T h e f i r s t o n e is a n i m m e d i a t e c o n s e q u e n c e of t h e h o m o m o r p h y p r o p e r t y of t h e l o g a r i t h m a n d of t h e binomial f o r m u l a : i
(1.64)
(dj(St)
=
j E N o , s , t E 10.+03[.
wk(s) b)j-k(t), k=O
T h e s e c o n d o n e is o b v i o u s : (1.65)
Ui(l/t) = ( - 1 ) J w j ( t ) ,
W e fix N E N .
Lemma1.48. Suppose that q o : X - C
i s almost quasihomogeneous o f degree m
and o f order 5 N . Then for ever-) k E N N the function qk appearing in (1.39) is uniquely determined and almost quasihomogeneous o f degree m and o f order 5 N - k
;
more precisely, we have N-k
I1.39)k
t-mqk(Mt,Y)
=s
o p t t ) qk+p(.Y),
Is,t l E X x , l O , + w c .
P= 0
I n particular, the s o - c a l l e d a l m o s t q u a s i h o m o g e n e o u s order of qo i s well-defined by
m.S i n c e
X x ~ l O , + m is c a n o p e n s u b s e t of X x l O , + ~ Ct h e u n i q u e n e s s of t h e
f u n c t i o n s qk i s a c o n s e q u e n c e of (1.42). F o r t h e proof of (1.39)k w e f i x ( x , s ) E X x M l O . + m C . S i n c e X is o p e n we f i n d a n
40
I.
( A l m o s t ) Quasihomogeneous F u n c t i o n s
T h e f u n c t i o n q k will a l s o be c a l l e d the k"'
order deficiencj, o f qo. I t is n o t
o n l y uniquely d e t e r m i n e d by qo b u t c a n also easily be c o m p u t e d o u t of q o . To s h o w t h i s w e m a k e u s e of t h e f o l l o w i n g m o r e p r e c i s e version of ( 1 . 4 2 ) :
Remark 1.49. Let (ti)ieuv, be a sequence of pairwise different numbers in the set 10.+~33L\lil. Then the N x N - m a t r h
w. W e set
si: =
log t i . I t t h e n f o l l o w s t h a t
w h e r e V ( s l , ., . , s N ) d e n o t e s t h e V a n d e r m o n d e d e t e r m i n a n t of s l , .. . , s N .
H
By s u b t r a c t i n g q O ( k ) f r o m b o t h s i d e s of ( 1 . 3 9 ) , i n s e r t i n g t = t i , a n d a p p l y i n g t h e i n v e r s e of t h e m a t r i x (1.66) o n e o b t a i n s
Remark 1.50. Let qo:X+
C be alniost quasihomogeneous o f degree m and of
order _< N . Then for ever). k E N N i t s k
th
order deficiency is given b),
N
qk = S L k i ( t j - l " q O o M ' i - q o ) on the set { . V E X : V I E " :
MtiSEX}
j=l
where (Lki)ck ,j ,
E"l
X"i
denotes the inverse of the matriv ( i . 6 6 ) .
8
41
1.e 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
S i n c e X x M I O , + m C is a n o p e n s u b s e t o f X x l O , + ~ Co n e o b t a i n s 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 R e m a r k 1 .SO.
Propodtion 1.51. Let qo :X -
C be a function which is almost quasihomogeneous
of degree m and o f order 5 N .
(i)
If qo belongs t o 2?:,,(X)
(ii)
Let r ENo u I a , o l : i f qo is a C'' function so are i t s deficiencies.
so do i t s deficiencies. #
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 o f d e g r e e 0 with c o n s t a n t d e f i c i e n c i e s a n d 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 of a r b i t r a r y d e g r e e a r e related in a s i m p l e way:
Remark 1.52. l i ) Let f : X -
C be almost quasihomogeneous o f degree 0 such
that its I s t order deficiencj is identicall) equal t o a constant P E C : then e.\pof is quasihomogeneous o f degree P . (ii) Converse/),, if q : X - l U .
+ w C is quasihomogeneous o f degree PEC then log q
is almost quasihomogeneous of degree 0 . its IS' order deficiencj being identicall)
equal t o P .
H o w to o b t a i n 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 of higher o r d e r is s h o w n in
Remark 1.53. Let v : X -
C be almost quasihomogeneous o f degree 0 and order
I such that its I s t order deficiencj, is identicallj. equal t o 1 . Then f o r ever)' j E N o
the function v j := v j / j ! is almost quasihomogeneous o f degree 0 and order j such that in case j ? I its Is'
order deficienq equals v,
-,.
w. By t h e b i n o m i a l f o r m u l a w e have f o r every ( x , t ) ~ X x ~ I O , + c o C "( X ) i - i
T h e p r e c e d i n g r e m a r k is a l s o a s p e c i a l c a s e of t h e f o l l o w i n g i m m e d i a t e c o n s e q u e n c e of Definition 1.32 a n d o f (1.38).
42
I.
( A l m o s t ) Quasihomogeneous
Functions
Remark 1.54. If f fresp. g ) is almost quasihomogeneous of degree m Iresp.P) and of order 5 N (resp. L ) then f g is almost quasihomogeneous OF degree m t i ? and OF
order 5 N + L such that f o r every
~ € N N + L its
k t h order deficiency is equal t o
N L
i=o
()!,
f/
gk - j
j2-k-L
where hl denotes the j t h order deficiency of h € l f , g ) .
I
In view of P r o p o s i t i o n l . S l . ( i i ) o n e o b t a i n s t h e f o l l o w i n g p r o p o s i 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 1.34.(iii) a n d t h e c o n d i t i o n s ( 1 . 4 9 ) ' a n d ( 1 . 3 8 ) .
Propoeltlon 1 . S . Let r E N , let q o : X d C be a C" Function which is almost quasihomogeneous OF degree m and of order 5 N , and let P: V * + C
be a polynomial
Function of degree 5 r which is almost quasihomogeneous of degree 'l and of type M'. Then P l d ) q c , is almost quasihomogeneous of degree m-l! and of t j pe
M. m
T h e t r e a t m e n t of a l m o s t q u a s i h o m o g e n e o u s d i f f e r e n t i a l o p e r a t o r s w i t h v a r i a b l e c o e f f i c i e n t s a p p l i e d to 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 is p o s t p o n e d u n t i l 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 have b e e n i n t r o d u c e d (see P r o p o s i t i o n 2.35 b e l o w ) . Likewise t h e infinitesimal c h a r a c t e r i z a t i o n of d i f f e r e n t i a b l e 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 will be c a r r i e d o u t in t h e s e t t i n g of 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 (see P r o p o s i t i o n 2.31 b e l o w ) . I n s t e a d w e s h o w t h a t ( 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 o n X c a n b e e x t e n d e d to t h e s m a l l e s t o p e n s u b s e t of V c o n t a i n i n g X which is invariant u n d e r t h e a c t i o n o f M , . T h i s r e s u l t is re-
q u i r e d f o r t h e p r o o f of P r o p o s i t i o n 2.37 b e l o w .
Defldtlon 1.56. I f Y is any s u b s e t of V t h e n ( i ) its quasihomogeneous hull (of tjpe
M1 Y,
is d e f i n e d by Y,
:=
u
M,Y
;
t E IO.+mC
( i i ) Y is called quasihomogeneous ( O F type
Y,
M ) if a n d o n l y if Y = Y M .
is obviously q u a s i h o m o g e n e o u s . N o t e t h a t Y M is o p e n if Y is so.
Roporltlon 1.57. IF q is almost quasihomogeneous of degree m and OF order
5 N
1.e
Almost Q u a s i h o m o g e n e o u s
43
Functions
then there exists a unique almost quasihomogeneous extension qM :X M
+C o f
the same degree and order. It is defined 6).
and has the following properties: ( i ) For every k€O\l, the k L h order deficiency o f qM is equal to (qk)M where qk denotes the
k t h order deficiency o f q .
( i i ) I f qEdPf,,(X)
then q M EdPloc(XM). /
I that q < C r ( X ) then q M E C r ( X M ) (iii) I f r < N u { ~ , iis~ such
P A . If q M is a n 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 o f q it m u s t s a t i s f y ( 1 . 6 7 ) . O n t h e o t h e r h a n d , let r , s . t E I O , + ~ aI n d x , y E X s u c h t h a t M , ( M , x ) = M , y . T h e n y = M,t,sx,
a n d it f o l l o w s by (1.3Y)k t h a t N
srn
N
C w k ( s ) q k ( y ) = sm 2 w k ( s )
N-k
(rt/s)'"
k =0
k=O
C
P=
( . ) p ( r t / s )q , + , ( x )
=
n
I=O'k=O
F r o m (1.64) a n d (1.65) w e d e d u c e t h a t
where I-i
Moreover, from (1.64) w e observe t h a t
I
q ( r t ) = c i = O w i ( r )w I - i ( t ) . C o m b i n i n g
t h i s w i t h t h e p r e c e d i n g e q u a t i o n s a n d c h a n g i n g t h e order o f s u m m a t i o n w e o b t a i n
T h e c a s e " r = l " s h o w s t h a t q M is w e l l - d e f i n e d by ( 1 . 6 7 ) . Applying t h i s to ( q k , N - k ) i n s t e a d of ( q . N ) a n d t a k i n g Lemma 1.48 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
44
I.
( A l m o s t ) Quasihomogeneous F u n c t i o n s
as well. The general case of the formula above can then be written as N
q M ( M r ( M t X ) ) = rm
Ui(r) ( q i ) M ( M t X ) , i=O
i.e. q M is almost quasihomogeneous of degree m and of order 5 N
;
more precisely,
(1.39)k holds with qk replaced by ( q k ) M
For the proof of the regularity properties (ii) and
m) it
suffices i n view of
Proposition 1.51 t o observe that for every z E X M there are t E l O , + a 1 and an open neighbourhood U of z in X M such that M , ( U ) C X and N -n'
w i ( l / t ) ( q io M t ) l u .
m
Under special assumptions on X and M the only quasihomogeneous C'n functions on X are polynomial functions. This follows from
Proposition 1.58. Suppose that o = o + . Let r 6 N be such that Rem/r " A m i n : = t n i n { R e A ;A g o , ]
(1.68)
And let f € C ' ' ( V ) be such that
f / c is
almost quasihomogeneous of degree m
If
f does not vanish identicall) then m € ? l ( M ) , and f is a poljnomial function of degree
5
I--1
.
The proof relies on a lemma for which the following important notation is required.
Since X is an open subset OF V so are X , and Xo. Moreover, for the same reason
Xo equals
x n M o ' ( Mo(x \ G M ( O ) ) ) .
Lemma 1.60. Suppose that 11.141 is valid. Let f 6 C o ( X ) . If the restriction of f t o X , is almost quasihomogeneous of degree m then f itself is almost quasihomogeneous of degree m , and one of the following conditions holds:
(a) f vanishes on X o ;
45
1.e A l m o s t Q u a s l h o r n o g e n e o u s F u n c t i o n s
Cp) rn = 0 , and f i s quasihomogeneous on X o ; ( y ) Rem 10,and
Proof. By
f vanishes on X \ X +
.
P r o p o s i t i o n 1.57 w e may 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 . Let g be
t h e r e s t r i c i t i o n of f to X + . W e set N : = o r d M ( g ) , a n d s u p p o s e t h a t f $ 0 o n Xo.
By R e m a r k 1.50 f o r e v e r y k E N N t h e kth order deficiency of g e x t e n d s to a c o n tinuous function qk:X-c.
Let x € X + such t h a t M o x € X . T h e equation (1.39)
c a n be w r i t t e n in t h e f o r m N
(1.39)'
f ( M t x ) = t m ( f ( x ) + R x ( l o g t ) ) w h e r e R , ( s ) : = Z - 1q k = 1 k!
( x ) sk, S E I R . k
S i n c e every p o i n t of X \ X + is in t h e c l o s u r e of X + it f o l l o w s f r o m t h e c o n t i n u i t y
of f , q k . a n d Mt t h a t ( l . 3 ( J ) ' is valid f o r every x € X \ X + , a s w e l l . T h i s s h o w s t h a t f 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 . T h e c o n t i n u i t y of f a n d 10. +a[3 t (1.69)
H
M t (see C o r o l l a r y 1.9) i m p l i e s t h a t
f ( M o x ) = lim f ( M , \ ) . t-0
If m = O t h e n it f o l l o w s f r o m ( 1 . 3 9 ) ' a n d (1.09) t h a t R,-O,
i.e. q k - 0 o n X
0
for
every k e N N . H e n c e , w e s u p p o s e t h a t in f 0 a n d f ( x ) # 0 . T h e n (1.39)' a n d ( 1 O(J) imply t h a t R e m > 0 and f ( M o X ) = O .
H
Proof of Proposition 1 . 5 8 . Let aE9I be s u c h t h a t la1 = r . S i n c e by R e m a r k l . 4 3 . ( i )
a"
is of t h e f o r m d " = P , ( 3 )
f o r s o m e polynomial f u n c t i o n P,:V*-@
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 a M a n d of t y p e M * P r o p o s i t i o n 1.55 s h o w s
that the function
f'"'lc is
a l m o s t quasihomogeneous of degree m - a M . Since
R e a M 2 la1 X m i n
it f o l l o w s f r o m (1.08) t h a t R e ( m - a M ) < 0 . H e n c e L e m m a 1.00
implies t h a t f(,'
v a n i s h e s identically. a n d in view of T a y l o r ' s f o r m u l a a s s t a t e d
in L e m m a 1.20 f is a p o l j n o m i a l f u n c t i o n o f d e g r e e 2 r - I , i n d e e d . T h e a s s e r t i o n a b o u t m is t h e n a c o n s e q u e n c e of P r o p o s i t i o n 1 . 3 4 . ( i i i ) .
H
T h e c o n v e r s e of Lemma 1.60 is valid in t h e f o l l o w i n g f o r m .
Roposltlon1.61. Suppose that (1.14) is valid. Let q O : X + - C
b e a continuous
function which is almost quasihomogeneous o f degree m and of order 5 N . i t s deficiencies being denoted bj 4 , . . . . . 4N . If R e m > 0 then f o r ever:, k EIOJuNN
46
I.
the extension
cjk
of
qk
to
x defined
( A l m o s t ) Quasihomogeneous Functions
by c j k / X , x + : =0 is continuous.
Proof. By P r o p o s i t i o n 1.57 w e may 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 . By Propos i t i o n l . S I . ( i i ) w e observe t h a t t h e deficiencies of q o are c o n t i n u o u s f u n c t i o n s o n X + , a s well. Now w e f i x x € X \ X + , i.e. P + ( x ) = 0 , a n d c h o o s e
E
> 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 < E , I M o ( y - x ) l S E } of x is c o n t a i n e d in X . T h e n L : = { y € V ; I P + y l = f , I M o ( y - x ) l < ~ } is a c o m p a c t s u b s e t of X + . Now l e t
(
~ be a s) e q u e n ~ c e in X~+ c o n v~e r g i n g to x a s 4 + a .
x
We choose t o E N
0
s u c h t h a t x p € K for every Q ? t o , a n d for every 4 > t o w e f i x t t E l O , + a C s u c h t h a t M
x E L . S i n c e P + ( x o ) c o n v e r g e s to P + ( x ) = O a s &‘+ait f o l l o w s t h a t t p t e n d s
t@ p
to +aas O + a .
By i n d u c t i o n o n N w e a r e g o i n g t o prove t h a t
(1.70)
for every k € N N u { O ) . N o t e t h a t in view of i k ( x ) = O t h e c o n d i t i o n ( 1 . 7 0 ) for E = O i s t h e desired continuity property.
S u p p o s e f i r s t t h a t (1.70) is a l r e a d y p r o v e d for every k E N , .
By (1.30) i t f o l l o w s
that N
Iqo(Xp)l-
2
llOgtpIk lqk(Xp)l 5 l t q ” ’ q o ( M t p x t ) I
-
t-Rem
e
C ,
e?e,,
k=l
w h e r e c : = s u p { ( q o ( x ) (; X E L } . S i n c e c is f i n i t e t h e i n d u c t i o n h y p o t h e s i s i m p l i e s t h a t (1.70) is valid for k = O , a s w e l l S i n c e for every i E N N t h e p a i r ( q i , N - i ) i n s t e a d of ( q o , N ) s a t i s f i e s t h e a s s u m p t i o n s , as well, a n i n d u c t i o n a r g u m e n t c o m p l e t e s t h e p r o o f of ( 1 . 7 0 ) . m
In t h i s s e c t i o n we discuss a s i m p l e b u t i m p o r t a n t m e t h o d for c o n s t r u c t i n g p o s i t i v e q u a s i h o m o g e n e o u s C‘ f u n c t i o n s o n X o u t of C‘ h y p e r s u r f a c e s in X a n d vice versa. W e s u p p o s e t h a t X is q u a s i h o m o g e n e o u s , f e e l i n g j u s t i f i e d by Proposi-
tion 1.57. F i r s t of all w e o b s e r v e
1.f
Remark 1.62. Let F : X and
47
The Hypersurfaces Sx
if Rem f
C be quasihomogeneous o f degree m . I f f has no zeros
O then by x ( s ) := / f l s ) / ' / R e m a function x : X + l O ,
+atis defined
which is quasihomogeneous o f degree 1 . m
So in t h e Following w e deal w i t h s u c h f u n c t i o n s x .
Propoaltlon 1.63. Let r E
N
u Iw,i d ) . Then for every subset N o f X the following
conditions are equivalent: ( a ) There exists a (unique) C" function x : X -IO.+OJC
which is quasi-
homogeneous o f degree 1 such that N = S x : = x - ' ( l ) : ( b l N is a C" submanifold o f X o f codimension 1 such that everj quasi-
homogeneous ray in X intersects N in ekact1-b one point. the intersection being transversal, i.e. f o r ever)' \ E X the map i , (defined in Proposition 1.10) is transversal t o N such that l i x - ' ( N ) / = 1 .
hoof.( a ) + ( b ) :Let
x E x - ' ( I ) . T h e n x o i , = Id,,,,,,
so t h a t 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 . M o r e o v e r , d i f f e r e n t i a t i n g t h e r e l a t i o n x ( M , x ) = t w i t h r e s p e c t to t y i e l d s
(1.71)
D x ( M , x ) - i L ( t )= I ,
tElO,+.oL.
C o n s e q u e n t l y , i, is i m m e r s i v e so t h a t its i m a g e , t h e q u a s i h o m o g e n e o u s r a y R M ( x ) t h r o u g h x . is a real a n a l y t i c s u b m a n i f o l d of X , t h e t a n g e n t s p a c e T , R M ( x ) R,(x)
a t x being g e n e r a t e d by i ; ( l ) .
of
M o r e o v e r , (1.71) s h o w s t h a t D x ( x ) f O
so t h a t S x is a C r h y p e r s u r f a c e a t x . S i n c e k e r D x ( x ) is e q u a l to T,SX o n e de-
d u c e s f r o m (1.71) t h a t V is t h e direct s u m of T,S"
and T,RM(x).
Finally, if
y E X a n d t E l O , + c o C t h e n 1 = x ( M , y ) = t x ( y ) if a n d o n l y if t = l / x ( y ) .
( b ) + ( a ) : We define x:X+lO,+mC
by t h e c o n d i t i o n
{M,x; tElO,+coC} n N = { M l / x ( x ) x ) l
XE
x
T h e n f o r any s E I0,+00C w e have
so t h a t x ( M s x ) = s x ( x ) , i . e . x is q u a s i h o m o g e n e o u s of degree 1 w i t h x-' ( 1 ) = N . To
see t h a t x is C' w e f i x x E X , set y : = M l / x ( x ) x E N , a n d c h o o s e a n o p e n neighb o u r h o o d U of y in X a n d a C'
submersion f : U + R
such that f-'(O)=U
nN.
48
I.
(Almost)
Quasihomogeneous Functions
W e set W : = { ( z , t ) E X x I O , + ~ CM ; I , t ~ E U } a n d d e f i n e a C' f u n c t i o n g:W+IR
by ( z , t )
H f(Ml,,z).
T h e n f o r a n y ( z , t ) E W w e h a v e : g ( z , t ) = O U Ml,,zEN
*
t = x ( z ) . Hence t h e i m p l i c i t f u n c t i o n t h e o r e m i m p l i e s t h a t x is C' n e a r x p r o v i d e d t h a t t&.g(x,x(x))
$ 0 . N o w , f r o m t h e e q u a t i o n g ( x . t ) = f ( i , ( x ( x ) / t ) ) we d e d u c e
I ii(1)). a2g(x,x(x)) = Df(y)*(-x(x) S i n c e in view of t h e e q u a l i t y k e r Df ( y ) = TYN t h e t r a n s v e r s a l i t y a s s u m p t i o n m e a n s t h a t i i ( 1 ) 6! k e r D f ( y ) it f o l l o w s t h a t d Z g ( x , x ( x ) ) does n o t v a n i s h , a s r e q u i r e d .
Recall f r o m t h e b e g i n n i n g of t h e p r o o f of P r o p o s i t i o n 1.03 t h a t t h e c o n d i t i o n ( a ) i m p l i e s t h a t 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 f o r e v e r y U E X . H e n c e , in view of P r o p o s i t i o n l . I O . ( i i ) a n e c e s s a r y c o n d i t i o n f o r t h e e x i s t e n c e of a Funct i o n x a s in c o n d i t i o n ( a ) of P r o p o s i t i o n 1.03 is t h a t X n E M ( ~ j t 3 ) is e m p t j . As w e s h a l l see l a t e r , in g e n e r a l t h i s c o n d i t i o n is by n o m e a n s s u f f i c i e n t . N e x t w e t u r n to t h e q u e s t i o n w h e n p o s i t i v e 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 x e x i s t locally.
Lemma 1.64. For ever:, x 6 V the following conditions are equivalent: l a ) there is a quasihomogeneous open neighbourhood U of x in a continuous (resp. real analqtic) function x: U
---?)lo.
+~121
V and
which is quasihonio-
geneous o f degree I , l b ) for some neighbourhood Y o f
A
in V the set { t E l O , + w C : Y n M , Y # @ ]
is relative/:, compact in I O , + L W C :
(which is well-defined b:, Proposition 1.10. l i i ) ) is continuous at
(A,
sl
Proof. ( a ) = +( b ) . Let x be a c o n t i n u o u s f u n c t i o n having t h e p r o p e r t i e s of c o n d i t i o n
( a ) , a n d let Y be any c o m p a c t n e i g h b o u r h o o d of x in U . T h e n I : = i n f x ( Y )
0
a n d S : = s u p x ( Y ) i+ w . H e n c e f o r a r b i t r a r y t € l O , + m l a n d y E Y s a t i s f y i n g M , y E Y w e c o n c l u d e t h a t t = x ( M i , y ) / x ( y ) 5 S/I a n d 2 I / S . ( b ) + ( c ) : Let Y be a n e i g h b o u r h o o d of x having t h e p r o p e r t i e s p o s t u l a t e d in
( b ) . First of a l l w e observe t h a t by P r o p o s i t i o n l.lO.(iii) t h e m a p i , : I O , + ~ C + V , x t + M t x , is injective a n d R M ( x ) is n o n - c o m p a c t . I f x w o u l d b e l o n g to E M ( a , )
l.f
49
The Hypersurfaces S x
t h e n t h e a s s e r t i o n s ( i i ) a n d ( i v . B ) of P r o p o s i t i o n 1.10 w o u l d i m p l y t h e e x i s t e n c e s u c h t h a t ( M x)~~,,,,c o n v e r g e s to x as j + w .
of a n u n b o u n d e d s e q u e n c e
ti
S i n c e t h i s c o n t r a d i c t s ( b ) w e c o n c l u d e t h a t x d E M ( o 0 ) . N o w , to d e d u c e t h e s e c o n d p a r t of ( c ) w e let ( x j ) j c N be a s e q u e n c e in X \ E M ( o , )
a n d ( t i ) j E Na s e q u e n c e
in I O , + a 3 C s u c h t h a t lim ( x i , M t j x i )= ( x , x ) . T h i s m e a n s , in p a r t i c u l a r , t h a t f o r j+m
s o m e N E N w e h a v e ( x j , M,.xj) E Y X Y f o r j 2 N . H e n c e , by c o n d i t i o n ( b ) ( t j ) s t a y s J
in a c o m p a c t s u b s e t of 10,+00[. H e n c e , t h e r e is a s u b s e q u e n c e of ( t j ) c o n v e r g i n g to s o m e t € I O , + w C . By t h e c o n t i n u i t y o f t h e m a p ( y , s ) H M , y t h i s i m p l i e s t h a t
x = M , x . S i n c e t h e m a p i, is injective w e c o n c l u d e t h a t t = l . By t h e s a m e a r g u m e n t w e see t h a t every s u b s e q u e n c e of ( t j ) 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 I . T h i s m e a n s t h a t t h e s e q u e n c e ( t i ) i t s e l f c o n v e r g e s to I . a s is to be s h o w n . (c)+
( a ) . Let W be real s u b s p a c e of V o f c o d i m e n s i o n I n o t c o n t a i n i n g M i ,
a n d let Y be a n o p e n n e i g h b o u r h o o d o f x in V \ E M ( o , ) (1.73)
such that
M y d W f o r every y c Y
M o r e o v e r , w e c l a i m t h a t by m a k i n g Y still s m a l l e r if n e c e s s a r y w e c a n a c h i e v e t h a t f o r every y E N : = Y n ( W + x ) t h e q u a s i h o m o g e n e o u s ray R M ( y ) i n t e r s e c t s N in p r e c i s e l y o n e p o i n t . I n d e e d , if n o t t h e n w e c o u l d find a s e q u e n c e ( y j ) j E N in W + i c o n v e r g i n g to x a n d a s e q u e n c e ( t i ) j E N in I O , + ~ l \ { l s) u c h t h a t x j : = M t i y j b e l o n g s to W + x f o r every j € N a n d s u c h t h a t ( x j ) i c N c o n v e r g e s to x , a s w e l l , as j + a . I t f o l l o w s f r o m ( c ) t h a t lim ti = 1 . T h e main t h e o r e m of c a l c u l u s a n d (1.18) s h o w j+m
t h a t f o r every j € N w e have I
w.j : = t i1- l ( M t j Y j - Y j ) = . I ' M T j ( s ) M Y j d s
ri(s)
0
w h e r e r j ( s ) : = s t i + ( l - s ) = t + s ( t j -Cl o) n. s e q u e n t l y ,
j+.m lim
w j = M x . Since w j € W f o r
every j E N a n d s i n c e W is c l o s e d t h i s c o n t r a d i c t s ( 1 . 7 3 ) . a n d t h e c l a i m is p r o v e d . N e x t w e o b s e r v e t h a t in view o f (1.18) a n d (1.73) t h e i n v e r s e f u n c t i o n t h e o r e m s h o w s t h a t by m a k i n g Y a n d h e n c e N s m a l l e r w e achieve t h a t t h e r e e x i s t a n o p e n n e i g h b o u r h o o d Z of x a n d s o m e E > 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 << * .
the generalized eigenspaces G , ( h ) .
A
60.
* *
>>
:
are pairwise orthogonal:
(ii) f o r every A E o we have
11.731
'
(ReA - E ) //s// 5 << s . M s >> 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 <<
-;a
>> 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 ) ~ <~ <- y' ( s ) . M y ( s ) > > so that by (1.74) one concludes that d ReX-E 5 -log ds
lly(s)ll C R e h + E .
Integrating this one obtains
Applying the same reasoning to s H y ( - s ) instead of y one deduces
Inserting s = log t one arrives at ( i i i ) .
Ptopoiltlon 1.70. Let c , d C l O , +wC be such that I / c < Rel < l / d for every A E U ,
.
Then there is a norm 11.11 on G , ( u + ) induced by a scalar product such that the
1.f
53
The Hypersurfaces S x
condition ( b ) OF Proposition 1.63 i s satisFied For ( X .N) = ( X + ,S , I where
and such that t h e corresponding real analytic function x+ deFined by condition ( a ) o f Proposition 1.63 satisFies the Following estimates:
Moreover, For ever)’ .uEG,(o+I\IOI
the map l O . + ~ C - - - - - ; ) l O , + ~ tCH. I / M , s / / . i s
strictlj. increasing.
Proof. S e t t i n g product <<
ci : =
l / c and
B : = l / d and applying Lemma
>> o n G,(a+)
a .;
1.69 we choose a scalar
such t h a t (1.74) is valid with (ReX-E,ReE+E) re-
placed by (ci,p) and such t h a t
Applying this to Ml,,x (1.78)‘
t’
I1x II
5
instead of x we deduce
I1 M, x 11
2 tn /I x I1
x€GM(cI,), t€l0,11
We fix S E X + . Since by (1.18) and ( 1 . 7 4 )
t h e function 1 0 , +a[3 t H IIP, M,xlI is strictly increasing so t h a t i [ ’ ( S + )
consists
o f precisely one e l e m e n t . Moreover, i, is transversal t o S , a t t h e point of intersection. In o r d e r t o prove t h e e s t i m a t e s (1.77) we observe t h a t by t h e definition of
x : = x+ we have l = l I M l , x ( x ~ P + x l / so t h a t in c a s e x ( x ) < l ( r e s p . > l ) w e deduce from (1.78) ( r e s p . (1.78)’) t h a t
x ( x ) - O ! ~ ~ p + x l1l <5 x(x)-’ I I P + ~, I I i.e.
I I P + ~5I xI( ~x )
<_
d
11~+xll
Since x ( x ) < 1 ( r e s p . > 1 ) if and o n l y i f IIP+xll< 1 ( r e s p . > 1) t h e proof is c o m p l e t e .
Note t h a t f o r X - o n e similarly obtains a real analytic quasihomogeneous function x _ : X - d I O , + c o C by applying Proposition 1.70 to - M instead of M and s e t t i n g x - := l / x + .
54
I.
( A l m o s t ) Quasihomogeneous Functions
For the sake of easy reference we formulate a special form of the situation of Proposition 1.70 as a separate condition: (1.79)
the scalar product
*
o n V is such that the generalized eigen-
X E a , are orthogonal to each other and such that the
spaces C , ( h ) ,
< *,
restriction of
<*, > >
to G,(a+
)
has the properties of Proposition 1.70
where c and d are given numbers satisfying the requirements of Proposition 1.70
;
and let x, be defined as in Proposition 1.70.
In section 3 . ( c ) below, in the context of quasihomogeneous averages, one finds necessary and sufficient conditions on X for the existence of functions x satisfying the condition ( a ) o f Proposition 1.03 ( s e e Theorem 3.39 and Proposition 3.34).
A t present we suppose that such a function x : X + l O , + c ~ [ is given and introduce a map which implicitly appeared i n the proof of Proposition 1.63, already.
Notatlon1.71. We define ( i ) p , : X + S x ( ii )
Ox :X
+S" x 10, + aC
by x
by x H M l , x ( x ) x and
H ( px ( x ) , x ( x )
Ropodtlon 1.72. ( i ) IF x i s continuous then
).
px is a continuous retraction OF X
o n t o S" s a t i s b i n g
and Ox is a homeomorphism, i t s inverse being given b y (9. t ) H M , 9 . ( i i ) Let r e N u (a, id); iF x is a C' function then p x is a C'submersion. and Ox is a C' diffeomorphism.
p r o O F . Since the map ( t , x ) t+ Ml,,x
is real analytic the regularity properties of
px follow by the chain rule. The condition (1.80) and the retraction property are
immediate. That px is a submersion follows from the facts that p x l s x is equal to the identity map of S x and that by (1.80) one has D p x ( M t 3 ) = Dpx(9) OM^,^ f o r arbitrary 9 E S X and t E IO,+mC. Finally, the assertions about
ax
are consequen-
ces of the fact that t h e restriction t o S x x l O , + m C of the real analytic map
XxlO,+aC+X,
( ( , t )H M , [ ,
is inverse to O x .
In view of Proposition 1.72 it is easy to describe all quasihomogeneous functions
55
1.f The Hypersurfaces S"
on
X
w i t h t h e h e l p of x :
Remark 1.73. ( i ) I f g: Sy-
C is any function then q := x m g o p x is quasihomo-
geneous of degree rn, and q1
S X
(ii) A function f : X - + C
=g ; i f
x and
g are C'
so is q .
is quasihomogeneous o f degree m i f and only i f
I t is n o t d i f f i c u l t to o b t a i n a s i m i l a r d e s c r i p 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 o u s f u n c t i o n s o n X . To t h i s e n d w e i n t r o d u c e c e r t a i n basic a l m o s t q u a s i h o m o g e n e o u s functions associated with x :
Remark 1.74. For every k €No the function q k := i d k
ox
is almost quasihomogeneous
o f degree 0 and of order i k such that k
(1.811
qk O M, = Z o .J ( t )
.
j=O
bf. By t h e
binomial f o r m u l a w e have
Remark 1.73'. ( i ) If g o , . . . , g N :S x 4 C are arbitrary Functions then bj
a sequence of functions is defined satisfr.ing qk lsx= gk such that qO is almost quasihomogeneous o f degree 5 m and o f order i N with k
''
order deficiency q k ;
moreover, i f x and the gi are C' so are the qk ( i i ) Suppose that qO is almost quasihomogeneous o f degree m and o f order 5 N ; then N
where ( q k ) is the sequence of deficiencies o f qo defined by (1.39) proOf. (il:T a k i n g (1.81) i n t o a c c o u n 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 w e deduce f o r every t E 1 0 , + ~ 0t [h a t
56
I.
N
( A l m o s t ) Quasihomogeneous Functions
N
j
2 o i (t ) q j - i
qo 0 M t = j=o i = O
tm xm gj o p x = t m
N
c c i=o wi( t )
q j - i Xm1 gj
.ex .
j=i
Since by definition the very last s u m is equal to qi the assertion follows.
(ii):
Let X C X and choose t € l O , + a ~ C s u c h that S : = M , , , x E S X . Then t = x ( x ) ,
w k ( t ) = q k ( x ) ,and 8 = p x ( x ) so that ( 1.83) follows from ( 1 . 3 9 ) . rn
In this section w e give a fairly explicit description of the set % ( M ) defined in Notation 1.65. This will be made use of i n the proof of Remark 2 . 3 0 and in $ 3 . ( c ) . only. The following lemma contains the basic part of the proof.
Lemma 1.75. Let dEW and C 6 @ " d . and let P be a pol-lnoniial o f degree ( i ) I f there are a sequence of p o l j nomials P,, and an unbounded sequence ( s , , ) , , ~ "
. n E N , o f degree
in C such that
5
5
d.
d converging t o P
lim ( P ! 1 ) ( s , , ) / j ! ) 0 2 1 L d=
n+ , ,<
<
then (a) d e g P > D . = d / Z ; (1.84)
(p) c1=Oforj>D,
( y ) i f d i s e v e n then
rD= ( - l ) D P ( D ) ( 0 ) / D ! .
(ii) Converse/), if the conditions (a).( p ) , and ( y ) are valid then there is an analjtic familq of poljnomials P, , s€@, o f degree 5 d such that
and such that the coefficients of P, are real for ever)' sElR provided that the coefficients o f P and [ are real.
For the proof of this and other assertions a simple formula is required:
Lemma1.76. Let j , k , P € N , such that k 5 j . P .
Then
57
1.g D e t e r m i n i n g the Set S ( M )
a f . By t h e b i n o m i a l f o r m u l a we o b t a i n t€R. D i f f e r e n t i a t i n g b o t h sides k - t i m e s w i t h r e s p e c t to t ( w i t h t h e r i g h t - h a n d
side
m a k i n g u s e o f Leibniz' f o r m u l a ) a n d e v a l u a t i n g t h e r e s u l t a t t = l w e d e d u c e
Dividing by k ! w e arrive a t t h e desired r e s u l t .
As a c o n s e q u e n c e w e are g o i n g to derive
Lemma1.77. Let L , n E N and r € N o be such that r + n 5 L . Then the matrix
is invertible. the first r o w o f its inverse being given bj rSiSr+n.
P r o o f . W e first deal w i t h the case " r = O " . T h a t B is invertible is readily s e e n by c o m p u t i n g its d e t e r m i n a n t : for every L+1 < j < _ L + ns u b t r a c t i n g t h e ( j - l ) t h c o l u m n f r o m t h e jth
o n e , m a k i n g u s e of t h e r e l a t i o n
(i)
-(I;')
=(/I;)
and expanding
w i t h r e s p e c t to t h e f i r s t r o w w e see t h a t d e t B is e q u a l to t h e d e t e r m i n a n t of t h e n x n - m a t r i x w i t h e n t r i e s b i j , O < i < n - 1 , L < j < L + n - 1 . H e n c e w e d e d u c e by i n d u c t i o n o n n t h a t det B = 1 . Applying (1.86) to ( j , L + n - j ) i n s t e a d of ( 4 , j ) a n d n o t i n g t h a t t h e i n e q u a l i t i e s j > L a n d k ? L i m p l y ( L + n - j ) + j - k = L + n - k < n ' L < j a n d L + n - k < n w e s e ethat t h e matrix
C with entries
Cki
:=
(-l)i(L+L-i)if l o
O
if L + n - k < i < n
h a s t h e p r o p e r t y t h a t A : = C B is a l o w e r t r i a n g u l a r m a t r i x , its d i a g o n a l e n t r i e s
Akk b e i n g e q u a l to 1 . T h i s s h o w s t h a t A ( a n d - b y t h e w a y - h e n c e B ) is i n v e r t i b l e
58
I.
a n d t h a t A-'
(Almost) Quasihomogeneous Functions
is a l o w e r t r i a n g u l a r m a t r i x s a t i s f y i n g
= I for L < k < L + n .
S i n c e B - l = A - l C t h i s i m p l i e s t h a t t h e f i r s t r o w of B-' c o i n c i d e s w i t h t h e f i r s t r o w o f C . T h i s p r o v e s (1.88) in case r = O . F o r t h e p r o o f of the case "r.0" o n e o b s e r v e s t h a t
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 case " r = 0" by t h e f o l l o w i n g s t a n d a r d f a c t .
rn
Remark 1.78. Let ( a i j ) be an invertible nxn-matriu with comples entries, and let b , c 6 ( C \ 101)". Then the nxn-matrix ( b i a i j c j ) is invertible. i t s inverse being given by
I (q dii
4)
where (diil denotes the inverse OF ( a i i ) .
ProoF OF Lemma 1.75.
m. W e m a y a s s u m e t h a t
I
l i m n + m l s , , l = + a .S e t t i n g a , , , j : =
PAi)(0)/j!a n d a j : = P t i ' ( 0 ) / j ! w e d e d u c e b y i n d u c t i o n o n k t h a t d
(1.89)
C a n , i sjn- k + l = 6k1 To-ak-1 j=k
~ i m I1+W
k€Nd.
1
'
I n d e e d , f o r k = l t h i s is d u e to t h e f a c t t h a t by t h e a s s u m p t i o n s Pn(sn) c o n v e r g e s to
To a n d a n , o
to a.
a s n + a ; a n d if t h e e q u a t i o n in (1.89) is valid f o r s o m e
k e N d - , t h e n w r i t i n g t h e s u m in t h e f o r m s n c n , k w h e r e d
a n d u s i n g t h e f a c t t h a t Isn( t e n d s to + a a s n + a w e c o n c l u d e t h a t lim
cn,k=
n+m
0
which i m p l i e s t h e e q u a l i t y in (1.89) f o r k + l i n s t e a d of k b e c a u s e ( a n , k ) n c N t e n d s to ak as n + m , Applying t h e s a m e r e a s o n i n g to t h e p o l y n o m i a l s
PAi', 0 < i < d ,
w e a r r i v e a t t h e s y s t e m of e q u a t i o n s
w h e r e bii is d e f i n e d in ( 1 . 8 7 . b ) . S u b s t i t u t i n g L = i + k a n d s e t t i n g a , , i , L : = a , , , i
j-L+l
s,
i+l< L < d . F r o m t h i s w e are g o i n g t o d e d u c e t h a t t h e s e q u e n c e s ( a n , j , L ) n e N c o n v e r g e f o r c e r t a i n indices j a n d L . To t h i s e n d w e have to invert c e r t a i n m a t r i c e s w i t h e n t r i e s b i i . W e f i x LENd a n d observe t h a t w i t h ( l . 8 9 ) i , O S i < L - 1 , w e have L
1.g
59
Determining t h e S e t Q ( M )
e q u a t i o n s for t h e d - ( L - l ) q u a n t i t i e s a,,j,L. So in order to be a b l e to f o r m s q u a r e m a t r i c e s o f s i z e d - L + l w e have to p o s t u l a t e L > d - L + l , i.e. 2 L > d + l or L > D . A s s u m i n g t h i s w e set n : = d - L a n d fix a n u m b e r r € N O s u c h t h a t r < L - n - 1 = 2 L - d - l t h e m a t r i x d e f i n e d i n L e m m a 1.77. S i n c e by t h i s l e m m a t h e
a n d d e n o t e b y B,
m a t r i x B, is i n v e r t i b l e w e d e d u c e f r o m (1.89), t h a t (1.90)
with q,:=
S i n c e Is,l
( s i , L - i ) , . s i 5 n + , .a n d
t e n d s to
+m
P,:= ( b i , L - i ) , . 5 i 5 n + r
as n + m t h i s i m p l i e s t h a t
ai = lim a n , , = 0 ,
L<_j
n+m
S i n c e t h i s is so, in p a r t i c u l a r , if L is e q u a l to t h e s m a l l e s t n a t u r a l n u m b e r Lo s t r i c t l y l a r g e r t h a n D w e have p r o v e d &).
M o r e o v e r , s i n c e t h e c o n d i t i o n (1.90)
f o r L = L o s h o w s t h a t f o r every L o < j c d t h e s e q u e n c e of n u m b e r s a,,,i,L,,
nEN,
is b o u n d e d w e d e d u c e t h a t in c a s e L 2 Lo+l w e even have lim a , , , j , L =0 f o r L < _j 5 d n+m
so t h a t ( l . 8 Y ) L - l in c o m b i n a t i o n w i t h ( a )s h o w s t h a t Cp) is valid.
In o r d e r to p r o v e (y) w e a s s u m e t h a t D E N a n d set L = D + 1 . T h e n o b s e r v i n g t h a t
r may be c h o s e n b e t w e e n 0 a n d 1 w e c o n c l u d e f r o m t h e f a c t t h a t t h e l e f t - h a n d side o f (1.00) does n o t d e p e n d o n r t h a t
S i n c e by o u r p r e s e n t c h o i c e of L w e have n = d - L = L - Z w e see t h a t q o = O a n d q l # O so t h a t
(1.91)
aD ( B1-'P1 - B,'
This already s h o w s t h a t
LD
PO) = 5~ B;'
ql .
a n d a D a r e p r o p o r t i o n a l by a c o n s t a n t o n l y d e p e n d i n g
o n D . To c o m p u t e t h e c o n s t a n t w e f i r s t o b s e r v e t h a t by L e m m a 1.77 w e have
Since
w e c a n a p p l y L e m m a 1.76 to ( n + l , L - r , D - r )
instead of ( j , k , 9 ) -observing t h a t
n + l + D - r > 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
Then by Lemma 1.77 and Remark 1.78 B ( s ) is invertible, the entries o f its inverse being given by sI-' d,I ,
d-nSj5d. O < I < n ,
where by d j r we denote the entries of B ( l ) - ' . Now we define a d - " ( s ) , . . . , a d ( s ) 6 CC by ( a j ( s ) ) d - n c j r d: = B ( s ) - ' . ( C 1 - a I -
bIJ a J S J - I ) O S I r n ,
s€C.
I+l<JCD
In view of what was observed about B ( s ) - ' t h i s means that n
Since n < D < d - n so that I - j < J - j < - l a j ( s ) = O ( l / l s l ) as I s I + a , i.e. lim Ps = P Isl+m
Moreover, for every 0 5 i 5 n we have
for I < J < D and j z d - n we observe that
1.g
01
D e t e r m i n i n g the Set Q ( M )
where d
(1.93),
Ri(s):=
n
c
bijsJ-i
n
y A
sl-idjI([I-aI-
sI-i ( c I - a l ) [
I=O
d
2
bIJa,sJ-'))
=
I+lcJsD
1=0
j=d-n
-
(c
bi,d,I]
j=d-n
-
J-1
2
aJ s J - i
x
bIJ
I=O
ISJSD
[
d
bi,d,r]
j=d-n
S i n c e by t h e d e f i n i t i o n o f ( d j l ) t h e s u m s in s q u a r e b r a c k e t s a r e e q u a l to h i , t h e l e f t - h a n d side c o i n c i d e s w i t h ci-ai
bij a j
-
SJ-i.
i+IzJlD
P:''(s)
Inserting this into t h e formula above w e obtain
=
ci,
0 < _5i n .
1 N o w , in c a s e d is e v e n w e o b t a i n t h a t jjj P A D ' ( s ) = a D + R D ( s ) w h e r e RD is d e f i n e d
by ( 1 . 9 3 ) D . S i n c e in t h i s c a s e w e have n = D - l w e observe t h a t
where d
n
I t follows that
Finally, f o r d - n C i < _ dw e have n-i < D-i < 0 so t h a t P:"(S)
ri
n
j= i
I=O
=Ib i j s J - i ( x s l - i d j l ( c l - a l -
blJaJSJ-I))
=o(l/lsl)
I+liJ'rD
a s I s l + m . H e n c e in view of t h e c o n d i t i o n ( B ) w e have p r o v e d t h a t f o r every j E N o s a t i s f y i n g 0 5 j 5d w e have
So in c a s e d is odd t h e p r o o f i s c o m p l e t e . H e n c e w e s u p p o s e t h a t d is e v e n .
To c o m p l e t e t h e p r o o f in t h i s case it s u f f i c e s , o f c o u r s e , to s h o w t h a t < = < . In view of (1.84) t h i s is d o n e o n c e w e have s h o w n t h a t (1.94)
C ( D ) =( - l ) D .
N o t e t h a t it does n o t s e e m to be e a s y to verify ( 1 . 9 4 ) by direct c o m p u t a t i o n . w
H o w e v e r , a p p l y i n g t h e a s s e r t i o n ( i ) to < E c C ' + d t h e s e q u e n c e o f p o l y n o m i a l s Pn : = P,,,,
i n s t e a d of
c,
thereby noting t h a t
n € ! N , s a t i s f i e s t h e r e q u i r e m e n t s o f ( i ) if
62
I.
(Almost) 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
t e n d s to + a as n + a , w e o b t a i n
( s , ) , ~is~any s e q u e n c e in C s u c h t h a t Is,i
C(D)aD = (-1)
D
a D . S i n c e for a m o m e n t w e may a s s u m e t h a t a D f O t h i s i m p l i e s
(1.94), i n d e e d . H e n c e t h e p r o o f of L e m m a 1.75 is c o m p l e t e .
N o w w e c o m e to t h e d e s c r i p t i o n of t h e set V \ Z ( M ) . For s i m p l i c i t y w e give it w i t h r e s p e c t to a f i x e d M-cyclic d e c o m p o s i t i o n 3 of GM(b,)
although, certainly,
V \ Z ( M ) does n o t d e p e n d o n t h e c h o i c e of s u c h a d e c o m p o s i t i o n . W e a s s u m e t h a t t h e d e c o m p o s i t i o n 3 c o n s i s t s of M-invariant s u b s p a c e s w h i c h a r e i n d e c o m p o s a b l e w i t h r e s p e c t to M ; in p a r t i c u l a r , f o r every Z E 3 t h e r e is a ( u n i q u e ) X E o , - d e n o t e d by A,
- such that Z C G M ( X ) .
Definition 1.79. Let Z E 3 a n d x E G M , ( o , ) . T h e n w e d e n o t e (i)
by dz t h e K x z - d i m e n s i o n o f Z ;
(ii)
by i z l s l t h e s m a l l e s t n o n - n e g a t i v e i n t e g e r m s u c h t h a t t h e p r o j e c t i o n sz
of x o n t o Z b e l o n g s to k e r ( M x z ) m ; (iii)
,S,(sJ
:= ( Z E . 3 ; iz(s)= ( d z + 1 ) / 2 ) , and
U(A)
:=
( i v ) by T ( p l t h e c l o s u r e of ( ( e x p ( X s ) ) x 6 p ; s E I R ]
subset
Q
in C'
f o r any n o n - e m p t y
of i IR .
Roposltion 1.80. The set V \ 2 ( M I consists o f a l l
such that i n case .;',(s)#@
11.96)
( X z ;Z E , ~ $ , ( X ) ] ;
WhZ=
(-1)
there exists w E
-.
6 G M ( o O ) satisfj,ing
T ( o ( A ) ) satisfving
i z (X )-f
z €3,( u l .
F o r l a t e r p u r p o s e s w e a r e g o i n g to p r o v e a s l i g h t l y m o r e g e n e r a l r e s u l t c o n c e r ning a n o t i o n to be i n t r o d u c e d in
Definition 1.81. Let x , y E G ~ ( b o ) . ( i ) T h e p o i n t x is s a i d to be M-connected t o y if a n d o n l y if t h e r e a r e s e q u e n c e s (t,),EN
(1.97)
in I O , + c o C a n d ( y m ) in G M ( b g ) s u c h t h a t ( a ) lim t, m+m
=+a;
( b ) y = l i m y,; m+m
and
( c ) x = lim M t m+m
m
ym.
1.g
63
Determining t h e Set B ( M )
( i i ) , , ,@ ,
d e n o t e s t h e set of p o i n t s X E G M ( O O ) which are M - c o n n e c t e d
to s o m e
y E G M 00).
Propodtion 1.82. ( i )
@M
c o n s i s t s o f all points s E G M M ( o ~ satisfving ) (1.95).
( i i ) Let X € @ M and y
proOf. W e f i x Z
Z E-3, (s).
for j = i z ( s ) ,
E a ~n d c h o o s e b z E Z s u c h t h e vectors bi : = ( M x z ) d z - i ( b z ) ,
BZ o f Z . F o r every z € Z w e d e n o t e t h e c o o r d i n a t e s
j€Ndz, f o r m a K,,-basis
of z w i t h r e s p e c t to (1.99)
if and only if j . € C M ,
&(x)#g there e x i s t s w E T ( o ( s ) ) such that
(yz)i = (-I)'-' w A z ( s Z l j
(1.98)
J
Bz
by ( z l , . . . , z d z ) . T h e d e f i n i t i o n o f i z ( x ) m e a n s t h a t
( a ) ( X = ) ~ = O j, > i z ( x ) :
and
( b ) ( x Z ) ~ # O if j = i z ( x ) a n d x # O .
*
M o r e o v e r , t h e m a t r i x o f N : = M x Z w i t h r e s p e c t to t h e b a s i s BZ is o f t h e f o r m (1.11)' so t h a t
w h e r e t h e m a t r i x o f P z ( s ) w i t h r e s p e c t to bZ is given b y (1.1O.b)'. T h i s m e a n s t h a t f o r every Z E Z t h e c o o r d i n a t e s o f e x p ( - s X Z ) M t z w i t h r e s p e c t to
Bz are
given by
cIZ-i
where
Now w e s u p p o s e t h a t x is M - c o n n e c t e d to y a n d c h o o s e a s e q u e n c e ( t m ) n , E N in IO,+coC a n d a s e q u e n c e ( y m ) m e N in G M ( a o ) s a t i s f y i n g ( 1 . 0 7 ) . In p a r t i c u l a r , t h e c o n d i t i o n (1.97.a) m e a n s t h a t lim s, =
+a
,+or>
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 +<1>
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
( x z ) ~,
1s1+-
By o u r c h o i c e o f ( s , , , ) , , , ~a ,n~d in view o f t h e d i s c u s s i o n a t t h e b e g i n n i n g o f t h e p r o o f 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 (1.07) is s a t i s f i e d f o r y m : = x yz(s,) Z €3
and
t m : =logs,.
rn
Proof of Proposition 1.80. F i r s t of a l l w e a r e g o i n g to see t h a t ( 1.100)
x c ? Q ( M ) W x is M - c o n n e c t e d to i t s e l f .
In f a c t , if (1.97) is valid f o r y = s t h e n t h e c o n d i t i o n ( c ) in L e m m a 1.64 is v i o l a t e d . C o n v e r s e l y , if x @ ! 2 ( M ) t h e n by t h e s a m e l e m m a w e c h o o s e s e q u e n c e s (t,,,),,EN in IO,+mC a n d (y,,,),€[~
in G M ( a o ) s u 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 Mt m y m
t e n d s to x , a s w e l l , and s u c h t h a t no s u b s e q u e n c e of (t,)
s t a y s in a c o m p a c t
s u b i n t e r v a l of I O , + c o C . By c h o o s i n g s u b s e q u e n c e s w e may assume t h a t (t,)
con-
1.a
65
D e t e r m i n i n g t h e Set B ( M )
v e r g e s to +a or to 0 . S i n c e in t h e latter case t h e c o n d i t i o n ( 1 . 9 7 . ~ )r e m a i n s valid w i t h t,
r e p l a c e d by l/t,
t h i s m e a n s t h a t x is M - c o n n e c t e d to i t s e l f , i n d e e d .
N o w , in view of (1.100) o n e o b t a i n s t h e a s s e r t i o n a s a n i m m e d i a t e c o n s e q u e n c e
of P r o p o s i t i o n 1.82, in t h e p r o o f of
'x', when deducing
(1.96) f r o m (1.98). t a k i n g
(1.99.b) i n t o a c c o u n t .
N o t e t h a t t h e c o n d i t i o n (1.96) is n o t a l w a y s s a t i s f i e d : f o r e x a m p l e , if t h e r e e x i s t s
Z C s s u c h t h a t h Z = O a n d s u c h t h a t d,
and ( d z - l ) / 2
a r e odd t h e n (1.96) does
n o t h o l d f o r every ~ € 6 ,s u c h t h a t i z ( x ) = ( d z + 1 ) / 2 . T h u s , in p a r t i c u l a r , in view of (1.100) P r o p o s i t i o n 1.80 s h o w s t h a t x is n o t a l w a y s M - c o n n e c t e d
to i t s e l f .
T h i s m e a n s t h a t , in g e n e r a l , M - c o n n e c t e d n e s s d o e s n o t d e f i n e a n e q u i v a l e n c e r e l a t i o n a s it is n o t n e c e s s a r i l y reflexive. T h a t t h i s , in f a c t , is t h e o n l y o b s t r u c t i o n i s s h o w n by
Corollary 1.83. ( i l Let y is M-connected t o
s , j ~ E G ~ f o O JThen . A is M-connected t o y i f and on/), i f
A .
fiil The following conditions are equivalent: l a ) M-connectedness defines an equivalence relation on
EM
;
( b l every x c E M is M-connected t o i t s e l f ; (c) i f the set , j ) , : = { Z E S : d z is o d d } is non-ernptj. there exists w E Tfo,,,)
- where (1.461,
aodd := { A z : Z E S , } - such that wxz = f - 1 )
IdZ - 1 ) / 3
zc3,.
(iiil The condition f c l in fii) is satisfied i f f o r ever-) Z E 3 either the number dz is even or dz - 1 is divisible bj 4 .
w. (i): 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 t h e f a c t t h a t T ( a ( x ) ) c a r r i e s
t h e s t r u c t u r e of a g r o u p w h e n e q u i p p e d w i t h c o m p o n e n t w i s e m u l t i p l i c a t i o n .
(ii).
( a ) * f b l : t h i s is trivial.
( b ) * f c ) : obviously, o n e can find an e l e m e n t ~ € 6 ,s u c h t h a t
s 0 ( x ) =so so
t h a t t h e c o n d i t i o n (1.96) b e c o m e s ( l . O h ) , .
( c ) + ( a l : in view of ( i ) it s u f f i c e s to s h o w t h a t M - c o n n e c t e d n e s s is t r a n s i t i v e . In view of (1.96), t h i s f o l l o w s w i t h t h e h e l p of t h e g r o u p s t r u c t u r e of T(a,,,)
66
I.
( A l m o s t ) Quasihomogeneous F u n c t i o n s
by an application of Proposition 1.82.
(iii): This
Q defined ~ by ~ w x : =~1 belongs ~ to~
is obvious since the element w E
T(~,,dd).
For the sake of easy reference w e reformulate a part of the proof of Proposition 1.82 as
Lemma1.84. Let s , y 6 F M such that W E
\
is M-connected to
-\'.
Then there e i i s t
T(oo) and a real analqtic function S : 1 O , + w C \ 1 1 1 ---+GM(oo)
such that f o r
every s E l O , + a ) (1.101)
where H i : (1.102)
( a ) lim S l t ) = - v , t +s
GMM(0g)--+G,~Mlog)
and
( b ) lim N , J f t ) = H , ( s ) t +s
is defined by
N , : = H,T(,, O M , , i . e . :
\
H
llwA
and where
M , =H,f(,, O N , .
with y : 10, +at--+Tloo) being defined b-t t H ( t A ) A
Edo.
proOf. By the equations following (1.09) i n the present notation we have
and the second equation i n (1.102) holds. Hence, setting S(t) :=
2 yz(log t ) ,
t E 10, +a[\ I ) ,
Z G 3
we see that the last part of the proof of Proposition 1.82 implies the assertion of the lemma. w
(h) Quaslhomogeneous IBol:lr Caordlnates
In this section we continue to suppose that X is quasihomogeneous (Definition 1.56). Let x : X+
l O , + a C be a C' function which is quasihomogeneous of degree I . Then
by Proposition 1.63 SX:=x-'(I) is a C' submanifold of X of codirnension 1 .
1. h
67
Q u a s i h o m o g e n e o u s Polar C o o r d i n a t e s
Notation 1.8s. By x" we denote the Radon measure on Sx induced by the differential form
UM
on V which is defined as follows: w M ( s )is
for every xEV volume form V o l
0
t h e contraction Vol'-~(Mx) of the canonical oriented
(associated t o the given scalar product on V ) by the tangent
vector Mx of t h e quasihomogeneous ray through x at the point x . More explicitly, i n coordinates t h i s means
Below the Lebesgue integral will be formed with respect to orthonormal coordinates (with repect to the given scalar product o n V ) ; in other words, integration will be carried out w i t h respect to the density on X defined by
Proposition 1.86. Let f : X--+ li)
Vole.
C be a function.
f is ( L e b e s g u e - ) i n t e g r a b l e i f and onlj i f the function defined b) It.$)
H
f f M , 8 )t p - l
belongs to 3 'f 1 0 , + mL'xSx: d t ex") where ,u := t r M - s e e 11.61: l i i ) i f this i s t h e case then
/'
/'flx)dx = X
10. + c17r Y
f ( M , 3 ) tP'-'dt@dx"($).
sx
( i i i ) f is Lebesgue-measurable i f and o n l j i f the function (t.31 H f ( M , 3 ) is measurable with respect t o d t e x " . Proof. For arbitrary x E X and
( & dx
A
(a,
'
Ix)
51,.
. . , < , - I E V w e have
) ( x ) - (Mx , < I
,. . . ,< , - I )
-- X ( X ) Dx(x) * M x Volo(Mx,fl , . . . ,<,,-I)
=
.
Applying Euler's equation for x ( s e e Proposition 1.22) an- denoting by fl the n-form
o n V defined by n : g V o l o one can write t h i s as
We define a Cm map
'p :
10,+ aE xX +X by ( t , x ) H M t ~ Since .
68
where
(Almost) Q u a s l h o r n o g e n e o u s F u n c t i o n s
I.
5c2 :
+X d e n o t e s t h e c a n o n i c a l p r o j e c t i o n o n t o t h e s e c o n d f a c t o r
IO,+~CxX
we deduce
= -tI d t h a +
= -xdo1( 9x o q ) A a
q*(;dxh(aMIX))
(1.104)
~ 1 0 rr:(dx) ~ 2ACL
where a : ='P'(WM Ix' .
N o w , by t h e d e f i n i t i o n of q * w e have f o r a r b i t r a r y z = ( t , x ) E I O , + a C x X a n d
rl,
E R X V that
(c1 , . . . , Cn-')
a ( z )*
=
M,x)
(
Dq(z 1
* r l , . . ,D q ( z ) .Cn-')
.
Writing C i = ( r i , c j ) where r i E R a n d c i E V , inserting t h e equality D q ( ~ ) - < ~ = r ~ d l +q D ( zq)( t , * ) ( x ) . < j ,
a n d m a k i n g u s e of t h e m u l t i l i n e a r i t y of w M ( M ~ x )w e f i n d a n ( n - Z ) - F o r m B o n IO.+aCxX such that U(Z)
=
[K$q(t,
) * ( U M ~+ ~d )t
*
A p ] ( Z )
.
From t h e d e f i n i t i o n o f U M a n d f r o m (1.4) a n d ( 1 . 0 ) w e derive f o r a r b i t r a r y t E IO,+coC, x E X , and
t i , .. . , <,,
-1 E
V that
I t f o l l o w s t h a t t h e r i g h t - h a n d side of (1.104) is e q u a l to
t" - d t
A
K; ( W M
1
) +
1
K
2" ( d x )
A
a.
C o m b i n i n g t h i s w i t h (1.103) a n d ( I . l 0 4 ) , i n t r o d u c i n g t h e C1 m a p Y : = q
I 3 0 . + d x S X ~
a n d t a k i n g t h e e q u a t i o n dxIsx
y * ( n l x )= t'-'dt
=0 A
into account w e obtain
(WMlsx).
S i n c e by P r o p o s i t i o n 1.72 Y is a C' d i f f e o m o r p h i s m t h e a s s e r t i o n s
a
and
lii)
f o l l o w by t h e t h e o r y of Radon m e a s u r e s i n d u c e d by d i f f e r e n t i a l f o r m s . Finally, t h e assertionliii)
is a c o n s e q u e n c e of ( i ) a n d ( i i ) .
1.h
69
Q u a s i h o m o g e n e o u s Polar C o o r d i n a t e s
For t h e f o l l o w i n g a s s e r t i o n o n h o w t o c o m p u t e dx" w e d e n o t e by
Volz
the
o r i e n t e d v o l u m e f o r m o n S x i n d u c e d by t h e given s c a l a r p r o d u c t o n V.
Lemma 1.87. (i1
idM
is,=
W,
Vol0 , where w,(H) := l / l g r a d x(L+)l, S 6 S X ; and con-
sequen t I),:
(iil I f U ( r e s p . W ) i s an open subset o f X ( r e s p . I?"-') and i f 9 :W+R a C
' function satisfying U n S x = graph ( p I
Proof. li):Let
is
then
x E S X . F i r s t of all o n e o b s e r v e s t h a t by Euler's e q u a t i o n g r a d x ( x )
is n o n - z e r o , i.e. w , ( x )
is w e l l - d e f i n e d a n d positive. M o r e o v e r , a n o t h e r way of
w r i t i n g Euler's e q u a t i o n f o r x a t x is (1.105)
where
< u ( x ) , M x >= w , ( x ) U(A)
: = w , ( x ) g r a d x ( x ) . C o n s e q u e n t l y , s i n c e w ( x ) is t h e o u t w a r d n o r m a l
unit vector to Sx at x a n d h e n c e is o r t h o g o n a l to T,SX o n e f i n d s <ET,SX satisfying
M k =
w,(w)u(x) +
c.
S i n c e t h e f a c t o r of w , ( x )
It f o l l o w s t h a t
o n t h e r i g h t - h a n d side is e q u a l to V o l z ( x ) t h e a s s e r -
tion follows.
(iil:
S i n c e x ( y , ' p ( y ) )= 1 w e have a i x ( y .( p ( y ) )+ a , , x ( y , cp(y)) ai'p(y) = 0 f o r a r b i t -
r a r y y E W a n d j € N n - , . C o n s e q u e n t l y , in view of Euler's e q u a t i o n f o r x t h e derivat i v e d , x ( y , ' p ( y ) ) does n o t vanish, a n d it f o l l o w s t h a t
I
+
I g r a d c p ( y ) 1 2 = Igradx(y.(p(y))12/la,x(y,'p(y))12,
YEW.
C o m b i n i n g t h i s w i t h t h e a s s e r t i o n ( i ) o n e o b t a i n s ( i i ) . rn
If t h e o u t w a r d n o r m a l u n i t vectors u ( x ) , x E S X , a r e explicitly k n o w n t h e n w , ( x ) c a n be c o m p u t e d f r o m (1.105) :
Example 1.88. Suppose that the condition (1.141 i s satisfied, and that x = x + i s
70
I.
( A l m o s t ) Quasihomogeneous Functions
defined a s in Proposition 1.70. Then I / l g r a d x M ) / =
.
BES".
P r o o f . S i n c e by t h e c h o i c e of t h e s c a l a r p r o d u c t in P r o p o s i t i o n 1.70 w e have w(9)=P + ( 9 ) t h e a s s e r t i o n f o l l o w s f r o m (1.105).
C o m b i n i n g P r o p o s i t i o n 1.86 w i t h t h e Fubini a n d t h e Fubini-Tonelli
theorem one
o b t a i n s a c o n v e n i e n t i n t e g r a b i l i t y c r i t e r i o n . H e r e t h e i n t e g r a b i l i t y of t h e f u n c t i o n s
9 H f ( M t 9 ) w i t h r e s p e c t to x" o f t e n easily f o l l o w s if t h e s u p p o r t of t h e s e f u n c t i o n s is c o m p a c t . T h e l a s t c o n d i t i o n c a n s o m e t i m e s be verified w i t h t h e h e l p of
Lemma1.89. ( i ) K M n S Y is compact for an) compact subset K of X . (ii) Suppose that X = V, and that ( t . t 4 ) is valid, and assume that (1.79) h o l d s . In be given a s in Proposition 1.70. Then K T n S "
particular, let x = x + : V + + l O , + w C
is compact for e v e p compact subset
m f .
u. This
K of V .
is a c o n s e q u e n c e of Lemma 3.20 a n d P r o p o s i t i o n 3.22 b e l o w . A
direct p r o o f is a s f o l l o w s : Let
(Xp)pso\l
b e any s e q u e n c e in K a n d ( t p ) o e N be
any s e q u e n c e in IO,+coC s u c h t h a t y p : = Mtpxp b e l o n g s to S" f o r every t ? € : 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 ( x p ) c o n v e r g e s to s o m e x € K . In view o f t e x ( x p ) = x ( y p ) = I it f o l l o w s t h a t
l i r n t p = lim l / x ( x q ) = I / x ( x ) = : t
p + ~ ,
e+m
so t h a t ( y e ) c o n v e r g e s to Mtx E K,
(ii).
n S". -
Let ( Y e ) p c N b e a s e q u e n c e in K,
n S X . We choose sequences ( t g ) in IO,+a3C
a n d ( x p ) in K s u c h t h a t lim ( y p - Mtpxp) = 0 .
P+m
By c h o o s i n g s u b s e q u e n c e s o n e achieves t h a t ( x p ) c o n v e r g e s to s o m e x € K . I t follows that lim M o x P = M o x p+m
so t h a t
lim M o y p = P+m
Max.
T h a t y p b e l o n g s to S" m e a n s t h a t ( P + ( y p ) l = l - s e e (1.76). 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 f o r a s e c o n d t i m e o n e achieves t h a t ( P + ( y o )) e e N c o n v e r g e s to P + ( z )
71
1.h Q u a s i h o m o g e n e o u s Polar Coordinates
for some z € V as J?+ + a .Consequently, ( y p ) converges t o y : = Mo(x) + P + ( z ) .
-
Since ( M t p x P ) t Econverges ~ to y , as well, one sees that y belongs t o K,
n Sx.
Combining Proposition 1.86 w i t h the Fubini theorem one sees that for any locally integrable function f : X + @
the function Sx+@,
3 H f ( M t B ) , is locally inte-
grable with respect t o x" for almost every t E l O . + a C . B u t , of course, generally one does n o t know whether t h i s is valid for t = l , i.e. whether fl,,
is locally
integrable. With almost quasihomogeneous functions t h i s is different:
Lemma 1.90. Let N E N , and let qo :X der
_<
C be almost quasihomogeneous o f or-
N . Then the following conditions are equivalent: ( a ) qo a s well a s i t s deficiencies q r
.. ..
I
c]N
belong t o
1 zlor (x):
I
( a ) ' 9o E . X , ~ , (= X ) :
(b) for arbitrary k ENN u (01 and t E 10. +wl the function qk 0 M , lSx is locall-\, integrable with respect t o x"
:
( b ) ' for ever} X E N N u (01 the function q k l s x i s locall} integrable with respect t o
2.
Proof. The equivalence ( a ) t--r. ( a ) ' and the fact that the measurability of qo im-
plies that of its deficiencies ql , . . . , q~ are consequences of Remark 1.50. Moreover, if q k l S X is 2-measurable then the function ( t . 9 ) H ( q k o p x ) ( M t 9 ) is (dtC3.x)measurable so that q k o p x is Lebesgue-measurable by Proposition 1.8O.(iii ) . Hence, Remark l.73'.(ii) shows that the functions 4 0 , . . . ,qN are Lebesgue-measurable if there restrictions to S" are 2-measurable. Consequently, for the rest of the proof we may assume that the functions q o , . . . , qN are Lebesgue-measurable. (a) 3 ( b ) : Let K be a compact subset of Sx. Then the set
(1.106)
L : = { M,x; t € I , x E K }
is compact for every compact subinterval I of I O , + a l , say I = C1/2.21. In view of condition ( a ) it follows from Proposition 1.86 and Fubini's theorem that for
every k E (0) u N,
there exists t k €1 such that t h e f u n c t i o n K-C
, 8 H q k ( M,,B),
is %-integrable. Since by Lemma 1.48 qN is quasihomogeneous this shows that
qNIK is 2-integrable. By induction one can now deduce that the same is valid for q N - l , q k - 2 , . . . , q l , and 4 0 . Indeed, let k c ( O ) u "-1
be s u c h that the restriction
72
I.
(Almost)
Ouaslhomoneneous Functions
o f q1 to K is 2 - i n t e g r a b l e for every jENN s a t i s f y i n g j > k . S i n c e t h e e q u a t i o n (1.39), in L e m m a 1.48 m e a n s t h a t N-k
(1.39);
qk=tkmqkoMtk-
qk+0
e=i
it f o l l o w s t h a t t h e r e s t r i c t i o n of q k to K is ;-integrable,
a s w e l l . A n o t h e r appli-
c a t i o n o f (1.39), n o w 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 q k O M t to K is 2 - i n t e g r a b l e u N N a n d t E 10,+a[. f o r a r b i t r a r y k E (0)
( b ) ' + f a ) ' . Let H be a c o m p a c t s u b s e t of X . T h e n by L e m m a 1 . 8 9 . ( i ) K : = H M n S X is a c o m p a c t s u b s e t of S x , a n d x ( H ) 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 1 o f
I O , + a C . S i n c e H is c o n t a i n e d in t h e set (1.100) t h e e q u a t i o n ( 1 . 3 9 ) , t h e FubiniT o n e l l i t h e o r e m , a n d P r o p o s i t i o n 1.80 immediately imply t h a t q o i s i n t e g r a b l e o n
H . rn S o m e t i m e s w i t h t h e h e l p of P r o p o s i t i o n 1.86 o n e o b t a i n s local i n t e g r a b i l i t y e v e n a t p o i n t s n o t b e l o n g i n g to U ( M ) . For t h e f o l l o w i n g p r o p o s i t i o n w e d r o p t h e a s -
s u m p t i o n c o n c e r n i n g t h e e x i s t e n c e of x .
Ropositlon 1.91. Suppose that 1 1 . 1 1 ) is valid. Let NEIN,,
. i t s deficiencies being qN . Let ik be the evtension o f qk t o X defined bJ,(.Ix, := 0 . +
be alniost quasihomogeneous o f degree denoted bj, q l , . . . ,
I
and let qo E 2 f I o r ( X + )
ni
and o f order 5 N
Then the following conditions are equivalent: (a)
ik ~ i , f i 'f o-r (ever)' ~ )~ E ~ o ~ L J / " :
f a ) ' cjo E
t lI o( , X ):
( b ) Rem-'
-/I.
o r s u p p i k C X + for e v e r j ' k 6 l O l ~ N ~ .
Pr o o f . f a ) ' + ( a ) : 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 R e m a r k 1.50. F o r t h e p r o o f of t h e o t h e r i m p l i c a t i o n s w e s u p p o s e t h a t (1.79) is valid a n d c h o o s e
x = x + as in Lemma 1 . 8 9 . ( i i ) . If supp:,CX+ n o t h i n g to be p r o v e d . So w e fix 4 E {O)U["
f o r every kE(OJu["
t h e n t h e r e is
and xE supp;Ie\X+ and choose
E
> 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
are integrable on Cl,+aland since the continuous functions qk are integrable on
t h e compact set S" with respect t o ii the Fubini-Tonelli shows that g is integrable on I l , + c o C x S " w i t h respect to d t @ G , a s desired.
rn
75
Chapter I1
(Almost) Quasihomogeneous Distributions. Definitions and Basic Properties
Section ( a ) of t h i s chapter contains the basic material on quasihomogeneous distributions. Among other things, distributions T which are quasihomogeneous of degree m are characterized as the solutions of Euler's equation (3, - m ) T = 0 where
, 3
denotes the directional derivative into the direction of the quasihomogeneous
rays (see Example 1.21). Moreover. we shall see that the restrictions of quasihomogeneous distributions to the hypersurfaces S" are well-defined for any Cm function
x:X-IO.+~I
which is quasihomogeneous of degree 1 . Section ( b ) contains some
basic facts on t h e behaviour of quasihomogeneous distributions under the Fourier transform. Section ( c ) is devoted to meromorphic functions of quasihomogeneous distributions.
I t turns o u t that their Laurent coefficients possess new invariance properties under composition with M t : they are almost quasihomogeneous in t h e sense of Definition 2.28 below. In section ( d ) the basic facts on almost quasihomogeneous distributions are presented. Many properties carry over from sections ( a ) and ( b ) . I n particular, we shall see that a distribution T is almost quasihomogeneous of degree m if and only if for some (d,-m)"'T
NEN, it solves the generalized Euler equation
= 0 . Section ( e ) is about meromorphic functions of almost quasihomo-
geneous distributions. I t turns out that the Laurent coefficients arising here do not have new invariance properties: they are almost quasihomogeneous themselves. Finally, an appendix -section ( f ) - contains the basic facts on distributions invariant under the action of a compact subgroup of G L ( V ) . As in Chapter 1 , there are quite a few results of a more technical nature required in later chapters.
76
11.
(Almost) Quasihomogeneous
Distributions
A s before, X is a n open s u b s e t of V which a t f i r s t is n o t required to b e quasihomogeneous. Considering X a s a n n-dimensional linear manifold w e recall t h a t t h e space a ' ( X ) of distributions o n X can be defined a s t h e dual of t h e s p a c e
of C F - d e n s i t i e s o n V (see for example $ 0 . 3 in Hormander 1111). In t h e following we shall identify % ' ( X ) with t h e dual of CFCX) via t h e strictly positive density (ax:=
Vol d e n o t e s t h e volume f o r m associated with t h e given
V o l J x where
scalar product o n V:
In particular, a s usual every locally integrable function f : X - @
(with respect
t o t h e Lebesgue measure d x induced by t h e given scalar p r o d u c t ) induces a distribution T f o n X defined by :=ff'pdx,
fp E
c;c x ) .
Multiplication by functions and t h e action of differential o p e r a t o r s a r e defined " a s in t h e s t a n d a r d way. For any T E a ' ( X ) by T we d e n o t e t h e distribution o n
" ( - X ) defined by < T , ' p > : =< T , G > where ( G ) ( s ): = c p ( - x ) .
We f i r s t observe t h a t if
'p :
V
t C is any function then tElO,+co[,
For any compact s u b s e t K of X we introduce t h e open s u b s e t
(2.2)
J M ( K ; X ) : = { t € l O , + ~ CM: , ( K ) C X }
of 1 0 , + ~ 1 I.t f o l l o w s f r o m ( 2 . 1 ) t h a t
qEc:(x)
OM,,^ belongs t o CTCX) for arbitrary
and t E J M ( S U p p ' p ; X ) . Consequently, f o r any f€6eio,(X)
a change of
variables s h o w s t h a t J f ( M t x ) q ( x ) d x = t-'J f(y)cp(M,,,y)dy X X where p = t r M
- see
( 1 . 6 ) . If
of degree m if and only if
,
t EJM(K;X),
f is continuous it follows t h a t f is quasihomogeneous
2.a
77
Quasihomogeneous Distributions
Definition 2.1. A distribution T € a ' ( X ) is said to be quasihomogeneous of degree m (and of type
MI if and o n l y i f
t - ' < T , q o M 1 / , > = t m
(2.3)
for arbitrary c p € C g ( X ) and t € J M ( s u p p c p ; X ) .
Of course, i f X is quasihomogeneous then t h e left-hand
(2.3) is by
side of
definition equal to < T o M , , y > , 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
'p,cp)o~,.>
=
sss
r€R
rm+" < ~ , ( c p , . ' p ) o ~ , >
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<5,x>)'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<
on V is transported t o V' by way of the Riesz
x , * > . A priori, for any 'pEY(V') the domain of de-
$ is V * * . However, via the canonical isomorphism
t(x)(<)=
< < , x > we shall identify V** w i t h V . So
linear map Y(V*)-Y(V).
'pH$
t:V-
V** defined
defines a continuous
Consequently, for any T E Y ' ( V ) it makes sense to
h
define T E Y'( V*) by the standard formula:
where Y " ( V ) denotes the space of temperate distributions on V . In fact, by
9 ~ sP'(:V ) -----?r sP'( V * ) . T I + ?. a n isomorphism is defined extending the Fourier transform o n F'P(V). The inverse of F V i s given according to the Fourier inversion formula:
85
2.b The Fourier Transform o f Q u a s i h o m o g e n e o u s Dlstributions
"
TE Y'( V*),
(VV)K1T= (2x)-"Vv+(T),
"
w h e r e < T , c p > : =< T . $ > a n d G ( x ) : = c p ( - x ) . B e l o w , m o s t l y w e s h a l l w r i t e F in-
stead of F V or 3 v + .
In order to be a b l e to f o r m u l a t e t h e r u l e for Fourier t r a n s f o r m s o f d e r i v a t i v e s we o b s e r v e t h a t f o r e v e r y polynomial f u n c t i o n P:V-C
a n d f o r e v e r y CcEC a
p o l y n o m i a l f u n c t i o n o n V d e n o t e d by x H P ( < x ) is w e l l - d e f i n e d via c o o r d i n a t e s s i n c e a l i n e a r c h a n g e of c o o r d i n a t e s l e a v e s t h e h o m o g e n e o u s p a r t s of p o l y n o m i a l s i n v a r i a n t . In p a s s i n g w e n o t e t h e f o l l o w i n g n o t c o m p l e t e l y trivial
Remark 2.18. Let P : V +
C be a pol-vnomial function. If P is quasihomogeneous
(resp. almost quasihomogeneous) of degree m then so is P I < ( - ) ) f o r every C E C .
Proof. W o r k i n g w i t h c o o r d i n a t e s a n d m a k i n g u s e of (1.29)' o n e sees t h a t (a,-m)
P ( < (*
))
-
= ((d,-m)P)(<(
)).
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 P r o p o s i t i o n 1.22 in t h e q u a s i h o m o g e n e o u s case a n d f r o m P r o p o s i t i o n 2.31 a n d R e m a r k 2.30 b e l o w in t h e a l m o s t q u a s i h o m o g e neous case.
H
Now let P : V * - + @
be a polynomial f u n c t i o n . T h e n t h e r u l e f o r F o u r i e r t r a n s -
f o r m s of d e r i v a t i v e s r e a d s a s u s u a l
w h e r e D : =- i d a n d w h e r e t h e polynomial f u n c t i o n P ( - i ( .
))
o n V* is d e f i n e d a s
a b o v e w i t h ( V , < ) r e p l a c e d by ( V * , - i ) ( c o m p a r e ( 1 . 2 s ) ) . M o r e o v e r , if Q : V - + @ is a polynomial f u n c t i o n t h e n
B ( Q T ) = Q( - D )
?
w h e r e for t h e d i f f e r e n t i a l o p e r a t o r Q ( - D ) o n V* to be w e l l - d e f i n e d w e have to c o n s i d e r Q a s a polynomial f u n c t i o n o n V'*
via t h e canonical i s o m o r p h i s m
L
de-
scribed a b o v e . Finally, w e r e c a l l t h a t by a t r a n s f o r m a t i o n of variables o n e o b t a i n s
(2.8)
B ( T o A ) = 1detAI-l ? o ( A - ' ) * ,
f o r every T E Y ' ( V ) . For A = M , , ,
this becomes
A € G L ( V ,V ) ,
86
11.
If T E Y Y V ) t h e n w e observe t h a t map
Y(V) h Y ( V ) , 'p H OM,,^,
(Almost) 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
CTCV) i s d e n s e in Y ( V ) a n d s i n c e t h e
is c o n t i n u o u s
-
T is q u a s i h o m o g e n e o u s of de-
g r e e m if a n d o n l y if (2.3) h o l d s f o r every v € Y ( V ) . W e s h a l l see b e l o w ( T h e o r e m 6 . 4 5 ) t h a t in case o = o + every 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 o n V is a u t o matically t e m p e r a t e . H e r e w e n o t e a s a n i m m e d i a t e c o n s e q u e n c e of ( 2 . 8 ) ' :
Propoaltion 2.19. Let T E Y ' ( V ) . Then T is quasihomogeneous o f degree m and o f I\
type M i f and only i f T is quasihomogeneous of degree m * := - m - p and o f type
M*. m
Corollary 2.20. Let Q be a polwomial function on V ' . and denote b-b So the Diracdistribution at 0 . Then Q(D)So is quasihomogeneous o f degree m' and o f t j p e M i f and on?, i f Q is quasihomogeneous o f degree m and o f tjpe M ' .
mf. Since
7 ( Q ( D ) S o ) = T Q t h i s f o l l o w s f r o m P r o p o s i t i o n 2.19 a n d f r o m w h a t
w a s said in t h e t e x t p r e c e d i n g Definition 2.1. rn
T h e a s s e r t i o n of P r o p o s i t i o n 2.10 is a l s o a c o n s e q u e n c e of t h e f o l l o w i n g l e m m a t e l l i n g u s w h a t b e c o m e s of t h e E u l e r o p e r a t o r d M u n d e r t h e F o u r i e r t r a n s f o r m . A
A
Lemma2.21. Let T E P * ( V ) .Then fdM4 - m * ) T = - d where d : = ( d M - m ) T
proof. C o m p u t i n g w i t h r e s p e c t to real o r t h o n o r m a l c o o r d i n a t e s w e o b t a i n h
A
<jdkT=-<ji7(xkT) = ( - i ) 2 %(dj(xkT))=-6,kT-%(xkdjT). F r o m t h i s w e d e d u c e by m a k i n g u s e of (1.20)' t h a t A
h
A
h
h
dM*T = -p T- % ( d M T ) = - p T - d - m T
.
T h e a s s e r t i o n of P r o p o s i t i o n 2.6. ( i i ) c a r r i e s over to w a v e f r o n t sets:
Propodtion 2.22. Let T E 3 '(XI be quasihomogeneous. Then
WF(T) = { (M,s, MI:,
5) ; IS,
f ) E W F ( T ) , t E J M I is); X)} .
2.b
a7
T h e Fourier T r a n s f o r m o f 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 close this section with
Propoeitlon 2.23. Suppose that o = o + and that M is semi-simple. Let T E Y ' ( V ) be such that the restriction o f T to V is induced 6>, 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
( d M - m ) a j ( z , ; h ) = i5l k+g(k)(zo) a j - k ( z , ; h l k= 1
(note that the sum is finite since o r d ( z o ; h ) > - 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
(2. Ih )'
> = +
N ( d k ( t )
k=l
f o r arbitrary cpECg(X) and t € J , ( s u p p p ; X )
-see ( 2 . 2 ) . By a s i m p l e m o d i f i c 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 . 5 ( c o m p a r e t h e p r o o f o f P r o p o s i t i o n 1.57) o n e c a n verify t h a t t h e a s s e r t i o n o f P r o p o s i t i o n 1.57 carries o v e r to 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 . H o w e v e r , w e s h a l l n o t m a k e u s e o f t h i s r e s u l t in t h e s e q u e l . N e x t w e are g o i n g to verify t h a t f o r locally i n t e g r a b l e f u n c t i o n s t h e t w o n o t i o n s of a l m o s t q u a s i h o m o g e n e i t y i n t r o d u c e d in Definitions 1.32 a n d 2.28 e s s e n t i a l l y coincide. For t h i s w e have to s h o w t h a t similarly a s in R e m a r k 1.50 o n e c a n c o m p u t e
t h e deficiencies by purely a l g e b r a i c m e a n s :
Remark 2.29. Let T E D ' f X ) be almost quasihomogeneous o f degree m and o f order 5
N . And let f t l ) 1 5 , s N be a sequence o f pairwise different numbers in 10, + m C \ ( 1 ) .
Then for every k € N N the k t h order deficiencj dk o f T is given bj N
12.17)
dk = ~ L k i l t i - r r ' T ~ M , j - T ) j=l
where ( L k j ) ( k . j )E N N X N N denotes the inverse o f the real NxN-matrix ( 1 . 6 b ) .
Proof.J u s t
as in t h e p r o o f of Remark 1.50 o n e h a s to s u b t r a c t T f r o m b o t h sides
of ( Z . l f ~ ) ,i n s e r t
ti
for t , and apply
(Lkj).
Remark 2.30. Let f ~ B e , b , ( X ) . (i)
I f f is almost quasihomogeneous o f degree m in the sense o f Definition 1.32
then the distribution TF induced 6-1. f is almost quasihomogeneous o f degree m in the sense o f Definition 2.211, and for ever,' k E N the k
Lh
order deficiency qk
o f f , being locall-1 integrable b), Proposition 1.Sl. f i ) , induces the k
th
order de-
ficiency dk o f T , . f i i ) Converselq, suppose that T f is almost quasihomogeneous o f degree m in the sense o f Definition 2.28; i f V f E M f o o ) then there is a function g:X*
C which
is almost quasihomogeneous o f degree m in the sense o f Definition 1.32 and which is equal t o f almost everywhere: i f f is continuous then f itself is almost quasihomogeneous o f degree m in the sense o f Definition 1.32.
2.d Almost Q u a s i h o m o g e n e o u s
9s
Distributions
proof. (i):in view of (1.39) a n d P r o p o s i t i o n l . S l . ( i ) a t r a n s f o r m a t i o n of v a r i a b l e s leads to t h e a s s e r t i o n .
(ii).
I t f o l l o w s by R e m a r k 2.29 t h a t f o r every k E N ,
t h a t t h e k t h order deficiency of map lO,+mIxX-X,
1
t h e r e is f k E z , o c ( X ) s u c h
T, is e q u a l to T f k . W e s e t f,:
= f . Since t h e
( t , x ) H M , x , is a s u b m e r s i o n t h e m a p h : l O , + c o C x X & C ,
( t , x ) H t - m f ( M t x ) , is m e a s u r a b l e . T h e a s s u m p t i o n o n Tf i m p l i e s t h a t f o r e v e r y t E IO,+mC w e h a v e N
h(t;)
=
hjk(t)
fk
a l m o s t everywhere.
k=O
If f is c o n t i n u o u s t h e n t h e f k c a n be c h o s e n to be c o n t i n u o u s , a s w e l l , so t h a t t h e p r e c e d i n g e q u a t i o n s h o l d e v e r y w h e r e , a n d t h e p r o o f is c o m p l e t e . If f is n o t c o n t i n o u s t h e n for any c o m p a c t s u b s e t K o f X a n d f o r every t € l O , + m C o n e c a n o n l y d e d u c e t h a t - s i n c e M , ( K ) is a c o m p a c t s u b s e t of X - t h e f u n c t i o n h ( t ; ) is i n t e g r a b l e o n K , a n d N
j ' ( h ( t , x ) l d x5 K
l
K
S i n c e a s a f u n c t i o n o f t t h e r i g h t - h a n d side is locally i n t e g r a b l e t h e FubiniT o n e l l i t h e o r e m s h o w s t h a t h is locally i n t e g r a b l e . a s w e l l . I t f o l l o w s by Fubini's t h e o r e m t h a t f o r every t ) E C ~ ( I O , + a C x X )w e have N
C o n s e q u e n t l y , h is e q u a l to ~ ~ = O c c ~ k a@ l mf oks t e v e r y w h e r e . N o w , w e c o m e to t h e c o n s t r u c t i o n o f g . F i r s t of a l l , by P r o p o s i t i o n 1.80 t h e a s s u m p t i o n o n V i m p l i e s t h a t X \ V ( M ) is a set o f m e a s u r e z e r o so t h a t w e may d e f i n e
I
g x,,i( M ) : 0 . M o r e o v e r . in view o f N o t a t i o n 1.6s a n d Lemma 1.64 w e f i n d a s u b s e t J of N , a n o p e n c o v e r i n g ( U i ) i e J of V ( M ) c o n s i s t i n g of n o n - e m p t y q u a s i h o m o g e n e o u s o p e n s u b s e t s o f X , a n d a family ( x ~ ) o f~ Cm ~ Jfunctions x j : U j - l O , + ~ C which 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 I . I t s u f f i c e s f o r every j E J to f i n d a f u n c tion g j : U i - C
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 m a n d c o i n c i d e s
w i t h f l u . a l m o s t e v e r y w h e r e . For t h e sets X j : = U i \ U i E J , i c j U i , j E J , a r e m e a s u r I
able, a n d d e f i n i n g g o n % ( M ) = U I E J X j by g l X i : = g i l x i ,j € J , c o m p l e t e s t h e c o n struction of g .
So w e may a s s u m e t h a t X is e q u i p p e d w i t h a C" f u n c t i o n x : X + l O . + c o C 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 1 . W e f i x s E I O . + a C a n d d e f i n e
which
96
II.
( A l m o s t ) Quasihomogeneous Distributions
N
g ( M t M s 8 ) := t m (
2 wk(t) fk(Ms8)) ,
t E
I O , + ~ C9,€ S X .
k=O
Applying R e m a r k l . 7 3 ' , ( i ) to X : = x o M g ' i n s t e a d of x a n d o b s e r v i n g t h a t Sx = M s ( S " ) w e deduce t h a t 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 . T h e r e m a i n i n g p r o b l e m is to f i n d s in s u c h a way t h a t t h e f u n c t i o n s g a n d f c o i n c i d e a l m o s t everyw h e r e . By u w e d e n o t e t h e m e a s u r e t p - ' d t o n I O , + a l . I t f o l l o w s f r o m P r o p o s i t i o n 1.86 t h a t t h e i n v e r s e o f t h e m a p Y : IO,+alxSXx I O , + a C +
IO,+alX X , ( t , 9 , s )H ( t ,M s 9 ) ,
( w h i c h is bijective by P r o p o s i t i o n 1.72, i n d e e d ) m a p s sets of d t 8 d x - m e a s u r e z e r o o n t o sets of d t 8 Z 8 u - m e a s u r e z e r o . H e n c e , f r o m t h e e q u a l i t y derived a t t h e beginning of t h e p r o o f if f o l l o w s t h a t N
hoY =
( ( d k 8 f k )O Y
d t @ x @ v - a l r n o s t everywhere
k=O
T h i s i m p l i e s t h a t o n e c a n find s E 1 0 , + w [ s u c h t h a t N
hoY( * , s )
(dk@(fkOMsIsX)
dt@x"-almost everywhere,
k=O
i.e.
N
f o r d t 8 G - a l m o s t every ( t , B ) E 1 0 , + ~ l x S xSince . by P r o p o s i t i o n 1.86 t h e m a p p i n g
1 0 , + ~ C x S x ~( tX, S.) H ( M t M s . R ) , m a p s sets o f d t 8 G - m e a s u r e z e r o o n t o sets of
d x - m e a s u r e zero t h e number s has t h e desired property.
m
T h e a s s u m p t i o n a b o u t V in a s s e r t i o n ( i i ) of Remark 2.30 is p e r h a p s n o t n e c e s s a r y
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 n infinitesimal c h a r a c t e r i z a t i o n f o r a l m o s t q u a s i h o m o g e n e i t y . I t is t h e a n a l o g u e o f P r o p o s i t i o n 2.11 a n d e s s e n t i a l l y is a r e f o r m u l a t i o n of P r o p o s i t i o n 2.26' f o r c = 1 .
Propodtion 2.31. Let T € D ' ( X ) . Then
(i)
T is almost quasihomogeneous o f degree m and OF order5 N i f and on]) i f
(dM-m)N*lT=O;
(iil i f this is the case then for every k € N N the k L h order deficiency dk o f T is equal t o ( J M - m I k r .
2.d
97
A l m o s t Q u a s i h o m o g e n e o u s Distributions
M.If
(2.16) holds then Proposition 2.26' (applied to TN : = T and TN-k: = d k ,
k€")
shows
(aM- m ) d N = 0 .
that
( d M - m ) T = d l , ( d M - m ) d k = d k + l for
Hence, by induction we obtain that
d k = ( d M - m ) k T . Conversely, if ( d M - m ) " + ' T = O
k€N,-,
,
and
(aM - m ) N + lT = 0
and
then the condition ( c ) of
Proposition 2.26' is valid for T j : =( d M - m ) N - i T and c = l . Consequently, (2.16) follows by Proposition 2.26' if we set dk :=TN-,
.
1
Proposition 2.31 s h o w s how to compute the polynomial functions P t , j associated to any polynomial function P in Remark 1.35:
r be a polynomial
Corollary 2.32. Let P: )I-
and k E N we have P p , k = ( d M -P)"(QpP).
function. Then f o r arbitrary k ' c X ( M )
8
As an immediate consequence of Proposition 2.31 one obtains the following asser-
tions t h e first two of which can also easily be derived directly from (2.16) (compare the proof of Lemma 1 . 4 8 ) .
Corollary 2.33. Suppose that T is almost quasihoniogeneous o f degree m and o f order 5 N . Then f o r ever) k € N N i t s k ' "
order deficiencj. dk has the following
properties :
li)
dk is almost quasihomogeneous o f degree m and o f order 5 N - k :
( i i ) supp dk C s u p p T : (iiil sing supp d , C sing supp T : ( i v ) W F ( d k l C WF( T ) .
I
In order to prove the analogue of Proposition 2.3 for almost quasihomogeneous distributions we require
Lemma 2.34. Let P : X x V * - - + C be a C"' function which is a polynomial with respect t o the second variable. Then f o r ever,. TEZ)'(XI we have
98
11.
( A l m o s t ) ~ u a a i h o r n o g e n e o u sD i s t r i b u t i o n s
proOf. Working with real coordinates, setting A : = N," , writing P(X,<)=
1 P,(X)
t',
X
, 5 E R"
&€A
and making use of X k a a 3 j T = a"(XkajT)
-Clk
a"-ekdjT
we obtain in view of (1.29)' that
Since the term in square brackets i s equal to d,P
where N : = M x ( - M ) * the first
part of the assertion is proved. Applying this t o ( ( a , - @ ) ' P , ( a M - m ) k - iT ) instead of ( P , T ) we obtain for arbitrary k E N and i E ( 0 ) u l N k
Since (i_kl)
+
( y ) = (';I)
for every i € N k + l the second part of the assertion fol-
l o w s by induction on k . H
As an immediate consequence of Lemma 2.34 one obtains
ProposlUon 2.35. Let TED'(XI be almost quasihomogeneous of degree m and o f order 5 N with deficiency d € D ' ( X ) .Moreover, let P:XxV*-C
be a C'" function
which is a polynoniial with respect t o the second variable. Suppose that P is almost quasihomogeneous o f degree l € @of, type M x ( - M ) ' , and of order S N ' E N , with deficiency Q . Then P ( x ,d ) T is almost quasihomogeneous of degree m+P of order 5 N + N ' with deficienq P ( h . d ) d + Q ( . \ , d ) T .
8
Two special cases of Proposition 2.35 are formulated a s
2.d 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
99
Comllary2.36. Let TEB'IX) be almost quasihomogeneous of degree m and o f order 5 N . let k E N N , and let d, be the k " order deficiencj. o f T . (i)
I f P: V*-
C is a pol-vnomial function which is quasihomogeneous o f degree
P E C and o f type M A then P I d ) T is almost quasihomogeneous o f degree m-P with
k t h order deficiency P ( d ) d k
.
(ii) I f q E C ' ? ' ( X )is quasihomogeneous of degree PEC then q T is almost quasihomogeneous o f degree m + t with k t h order deficiency q d , . I
T h e next result generalizes Proposition 2 . 0 ,
Pmposltlon 2.37. Let T E B ' ( X ) be almost quasihomogeneous. Then (i)
supp T = (supp T I , :
( ii) sing supp T = (sing supp T ) , : (ii)' more generally:
W F ( T )= I ( M , ~ M,:, .
E ) : (1.t) E
W F ~ T ) t.E 1 0 , + a c }
P E E . W e set d o : = T a n d f o r k € N N d e n o t e by d k t h e k t h order deficiency o f T.
li):
Let Y be 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 < Y M 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 a n d a n u m b e r t E l O , + o J C s u c h t h a t M , / , U
C
Y.
H e n c e f o r every q C C F ( U ) w e d e d u c e t h a t s u p p ( q o M , ) C Y so t h a t in view of
Y c x \ s u p p d k (see C o r o l l a r y 2 . 3 3 . ( i i ) )
< T , 'p > = t" < T
0
,
M ,q 0 M
it f o l l o w s f r o m (2.16) t h a t
,> = t
N "' + "
Uk(t)
=O.
k=O
T h i s m e a n s t h a t YM C Y . S i n c e t h e c o n v e r s e i n c l u s i o n is trivial t h e p r o o f is c o m plete.
(ii). Here
w e set Y : = X \ s i n g s u p p T , a n d let f € C m ( Y ) b e s u c h t h a t TI, = T,.
S i n c e f is a l m o s t 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 n 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 f,
: YM
-+
T , f M , a n d T coincide o n
m'.By
C by P r o p o s i t i o n 1.57. T h e a s s e r t i o n ( i ) i m p l i e s t h a t YM
(2.16)a n d by ( 2 . 8 ) '
.
Hence Y M C Y , i.e. YM = Y . w e have N
2 h ) k ( t ) F ( ( q d k ) O M l / t ) (M:/,<)
F('pOM1/, T ) ( M & < ) = tm
=
k=O
N
= tm+"
b&(t)
k=O
q('pdk)
(c),
<EV*, v E C T ( X ) , and t ~ l O , + m C .
100
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
For arbitrary x e X ,
(Ev),
and t E l O , + a C this implies t h a t (Mtx,M:,,()
f o r every k E ( O ) u N N . In
belong to W F ( T ) if ( x , ( ) does n o t belong t o WF(d,) view of Corollary 2.33.(iv) this proves the inclusion
does n o t
'2". As t h e
inverse inclusion
is trivial t h e proof is complete.
Carrying over t h e proof of Corollary 2.7 one deduces from Proposition 2.37.(ii):
Corollary 2.38. Suppose that (1.14) holds. Let TEZJ'IV , I be almost quasihomogeneous. I f singsupp T n U is empty for some neighbourhood U of ker M then T is induced by a C" function.
I
Combining this with Proposition 1.58 one obtains
Ropositlon2.39. Suppose that o = o , . to
Let T€.!BD'(V) be such that i t s restriction
i s almost quasihomogeneous. Then either 0 E sing supp T or T is induced by
a polynomial function.
I
As for t h e Fourier transform of almost quasihomogeneous distributions, Lemma 2.21
immediately leads to
Proposition 2.40. Let T E Y " ( V I . Then (i)
T i s almost quasihomogeneous of degree m and of order 5 N i f and only if
A
T is almost quasihomogeneous of degree m * = - m - p
.
of order 5 N and of type
M *: A
(iil i f this is the case then f o r everj k € N , the k r t ' order deficienc) of T is equal t o ( - l l k 9 1 d k ) where d , denotes the h
th
order deficiency of T .
I
Next we generalize Proposition 2.14 and Theorem 2.16 t o a l m o s t quasihomogeneous distributions.
R o p o r i t l o n 2.41. If T € B ' ( X I is almost quasihomogeneous then i t s analytic wave front set WF,(T)
&f.
is contained in the set r M ( X ) defined in (2.6).
The principal symbol of t h e differential operator ( a M - m ) N + 1 is equal t o
101
2.d A l m o s t Q u a s i h o m o g e n e o u s Distributions
t h e f u n c t i o n (iPMM)N+lw h e r e PM is d e f i n e d as in E x a m p l e 1.21. H e n c e t h e c h a r a c teristic set o f t h e o p e r a t o r ( a , - m )
N+l
is e q u a l to
rM(x). C o n s e q u e n t l y ,
the
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 . 3 l . ( i ) a n d T h e o r e m 8.6.1 in H o r m a n d e r [ I l l .
C o m b i n i n g P r o p o s i t i o n 2.41 a n d Lemma 2.15 w i t h C o r o l l a r y 8 . 2 . 7 in H o r m a n d e r 111 1 one obtains
Theorem 2.42. I f T E B ' f X ) is almost quasihomogeneous then i t s restriction TIS.. to S y is well-defined as the pullback b1. the inclusion map J : S " + X .
I
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 it is p o s s i b l e to p r e s e n t every 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 a s a linear c o m b i n a t i o n o f 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 where t h e coefficients a r e certain distinguished a l m o s t quasihomogeneous distribut i o n s . F o r a s i m p l e r e m a r k i l l u s t r a t i n g t h i s w e s u p p o s e t h a t w e c a n f i x a distrib u t i o n u € B ' ( 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 0 w i t h deficiency 1 . T h e said r e m a r k s h o w s I 0 h o w to c o n s t r u c t f u r t h e r 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 a n d 20 t h a t in case " s i n g s u p p u = # " every 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 o n X of o r d e r 5 I h a s a " q u a s i h o m o g e n e o u s p a r t " f r o m w h i c h it c a n be r e c o n s t r u c t e d .
Remark 2.43. Let d 6 B a ' ( X Ibe quasihomogeneous o f degree m . Suppose that the singular supports o f d and v do not intersect so that v d is a well-defined distribution on X . Moreover, let T, u 6 3 ' I X ) be distributions defining each other bj the equation
(2.I#)
T=u+vd
Then under the preceding h.vpothesis on u T is almost quasihomogeneous o f degree m with deficiency d i f and on/) i f u is quasihomogeneous o f degree m .
mf. From
(2.18) o n e d e d u c e s by t h e Leibniz r u l e t h a t
C o n s e q u e n t l y , ( a M - m ) T = d if a n d o n l y if ( d M - m ) u = O . rn
U n d e r w h a t c o n d i t i o n s o n X do d i s t r i b u t i o n s u s a t i s f y i n g t h e a s s u m p t i o n s o f Re-
102
11.
( A l m o s t ) Quasihomogeneous Distributions
m a r k 2.43 e x i s t ? C o m b i n i n g R e m a r k 1.52 w i t h T h e o r e m 3.39 b e l o w o n e sees t h a t s u c h a u having t h e a d d i t i o n a l p r o p e r t y “ s i n g s u p p u = 0
”
e x i s t s if a n d o n l y if X
is l o c a l l y M - b o u n d e d (see Definition 3.30 a n d P r o p o s i t i o n 3.31 b e l o w ) . M o r e o v e r ,
if (1.14) h o l d s t h e n s u c h a u a l w a y s e x i s t s w i t h s i n g u l a r s u p p o r t c o n t a i n e d in
X \ X,
:
to see t h i s o n e h a s to c o m b i n e P r o p o s i t i o n 1.70 w i t h t h e f o l l o w i n g corol-
l a r y to P r o p o s i t i o n 1.91.
Remark 2.44. Suppose that
( 1 . 1 4 ) holds. I f x : X + + l O . + ~ C
quasihomogeneous o f degree I then TIog e\tends
is continuous and
to an almost quasihomogeneous
distribution o f degree 0 with deficient) equal to the constant function I .
I
U n d e r s t r o n g e r a s s u m p t i o n s o n u t h e a s s e r t i o n of Remark 2 . 4 3 c a n b e p a r t i a l l y generalized:
Propoaltion 2.45. Suppose that v : X -
Q‘ is a C“’ function which is almost quasi-
homogeneous o f degree 0 with deficienq / . For ever)’ j c N O set vi : = v J / j ! as in Remark 1 . 5 3 . Let TED’IXI. Then (i)
T is almost quasihomogeneous of degree m and of order
there is a sequence o f distributions S , ,
05
5
N i f and on11 i f
P 5 N . which are quasihomogeneous
o f degree m such that N
T=
5 vg S g :
P=O
f i i l i f this is the case then the distributions Sg satiscying (2.19) are unique/>
determined: in f a c t , they are given b> N-P
(2.20)
S, =
f -1)
vi Id,
T
O5P3N:
i =0
moreover. f o r ever,. k C N , we have N
f2.lY)k
I-=
vk+g
sp).
P=O
Proof. f i ) , “ e ”t : h i s is a c o n s e q u e n c e of Remark 1.53 a n d P r o p o s i t i o n 2.35 ( i i ) and ( i l . ’*+”: N o t e t h a t R e m a r k 1.53 a n d P r o p o s i t i o n 2.31.(ii) imply
(2.21)
To prove u n i q u e n e s s w e d e d u c e f r o m (2.19) m a k i n g use of Lemma 2.34
103
2.d 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
I n s e r t i n g t h i s f o r k = P + i i n t o t h e r i g h t - h a n d side of ( 2 . 2 0 ) a n d c h a n g i n g t h e order of summation w e obtain
Taking (2.22)
wi WI-i
=
(i)
O
WI,
f o r I = L - P i n t o a c c o u n t w e d e d u c e t h a t t h e t e r m in s q u a r e b r a c k e t s is e q u a l to ( l - l ) L - p w L - p = S L p , i . e . ( 2 . 2 0 ) is valid. C o n v e r s e l y , if w e d e f i n e S p by ( 2 . 2 0 ) w e d e d u c e by L e m m a 2 . 3 4 a n d ( 2 . 2 1 ) t h a t ( d M - m ) S o is e q u a l to
By t h e a s s u m p t i o n o n T t h i s i m p l i e s t h a t
SO
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 .
In view o f C o r o l l a r y 2 . 3 3 . ( i ) a n d P r o p o s i t i o n 2.31.(ii) w e may a p p l y t h i s a r g u m e n t to ( d M - m ) ' T i n s t e a d o f T . H e n c e S p 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 f o r every
P E N ,as w e l l . Applying Lemma 2.34 a n d (2.21),i n s e r t i n g (2.20),s u b s t i t u t i n g I = P + i , 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 t a k i n g ( 2 . 2 2 ) i n t o a c c o u n t w e o b t a i n f o r e v e r y k € l N o that N
(dM-m)k(
=
c
Uk+pSe) =
N V p s p =
e=o
N I 2 ( - 1 ) ' W i ( d ~ - m ) ~ + ' Tc = ( 2
~ - e
N
P=O
s e=o
Up
l=n
(l) ( - l ) l - e ) U ~( 3 M - m ) T . I
i=n e=n
S i n c e by t h e binomial f o r m u l a t h e t e r m in b r a c k e t s is e q u a l to ( 1 - 1 ) '
the right-
h a n d side is e q u a l to T , i.e. t h e e q u a t i o n ( 2 . 1 9 ) k is p r o v e d . rn
N o t e t h a t v : = log s a t i s f i e s t h e a s s u m p t i o n s of P r o p o s i t i o n 2 . 4 5 in case M = Id, and
X = I O , + c o C . H e n c e , c o m b i n i n g P r o p o s i t i o n 2.45 a n d Lemma 2.12 o n e o b t a i n s
Corollary 2.46. A basis f o r rhe space o f distributions T E B ' l l O , + ~ C )which are almost homogeneous o f degree m and o f order t
H
t m c ~ i l t I ,O l j S N . m
S
N is given by the functions
104
11.
( A l m o s t ) Quasihomogeneous
Dlstributions
Next w e formulate a variant of Proposition 1 . 5 8 .
Proposition 2.47. Suppose that a = U + . Let u E & ' ( V ) ( o r more generally D);7(V)) be almost quasihomogeneous o f degree m . Then there is a polynomial function Q: V * +C which is almost quasihomogeneous o f degree m* and type
M * such
that u = Qfd) 8 , .
PrOofF. In view of Proposition 2 . 4 0 . ( i ) f : =
6
satisfies the assumptions o f Proposi-
tion 1.58 with M replaced by M * so that the assertion f o l l o w s . w
As another consequence
of Proposition 1.58 one can s t a t e
Propodtion 2.48. Suppose that ( 1 . 1 4 ) holds. Let q : V ' --+ C' be a hjpoelliptic pol-),nomial Function. Then the Following assertions hold. (i)
Every distribution u €a)'( V ) which is almost quasihomogeneous o f degree m
such that q l D l u E C ~ ( Vis) a C"Function
being a pol-lnomial with respect ro
the variables in GMMlo+); moreover, the almost quasihomogeneit) order of
11
is
bounded b) a constant depending on m . onlj; in particular, iF M is semi-simple then u is quasihomogeneous. fii)
IF M
is semi-simple then For ever) T6.3'IV) all the almost quasihomogeneous
solutions S E a ) ' ( V ) OF degree m o f the equation q ( D ) S = T have one and the same deficienc) .
8
Finally, w e c o m e back to Example 2.27:
Remark 2.49. Under the assumptions OF Esample 2.27 the Following holds: (i)
Bernstein's fundamental solution E p OF P I D ) is almost quasihomogeneous
o f degree P - p and OF order i - o r d ( - l ; $ j ) ; its Is'
order deFicienc) is equal t o
- l a - , f - l ; $ g ) , it is a solution OF the equation P f D ) u = O . ( i i ) i f P is hypoelliptic and i F (1.14) is valid then the deficiency OF Ep is a C'"
Function which is a polynomial with respect t o the variables f r o m G M ( o +I; in particular, i f
t
< q then Ep is quasihomogeneous o f degree 8-p .
2.e
105
Merornorphic Functions of Almost Q u a s i h o r n o g e n e o u s D i s t r i b u t i o n s
-ProoF. f i ) : t h i s is a reformulation of part of t h e assertion ( i i ) of Example 2.27.
(ii):
I t f o l l o w s from Example 2.27.(ii) and Corollary 2.33.(i) t h a t u:= a - l ( - l ; p )
is a l m o s t quasihomogeneous of degree P - V of o r d e r < - o r d ( - l ;
Proposition 2 . 4 8 . ( i ) implies t h a t a - l ( - l ; p ) is a C"
p ) .Since
P(D)u = 0
function, being a polynomial
with r e s p e c t to t h e variables f r o m G M ( b + 1 which m u s t vanish if P - p < 0 .
In general, Ep need n o t be quasihomogeneous; in f a c t , i t s quasihomogeneity o r d e r can be arbitrarily large:
Example 2.50. Let k E 3
Q(s.tl := - s - - t -
3
N be such
that n = 2 k
.
and suppose that
M=Idv.
and P : = Q @... @ Q ( h t i m e s ) . Then EO is almost homogeneous
o f order 1 but not homogeneous (see e . g . E.\arnple 4.13 in Grudzinshi I 7 1 1 ord(-l;
p3) = - k .
Set
so that
I
te) Mei-omor-phi<:Functions of' A l m o s l Quaslhomogvntwus Distr*ibuLions
In this section, in analogy to Proposition 2.25 we describe meromorphic functions
of a l m o s t quasihomogeneous distributions by their Laurent series. To this end we l e t fl be a connected open s u b s e t of C , let h : n +
% ' ( X ) b e a meromorphic
function, and let z o E n . Moreover, we fix N E N , a € @ , and b E C .
Propooltion 2.Si. The Following conditions are equivalent: l a ) h f z l i s almost quasihomogeneous o f degree a z + b and of order 5 N For every
ten\ ({zolu$W(h3)); f b ) for every k 6 E we have N +I
(2.233)
akfzo;h) =
-1 (N + I)
(t)M-azo-b3)iak+i(zo;h3)
i= 1
proOf. Applying t h e binomial formula to t h e s u m
106
( A l m o s t ) Quasihornoneneous
11.
Distributions
- (az-az,)
aM-az-b = (aM-azo-b)
of commuting o p e r a t o r s we obtain f o r every Z E n \ v o l ( h )
Obviously, f o r every i EN,
t h e function g i : n-+
a ' ( X ) defined by
fl\'p)ol(h)3 2 H ( a M - a z , - b ) ' h ( z ) is rneromorphic, and from (2.10) one deduces t h a t
a k ( z o ; g i )= ( a M - a Z , - b ) ' a k ( Z , ; h ) ,
I t follows t h a t by
-
n\CpoL(h) 3 z H ( d M - a z - b ) N + l h ( z ) a meromor-phic function h N + ,
:
n
a'(x )
is defined s u c h t h a t
Since a meromorphic function o n t h e connected open s e t Q vanishes identically if and only if its Laurent expansion a t z,
d o e s so t h e condition ( b ) is equivalent
to t h e condition
-az-b)"'h(z)
=0,
2
E Q \ (Cpol ( h ) u
(2,
) ),
which in view of Proposition 2.31 is equivalent t o ( a ) .
In c o n t r a s t to meromorphic functions of quasihomogeneous distributions t h e Laurent coefficients of meromorphic functions of a l m o s t quasihomogeneous distributions d o n o t have new invariance properties with respect to t h e action of M,. In f a c t , they a r e a l m o s t quasihomogeneous:
Remewk 2.52. If o n e ( a n d hence b o t h ) of t h e conditions of Proposition -7.51 a r e satisfied then for everj k € Z s a t i s f , i n g k 1 v : = ord(z,;h)
t h e distribution a k ( z , ; h l
is a l m o s t quasihomogeneous of degree a z , + b a n d of order
A-oof. Employing
5
N+k-v.
(2.23) with k replaced by k - N - 1 . multiplying by ( - a ) N + i , re-
arranging t h e s u m m a n d s , and applying ( d M - a z o - b ) k - w w e obtain
2.e
Merornorphic F u n c t i o n s of A l m o s t Q u a s i h o m o g e n e o u s
Distributions
107
S i n c e N + ( k - N - l + i ) - w < k-w+i a n d k - N - l + i 5 k-1 f o r i C N it f o l l o w s by i n d u c t i o n o n k - starting with k = w - t h a t (aM-az,
+)N+k-u+l
ak(zo ;h ) = 0 ,
k€Z. m
If a c e r t a i n p a r t o f t h e L a u r e n t series o f h a t zo v a n i s h e s t h e n t h e c o n d i t i o n ( 2 . 2 3 ) t a k e s on a much simpler form f o r certain values of k :
Propositlon 2.53. Let j C Z such that (2.24)
a,-i(z,;h)=O
f o r ever) i C N N .
Then for k = P : = j - N - 1 the condition (2.23) amounts to ap(z,;h) = - (-a)-N-' ( ~ M - a z , - b ) N ' l a i ( ~ o ; h ) ,
(2.2.5I
and the fol l owi n g conditions are equivalent: ( a ) the condition (2.23) holds for ever) k C Z such that X cf': ( b ) for every iCINo we have (2.26)
a p - i (2,
:h )
'Ti)a - ' ~ d M - a z o - b ) ' a p ( z o : h ~ :
=(
Proof. T h e f i r s t part of t h e a s s e r t i o n is o b v i o u s . For t h e r e s t o f t h e p r o o f t h e f o l l o w i n g e q u a l i t y is r e q u i r e d . I t is o b t a i n e d by a n a p p l i c a t i o n of L e m m a 1.76 t o ( j , k , P )= ( L - l , N . N + t ) : rnin{N+l.L)
(2.27)
L i =1
N
LEN.
( a ) = + ( b ) :f o r i = O t h e c o n d i t i o n ( 2 . 2 6 ) is trivially valid. For i ? l ( 2 . 2 6 ) i s p r o v e d by i n d u c t i o n : f o r I E N appylying ( 2 . 2 3 ) to k = P - 1 a n d t a k i n g ( 2 . 2 4 ) i n t o a c c o u n t one obtains
108
II. ( 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
By t h e i n d u c t i o n h y p o t h e s i s t h i s is e q u a l to
S i n c e by ( 2 . 2 7 ) t h e t e r m in b r a c k e t s is e q u a l to
t h e c o n d i t i o n ( 2 . 2 6 ) is
valid f o r i = I . We fix k € E s u c h t h a t k < P . As a n i m m e d i a t e c o n s e q u e n c e of ( 2 . 2 4 )
(b)+(a):
a n d ( 2 . 2 6 ) ( a p p l i e d to i = t - k - l ) w e o b t a i n
c(
niin N + I . O
N+I
N+ I l ) ( - a ) ~ ' ( a M - a z o - b ) l a k + l ( z o ; h=)
minl N + I , P - k
c
.... I=l
I=l
=
-k )
)
(N;1)
( -1) -I
a - I- ( e -
k - I)
(
N + P-k-1) N
(
aM
- a zo- b ) I +( e - k -
I)
a p ( z o ;h ) .
1=1
Applying ( 2 . 2 7 ) to L = P - k w e see t h a t t h i s is e q u a l to -a-Y+k
(NtA-k) ( a M - a z o - b )
Y-k
ap(zo;h ) .
In view o f ( 2 . 2 6 ) ( f o r i = 0 - k ) t h i s , in t u r n , e q u a l s - a k ( z 0 ; h ) . a s desired ( b ) * ( c ) ; Since (
N i i )"$I)/ (
=
t h e c o n d i t i o n ( 2 . 2 6 ) ' is a c o m b i n a t i o n o f
(2.26) f o r i and i - 1 .
(c)*lb)
;
t h i s i m m e d i a t e l y f o l l o w s by i n d u c t i o n .
H
Of c o u r s e , o n c e o n e k n o w s t h a t t h e c o n d i t i o n s of P r o p o s i t i o n 2.51 are s a t i s f i e d
a n d h e n c e f o r e a c h k € Z t h e L a u r e n t c o e f f i c i e n t a k ( z o ; h ) is a l m o s t q u a s i h o m o g e n e o u s t h e n in order to e x h i b i t t h e p r e c i s e b e h a v i o u r of a k ( z o ; h ) u n d e r t h e a c t i o n of M, o n e w a n t s to d e t e r m i n e its d e f i c i e n c i e s , i.e. - b y
d i s t r i b u t i o n s (3, - a z , - b ) ' a k ( z , ; h ) .
P r o p o s i t i o n 2.31 - t h e
I f h ( z ) is e v e n q u a s i h o m o g e n e o u s o f d e g r e e
a z + b f o r every z ~ n \ ~ o l ( ht h)e n P r o p o s i t i o n 2.25 s a y s t h a t o n e j u s t h a s to c o m p u t e c e r t a i n o t h e r L a u r e n t c o e f f i c i e n t s w i t h o u t having to b o t h e r a b o u t d e r i v a t i v e s . H o w e v e r , in g e n e r a l , c o m p u t i n g derivatives s e e m s to be u n a v o i d a b l e as a l o o k a t ( 2 . 2 3 ) s h o w s . N e v e r t h e l e s s , if j € Z is s u c h t h a t ( 2 . 2 4 ) h o l d s t h e n f o r k = j o n e c a n do w i t h o u t d e r i v a t i v e s a t l e a s t f o r s u f f i c i e n t l y large I by c o m b i n i n g ( 2 . 2 5 ) and (2.26):
2.e
M c r o m o r p h i c F u n c t i o n s of A l m o s t Q u a s i h o m o g e n e o u s
Distributions
109
By c o m b i n i n g ( 2 . 2 3 ) f o r k = j - I w i t h ( 2 . 2 4 ) a n d s u b s t i t u t i n g i f o r i-I o n e o b t a i n s
H o w e v e r , h e r e t h e a b o v e m e n t i o n e d d i f f i c u l t y c o n t i n u e s to a p p e a r as o n e h a s to k n o w n o t o n l y t h e d i s t r i b u t i o n s a k ( z o ; h ) f o r j < k 5 j + N , b u t also t h e i r derivatives u p to t h e order N + l . S o m e t i m e s it is p o s s i b l e to c i r c u m v e n t t h i s d i f f i c u l t y . Namely w h e n o n e is a b l e to c o m p u t e t h e L a u r e n t c o e f f i c i e n t s o f t h e m e r o m o r p h i c f u n c t i o n s n\VOt(h)3z
In f a c t , t h e l e f t - h a n d side o f ( 2 . 2 9 ) c a n b e de-
H(dM-az-b)'h(z).
scribed in t e r m s o f t h e i r L a u r e n t c o e f f i c i e n t s . a s f o l l o w s .
Remark 2 3 . Suppose that one (and hence e ac h) o f t h e conditions o f Proposition 2.51 is valid. Moreover. let j 6 Z b e such that 12.24) holds. Then for ever)
I E N , we have ( 3 , -a.z,- b)' ai
12,
:h )
= ai
(2,.
h, )
where h , :fl -+ . 3 ' ( X ) de note s the meromorphic function defined bj
proOf. A s a t t h e b e g i n n i n g of t h e p r o o f of P r o p o s i t i o n 2.51 w e a p p l y t h e b i n o m i a l f o r m u l a to o b t a i n f o r every z € n \ V o l ( h ) I
hI(z)=
(1) ( - a ) ' - i
(dM-azO-b)'h(z)
i=O
so t h a t
If k = j a n d I c N N t h e n ( 2 . 2 4 ) i m p l i e s t h a t o n t h e r i g h t - h a n d side o f (2.31) t h e s u m m a n d s i n d e x e d by i < I v a n i s h . H e n c e t h e a s s e r t i o n is p r o v e d .
H
N o w , in c o n c l u s i o n w e c o m e back to E x a m p l e 2 . 2 7 . W e w o u l d l i k e to w e a k e n t h e a s s u m p t i o n in t h a t w e a s s u m e of P to be a l m o s t q u a s i h o m o g e n e o u s , o n l y . H e r e t h e f o l l o w i n g a n a l o g u e o f R e m a r k 2.24 will be u s e f u l .
110
( A l m o s t ) Quasihomogeneous
11.
Remark 2.24'. Let g:n+
Distributions
C be a holomorphic function. Suppose that there is
an uncountable subset D o f 0 such that h ( z ) is almost quasihomogeneous of degree g ( z ) f o r every Z E D . Then there is N E N such that h ( z ) is almost quasihomogeneous o f degree g ( z ) and of order
_<
N for every
z €
n \ POL( h l .
b f . For every N E N , by h N ( Z ) : = ( d M - g ( Z ) ) N + l h ( Z ) , z e f l \ p o l ( h ) , a m e r o m o r phic function h N : f l - - + 3 ' ( X )
is d e f i n e d . Let
of c o m p a c t s u b s e t s o f 0 s u c h t h a t of t h e sets DN : = { z E K N
;
( K i ) j E ~be
UjEINKi = fl . S i n c e D
an increasing sequence
is c o n t a i n e d in t h e union
h,(z) = 01, N E I N , a n d s i n c e D is u n c o u n t a b l e w e f i n d
s o m e N E N s u c h t h a t DN is u n c o u n t a b l e . 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 it f o l l o w s t h a t hN v a n i s h e s identically.
T h e f o l l o w i n g e x a m p l e s h o w s t h a t in c a s e P is only 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 P b u t n o t q u a s i h o m o g e n e o u s t h e n p ( z ) is n o t n e c e s s a r i l y a l m o s t q u a s i h o m o g e n e o u s w h e r e 1, is d e f i n e d in t h e t e x t preceding E x a m p l e 2 . 2 7 .
Example 2-55. Let P : v'+
co.+w[
be a non-negative pol)nomia/ function which
is almost quasihomogeneous OF degree P E C and o f order
11.
Then f o r arbitrarj
,k€N and Z E C such that Rez > 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 = (TI<+, * ; (iii) T,, = T if and onlj if T is @-invariant.
proOf. T h e a s s e r t i o n s ( i ) a n d ( i i i ) of P r o p o s i t i o n 2.50 imply t h a t by
c o n t i n u o u s linear m a p C T ( K ) m a 2 . 5 8 . ( i i ) T,
+C g ( K G ) is w e l l - d e f i n e d .
'p H ' p C - 0
a
H e n c e in view of Lem-
is w e l l - d e f i n e d a n d b e l o n g s to a ' ( X ) . T h a t T,
is @ - i n v a r i a n t
f o l l o w s f r o m t h e @ ' - i n v a r i a n c e of y a - or f r o m ( 2 . 3 5 ) .
li): If
v E C F ( X ) is s u c h t h a t s u p p ? n n w p p T ) , = @
s u p p 'p oA = A-'
(
supp
'p )
does n o t i n t e r s e c t s u p p T
~
then
for
every
AEG
i.e. ( s u p p 'p ) G n s u p p T = @ ,
a n d 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 2.50. ( i ) .
(ii):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 2 . 6 2 . M o r e o v e r . u s i n g (2.34) we deduce t h a t
115
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
(iiil:
In view of pG(G)=l t h i s f o l l o w s f r o m (2.35).
T h e a n a l o g u e of P r o p o s i t i o n 2.01 h o l d s :
Propoeltlon 2.65. Let
5
and .$ be as in Proposition -7.61. Then
(i)
P ( d ) T e = ( P ( d I T ) ( G + co) f o r ever) .$'.-invariant poljnomial function P : V * - + Q ' ;
(iil
q T @ = ( q T ) ( , , , , ) for ever) $-invariant Cmfiinction q . X + C
m f . F o r a r b i t r a r y ' p € C T ( X ) o n e o b t a i n s by P r o p o s i t i o n 2.01 t h a t
Finally, w e e x a m i n e w h e n q u a s i h o m o g e n e i t y is p r e s e r v e d by t h e o p e r a t i o n T HTc+,.
Remark 2.66. Let A E GL ( V. V ). Then (i)
A commutes with M i f and onlj i f
(-7.37) lii)
AoM, = M,oA
f o r every t € 1 0 . + w C :
i f M is o f the f o r m (1.l.a) then this is the case i f and on/), i f A i j = O f o r
arbitrar), i, j
6 N,,
satisf)ing p i
f
pi .
proof. (i): "j" is c l e a r f r o m t h e p o w e r s e r i e s p r e s e n t a t i o n of e x p ( ( log t ) M ) ,
x' is o b t a i n e d lii):
through differentiating (2.37) a t t = l a n d making use of ( 1 , s ) .
F o r a r b i t r a r y i , j E N n w e have: ( A o M ) i i = A i i p ja n d ( M O A )=~p ~i A i j ,
N o t e t h a t A - ' s a t i s f i e s ( 2 . 3 7 ) if A does so. M o r e o v e r , ( 2 . 3 7 ) r e m a i n s valid if ( M , A ) is r e p l a c e d by ( A * , M * ) .
Remark 2.67. l i ) Suppose that AEGL(V. V ) commutes with M . Then (-7.38)
3 , ( T OA ) = ( 3 , T ) 0 A ,
TE S'f X ) :
116
consequently,
11.
if
( A l m o s t ) Quasihomogeneous Distributions
T E a ' ( X 1 is almost quasihomogeneous of degree m with deficiencj,
d c a ' ( X ) then T o A is almost quasihomogeneous of degree m with deficiency d o A : in particular, if T is quasihomogeneous of degree m so i s T o A .
(ii) Suppose that every A C G commutes with M . Then
consequently, if T E D ' I X ) is almost quasihomogeneous of degree m with deficiency d C B ' ( X 1 then T ,
is almost quasihomogeneous of degree m with deficiencj d,
in particular, if T is quasihomogeneous of degree m so is T e .
(iil:t h i s f o l l o w s from ( 2 . 3 8 ) and ( 2 . 3 5 ) . H
;
117
Chapter I11
Constructing (Almost) Quasihomogeneous Functions by Taking Quasihomogeneous Averages of Functions with M-bounded Support
S u p p o s e f r o m n o w o n t h a t X is q u a s i h o m o g e n e o u s . Similarly as in s e c t i o n 2 . ( f ) t h e s i m p l e idea f o r c o n s t r u c t i n g 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 a given f u n c t i o n f : X -+ C is to t a k e t h e a v e r a g e o f t h e f u n c t i o n s
t-ln
f 0 M , , t C l 0 , +a[, with
r e s p e c t to t h e H a a r m e a s u r e o f t h e g r o u p { M,; t C l O , + a C 1 . S i n c e u n d e r t h e c a n o nical i s o m o r p h i s m of t h i s g r o u p o n t o 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 t h e H a a r dt m e a s u r e is i n d u c e d by F, o n e is lead to i n t r o d u c e t h e f u n c t i o n f, : X -+ C f o r m a l l y given by XEX. In o r d e r to g u a r a n t e e t h a t t h e i n t e g r a l is w e l l - d e f i n e d , in t h e p r e s e n t p a r a g r a p h w e i m p o s e r a t h e r s t r o n g c o n d i t i o n s , namely t h a t f o r every X C X t h e s u p p o r t of t h e m a p d e f i n e d by t H t - m f ( M t x ) is a c o m p a c t s u b s e t o f I O , + m C ; m o r e p r e c i s e l y , w e p o s t u l a t e t h a t s u p p f is a n M-bounded subset o f X in t h e s e n s e o f Definition 3.1 b e l o w . In C h a p t e r 5 t h i s a s s u m p t i o n will be r e l a x e d . In s e c t i o n ( a ) b e l o w t h e t h e o r y of t h e f u n c t i o n s f,
i s d e v e l o p e d . For t h e s a k e
o f a p p l i c a t i o n s in C h a p t e r 4 w e p r o c e e d m o r e g e n e r a l l y by 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 w i t h r e s p e c t to m e a s u r e s o f t h e f o r m w ( t d) ti w h e r e w : IO.+coC+
C is any locally i n t e g r a b l e w e i g h t f u n c t i o n : f o r m a l l y t h i s m e a n s
that +m
(3.1)'
f,,,,(,\)
:=
J' 0
dt tCm f ( M , x ) w ( t ) t ,
X€X.
118
111. Quasihomogeneous Averages of
Functions.
Part 1
Section ( b ) deals with criteria for the M-boundedness of subsets of X . I n particular, i t turns out that for some X n o t every compact subset of X is M-bounded.
If X does not have t h i s flaw it is called locally M-bounded. Necessary and and sufficient conditions for X t o have t h i s property are given in section ( c ) . One of them is the existence of Cm f u n c t i o n s x:X+IO,+mC
which are quasi-
homogeneous of degree 1 . Another equivalent condition is the existence of a COD function
4:X +C w i t h
4o of degree 0
M-bounded support whose quasihomogeneous average
is identically equal to I .
If X is locally M-bounded then every quasihomogeneous continuous function on X can be written as the quasihomogeneous average of some continuous function f with M-bounded support (section ( d ) ) ; in fact, finitely many quasihomogeneous
functions of pairwise different degrees are described i n t h i s manner by means of a single f .
Let w:IO,+mC+@
1
be a weight function belonging to Z,oc(lO,+mC), and s e t
I : = supp w . I f f : X + @ support of the map l + C ,
is a continuous function such that for every X E X t h e
t H f ( M , x ) , is contained in a compact subinterval
of IO,+mC then a function f,,,:X+C
Jx
is well-defined by (3.1)'. In order to
be able to derive regularity properties of f,,,
we require that the intervals J,
can be chosen not to depend too wildly o n x ; in fact, we are going to postulate that locally one can choose them to be independent from x .
Deflnltion 3.1. A subset L of X is said to be an (M.1)-bounded subset of X i f and only i f for every compact subset K of X t h e set
is a relatively compact subset of 10,+00C (hence of I since I is closed in lO,+mC ) .
If I = 10,+03C one also says M-bounded instead of (M,I)-bounded. Note that i f , in addition, L is closed in X then the sets ( 3 . 2 ) are compact.
3.a
119
Introduclng Quasihomogeneous Averages
We fix r € N o u ( m ) and introduce
NotatJon 3.2. By C i ( X ) we d e n o t e t h e set of C' functions f : X-C
such t h a t
s u p p f is a n ( M , I ) - b o u n d e d s u b s e t of X . If I = l O , + m C we a l s o write C h ( X ) .
Since finite unions of ( M , I ) - b o u n d e d s u b s e t s of X a r e (M.1)-bounded themselves i t f o l l o w s t h a t CYCX) is a linear subspace of C'(X). Now we fix f € C ; ( X ) .
Propodtion 3.3. ( i ) By (3.1)' a C" function f , , , , ( i i ) The function
Gf,,
is w e l l - d e f i n e d .
C " ( X ) , m H f , , , , , is holomorphic satisfling
:C '+
kEN.
Proof.
a:If I
is c o m p a c t t h e assertion follows by t h e rule f o r differentiating
a n integral depending o n a parameter. In particular, one has ( d a f m , w ) ( x )= f t - ' n 3 a ( f O M t ) ( x ) dTt ,
(3.3)
x € X , IcLI>r.
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<
Definition 3.30. X i s c a l l e d locallj, M-bounded if a n d only if every c o m p a c t s u b s e t o f X is an M - b o u n d e d s u b s e t o f X .
Fropodtlon 3.31. Let I b e a non-compact closed subset o f 1 0 , t w C . Then the f o l lowing conditions are equivalent:
136
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
( a ) X is locally M-bounded; ( b ) every compact subset o f X is (M,I)-bounded; (c)
the condition 13.16) holds, X I : = { ( x , M , s ) ; s € X ,t 6 l ) is a closed sub-
set o f X x X , and the map X, -10,
+a[, ( x , M , x ) H t , is continuous;
( d ) for every compact subset K o f X there exists an M-bounded subset L of X such that LM 3 K
;
( e ) there is an M-bounded subset L o f X satisKving L , = X
F o r t h e p r o o f of t h e implication " ( b ) * ( a ) "
.
in c a s e I is n o t a n i n t e r v a l w e r e q u i r e
t w o lemmata, t h e first o n e elaborating o n Example 3.24.
Lemma 3.32. Suppose that oo = 0.Let
-
EG,(o-)
and
J+
E G , ( o + I , and let U,
be a neighbourhood o f )_+ . Then there e\ists a ~ 1 0+a[ . such that M , ( U -I n U + # Q f o r every t€Ca,+wC.In particular, there is a compact subset K o f UM such that
M , ( K l n U # @ f o r ever) t € l O , + w l where U : = U + u U _ .
pI-oof. W e fix E > 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
proOf. F i r s t o f a l l w e r e p l a c e U by a n o p e n c o v e r i n g ?B o f Y : =
uU
( V,
I,.
such that for
every W E % t h e r e is U E U s a t i s f y i n g W C U . T h e n w e f i x a s e q u e n c e o f c o m p a c t m € N . s a t i s f y i n g ( 3 . 2 2 ) . F o r every m € l N w e d e f i n e L,, by ( 3 . 2 3 ) . F o r every y € Y w e c h o o s e W, E ' W a n d E, > 0 s a t i s f y i n g K ( y , 2 ~ C ~W,. ) S i n c e
subsets X,,
u
W,
= X a n d s i n c e L,
is c o m p a c t w e f i n d a f i n i t e s u b s e t A,
of l O . + 0 0 C X Y
such that
By t h e a s s u m p t i o n a n d by Lemma 3.9 it f o l l o w s t h a t ( X m - , ) M is a c l o s e d s u b s e t of X . H e n c e Vm,a is a n o p e n n e i g h b o u r h o o d o f t h e c o m p a c t s u b s e t Km,,. S i n c e ( K m , a ) M = ( M , , , ( K ( y , €,))nL,), it f o l l o w s f r o m ( 3 . 3 3 ) a n d ( 3 . 2 4 ) t h a t
U
U
( K m , a ) ~3
m € N a€A,
U
(L,),=X.
mE N 0
S i n c e ( V m , a ) M C Z , ~ : = ( X m + 2 ) M \ ( X , , - ~ ) M s, i n c e { k E N ; Z , n Z , f @ } t a i n e d in t h e set { k € N ; m-2 5 k 5 m + 2 } , a n d s i n c e t h e sets A,,
is c o n -
mE N , a r e finite
144
1 1 1 . 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 I
i t f o l l o w s t h a t e v e r y set ( V m , a ) ~i n t e r s e c t s w i t h finitely many of t h e sets ( V k , b ) M ,
k € N a n d bCA,,
o n l y . S i n c e t h e i n d e x set
UIneN { m ) xA,
is c o u n t a b l e w e c a n
replace it b y N a n d a r r i v e a t t h e desired a s s e r t i o n . rn
V,
Proof o f Theorem 3.39. ( a ) + ( c ) ' : W e c h o o s e K,, T h e n w e f i x rp,EC;;)(V,) set
xu
:=
and U ,
s u c h t h a t q m t 0 a n d q m - l o n K,
a s in L e m m a 3 . 4 0 . a n d for every U E U
c
u k =LI q k
uk=u}
S i n c e b y (3.32) f o r e v e r y n i E l N t h e s e t I L I , , : = { k E I N ; ( v k ) M n ( v , , , ) M + @ , is f i n i t e i t f o l l o w s t h a t f o r every m € N t h e s u m is f i n i t e o n ( V m ) ,
so t h a t
xu
is a w e l l - d e f i n e d C m f u n c t i o n o n X s a t i s f y i n g
SUPP xu
(vm)M
u I\ 6
s'PP~k~
ILI ,111
S i n c e t h e r i g h t - h a n d s i d e is c o m p a c t t h e c o n d i t i o n ( a ) i m p l i e s t h a t it is a n M - b o u n -
ded s u b s e t of X . H e n c e Remark 3 . 2 8 . ( i i ) ( a p p l i e d to t h e c o v e r i n g c o n s i s t i n g of i m p l i e s t h a t s u p p x L I is a n M - b o u n d e d s u b s e t of X . M o r e o v e r ,
t h e sets (V,)M)
f r o m ( 3 . 3 0 ) it f o l l o w s t h a t s u p p xu C U . Finally, f o r every k E N s a t i s f y i n g
U k = U w e have
xu
L Cpk 2 0 so
that
which is p o s i t i v e o n ( K k ) M .
1
Consequently,
(xLl)o
q :=
u (Kk)M = X
is positive o n
u e 11
( s e e (3.31))
kPN
S i n c e by P r o p o s i t i o n 3.10 s u p p ( ~ is~ c~o n)t a~i n e d in U,
a n d s i n c e ( U M I L I G u is
locally f i n i t e q is a w e l l - d e f i n e d C m f u n c t i o n o n X 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 0 . H e n c e in view of C o r o l l a r y 3.5 t h e f u n c t i o n s QL1 : = x u / q
have t h e
desired properties. ( c ) ' * ( c ) : we apply ( c ) ' to U = { X ) .
(c)* (c)": is trivial.
( c ) " * ( a ) : If ( J , K ) O # 0 o n K t h e n ( s u P P J , ~ 3) ~K . 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 t h e i m p l i c a t i o n " ( d )= + ( a )
Ic)
*( b )
:
"
in P r o p o s i t i o n 3.31.
Let J, be a s in ( c ) . T h e n by P r o p o s i t i o n 3.13 x : = J,, is a C'" 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 1 . I f
JI
is n o n - n e g a t i v e t h e n x is p o s i t i v e
o n t h e set ( J I - ' ( l O , + ~ C ) ) M which in view of t h e c o n d i t i o n " J l 0 = l " c o i n c i d e s with X .
To s h o w t h a t w e may a s s u m e t h a t J, is n o n - n e g a t i v e w e f i r s t observe t h a t J, c a n
145
3.c W h e n is Every C o m p a c t S u b s e t of X M - b o u n d e d ?
be replaced by Re$ so that we may suppose that J, is real-valued. Then we set
x := J,2. Since the condition Jlo = I implies that xo is positive everywhere the function Y : = x / x o is non-negative and belongs to CGCX). Moreover, since by "
xo
Proposition 3.13
Yo
=1 ,
"
is quasihomogeneous of degree 0 Corollary 3.5 implies that
indeed.
(bj+fa)
: since
x is continuous the set x ( K ) is compact in 10,+00C for every com-
pact subset K of X . Hence t h e assertion follows by Lemma 3.29. m
In view of Notation 1.05 and Lemma 1.04 the following assertion is a consequence of Theorem 3.39.
Corollary 3.41. Everj point in P ( M I hood which i s locall-s M-bounded.
possesses
a quasihoinogeneous open neighbour-
I
We are now ready to prove the following supplement t o Proposition 3.31.
Propodtion 3.42. Suppose that I is a non-compact closed subset o f 1 0 , + m y . Then X i s 1ocall.s M-bounded i E and on!,, i f everj' point o f X is ( M . 1 ) - b o u n d e d in X and ever, ( M , I I - b o u n d e d subset o f X has an open neighbourhood which is an ( M . I ) - b o u n d e d s u bs et o f X . a s w e ll. Proof.
E'; The
assumptions mean that every point of X has an (M.1) -bounded
open neighbourhood. Since every compact subset of X is covered by finitely many of such neighbourhoods it is (M.1)-bounded as finite unions of ( M , I ) - b o u n ded subsets of X are (M.1)-bounded. " j "We :
choose a continuous function x as in Theorem 3.39.(b) and by Lemma 3.40
fix a sequence o f relatively compact open subsets V,
(3.34)
u
(vM)M
of X satisfying (3.32) and
=x.
M€N
Now, let L be an ( M , I ) -bounded subset of X . For every m e "
-
]1/2.2C.x(Ln ( V m ) M ) is an open neighbourhood of x ( L n
the s e t I , : =
(K)M) in IO,+00C.
By Lemma 3.29 and in view of I n JI,, C 1 n ( J . E 1 / 2 , 2 l . x ( L n ( c ) M ) ) the s e t
I n J I M is a relatively compact subset of I for every compact subset J of 10,+00C. We set
146
111. O u a s i h o m o e e n e o u s A v e r a e e s of F u n c t i o n s . Part 1
u ~-'(l,)~(vm)M.
u:=
rnEN
Let K be any compact subset of X . Then one finds a finite subset A of IN such that
K C
U (Ve)M. OEA
Moreover, let J be a compact subset of I O , + a [ . Then
InJx(UnKM)
C
u
InJx(Un(V,),)
C
@€A
C
u u
4 E A
InJx(x-'(l,)n(V,),n(V,)~)
C
m€Bp
u u
O E A
InJ1,
mEBp
where B p : = { m E l N ; ( V m ) M n ( V e ) M # @ } . Since by (3.32) the sets B e , @ € I N ,are finite it follows that I n J x ( Un K M ) is relatively compact in I . Hence, by Lemma 3.2')
U is an (M,I)-bounded subset of X . Since U contains the set U m e N L n ( V , , l ) , which is equal to L by (3.34) we see that U is the desired open neighbourhood of L .
m
In the proof of the implication " ( c ) * ( b ) " of Theorem 3.3') the function x is explicitly constructed out of
+.
Here is a different way to do this:
Remark 3.43. Let $ E C , & ( X ) be such that (cl0 = I . Then b-) (3.351
x .' = e s p 0 ( - Go, l o g 1
a nowhere vanishing C I' function x : X neous o f degree 1
;
+c' i s
defined which i s quasihomoge-
x is positive everj.where i f $ i s real-valued.
proOf. That x is a C" function follows from Proposition 3 . 3 . ( i ) . A s a special case of (3.11) we note that
Since J l o = l it follows by Remark l.SZ.(i) that x is quasihomogeneous of degree 1 , indeed.
The preceding relation between x and $ will be relevant i n Chapter 8 below. Here we use it i n combination with Theorem 3.30 to derive the following supplement t o Lemma 3.29.
147
3.c W h e n is Every C o m p a c t Subset of X M - b o u n d e d 7
Lemma 3.44. Suppose that X is locally M-bounded. Let I be a non-compact proper closed subinterval o f I O . + a C , and let L be a subset o f X . Then L is an
( M , I I -bounded subset o f X iE and only i f there is a C ^ function x : X
+lo9+a[
which is quasihomogeneous o f degree 1 such that x ( L )n I is a relatively compact subset o f IO,+wC.
A part of the proof is formulated as
Lemma3.45. Suppose that X is locallj, M-bounded. Let I be a non-compact closed subinterval o f 10.+~mC. and let L be an (M.1)-bounded subset o f X . Then there exists an M-bounded open subset Y o f X such that Y M = X and
proOf. By Lemma 3.40 we choose a sequence (V,),,, implication
"j" in Proposition
as in the proof of the
3.42.
C a s e l : I = C a , + d for some a > O . Since L is (M,Cl/a,+mC)-bounded, a s well, and
-
since for every mCN V,
is a compact subset of X w e find E,EIO,II such that
{ s E I O , a l ; M , ( L ) n K , # @ ) C C ~ , , a , a l . It follows that { t € l O , + a C ; M , ( L ) nY,#
(3.38) where Y,
:=
M,,,,(V,),
me".
@}
C
Ca,+coC
Since X is locally M-bounded the sets , Y
are
(M.1)-bounded open subsets of X . Since i n view of (3.32) and (3.34) the covering consisting of the sets ( V m ) M satisfies the requirements of Remark 3.28.(i) it fol-
l o w s that Y : = U r n G N Y m is an M-bounded subset of X . By (3.34) and (3.38) Y has the desired properties.
Case?: I = I O , b l for some b € l O , + a C . Since L is ( M , I O , l / b l ) - b o u n d e d , as well, one finds ~ , E l O , l l such that { s € C b , + m C ;M , ( L ) n ~ , # Q ) } C C b , b / E m l . Consequently, (3.38)' where Y,
{ tE I O , + a C ; M , ( L ) nY, : = M,,(V,).
me".
#
@ } c 10, b l
Similarly as in Case 1 it follows that the set
Y : = U m E N Y m has the desired properties. w Proof o f Lemma 3.44 . T h e implication
'='* is a
special case of Lemma 3.29. To
148
prove
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
'3" we
choose Y as i n Lemma 3.45, and by Theorem 3.39 we fix $ E C ~ ( X )
w i t h support contained i n Y such that J, t 0 and JlO=I. Then we define x by (3.35). I n view of what was said a t the beginning of section ( b ) we may assume that 1 = 1 0 , 1 1 or I = C l , + m C . If I = l O , 1 1 ( r e s p . C l , + ~ C1 then by (3.37) it follows that -J,o, log t 0 (resp. 5 0 ) on L so that x t 1 (resp. 5 I mark 3.43 the proof is complete.
)
on L . I n view of Re-
H
Throughout t h i s section we suppose that X is locally M-bounded so that each of the conditions of Theorem 3.30 is valid. Then families of functions $ K , as re-
quired in Proposition 3.14, exist. I n fact, one can find a single non-negative funct i o n $ E C G ( X ) s u c h that $ , , = I
o n the whole of X . Moreover, choosing any family
U of open subsets of X such that the sets U,
, U E U , constitute a locally f i n i t e
covering of X one can fix a family ($u)ucuin C m ( X ) having the properties of condition ( c ) ' in Theorem 3.30. With these data one obtains the following variant
of Proposition 3.14.
Propodtion 3.46. Let qE C O I X ) . Then q is quasihomogeneous of degree m i f and only i f
s
4 =
lI€ll
($11
q),,
'
Here the sum is locall-1. finite since supp ( $ L I q ) , , , C U,
for every U E l l . rn
A n immediate consequence is
Theorem 3.47. Let r E N o u I m l . and let q E C ' ( X ) be quasihomogeneous of degree m. Then there exists a family ( f u ) u E 1 l in C,&;;tX) such that s u p p f U C U for every U E l l and q =
.y (fUjm. rn
U E 11
149
3.d 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 are Q u a s i h o m o g e n e o u s Averages
The following r e s u l t s h o w s t h a t finitely many ( a l m o s t ) quasihomogeneous functions of pairwise different degrees can b e described a s quasihomogeneous averages of a s i n g l e function.
Theorem 3.48. Let A be a finite subset o f C , and let ( j m ) m E A be a farnilj. in N o . For every m € A let functions qrrl,o,. . . , qm,ImE C ' i X ) be given. Moreover, let Z
be
a
closed quasihomogeneous subset o f X such that SUPP 4rn.i C
(3.39)
z
m E A and j € N j l n u { O ) .
9
Finall-v, let Y be an open subset o f X such that Y , = X . Then the following conditions are equivalent: ( a ) there exists F ' E C L ( X ) with support contained in Y n Z such that
( b ) f o r ever) m E A q , , l , o is quasihomogeneous OF degree m and i (3.41)
t-mq,,,oMt
( - l ) i u i ( t )q ,,,,i - i .
=
tElO.+wC, j E N j I n .
i=O
Proof. ( a ) + ( b ) : this is valid by Proposition 3.13. ( b ) * ( a ) : We fix a C- function x a s in condition ( b ) of Theorem 3.30 and proceed in t w o s t e p s .
Step 1 : w e s u p p o s e f i r s t t h a t Y = x - ' ( I ) f o r s o m e relatively compact open s u b s e t 1 of l O . + a - . I . We define ~ , , , ~ : l O , + ~ C + c tby t H t - , W j ( t )
s u c h t h a t ~ $ 0 I .t follows from (1.42) t h a t t h e functions and jEIN,,
a r e linearly independent over
rated by t h e functions
x
~
,
(r
x
and fix XEC;(I)
~:
~
~
,
~
= X W ~ ~ m , ~E ,@
. Let H be t h e s u b s p a c e of C F ( l ) gene-
and m E A . Equipped with t h e s c a l a r p r o d u c t
0~<_ j, 5 , j
defined by +
mi
T ) t ( < lc p J, J I)> : =t ( J'J q
dt
0
H becomes a Hilbert space. By (b"") which is dual t o t h e basis
4111 J. E H such I
that
(x,,~)
< ,JI m.J. > = fin'".
belonging to CgC I ) , satisfy
we d e n o t e t h e basis of t h e dual space H'
of H . Then by t h e Riesz lemma we can c h o o s e This means t h a t t h e functions cpm,j : =
XK,
150
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
+a
J t - m c p z , e ( t ) a , (dtt) T- L , , S ~ ~ ,
(3.42)
z , m ~ l \ P, E N ~ ~ ~ ( oj E) N, j m u ( 0 ) .
0
Now w e can define ir
(3.43)
f:=
C C
ZEA
e=o
qz,eOQx ( ~ z , e o x
w h e r e t h e C m f u n c t i o n px is d e f i n e d in N o t a t i o n 1.71. T h e n f is a C ' f u n c t i o n o n X w i t h s u p p o r t c o n t a i n e d in Z l n Z , w h e r e Z l : = p,'(Z)
x
-1
( s u p p x ) C x - ' ( I ) = Y . S i n c e by Lemma 3.20 x - ' ( I )
= Z,
= Z a n d Z,
is a n M - b o u n d e d
:=
subset
of X f b e l o n g s to C L C X ) . To p r o v e ( 3 . 4 0 ) w e f i x x E S X . T h e n p x ( M , x ) = x a n d
x ( M t x ) = t , t E I O , + a I , so t h a t I,
+ rr,
C 2 f zeA e = o b
(x) = 'ms~j
dt t - m q , , e ( x ) V,,e(t) w j ( t ) T ,
mEA, 0 5 j 5
in,.
By ( 3 . 4 2 ) t h i s c o i n c i d e s w i t h q m , i ( x ) , i.e. ( 3 . 4 0 ) h o l d s o n Sx. By (3.11) a n d ( b ) t h e e q u a l i t y ( 3 . 4 0 ) is t h e n valid a t t h e p o i n t s M , x , x E S x a n d t E IO,+mC S t e p 9 : t h e g e n e r a l c a s e . Let x E Y . S i n c e by P r o p o s i t i o n 1.72 t h e m a p
ax (see
N o t a t i o n 1.71 ) is a h o m e o m o r p h i s m w e f i n d a relatively c o m p a c t o p e n s u b i n t e r v a l
J, of I O , + ~ aI n d a n o p e n s u b s e t V, of S x s u c h t h a t Y,:= ( M , 1 9 ; ~ E V , , t c J , } is a relatively c o m p a c t o p e n n e i g h b o u r h o o d o f x in Y. O n e observes t h a t Y, =
= ( Y , ) M n x - l ( J x ) . Applying Lemma 3 . 4 0 to t h e family u:=(Y,),Gy
one obtains a
s e q u e n c e of relatively c o m p a c t o p e n s u b s e t s Ve, P E IN, of Y s u c h t h a t t h e sets ( Ve) M , PEN, f o r m a locally f i n i t e o p e n c o v e r i n g of X a n d s u c h t h a t f o r every P E N o n e c a n fix XeEY s a t i s f y i n g Ve CY,,. We define l e : = J X e a n d U e : = ( V e ) M n x - ' ( l , ) .
T h e n Ue is o p e n in Yxe s a t i s f y i n g ( U e ) M = ( V e ) M . C o n s e q u e n t l y , U : = ( U p ; PEN} s a t i s f i e s t h e a s s u m p t i o n s of c o n d i t i o n ( c ) ' of T h e o r e m 3.39. H e n c e w e f i n d a s e q u e n c e of f u n c t i o n s + e E C E ( X ) w i t h s u p p o r t c o n t a i n e d in Ue s u c h t h a t m
(3.44)
C
(+e)o = I
.
P=l
By t h e f i r s t s t e p of t h e p r o o f , f o r every O E N w e c h o o s e a f u n c t i o n f , E C L ( X ) w i t h s u p p o r t c o n t a i n e d in x - l ( l p ) s u c h t h a t ( 3 . 4 0 ) is valid f o r fe i n s t e a d of f . W e define m
f : = 2 ( + e ) of e . e=i
S i n c e by t h e d e f i n i t i o n of Up w e have ( U e ) M n x - ' ( l o ) = U e it follows b y Propos i t i o n 3.10 a n d by t h e local f i n i t e n e s s of t h e family ( U e ) t h a t f is a w e l l - d e f i n e d
3.d Q u a s l h o r n o g e n e o u s F u n c t i o n s a r 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
151
' function on X with support contained i n C
U
c
U,nZ
YnZ.
P €IN
Since by Lemma 3.29 every U, is an M-bounded subset of X and since the s e t s
( U P ) M ,( E N , form a locally finite open covering of X Remark 3.28.(ii) implies that suppf is an M-bounded subset of X , a s well. Moreover, since by Proposition 3.13 the functions (+P)oare quasihomogeneous of degree 0 it follows by Corollary3.5 and by (3.44) that f satisfies ( 3 . 4 0 ) .
H
A special case of Theorem 3.48 is the fact that every function x as in Theorem
3.39.(b) is of the form (3.35):
Corollary 3.49. Let x : X
+10. +a[ be
d
C" function which is quasihomogeneous
OF degree I . I f Y is as in Theorem 3.48 then there esists a function + E C L ( X ) with support contained in Y such that
and (3.35) holds.
proOF. Since -log x(M,x) = -log x ( x ) - log t one obtains the assertion by applying
Theorem 3.48 to q o . o : = l and q o , , : = - 1 o g x .
H
Next we are going t o have a look at certain spaces of almost quasihomogeneous functions required in section 4 . ( e ) . To this end we f i x r ~ N o u ( ~and ) keNo and introduce
Notatlon 3.50. By X & ? A , k ( X ) we denote the space of all C' functions q : X-C which are almost quasihomogeneous of degree m and of order 5 k such that (3.45)
suppq
C
K,
for some compact subset K of X
Proporitlon 3.51. ( i ) 2 1 C A , k ( X ) = ( ii) Let x : X + l O , + w [
{fm,,,k;
fEC&(X)}
be a C"function which is quasihomogeneous of degree I .
Then. S x being a CCTsubmanifoldo f X , the space CztSx) is well-defined, and b-k (3.46)
4
H (((~M-m)idlsX)05,5k
a linear isomorphism
given by
IX)-+ Civ(Sx)k*' is well-defined, i t s inverse being
152
111.
Quasihomogeneous Averages of Functions. Part 1
k
(3.46)' where the Proof.
(gi)olisk
vi
*i.Z 111 =o
XI''
giopx
are defined in Remark 1.71 and p x is defined in Notation 1.71.
u.*z": t h i s 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 s 3.13 a n d 3.10.
" C *': By P r o p o s i t i o n s 3.31 a n d 3 . 4 2 w e f i n d a n M - b o u n d e d o p e n s u b s e t Y of X s u c h
t h a t Y M = X . W e f i x q E a a L , k ( X ) . T h e n by P r o p o s i t i o n I . S l . ( i i ) its d e f i c i e n c i e s q j , j < N k , are Cr f u n c t i o n s , a s w e l l . M o r e o v e r , s i n c e t h e e q u a t i o n s ( 1 . 3 9 ) a n d ( l . 3 0 ) k
(see L e m m a 1.48) are valid f o r q o : = q t h e c o n d i t i o n ( b ) in T h e o r e m 3 . 4 8 is s a t i s f i e d
f o r A = ( m ) , j,,:=k
a n d q m , i : =( - l ) k - i q k - i . C o n s e q u e n t l y , by T h e o r e m 3 . 4 8 w e such that q = fm,wk and such that supp f C Y n supp qo.
f i n d a Cr f u n c t i o n f : X-@
C h o o s i n g a c o m p a c t s u b s e t K of X s a t i s f y i n g ( 3 . 4 5 ) w e c o n c l u d e t h a t s u p p f is c o n t a i n e d in t h e set Y n , K
(ii): S i n c e
which is relatively c o m p a c t in X by P r o p o s i t i o n 3 . 2 2 .
by L e m m a 3.20 S x is a n M - b o u n d e d s u b s e t of X it f o l l o w s by Proposi-
t i o n 3.22 t h a t S"n KM is c o m p a c t f o r every c o m p a c t s u b s e t K of X . C o n s e q u e n t l y , t h e m a p given b y ( 3 . 4 6 ) h a s v a l u e s in C F ( S " ) " k .
O n t h e o t h e r hand, since
s u p p g i O p r = ( s u p p g i ) M it follows by R e m a r k l . 7 3 ' . ( i ) t h a t t h e m a p ( 3 . 4 6 ) ' is w e l l d e f i n e d a n d h a s v a l u e s in X n z , k ( X ) . T h a t t h e s e m a p s a r e i n v e r s e to e a c h o t h e r is also a c o n s e q u e n c e of Remark 1 . 7 3 ' .
T h e f a c t t h a t in T h e o r e m 3 . 4 8 t h e case I A I ? 2 is a d m i t t e d , a s w e l l , m i g h t be u s e f u l , f o r e x a m p l e , f o r recovering t h e q u a s i h o m o g e n e o u s p a r t s of f i n i t e s u m s
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 :
Remark 3.52. Let A be a finite subset o f
C , and f o r ever) m € A let 9- c C 0 ( X I
be quasihomogeneous of degree m . We set 9 :=
-? &
m €A
qrn
.
I f + 6 C E : ( X Ii s s u c h that +z'Soz
Proof. C o r o l l a r y 3.5
for ever).zEA-A then ( + q I , = q ,
s h o w s that (+q,)p
forevery m e n .
= + e - m q m for arbitrary m , P E A .
D
F o r e x a m p l e , if o = a + a n d if M is s e m i - s i m p l e t h e n R e m a r k 3.52 c a n be a p p l i e d
to any p o l y n o m i a l f u n c t i o n q a n d to A = { m € : U ( M ) ; Q,q
-f 0 } a n d q,:
= Q,q.
153
Chapter IV
Constructing (Almost) Quasihomogeneous Distributions by Taking Quasihomogeneous Averages. The Case: X is Locally M-bounded
In section ( a ) the method of taking quasihomogeneous averages is extended to distributions. Here it is required that the support of every test function cpE CFCX) is an M-bounded subset of X . Consequently, we assume throughout the present
chapter that X is locally M-bounded in the sense of Definition 3.30. A few basic facts about the singular support and the wave front set of the quasihomogeneous averages appear i n section ( d ) .
In section ( b ) it is established that every quasihomogeneous distribution T on X appears as the quasihomogeneous average u,,,
of some distribution u w i t h M -
bounded support. In fact, as w i t h functions, any finite collection of quasihomogeneous distributions of different degrees can be written in t h i s way by means of a single u . The corresponding result for almost quasihomogeneous distributions
is valid, as well ( s e e Theorem 4.14). A consequence of these facts is the result
that for every quasihomogeneous real analytic function q : X
+C
the equation
q S = T has an ( a l m o s t ) quasihomogeneous solution S E a ' ( X ) for any ( a l m o s t ) quasihomogeneous right-hand side T E a ' ( X ) ( s e e Theorem 4.15 below).
In section ( c ) the inhomogeneous Euler equation
(aM- m ) S = T
is shown to be
solvable f o r every TE a'(X);in particular, every (almost) quasihomogeneous distribution on X appears as the deficiency of some other almost quasihomogeneous distribution. Finally, in section ( e ) it will be proved that every quasihomogeneous distribution on X is of the form xrnp:v
for some v E a ) ' ( S x ) where x : X d l O , + ~ C is any
I54
I V . Q u a s i h o m o g e n e o u s Averages o f D i s t r l b u t i o n s . Part 1
C m function which is quasihomogeneous of degree 1 and where px i s t h e canonical submersion from X o n t o Sx associated with x (see Notation 1.71). The proof
relies o n t h e quasihomogeneous continuations urn E
a'(X )
which are introduced in
section ( e ) . a s well.
( a ) Introducing the L)u:islhomogc.ncous Avc?ragcs
Before coming to t h e main definition we recall an elementary fact from t h e theory of distributions (see f o r example Theorem 2.2.5 in Hormander 1111).
Remark 4.1. Let u € . ? J ' ( X )Then . f o r any open subset Y of V there exists a unique linear extension u" of u I X n y t o the space
+ n supp u i s compact} supp + n supp u = 0 1 . The following proper-
D,( Y ) := { +t-C'"( Y ) : supp vanishing on the subspace { +cC"'( Y ) ; ties hold f o r every
+ € D,, ( Y ) :
(iii) if Y n s u p p u is a closed subset o f Y then C P l Y ) is contained in D , , ( Y ) , and the restriction of u" t o CPCY) is a well-defined distribution on Y . Below we shall omit t h e superscript and denote
I
by u, a s well. Moreover, we
shall not specify t h e choice of Y; mostly, we have to deal with t h e case Y = X .
As in C h a p t e r 3 we fix a closed s u b s e t I of IO,+wC and a locally integrable
function w : l O , + ~ C with ~ @ support contained in I . We s e t
We observe t h a t v is locally integrable on 10,+00C, a s well, and t h a t t h e supp o r t of v is contained in 1 1 1 . To motivate t h e main definition we let f € C p ( X ) and
cpEC;(X)
and observe t h a t
( s u p p f I M , , n suppcp ( b y
Lemma 3 . 9 ) and
I55
4 . a 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 u,
supp f n (supp 'p)M,l/I
( b y P r o p o s i t i o n 3.22
)
are c o m p a c t so t h a t L e m m a 3.11
implies (4.2)
= J f ( x ) rp,*,,(x)dx
Jf,,w(x)'p(x)dx X
.
X
Notation 4.2. By 3;tX) 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 3 ' ( X ) s u c h t h a t s u p p u is a n ( M , I ) - b o u n d e d s u b s e t of X . I f I = l O , + ~ [ :we a l s o w r i t e 3 g t X ) .
We n o w fix u E D ; ( X ) . T h e n by P r o p o s i t i o n s 3.10 a n d 3.22 a n d by R e m a r k 4.1 it
m a k e s s e n s e to d e f i n e
N o t e t h a t ( 4 . 2 ) now r e a d s as
f E cpc X ) .
(4.2)'
Propoeltion 4.3.
(il u,,,,,
is a well-defined distribution on X with support
contained in (supp u J M , I . liil IF p EC;> ( X I is such that supp p n ( s u p p ~ ) ~is, ,compact then the set supp pm*, n supp u is compact, as well, and the defining equation in ( 4 . 3 ) remains valid : moreover.
(here the integrand on the right-hand side is a well-deFined continuous Function on I with compact support).
3'(X).m
(iii) The function bju. ,: @--+ ( k)
Q,,,(ml
= k! ( - 1 )
proOf. fi); W e set L : = s u p p u
.
k
H
urn,+, , is holomorphic satisfying kEN.
u,,,,~~.
L e t K be a c o m p a c t s u b s e t of X . T h e n b y P r o p o -
s i t i o n 3.22 L n K M , l / l is a c o m p a c t s u b s e t of X . We let W be a c o m p a c t neighb o u r h o o d of t h i s set in X a n d fix XECFCW ) s u c h t h a t x = 1 n e a r L n K M , , , I . by P r o p o s i t i o n 3.10 t h e s u p p o r t of
'prn*,"
one sees t h a t t h e d e f i n i t i o n of urn.,
i s c o n t a i n e d in KM,,,,
a m o u n t s to
Since
f o r every ' p E C g ( K )
156
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
Since K is an M-bounded s u b s e t of X there is a compact subinterval J of 10,+00C such t h a t { t ~ l O , + a CM; t ( W ) n K f @ }C J . Hence for every ' p E C g ( K ) t h e functions Tm*,
and 'Pm*.vy, coincide on W so t h a t in view of ( 3 . 5 ) it follows t h a t CgC W ) ,
t h e linear map C g ( K )
(4.5)
'p H
x 'p,*,,
, is continuous. Consequently, by
is continuous on C F ( K ) .
To prove t h e assertion a b o u t s u p p u,,,,
we first observe t h a t K n ( s u p p u ) M , I = @
if and only if s u p p u I I K M , l / l = @ . Since s u p p ~ p , * , ~ C K M , ~ / , f o r every ' p E C F ( K ) this implies in view of Remark 4.1 that urn,,
vanishes o n X \ ( s u p p u ) M , I . Since
by Lemma 3.9 the last set is open in X t h e proof of ( i ) is complete. (iiil.Choosing K , W , and
x
a s above and fixing rp€C;(K)
we deduce from Propo-
C F C W ) , m Hx'p,,*,, , is holomorphic. Since u
sition 3.3.(ii) t h a t t h e map Q1-
maps CZCW) continuously into C this implies in view of (3.5) t h a t t h e map C d C , mH
<
'p
> , 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*,,>=<~,(X'P),*,v>= =.
To p r o v e t h e l a s t p a r t of t h e a s s e r t i o n w e d e f i n e J to b e t h e c l o s u r e i n I of t h e set { t e l ; M l / , x e @ f o r some x E U ) . A s a s u b s e t of { t E l ; M , K n E # @ } J is c o m p a c t . H e n c e w e c a n fix a f u n c t i o n X E C FC IO,+mOC) e q u a l to I n e a r J . W e s e t J , ( x , t ) := t " * X ( t )
(x'p)(M,/,x),
S i n c e t h e s u p p o r t of J, is c o n t a i n e d in
(X,t)€
XxIO,+ml.
( s u p p ~ ) ~ , x~J '/ , w~h e r e J ' : = s u p p X
J , b e l o n g s to C r C X x I O , + m l ) . Now w e o b s e r v e t h a t by t h e d e f i n i t i o n of J w e have
'p
u o M t ~0 f o r every t E I \ J .
M o r e o v e r , From t h e d e f i n i t i o n of K w e d e d u c e t h a t s u p p ( y , uoM,) C K f o r every t E J . Consequently,
w h e r e TE 3'
O n t h e o t h e r h a n d , s i n c e for every X E U t h e s u p p o r t OF t h e Function I + C , t H ( x r p ) ( M t , , x ) , is c o n t a i n e d in J a n d s i n c e x I = X o n J it f o l l o w s t h a t
< T . $(x:
)
>=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 =
-c
wi(t) <(dM-m)'T,(r,>,
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,,,,)(<)=
-i
.>
)m*.v
In o r d e r to derive s u i t a b l e e s t i m a t e s for ( c p e-'"' E
>
S E V . cpECT(K).
1
.>
)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
<<, M,Mx>
g is w e l l - d e f i n e d , all of w h o s e derivatives are b o u n d e d o n D .
Note that
e x p ( - i < r 5 , M t x > ) = 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, <€HI
i s a b o u n d e d s u b s e t o f CTCW). In view of ( 4 . 2 7 ) t h i s s h o w s t h a t if x N is t h e s e m i - n o r m o n 3 ( T M ( X ) ( X ) d e f i n e d by ( 4 . 2 5 ) t h e n
u++K~(u,.,,)
defines a con-
tinuous semi-norm o n D ' ( X ; L ) . The assertion follows.
Corollary 4.22. The sequence ( q i l in Corollary 4.11 can be chosen in such a that ( T
qi
I converges t o T in t h e topologj of D;M(x)( X )
as
was
j+".
proOf. Let f i a n d J, be a s in t h e p r o o f of C o r o l l a r y 4.11. T h e n f o r e v e r y i < k 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 i n d u c e d by t h e f u n c t i o n s J,
JI (aM - m ) ' T
( a M - m ) i fi
c o n v e r g e s to
in t h e s p a c e 3 ' ( X ; L ) w h e r e L : = s u p p J , . C o n s e q u e n t l y , in view of
t h e p r o o f of C o r o l l a r y 4.11 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 o r e m 4.21.
174
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
(e) The Quaslhomogeneous Conlinuallons urn and the I~lstrlhuLlonsxrn p:(v),
u E 3'(Sy)
W e f i x a Cm f u n c t i o n x : X - l O , + a C
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
(see T h e o r e m 3.39 1 . To m o t i v a t e t h e main d e f i n i t i o n w e f i r s t p r o v e t h e f o l l o w i n g l e m m a in a f o r m which is s u i t a b l e f o r a p p l i c a t i o n s in C h a p t e t - 8 , a s w e l l .
Lemma 4.23. Let k E N , let qO 6 2 , b , ( X l be almost quasihomogeneous o f degree m and of order i k , and let f € C o ( X ) be such that 14.301
( s u p p F,,,,)
n s u p p qi n S x is compact for ever! i E N, u / O /
where qi denotes the i e h order def'iciencj OF q O . Suppose.
/ElRer,,*.
Iwh,kI
moreover.
that
is well-defined b-b (3.1)' (with I m , w ) replaced by IRem'. / w o k / ) )
and continuous. Then for every i E N k ~ l O flI , , * , w L J i is well-defined bj, (3.1)' (with m and w replaced bj m' and wrc)iI and is measurable
on S x .
and the function
qi f (wi+I o x belongs to X ' I X ) such that k
(4.31)
j
j ' q O ( s l f ( s l ( w o x ) ( s )dx = i=O
X
,
qi(B) fI,,..wuiIt?) dx"I-9)
sx
and k
(4.31)'
j ' q O ( 9 1fm1,(91
1-11' J ' q i ( s )f l s l ( I W ( J ~ ) O X ) ( . dYs)
dX(9)=
x
i=O
S X
Proof. W e d e n o t e t h e u n i o n of t h e c o m p a c t sets in ( 4 . 3 0 ) by K . S i n c e by L e m m a 1.90 t h e f u n c t i o n s qiISr b e l o n g to Lf,,(?)
and since t h e functions
I f IRe
m*,
are
w e l l - d e f i n e d a n d c o n t i n u o u s w i t h s u p p o r t c o n t a i n e d in ( s u p p f )M it f o l l o w s b y ( 4 . 3 0 ) t h a t t h e f u n c t i o n s h i : = ( ( q i l If
IR~[, *,
Iwwil)lsx
b e l o n g to L ' t z ) . H e n c e ,
in p a r t i c u l a r , t h e r i g h t - h a n d side of ( 4 . 3 1 ) is w e l l - d e f i n e d . In o r d e r to p r o v e t h e s a m e a b o u t t h e l e f t - h a n d side of (4.31) w e f i r s t observe t h a t in view of P r o p o s i t i o n 1.86 a n d by t h e Fubini a n d t h e Fubini-Tonelli t h e o r e m it s u f f i c e s to s h o w t h a t t h e function
b e l o n g s to Z ' ( d t 8 ; ) .
In view of t h e e q u a l i t y
4.e
175
The Q u a s i h o m o g e n e o u s Continuations v,
t h e assumptions o n f imply t h a t the function S x - J R ,
8-
J,'
+m
l g ( t , 8 ) 1 d t , is
well-defined and majorired by t h e s u m of t h e functions hi defined above. Consequently, t h e Fubini-Tonelli theorem leads to t h e desired conclusion a n d , moreover, to t h e equation (4.31)
For t h e proof of (4.31)' we first observe t h a t by what was proved above t h e integrands of b o t h s i d e s of (4.31)' a r e absolutely integrable. Hence in view of Proposition 1.86 and t h e equation q,(B) t-,*f(M,S)
w ( t ) =qo(M1,,(MtS)) tm+' f ( M , S ) ( w o x ) ( M , S ) =
k
=
I
q i ( M t H ) t'
( q o x ) ( M , 9 ) f ( M , 9 ) (wox)(M,S)
i=O
(which is valid f o r arbitrary t E I O , + a [ and 9 E S y ) t h e equation ( 4 . 3 1 ) ' f o l l o w s by t h e Fubini theorem.
w
Recall now t h a t S x is a C"'submanifo1d of X ( s e e section l . ( f ) ) and t h a t t h e space a'(S")of distributions on S x can be defined a s t h e dual of t h e space of
C g densities o n Sy ( s e e f o r example
b.3 in Hormander C111). In t h e following
we shall identify a ) ' ( S x ) with t h e dual o f C F ( S x ) via t h e strictly positive densit) ( s e e Notation 1.85) :
wMlsx
Now we fix k E N , a ( l + k ) - t u p l e w = ( w o . . . . , w k ) E ~ ' ( S x ) " " , and m € C . In view of t h e preceding identification it makes s e n s e to define
Note t h a t when Lemma 4.23 is applied t o f = ' p E C g C X ) and w = 1 t h e f o r m u l a (4.31) reads a s
(4.34)
h p O d ~ 0 1 4.24. 1
in (ii)
k
U,=, v,
(i) v,
is a well-defined distribution on X with support contained
(supp v i ) M ;
is almost quasihomogeneous of degree m and o f order 5 k
;
176
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 o f D i o t r l b u t i o n s . Part 1
(iii) for every j E /01 U N k we have k -1 rn I. (4.35) (aM-m)lv, = x qi p x ( v i + i ) I
i=O
where pr i s deFined in Notation 1.71 and the qi are defined in Lemma 1.74 ; ( i v ) the Function C
mf. (i): Let
+D ' ( X ) , m H urn ,
is holomorphic.
K be a compact subset of X . Since by Lemma 3.29 S x is an M -
bounded subset of X it follows by Proposition 3.22 that L : = K,
n S x is a compact
subset of S x . By Proposition 3.10 we see that i f q J € C g ( X ) t h e n the f u n c t i o n s Trn*,tq I s x belong to C g ( L ) . Since K is an M-bounded subset of X we find a com-
pact subinterval J of lO,+mC such that q~,,,ml,~,,~(B) =
[ t m + ' q ( M , B ) w i ( t ) T~ dt
B E L , cpECF(K).
J
Consequently, i n view of (3.5) the linear maps C;(K)-C:(L), are well-defined and continuous. Hence ,v
'p
Hcpmr*,cJiIS,,
is well-defined and continuous o n
the space C:(K). The assertion about suppvlll is proved in a similar way as the corresponding part of Proposition 4.3.(i) .
0: Let K
-
and L be as above. We fix c p E C r ( K ) and i E ( 0 ) u I N k . Since the re-
striction map
c ' ~ ( x ) c ' " ( s ~f)H, f I s x ,
3.3.(ii) implies that the map C-
is linear and continuous Proposition
CZC L ) , m H cpl,,l,wi
Isx
, is holomorphic. Since
v i : C z ( S x )-+ C is linear and continuous the assertion follows.
m :t h i s
is a consequence of ( i i i ) or can be proved directly w i t h the help of the
almost quasihomogeneity properties of the functions cpn,,,c,,i (iiil:I f u = (Tg,. . . . , T
gk
)
,
for s o m e g o , . . . , g l , € C o ( S X ) then by Remark 1.73' the
functions qk defined by (1.82) satisfy (1.30) (with N replaced by k ) s o that in view of (1.83) and because of q i I s x = g i the equality (4.35) for j = O is identical t o (4.34) i n t h i s case. In general ui can be approximated ( i n the weak topology) by a sequence of functions g i , i E C ; ( S X ) , j € N . Since the maps 9,) and
are (weakly) continuous the equality (4.35) follows for j = O . I n order to derive (4.35) for j C N k we conclude by Remark 1.74 and Theorem 2 , 1 7 . ( i ) and by changing the order of summation and substituting I = i - j that
4.e
177
T h e Q u a s i h o m o g e n e o u s C o n t i n u a t i o n s v,
k wmo
M, = t m x m
i
w j ( t ) q i - j ) p:(Vi) i=o j=o
k
= tm
(
=
k-j
c wj(t)( c xm
71 p:(wj*I))
t E 10,+001.
,
I=O
j=O
By Proposition 2.31 t h i s implies (4.35). We can n o w complete the description of almost quasihomogeneous distributions on X .
Theorem 4.25. Let T E D ' l X I be almost quasihomogeneous o f degree
m and of
order _ < k . Then the ( l + k l - t u p l e d ( T ) o f distributions ( d , - n ~ l ' T / ~ O ~Z . iLk. (which are well-defined bj Proposition 2.31. ( i i ) . Corollarj 2.33. ( i ) . and Theorem 2.42 I has the following properties:
(i)
diT),,, = T ;
( i i ) i f v 6 D D ' I S X l ' + "satisfies the equation v,=T
then
v =dlTI
Proof. -(i): If forevery i€(O)ulN, there exists q i € C o ( X ) such that ( a M - m ) ' T = T q i then the assertion is identical with ( 4 . 3 4 ) . In general, by Corollary 4.22 we choose a sequence of functions qiE C m ( X ) which are almost quasihomogeneous of degree m and of order 5 k such that ( T q i ) converges to T in the topology of the space
a',,,,,
( X )as
j+W.
Then (by Theorem 2.42 and Lemma 2.15 and bq Theorem 8.2.4
in Hormander 1111) the sequence of distributions d ( T ) converges to d ( T ) a s qi j + m , and hence Tqi =d(Tqi),,, converges to d ( T ) , as j + a . ( i i ) : Since q i I s x = S i 0
this follows from (4.35) and Theorem 2.17.(iv).
Corollary 4.26. Let T € D ' ( X ) be almost quasihomogeneous of degree m and o f order 5 k , and let So,. . . , S, be the quasihomogeneous (of degree m ) distributions defined bj, (2.20) with ( N , v i l replaced bj ( k , q i ) . Then we have (4.36)
s, = x m p:
Proof. Inserting
((
(a,,,,
-m)
@
T ) / ~ ~ ) ,
eEIOIUNk.
the equation ( i ) in Theorem 4.25 into the left-hand side of the
equation (4.35)for w = d ( T ) and j = O we see that the equation (2.19) is valid for u i = q i if the distributions S, are given by ( 4 . 3 6 ) . Since by Theorem 2.17.(i) and
178
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
C o r o l l a r y 2 . 4 . ( i i ) t h e d i s t r i b u t i o n s given by ( 4 . 3 6 ) are q u a s i h o m o g e n e o u s of deg r e e m t h e a s s e r t i o n f o l l o w s by t h e u n i q u e n e s s p a r t o f P r o p o s i t i o n 2 . 4 5 .
H
As a n o t h e r c o n s e q u e n c e o f T h e o r e m 4.25 w e n o t e in p a s s i n g :
Corollary 4.27. Let T E . ~ ' I X Ibe almost quasihomogeneous o f degree m and of order s k . Let ffjJjcW be a sequence in C " I X ) n 3 ; ; M f x ) f X )converging t o T in the topology of . 3 ; M f x ) ( X )
as j + "
then the sequence of distributioris induced
bj, the furictions k
x'"((dM - m ) ' f j ) "px i=O
converges fweaklj,) t o T as j + & >
Proof. W e set g i , j : = ( ( 3 M - m ) i f i ) l S x . T h e a s s u m p t i o n o n ( f i ) i m p l i e s t h a t (Tg.
.)
1.1
c o n v e r g e s to ( d M - n i ) ' T l s X
c o n v e r g e s to (TIsx
,.
..,
a s j+"
SO
that
k
(3, - n i ) TIsx),,, = d(T),, = T as j + m .
H
C o m b i n i n g T h e o r e m 4.25 a n d P r o p o s i t i o n 4 . 2 4 . ( i i ) o n e sees t h a t a d i s t r i b u t i o n
T E ~ ' ( Xis) 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 5 k if a n d o n l y if T = u m f o r s o m e v E D'(SX)''k. I n t r o d u c i n g
NotatJon 4.28. By 21,i7,i,,kfX) w e d e n o t e t h e c o m p l e x v e c t o r s p a c e of a l l d i s t r i b u t i o n s T E 3 ' ( X ) w h i c h a r e 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 5 k .
w e r e f o r m u l a t e T h e o r e m 4.25 a n d P r o p o s i t i o n 4 . 2 4 . ( i i ) a s f o l l o w s .
Theorem 4.25'. The map XL2~,,kfX) --+ 3 ' ( S x I '+'. phism. its inverse being given bj v H v,,, .
T + + d ( T )is , a linear isomor-
#
W e e n d t h i s s e c t i o n by r e w r i t i n g t h e e q u a l i t y ( i ) in T h e o r e m 4 . 2 5
4.e
I79
T h e Q u a d h o m o g e n e o u s C o n t i n u a t i o n s vm
Theorem 4.29. Let T € X D A , k ( X ) and (q5j)osi5k
q €2lD:*.kIX)
(see Notation 3.50). And let
be any sequence o f functions in C G l X ) satistving ( 4 . 1 8 ) f o r 3; = CO/ uNk
and ( 4 . 1 9 ) . Then
where the left-hand side i s well-defined since by Proposition 3.51. ( i i ) the functions ((d)M-rn)iq)lsx. i e l O l u N , .
belong t o C g ( S x ) .
In particular. it follows that the left-hand (resp. right-hand) side o f (4.371 is independent o f the choice o f x (resp.
Go.. .. .G k ) .
Proof. Let K b e a c o m p a c t s u b s e t of X s u c h t h a t ( 3 . 4 5 ) is valid. By T h e o r e m 3 . 4 8 w e f i n d a C"
f u n c t i o n p:X--+cf
s u c h t h a t q = rqm*,c,,k.
w i t h s u p p o r t c o n t a i n e d in L : = K ~ n X - ' ( [ l , ? l )
By Lemma 3.29 a n d P r o p o s i t i o n 3.22 t h e set L is c o m p a c t .
H e n c e y b e l o n g s to C F C X ) . By P r o p o s i t i o n 4.13 w e have k
< T , y >= ( - l ) k
<(dM-rn)'T,+,q>. 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~~>.
<((d,-m)'T)lsX, i=O
S i n c e by P r o p o s i t i o n s 3.13 a n d 2.31 w e have p,,,,,c,,i=( - l ) k - i
( a - m~* ) k - i q
the
assertion follows.
T h i s r e s u l t c a n be i n t e r p r e t e d in t e r m s OF d u a l i t y b r a c k e t s a s f o l l o w s .
Remark 4.30. Defining < T , q > by the right-hand side o f 14.37) one obtains a ( n o n degenerate) dualit, bracket ?l&?A,k(X)x ?lk2z*,k(X)--+
@. In view o f (4.371 this
duality bracket is isomorphic - via the isomorphisms in Theorem 4.25' and Propo-
-
sition 3.51. ( i i ) - to the dualit), bracket (4.311)
where
c,
6 ) ' ( s x ) ' + k xc ; - ( s x ) ' + ~
<-, >
is defined bj. ( 4 . 3 - 3 ) .
k
(v
I
x ) H 5 (-1)
I
Xk - i
>.
i=O
I
Of c o u r s e , via t h e i s o m o r p h i s m s in T h e o r e m 4.25' a n d P r o p o s i t i o n 3.51.( i i ) t h e s p a c e s
2[QA,k(X) a n d U R z , k ( X ) c a n be e q u i p p e d w i t h locally c o n v e x t o p o l o g i e s in s u c h
180
I V . 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
a way that each of these spaces is identified as the topological dual of the other via the duality bracket defined i n Remark 4.30. A t present we refrain from giving an intrinsic description o f these locally convex topologies. However, under t h e assumptions of Chapter5 we shall have t o do this for the analogous spaces defined there (see
5 5.g
and
5 0.f).
181
Chapter V
Constructing (Almost) Quasihomogeneous Functions by Taking Quasihomogeneous Averages of Functions Not Necessarily Having M-bounded Support
The purpose of t h e present chapter is to extend part o f t h e theory of C h a p t e r 3 to t h e c a s e where X is not locally M-bounded. Here o n e w a n t s to a d m i t func-
tions f s u c h t h a t the s u p p o r t of t h e functions 10,+03C+@,
t H f ( M , x ) , is n o
longer a c o m p a c t s u b s e t of 1 0 , + ~ 0 1 This . leads t o t w o kinds of difficulties: it has t o be ensured t h a t t h e integrand on t h e right-hand side of ( 3 . 1 ) ' is integrable
I.'
near t = + a and 2: near t = O . The first difficulty is easily overcome by suitable
g r o w t h conditions o n f and w a t infinity. However, t h e second difficulty is more serious: since t
e t-,-'
'I S
o n l y integrable near t = O i f R e m < 0 , in general it is
possible to define f m , w by (3.1)' only if m belongs to t h e half plane @ ( - w . d ) : = { z E C ; R e z < d ) f o r s o m e d E R . This is elaborated in section ( a ) - e v e n t u a l l y
under t h e assumption t h a t (1.14) holds. Under very special assumptions o n w , i n section ( b ) for every f E C m ( X ) a func-
tional equation is derived which makes it possible t o extend t h e holomorphic function @ ( - a , d ) + C m ( X ) . o f 43.
mef,,,,,
,
t o a meromorphic function on t h e whole
This leads t o a natural definition o f f m m Wfor every m E C . I t is f o r t h e
derivation of t h e functional equation t h a t t h e assumption (1.14) is e s s e n t i a l . I n section ( c ) t h e negative Laurent coefficients a t t h e poles of t h e meromorphic extension a r e c o m p u t e d while section ( d ) contains explicit formulas for f m , w . In section ( e ) . a s a combination of t h e r e s u l t s of the preceding sections a reasonable definition of f, f u n c t i o n s ,f
,
for arbitrary f E f P ( V ) and arbitrary m E C is obtained. These
however, d o not always have t h e desired properties: in general they
182
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
are d e f i n e d o n X , , o n l y , a n d if m € ' U ( M ) t h e n it h a p p e n s t h a t f m is n o t q u a s i -
h o m o g e n e o u s b u t o n l y 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 , its deficiency e x t e n d i n g to a n ( a l m o s t ) q u a s i h o m o g e n e o u s C m f u n c t i o n o n t h e w h o l e of X . In s e c t i o n ( f ) locally c o n v e x s p a c e s of f u n c t i o n s of t h e latter k i n d are i n t r o d u c e d , w h i l e s e c t i o n ( g ) is devoted to locally c o n v e x s p a c e s of f u n c t i o n s of t h e f o r m
f,
.
T h e i r t o p o l o g i c a l s t a n d a r d p r o p e r t i e s a r e of i m p o r t a n c e in C h a p t e r 7 .
As in C h a p t e r s 3 a n d 4 let w : I O , + a 3 C +
6 b e a locally i n t e g r a b l e w e i g h t f u n c t i o n .
F o r v a r i o u s c h o i c e s of s u b s p a c e s E a n d F of C o ( X ) a n d o p e n s u b s e t s D of 6 w e a r e g o i n g to verify t h e f o l l o w i n g c o m b i n a t i o n of a s s e r t i o n s : 15.1)
1i) f o r arbitrary mED and f C E a function f , , , E F ( i i ) the map D x E -
F . 1 m . f )H f,,,
is well-defined b> ( 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 <
+m
t h e n b y c o n t i n u i t y w e o b t a i n t h a t x = M,k . T h i s m e a n s : if s = 0 t h e n
x c M o ( K ) a n d 1 is u n b o u n d e d ; if 0 < s <
+m
t h e n s E 1 / 1 a n d X E K M , 1 . If s = + a
t h e n 1/1 is u n b o u n d e d , a n d t h e c o n d i t i o n lim t j G + ( k i ) =
j+ m
( w h i c h is valid s i n c e
+ ;
;+(XI
is c o n t i n u o u s - see P r o p o s i t i o n s 1.70 a n d 1.61) i m p l i e s t h a t
i.e. k E K n M o ( V ) = K n M o ( K ) . S i n c e M o is c o n t i n u o u s it f o l l o w s t h a t
188
V . Quasihomogeneous A v e r a g e s of Functions. P a r t 2
M o x = lim M o ( M t j k j ) = lim M k - M o k = k , i.e. x E M , ' ( K n M 0 ( K ) ) . I+j+m O jFrom now o n w e s u p p o s e t h a t (1.14) holds. Moreover, w e a s s u m e t h a t t h e n o r m
1 . 1 is induced by
a s c a l a r p r o d u c t o n V which s a t i s f i e s t h e condition ( 1 . 7 9 ) . In
order to apply t h e preceding r e s u l t s w e a r e going to c h o o s e special W , U , a n d w satisfying (5.2). O u r choices d e p e n d o n w h e t h e r t h e s u p p o r t of w s t a y s away f r o m 0 or f r o m + a .In o r d e r to deal with t h e first case w e f i x a c o n t i n u o u s function u : M o ( X ) - l O , + m C , WN: X+
a n d for N E R w e define a c o n t i n u o u s f u n c t i o n
IO,+mC by W N ( x ) : = u ( M o x ) ( 1 + I P + ( x ) l) N ,
(5.10)
X€X,
w h e r e P+ is defined in Notation l.O.B.(iii). Moreover, for every q > O w e d e f i n e WN,rl: X + +
I O , + m l by
employing t h e following
NotationS.10. For any s u b s e t J of IO,+COC we set n , : = { x E V ; I P + ( x ) l < J } w h e r e P+ is defined in Notation l.O.B.(iii).
Proporltion s.11. Suppose that a := inf supp w
>0
( i ) Let L E l R be such that (5.12)
T,w(t), tL$
< +a
a
Then for ever.+ NECO,+mu[ the assertion (5.1) is valid for the Banach space 0
E = C ( W - N / ~ +for ) , the Frechet space F = n , e , o s , ,
C o ( W _ N , , ) and f o r the set
D = C I - N A m I , - L , + a l where X m i n : = m i n { R e A; A E o + } ; moreover, f o r arbitrary f ~ C o ( W - N I , + ) and r n E C ( - ~ A , ~ , - L + t . + r n ) w e have (5.13)
lim f m , w E = f m in , wthe topology of F where w E ( t ) : = e - " w ( t ) .
E+O
(ii) I f (5.1-7) is satisfied f o r ever:,' L E I O , + w C then f o r every N E R the assertion (5.1) holds f o r E = F = Co( WN I and D = C.
5.a
Defining f,
189
by ( 3 . 1 ) '
m f . (i):W e fix AE1O,XminC a n d set c : = - N A - L a n d S : = X m i n - A .
Moreover,
w e c h o o s e BElO,+coC s u c h t h a t
Since by Lemma 1.69 w e may a s s u m e t h a t I P + ( M , x ) ( 2 I ( t ) ( P + ( x ) l it f o l l o w s t h a t
Since
w e d e d u c e t h a t ( 5 . 2 ) ( w i t h X replaced b j X + ) is satisfied for W=W-,I
U(x)=I P + ( x ) I - N ~ ( M o x ) . and
u5
(q/2
)-N
w , , ~ on
C
I
~
y ( t ) = B - N m a x { l , a C - C ' }I w ( t ) I t L .
Note
X +,
that
,Since + ~ by ~ (5.12) y b e l o n g s to Z 1 (I O , + a l ; $ )
~
t h e first p a r t of t h e a s s e r t i o n f o l l o w s by Lemma 5 . 2 . In view of le-et-ll 5
E t ,
t E l O , + m l , w e have
Since 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 remains valid if w is replaced by IwI t h e a s s e r t i o n (5.13) f o l l o w s .
(ii):
h
W e set N : = m a x { N X , , , , , O }
w h e r e A,,
:=niax{ReX; XEo,}.
Then by
Lemma 1.60 w e n i a j a s s u m e t h a t t h e r e ekists a c o n s t a n t CN s u c h t h a t
,.
( I + I P + ( M , X ) lN ) 5 C N t N c l( 1
+
x E V , t E Ca,+mC .
IP+(
W e fix c < 0 < d . Then ( 5 . 2 ) is satisfied f o r W = U = W,
and - in view of (5.14) -
h
y ( t ) = m a x { l , a C - d }CN I w ( t ) I t N + ' - c. m
N e x t w e deal with t h e case t h a t t h e s u p p o r t of w is bounded.
Propoeftlon 5.12. Suppose that b
(5.161
J'lwttll t
-d
b := sup supp w
is finite and that
$ < +a' .
0
Then for ever). N E C O , + m C the assertion ( 5 . 1 ) is valid for E = F = C o ( W N ) and D = @ l - a , d ) . Moreover, for ever, F E C o I W N ) the functions
f m S win t h e norm o f C o ( W N ) a s E + O where ~ , I t ) : = e - 'w~ ( t ) .
converge t o
190
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
proOf. Since N 2 0 and b < + a it follows from t h e last assertion of Proposition 1.70 and from Lemma1.69 t h a t there is a constant C such t h a t W N ( M t x ) 5 C W N ( x )
for arbitrary xCX and t € I O , b l . Consequently, for any c € l - a , d C t h e estimate (5.2) holds f o r W = U = W N and y ( t ) = C m a x { l , b d - C }I w ( t ) J t - d , and t h e f i r s t assertion follows by Lemma 5.2. The second assertion is a n immediate consequence of (5.15)
.
m
Working with general weight functions W requires an additional assumption on X
:
f r o m now on f o r t h e rest of Chapter 5 with t h e exception of section ( g ) we assume t h a t X satisfies
X 3 M o ( X ) , i.e. X = G M ( b + ) + X " where
(5.17)
X " : = { x"E ker M ; x ' + x " E X f o r some x'EG,(n+)
}.
Proposition 5.13. S u p p o s e . in addition, that the assumptions of Proposition 5.12 are valid. Then t h e assertion ( 5 . 1 ) holds f o r E = F = C o I X ) and D = C ( - w , d ) . More precisel) , i f W : X
+LO,
is an) continuous weight function then b-t
a continuous weight function W,,,
:X
+
LO, +a[ i s defined such that the asser-
tion ( 5 . 1 ) holds for E = C o ( W ) , F = C o ( W ( b ) ) ,and D = C ( - w , d ) . Proof. T h a t
w(b) i s
well-defined and continuous follows from t h e continuity of
W, from (5.17) and from the continuity of C O , b l x V + V ,
( t , x ) H M t x , (see
Corollary 1.9). Obviously, for any c € l - a , d C t h e condition ( 5 . 2 ) is satisfied f o r V=W,,,
and y ( t ) : = m a x { l , b d - C }t-d w ( t ) . Since by
Xi(IO,+aC;
q )t h e second part of
(5.16) y
belongs
to
the assertion follows from Lemma 5 . 2 . One
obtains t h e first part by first fixing f € C o ( X ) and then applying t h e second part to W : = l f l + l . m
If the support of f s t a y s away from kerM then f,,,,
Remark 5.14. Suppose that b := s u p supp w (5.19)
supp f
c x,
< + 03
.
is well-defined by (3.1)':
Let f E C o ( X ) be such that
5.b M e r o m o r p h l c E x t e n s i o n s of f
Then f , , , , : X + C
H
191
f,,.w
is well-defined by 13.11' for every m e @ , a n d by m H f , , ,
a holomorphic function on
a?
with values in C o l X l is defined.
mf. By Proposition 3.3 it suffices t o show t h a t s u p p f is an s u b s e t of
( M I1 0 , b 1 ) - b o u n d e d
X.
Let Y E X . Since s u p p f is a closed s u b s e t of X t h e a s s u m p t i o n s (5.17) and (5.19) imply t h a t d i s t ( M o y , s u p p f ) is positive. Hence, by t h e continuity of t h e map a t ( 0 , ~(see ) Corollary 1.9) we find z y > 0 such t h a t M , x kfsupp f
( t , x ) H M,x
f o r arbitrary t E I O , E y l and X E K ( Y , E ~I )f . K is a compact s u b s e t of X then we find a finite s u b s e t N of K s u c h t h a t K is covered by t h e balls K ( y , z , ) ,
YEN.
Consequently,
where
E
:= min{ E~
;
{ t € I O . b l ; M,(K)nsuppf # @ }
y E N } is positive.
is
contained
in
Cz,bl
H
In t h e following section o u r aim is t o g e t rid of t h e assumption (5.10)
We continue t o s u p p o s e t h a t (1.14) and (5.17) a r e valid. Moreover, w e a s s u m e t h a t b : = s u p s u p p w is finite and t h a t
We fix a function f € C o ( X ) . Recall t h a t by Proposition 5.13 t h e prescription mHf,,,,
defines a holomorphic function from t h e half plane @ ( - a , O )
into
C o ( X ) . Under additional assumptions o n w and f we a r e going to e x t e n d it t o a meromorphic function o n t h e whole of @ . In view of (5.21)
t-rn-~
--
- m1
3,t-m
t h e idea is to d o partial integration with t h e defining integral of f m , w in (3.1)'. Consequently, w e a r e lead t o p o s t u l a t e conditions ensuring t h a t t h e integrand in (3.1)' is differentiable with respect t o t . So f i r s t of all we s u p p o s e t h a t
wI,~,~,
is a C' function. Moreover, we require f t o be continuously differentiable with respect to t h e variables i n G M ( o o ) ; more precisely, we s u p p o s e t h a t f belongs to
192
V. Quasihomogeneous A v e r a g e s
o f F u n c t i o n s . Part 2
C1'O(X) where from now o n we take the decomposition (1.14) as the basis for the definition of C r ' s ( X ) , i.e. in Definition 1.18 we have
V1 = G M ( d + ) and V 2 = ker M Then it follows by the chain rule and by (1.5) that for every x € X the function 10,bI+C, (5.22.a)
t H f ( M t x ) w ( t ) , is differentiable satisfying
at( f ( M , x ) w ( t ) ) = 1t ( D 1 f ( M t x ) * P + M M , xw) ( t ) + f ( M , x ) w ' ( t )
for every t E l 0 , b l where by D, w e denote differentiation with respect to the variables i n G M ( o + ) . I n order to compute the derivative w i t h respect t o coordinates we suppose that mixed real-complex coordinates are chosen according to Convention 1.24. This means, in particular, that the basis A of V,
contains a basis of
ker M . By lowe are going to denote the set of indices describing the coordinates of the points in ker M . More precisely, employing the notation from Convention 1.24 w e introduce
NotationS.1S. ( i ) J o : = { j € N , ; p i = O } ; (ii)'
?[+:={a= C ~ . ~ , S ) C ~P iL= O; for every j € J o )
(ii)'
2 1 ° : = { a = ( p , y , S ) E X ; P1. = O for every j E I N , \ J o ,
(iii)
For any ~ € 2 1we define (r+E2L'
(iv)
For arbitrary r , s C I N o ~ ~ mwe) set 2 1 ' I ' s S : = { a E X ; l a + l S r ,
;
and a o € ' u o
y=O=S} ;
by the equation a = a + + a O . laolSs}.
Note that i n view of (1.32) and Remark 1.30 we have (5.22.b)
D,f(M,x)-P+MM,x =
2 taM 2 o i ( t ) R,,j(x) ( d U f ) ( M t x )
UE'U~
jeN,
where X , : = { ~ c ' U ' ;I a l = l } and where the R , , i : V - @
are suitable polynomial
functions of degree C ( 1 , O ) which are almost quasihomogeneous of degree a M ; of course, only finitely many of them do not vanish identically. Note that in case
M is semi-simple we have
Combining (5.22.a) and (S.22.b) we see, in particular, that the derivative of t i + f ( M , x ) w ( t ) belongs to 2 ' ( 1 0 , b l , t - d d t ) for every del-m,OC provided that
5.b Merornorphic Extensions of f
(5.20)'
S(W') 2
1-3
193
f,,.w
.
-1
C o n s e q u e n t l y , if w e a s s u m e , in a d d i t i o n , t h a t (5.24)
lim w ( t ) t-d = 0
f o r every d ~ l - m , O C
t+O
t h e n f o r every m € C ( - a J , O ) t h e e q u a t i o n (5.21) a n d p a r t i a l i n t e g r a t i o n l e a d to b
(5.25)
fm,,(x)
=m 1
tCmat(f(M,x) w ( t ) ) dt
- m1 b - m f ( M b x )
w(b)
0
w h i c h in view o f ( 5 . 2 2 . a ) a n d (5.22.b) c a n be w r i t t e n as (5.26)
fm,w =
$,(
C C
Ra,j ( a a f ) m - a M , w u ,
+
frn-1.w'-
ac'U, jeN,
- b-'"
w(b)foMb).
N o w , a s s u m i n g t h a t n o t only ( 5 . 2 0 ) ' b u t even
is valid, t h e n in view o f S ( w w j ) = S ( w ) 2 0 w e observe t h a t t h e r i g h t - h a n d
of ( 5 . 2 6 ) is w e l l - d e f i n e d if R e m < c,
side
where
in f a c t , it d e f i n e s a m e r o m o r p h i c f u n c t i o n o n @ ( - ~ , c , ) w i t h v a l u e s in C o ( X ) a n d possibly with a ( s i m p l e ) pole a t m = 0. Consequently, (5.26) s h o w s t h a t m
H
fm,,
e x t e n d s to a m e r o m o r p h i c f u n c t i o n o n @ ( - a , c , ) w i t h v a l u e s in C o ( X ) . In order to e x t e n d t h i s m e r o m o r p h i c e x t e n s i o n f u r t h e r o n e w a n t s to iterate t h e a b o v e p r o c e d u r e . For t h i s a good c h o i c e for w s e e m s tci be t h e f u n c t i o n wo d e f i n e d by
w i t h E E C being a fixed c o n s t a n t . Since o n 1 0 , b l w e have w ~ = - E w t h e~ a s s u m p t i o n s (5.20) a n d ( 5 . 2 4 ) are s a t i s f i e d , i n d e e d . H o w e v e r , w h e n g o i n g to r e p e a t t h e e x t e n s i o n p r o c e d u r e w e have to w o r k w i t h t h e w e i g h t f u n c t i o n s w k : = w o wk , k E N , a s w e l l , w h i c h in view o f (5.29)
w k ( t ) = - E w k ( t ) + -tI w k - l ( t )
(where W - ~ : = O ) .
tEIO,bl,
n o l o n g e r s a t i s f y ( 5 . 2 0 ) " . To o v e r c o m e t h i s d i f f i c u l t y w e o b s e r v e t h a t t h e w k
194
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 t i l l s a t i s f y (5.20) a n d (5.24) so t h a t ( 5 . 2 6 ) may be a p p l i e d to w = w k . In view
of ( 5 . 2 9 ) a n d (1.38) t h e e q u a t i o n ( 5 . 2 6 ) t h e n b e c o m e s
T h e o n l y t e r m o n t h e r i g h t - h a n d side which is n o t d e f i n e d f o r e v e r y m E @ ( - m , c , ) is f m , w k - l .
(5.26),-2,.
Consequently, we insert
. . , (S.Z6)1 a n d
first t h e equation
(5.26),-,
and then
finally ( 5 . 2 6 ) i n t o ( 5 . 2 6 ) k until a l l t h e t e r m s c o n t a i n i n g
f m , w i , i E ( 0 )U N k , a r e e l i m i n a t e d . T h i s leads to k
S i n c e h e r e t h e r i g h t - h a n d side is w e l l - d e f i n e d f o r every m E @ ( - c o , c , ) t h a t t h e function C ( - a , O ) d C o ( X ) , m Hf,,,k,
w e see
e x t e n d s to a m e r o m o r p h i c
f u n c t i o n o n @ ( - a , c , ) w i t h v a l u e s in C o ( X ) a n d p o s s i b l y w i t h a p o l e of order 5 k + l at m = O .
In order to e m p l o y t h e above e x t e n s i o n p r o c e d u r e f o r a s e c o n d t i m e w e have to a s s u m e t h a t t h e d e r i v a t i v e s a a f , u E X 1 , b e l o n g to C ' ' o ( X ) t h e m s e l v e s . In f a c t , if w e fix r E [ N u ( m ) a n d a s s u m e t h a t f b e l o n g s to C r ' o ( X ) w e o b t a i n by i n d u c t i o n a m e r o m o r p h i c e x t e n s i o n to C ( - a , r c , ) (5.31)
xc( M I
:=
1
w i t h p o l e s lying in t h e set
X(M)
if E = O
'U(M)+IN,
if E # O
I t is desirable to k e e p t r a c k of t h e derivatives w i t h r e s p e c t to t h e variables in k e r M , as w e l l . To t h i s e n d f o r t h e r e s t of t h i s a n d t h e f o l l o w i n g t w o s e c t i o n s w e fix r E [ N u ( a ) , s E N , u ( a ) , set w = w k . I n s t e a d of c,
and f E C r ' S ( X ) . Moreover, w e c h o o s e k E N O a n d
(see ( 5 . 2 7 ) ) f r o m n o w o n w e w r i t e c,.
Proporltlon 5.16. The function C( - m , 01 3 m extension Di'f,, (i)
:C
the p o l e s of
( - w , r c ,I
mf,,
f,,
+C r S s ( X )having
has a (unique) meromorphic the following properties:
are contained in the s e t 21E,T,(M)defined b) 15.311; thej,
are simple if k = O and i f M is s e m i - s i m p l e ; (ii) s u p p a i ( m ; W f , , ) C I s u p p f ) ~ , 1 0 , b fl o r arbitrary r n E C ( - m , r c , ) and i E Z ;
S.b M e r o m o r p h i c E x t e n s i o n s of f
H
195
fm,,
(iii) For every polynomial Function P : V ' d C OF degree 5 f r , s ) we have
For arbitrary m E C I - w , rc,) and i €2' where g p , j := P:i (8) f , the pol-b,nomial functions P& being defined in the test preceding Remark 1.42.
N o t e t h a t f o r a r b i t r a r y m E @ ( - a , r c , ) a n d 4 E A ( P ) t h e p o i n t m - 4 b e l o n g s to t h e d o m a i n of d e f i n i t i o n of
~ g p , j , w , + j,
i n d e e d . In f a c t , s i n c e P:,i
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 d e g r e e 4 t h i s is s e e n f r o m t h e f o l l o w i n g s i m p l e r e m a r k w h i c h is given a s e p a r a t e f o r m u l a t i o n f o r t h e s a k e of e a s y r e f e r e n c e .
Remark 5.17. Let P E X ( M ) be such that the set A : = {
C Y E ~ K I ' . ~a; M = P }
is nori-
empt-v. and define p : = m a x { l a + Ia: E A } fi)
and
n : = m a x { l a o l ;a E A } .
For every po1)nomial function P : V * - - + C o f degree L f r . s ) which is almost
quasihomogeneous o f degree P and o f t-vpe M ' C'
-prs
m f .
the function P ( d ) f belongs t o
(XI.
u: By t h e very d e f i n i t i o n of
Q
and a the functions a"f, a E A . belong to
Cr-p*s-o ( X ) . S i n c e t h e a s s u m p t i o n o n P m e a n s t h a t P ( a ) f is a linear c o m b i n a t i o n of t h e d"f. ~ E A t h, e a s s e r t i o n f o l l o w s . (iil:W e c h o o s e a E A s u c h t h a t I a + I = p . T h e n
R e 4 = R e a M ? l a t l h m i n= p X m i , 3 p c r so t h a t
Re(m-4) < rc,-Re4 5 rc,-pc,
f o r every m E @ ( - c o , r c E ) . w
Proof OF Proposition 5.16. By t h e r e a s o n i n g p r e c e d i n g t h e f o r m u l a t i o n of t h e propos i t i o n w e o b t a i n a m e r o m o r p h i c e x t e n s i o n F f , w: @ ( - c o , r c , ) +
Co(X)of
m Hf,,,
s a t i s f y i n g ( i ) . In order to verify t h e d i f f e r e n t i a b i l i t y p r o p e r t i e s of t h e f u n c t i o n s in t h e i m a g e of Ff., w e f i r s t d e a l w i t h t h e c a s e
(5.33)
P = < - , y > for some y E G M ( a + ) u k e r M .
H e r e t h e f u n c t i o n s g e , j b e l o n g to C
r-q,s+l-q
( X )w h e r e q : = 1 if y E G M ( a + ) a n d
196
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
q : = 0 if y E k e r M
. It follows that
9g Q , j , w k + j ,P E A ( P ) a n d
meromorphic functions f r o m C ( - a , ( r - q ) c , )
of
jENo,
F u n c t i o n s . Part
2
are w e l l - d e f i n e d
i n t o C o ( X ) . If, in addition, w e fix
x E X a n d h c k s u f f i c i e n t l y s m a l l t h e n in view o f (1.38)t h e f o r m u l a ( 5 . 9 ) reads as
f o r every m E C ( - a , O ) . N o t e t h a t m-P b e l o n g s to C ( - a , ( r - q ) c = ) if m E @ ( - m , r c , ) and
e E A( PI.
H e n c e , 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 t h e f o r m u l a ( 5 . 9 ) '
r e m a i n s valid f o r every m
E
D,:=@ ( - a , r c , ) \ U , ( M ) . S i n c e 7g e , j , W k + j ( m - P )is c o n -
tinuous i t follows f o r every m E D r t h a t P ( a ) 7 f , , ( m )
By i n d u c t i o n o n e derives t h a t 7f,,
e x i s t s a n d is e q u a l to
i n d u c e s a m e r o m o r p h i c f u n c t i o n 3nf,,
defined
o n C ( - m . r c , ) w i t h v a l u e s in C ' ' ' s ( X ) . M o r e o v e r , t h e f o r m u l a
is a l r e a d y p r o v e d for t h e polynomial f u n c t i o n s P o f t h e f o r m ( 5 . 3 3 ) . C o n s e q u e n t l y ,
m a k i n g u s e o f C o r o l l a r y 1.37.(ii) w e d e d u c e by i n d u c t i o n t h a t ( 5 . 3 2 ) ' is valid if p is a p r o d u c t o f polynomial f u n c t i o n s of t h e f o r m ( 5 . 3 3 ) . S i n c e a n a r b i t r a r y P is a l i n e a r c o m b i n a t i o n o f s u c h p r o d u c t s w e c o n c l u d e t h a t ( 5 . 3 2 ) ' h o l d s for a r b i t -
rary P . In order to c o m p l e t e t h e p r o o f of (iiil w e fix m E @ ( - a , r c , ) , c h o o s e a > 0 so small that K ( m , a ) C @ ( - a , r c , ) and K ( z , a ) n U , ( M ) C ( z ) , z E ( m ) u ( m - A ( P ) ) , and d e f i n e y : C0,25cI+@ ( 5.32 )"
by y ( t ) : = a e i t . T h e n for a r b i t r a r y x E X a n d I E Z w e o b t a i n 1
2ni J P ( a ) [9Zf,,(m+z)] ( x ) z-'-'dz [P(a)aI(m;3nf,,)] ( x ) = -
Y and 1
' 3 n g p , j , w k + j ( m - 4 + z ) ( x 2) - I - I d z = a l ( m - 9 ; 3 n g p , j , w k + j
)(X)
Y Herce, t h e equality (5.32) follows from (5.32)'. w e o b s e r v e (see Remark 5 . 7 ) t h a t in c a s e R e m < 0 w e have F o r t h e p r o o f o f (ii) mf,,,(m)(x) = O ,
xEX\(suPPf)M,,o,bl.
197
5 . b Merornorphic E x t e n s i o n s of f H f,
By 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 t h i s r e m a i n s valid f o r every m e D,. F r o m t h i s t h e a s s e r t i o n f o l l o w s s i n c e f o r m € X , ( M ) n @ ( - a , r c , ) a n d I e Z w e have al(m;3nf,,)(x) =
1
J3Jtf,,(m+z)(x)
2-I-l
dz
Y w h e r e y is c h o s e n as a b o v e .
H
An e x p l i c i t f o r m u l a f o r
mf,.,
Notatlon S.18.
a o ( m ; 3nf,,),
f,,,,:=
is given in P r o p o s i t i o n 5.28 b e l o w
m E C( -a, rc,) .
If f s a t i s f i e s (5.19) t h e n t h i s n o t a t i o n is c o m p a t i b l e w i t h (3.1)' as is s e e n f r o m R e m a r k 5.14 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 t f i r s t g l a n c e it may s e e m to be s u r p r i s i n g t h a t f,,,,,
is d i f f e r e n t i a b l e o f t h e s a m e
orders a s f a l t h o u g h f o r its very d e f i n i t i o n derivatives are u s e d u p . H o w e v e r , a l o o k a t f o r m u l a ( 5 . 3 2 ) a n d a t R e m a r k 5 . 1 7 s h o w s t h a t f o r ( g 4 , j ) t n - P . w k to be +i
w e l l - d e f i n e d f o r m € C ( - a , r c , ) r e q u i r e s l o w e r order d i f f e r e n t i a b i l i t j of g p , js i n c e m - t lies to t h e l e f t of m .
A s f o r a l m o s t q u a s i h o m o g e n e o u s d i f f e r e n t i a l o p e r a t o r s w i t h variable c o e f f i c i e n t s t h e a n a l o g u e o f P r o p o s i t i o n 3.4 is valid. A s f o r t h e a s s u m p t i o n s c o m p a r e a l s o t h e t e x t preceding t h a t proposition.
Proposition S.19. Let t'6C and N E N . and let Po:XxVL+Q'
be a continuous co-
poljwomial function of degree 5 Ir,sl which is almost quasihomogeneous o f degreet', of order 5 N , and o f type M x l - M ) + . For every j 6 N N let P i : X x V * - @ be its j t t ' order deficiencj- which is a continuous copol),nomial function o f degree 5 ( r , s ) , a s well. Moreover, let p E N ,
be such that P i ( , \ , d l f E C p V s ( X )f o r ever)
j E { O l u G V N . Finall-v. let m E @ ( - m , m i n I r c , , p c , - R e e l ) . Then f o r ever-v i E Z we
have
and
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
198
In view o f N o t a t i o n 5.18, for i = 0 t h e e q u a t i o n s ( 5 . 3 4 ) a n d ( 5 . 3 4 ) ' t a k e t h e familiar form (3.4) and (3.4)' with w = w k .
m.If Re m is sufficiently s m a l l t h e n by Lemma 5 . 4 in
t h e defining i n t e g r a l for
f m , w w e may i n t e r c h a n g e differentiation a n d i n t e g r a t i o n . H e n c e p e r f o r m i n g t h e c o m p u t a t i o n s d o n e in t h e proof of Proposition 3.4 a n d taking (1.38) i n t o a c c o u n t w e o b t a i n for sufficiently s m a l l R e m N
1 ( k ~ j )( - i ) i ( ~ ~ ( x . a ) f ) , + , +, I~ ~
PO(x,a)fm,,=
j=O
and
N
c pi')P j ( x , a ) f m , w , + j
-
(PO(x,a)f),+,,,-
j=O
By t h e principle of @(-a, R ) \(&(
analytic continuation
M ) n ( X s (M ) - P
t h i s remains valid
for every m E
w h e r e R : = min{ rc, , p c , - R e @ ).
))
Now, for any m E C ( - a , R ) let a > O b e s u c h t h a t
K ( m , a ) C C ( - w , R ) and
-
K ( z , a ) n ' U , ( M ) C l z ) for every z E ( m , m + P ) ,a n d define y : C 0 , 2 ~ 1 - @ by t H a e i t .
T h e n f o r arbitrary I C Z a n d x E X it f o l l o w s t h a t [ P i ( x , ~ ) a I ( m ; 3 ~ f , w ) ] =( x )I
.I' [ P j ( x , a ) f , + z , w l ( x )
dz
Y
and aI(m+P;(Mpi(x.3)f,wk+i)(x) =
1
'
[ P i ( x , d )f l m + P + z ,+ Jw . ( xk) z
-1-1
dz.
Y Combining t h i s with t h e e q u a t i o n s above o n e o b t a i n s ( 5 . 3 4 ) a n d ( 5 . 3 4 ) ' . rn
A special c a s e is t h e following a n a l o g u e of Corollary 3 . 5
Corollary 5 . 2 0 . Let t < C and N E N , and let qo: X + C neous of degree P and o f order i N . Moreover, let
be almost quasihomoge-
PEN,
be such that for ever)'
j E { O l u N N the function q i f belongs t o C p ' o ( X ) where in case j 2 1 q i : X - C be the j
'*'order deficiency of qo .
Then for arbitrary m E C(- m , min ( r c , ,p c, - Re l I 1
and i g k we have N
15.351
=I
qo a i ( m ; f i l f e w )
( - ~ ) j ( ~ ; j )
a i ( m + e ; 3 r qI . f , W k+i
j=O
and N
(5.351'
=1ik;j)
ai(m+P;3zq,f,w)
j= 0
qi ai(rn;fiZf,Wk ) . m
+I
5.b M e r o m o r p h i c E x t e n s i o n s of f
H
199
fm,,
N e x t w e p r o v e t h e a n a l o g u e of P r o p o s i t i o n 3 . 7 , r e s p . R e m a r k s 5.5 a n d 5.6
Roporltlon 5.21. ( i ) I f A € L ( V . V ) c om m ute s with M then f o A E C r ' S ( A - l I X ) ) , and for arbitrary m E @ l - ~ , r c , a) n d i € E w e have a i ( m ; % ' f o A , w l = a i l m ; $ ~ f , , ) o A . ( i i ) I f @ s at i s fi es the assumptions o f Remark -3.67.( i i ) then fH E C r ' s I X ) , and f o r arbitrary m E C ( - a , r c , ) and i c Z w e have aj(m;9Rfm,,,) =(ailm;Tlf,,))H,. proOf. F i r s t o f a l l w e n o t e t h a t every A C L ( V , V ) c o m m u t i n g w i t h M l e a v e s every s u b s p a c e of t h e f o r m k e r Q ( M ) invariant w h e r e Q is a n y c o m p l e x p o l y n o m i a l in o n e variable. C o n s e q u e n t l y , t h e g e n e r a l i z e d e i g e n s p a c e s G M ( X ) , X E 0
and hence
G M ( b + ) a n d k e r M a r e A - i n v a r i a n t . T h i s s h o w s t h a t f o A E C r . S (A - ' ( X ) ) a n d - u n d e r
t h e assumptions of ( i i ) - f,EC"'"(X).
M o r e o v e r , every A c o m m u t i n g w i t h M
c o m m u t e s w i t h Mo so t h a t A - ' ( X ) s a t i s f i e s t h e c o n d i t i o n (4.171, as w e l l . N o w , by R e m a r k s 5.5 a n d 5.6 t h e a s s e r t i o n s ( i ) a n d ( i i ) are valid if m e @ ( - m , O ) . By t h e p r i n c i p l e o f a n a l y t i c c o n t i n u a t i o n t h e y remain valid f o r every m E @ ( - a , r c , ) n o t b e l o n g i n g to 2 I , ( M ) . W e f i x a n a r b i t r a r y m C
@(-a, , r c , ) ,
choose a > 0 such
t h a t K ( m , a ) C C ( - m , r c , ) and K ( m , a ) n X , ( M ) C ( m ) , and define y : C0,2~tl-+@
by y ( t ) : = a e i t . T h e n f o r a r b i t r a r q I E Z . a n d x C X w e c o n c l u d e t h a t
200
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
t c ) Computing l h e H e s l d u e s o f ’ ~ Z J D , ~
T h e following e x a m p l e is basic.
Example 5.22. Suppose that
E
= 0 . Let P E C and
N E N , and suppose that f is al-
most quasihomogeneous of degree P and of order
5
N . Let (ql)l_ci_cN be the se-
quence of i t s deficiencies. li)
If b = l then fm,,, is equal t o
(ii) In general,
!V?f, ,is holomorphic on C \ {PI. and the negative part o f its Lau-
rent series a t rn = P is given by ( 5 . 3 6 ) .
hoof.(i): S u p p o s e
t h a t R e m < m i n { O , R e P ) . Inserting (1.39) (for q o = f ) a n d
taking (1.38) i n t o a c c o u n t w e o b t a i n
where 1.
J t-m GJj(t)
I j ( m ): =
d t t ’
0
Performing partial integration o n t h e basis of b)j(t)
(5.21) a n d using t h e e q u a t i o n
= w i W l ( t ) / t w e o b t a i n l i ( m ) = l i - l ( m ) / m so t h a t in view of I,(m) = - 1 / m
w e conclude that
Inserting t h i s i n t o t h e equation above l e a d s t o (5.36).
( i i ) :Since
@-CO(X),
mHF,,,vk,
i s holomorphic w h e r e v : =
x
~ t h e a~ s s e r t,i o n ~
follows from ( i ) .
I t is easy to c o m p u t e t h e derivatives of f,,,,
a t t h e p o i n t s of M,(X)
if E = O :
RemmkS.23. Suppose that F = O . Let P E C , N E N , and P i , O S i S N , satisfj the
~
20 1
5.c C o m p u t i n g t h e R e s i d u e s of %f,-
assumptions of Proposition 5.19. Then for every x E M O ( X ) the Following assertions hold:
(i)
I f b = 1 then
(iil
The meromorphic Function on C ( - a , r A r J J i , ,defined ) by
estends t o a meromorphic Function on C which is holomorphic outside the point m = - P , and the negative part o f its Laurent expansion at this point is given bj (5.38).
proOf. jiJ: S u p p o s e t h a t R e m
mint 0 , -Re
J?
} a n d b = 1 . S i n c e by t h e s e c o n d p a r t
o f L e m m a 5 . 4 o n e c a n i n t e r c h a n g e d i f f e r e n t i a t i o n a n d i n t e g r a t i o n o n e o b t a i n s by m a k i n g u s e o f ( 3 . 4 ) a n d ( 1 . 3 8 ) a n d in view of M , x = x t h a t 1
N
=
1( - I ) ,
( k ~ i [) ~ , ( x , d ) f ] ( x I), , + , ( r n + t )
i=O
w h e r e l k , , ( m + 4 ) is d e f i n e d a s in t h e p r o o f o f E x a m p l e 5 . 2 2 . H e n c e t h e f i r s t a s s e r t i o n f o l l o w s by (5.37) a n d 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 .
m:S i n c e @ - C o ( X ) ,
m H f , , , , V k , is h o l o m o r p h i c w h e r e v : =
x
~
~
t h, e ~ a sIs e r -
tion follows from ( i ) .
Notation5.24. ( i ) For any m € C w e set of
U'
see N o t a t i o n 5.15). Of c o u r s e ,
( i i ) For e v e r y m E C ( - a , r A,,,) Q,
?[A:= ( a E U + ;a M = m }
U k #0
t h e so-called
( f o r t h e definition
if a n d o n l y if m c X ( M ) .
(almost) quasihomogeneous part
f OF degree m (and of t j p MI in F is d e f i n e d by X€X.
202
V.
Quasihomogeneous Averages
of F u n c t i o n s . Part 2
Proof. By f r e e z i n g t h e variables in k e r M o n e r e d u c e s t h e p r o o f to t h e case
X = G M ( o + ) . I n s t e a d of giving a direct c o m p u t a t i o n a l p r o o f by i n d u c t i o n based o n t h e m a i n t h e o r e m o f c a l c u l u s , I p r e f e r to give a s l i g h t l y less e l e m e n t a r y p r o o f which s e e m s to be m o r e s u i t a b l e to e x p l a i n t h e r e s u l t .
The case " E = O " : W e f i x i E N , set q : = a - i ( m ; 3 n f , w ) , a n d let d E l O , + a C . T h e n s u b s t i t u t i n g t = r / d a n d m a k i n g u s e of (1.64) a n d (1.65) o n e o b t a i n s f o r e v e r y z ~ @ ( - a , O )t h a t b
f,,,(MdX)
=
J' t - L f ( M , , l x )
dt t =
Uk(t)-
0
bd
= dz . \ ' r - L f ( M , x ) c d k ( r / d ) 7 dr = d Z
(X)
I=O
0
w h e r e v : = xj0 b d
k 1 ( - l ) l m l ( d )f L , V k - l
,. 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 e q u a t i o n r e m a i n s
valid f o r every z E C ( - a , r c , ) \ ' U ( M ) . C o n s e q u e n t l y , e x p a n d i n g dL-",
comparing
c o e f f i c i e n t s , 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 , t a k i n g ( 1 . 3 8 ) i n t o a c c o u n t , s u b s t i t u t i n g j = I + 4 , a n d r e - 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 o n e d e d u c e s k
m
S i n c e s u p p ( v - w ) is a c o m p a c t s u b s e t of I O , + a 3 C t h e f u n c t i o n 9 R f , v k - w k is h o l o m o r p h i c o n C ( - a , r c , ) so t h a t (5.40)
a - 0 ( m ; 312f, v j ) = a - p ( m ; 9Rf, w j ) ,
@ E Nj,E N o .
I n s e r t i n g t h i s i n t o t h e e q u a t i o n a b o v e o n e sees t h a t t h e s u m m a n d f o r j = 0 is e q u a l to q . H e n c e , ( 5 . 3 0 ) m e a n s t h a t q 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 . S i n c e
q is a Cr f u n c t i o n it m u s t be a polynomial by P r o p o s i t i o n 1 . 5 8 , i.e.
N o t e t h a t by ( 5 . 3 2 ) " a n d by t h e i n t e g r a l f o r m u l a f o r t h e L a u r e n t c o e f f i c i e n t s w e have
5.c C o m p u t i n g t h e R e s i d u e s of ntf.-
203
where t h e meromorphic function h,:@(-w,rXmi,)
(fz,,,)
-QI
is d e f i n e d b y h , ( z ) : =
(a)
(0)for z € @ ( - a , r X m i , ) \ ' U ( M ) . C o n s e q u e n t l y , R e m a r k 5.23 s h o w s t h a t
q'"'(0) =
lo
if i < k + l
-f'"'(O)
(-l)i-k
if i = k + l
( i k l ) [P,*,i-k_1(3) f](O)
if i > 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)
<
f o r R e m < 0 . W e fix q E { O ) u N , . a n d a f i n i t e
in I O , + ~ Is u c h t h a t 4, < c,
4, 5
+CE
P,
= 0 , and
iEINq.
,
T h e p r o o f is d o n e by i n d u c t i o n o n q . s t a r t i n g w i t h q = 0, t h e a s s e r t i o n t h e n being valid by ( 5 . 3 0 ) . In f a c t , if q = O t h e n P < c , so t h a t I p = ( l ) x X 1 * ao n d J p = # . a n d via t h e c h a n g e of s u m m a t i o n v a r i a b l e s
G,
= k + l - i a n d i + j = G~ t h e e q u a t i o n ( 5 . 3 0 )
b e c o m e s t h e f o r m u l a in t h e p r o p o s i t i o n i f w e set
For t h e i n d u c t i o n s t e p w e fix i E I N q - l u ( O ) , a n d f o r every ( N , p ) E l t i \ l t i + ,
.-
~ , E N , w e a p p l y ( 5 . 3 0 ) to ( a p f , m - h
- . . . - h p N , b o ) i n s t e a d of ( f , m , k ) . Defining
p, p o l y n o m i a l f u n c t i o n s P i , a , i of d e g r e e 5 ( 1 , O ) by
we obtain
and
S . d A Formula f o r f,
207
If R e m Z O
N o t e t h a t t h e f u n c t i o n s Pi,a,j are 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 X u a n d t h a t o n l y finitely m a n y o f t h e m do n o t vanish identically. I n s e r t i n g (5.45)i n t o t h e f o r m u l a in a s s e r t i o n ( i ) f o r 4 = Pi o n e o b t a i n s t h e f o r m u l a f o r ='k
ti+, by t a k i n g i n t o a c c o u n t t h e c o n d i t i o n s Soa=1 - la1
(5.46)
J e i + l \ Je,
lei\
=
It,+
\ lei = 1 ( N
,
I
and
(5.47)
+
1,
. . . ,[
3 ,a ~ ) ;
( N. P )
E I t i \ 1 p i + , a E .U"O
1
to be p r o v e d b e l o w . In f a c t , d e n o t i n g t h e s u m m a n d s in ( i ) f o l l o w i n g t h e s u m m a tion over ( N , B ) E l , (resp.
Jp)
by T , ( N , p ) ( r e s p . U , ( N , B I ) w e o b t a i n by i n s e r t i n g
(5.45)t h a t
1
T,,(N,P) = B - C
(N.P)E Ifi\ lei+
where
and
In order to c o m p u t e B w e i n t r o d u c e n e w s u m m a t i o n v a r i a b l e s o
ab:= i + j = ~
~
+
l
-
~
~ : = +0 0 + 1~- i
and
a~n d+ verify ~ + jt h e f o l l o w i n g s u m m a t i o n r u l e :
c c . . . = c1 c . . . = 2 . . . = c1 = c c m
r
0,+1
m
Y
oO
ON+1 = 1
j=O
W
Oo'oN+l-l
Consequently, defining
a o c N o 0 ~ + 1 = 1j = O m
ob=ao+l-oN+1
W
dN+1=1
co
Ob+ON+1-l
1
o,=o O o = d N + l - l
....
208
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
N
(pl,. . . ,p N , a ) =
and taking
i+, a n d
-"
N - I p I +So,
= N +1- I
--
B +u I into account
o n e o b t a i n s in view of (5.47) t h a t
F o r t h e c o m p u t a t i o n of C w e i n t r o d u c e t h e s u m m a t i o n variable o rename i as
oh,
~ : = +o 0 +~1 - i ,
a n d verify
c
OO
2 ... =
ooENo i = O
m
m
m
2 . . . = o b2= o 2
i = O oo=i
ON+1'1
Hence, defining
w e o b t a i n in view of ( 5 . 4 6 ) t h a t
is c o m p l e t e .
The assertion
is i m m e d i a t e l y o b t a i n e d f r o m ( S . 4 3 ) , ( 5 . 4 4 ) . ( 5 . 4 8 ) a n d ( 5 . 4 9 )
by i n d u c t i o n . U n d e r t h e a s s u m p t i o n of (iiil w e d e d u c e f r o m ( 5 . 4 3 ) a n d ( 5 . 2 3 ) t h a t
Similarly, o n e o b t a i n s t h a t
F o l l o w i n g n o w t h e l i n e s of t h e i n d u c t i o n s t e p a b o v e , o n e derives t h e a s s e r t i o n ( i i i ) in its f u l l g e n e r a l i t y f r o m ( 5 . 4 8 ) a n d (5.40) by i n d u c t i o n .
5 . d A Formula for f,,
w e deduce t h a t
p
p1 +
209
if Rem 2 0
. . . + pBN- l
- c , 5 t i . If, in addition, ( N , P ) d J p i t h e n
<
t h i s implies t h a t ( N , b ) f l e i .
Proof o f ( 5 . 4 7 ) . In view of ( 5 . 4 6 ) we f i r s t observe t h a t o n the right-hand side of ( 5 . 4 7 ) t h e s e t I p i \ I p i + l can be replaced by J
e i + l\ J p i .
x: I f ( N , P ) E l p i + l \ l p i then ( a ) p p l + , , . + p p N_> t i + 1,
(5.50)
and
( b ) pol+. . . + p p N - , < e i + l
9
and ( a ) p p , + . . . + ppN < ti ,
(5.51)
or
( b ) p p l + . . . + P ~ >~ P- i ~> O -
> t i it f o l l o w s i n view of ( 5 . 5 0 . a ) t h a t (5.51.a) c a n n o t hold so t h a t
Since
(5.51.b) is valid which implies t h a t N - l 2 1 . Moreover, from ( 5 . 5 0 . b ) w e s e e t h a t
( N - l , ( p l , . . . , p N - 1 ) ) belongs to
J e i + t\ J e i .
( a ) p p l + . . . + ppN < P i , l ,
(5.52)
and
( b ) p p l + . . . + ppN 2 t i .
From ( 5 . 5 2 . b ) and ( 5 . 4 2 ) it follows f o r every a € X 1 ' o t h a t ppl + . . . + p B N+ pa 2 P i
+
c, 2 t i + ,
which in combination with ( 5 . 5 2 . a ) implies t h a t ( N + l,(pl,.. . , O N , a )) belongs t o
I%+,
,
In view of (5.52.b) it does not belong t o l e i .
Combining Proposition 5.28 and Remark 5.17 with Propositions 5.13 and 5.16 o n e obtains
Corollary 5.29. The condition ( i i ) in assertion (5.1) is satisfied f o r E = F = C'"s(X ) and D = C(- m , r c , ) \ 21, ( M ) . More p r e c i s e l y , let W :X +LO. + a t be an) continuous weight function; then the condition ( i i ) in assertion ( 5 . 1 ) holds for E = C " S ( W ) , D = C ( - w , r c , I \ 2 1 z ( M ) , and F=C"S(UI where U:=(l+/P+(.)IIw W(b, with
and where W,,, i s defined b-v ( 5 . 1 8 ) .
I
210
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
Our n e x t aim is to f i n d e x p l i c i t f o r m u l a s f o r f z , w if z b e l o n g s to @ ( - m , r c E )n2l,(M). W e are g o i n g to d e a l w i t h t h e case z = O f i r s t .
Propomltlon 5.30. (i) For arbitrary k € N O and S E X w e have
(ii)
the integral in l i ) is equal t o
where Ra,j are the pol-vnomial Functions f r o m (5.22.6)
Proof.(ii): We
have to a p p l y (5.22) to w = l a n d t a k e (1.38) i n t o a c c o u n t
l i ) : C o m b i n i n g (5.25) a n d (5.2Y) w e o b t a i n f o r a r b i t r a r y m E @ ( - a , O ) a n d ~ E t X hat b
f,,wk(x)
= m1 ,\' t - m 3 t ( f ( M ~ x ) e - ' t ) c , ~ k ( t )1 d t + ~ f , ~ w- k1 _bl- m f ( M b x )
wk(b).
0
By i n d u c t i o n o n k t h i s l e a d s to
where
k
S , ( t , m ) : = t-'"
2 q ( t )m i . i=O
S i n c e by t h e main t h e o r e m of c a l c u l u s a n d by t h e c o n t i n u i t y o f t H f ( M , x ) w e h a v e
0
it f o l l o w s t h a t
5.d
A F o r m u l a for f m . , , if R e m
211
2 0
1
= m
k+l
w k + l ( t ) J(k.1)
Tk
tCTmdT.
0
Since
5 max{l,b-Rem}
w h e r e t h e r i g h t - h a n d side t e n d s to 1 a s m + O
t h e i n t e g r a l c o n v e r g e s to 1 u n i f o r m l y f o r t E 1 0 , b l as m + O .
S i n c e in view o f
(5.22) t h e f u n c t i o n I O , b l 3 t ~ a , ( f ( M , x ) e - f t ) c h k + l ( t ) is i n t e g r a b l e i t f o l l o w s by Lebesgue's D o m i n a t e d C o n v e r g e n c e T h e o r e m f r o m (5.54) t h a t lim
( fm,wk(x)
+
m-k-l
f(Mox)) =
m+m Rem < O b
=
-J'a,(
f ( M , x ) e-',)
( d k + l ( t )d t +
f(Mbx)e-EbWk+j(b),
X € X .
0
T h i s i m p l i e s ( i ) . N o t e t h a t it also gives a n i n d e p e n d e n t p r o o f of t h e a s s e r t i o n o f P r o p o s i t i o n 5.25 in case m = O .
Combining Proposition 5.30 with Proposition S.l6.(iii) and taking R,,joMo a 0 i n t o account one obtains
Corollary 5.31. For ever) P € X ( M ) and ever-) poljnomial function P : V ' -
C which
is a l mos t quasihornogeneous o f degree P and o f tqpe M u we have
where Pi :=
(aM* - [ l i p .
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 n e x p l i c i t f o r m u l a f o r f z , w in case z b e l o n g s to U , ( M ) . I t g e n e r a l i z e s t h e o n e f o r z = O in P r o p o s i t i o n 5.30.
Propodtion S.32. Let z E C ( - c o , rct I n # , ( M I . Then there is a (unique) famil) o f pol-vnornial functions P , , a , p : V - - C .
j , p E I N , , and
cr€U', o f degree i ( r , O ) , o n / )
finitel-\, many o f them not vanishing identicall) , having the following properties: ( i ) for every c c C we have +a fZ,+,
(-f)I
= j=O
where
I - Hi + T i - B j )
212
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
with
X j ( z 1 : = { a 6 2 1 +a; M f z - j , R e a M ? P - j }
with
P:=Rez,
and where
S
BJ . : -=
- r crEW+ ReaM
pj.a.p b
(aM+j-z)
wp(bi
(aaf)
O
~
;b
pEN,
(iil For arbitrary j . p E N O and a € # '
the function P,,cr,p is almost quasihomoge-
neous OF degree aM ; (iiil let
(j,a1ENOX21f
be such that a M + j = z and let P E N * ; i f p s k - 1 or j > r + l
then Pi,rr,p G O ; i f j S r and i f , in addition, M i s semi-simple then
N o t e that by Proposition 5 . 3 0 . ( i i ) w e have b
(5.551
/'at ( ( aa F )
( Mt
7
up+ ( t 1 d t =
s1 c -
0
p r o o f . The c a s e
L = 0 is
an obvious reformulation o f Proposition 5 . 3 0 . So w e suppose
t h a t z f 0 . Then 4 > 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
<
P so t h a t for aElN"'
w e have
21 4
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
C o m b i n i n g t h e c o m p u t a t i o n s a b o v e w i t h P r o p o s i t i o n 5 . 2 8 . ( i ) w e f i r s t of a l l d e d u c e t h a t t h e n e g a t i v e L a u r e n t c o e f f i c i e n t s of
mf,+,a t
m = z are given by
M o r e o v e r , f r o m t h e f o r m u l a in P r o p o s i t i o n 5 . 2 8 . ( i ) w e o b t a i n t h e main a s s e r t i o n o f t h e p r o p o s i t i o n by s u b t r a c t i n g t h e n e g a t i v e p a r t of t h e L a u r e n t series of Mf,w
at m = z and t a k i n g t h e l i m i t as m+z . Finally, c o m p a r i n g (5.57) w i t h t h e f o r m u l a s in P r o p o s i t i o n 5.25 w e c o n c l u d e t h a t f o r every pair ( j , u ) s a t i s f y i n g U M + j = z t h e polynomial f u n c t i o n
vanishes
identically in case p < k - 1 or j l r + l a n d is given by if p z k a n d j < r .
If M is s e m i - s i m p l e t h e n
(*)"
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 u M = z - j so t h a t
Pj .a ,p v a n i s h e s f o r p > 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 /
},
PEIN,
becomes a Frdchet space.
( i i ) Let L be any compact subset of V , s a t i s f , i n g 15.68). and l e t U be anj open neighbourhood of ( K
fl
Mo ( K ) ) u L ; then we have
F o r t h e p r o o f 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 .
Lemma5.58. L n X C X , for everj I M , I O . t I ) - b o u n d e d subset L of X , proOf. W e fix x E X \ X + a n d c h o o s e
E
> 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 )
Now we fix f€x:,k(K;@),
we choose @ € C T ( W ) such t h a t Q m @ = R . Since by Theorem 5.37.(iii) a l m o s t quasihomogeneous of degree m s u c h that ( 3 M - m ) it follows t h a t q : = f + ( - 1 )
k
k+l
@m,wk=( - 1 )
+m,wk
k+l
is
R(,+
is almost quasihomogeneous of degree m and
of order 5 k . Moreover, by Theorem 5.37.(ii) t h e support of
@m,wk
is contained
in W M . Consequently, in view of Proposition 2.31 we can apply Theorem 3.48 to X = V , , A = ( m l . j m = k , q m , j ..-
(-l)k-i (aM-m)k-i
q , Z = V , n ( K u W M ) , and
Y = ( V + \ Z ) u ( U n V + )( n o t e t h a t in view of (5.68) and U M 3 L M U W M we have
Y,=V+,
indeed). Hence we obtain a function + E C G ( V + ) such t h a t +m,L,k=q
and supp 4 C Z n Y C U . Since we may assume t h a t U is relatively compact in X and since supp $ is an M-bounded subset of V, Lemma 5.58 and t h e last inclusion imply t h a t s u p p + is a compact s u b s e t of V, so that by setting extends
+ to a
I t follows that
+Iv,,+:%
0 one
Cm function defined on the whole of V and again denoted by rp:=+-
(-1)
k
@
4.
belongs to CTCU). Since f equals qm,cs,kand is
@-invariant an application of Theorem 5.37.(viii) and Proposition 2.59 completes t h e proof of ( i i )
.
(i): That t h e semi-norms
T C a ~ re
well-defined follows in view of t h e assertion ( i i )
from Proposition 5.41.(ii). For t h e proof of completeness we f i r s t observe t h a t t h e assumption (1.14) implies t h a t Mx = P+ Mx = M P + x , x~ V , so t h a t by means of (1.32) one obtains
5.g
220
The L o c a l l y C o n v e x Spaces X Z , k ( E m )
M o r e o v e r , w e observe t h a t t h e s e m i n o r m s IIB d e f i n e d in L e m m a S.50 s a t i s f y
II@(R) = s u p { I R ' " + " ( ~ ) I ; a E x L ,
REQ (:
X E ~ ( ~ , } ,
K ;@) .
S i n c e ( 5 . 6 9 ) s h o w s t h a t t h e c o e f f i c i e n t s in t h e d i f f e r e n t i a l o p e r a t o r a M a r e f u n c t i o n s of P + x , o n l y , w e c o n c l u d e t h a t f o r every BEU' t h e r e are c o n s t a n t s Cg > 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
S i n c e by t h e c o m p a c t n e s s of P,
(n,),
X € Y .
a g a i n , w e have 1 + I Mo( x ) I ? c o n s t ( 1 + I x I ) ,
X E Y , w e c o n c l u d e t h a t f b e l o n g s to Y'(V), i n d e e d .
Proof o f Proposition 5 . 6 / .
u; If f = ( q m ) m , b , kfor s o m e v E Y ' ( V )
then Theorem
S.37.(iii) a n d P r o p o s i t i o n 5 . 4 1 . ( i i ) s h o w t h a t f h a s t h e d e s i r e d p r o p e r t i e s . C o n v e r s e l y , l e t f be a f u n c t i o n having t h e s e p r o p e r t i e s . T h e n w e c a n f i n d @ E Y ( V ) such that R : = Q , @ @ q : = f + ( -l)k@m,uk
e x t e n d s (3, - m ) k "
f . C o n s e q u e n t l y , by T h e o r e m 5 . 3 7 . ( i i 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 m a n d of order
W e s e t A = ( m ) a n d j m = k . Since t h e f u n c t i o n s q m , i : = ( - l ) k - i ( d M - m ) k - i q s a t i s f y t h e e s t i m a t e s in t h e a s s u m p t i o n of Lemma 5.62 a n d s i n c e in view of Proposit i o n 2.31 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 valid w e o b t a i n a f u n c t i o n + E Y ( V ) s u c h t h a t s u p p + is a n M - b o u n d e d s u b s e t of V + a n d s u c h t h a t + m , w k = q , i.e.
= 'PITl.Uk w h e r e
'p:=
+ - (-1)
k
@ E
Y ' ( V ) . S i n c e f is @ - i n v a r i a n t a n a p p l i c a t i o n of
T h e o r e m 5.37.(viii) a n d P r o p o s i t i o n s 2.SY.(ii) a n d 2.62 c o m p l e t e s t h e p r o o f of ( i ) .
(ii) : T h a t
t h e s e m i - n o r m s n p are w e l l - d e f i n e d f o l l o w s in view o f t h e a s s e r t i o n
( i ) f r o m P r o p o s i t i o n S . 4 1 . ( i i ) . W e observe t h a t f o r every R E Q , ( V ( V ) @ )
t h e semi-
n o r m s n o ( R ) d e f i n e d in P r o p o s i t i o n S . S 2 . ( i ) s a t i s f y 0
l l e ( R ) = s u p { l R ( a + ' ) ( x ) I ( I + ~ M O ( X ) ~x) E~ f: l ( ~ ) , c l E x @ ~ ,E X , I @ l S e } .
232
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
In view of (5.69) i t f o l l o w s that for every 4 E N we find constants C, and j a c N such that
I I o ( R ) 5 C, x j a ( f ) , N o w , let
(fi)jEN
fE‘Uz,k(y(V)@,), ~ = ( a , - m ) ~ + ’ f .
be a Cauchy sequence in the space F : = X g , k ( Y ’ ( V ) w ) , and
choose RIEQ,(Y’(V),)
such that Rj extends ( a M - m )k + i f j . Then ( R j ) is a
Cauchy sequence i n Q,,,(Y’(V)W).By Proposition 5.52 it converges to a limit R in t h i s space. Since (5.67) is valid w i t h ( f , R ) replaced by ( f i . R i ) we conclude that (5.67) remains valid for ( f , R ) if we denote by f the limit of ( f j ) in the space C-(V+
).
Hence f is almost quasihomogeneous of degree m satisfying the
equation R l v = (3, - m ) k ’ ’ f .
I t follows that f is the l i m i t of ( f i ) in the topo-
logy of F. To prove the other assertions we observe that the mapping P : Y ( V ) + fined by
‘p H ( ‘ p ~ ) , ~ ,is” ~ linear,
F de-
continuous and surjective, hence open by Banach’s
open mapping theorem. The rest of the proof is done as in the proof of Proposition 5.52.
233
Chapter VI
Constructing (Almost) Quasi homogeneous Distributions by Taking Quasihomogeneous Averages.
The Case: (1.14)holds
In this chapter we continue to assume that (1.14) holds. Instead of (5.17) we o n l y suppose that X \ X + f
0 .Our
aim is to carry over the theory of Chapter 4 .
For very special weight functions w the distributions u,,
( 4 . 3 ) , as before. This time one has t o employ the functions
will be defined by q
~
~
c p, E C~ T ( ,X ) ,
introduced i n Chapter S . Moreover, the support of u has to be a weaklj, ( M , l ) -
bounded subset of X in the sense of Definition 6.1 below. If m does not belong to the set ( - X ( M ) - p ) the results are parallel to those of Chapter 4 . However, if m lies in the exceptional set ( - X ( M ) - p ) t h e n complications occur w h i c h are
mainly due to the fact that the quasihomogeneity order of the functions
(P~,',,~
may exceed k , the support of its ( k + l ) t h order deficiency being contained in
X \ X + . So, i n particular, i n general ,u
will only be almost quasihomogeneous
of degree m , the support of its deficiency being contained in X \ X + . In order later to be able to apply the Fourier transform we develop the theory for temperate distributions, as well.
In section
(a)
t h e weakly (M,I)-bounded subsets of X are introduced and studied.
In section ( b ) f o r every mCQ3 the distributions urn,, are defined, and their basic properties are established. I n section ( c ) the almost quasihomogeneous distribu-
t i o n s o n X w i t h support contained in X \ X , are determined and described in a way which serves the purposes of the following sections. In section ( d ) those almost quasihomogeneous distributions T such that supp
(aM - m ) k + l T is
contained
in X \ X , are described in terms of quasihomogeneous averages of distributions with weakly M-bounded support. For these averages in section ( f ) a duality bracket
234
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 introduced required in Chapter 7 . Section ( e ) deals w i t h the solvability of the
equation ( a M - m ) S = T for arbitrary T E D ' ( X ) .
ta) W e a k l y t M . I)-bounded S u b s e t s 01' X
L e t 1 be a closed subinterval of 10,+a~C.And let L be a subset of X .
Deflnltlon 6.1. L is called a weak?,. IM,I)-bounded subset of X if for every comis a compact subset of X,,,
pact subset K of X the set < n X n K,,,,,
X
x, : =
(6.1)
where
i f I is bounded
X , i f I is unbounded If I = I O , + c o C we also say weak!,, M-bounded instead of weak!,, IM,I)-bounded.
Obviously, every compact subset of X,,,
is a weakly (M,I)-bounded subset of X .
Moreover, if I is compact then every subset of X is weakly (M,I)-bounded so that in this case the notions of weakly (M,I)-boundedness and (M,I)-boundedn e s s coincide. The same turns out to be true i f I is merely bounded. This follows
from part A of the proposition below characterizing weakly ( M , I ) -bounded subsets of X . For its formulation we require
Notatlon 6.2. By Note that
2
2 we
denote the set of all interior points of X u ker M
,
is the set of all interior points of X u M 0 ( X ) and that -"
(a) ( X ) + = X + ,
(6.2)
and
(b) (X+)"=?.
-"
Moreover, X is a quasihomogeneous open subset of V . Note that there are two other important quasihomogeneous open subsets of V related to X , namely, the set X o = M G 1 ( X ) already introduced i n Notation 1.59 and the s e t M i l ( M o ( X ) ) . Obviously, we have (6.3)
6.a
235
Weakly ( M , I ) - b o u n d e d Subsets of X
The s e t
Xo will play an i m p o r t a n t r o l e l a t e r in this chapter. The s e t M,'(Mo(X))
already appeared in C h a p t e r 5 , though implicitly, i n t h a t t h e condition (5.17) a m o u n t s to equality being valid everywhere in ( 6 . 3 ) .
Proporltlon 6.3.A. S u p p o s e t h a t I =lo,b l for s o m e b € 1 0 ,+a[. Then (il
t h e following conditions a r e equivalent: (a)
L i s a weakly ( M , I ) - b o u n d e d s u b s e t of X ;
(b)
for s o m e (resp. ever)) Y € { X , X + , X ]L is a n ( M , I ) - b o u n d e d s u b -
Y;
set of
(c)
LnK,,,,,
C X , f o r everj' c o m p a c t s u b s e t K of Y for s o m e Iresp.
ever).) Y C ( X , X + , ~ I ; ( i i ) each of t h e conditions of l i ) implies t h a t
(d)
i n 2 c x,:
t h e converse i s valid provided t h a t (5.17) holds.
However, if I is an unbounded proper subinterval of 10,+031 t h e n t h e description of
t h e weakly ( M , I ) - b o u n d e d s u b s e t s of X looks different:
Proporltlon 6.3.B. S u p p o s e t h a t I=Ca.+a.C for s o m e aE1O,+LvC. Then L i s a weakly ( M . 1 ) - b o u n d e d s u b s e t of X if a n d on1.k. if L n X , i s a n ( M , I ) - b o u n d e d s u b s e t
of
x,.
So, if I is a n unbounded proper subinterval of 10,+aCthen weakly ( M , I ) - bounded s u b s e t s of X may contain points of X \ X,
;
and since by Remark 3.17 t h e s e points
a r e not ( M , I ) - b o u n d e d i n X one s e e s t h a t not every weakly ( M , I ) - b o u n d e d subset of X is ( M , I ) - b o u n d e d .
Combining Propositions 0.3.A and (1.3.8 one obtains
Propoeitlon 6.3.C. L is a weak/), M - b o u n d e d s u b s e t of X if a n d on1-t. if L is a n M - b o u n d e d s u b s e t of X , .
H
Note t h a t if L is a n ( M , I ) - b o u n d e d s u b s e t of X then L n X , is a n ( M , I ) - b o u n d e 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 D i s t r i b u t l o n s . Part 2
236
s u b s e t o f X,
.
H e n c e , f r o m P r o p o s i t i o n 6.3 o n e o b t a i n s
Corollary 6 . 4 . I f L is an (M,II-bounded subset o f X it is a weakly IM,I)-bounded subset o f X . 8
A n o t h e r c o n s e q u e n c e o f P r o p o s i t i o n 6.3 is
Corollary 6.5. Let Y be a quasihomogeneous open subset o f V containing X . I f L'is a weakly (M.I)-bounded subset o f Y then L ' n X is a weakl-\ lM,I)-bounded subset OF X .
W fT.h i s
f o l l o w s f r o m P r o p o s i t i o n 6.3 in view of t h e f a c t t h a t if L ' n Y , is a n
( M , I ) - b o u n d e d s u b s e t o f Y, t h e n L ' n X , is a n ( M , I ) - b o u n d e d s u b s e t o f X , .
F o r t h e p r o o f o f P r o p o s i t i o n 6 . 3 . A w e require
Lemma6.6. Suppose that 067. Then f o r ever) compact subset K o f X there is a compact subset H of X , such that (6.4)
proOf. W e fix R E I O , + m C s u c h t h a t K C n , , , , ] .
T h e n H : = K M , l o , l l n f I ~ , , is c o m -
p a c t f o r it is closed in V a n d - by Lemma 5.9 - c o n t a i n e d in t h e b o u n d e d s u b s e t {
XE
V ; IP+x I = R ,
1 Mox I 5 m a x ( I k l ; k € K 1 1 . M o r e o v e r , by L e m m a 5.Y a n d
H lies in X + . Now let x € X + n K , , , / , .
(5.70)
W e fix k € K a n d t E I s u c h t h a t x = M , k .
T h e n P + (k ) 2 0 , a n d by t h e c h o i c e o f R a n d by t h e last p r o p e r t y o f P r o p o s i t i o n 1.70 w e find s ~ 1 0 , l Is u c h t h a t M 1 / s k € C l i R , . S i n c e s t E I , as w e l l , it f o l l o w s t h a t X=MstMl/skEHM,l/I.
Proof o f Proposition h . 3 . A . l a )
* (c)f o r Y = X :
t h i s is a d i r e c t c o n s e q u e n c e of
Definition6.1 since under t h e present assumption o n I w e have X , , , = X + s i n c e f o r every c o m p a c t s u b s e t K o f
x the
and
set L n x n K ~ , , , , c o n t a i n s LnK,,,,,
a n d h e n c e t h e c l o s u r e o f t h e l a s t set if it is c l o s e d i t s e l f . f c ) " ( ( d l 4 (5.17))
* ( b ) Let K be a c o m p a c t s u b s e t o f Y. If J : = { tcI ; M , ( K ) n L # (B} :
is n o t e m p t y w e c h o o s e a s e q u e n c e
in J c o n v e r g i n g to s : = i n f J E C 0 , b I .
237
6.a Weakly ( M . 1 ) - b o u n d e d S u b s e t s of X
We f i x a s e q u e n c e ( k i ) in K s u c h t h a t ti : = M t j k j E L f o r every
j 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 i ) c o n v e r g e s to s o m e x E K . By c o n t i n u i t y (see C o r o l l a r y 1.9) it f o l l o w s t h a t N o w , if ( c ) h o l d s t h e n M,x
c o n v e r g e s to Msx E L n KM,,,,
([,)
.
does n o t b e l o n g to k e r M so t h a t s > 0 . O n t h e
o t h e r h a n d , if (5.17) is valid t h e n t h e a s s u m p t i o n s = O i m p l i e s t h a t M,x b e l o n g s to L n M , ( K ) C X \ X ,
f b ) for Y = ?
+
so t h a t ( d ) is v i o l a t e d . T h i s e s t a b l i s h e s ( b ) .
( b ) for Y = X
* ( b ) for Y = X +
:
t h i s is trivial s i n c e t h e p o i n t s o f
k e r M are n o t ( M , I ) - b o u n d e d in X or X . w
f b ) for Y = X +
( d l n ( c ) for Y = X : In ( b ) w e m a y a s s u m e t h a t 1=10,11. H e n c e , N
a p p l y i n g L e m m a 5 . 5 8 to X i n s t e a d o f X a n d t a k i n g ( 6 . 2 . a ) i n t o a c c o u n t w e d e duce (d). m
To derive ( c ) w e f i x a c o m p a c t s u b s e t K o f X . By Lemma 6 . 6 w e f i n d a c o m p a c t s u b s e t H o f X , s u c h t h a t ( 6 . 4 ) h o l d s . C o n s e q u e n t l y . t h e set LnK,,,,,
is c o n -
-
t a i n e d in L n H M , , / 1 . S i n c e by P r o p o s i t i o n 3.32 t h e l a s t set is a c o m p a c t s u b s e t m
o f X, t h e c o n d i t i o n ( c ) f o l l o w s f o r Y = X . ( b ) n ( d ) + ( a ) : Let K be a c o m p a c t s u b s e t of X . T h e n it f o l l o w s b y Proposi-
t i o n 3.22 f r o m ( b ) f o r Y = X t h a t i n K , , , , ,
is a c o m p a c t s u b s e t o f X . H e n c e
in view o f ( d ) it s u f f i c e s to s h o w t h a t (6.5) S i n c e l / ( l / I ) = I is b o u n d e d t h i s f o l l o w s f r o m Lemma 5 . 9 .
Proof of Proposition 6 . 3 . 8 .
"3": Let K be a c o m p a c t s u b s e t o f X , .
S i n c e 1 / I is
b o u n d e d a n d s i n c e K n M o ( K ) = @ it f o l l o w s by Lemma 5 . 9 t h a t K M , , / I is a closed _ _ _ subset of V . Consequently, L I I X n K M , l / l = ( L n X + ) n K M , , / 1 , and the assumpt i o n o n L i m p l i e s t h a t t h e set o n 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 o f X , . H e n c e P r o p o s i t i o n 3.22 s h o w s ( n o t e t h a t ( 3 . 1 8 . b ) is s a t i s f i e d f o r X,
i n s t e a d of
X in view o f R e m a r k 1 . 6 7 ) t h a t L n X , is a n ( M . 1 ) - b o u n d e d s u b s e t o f X,
"e": Let K b e a c o m p a c t s u b s e t of X , a n d let ( t m ) m E Nbe a s e q u e n c e in L n X n K M , l / l . W e have to s h o w t h a t a s u b s e q u e n c e c o n v e r g e s to s o m e ( E X . To t h i s e n d w e f i x s e q u e n c e s (k",) in K a n d (t,)
in I s u c h t h a t (P,-M,,k,)
c o n v e r g e s to 0 as m + 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,)
c o n v e r g e s to s o m e X E K . W e f i x
E >
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) <
E
into account - we
for m ? N . By combining this
with ( 6 . 6 ) and the assumption ' * x @ X , " we conclude that l P + ( M l / a t m ) l>
E
for
every m t N . In view of the last property i n Proposition 1.70 we can therefore choose s,ECa,+mC
such that y,,,:=MI/s,t,
-
lies in f l : = { z E P ( x , ~ I) P ; +zI=E}.
A
Since K is a compact subset of X + the requirements above are satisfied, again. rn
From Proposition 6 . 3 we are going to deduce that Lemma 3.29 carries over to weakly (M,I)-bounded subsets of X . To t h i s end (and for other purposes) we once and for all fix a continuous function x : X + + l O , + m C
which is quasihomo-
geneous of degree 1 . This is possible by Corollary 3.36 and Theorem 3 . 3 9 . Recall
6.a
239
Weakly ( M . 1 ) - b o u n d e d S u b s e t s of X
that by Proposition 1.61 x extends to a continuous quasihomogeneous function
i : X+
-' ( 0 )= X \ X,
CO, +aC such that
.
Lemma6.7. Let x b e a s above. Then L is a weakly ( M , I ) - b o u n d e d subset of X i f and on/),i f 16.8)
i ( LnKMln
is a relative]-v compact subset of
for every compact s ubse t K of X I ( r e s p . X ,
I
).
proOf. If I is an unbounded proper subinterval of 10,+0>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 <
M,x = lim M t j y j = lim t j = z j+cc
j i m
( t i ) converges t o s o m e
+a.
If s = O t h e n 1 / 1 is unbounded, b u t t h e n X , / I = X + so t h a t Z E X , which c o n t r a d i c t s ( 6 . 1 0 ) . C o n s e q u e n t l y , 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
_<
N(m*) (for
the definition o f N ( m * ) see Proposition 5.45): ( i i ) the support o f Q& u is contained in M o ( s u p p u ) n X , the latter being a closed subset o f X which is contained in X \ X,;
liii) i f X = V and if u is temperate with ( M , l l , + ~ L I - t e m p e r a t esupport then
Q A u is temperate, as well. and (6.18) holds for every rpESP(V); QAu =
(iv)
n
r
(-a)axa~ a!
i )
ae2Clfir
where &' is defined by
Note t h a t in c a s e X = V and k e r M = ( O ) t h e equation i n ( i v ) re a ds a s
Q
(iv)'
~ =uA -c a! 1. < u , s a > 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
<(
Q ~ u0 )M,
,'p
> = t-" < Q
~ u , 'M1/, ~ o
> = t-'I
< u . (Q,*
(p) O M l / ,
>=
e = t-' ( l / t ) m *
wi(l/t) i=O
so t h a t in view of (1.65) and ( 2 . 5 ) we obtain
e ( Q k U ) O M t = trn
w i ( t ) Qk((aM-lll)'U).
i=O
(ii) ; We
f i r s t n o t e t h a t t h e condition " K n Mo ( supp u ) = #
"
implies t h a t Z = #
250
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
so t h a t by Remark 4.1 we have
'p
> = 0 for
every
'p E C g ( K ) .
Since by Lem-
ma 6.9.(ii) Mo( s u p p u ) n X is a closed s u b s e t of X t h e assertion is proved.
(iii):
Since by t h e assumption s u p p u is ( M , C l , + m C ) - t e m p e r a t e it follows from
Proposition 5.52 t h a t Q m * y belongs to Y ( s u p p u ; V ) f o r every ' p E Y ( V ) . More precisely, let R E I O , + m I be such t h a t supp u C function x E C r n ( V ) equal to 1 on such t h a t all t h e derivatives of
x
flLo,R+,]
n]O,R].
By Lemma 5.53 w e find a
with s u p p o r t contained in
OLo,R+2,
are bounded. Then it follows by t h e Leibniz
rule and by Proposition 5.52 t h a t t h e map Y ( V ) + Y ( V ) ,
'p
H x Q m * ' p , is well-
defined and continuous. In view of Lemma 6.19 and (6.17) this proves t h e assertion.
( i v ) : is
immediately deduced from t h e definitions.
We can now write down a convenient formula for the negative Laurent coefficients of t h e meromorphic functions %lu,w :
Remark 6.25. Suppose that I is unbounded. Let
$?u,w
be the meromorphic func-
tion defined in Proposition 6 . 1 5 . Then for every m € ( - 2 l ( M ) - p ) we have
&F.
One has to combine (6.14) and (6.12)' with Proposition 5.25 and Theorem
5.37.(iv), respectively, and with Proposition 5.48.(iii) and ( 2 . 5 ) .
A s f o r t h e computational properties of QAu , making use of Corollary 5.47 ( i n
order to cope with t h e fact t h a t (5.17) may not be valid) one immediately derives t h e following result from Proposition 5 . 4 8 .
Propodtion 6.26. Let u
E
ail,+ m T ( X ) .
( i ) a"Q'rn u = Q k . . < a , p > ( a a u )
Then we have
for every a E X ;
( i i ) q Q A u = Q k , / ( q u ) for every P E C and every q € C " ( X ) which is almost quasihomogeneous o f degree C; (iii) P ( s , d ) [ Q , ' , u ] =Q ~ + p ( P ( s , d ) u For ) every P E C and every Cm copolynomial
Function P:XxV*+ type M X ~ - M I * :
C which is almost quasihomogeneous o f degree P and o f
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 i n X \ X +
251
(iv) ( Q A u ) o A = Q A ( u o A ) for every A E G L ( V , V ) commuting with M ; (v)
(QA u
) = ~QA ( u o ) i f B satisfies the assumptions of Remark 2.67. ( i i l ; in
particular, i f u is @-invariant so is Q A u .
I
The next proposition characterizes t h e almost quasihomogeneous distributions w i t h s u p p o r t c o n t a i n e d in X \ X + .
Pmpodtlon 6.27. Let d E 3 ' ( X ) ( i ) The following conditions are equivalent:
( a ) d is almost quasihomogeneous o f degree m with support contained in
x\x+; (6) d = Q A d ;
( c ) there is a famil.). of distributions d , 6 3 ' ( X ) , cr62{:,* , with support contained in X \ X, which are quasihomogeneous of degree - p such that
(iil The distributions d , in condition (cl of ( i ) are uniquel). determined, in f a c t , w e have i?
(6.20.6)
d, =x'd = s'd
(see (6.10));
in particular, d vanishes i f and only i f x'd = 0 f o r evecv a€#:*
Proof. ( a ) * ( b l :
.
F o r every j € N O w e set w i : = w i x l o , l l . Let q € C r ( X ) . T h e n
: = q ~ , , , * , ~is ~ a w e l l - d e f i n e d C- f u n c t i o n o n V w i t h s u p p o r t c o n t a i n e d in t h e
set ( s u p p 'p ) M , l O , l l
.
By P r o p o s i t i o n 5.27 w e have
(dM-m+)Qj =
'p-Qm*'p
if j = O
- aj-1
if j e l N
In view of ( 2 . 5 ) t h i s i m p l i e s by i n d u c t i o n t h a t (6.21)
'p=
~,*'p-~(a,-m)j
+1
aj,
j€[N,.
N o t e t h a t b y C o r o l l a r y 5 . 4 7 a n d Lemma 6 . 2 3 s u p p Qm*'p is c o n t a i n e d in L M , j 0 , i , f o r s o m e c o m p a c t subset L o f X . C o n s e q u e n t l y , s i n c e by P r o p o s i t i o n 6 . 3 . B s u p p d is a w e a k l y ( M , C l , + m 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 d is w e l l - d e f i n e d
252
VI. Quasihomogeneous Averages OF Distributions. Part
2
o n each summand of t h e right-hand side of (6.21) separately (see Remark 4.1) so t h a t
< d , 'p > = < d , Q,*
'p
> - < (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
~<(-i))ada,xpyoMo>=
cr€'u;*
5
=
A
( a a x P ) cpoM,
> = < 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>=. U
In P r o p o s i t i o n 6.27 w e s a w t h a t t h e d i s t r i b u t i o n s o n 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 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 r e p r e c i s e l y t h e distrib u t i o n s OF t h e f o r m Q L u w h e r e U E ~ ~ , , + ~ ~For ( Xs o)m.e o f t h e p r o o f s in sect i o n s ( d ) a n d ( e ) it is c r u c i a l to k n o w t h a t u c a n be c h o s e n to b e l o n g to 9 L ( X ) . w
A s w e s h a l l see in a m o m e n t , t h i s is n o t a l w a y s p o s s i b l e e x c e p t w h e n X = X .
Lemma 6.31. Let d E D ' ( X ) be almost quasihomogeneous o f degree m with support contained in X \ X , . Then there esists uED;,~+,,,(X) with support contained in
X , satisfjing the equation d = QA u and having the following properties: i f O E i t h e n (suppu)nKMM.i,l n k e r M C
(i) of
g\X
f o r every compact subset K
2:
(ii) i f s E No u {aI and i f d = Q,: T, f o r some f E C o ' s ( X I then u is induced b),
a
c
LY'.
s
(iii) i f
function on X ;
X = V and i f d
69"(
V ) then u be/ongs t o Y,&( V ) , more precisel-), u is
temperate with support contained in f l r l , a I .
255
6.c Almost Q u r 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 ,
h.oof.In view o f
P r o p o s i t i o n s 6.27 a n d 6.26. ( i ) it s u f f i c e s to p r o v e t h e case m = - p .
N o t e t h a t by C o r o l l a r y 6.28 d is t h e n q u a s i h o m o g e n e o u s o f d e g r e e - p . By x w e d e n o t e t h e f u n c t i o n x , : V,
-----;,10,+00C
given in P r o p o s i t i o n 1.70 f o r X = V .
C a s e f : k e r M = ( O ) . T h e n X = V a n d X \ X + = (01 so t h a t by C o r o l l a r y 2.20 w e have d = c 8 for s o m e c E C . W e c h o o s e + E C ; ( K ( O , 2 ) \ K ( O . l ) ) a n d set u : = c T + E & ' ( V ) C Y ' ( V ) . F o r every cpEC;(V)
such that $(0)=1
w e have
A
< G , c p > = < u , ' p o M o > =c p ( O ) c + ( O ) = < d , c p > . Case -3: k e r M # (0). U s i n g t h e n o t a t i o n i n t r o d u c e d in ( 6 . 2 2 ) w e f i r s t s u p p o s e t h a t
c h o o s e +EC;(V,)
&V,
w i t h s u p p o r t c o n t a i n e d in K ' ( O , Z ) \ K ' ( O , l ) s a t i s f y i n g
"
U
$ ( 0 ) = 1 , a n d set u : = T + @ d w h e r e d is d e f i n e d a c c o r d i n g to ( 0 . 2 4 ) . T h e n by
Remark 6 . 2 0 t h e set x ( s u p p u ) is a c o m p a c t s u b s e t of IO.+mC. H e n c e by Lemm a 6 . 7 s u p p u is a w e a k l y M - b o u n d e d s u b s e t of V. For a r b i t r a r y
'pi
ECF(V1)
a n d r p p , ~ C r ( V o )w e have by ( 6 . 2 4 ) a n d ( 6 . 2 5 ) t h a t < ~ , ' p 1 8 ' p 2= >
"
< T + . ' p l( 0 ) cl > < 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
=
< G k , 'p
>=
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 , @ > +
=
7
<(a,-m)'T,+i'p1,,.,,,)
+
<X+T,Ql,,+'p>,
i-0
and (6.43) follows. For an alternative proof of (6.43) in case
x+-1
o n a neighbourhood of X n M o ( X )
(compare Lemma 6.47 below) one proceeds as follows: Denoting the distribution
zF=o(
+i
( 0 , - m ) ' T ) m , w k by S one deduces from Proposition 4.13 that the support
o f T-S is contained in X \ X , so that the assumption o n
x9
and Proposition 6.27
imply that T - S = Q:,(x+,T- x + S ) . Consequently, to prove (6.43) one has to show that Q L ( x + S )= O . This can be carried out by way of similar computations as above. For the proof of (0.44) one easily checks that the computations done in the End
of the p r o o f o f 7heorem 1.8 and in the derivation of (4.14)' remain valid i n the present context.
As our final preparation for t h e end of the proof of Theorem6.37 we derive the
following formula which is a variant of (0.43) i n case T is of the form T = u ~ , ~ ~ .
Remark 6.43. For
every u E a h ( X ) the distribution urn,,k
is equal t o
proof. As before, we define N by (6.35). Let q E C T ( X ) .Then from (6.12 and (6.4.5) we deduce for arbitrary x C C 6 ( X ) and q € N o that
6.d Characterizing
(Almost)
Quasihomogeneous
269
Distributions
N
By (1.38) a n d (2.5) t h i s i m p l i e s t h a t
Inserting the equations (dM-m)'um,,k=
( - l ) ' ~ ~ , , , ~O- <~i < k , a n d a p p l y i n g
(6.46) to ( + i , k - i ) i n s t e a d of ( x , q ) w e o b t a i n t h a t
and
Inserting (4.18) w e deduce t h a t
By ( 0 . 3 9 ) a n d ( 4 . 1 9 ) t h i s is e q u a l to urn ,,,,k
.
M o r e o v e r , i n s e r t i n g ( 4 . 1 8 ) a n d i n t e r c h a n g i n g t h e order of s u m m a t i o n w e o b t a i n
S i n c e by ( 6 . 3 9 ) t h e s u m in s q u a r e b r a c k e t s is e q u a l to
(kLj)
t h e p r o o f is c o m -
plete. F o r a n a l t e r n a t i v e p r o o f o n e c a n a p p l y T h e o r e m 6 . 4 2 to T = u,,,,~.,~ a n d use of (6.41) - verify t h e e q u a l i t y
- making
270
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
As a c o n s e q u e n c e of t h e c a s e " k = O " o f R e m a r k 6 . 4 3 w e d e d u c e t h e f o l l o w i n g
s t e p in t h e p r o o f of T h e o r e m 6 . 3 7 .
Lemma 6.44. Let d E a ' f X ) b e almost quasihomogeneous of degree m with support contained in X \ X +
.
Then w
(i)
w
( X I if and on]), if d e s t e n d s t o a distribution d E S ' ( X )
d = u , ~ ,For s o m e u
which is almost quasihomogeneous OF degree m , a s well; ( i i ) if X = V and i f T is temperate then in fi) one can choose u t o be temperate, a s well, with M-temperate s u p p o r t .
Proof. fi ) , j : In view -
of ( 6 . 2 . a ) R e m a r k 6.14 s h o w s t h a t u e x t e n d s to a d i s t r i b u -
I<
t i o n C E ~ ~ ( XS i)n c. e (C),,,
extends u ,
it f o l l o w s t h a t it e x t e n d s d . S i n c e by
P r o p o s i t i o n 0.35 it 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 t h e i m p l i c a t i o n
''a "
is p r o v e d . ,<
( i ) . (r: O b v i o u s l y , w e may s u p p o s e t h a t X = X . T h e n by Lemma 6.31 w e c a n c h o o s e
v ~ a h ( X s)u c h t h a t d=Q:,v.
By R e m a r k 6 . 1 4 a n d by P r o p o s i t i o n 3 . 4 2 w e f i n d a n
M - b o u n d e d o p e n s u b s e t L of X, s u c h t h a t L 3 s u p p v . M o r e o v e r , a p p l y i n g Lemm a 3 . 4 5 to I : = l e , + i u C w e o b t a i n a n o t h e r M - b o u n d e d o p e n s u b s e t Y of X , having t h e p r o p e r t i e s l i s t e d in t h e c o n c l u s i o n o f Lemma 3 . 4 5 . By T h e o r e m 3 . 3 9 a n d Rem a r k 6.12 w e f i x a n o n - n e g a t i v e C m f u n c t i o n
JI
o n X 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 ( 6 . 3 6 ) h o l d s . T h e n f r o m ( 3 . 1 ) ' a n d ( 3 . 3 7 ) w e d e d u c e t h a t +o,o,l> 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<
( b l + f ( c l . I f + E C K ( X + ) is s u c h t h a t
x+:=x-+
t h e n in view of (5.63) a n d ( 4 . 1 0 ) ( $ ) o,,,,k + h
+
Of
we deduce (6.41) f o r
satisfies (6.41) for 3 = ( 0 ) u N h
( $ O ) O . < S >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 ; * -<
-
( i v ) iF q 6 C m ( X ) is quasihomogeneous OF degree P 6 C then qT,,,,,=(qT),,,+,~,
;
m
( v ) suppose that X = V and x = x + : i f T is temperate so i s T r n , k .
ProoF. (i): Let q E C T ( X ) . By ( 4 . 1 0 ) and Proposition 5.27 we have ( ( a M - m * ) c p ) r n * s W k + l=, oSince
Ynnl*,mk,m
'Pm*.c.>k,O
= vY*.ldk- vni*,c.,k,O on X , , t h e preceding equation combined with
( 4 . 1 0 ) and (5.63) leads t o
l i i ) : The
f i r s t equality sign follows froni ( i ) by induction. For t h e proof of t h e
second equality we fix q€EC;(X).
Since q m * , C l . + a n =C ym+ - q,,,+,,0,11 (4.10) and ( 5 . 6 3 ) leads to
Bq (4.10) and Proposition5.27 we have
on X + , t h e preceding equation combined with
278
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
(iii): t h i s f o l l o w s f r o m Lemma 6.17.
0: t h i s is (v):
a c o n s e q u e n c e of P r o p o s i t i o n b . l b . ( i i )
see P r o p o s i t i o n 0 . 2 2 . rn
Proof of T h e o r e m 6 . 4 9 . If m f t ( - X ( M ) - p ) t h e n Lemma 6 . 5 0 a n d C o r o l l a r y 0.28 N
show that S:=T,
is a s o l u t i o n s a t i s f y i n g ( i ) a n d ( i i ) . H e n c e w e s u p p o s e t h a t I"
m G ( - Z ( M ) - p ) , i . e . m * € E ? I ( M ) , a n d X = X . T h e n by L e m m a 0.31 w e c a n c h o o s e w ~ a ' , ( X ) such that (6.56)
Q;(xT)
=Qkw. 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 d e g r e e m
By P r o p o s i t i o n 0 . 3 5 w, (d,-m)w,=QLw.
In view of ( 0 . 5 6 ) Lemma 6 . 5 0 i m p l i e s t h a t
such S:=T,+w,
that is
t h e desired s o l u t i o n of (6.53).
If T is i n d u c e d by a C r f u n c t i o n so is w by Lemma 6.31, a n d by Lemma 6.17 t h i s r e m a i n s valid f o r w,,,.
H e n c e t h e p r o p e r t y l i ) f o l l o w s by Lemma h . S O . ( i i i ) .
If X = V a n d if T is t e m p e r a t e t h e n by Lemma b.31.(iii) w is t e m p e r a t e , a s w e l l , w i t h M - t e m p e r a t e s u p p o r t . C o n s e q u e n t l y , by P r o p o s i t i o n 6.22 w, a n d p r o p e r t y (ii) r e s u l t s f r o m L e m m a b.SO.(v)
is t e m p e r a t e ,
.
Finally, w e o b s e r v e t h a t if S is a s o l u t i o n of (6.53) s a t i s f y i n g ( i ) a n d ( i i ) a n d if T is @ - i n v a r i a n t t h e n by P r o p o s i t i o n s 2.64 a n d 2.59 a n d by ( 2 . 3 6 ) a n d ( 2 . 3 9 )
,S
is a @ - i n v a r i a n t s o l u t i o n of ( 6 . 5 3 ) s a t i s f y i n g ( i ) a n d ( i i ) , as w e l l .
rn
Corollary 6.51. S u p p o s e that ( 6 . 5 2 ) h o l d s . Then for ever,' d E.21D'(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 t h e r e exists a distribution T E a , ' ( X ) h a v i n g the properties (i) - (iii) of Theorem 6.49 and being a l m o s t q u a s i h o m o g e n e o u s of de-
gree m w i t h d e f i c i e n c y d .
279
6.e S o l v i n g t h e Equation ( 3 M - m ) S = T
Is t h e a s s u m p t i o n ( 6 . 5 2 ) n e c e s s a r y ? To e x a m i n e t h i s w e let S E a ' ( X ) be a s o l u t i o n of t h e e q u a t i o n (6.53). a s s u m i n g t h a t m E ( - U ( M ) - p ) a n d t h a t T 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 d e g r e e m w i t h s u p p o r t c o n t a i n e d in X \ X, . T h e n 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 degree m . H e n c e by T h e o r e m 6 . 3 7 w e f i n d u ~ a h ( X )a n d
a d i s t r i b u t i o n d E a ' ( X ) s a t i s f y i n g t h e c o n d i t i o n ( a ) of P r o p o s i t i o n 6 . 2 7 . ( i ) s u c h that S=u,+d.
I n s e r t i n g t h i s i n t o ( 6 . 5 3 ) a n d t a k i n g (4.10) i n t o a c c o u n t w e o b t a i n
( 3 M - m ) k T = v m where k : = o r d M ( d ) and v : = ( d M - m ) k " u €
a L ( X ) . S i n c e by
1_<
3 h ( X ) it f o l l o w s t h a t (?),,,
R e m a r k 6 . 1 4 v e x t e n d s to a d i s t r i b u t i o n (3,-m)
k
extends
T . S i n c e k < N ( m * ) a n d s i n c e N ( m * ) = O in case m = p o r M is s e m i -
s i m p l e w e have p r o v e d t h e a s s e r t i o n ( i i ) o f t h e f o l l o w i n g r e m a r k t h e f i r s t p a r t
of which is a direct c o n s e q u e n c e of T h e o r e m 6 . 4 9 .
Remark 6.52. Let d < B ' ( X ) be a distributions s a t i s f , i n g the condition ( a ) OF Proposition 6.27. ( i ) . Y
(i)
I f d e x t e n d s t o a distribution on X which i s almost quasihomogeneous o f
d e g r e e m then the equation 16.531 has a solution S < D ' ( X ) f o r T = d . ( i i ) The converse OF ( i ) is valid provided that m = p or M is s e m i - s i m p l e .
I
T h e o r e m 6.40 also l e a d s to t h e f o l l o w i n g i m p r o v e m e n t of T h e o r e m 6 . 4 5 ,
Theorem 6.53. Suppose that k e r M = 101. Let T E B ' ( V ) .If ( 3 , - m )
T i s temperate
then T is temperate i t s e l f . In particular, i F T I v , , o , is almost quasihomogeneous then T is ternperate. Proof. By T h e o r e m 6 . 4 9 o n e f i n d s a t e m p e r a t e d i s t r i b u t i o n w € Y ' ( V ) s u c h t h a t ( d M - m ) w = ( d M - m ) T . By P r o p o s i t i o n 2.11 t h i s m e a n s t h a t T - w is q u a s i h o m o g e n e o u s o f degree m , h e n c e t e m p e r a t e by T h e o r e m 0 . 4 5 , a n d t h e f i r s t a s s e r t i o n follows.
If
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 d e g r e e m t h e n f o r s o m e N E N t h e
s u p p o r t o f S : = ( d M - m ) N T is c o n t a i n e d in t h e set { O ) , i.e. S E & ' ( V ) C Y ' ( V ) . T h u s it f o l l o w s f r o m t h e f i r s t a s s e r t i o n by i n d u c t i o n t h a t T is t e m p e r a t e .
280
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
Let @ satisfy t h e a s s u mp t i o n s of Remark 2.67.(ii). Let E be one of t h e spa c e s
3 t X ) or Y ( V ) .T h e condition ( b ) in Proposition 6.27.(i) gives rise to a duality bracket f o r Q m ( E w ) . We employ t h e following
Notatlon6.54. ( i ) I f F is a s u b s e t of 3 ' ( X ) we write F m : = { T m T: E F } . (ii)
By Q,;CE) we den o t e t h e set of all distributions of t h e form QL,u where
u ~ 3 ; , , + , ~ ( X )and where in case E = Y ' ( V ) u is t em pe ra te with ( M , C l , + a C ) - t e m perate s u p p o r t .
Remark 6.55. Q L ( E ) @ i s equal t o the space of all @-invariant distributions d c E ' with support contained in X \ X , which are almost quasihomogeneous of degree m. It i s a weakl) closed subspace o f E'.
Proof. The f i r st assertion follows by Propositions 0 . 2 4 , 11.27.(i ) and 2.64 and Remark 2.67.( i i ) . The second assertion is an immediate consequence of Proposit i o n 2.31 and Definition 2.56.
Ropodtion 6.56. B-b, Q A ( E ) e xQ,,
(Eel)+
@ , ( d , R ) H ( d . R > , a dualit), brak-
k e t is well-defined eshibiting Q , ; I ( E ) , a s the ( t o p o l o g i c a l ) dual of the space Q,,. (Em' ) i f the latter i s equipped with the locall-t, c0nve.v topolog), described in Propositions 5 . 5 t and 5 . 5 2 respective/) .
proof. Let
d E Q A ( E ) c e and qpEE. Recall t h at t h e condition b ) of Proposition
6.27. ( i ) means t h a t (6.57)
< d .Qm*
> = < 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 = <(-a)"s,,
Q,*((Q,.~,~),~)>
=
aK(~,*'p,l)(y).
Hence w e c o n c l u d e ( i n t h e case E = D ( X ) using Remark 5.60 a n d (5.70)) t h a t
Q,*
I
0.
To prove t h e last a s s e r t i o n w e recall f r o m t h e p r o o f s of P r o p o s i t i o n s 5.51 a n d 5.52, respectively, t h a t t h e m a p P : E
+Q,+(E,o),
defined by
'p t+
Q,*
'pea,
is
a surjective c o n t i n u o u s linear h o m o m o r p h i s m . Consequently, every d E QA( E ) @ induces a c o n t i n u o u s linear functional o n Q,+(E,I),
a n d , conversely, every con-
is induced by t h e distribution d : = u o P . m
t i n u o u s linear functional w o n Q,+(E,*)
Now w e c o m e to t h e s p a c e s W z , k ( E @ )
Notatlon6.57. W e set X A , k ( E ) : = { T E E ' ; ( d M - m )
k+l
TEQk(E)}
Remark 6.58. ( i l y [ A . k ( E ) @ ( s e e Notation 6.54.( i ) ) consists of all @-invariant distributions T E E ' s atisfjing the condition ( a ) of Theorem 6.37. ( i ) . ( i i ) IF (6.5-7) hol ds then ? L A . k ( E ) , i s equal to the space of all distributions of the form
( ~ m t )where ~ , u~ € B~ - (~X I
and where in case E = Y ' ( V ) u is temperate
with M-t emper at e s upport. ( i ii )
XA,k(E)e is a weak!, c lose d subspace of E ' .
m f . (il:This is a n immediate consequence of P r o p o s i t i o n s 2 . 0 4 a n d 6.27 a n d of (2.39).
(ii):
O n e o b t a i n s t h i s by combining T h e o r e m 6.37 a n d Proposition b . l b . ( i v ) .
liii): T h i s tor a,-m,
is deduced f r o m Definition 2.63, f r o m t h e weak continuity of t h e opera-
a n d f r o m t h e s e c o n d a s s e r t i o n of Remark 6.55.
T h e condition ( b ) of Theorem f ~ . 3 7 . ( i )leads to a canonical duality b r a c k e t f o r t h e space W A . k ( E ) ~ :
282
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
space 2 1 A , k ( E ) ~a s the topological dual o f the space X Z * , k ( E m , ) i f the latter is equipped with the locallj, convex topology described in Propositions 5.59 and 5.61, respectively.
=
prooF. If T E E ' is @ - i n v a r i a n t t h e n by P r o p o s i t i o n 2.64.(iii) w e have
< T , ' p @ * >H. e n c e 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 ) ( s e e a l s o T h e o r e m 6 . 3 7 . ( i i i ) in case E = Y ( V ) ) s h o w s t h a t
<*;*
> M is w e l l - d e f i n e d .
T = O by (6.58). I f , o n t h e o t h e r h a n d ,
< *
If
then
, ( ' ~ B ~ ) ~ * , , , ~t h >e n~ it ~ Of o l l o w s
by T h e o r e m 5 . 3 7 . ( v i i i ) t h a t f o r every y E X , w e have
o=
<((Sy)@)m.,,,,
('P@J,')m*,wk>M =
(-1) k <((8y)@j)m,mk,v =)
= < S Y , ( v c * L * , G , k > = ((Pc*')m*,',k(Y)
In view o f R e m a r k 5 . 6 0 . ( i ) t h i s i m p l i e s t h a t ( c p w * ) , n w , c d k = 0 . C o n s e q u e n t l y , t h e bracket
< -,* * >M
is n o n - d e g e n e r a t e .
F o r t h e p r o o f o f t h e l a s t p a r t w e r e c a l l f r o m t h e p r o o f s o f P r o p o s i t i o n s 5.59 a n d 5.61, r e s p e c t i v e l y , t h a t t h e m a p P : E - + U , " , , k ( E e H i )
d e f i n e d by
'p H ( ( P ~ ~ * ) ~ , ,
is a s u r j e c t i v e c o n t i n u o u s linear h o m o m o r p h i s m . C o n s e q u e n t l y , f o r every d i s t r i b u -
tion T€'Uk,,k(E), m
the map
>M
is a c o n t i n u o u s l i n e a r f u n c t i o n a l o n t h e s p a c e m
E B ~ ) ,a n d , c o n v e r s e l y , e v e r y c o n t i n u o u s linear f u n c t i o n a l v o n 3 m * , k ( E e * )
d e f i n e s a d i s t r i b u t i o n T : = u o P E E' which b y T h e o r e m 6 . 3 7 . ( i ) a n d P r o p o s i t i o n s 2 . 6 4 a n d 2.59 b e l o n g s to U A , k ( E ) ( f i a n d obviously s a t i s f i e s < T . * )M = v . rn
283
Chapter VII
Solvability of Qua sihomo geneous Multiplication Equations and Partial Differential Equations
As in t h e preceding c h a p t e r s , w e continue to a s s u m e t h a t (1.14) holds and t h a t
X \ X,
#
@ . In s e c t i o n s ( a ) , ( b ) , and ( c ) we a r e going t o solve quasihomogeneous
multiplication equations in t h e s e t of ( a l m o s t ) quasihomogeneous distributions o n X . T h e main results a r e Theorems 7 . 3 , 7 . 5 , and 7.14 in section ( a ) , Theorem 7.31 and Corollary 7.32 i n section ( b ) , and Theorem 7.35 and Corollary 7.30 i n section (c). If X = V then via t h e Fourier t r a n s f o r m t h e r e s u l t s can be r e s t a t e d a s a s s e r t i o n s
o n t h e solvability of partial differential equations with c o n s t a n t coefficients in t h e s p a c e of ( a l m o s t ) quasihomogeneous temperate distributions. This is d o n e in section ( d ) . Here t h e main r e s u l t s a r e Theorems 7 . 4 4 , 7 . 4 5 , and 7.51 and Supplem e n t s 7.53 and 7.54.
I t t u r n s o u t t h a t f o r exceptional m one d o e s n o t always have solvability. In f a c t , f o r t h e s e m solvability is characterized completely i n t e r m s of t h e function defining t h e o p e r a t o r , only. This is illustrated by way of several examples given in section ( e ) . Finally, in section ( f ) , f o r t h e heat and f o r t h e Schrodinger o p e r a t o r we d e t e r mine t h o s e fundamental s o l u t i o n s which a r e ( a l m o s t ) quasihomogeneous o f type ( 2 , 1 , .. . , 1 ) and which a r e invariant under orthogonal t r a n s f o r m a t i o n s o f t h e space variables. For t h e whole c h a p t e r , in order t o handle additional invariance properties we fix a c o m p a c t s u b g r o u p G of G L ( V ) satisfying t h e assumptions choose t w o continuous homomorphisms a , r : G @ : = ( G , a ) and .$:= ( G , r ) .
+C
of Remark 2.67,
a s in section 2. ( f ) , and set
284
V11. Solvability o f Q u a s i h o m o g e n e o u s Equations
ta) Quaslhornogeneous MulLLpllcaCIon Opcralors
For this and t h e following t w o sections we fix a number 4 E C and a C m f u n c t i o n q:X+C
which is quasihomogeneous of degree 4 and which is (G,T/a)-invariant.
The r e s u l t s we are going to derive a r e valid f o r t h e spaces a ' ( X ) and Y ' ( V ) . To unify t h e t r e a t m e n t we denote by E either a ( X ) or case we a s s u m e t h a t q belongs to t h e space 6,(V)
- if
X = V - Y ( V ) ; in t h e l a t t e r
of continuous multiplication
operators o n Y ' ( V ) . O u r aim is f o r every $-invariant
distribution TE E' which is ( a l m o s t ) quasihomo-
geneous of degree m to find a @-invariant distribution S E E ' which is ( a l m o s t ) quasihomogeneous of degree m - 4 and solves the equation (7.1)
qS = T.
If k E N and if d denotes t h e kth order deficiency of T then Corollary 2.36.(ii) implies t h a t t h e k t h o r d e r deficiency c of S satisfies t h e equation (7.2)
qc=d.
Note that by Proposition 2.64.(iii) and Remark 2.67.(ii) d (resp. c ) is $-invariant ( r e s p . @-invariant). Thus we are lead t o t h e aim of solving (7.1) in t h e following more precise way: given a @-invariant solution c of ( 7 . 2 ) which is a l m o s t quasihomogeneous of degree m - 4 we would like t o find a solution S of (7.1) such t h a t its kth order deficiency is equal t o t h e prescribed distribution c . The solvability results given below are based on t h e solvability of (7.1) in t h e whole space E'; more precisely, throughout sections
(a),
( b ) and ( c ) we a s s u m e t h a t
(7.3)
qE'=E';
(7.4)
the ideal generated by q i
(7.5)
if E = Y ( V ) then every fECCO(V) such t h a t q f E Y ( V )
X+
in C m ( X + ) is closed;
belongs to YP(V). Note t h a t (7.3) implies t h a t
(7.6)
s u p p q = X , i.e. X \ q - ' ( O ) is dense in X .
285
7 . a Multiplication Equations
S i n c e by P r o p o s i t i o n 5 . 4 5 t h e r e s t r i c t i o n o f q to t h e n o n - e m p t y set Xo is a polynomial w i t h r e s p e c t to t h e v a r i a b l e s in G M ( o + ) , t h i s i m p l i e s , in p a r t i c u l a r , t h a t
4 b e l o n g s to X ( M).
Remark 7.1. ( i ) If q is real analytic then the condition (7.31 f o r E=DD(XI and the condition ( 7 . 4 ) are valid by the Lojasiewicz division theorem in t14.l (see also Remarque VI.1.9 in Tougeron Li711. ( i i ) If q is a pol-vnomial function and i f E = .V( VI then the condition (7.3) holds by Hormander's division theorem in [ 8 1 while the condition (7.51 is an immediate consequence of the estimate (4.3) in Hormander [ H I .
I
W e first deal with t h e equation (7.2) f o r a l m o s t quasihomogeneous distributions w i t h s u p p o r t c o n t a i n e d in X \ X , .
Propodtion 7.2. Let d c E ' be .$-invariant and almost quasihomogeneous o f degree m . If supp d
C
X \ X + then the equation (7.2) has a @-invariant solution
c6 E' which is almost quasihomogeneous of degree m -
such that supp c
C
X \X, .
In o t h e r w o r d s , m u l t i p l i c a t i o n by q d e f i n e s a s u r j e c t i v e o p e r a t o r f r o m t h e s p a c e Qkn-p(E)c, onto Qk(E),
(see N o t a t i o n 6 . 5 4 ) .
m f . By ( 7 . 3 ) w e find v € E ' s u c h t h a t q v = d . If E = D ( X ) t h e n m u l t i p l y i n g b y a c u t - o f f f u n c t i o n w e a c h i e v e t h a t t h e s u p p o r t o f v is c o n t a i n e d in
x
-'(
C0,l C )
s o t h a t by L e m m a 6 . 7 v b e l o n g s to 3 i l , + m c ( X ) . H e n c e by P r o p o s i t i o n s 6 . 2 6 . ( i i )
a n d 6 . 2 7 . ( i ) . ( b ) a n d by R e m a r k 2 . 6 7 . ( i i ) a n d P r o p o s i t i o n 2 . 6 S . ( i i ) t h e d i s t r i b u t i o n C : = Q ~ - ~ ( Vis) t@ h e desired s o l u t i o n o f ( 7 . 2 ) .
If E = Y ( V ) t h e n m u l t i p l y i n g by a c u t - o f f f u n c t i o n c h o s e n a c c o r d i n g to L e m m a 5.53 w e achieve t h a t s u p p v C
n,,,,,.
T h e n t h e s o l u t i o n c o f ( 7 . 2 ) d e f i n e d above is
t e m p e r a t e by P r o p o s i t i o n 6 . 2 4 . ( i i i ) .
N e x t w e c o m e to t h e e q u a t i o n (7.1) f o r T € X A , k ( E ) 6 (see N o t a t i o n s 6.57 a n d 6 . 5 4 ) .
Theorem 7.3. Let T 6 E'be Sj -invariant and almost quasihomogeneous of degree m. Then the equation (7.1) has a @-invariant solution S6 E' which is almost quasihomo-
286
VII. S o l v a b l l i t y o f Q u a s i h o m o g e n e o u s Equations
geneous o f degree m - P . M o r e precisely, f o r every k C N , induces a surjective operator f r o m the space % ;
proOf. Let k E No
. By T h e o r e m
the multiplication by q
- e , k ( E ) , o n t o 21,',,k(E)a.
9L(X ) n E'
h.37 w e find S j - invariant d i s t r i b u t i o n s u E
a n d d E E ' s a t i s f y i n g T = ~ , , , ~ + ds u c h t h a t s u p p d C X \ X + a n d s u c h t h a t in case
E = Y ( V ) s u p p u is M - t e m p e r a t e . An a p p l i c a t i o n of P r o p o s i t i o n 7.2 r e d u c e s t h e p r o o f t o t h e c a s e d = O . B y ( 7 . 3 ) w e f i n d v E E ' s u c h t h a t q v = u . If E = 9 ( X ) t h e n m u l t i p l y ing 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 e achieve t h a t v b e l o n g s to Z ) k ( X ) , as w e l l . H e n c e , in view of P r o p o s i t i o n s 6 . 1 6 . ( i i ) a n d 2 . 6 S . ( i i ) t h e d i s t r i b u t i o n S : = ( v ~ - ~ , , ~ , ) ~ ~ is t h e desired s o l u t i o n . If E = Y ( V ) t h e n by e m p l o y i n g a c u t - o f f f u n c t i o n given
by Lemma 5 . 5 3 w e achieve t h a t s u p p v is M - t e m p e r a t e , a s w e l l . 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 6 . 2 2 a n d 2 . 6 4 . ( i i ) t h a t t h e s o l u t i o n S d e f i n e d a b o v e is t e m p e r a t e . N o t e t h a t in view of C o r o l l a r y 6 . 2 8 in case m* Q ' U ( M ) t h e s p a c e Q:,(E)
is t r i v i a l .
C o n s e q u e n t l y , in case m ' + t Q 4 L ( M ) , i . e . m - 4 Q ( - X ( M ) - p ) , t h e a s s e r t i o n of T h e o r e m 7.3 m e a n s t h a t t h e f o l l o w i n g solvability c o n d i t i o n h o l d s f o r every k EN, (7.71,
f o r everj &-invariant
:
T E E ' which is almost quasihomogeneous of
degree ni and o f order 5 k the equation 17.1) has a ((%-invariant) solution S E E ' which is almost quasihomogeneous and o f order
5
o f degree m - P
k.
In t h e e x c e p t i o n a l c a s e " m * + 4 E 4L(M ) " . fixing a n a l m o s t q u a s i h o m o g e n e o u s distrib u t i o n T w e give a f i r s t a n s w e r to t h e q u e s t i o n u n d e r which c o n d i t i o n s t h e e q u a t i o n (7.1) h a s a s o l u t i o n S 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 order n o t e x c e e d i n g t h e order of T .
Propositlon 7.4. Suppose that n7 € ( - # ( M ) - p + P ) and that X = ? . Let k E N be such that k > N : = N ( m * + P ) (see ( 5 . 6 5 ) ) . and let
T E E ' be &-invariant
quasihomogeneous o f degree m and of order 5 k
and almost
.
( i ) Then the f o l l o w i n g conditions are equivalent:
( a ) the equation (7.1) has a ( @ - i n v a r i a n t ) solution S E E ' which is almost quasihomogeneous of degree m - P and of order 5 k ;
( b ) f o r some (resp. every) j € E V k - N
the equation (7.2) w i t h d = ( d M - m l i T
has a ( @ - i n v a r i a n t ) solution cE E' which is almost quasihomogeneous o f degree
m - P and of order 5 k - j .
287
7.8 M u l t i p l i c a t i o n E q u a t i o n s
(iil If c is a solution as in ( b ) then in ( a ) S can be chosen in such a way that
.
( d M - r n +!)' S and c coincide on X , proOf.
'z': Corollary2.36.(ii) shows that
c : = ( 3 M - m + t ) J S is t h e d e s i r e d solu-
t i o n of ( 7 . 2 ) .
E:By
C o r o l l a r y 6.Sl w e f i n d a d i s t r i b u t i o n R E E ' s u c h t h a t c = ( a M - m + t ) ' R .
S i n c e t h e n , in p a r t i c u l a r , R 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 - 4 a n d of
order 5 k it f o l l o w s by C o r o l l a r y 2 . 3 6 . ( i i ) , a g a i n , t h a t q R 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 n d of order 5 k s u c h t h a t ( a M - m ) ' ( q R ) = q c = d . H e n c e 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 n d o f order 5 j - 1 .
T-qR
by T h e o r e m 7 . 3 w e find a s o l u t i o n U E E ' of t h e e q u a t i o n
Therefore
q U = T - q R which is
almost quasihomogeneous of degree m-P such t h a t the support of a : = ( a M - m + 4 ) j U is c o n t a i n e d in X \ X , . S i n c e by t h e d e f i n i t i o n of N w e have ( 3 M - m + 4 ) N ' i ' '
U =0
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 U 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 - t a n d of o r d e r 5 k . H e n c e S : = R + U is t h e d e s i r e d s o l u t i o n of
(7.1) ( n o t e t h a t
( 3 M - m + 4 ) ' S is e q u a l to c + a which may be d i f f e r e n t f r o m c ! ) .
F o r t h e e x c e p t i o n a l case " m * + 4 E X ( M ) q u e s t i o n w h e t h e r f o r e v e r y TEE,;
"
t h e preceding r e s u l t s do n o t a n s w e r t h e
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 t h e
e q u a t i o n (7.1) h a s a s o l u t i o n S E E ' 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 m - 4 .
Theorem 7.5. Suppose that ( a ) m d ( - ? l ( M ) - p + P ) , or
17.81
(b) X=?.
Then the Following conditions are equivalent:
( a ) the solvabilit:, condition (7.71, is valid For some k E N o . ( a ) ' the condition ( 7 . 7 ) , is valid For every k E N , , ; in particular, For ever:,
,$-invariant
T E E ' which is quasihomogeneous OF degree m the equation (7.11
has a (@-invariant) solution S C E ' which is quasihomogeneous OF degree m - t :
( b ) For some (resp. every) k E N o the Following condition holds: (7.9),
For every pair ( T , c )E E ' x E ' such that T (resp. c ) is $-invariant (resp. @-invariant) satistving the equation q c = ( d M M - m ) k +T*there esists a (@-invariant) solution S E E ' of the equation (7.11 satisFying (aM-m+P)k+ls=c.
288
VII. Solvabillty of Q u a s i h o m o g e n e o u s Equations
(c) For some Iresp. every) kENo the following condition holds:
for every @-invariant distribution c E E' with support contained in
(7.10)k
X \ X + which solves the equation q c = O and is almost quasihomogeneous of degree m-P there exists a @-invariant solution S E E ' of the equation q S = O such that ( d M - m + P l k + l S = c .
T h e p r o o f is easily r e d u c e d to T h e o r e m s 7.3 a n d 6.49. The r e d u c t i o n s t e p is c o m p l e t e l y e l e m e n t a r y a n d is f o r m u l a t e d a s
Lemma 7.6. Let V , , V , , W , , and W , be complex vector spaces, and for every j E ( 1 , -31 let A; : Vj
-+Wi
be a linear map. Moreover, let B: V, --ir V, and C : W, --ir W ,
be linear maps such that
(i)
We define linear maps P : V , --ir V, x W l by vl
y : V,xW,
+W,
by (v,,w,)
H
C(w,)-A,1v2).
0:( k e r A , ) / B ( k e r A , ) - - i r k e r y / i m P
,
H
( B ( v l ) .A l ( v l ) ) and
Then i m p C k e r y . and b v v 2 + B ( k e r A l lH ( v , . O ) + i m p .
a linear monomorphism i s defined which is surjective i f A , is surjective. (iil Let LI, be a subspace of V , such that
( a ) U,
( 7.121
3 kerAl ,
and
( b ) B ( U , )3 kerA,.
Then by
Y : ( A l ( U , ) n k e r C ) / A l ( U , n k e r BI
*(kerA2 ) / B ( k e r A, ) ,
A, ( u l ) + A ,( U , n k e r B ) H Blu, I + Blker A , ) ,
a linear isomorphism i s well-defined.
a f . g: T h e c o n d i t i o n (7.11) m e a n s t h a t
yoB
= O , i.e. i m p C k e r y .
If v 2 E k e r A 2 t h e n y ( v 2 , 0 ) = - A 2 ( ~ 2 ) = 0 a; n d i f , in a d d i t i o n , v z = B ( v l ) f o r s o m e vl
E ker
Al t h e n ( v z , 0 ) = B ( v l )
E
i m B . H e n c e CD is w e l l - d e f i n e d .
If C D ( v 2 + B ( k e r A 1 ) ) = i m p , i.e. ( v 2 , 0 ) = ( B ( v l ) , A l ( v l ) ) f o r s o m e v l E V 1 , t h e n v2 E B ( k e r A , ) . T h i s s h o w s t h a t CD is injective.
Let ( v 2 , w 1 ) ~ k e r y i, . e . C ( w 1 ) = A 2 ( v 2 ) . If A, is s u r j e c t i v e w e f i n d v l ~ V ls u c h t h a t A l ( v l ) = w l . I t f o l l o w s by
(7.11) t h a t
A 2 ( v Z ) = C ( A 1 ( v 1 ) ) = A 2 ( B ( v 1 ) ) , i.e.
7.a M u l t l p l l c a t l o n E q u a t i o n s
289
v 2 - B ( v l ) E k e r A 2 . S i n c e ( B ( v l ) , w l ) = B ( v l ) E i m B w e c o n c l u d e t h a t t h e i m a g e of v 2 - B ( v 1 ) + B ( k e r A 1 ) u n d e r Q, is e q u a l to ( v 2 . w 1 ) + im p . H e n c e 0 i s s u r j e c t i v e
provided A, is so.
(iil:
If u l E U l is s u c h t h a t A l ( u l ) E k e r C t h e n (7.11) i m p l i e s t h a t B ( u l ) E k e r A 2 ;
i f , in a d d i t i o n , A , ( u i ) = A l ( u ; )
f o r s o m e u; E Ul fl k e r B t h e n
u l - ui
E
k e r A, a n d
B ( u l ) = B ( u l - ti;) E B ( k e r A , ) , i.e. T is w e l l - d e f i n e d .
If B ( u l ) E B ( k e r A l ) t h e n B ( u l ) = B ( u ; ) f o r s o m e u ; E k e r A l , i.e. u l - u ; E k e r B , a n d A l ( u l ) = A l ( u l - u i ) E A l ( k e r B ) . S i n c e by (7.12.a) u l - u ;
b e l o n g s to U, t h i s
s h o w s t h a t Y is injective.
If v 2 ~ k e r A 2t h e n by ( 7 . 1 2 . b ) w e f i n d u l E U 1 s u c h t h a t B ( u l ) = v 2 . By (7.11) it follows t h a t C ( A l ( u , ) ) = A 2 ( B ( u l ) )= 0 , i . e . A l ( u l ) E A l ( U , ) n k e r C . C o n s e q u e n t l y , 'i'i s s u r j e c t i v e .
Proof of Theorem 7.5. Let k € N o . W e a r e g o i n g to a p p l y L e m m a 7 . 6 to V, = W , = E & , V ~ = W , = E & , ~ ~ : = ( a M - m + k! )+ 1 , ~ ~ : = ( a ~ - m )u~, :+=l xm , '- e , k ( E ) a t , a n d B = C : = m u l t i p l i c a t i o n by q . T h e n A1 ( a s well a s A,)
is s u r j e c t i v e by T h e o r e m
6 . 4 9 : t h e c o n d i t i o n (7.11) f o l l o w s f r o m Lemma 2 . 3 4 ; (7.12.a) is valid by Proposit i o n 2 . 3 l . ( i ) ; a n d ( 7 . 1 2 . b ) h o l d s by T h e o r e m 7 . 3 a n d P r o p o s i t i o n 2 . 3 1 . ( i ) . N o t e t h a t by C o r o l l a r y 6.51 A , ( U l ) is e q u a l to Q k - e ( E ) c v .
Now, in view of P r o p o s i t i o n 2 . 3 l . ( i ) t h e c o n d i t i o n ( 7 . 7 ) k c a n be r e p h r a s e d a s " k e r A 2 = B ( k e r A l ) " . By
Lemma 7 . 6 . ( i )
this
is
equivalent
to
the
condition
" k e r y = i m p " which a m o u n t s to ( 7 . 9 ) k . M o r e o v e r , by Lemma 7 . 6 . ( i i ) t h e c o n d i t i o n ( 7 . 7 ) , is e q u i v a l e n t to t h e c o n d i t i o n " A l ( U l ) n k e r C = A 1 ( U i n k e r B ) " which r e a d s a s (7.10), . S i n c e by C o r o l l a r y 2 . 3 6 . ( i i ) w e have q ( a M - m + 4 ) k S = ( 3 M - m ) k( q S ) w e see t h a t (7.10), i m p l i e s ( 7 . 1 0 ) 0 . H e n c e , in order to c o m p l e t e t h e p r o o f w e have to verify t h e
c o n v e r s e i m p l i c a t i o n . I n d e e d , s u p p o s e t h a t (7.10), is valid, a n d let c be given having t h e p r o p e r t i e s listed in ( 7 . 1 0 ) k . By (7.10), w e f i n d S o E E ' s u c h t h a t q S o = O a n d
( a M - m + 4 ) S o = c . Applying ( 7 . 9 ) , to ( 0 , S i - , )i n s t e a d of ( T , c ) w e recursively c o n s t r u c t a s e q u e n c e of d i s t r i b u t i o n s Si € E' s u c h t h a t qSi = 0 a n d ( a M - m + 4 ) Si = S j - l , j E N , I t f o l l o w s t h a t S : = sk h a s t h e p r o p e r t i e s r e q u i r e d in ( 7 . I O ) k .
From n o w o n f o r t h e r e s t of t h e p r e s e n t s e c t i o n a n d f o r s e c t i o n s ( b ) a n d ( c ) w e s u p p o s e t h a t ( 7 . 8 ) h o l d s . A c t u a l l y , m o s t of t h e f o l l o w i n g r e s u l t s a r e trivial
200
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
in case (7.8.a) i s valid but a few are not. I n order to examine t h e validity of the conditions of Theorem 7.5 we fix k E N , and rephrase the condition (7.10), in the form
;
(7.10)
= A,
ker C,
-
(ker ) ,B
employing the following
Notatlon 7 . 7 . ( i ) By A , , : 2 f A 1 , k ( E ) e+ Q ; f E I c 9
we denote the map defined by
T H (dM-m)k+'T; (ii)
by B m : X A - p , k f E ) c s +Xr;,,k(E).i,
( i i i ) by C , : Q , ~ _ p f E ) , - - ? r Q r : , ( E ) S ,
We observe that C,,
we denote the map defined by T H q T ; we denote the map defined by c H q c .
and hence the left-hand side of (7.10); d o not depend on
k . In order to see how the right-hand side of (7.10); depends on k we write
f A m , k , B r n , k ) instead of (A,,B,)
in the following
Remark7.8. ( i ) Wk : = A r , - t , k f k e r B , , , , k ) 3 W k + l for e v e r y k 6 N J , . f i i ) I f k o 6 N o i s s u c h t h a t W k o + l =Wk, then Wk = Wk, For e v e r y k > 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
Q( 3 ) 6 ,
[p
E
c;c x ) ,
this amounts t o (7.20)‘
(Q ( a )3)f )
y)
=o,
a€?[.
( i i ) By J m , k ( 9 1 we d e n o t e t h e s u b s p a c e of
tions ( Q ( - 3 ) 6 , ) ,,,,,,,k
Remark 7.22. ( i )
s,,
, (y,Q)
-
ETS(q1,
+
? l , k , k ( E ) generated by t h e distribu-
).
(,k(q)
.
(iil The space (d, - m I k * l , j , n , k ( q I consisting , o f the I!+kI t 11 order deficiencies of the elements of j m . k ( q ) ,is generated b-v the distributions of the form (6.32) where (y,QI 6Z?(q/, I : it does not depend on the choice of k . +
Proof. (i): this we have
( i i l : see
f o l l o w s from ( 7 . 2 0 ) since by Propositions 6.16.(ii) and 2.hS.(ii)
q ( ( Q( - 3 ) sy )elln - y , W k = ( ( 4 Q ( - 3 ) 8, ) Q lm,Wk .
( 6 . 3 0 ) a n d Example 6.36.
T h a t t h e ( l + k ) t h o r d e r deficiencies of t h e distributions in 3 m - o , k ( q ) 0 play 1 a decisive role when one w a n t s to decide whether or not t h e condition (7.101, h o l d s becomes clear by
301
7 . b Reformulating ( 7 . 1 8 )
Lemma 7.23. If k 1 N ( m *+ and A, - p
(Jm
-P
then t h e weak closures o f the subspaces A,
-
( k e r 8,)
in the space E' are equal.
, (41,) ~
For the proof of t h i s lemma, employing the duality brackets of section 6 . ( f ) , again, by the bipolar theorem we have to show that the polar sets of both subspaces are equal. To prepare t h i s we are going to compute the polar set of sm,k(q)(y first. This requires the following reformulation of a special case of Whitney's spectral theorem on the divisibility of Cm functions.
Lemma 7.24. Let
J generated
be an open subset o f V , let f 6 C"'(n) be such that the ideal
b v f in C"(n) is c l o s e d . and let g E C"'(n). Then the equation g = f h
) and onlj i f B(g) 3 2 ? ( f ) . has a solution h E C " ( ~ if
proOf.
2": t h i s is
x': Choosing
an immediate consequence of the Leibniz formula.
real coordinates we may assume that V = R " . Moreover, as the
definition of % ( f ) does not depend on M , we may assume that M=ld, 'u = N I .:
so that
We note first that the formulation of Whitney's spectral theorem in Tou-
geron 1177 ( s e e Corollaire V.l.0 there) is valid for complex valued functions, a s well: one reduces the proof to the real valued case by introducing the C m ( n ; l R ) submodule of C C n ( f l R ; ) 2 generated by
(
Re f , In1 f ) and
(-
Im f , Re f ) . Consequently,
since .J is closed in C'"(fl), i n order to verify that g belongs to 3 o n e has to check that for every x € X the Ta),lor series T x ~ gof g a t x belongs t o the set 7 ; . , 7 : = {T,h; h € . J } . If f ( x ) f O then this is the case for trivial reasons. Hence
we may suppose that f ( x ) = O .
By Borel's theorem ( s e e Remarque IV.3.5 in 1171) it suffices to find a formal power series d = x , E ' U d a Z" such that T,g = d T,f . This equation can be written as the infinite system of linear equations
f , - y d , = L u ! g' ' ' ( x ) = : g , ,
L,(d) : =
aEX,
where
f,:= &f'"'(s).
y 5,
By Lemma6.3.7 i n Hormander C O I there exists a solution (d,),,, family
c , ) , ~ such ~ ~ that c, is non-zero for finitely many u , only, the condition
cu L,
(7.21.A
0
ae'U
implies that
(7.31.B
if for every
1 c,g, 0:
E'U
=o.
302
V I I . 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 Equations
By t h e d e f i n i t i o n o f L, t h e c o n d i t i o n ( 7 . 2 t . A ) a m o u n t s to
N o w , if w e set
t h e n a s i m p l e c o m p u t a t i o n shows t h a t f o r a r b i t r a r y h E C - ( f l )
EX
and
w e have
H e n c e t h e c o n d i t i o n (7.21.A)' m e a n s t h a t ( x , Q ) E ' L l ( f ) . By t h e a s s u m p t i o n o n g it f o l l o w s t h a t ( Q ( d ) g ) ( x ) = O . T h i s is identical w i t h (7.21.B) a s a look a t (7.22)
- a p p l i e d t o ( h , y ) = ( g . 0 )- s h o w s .
Now w e come to t h e d e s c r i p t i o n of t h e p o l a r set o f 3 m - u , k ( q ) c e
Lemma 7 . 2 5 . The polar s e t
(3,,
f*k(q), )
0
consists o f all g E
XI:+
+
P.k(Em#)
such that for s o m e h 6 C c " ( X , ) w e have (7.23)
Proof. We fix a @'-invariant f u n c t i o n
'p E
E s u c h t h a t g = 'p,,,v
0
to ( 3 m - P , k ( q ) C H ) m e a n s t h a t f o r every ( y , Q ) E ' L l ( q l x 0 = ( - l ) k< ( ( Q ( - a ) s , ) , ) , _ , , , , , , g
)
+
p,L,k.
That g belongs
w e have
>M =
= ( - 1 1 ~ < ( ( ~ ( - d ) ~ ~ ) ~ ) ~ - =e , ~ ~ , ' p > = < Q ( - a ) S y ,( ~ m + + g , c , , k ) ~ t ) = ( Q ( d ) g ) ( y ) . In view o f ( 7 . 4 ) a n d L e m m a 7 . 2 4 t h i s is e q u i v a l e n t to t h e e x i s t e n c e of a f u n c t i o n h E Cm(X+) satisfying ( 7 . 2 3 ) .
N o t e t h a t by ( 7 . 6 ) any f u n c t i o n h s a t i s f y i n g (7.23) is uniquely d e t e r m i n e d by t h a t p r o p e r t y . W h a t o t h e r p r o p e r t i e s does it have? In p a r t i c u l a r , w h e n does it b e l o n g
to U Z a , k ( E G * ) ?To f i n d a n a n s w e r , m a k i n g use o f L e m m a 2 . 3 4 w e d e d u c e f r o m (7.23) t h a t
303
7.b R e f o r m u l a t i n g ( 7 . 1 8 )
Since g i s 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 * + t and s i n c e X , \ q - ' ( O )
is
d e n s e in X, t h i s implies f i r s t of all t h a t 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 d e g r e e m * a n d of order 5 o r d M ( g ) . Moreover, s i n c e t h e ( l + k ) ' h o r d e r deficiencies of t h e e l e m e n t s of W z * , k ( E & * ) are r e s t r i c t i o n s of f u n c t i o n s f r o m Q,*(E.+*)
w e deduce
f r o m ( 7 . 2 3 ) ' t h a t a necessary condition for h to belong to U g * , , ( E Q * ) is t h a t ( a M - m * ) k " h e x t e n d s to a Cm function P o n X , belonging to Q,,,*(Etl).
Denoting
by R E Q m + , e ( E ~ * ) t h e e x t e n s i o n of ( a M - m * - l ) k + ' g to X w e c a n r e p h r a s e t h i s by s t a t i n g t h a t t h e e q u a t i o n R = q P b e valid. This condition is a l s o sufficient:
Lemma 7.26. Let gE21z*+p,k(Ee*). Then the equation (7.23) is valid for s o m e h E U E + , k ( E G ~if) a n d on/ > if g b e l o n g s t o ( * q , n - t . k ( q ) e0) a n d 17.23)
( d M - n , * - e ) k + l g= (qP)/,,*
for s o m e P E Q , , , * ( E ~ , ~ ) .
Note t h a t t h e f u n c t i o n s described in Lemma 7.26 c o n s t i t u t e t h e image of tB,,,,
m.'z': see t h e t e x t
preceding t h e s t a t e m e n t of t h e l e m m a .
"(m-: F i r s t of all w e fix a @'-invariant function y E E s u c h t h a t g = y m + + O , c , , k .
Moreover, we c h o o s e
@ € EG*
s u c h t h a t ( 7 . 2 4 ) is valid f o r P = ( - l ) , + ' Q,,,*(@).
Since X + i s d e n s e in X it f o l l o w s f r o m (5.63) t h a t (7.24 )'
Q,*+e(y) = q Q m * ( @ ) .
By L e m m a 7 . 2 5 w e find h E C C O ( X + ) s u c h t h a t ( 7 . 2 3 ) h o l d s . Since X + \ q - ' ( O )
is
d e n s e in X , w e observe t h a t s u p p h C s u p p g . W e fix f u n c t i o n s $O,...,$kE C ~ ( X ) satisfying (4.18) f o r 3 = N 0 a n d ( 4 . 1 0 ) . If E = Y ' ( V ) t h e n in view of C o r o l l a r y 6 . 3 9 t h e s e f u n c t i o n s c a n be c h o s e n to have t h e properties ( i ) - ( i i i )
of L e m m a 6 . 3 8 .
W e claim t h a t f o r every i E ( O ) u N , t h e f u n c t i o n s (7.25)
Jli ( d M - m * ) ' h
and
+i
( 3 M - m * ) i @ m * . c , , kbelong to E .
Indeed, w e f i r s t s u p p o s e t h a t E = a ( X ) . Then t h e set L : = s u p p 'p is c o m p a c t , a n d by Remark 6 . 8 w e c a n c h o o s e a c o m p a c t s u b s e t K of X, s u c h t h a t
n X, = KM .
Since by T h e o r e m 5.37.(ii) t h e s u p p o r t of g and hence t h a t of h a n d e a c h of its derivatives is contained in t h i s set a n d s i n c e by Proposition 3.22 KM n s u p p J I , is
a c o m p a c t s u b s e t of X, w e c o n c l u d e t h a t t h e s u p p o r t of $ i ( 3 M - m * ) i h , is so, as well. T h a t t h e s a m e is valid f o r t h e s u p p o r t of J I i ( 3 M - m * ) ' @ m + , w k f o l l o w s similarly.
304
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
Supposing now t h a t E = Y (V ) we deduce f r o m Proposition 5.41 and Remark 5.40 to Y'(V). t h a t + i ( 3 M - m * - t ) i g and + 1 ( 3 ~ - m * ) ' @ ~ * belong ,.,~
we deduce f r o m ( 7 . 5 ) t h a t +,(a,-m*)'h
In view of (7.23)'
belongs to Y'(V), a s well. Hence t h e
proof of (7.25) is complete. Now, applying Proposition 4.13 t o ( g , m * + Q )instead of ( T , m ) a nd taking ( 4 . 2 ) ' and (5.63) i n t o a c c o u n t we deduce k
In view of (7.23)' and (7.25) it f o l l o ws by ( 3 . 9 ) tha t
Since by L e m m a 2 . 3 4 , again, we derive f r o m ( 7 . 2 4)' tha t
we conclude t h a t g = 91
f where
X+
Similarly a s above, applying Proposition 4.13 to ( @ m * , , k , m * ) instead of ( T . m ) and taking ( 4 . 2 ) ' and (5.63) into account we s e e tha t t h e second s u m is equal t o
Consequently, f = x,,,',"~
where
k
Note t h a t by ( 7 . 2 5 )
x
belongs t o E . Since by P r o p o s i t i o n s 3 , 7 . ( i i ) , 2.f)l.(ii), and
2.59.(ii), t h e equation g = q l
f remains valid i F x is replaced by
x+
xG+
t h e proof
i s complete. rn
Note that by Remark 7.10 t h e condition ( 7 . 2 4 ) in Lemma 7.20 is slightly weaker
than i t looks. Making use of Lemma7.26 we now give a description of t h e polar sets of t h e subspaces A m - , ( 3 , , , - e , k ( q ) ~ ) and A, - @ ( k er B,,,). To this end we introduce
305
7.b Reforrnulatina (7.18)
Notatlon 7.27. ( i ) ~ , " ( q ; E g t , ) : = { R E Q , ( E B , ) ; R(,+=q(,+h (ii)
3 1 , , ; ( q ; E , ~ ) : = ( R E X r n ( q ; E ~ l ) (; a , - m ) ' R = q P
for some hrzCm(X+)};
f o r s o m e PEQ,-e(Es,#)}
f o r every i E No. W e also w r i t e X,-,, ( r e s p .
s,,;)instead
of X,(q;E,i)
( r e s p . X,,i(q;E,t)).
In view of t h e c o n d i t i o n s ( v ) a n d ( v i ) t h e conditions ( i ) - ( i i i )
of t h e f o l l o w i n g
Lemma c o r r e s p o n d to t h e conditions ( i ) , ( i i ) , a n d ( v ) of Remark 7 . 8 .
w.(i): Let
REX,,,,' a n d c h o o s e P E Q , , - p ( E ~ ~ )s u c h t h a t ( d M - m ) ' R = q P . It
t h e n f o l l o w s by Proposition 2.30.(ii) t h a t ( a , - m ) " ' R =
q ( d M - m + 0 P . Since by
Proposition 5 . 4 8 . ( i i i ) - a p p l i e d to d M - m + t instead of P o ( x , d ) - and by ( 2 . 3 0 ) t h e function (a,-m+P)P
(ii): Let i c N X,,'. H e n c e
b e l o n g s to Q,,,-e(EQ*)
we conclude that R C X m , i + l .
be s u c h t h a t R r n , i = X m , i - lI .t suffices to verify t h e eqality X m , i + l=
w e fix R E % ,
such that ( a M - m ) ' + ' R = q P for some P E Q , - ~ ( E Q ~ ) .
Since by Lemma 2.34 ( a M - m ) R b e l o n g s to X r n , a s w e l l , t h i s m e a n s t h a t ( d M - m ) R
-
b e l o n g s to X m , i a n d hence, by t h e a s s u m p t i o n , to Xrn,'-,. T h i s m e a n s t h a t (aM-m)'-'(aM-rn)R=qP
rn
f o r s o m e PEQ,-e(E+*),
(iii):t h i s is c l e a r since ( 3 , - m ) ' + ' R =
( i v ) : In
i.e.
0 f o r arbitrary R E Q,(E,*) t
-1
view of ( ~ , _ g ( 3 , - ~ , k ( q ) B ) ) O = ~ , - e ( ( 3 , , , - P , k ( q ) @ ) O )
as desired. and i ? N ( m ) . t h i s is a con-
s e q u e n c e of Remark 7.12.(i) a n d Lemma 7.25.
( v ) : in
view o f (7.17) a n d Remark 7.12.(i) t h e a s s e r t i o n is a c o n s e q u e n c e of Lem-
m a 7.26 a n d of t h e f a c t t h a t
X + is d e n s e in X .
306
V I I . S o l v a b l 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
w:S i n c e in view o f R e m a r k 7.12.(iii) t h e s u b s p a c e i m t C m e q u a l s X m * + e , o ( q ; E B t ) t h e a s s e r t i o n is a c o n s e q u e n c e o f ( 7 . l h . b ) .
0: Let R E Q m x + q ( E ~ ~By) . (2.5) if ( d M - r n * - t ) ' R E (kerC,)O.
R b e l o n g s to t h e p o l a r set ( Q i ) O if a n d o n l y
Hence t h e assertion follows from ( v i ) .
P r o o f o f L e m m a 7.23. By c o m b i n i n g t h e a s s e r t i o n s ( i v ) , ( i i i ) , a n d ( v ) o f L e m m a 7 . 2 8 one obtains ( A r n - , ( 3 m - p , k ( q ) ( y ) ) 0 = (A,-p(kerB,,,))
(7.27)
0
if k > N ( m ' + t ) .
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 t h e b i p o l a r t h e o r e m .
T h e f o l l o w i n g p r o p o s i t i o n is t h e desired d e s c r i p t i o n of t h e c o n d i t i o n ( 7 . 1 8 )
Ropositlon 7.29. Each of t h e f o l l o w i n g c o n d i t i o n s i s equivalent t o (7.18) : ( a ) t h e weak c l o s u r e o f A I r l - p ( k e r B l r l )in Q l i l - p ( E ) c + i s equal t o k e r C , , ;
( b ) t h e w e a k c l o s u r e o f A , - p ( , 3 m - P , k ( q ) m ) in Q A - p ( E ) e e q u a l s k e r C , ; ( c ) t h e weak c l o s u r e o f , s I , , . p . k ( q ) e j in Xr; , - p ( E) c qi s equal t o k e r B , , , :
I d ) : K , + + p , o ( q : E c 9 ~ =) X l r l + + p ( q ; E c a ~i .)e,. : for e v e n R E Q m t , , ( E c + ~ ) such t h a t R l x = q I x + h for some h
C"'(X+ I t h e r e e x i s t s a f u n c t i o n P E Q m . (E*-,l) ( r e s p .
CLnlX)) s a t i s f y i n g R = q P . P r o o f . 17.18) H ( a ) : T h i s f o l l o w s by t h e p r o o f of T h e o r e m 7 . 1 4 .
( a ) H ( d ) : In view of t h e a s s e r t i o n s ( v ) a n d ( v i ) o f Lemma 7 . 2 8 a n d s i n c e t h e i n c l u s i o n "A,,
( k e r B",) C k e r C,"
-
is a l w a y s valid t h e b i p o l a r t h e o r e m s h o w s
t h a t t h e c o n d i t i o n ( a ) is e q u i v a l e n t to
By t h e a s s e r t i o n s ( i ) , ( i i ) , a n d ( i i i ) of L e m m a 7 2 8 t h i s is e q u i v a l e n t to ( d ) .
( b ) e ( d ) :In view of t h e a s s e r t i o n s ( v i ) a n d ( i v ) o f L e m m a 7 . 2 8 a n d s i n c e t h e i n c l u s i o n "A,-
p(
j m - e , k ( q ) m ) C k e r C,,"
a l w a y s h o l d s t h e c o n d i t i o n ( d ) c a n be
r e p h r a s e d as
(d)'
(A,-p(3n,-p,k(q)@))
0
= (kerC,)O.
T h e b i p o l a r t h e o r e m , a g a i n , s h o w s t h a t ( b ) is e q u i v a l e n t to ( d ) ' .
307
7.b R e f o r m u l a t i n g ( 7 . 1 8 )
To p r o v e t h e e q u i v a l e n c e o f ( c ) a n d ( d ) w e f i r s t n o t e t h a t in view o f ( 7 . 1 b . b ) a n d R e m a r k 7 . 2 2 . ( i ) t h e c o n d i t i o n ( c ) is e q u i v a l e n t to (S,,,-e,k(q
(C)'
)@ ) O
c im 'Bm.
( c ) ' + ( d ) ; Let R be a s in c o n d i t i o n ( d ) , a n d c h o o s e a f u n c t i o n R=Q,+,e('p). jcbJ,,,
'pE
Em* s u c h t h a t
I t f o l l o w s by L e m m a 2.34 t h a t t h e f u n c t i o n s ( a M - m * - B ) J R l x + ,
are divided by ql,+.
C h o o s i n g f u n c t i o n s + o , . . . , + k E C G ( X ) a s in t h e p r o o f
o f L e m m a 7.26 a n d a p p l y i n g ( 7 . 2 6 ) to g : = ' p m * + e , b , kw e see t h a t t h e f u n c t i o n k
@ : = 'p -
+i i=O
(aM-m*-t)'g
h a s t h e p r o p e r t y t h a t @ m * + e , w kis divisible by ql,+ m a 7.25 b e l o n g s to (3,-e,k(q)cr)
0
a n d , c o n s e q u e n t l y , by Lem-
a n d h e n c e , by ( c ) ' , to i m t B m . In view of
L e m m a 7 . 2 6 , R e m a r k 7 . 1 2 . ( i i ) , a n d ( S . b 3 ) , a n d s i n c e X + is d e n s e in X t h i s m e a n s t h a t Q m . + e ( @ =) q P f o r s o m e P E Q m * ( E Q * ) .S i n c e s u p p
Jli
C X, w e have Q m . + p ( @=)
Q m * + e ( ' p ) so t h a t R = q P , as desired ( d ) a ( c ) ' ; Let g € ( , 3 m - e , k ( q ) ~ By r ) 0Lemma . 7.25 t h i s m e a n s t h a t g s a t i s f i e s
( 7 . 2 3 ) f o r s o m e h E C m ( X + ) . S i n c e t h e n f o r i = k + l t h e e q u a t i o n in ( 7 . 2 3 ) ' is valid a n d s i n c e by ( 5 . 6 3 ) w e have ( a M - m * - P ) k " g = RI,+ i t f o l l o w s f r o m ( d ) t h a t R = q P f o r s o m e PEQm*(Es,i).
7.12.(ii) L e m m a 7.26 s h o w s t h a t g
E
for s o m e REQm*+e(Ec31-) H e n c e in view o f R e m a r k
imtBm.
S i n c e by Remark 7.16 b o t h v e r s i o n s of t h e " i . e . ..." p a r t o f c o n d i t i o n ( d ) a r e equiv a l e n t t h e p r o o f is c o m p l e t e .
In p a s s i n g w e n o t e
Remark 7 . 3 0 . Let M' be the se m i-sim ple part o f M . i . e . the unique semi-simple linear endomorphism o f V in R C M I such that M - M ' is nilpotent. Then the r e a l cornple.\ coordinates chosen for M in Conventions 1.24.A and 1.24.8. ( i ) are suitable
For M
I,
a s wel l . Consequently, the spaces Q,,, ( E l remain the same i f M is replaced bj
M ' . Hence, the condition ( d ) o f Proposition 7.29 does not change i f M is replaced bj
M ' . Not e that i f X o = X then q is quasihomogeneous o f t j p e M ' . a s well.
I
P u t t i n g T h e o r e m 7 . 1 4 , P r o p o s i t i o n 7 . 2 0 , a n d Lemma 7.19 t o g e t h e r o n e c a n f o r m u l a t e t h e main t h e o r e m o n t h e solvability o f ( 7 . 1 ) in t h e set of a l m o s t q u a s i h o m o g e neous distributions a s follows.
308
V11. S o l v a b i l i t y of Quasihomogeneous E q u a t i o n s
Theoram 7.31. One (and hence each) o f the conditions of Theorem 7.5 is valid i f and only if one (and hence each) o f the conditions o f Proposition 7.29 and one (and hence both) of the conditions o f Lemma 7.19 hold. m
In view of Remark 7.0 t h e special case that q satisfies (6.28) reads as follows ( i f G = { Idv} see also Proposition 7.11 above):
Corollary 7.32. Suppose that q-‘(O) n X + =
a . Then the conditions o f Lemma 7.19
are automatically satisfied, and the conditions o f Theorem 7.5 are valid if and on14 i f one (and hence each) of the following equivalent conditions holds:
( a ) C,,, is injective: ( b ) B,,, is injective; (c) for ever, R E Q , , , L + P ( E there ~ ~ ~ )e d s t s a function PEQm*(E.c,t) satis-
tjing R = q P . m The simplifications which occur in the formulation of Theorem 7.31 when ker M = ( 0 ) are discussed in section ( d ) below.
(c?)
S o l v a b i l i t y o f ’ ( 1 . 1 ) f’or Incliviclual 1’
In t h i s section w e fix k € l N , .
If the conditions of Theorem 7.5 do not hold one
would l i k e to be able to decide for individual T € x o k , k ( E ) Qwhether or not the equation (7.1) has a solution S € ~ D ~ - t , k ( EFrom ) ~ . the proof of Theorem 7.14 we are going to extract a necessary condition which is often also sufficient. It requires the introduction of a special set of test functions appearing in (7.18)’:
Notatlon7.33. B y S m , k ( q : E ) we denote the s e t of all functions f E U z , k ( E ) such that q1
f extends to a function REQ,+p(E).
X+
This extension R is unique
and will be denoted by e q ( f l . If E = F ( V ) we also write S , n , k f q ) .
Remark 7.34.
( i ) Every f € S - , k ( q ; El is almost quasihomogeneous of degree m
and of order 5 N ( m +t).
309
7.c Solvability of ( 7 . 1 ) f o r Individual T
( v i i ) If X = V then Z , , k ( q ; 3 f V ) ) , * is dense in Sr,,.k(q),c,t in the topologj of the space X , r * k ( Y ( V ) ) o 4 . proOf. (i): Let f E X , , , k ( q ; E ) . S i n c e by L e m m a 2 . 3 4 w e have ql,+(aM -m)"' a n d s i n c e by ( 7 . 6 ) X \ q - ' ( O ) that (aM-m)"'f
f = (aM-m-O
N+1
9,:)
is d e n s e in X w e c o n c l u d e f r o m P r o p o s i t i o n 5 . 4 5
= O for N = N ( m + O .
(ii): T h i s f o l l o w s f r o m P r o p o s i t i o n s 5 . 4 8 . ( i i ) a n d 2.61.(ii) (iii) f o l l o w s f r o m (7.15) a n d P r o p o s i t i o n 5 . 4 8 . ( i i ) . (ivl: is c l e a r .
( v ) : The
inclusion
"L" is trivial.
For t h e p r o o f of
S i n c e by R e m a r k 7 . 9 . ( i ) M is s e m i - s i m p l e t h e C- f u n c t i o n f : = ( R ( , + ) / ( q
),I
"2" w e fix
REQ,+p(Em#).
it f o l l o w s f r o m C o r o l l a r y 6.28 t h a t
is $ ' - i n v a r i a n t
a n d q u a s i h o m o g e n e o u s of
degree m . Now w e f i x + E C E ( X ) s a t i s f y i n g + o s l . By P r o p o s i t i o n s 3.14 a n d h . 3 . B w e t h e n have f = ( + f ) , .
In case E = 3 ( X ) it follows f r o m C o r o l l a r y 5 . 4 7 , Lemma 6.23 a n d R e m a r k 5 . 5 6 t h a t s u p p f is c o n t a i n e d in K,
Cf
f o r s o m e c o m p a c t s u b s e t K o f X, so t h a t t h e f u n c t i o n
b e l o n g s to 9 ( X ) by P r o p o s i t i o n 3 . 2 2 . In c a s e E = Y P ( V ) w e may a s s u m e t h a t
t h e function
+ has
t h e p r o p e r t i e s ( i ) - ( i i i ) of L e m m a 6 . 3 8 . H e n c e w e d e d u c e f r o m
P r o p o s i t i o n 5 . 5 2 t h a t q + f = + R E Y ( V ) . C o n s e q u e n t l y , it f o l l o w s f r o m ( 7 . 5 ) t h a t t h e f u n c t i o n + f b e l o n g to Y ( V ) , a s w e l l .
So in e a c h case w e c o n c l u d e t h a t f b e l o n g s to X Z ( E ) a n d - s i n c e it is $'-invar i a n t - to X;(EhG*.
( v i ) : By
S i n c e X",E),,o
C 'Il",k(E)G* t h e p r o o f of ( v ) is c o m p l e t e .
( 7 . 6 ) eq is injective. In view of (7.17) t h e rest of t h e f i r s t p a r t of t h e
310
VII. Solvability of Q u a s i h o m o g e n e o u s Equations
assertion is a n immediate consequence of t h e a s s e r t i o n s ( i ) and ( i i ) of Remark 7.12. In view of Proposition 6.56 t h e second p a r t is obvious.
( v i i ) : We
fix a G-invariant
function r p E C g ( V ) with values in C 0 , l l equal t o 1 where @ € C g ( V )
o n t h e polycircle P : = { x € V ; P + ( x ) , M o ( x ) ~ K ( 0 , 1 )( }t a k e is equal to 1 o n P,)
Then
xi
and f o r every j € N define x j € C m ( V ) by x j ( x ) : = y , (J S M o x ) .
is G-invariant, quasihomogeneous of degree 0 and equal t o 1 o n t h e set
Li := { x E V
;
I Mox I 5 j 1 .
Now l e t f E 2 i z , k ( y ( V ) Q ' ) b e such t h a t ql,+f
e x t e n d s to s o m e R E Q m , p ( y P ( V ) ~ ~ ) .
Since s u p p x ) C M o ' ( j supprp) it follows by Lemma6.23 and Lemma S.S7.(ii) t h a t x i f belongs t o ltg,k(3(v)@,*).
By t h e Leibniz r u l e we obtain f o r every ci€lI t h a t
( f X j - f ) ' u ' ( x ) = f ( a ) ( x ) ( x j ( x ) - l )+
C
(i)f'OL-P)(x)
j-'"rp'')(:
1 M ~ X ) .
P €210 \ 0 ) p 5 U
Since x j - 1 has values in C0,ll and vanishes o n L, and since t h e functions rp") a r e bounded it follows t h a t f x j converges t o f i n t h e topology of X z , k ( Y ( V ) G ~ ) as j + a
(see Proposition 5.61). Finally, since
xj R
is a l m o s t quasihomogeneous of
degree m + e with s u p p o r t contained in M o ' ( j suppcp) it f o l l o w s by Lemma S.SO.(ii) t h a t x i R belongs to Q m , e ( 3 ( V ) , * ) .
Hence x j f € ~ m , k ( q : ~ ( V ) ) + - , * .
We now c o m e to a r e s u l t which f o r individual distributions T gives a necessary and sufficient conditions f o r t h e equation (7.1) t o have a solution S with prescribed ( l + k ) t h o r d e r deficiency c .
Theorem 7.35. Let T E U ~ , ~ ( Eand ) ~C~E, Q L _ ~ ( E ) ~ . ( i ) I f the equation ( 7 . 1 ) has a @-invariant solution S E E ' which is almost quasi-
homogeneous of degree m - P such that ( d , - m + k ' ) k + ' S = c
then the following
condition holds:
(ii) The converse implication is valid provided that ( 7 . N ) holds.
The special c a s e c = 0 is formulated a s
Corollary 7.36. Let TEE' be &-invariant and almost quasihomogeneous o f degree m and of order 5 k . Then the following holds.
311
7.c Solvability o f ( 7 . 1 ) f o r Individual T
lil I f ( 7 . 1 ) has a @-invariant solution S E E ' which is almost quasihomogeneous
M = 0
of degree m -! and o f order 5 k then
for ever, f E Sm* ( q ; E)&t .
( ii l The converse implication is valid provided that (7.19) holds.
I
Proof o f Theorem 7 . 3 5 . W e c l a i m : if RE E' 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-P s u c h t h a t d : = ( 3 M - m + t ) k + 1 RE Q A - t ( E ) ( y t h e n
In f a c t , w e c h o o s e c p E E w . s u c h t h a t e q ( f ) = Q , n f + t ( c p ) ;a n d s i n c e by P r o p o s i t i o n 0 . 7 7 w e have Q A - e ( d ) = d w e c o n c l u d e by
(().la), ( 2 . 5 ) , ( 0 . 5 8 ) , a n d
(2.39) that
< d , e q ( f ) > = < Q : , - p ( d ) , c p > = < d , c p >= < ( d M - m + t ) k " R , c p > =
=
( 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
<(~i(aM-m)iT),,,Jk,P>= i=O
k
=
2 i=O
< + i( d M - m ) ' T ,
f
k
>=
< T , ( d M - m * ) ' ( + i f )> =
1=0
k
=
c < T , [ ( - d ~ ) ~ $f j>]
=
Ck
f
)
i=O
w h e r e by a n a p p l i c a t i o n of ( 6 . 3 9 ) to j = O w e see t h a t
Ck:=
(-l)k
ci=ock,i = 1. k
313
7.c S o l v a b i l i t y of ( 7 . 1 ) for I n d i v i d u a l T
H e n c e , t h e f i r s t e q u a l i t y i s p r o v e d . For t h e p r o o f of t h e s e c o n d e q u a l i t y w e ob-
serve t h a t by T h e o r e m 4.29 w e have k
k
a n d t h a t ( a M - m * l k - ' f = 8i.k f .
If q s a t i s f i e s ( 6 . 2 8 ) i t is s o m e t i m e s u s e f u l to c h o o s e x a s f o l l o w s :
Remark 7.38. Suppose that C f 0 and that q-'(O) n X , of /ql''ReP
t o X, is a Cmfunction x q : X + + l O ,
neous of degree 1 such that q Q = l on S x q .
= @ . Then the restriction
+mt
which is quasihomoge-
I
Bxample 7.39. Suppose that C f 0 . that 9 is non-negative and has no zeros in X , , and that there exists c r E X L . + p such that each component of
(Y
is even.
Then for every T E B ' I X ) which is almost quasihomogeneous of degree m such that on X , its k t h order deficiency is induced bj. a positive continuous function the equation 9s = T has no solution S E B ' I X ) which is almost quasihomogeneous of degree m-t' and of order 5 k .
m f . W e a s s u m e t h a t E = % ( X ) .In view
of t h e d i s c u s s i o n c o n c e r n i n g t h e e q u a t i o n s
( 7 . 1 ) a n d ( 7 . 2 ) a t t h e beginning of s e c t i o n ( a ) t h e p r o o f is r e d u c e d to t h e case k = O . H e n c e , w e a s s u m e t h a t TI,+=T,
for a suitable continuous function
g : X + d l O , + ~ CW. e set x : = x q (see R e m a r k 7 . 3 8 ) . M o r e o v e r , w e f i x a p o i n t x in X \ X ,
a n d c h o o s e E , K , a n d L as in t h e p r o o f of t h e i m p l i c a t i o n " ( a ) ' J ( a ) "
of P r o p o s i t i o n 1.91. T h e n w e fix a n o n - n e g a t i v e f u n c t i o n y E C T ( X ) e q u a l to 1 o n
K . By (1.107) y o M O is e q u a l to 1 o n L . Finally, by t h e a s s u m p t i o n w e c h o o s e CLEW;*,@
s u c h t h a t e a c h c o m p o n e n t of a i s e v e n a n d d e f i n e a n o n - n e g a t i v e f u n c -
t i o n f : X , d @ by f ( x ) : = x " y ( M o x ) / q ( x ) . S i n c e by our c h o i c e of x w e have f ( 8 ) = 3 " ,B E L , i t f o l l o w s t h a t
J' g ( B ) f ( 8 )dx"(8) t
1' g ( 8 )3" dx"(3).
SX
L
H e r e t h e r i g h t - h a n d side is positive: t h i s f o l l o w s by P r o p o s i t i o n 1.86 s i n c e t h e s e t Z : = { M t 9 ; B E L , tCC1,21} isofpositiveLebesgue m e a s u r e s o t h a t J Z g ( x ) x a d x is p o s i t i v e , H e n c e , in view of R e m a r k 7 . 3 7 . ( i i ) ( t h e a s s u m p t i o n of w h i c h is valid
314
VII. Solvability of Q u a s i h o m o g e n e o u s Equations
by Remark 7.9.i) and Corollary 7.36 . ( i ) t h e proof will be complete once i t is verified t h a t f belongs to X,*(q;E).
But this follows from R e m a r k 7 . 3 4 . ( v ) since
by Corollary 5.47 and Proposition S.48.(ii) t h e function x Qm*+p(E).
t+
xacpoMo belongs to
H
The following example illustrates in a less simple situation t h a t it is sometimes possible to verify t h e condition ( 7 . 2 9 ) by direct computation.
Example 7.40. Suppose that V = R x R " - ' . Let W E C\ iR,define 17.31)
P,,,(t.o= i w t + I [ / ' ,
and s e t p : = ( . ? . l .
It.{) ERxR"-',
, l ) . G : = i l d , I , and o : = r : ~ l Then . f o r ever) h E N q : = P , h
satisFies t h e condition OF Corollar) 7.36 for T - 1 , m = 0 , and k = 0 . C o n s e q u e n t l ) , the equation q S = l has a solution S E Y ' ( R " ) which i s quasihomogeneous o f d e gree - P . proOf. Indirectly this follows from Corollary 7.30 and t h e existence of quasiho-
mogeneous fundamental solutions for q ( D ) ( s e e t h e references in O r t n e r C151).
We sketch a direct computation. L e t u , v E IR be such t h a t i w = u + i v . We assume t h a t IwI = I and set x : = lP,l
1/2
.
Then x ( r , < ) = lif and o n l y i f 1 < I 2 S R : = $ and r € { r ? ( < ) } where r,([):=
-U1
t j 3 - g .
Moreover, if x h ,5 ) = 1 then
We choose j E N o and PEN:-' (7.32)
such that
Zj+lBI = 2h-n-1.
In view of Remark 7 . 3 4 . ( v ) , Remark 7.37.(ii) and Lemma 1.87 and since o n S x we have l/q
= 4 it
suffices to show that
1'
( q ( r + ( [ ) , < )r + ( < ) J+ q ( r - ( < ) , < ) r - ( < ) j
2dS
= 0.
KtO.6)
Introducing polar coordinates with respect to the space variables [ one sees t h a t t h e left-hand side is equal to
315
7.c Solvability of ( 7 . 1 ) f o r Individual T
i n t o account where
w i t h qj(r) : = r - i ? . Obviously, Gj is a polynomial of degree 5 j satisfying
Hence it suffices to prove that G i ( 0 ) = O . Observing that r
h-j-2
- d rt1-i-l dr h-1-1
We
do partial integration and i n view of (7.33)
(‘ph
?’)‘(r) = y hi
T Fh)(r)
This reduces the proof t o the verification of G , ( O ) = 0 . To this end we set 1
h
-
h
bj : = s ( c p ( r ) + q ( r ) ) ri 0
e. r
Doing partial integration first on the basis of
and then in the first of the resulting integrals on the basis of
d rJ.2 r J + ‘ =; i ~ j + 2
we obtain if O < j < h - 2 .
If h is even then it follows from (7.33) that bo=O and hence by induction that G , ( O ) = b h - 2 = O . If h is odd then first substituting p = ? , i.e. d p = ( r / ? ) d r , and then performing partial integration based on
316
VII. Solvability of Quasihomogeneous Equations
( s e e (7.33)) we obtain 1
bl = i h J (rph - ( p h ) ( p ) d g =
ih-' b,.
0
Since by (7.32) and because of n t 2 we have h r 2 t h i s shows that b l = O so that by induction, again, we deduce that Go(0)= b h - 2 = 0 .
In order t o avoid repetitions w e postpone the discussion of further examples after having reformulated Theorem 7.31 for the case E = Y ( V ) by employing the Fourier transform in the following section.
(d b
Quasihomogc-n e o us IA1n ea I- 1% 1.1 1a I I)i I ' f ' c v * t w 1 1a I Ey ua 11o n s
w 11h
C:o
n sl an 1
Cot?1'1'1 1e n 1s
In t h i s and the following section we suppose that q is a non-trivial complex-valued polynomial function defined on V' which is quasihomogeneous of degree 0 EX(M ) and of type M' and is (G,r/a)*-invariant (see Notation 2 . 0 0 ) . Then for E = Y ( V * ) the conditions (7.3).(7.4)and (7.5)are valid ( s e e Remark 7.1). Via the Fourier transform the results ( for E = Y(V * ) and for ( M * , G * , a*,
t*.
m* ) instead of
( M, G , a , t , m ) ) of the preceding sections turn into assertions o n the solvability of the equation (7.34)
q ( D ) S= T
in the s e t of (almost) quasihomogeneous temperate distributions. For the k t h
order deficiencies d of T and c of S t h i s involves the equation
(7.35)
q(D)c= d
as is seen by Corollary 2.36.(i).We fix kCN, and define
Notatlon 7.41. ( i ) &W'A(V) to be replaced by M * ;
: = F - ' ( Q k t ( Y ' ( V * ) ) ) where i n Notation 6.54 M has
7.
317
Partial D i f f e r e n t i a l E q u a t i o n s
( i i ) Z E P A , k ( V ) : = F - ’ ( Xk*,,(Y’P(V*))) where in Notation 6.57 M has t o be re-
placed by M*.
Remark 7.42. (i.1) IF k e r M = (01 then f l Y $ , ( V ) ,
is equal t o the space
‘p,CV),
OF all @-invariant complex- valued poljmomial Functions on V which are almost quasihomogeneous OF degree m ;
(i.2) if ker M f (01 then making use o f the notation introduced in (6.22) we have f l Y ’ A ( V ) m is equal t o the space OF all @-invariant distributions OF the Form
,y ( i i ) X9,L.k I V ) ,
XUdS,
where S, 6 Y”(V , 1 .
is the space of all @-invariant temperate distributions on V
which are almost quasihomogeneous o f degree m with ( l + k )t h order deficienq belonging t o ,tW&( V ) , . proOF. Everything f o l l o w s from Propositions 2 . 4 0 , 6 . 2 4 . ( i v ) ,2 . 6 4 . ( i i ) . and 6 . 2 9 . m
As an immediate consequence o f Propositions 7.2,2 . 6 4 . ( i i ) , and 2 . 4 0 o n e obtains
Ropoeitlon 7.43. For ever) d € i W i , ( V),c, the equation ( 7 . 3 5 ) has a solution cE W A
+
p(
VI, .
#
Likewise, Theorem 7 . 3 transforms i n t o
Theorem 7.44. For any J:, -invariant distribution T€XY’A,,k( V ) $ , the equation 17.341 has a @-invariant solution S€2K.9’A+p.k(V)c*.
A s a special c a s e w e obtain the assertion ( i ) of
Theorem 7.45. Suppose that r = I . Then (i)
q ( D ) has an almost quasihomogeneous ( O F degree l - p ) @-invariant temperate
Fundamental solution whose deficiencj belongs t o f?Y’i-p ( VJC3 ; note that such
a Fundamental solution is quasihomogeneous iF P < p . (ii.1) i f q ( D I has a quasihomogeneous temperate fundamental solution then
(7.36)
( I , F>,=
0
For ever). f € R - , , ( q : 3 ( V f ) ) , & ,
(see Notation 7.331
318
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
where t$:=(G*,ts)
with
‘r(A):=r(A*),
i.e. ‘ r = l / r 4 .
(ii.2) the converse is valid provided that q -‘(O) n ( V * ) = 0 or ker M = ( 0 ) . +
Note that by Remark 7.37 in case M is semi-simple the left-hand side of (7.36) is equal t o
Jixf(31 d x “ f 8 ) .
prooE. S i n c e
t h e Dirac d i s t r i b u t i o n S is G-invariant a n d q u a s i h o m o g e n e o u s of de-
gree - p t h e a s s e r t i o n (i) i s a c o r o l l a r y of T h e o r e m 7 . 4 4 . T h e a s s e r t i o n (ii) follows f r o m C o r o l l a r y 7.36 a n d Remark 7 . 3 4 . ( v i i ) .
W e r e f r a i n From c a r r y i n g o v e r t h e f u l l s t a t e m e n t o f T h e o r e m 7.35 via t h e F o u r i e r
t r a n s f o r m . N o t e t h a t in case m + @ @ ‘ u ( M )t h e s p a c e s n Y A ( V ) a n d , f W A + e ( V )
are trivial b y C o r o l l a r y 6.28 so t h a t ‘ u Y A , k ( V )= ‘uD;,,(Y(V))
(see N o t a t i o n 7 . 1 8 ) .
H e n c e , in t h i s case t h e a s s e r t i o n of T h e o r e m 7 . 4 4 b e c o m e s
(7.37),
for every .):,-invariant T 6 9 ’( V ) which is almost quasihomogeneous
of degree m and of order
2
k the equation (7.341 has a (&-invariant)
solution S 6 Y ’ ( V ) which is almost quasihomogeneous of degree m+P arid of order- 5 A .
A s for t h e e x c e p t i o n a l c a s e “ m + 4 E X ( M ) ” , in view o f L e m m a 2.21 P r o p o s i t i o n 7.4 becomes
Roposltlon7.46. Suppose that m + P E U ( M ) and that k > N : = N ( m + P ) Let . T be an
6-invariant
temperate distribution on V which is almost quasihomogeneous
of degree m and of order- i A .
(i) Then the following conditions are equivalent. ( a ) the equation (7.341 has a (@-invariant) temperate solution S which is almost quasihomogeneous of degree m +P and of order 5 k
;
( b ) for some (resp. every) j 6 N k - N the equation 17.351 for d = ( d M - m l J T
has a (@-invariant) temperate solution c which is almost quasihomogeneous of degree m +t and of order 5 A - j .
(ii) If c is such a solution as in ( b ) then in ( a ) S can be chosen in such a wa-t that ( d M - m - P ) ’ S differs From c b, an element of
A ~ Y ~ , + ~I ( V ) ~ .
7.
319
Partial Differential Equations
In order to d e a l with t h e case “m+k‘EfU(M)” f u r t h e r w e i n t r o d u c e
Notation 7.47. ( i ) Xrn.k(q) : = ? - ‘ ( 3 , * , , ( q ) ) (ii)
a,n(q) : = ( 3 M - m ) k + ’x r n , k ( q ) =
(see Notation 7.21).
v-’((a,*
-m*)
k+l
s m * , k ( q ) ) ; this space
does n o t d e p e n d o n k , indeed, a s can be seen f r o m Remark 7.22.(ii) o r f r o m Remark 7 . 4 8 . ( i ) below.
I t d o e s n o t s e e m to b e easy to give a fairly explicit description ( n o t involving t h e Fourier t r a n s f o r m ) of t h e e l e m e n t s o f t h e s p a c e % , , k ( q )
( c o m p a r e sec-
t i o n 8 . ( d ) b e l o w ) . However, f o r t h e s p a c e ; D r n ( q ) o f its ( I + k ) t ” o r d e r deficiencies it is easy:
Remark 7.48. (i)
prooF.
,arII ( q ) is
g e n e r a t e d b-, t h e f u n c t i o n s of t h e form
(il: t h i s f o l l o w s f r o m Lemma 2.21 and Example 0 . 3 6 ,
(iil: t h i s is a c o n s e q u e n c e o f Lemma 2.21 and Lemma 7.23.
rn
Proposition 7.29 t u r n s i n t o
Ropoeltion 7.49. The f o l l o w i n g conditions are equivalent: ( a ) t h e weak c l o s u r e of . a t r l + P ( q ) ( ~in , L 2 p A , + p ( V ) c 9 is e q u a l to t h e s u b space k W ’ k + p ( V ) c S , n k e r q ( D 1 :
( b ) t h e weak c l o s u r e of JZ,,+p,k(q)m
i n 2 W k + p , k ( V ) m is e q u a l to t h e
s u b s p a c e 2 1 Y A + p , k ( V ) ( qn k e r q ( D 1 ; (c) t h e condition (d) of Proposition 7.29 is valid
with
( V ,M , m * ,
@ ’ , $ I )
replaced b., ( V * , M ’ , m , t @ . t , o ) . i.e. Zrrl.k(q)tac, (see N o t a t i o n 7.331 is e q u a l t o
)I
Q,,l(.YP(VX)t-
.v
(V”)+
Lemma 7.19 b e c o m e s
w h e r e t,!.j:=(G*,‘r) with ‘ r ( A ) : = r ( A * ) .
I
320
V I I . 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
Lemma 7.50. The space q ( D ) U ~ ~ + P , k ( Y P ( (see V))~ Notation 7.18) is weakly closed in
Y'( V ) i f and only i f
closed in
*Qs/A
+
(dM - m - L')
UYA
+
t,k
I V ) m n ker q ( D ) ) is weaklq
p(V),.
Finally, Theorem 7.31 transforms into our main solvability theorem for ( 7 . 3 4 ) :
Theorem 7.51. The following conditions are equivalent: ( a ) the solvability condition 17.371, is valid f or some k
for every pair ( T ,c ) o f temperate distributions on V such that T (resp. c ) is ,$-invariant
(resp. @-invariant) satistving q ( D ) c = ( d M - m ) " + lT
there exists a (@-invariant) temperate solution S o f the equation (7.34) such that ( ~ M - I T I - P ) ~ + ' S = C ; ( c ) f o r some (resp. every) k 6 N o the following condition holds. (7.40),
for every solution C E ~ Y ' , ~ + ~ (o V f the ) ~ equation q ( D ) c = 0 there exists a (@-invariant) solution S E Y ' I V ) of the equation q ( D ) S = O which is almost quasihomogeneous o f degree m+P with ( l + k )t h order deficient) c ;
I d ) for some (resp. ever).) k 6 N o one (and hence each) o f the conditions o f Proposition 7.49 and one (and hence both) o f the conditions o f Lemma 7.50 are valid.
I
There are t w o important general situations where the condition ( d ) o f Theorem 7.51 can be simplified, namely if ( 6 . 2 8 ) holds or if ker M = (0).T o handle the second o n e we require the following
Remark 7.52. Suppose that ker M = 101. l i ) Then the space V,,,CV)
o f complex-valued polynomial functions on V which
are almost quasihomogeneous o f degree m is finite -dimensional, and
7.
321
Partial Differential Equations
dim prn( V " )
(7.41)
tB
= dim pm (V)q,.
(ii) d i m ( ~ , , p ( V ) g n k e r q ( D ) ) = d i m 9 , + p ( V ) g -dimP,IV),.
(iii) amI ( q ) is the subspace OF pm8+ +
(V)
generated by the polynomial Functions
( i v ) For the subspace S r n , k ( q )o f 2 K z , k ( y ( V * ) )introduced in Notation 7.33 we have d i m B m + p ( q ) e5 dim P , + p l V ) ~ - d i m F,,k(q)Lsc,
(7.13)
with equality being valid provided that k 2 N ( m + e ) .
m: t h e
proOF.
f i r s t a s s e r t i o n is c l e a r . For t h e proof of (7.41) o n e o b s e r v e s t h a t
p,(V)=QY'k(V)
( b y R e m a r k 7 . 4 2 ) a n d Q,(YPV*))=Cp,(V*)
a n d t h a t by t h e
i s o m o r p h i s m property of t h e Fourier t r a n s f o r m a n d by Propositions 2 . 6 4 . ( i i ) a n d 6.56 o n e has:
d i m Q Y ' P ; , ( V ) b = d i m Q ~ + ( Y ' P V * ) ) b *= dimQ,(Y'(V*)),Q.
(ii): in view of Remark 7 . 4 2 . ( i ) t h i s is a consequence of Proposition 7 . 4 3 .
liii): is a special case o f Remark 7 . 4 8 . ( i ) .
(iv):
f o l l o w s f r o m ( i i ) - a p p l i e d to (m+P,(H) i n s t e a d of ( S 2 , m ) - a n d f r o m Pro-
positions 2 . 6 4 . ( i i ) a n d Lemma 7 . 2 3 .
T h e following s u p p l e m e n t to Theorem 7.51 d e a l s with t h e c a s e t h a t ( 6 . 2 8 ) h o l d s .
Supplement 7.53. Suppose that q - ' ( O ) n I V * I + = @ (i)
.
Then the conditions o f Lemma 7.50 are automaticallq satisfied, and the con-
ditions OF Proposition 7.49 amount t o (7.44)
(ifm
n ker q ( D ) = 10)
For
@ = ,QYrA +
(V
) or
(if =
XYA
+
P.k ( V
);
resp . 7.44)'
Qm+p(SP(Vx)t,) = q Q m I P ( V * ) t S ) .
IF ker M = 10) then ( 7 . 4 4 ) is equivalent t o
(ii) In particular, i f G = { l d v } then the conditions OF Theorem 7.51 are violated iF and only i f r n + e € I ( M ) , i f e # O . and i f R e m < O or i F dim'GMM(o+)?9.
322
V I I . Solvability of Q u a s i h o m o g e n e o u s Equations
Proof. Ij): In view of t h e as s er t i o n s l i i i ) and ( i v ) of Remark 7.34 t h e f i r s t pa rt is
a reformulation of Corollary 7.32 : by Remark 7 . 4 2 . ( i ) t h e second pa rt follows
f r o m Remark 7 . 5 2 . ( i i ).
(ii)
is a consequence of Proposition 7.11.
Concerning t h e assertion ( i i ) of Supplement 7.53, it should be pointed o u t tha t if @-invariance is taken into account then the chances a r e b e t t e r tha t the re a r e posit i v e r e s u l t s. This is illustrated by t h e f i r s t t h r ee examples in section ( e ) below.
Next we formulate t h e simplifications in t h e condition ( d ) of Theorem 7.51 which follow from t h e assumption k er M = ( 0 ) .
Supplement 7.54. Suppose that ke r M = 101. Then the conditions o f Lemma 7.50 are automaticallj s at isfie d, and the condition ( a ) o f Proposition 7.44, becomes
which i s equivalent t o
the converse inequality being alwajs valid. Another equivalent condition is
the converse inequality being always valid.
mf. By Remark 7 . 4 2 . ( i ) t h e assumption o n M implies t h a t nY';,e(V)ce
is equal
to 'p,,,,e(V) and hence finite-dimensional so t h a t i t s subspaces X ) r r l + P ( q ) wa nd (
aM- m - t ) k + ' ( X Y 'm + P , k ( V ) c r in k e r q ( D ) ) c e ar e so, a s well.
In particular, all the se
s p a c e s a r e closed. Hence, t h e second condition of Lemma 7.50 holds, and t h e condition ( a ) ( r e s p . ( c ) ) of Proposition 7.40 is identical with (7.45) ( r e s p . ( 7 . 4 6 ) ) . Moreover, t h e equivalence of ( 7 . 4 5 ) and ( 7 . 4 5 ) ' ( r e s p . of (7.46) and ( 7 . 4 0 ) ' ) is a consequence of Remark 7.52.(ii) ( r e s p . (7.41) and Remark 7 . 3 4 . ( i i ) ) . rn
In passing we observe
7.
323
Partial D i f f e r e n t i a l E q u a t i o n s
Remark 7.55. Suppose that kerM = 101. Let M' be ( a s in Remark 7.30) the semisimple part OF M . Then the spaces , Z ) , , , + ~ ( qand ) pI,,+p(V) remain the same i f
M is replaced by M'. Hence, in particular, the conditions o f Theorem 7.51 are valid i f and only i f they are so when M is replaced by M'.
I
As a f i r s t application of Supplement 7.54 we obtain
Ropoaltlon 7.56. Suppose that ker M = {0).Then q satisFies the conditions
of Theo-
rem 7.51 f o r G = 1IdVl provided that one o f the Following conditions holds:
( a ) 3 , q = 0 For some VEV'\COI: ( b ) there e,\ist two non-trivial M-invariant subspaces V , and V , of V satisfying V = Vl @V2 such that q is OF the Form q = q l B q , where for ever, j 6 1 1 . 3 ) q i : Vi* - C
is a poljmomial Function which is quasihomogeneous o f t j p e
(M
vi
) I.
proOf. Under t h e assumption (a) it follows in view of Remark 7 . 3 4 . ( i v ) from Re-
mark 7 , 1 7 , ( i ) t h a t t h e condition (7.4(1) is satisfied. If under t h e assumption&) q j = l f o r s o m e j e ( 1 . 2 ) then ( a ) is valid, a s well. Since q l @ q 2 = ( q 1 @ 1 ) ( 1 @ q 2 ) o n e obtains t h e general assertion from t h e special one by t h e following obvious remark on t h e surjectivity of composites of multiplication o p e r a t o r s .
Remark 7.57. For ever., j€C1,2} let ql be a pol.bnomia1 Functioii on V * which is quasihomogeneous o f degree PI and o f t j p e M
'.
Suppose that q = q , q, . For sim-
plicitj we assume that G = {Id, I . Then (i)
q satisfies the condition ( a ) ' o f Theorem 7.51 provided that q1 does so f o r
P I instead of P and q2 does so with ( m . P) replaced bj
(m+Pl,
P,
):
( i i ) converselj, if q satisFies the condition f a ) ' of Theorem 7.51 then q , and q2
d o so with P replaced by P , and P,
.
respectivelj.
I
As a simple consequence of Proposition 7.56 we a r e going t o derive
Ropoaltlon 7.58. Suppose that M = Idv and P = 1 . I F q has a zero
in V
* \ 10) - which
is always the case it" n 2 3 - then q satisFies each o f the conditions of Theorem 7.51,
proOF. By the assumption o n q we find V E V ' \ { O ) such t h a t q ( v ) = O . Since t h e
324
VII. S o l v a b i l i t y of Quasihomogeneous E q u a t i o n s
t h e assumptions on M and .t imply t h a t q is linear w e conclude t h a t a,q = O t h e assertion follows by Proposition 7.56.
.
Hence
rn
We close this section by observing t h a t t h e condition ( 6 . 2 8 ) is not necessary for t h e non-solvability of (7.34)in t h e set of quasihomogeneous distributions. Namely, in view of Supplement 7.53.(ii) t h e contraposition of Remark 7.57.(ii) leads to
Remark 7.59. Suppose that d i m ' G M ( o + ) ? 2 and m + P E U I M ) . Let qr be a pol-vnowhich is quasihomogeneous o f degree Pl # 0 . Suppose that
mial function on V'
q , - ' ( O ) n (V*)+= @ . Then q := q 1 qz does not satis[), the condition ( a ) ' o f Theorem 7.51 for every polynomial function q2 on V x which is quasihomogeneous o f de-
gree P - P , .
I
To see how under t h e assumptions of Remark 7.50 t h e condition ( 7 . 4 5 ) is violated we require
Lemma 7.60. Let g , h E C ' " ( X ) and ) , E X , and let Q be a polynomial function on V'. Then
( i ) ( y , Q) E S ( g )
(j,,Q)
E SU(gh):
( i i ) the converse is valid i f h ( j . ) # 0 .
Proof. (i) is an immediate consequence of t h e Leibniz formula and of t h e f a c t t h a t ( y , Q t a ' ) € z ' \ ( g ) for arbitrary ( y , Q ) and ole'u.
(ii) is
obtained by an application of ( i ) to ( g h , l / h ) instead of ( g , h ) . m
Remark 7.61. Suppose that k e r M = 101 and m + P E # ( M ) . Let q1 and PI be as in Remark 7.59, and let q p be a polynomial function on V' neous o f degree Pz :=
&-el such
which is quasihomoge-
that q2 satisfies the condition ( a ) ' o f Theorem 7.5/
for ( l l d v l , m + P , ) instead o f ( G . m ) . Then f o r q : = q ,q2 we have dim p,r, +
(V
) - dim ' p , ( V ) - dim
+p
( q ) = dim p ' ,,
+
(
V ) - dim pn, ( V ).
proOf. By Lemma 7.60 we have ,23m+,(q) = B m + p ( q z ) . Hence t h e assertion follows
by Supplement 7.54 applied to ( q 2 , m + 4 , ) instead of ( q , m ) . rn
Example 7.62. Suppose that M = l d v , V = R ' , and q ( s , y ) = ( x . - i y ) ( x - ~ ,( )x, , y ) ~ l R ~ . Then d i m D a , + 2 ( q ) = l= ~1 ( d i m ' p , + z ( V )- d i m p m ( V ) ) f o r every m 6 N o . I
I
7.e
325
Examples
te) Examples
In t h i s s e c t i o n w e a r e g o i n g to treat s e v e r a l e x a m p l e s . T h e f i r s t o n e d i s c u s s e s t h e e l e m e n t a r y case " d i m ' V = l " . For t h e s a k e of c l a r i t y w e f o r m u l a t e t h e c a s e s
" V = IR" a n d " V = @ " s e p a r a t e l y . Example 7.63,.
Suppose that V = R and M = I d R .
I f G = ( I d R I then the conditions o f Theorem 7.51 hold i f and only i f m is
lil
not an integer satistying -t 5 m 5 - 1 . Suppose that G = { _ + I d ,I ; then the assumptions on @. J j , and q mean that
liil P
o = p r where p ( - + l d R ) : = _ + I ; consequently, i f m belongs t o the exceptional s e t
o f assertion ( i l then the conditions o f Theorem 7.51 hold i f and on/> 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
?
(
in
t h e subset o f C" c o n s i s t i n g o f
a l l t h e n - t u p l e s of p s e u d o - r e a l c o o r d i n a t e s o f t h e p o i n t s of T , w e c l a i m t h a t w
N
t h e T a y l o r c o e f f i c i e n t s o f P a t any p o i n t o f t h e set T A c a n be c o m p u t e d f r o m N
t h e k n o w l e d g e of t h e r e s t r i c t i o n o f
to T A . To see t h i s 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 f o r any h o l o m o r p h i c f u n c t i o n h d e f i n e d o n a n o p e n s u b s e t o f C2 w e have
w h e r e f o r any z in t h e d o m a i n o f d e f i n i t i o n o f h by d + h ( z ) ( d - h ( z ) ) w e d e n o t e t h e derivative a t t = O of t h e real f u n c t i o n t S i n c e t h e family
H h ( z l + t , z 2 + t ) (resp.
h(zl+it,z2-it)).
h a s t h e desired p r o p e r t i e s t h e p r o o f o f t h e m a i n p a r t
of the a s s e r t i o n is c o m p l e t e .
330
V11. 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
S u p p o s e n o w t h a t M is s e m i - s i m p l e . If
C o n s e q u e n t l y , w e c a n r e p l a c e E,,
is any family i n I O , + ~ Ct h e n
(t,),EA
by M,&
a n d - in view o f P r o p o s i t i o n 1.70 - p r e -
scribe t h e m o d u l u s o f E,, as w e w a n t . In p a r t i c u l a r , w e c a n achieve t h a t E,, E S v .
T h e following example concerns t h e Schrodinger operator.
Example 7.67. Let u 6 k , and suppose that qtt,EI = u t + I t / ' , t t , E ) ~ l R x l R " - ' . Then q is quasihomogeneous o f t j p e p := ( 2 ,I , . . . , I ) and o f degree I := 2 , and f o r ever-) m the conditions o f Theorem 7.51 are valid f o r G = I l d , 1 . In particular, f o r ever)
h E N the operator q"(DI has an O ( n - 1 ) -invariant fundamental solution which is quasihomogeneous o f degree 2 h - n - I (Bernstein's fundamental solution has these properties - see t 71I . proOf. F o r every kCN, w e set I n ( k ) : = { a E N t ; l a l = k } . W e f i r s t o b s e r v e t h a t b y
a bijective m a p 4 : ' U ~ ~ I , - ~ ( m + 2 ) + ' U ~ + 2 is d e f i n e d . S i n c e d i m ' p m ( l H " ) = this implies
W e set p:=-,, 1
fix E , E S n - 2 a n d set q : = ( p , [ ) . T h e n q ( q ) = O , a n d t h e p o l y n o m i a l
f u n c t i o n in ( 7 . 4 2 ) f o r Q
3
1 becomes +m
RX!R"-'
3x=(Xi,X')
Hd,(x):=
j=o
(ipx1)jRi,€(x') J '
where
in S"-'
s u c h t h a t t h e poly-
n o m i a l s Ro,es, S C I , - , ( m + Z ) , a r e linearly i n d e p e n d e n t . I t f o l l o w s t h a t t h e polynom i a l s dc8, S E l , - , ( m + 2 ) ,
a r e linearly i n d e p e n d e n t . In view of ( 7 . 5 0 ) t h i s i m p l i e s
t h e c o n d i t i o n ( 7 . 4 5 ) ' of S u p p l e m e n t 7 . 5 4 .
T h e n e s t e x a m p l e is about t h e w a v e o p e r a t o r .
331
7.e Examples
Example 7 . 6 8 . Suppose that q ( t , E ) =( c t ) 2 - ( f , c > .( t , € l E R x R r ' - ' . f o r s o m e p o s i tive constant c . Then q is homogeneous OF degree 2 , and For ever, m the conditions o f Theorem 7.51 are valid For M = l d R n and G = ( l d R n 1 . In particular, for every h E N the operator q h ( D ) has an O ( n - 1 ) -invariant homogeneous fundamental solution.
Proof.
We define I,(k)
a s in t h e proof of Example 7.67 and set B
:=
I,_,(m+2)
a n d T : = l , - l ( m + l ) . Since by
a bijection @ : I , ( n i + 2 ) - +
(7.51 1
I , ( m ) u B u T is defined w e obtain
~ l l l ( m + 2 )I ~l , (- m ) l = I B I +
lrl.
W e now fix < E S n - 2 . Then q t : = ( ? l / c , < ) is a non-trivial real z e r o o f q so t h a t
for Q e l a n d q = q + t h e polynomial function in ( 7 . 4 2 ) b e c o m e s
where
R J.E . (x') :=
5-
ED (ix')'
A
P E I,
- ( in
+
2- j
)
If m = - 2 t h e n d , + _ = R , , E , ~ l a. n d since dim
Po(IR1')= 1
t h e condition ( 7 . 4 5 ) ' 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 m 2 - 1 . Then w e s e t Q,+ : = - l( d , + ? d , 2
By Lemma 7.06 w e can c h o o s e families (f,B)BEBand t h e polynomials Ro.eB, P E B , ( r e s p . R i , c
Y
, YET)
I t f o l l o w s t h a t t h e polynomials Q & , D E B , and
-)
so t h a t
(
in
$-2
such that
a r e linearly i n d e p e n d e n t over C .
QCy,
Y E T , a r e linearly i n d e p e n d e n t
o v e r @ . Since they belong to a,,2(q) w e c o n c l u d e t h a t
In view of (7.51) t h i s implies t h a t t h e condition ( 7 . 4 5 ) ' of S u p p l e m e n t 7.54 is s a t i s f i e d . rn
W e are now g o i n g to d i s c u s s a f e w e x a m p l e s in c a s e
M is n o t s e m i - s i m p l e
332
VI1. 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
Propodtion 7.69. Suppose that V is M - c ~ c l i cand that dim' V = 3 ( r e s p . -7) in case the only eigenvalue A of M is real ( r e s p . non-real). Then the conditions of Tbeorem 7.51 are always valid for G = (Id"). Proof. The case A E R . W e set d := e / X and
Q
:=
m/X
. We
may s u p p o s e t h a t V* = R3
a n d t h a t M * . ( x , y , z ) = ( X x + y , X y + z , A z ) . By Proposition 1.46 o n e f i n d s c E N , , s u c h t h a t c S d / 2 a n d c o m p l e x n u m b e r s a o , . . . , a, s u c h t h a t a,# 0 a n d q ( x , y , z )= zd P ( ( y 2 - 2 x z ) / z 2 ) w h e r e P is t h e polynomial defined by P : =
xrz0a j T i . W e f i r s t o b s e r v e t h a t by Pro-
position 7.56 t h e polynomial f u n c t i o n qo defined by q o ( x , y , z ) : = aczd-"
satisfies
t h e conditions of T h e o r e m 7.51. Hence w e may s u p p o s e t h a t c > 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 <_ k + l it f o l l o w s by i n d u c t i o n by a s i m i l a r a r g u m e n t f r o m ( 7 . 5 6 . ~ )t h a t Bk is a p o l y n o m i a l
f u n c t i o n o f d e g r e e 5 k . C o n s e q u e n t l y , by ( y , z ) H B k ( Z / y ) y k a p o l y n o m i a l f u n c t i o n o n IR2 is w e l l - d e f i n e d . T h i s i m p l i e s ( 7 . 5 4 ) . as desired.
The
case A
I R . H e r e w e may s u p p o s e t h a t V* = C2 a n d t h a t M * - ( Z ~ , Z=~( )h z 1 + z 2 , X z 2 ) .
By P r o p o s i t i o n 1.47 w e f i n d c € N o s u c h t h a t c 5 m i n { m R , m l } a n d c o m p l e x n u m -
bers a o , . . . ,a, s u c h t h a t a, # 0 a n d q ( z ) = ( z 2 ) r n R ( Z . , ) r n 'P ( ( q G - T l z z ) 1z21-2) w h e r e P is t h e polynomial d e f i n e d by P : = ~ ~ = o a j T W i .e f i r s t observe t h a t in view of P r o p o s i t i o n 7.5h a n d Remark 7.57 t h e polynomial f u n c t i o n q n d e f i n e d bj
qo(z) =a, (z2)
mR-c
( G )m 1- c
s a t i s f i e s t h e c o n d i t i o n s of T h e o r e m 7.51. H e n c e w e may s u p p o s e t h a t c > o . I f q,...,u,€C
are such that P=a,fl,c=l(T-wi)
we conclude that q =n;=oqi
where
q j ( x , y . z ) : =( z 1 2 - q z 2 ) - u i 2 2 2 ,
j€N.
In view o f R e m a r k 7.57 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 t h e q i . In view o f w h a t w a s s a i d a b o v e a b o u t 90 t h i s leaves us w i t h t h e c a s e q ( z ) = ( z l c for some u€@.
-
+w z 2 g
N o t e t h a t if u € l R t h e n q ( z ) v a n i s h e s if a n d o n l y if z 2 = 0 .
F i r s t o f all w e o b s e r v e t h a t t = X + X = Z R e X . W e set b : = ( m + 4 ) , a n d ~ : = ( m + t ) ~ . I t f o l l o w s f r o m Remark 1.40 t h a t
C o n s e q u e n t l y , if m a n d m + 4 b o t h b e l o n g to X ( M ) t h e n b = m R + l a n d c = m I + l so t h a t (7.57)
dirn13,,+e(V)-dim?),,,(V) = ( b + l ) ( c + l -) b c = b + c + l
M o r e o v e r , if m + 4 b e l o n g s to X ( M ) b u t m does n o t t h e n by L e m m a 1 . 4 1 w e have
b = O or c = O , a n d d i r n ' ) 3 , + e ( V ) = b + c + l , a n d ( 7 . 5 7 ) r e m a i n s valid. For t h e computation of B m + p ( q ) w e first observe t h a t q ( z I . O ) = O f o r every
z , E C . F o r every j € N w e set i Q~: = Ca i , i ( ~ ~ (z,+Z,)' ) i - ~ i=l
336
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
where a j , j: = 1 and where a j ,1 , . . . , aj , j - I are constants to be determined below in
s u c h a way that ( ( l , O ) , Q j )belongs to B ( q ) . To t h i s end we fix a = ( ( r , s ) , ( t , u ) E ) X and first observe that in case IalSj-3 or l a l > j + l or r 2 j o r t t l we have Q:a'(3)q
10.Moreover, if I a l = j then Q{a' is constant so that Q i a ' ( 3 ) q vanishes
a t z = ( 1 , O ) . Hence we suppose that j > 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
d-1
n , ( i + l=) i = j
gI ! , in= j ( d - i ) = ( d - j ) !
cl - 1
,
and
d-j-I
n,(n+Z(d-i-l)) =
(n+Zi) i=O
i = j
t h e proof is c o m p l e t e .
In view of t h e f a c t t h a t by Proposition 2.1cj a n d by ( 2 . 8 ) a t e m p e r a t e d i s t r i b u t i o n h
is invariant o f d e g r e e m if and only if its Fourier t r a n s f o r m T is invariant of de-
g r e e m* = - m - n - 2
o n e o b t a i n s t h e following version o f Proposition 7 . 7 4 .
Proporitlon 7.74'. The equation 17.64) '
q s =0
has a non-trivial solution S E E ' ( i 0 ) ) which is invariant o f degree m i f and on1-v i f k : = -Im+n+.-3)/2is a non-negative integer. In this case every such solution is a complex multiple o f Qk(-D)cSo where
Qk
is defined in Proposition 7.71.
I
I t is now easy to d e t e r m i n e a l l a l m o s t invariant s o l u t i o n s o f (7.64) if v f O :
Theorem 7.75. Suppose that v f O . Then the equation q(DI T = 0 has a non-trivial solution TE3'(RxIR"I which is almost invariant o f degree m i f and only i f m is a non-negative even integer. All such solutions constitute a 1-dimensional space
consisting of the complex multiples o f the polynomial functions Qm,= in Proposition 7.71 ).
(defined
343
7.f T h e Heat a n d t h e Schrodinger E q u a t i o n
w. By Theorem 6.45 it suffices to s t u d y temperate s o l u t i o n s of
( 7 . 6 4 ) . Since
then t h e equation ( 7 . 6 4 ) is equivalent to q T = 0 and since v f O implies t h a t q - ' ( 0 ) n R x R n = (0)w e have to consider only t h o s e t e m p e r a t e s o l u t i o n s T s u c h t h a t s u p p T C ( O ) ,i.e. T is induced by a polynomial function Q . Consequently, since a l l a l m o s t invariant polynomial functions a r e invariant t h e a s s e r t i o n f o l l o w s by Proposition 7 . 7 4 . m
We now come to the c a s e v = O . Here we f i r s t of all determine a sequence o f
distinguished a l m o s t invariant s o l u t i o n s of ( 7 . 6 4 ) ' . To this end we observe t h a t by (7.68)
< S .'p > : = *I'
'p( - l / u
, 8 )dlCf ,
'pECrn(lRX (lR"\{O})),
sn-1
a distribution S E E ' ( I R x ( l R " \ ( O ) ) )
is well-defined
solving t h e equation ( 7 . 0 4 ) '
and being invariant under t h e action of O ( n ) o n t h e space variables
5 . I t follows
by t h e a s s e r t i o n s ( i i ) and ( i v ) of Proposition 6.16 and by Proposition 6.35 t h a t f o r every k E No t h e distribution (7.69)
S,,k
: = 'm,"k
o n IRxlR" is a solution of ( 7 . 6 4 ) ' which is a l m o s t invariant such t h a t t h e s u p p o r t of (d,-m)k''S,,k
is contained in (0).Moreover. ( 6 . 3 0 ) implies t h a t
Since in c a s e O@suppcy o r Rem < 0 we have +m
we obtain by introducing polar coordinates t h a t in case O d s u p p ' p o r Rem > - n - 2 we have
Propodtion 7.76. ( i ) S,,,,o is invariant if and only i f -m-n-.?@2/NO lii)
If
P : = - ( m + n + 2 ) / 2 is a non-negative integer then
344
(7.71)
V I I . Solvability OF Q u a s i h o m o g e n e o u s
Equations
( 3 M - m ) S m , o= d pQ,(-D)&,
where Qp i s defined in Proposition 7.74 and where
d , := I S " - ' / ( - ~ ) - ~ / t !
mf.By Proposition6.35 we have R : = ( d M - r n ) S , , , = Q A ( S ) ,
i.e. R = Q ( - D ) G ,
where
with
and
Since the last integral vanishes i f pi is o d d f o r s o m e i E N n it f o l l o w s t h a t Q = O in c a s e - m - n - 2
is odd. So in t h i s case S,,,,O is quasihomogeneous. S u p p o s e now
t h a t t h e assumption of ( i i ) is satisfied. I t follows from Corollary 2.3h.(ii) t h a t
R is a solution of ( 7 . 6 4 ) ' , a s well. Moreover, by Propositions h . 2 4 . ( i ) and 6 . 2 6 . ( i v ) we conclude t h a t R is invariant of degree m . Hence, by Proposition 7.74' t h e r e is a c o n s t a n t d , E @ such t h a t Q = d , Q , .
TO c o m p u t e dp we fix x E C ~ ( R x l R " )
s u c h t h a t x - 1 o n a neighbourhood of 0 and define c p ~ C ~ ( l R x l R "by ) T(T,<):= T ' x ( T , ~ )Then .
< Q ( - D ) G , , , ~ ~=> ( Q ( D ) ~ ) ( O =) + i p ( - u ) - '
I S - 'i-,t! I
and
=(Qe(D)cp)(O) =i-'P! . This implies t h e assertion o n d , , and t h e proof is c o m p l e t e .
Now we can f o r m u l a t e
Theorem?.??. Suppose that v = O . Then For every N E N , the space OF solutions T ~ d ) ' ( l R x l ? " ) of (7.b-l)' which are invariant of degree m and of order 5 N is (N+l)-dimensional, a basis being given by the distributions Sm,,, O l j _ < N , in case P : = -(m+n+21/.2 does not belong t o 2GVO and by the distributions Qp(-DISo and S m , i , O-CjTN-1, in case P is a non-negative integer.
For t h e proof we require t h e following lemma describing t h e invariant d i s t r i b u t i o n s
7.f
345
T h e H e a t a n d t h e Schrodinger Equation
o n t h e set X : = I R x ( l R " \ ( O ) ) . I t is a l s o valid in case v Z 0 . Here we make use of t h e C" diffeomorphism Y : X h X defined by
Note t h a t if f:X-C
is a function which is invariant of degree m then - c o m p a r e
t h e proof of ( 7 . 6 6 ) - we have IxI-m(foY)(t,x)= f(t,el),
( t , x )E X ,
where el is any unit vector in R". That this remains valid f o r arbitrary distributions o n X is t h e c o n t e n t of
Lemma 7.78. If X = l R x ( i R n \ l O / l then f o r ever, T c B ' ( X 1 the following conditions are equivalent: ( a ) T i s invariant OF degree r n :
( b l 3 , , ~ I x I - ' " T o Y1 = O f o r everl jCEV,,,
and i f n = l then T is invariant
under t h e transformation (t..\1 + j ( t . - A ) .
( c l there i s a distribution R C B ' ( l R 1 such that T = (RdlvI'"1o Y - ' .
(7.73)
The distribution R satistving (7.731 is unique and will be denoted by T' Proof. (a1
* (b). First
( 7.74)
Xj
axkT =
we a r e going to show t h a t Xk
axiT,
We may suppose t h a t j < k
.
j,kEINI,.
Then f o r 8~ IR we define S,ESO( I + n ) by
X k - l , x js i n s + + C O S 8 , X k + l , . .. x n ) . ( t , x ) e ( t . x l , .. . , x j - l , x j c o s 9 -xk s i n 9 , ~ .~. .+. . ~
Since To (S,)-'= T we obtain
O=-I d d9
= , 8=0
'p
which immediately implies ( 7 . 7 4 ) . Now, by the chain rule we have
a =
xi
( I X I - ~ T O Y= )
axi(IX1z)-m'z
T O Y+ lx I - m
2 t x j (a,T) oY + I x I - m ( d x i T )oY =
E
c;c
X),
346
V I I . 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
Since t l x I 2 is equal to t h e f i r s t component of Y ( t , x ) inserting ( 7 . 7 4 ) leads to n
Since by Euler's equation t h e t e r m in b r a c k e t s vanishes t h e derivation of ( b ) is complete.
( b ) * ( c ) . We f i r s t s u p p o s e t h a t n ? 2 . By t h e theory of distributions t h e condition ( b ) implies t h a t t h e r e is a unique distribution R~a'clR)s u c h t h a t
(7.73)'
IxI-"'ToY
= RBI
where I d e n o t e s t h e distribution o n l R " \ { O )
induced by t h e c o n s t a n t function 1 .
Obviously, t h i s is equivalent to ( 7 . 7 3 ) . We s u p p o s e now t h a t n = 1 . Then t h e theory of distributions yields t h e e x i s t e n c e of unique distributions R, , R - E 1 x 1 -"I
7-0 y
I
IR
~
a'(lR) satisfying
3 0 . 2 a,[
= Rt@(q,o,+m[)'
Since T and hence l x I - m T O Y ~ I R x x o , + ma rCe invariant under t h e t r a n s f o r m a t i o n ( t , x ) H ( t , - x ) it f o l l o w s t h a t R + = R - .
(~)+(a)T , h a t T is invariant under t h e action of O ( n ) on t h e x-variables is obvious. For arbitrary s ~ l O , + aaln d r p E C T ( X ) we obtain < ~ o ~ , . r p )=
s - " - ~ < T , ~ ~ o M ~ ,=, >
= s - n - 2 < R ~ I x I n ' , c p ~ M 1 , ~ o Y d e t Y=' > = sm
< R,
,I (
Ixl/s)m'2rp(t
(
l x l / ~ ) x~/ s, ) d ( x / s ) ) =
IR"
= s m < R t , J l y l m + 2 r p ( t l y 2l , y ) d y > = s m < T , c p > . rn !R"
A s a consequence of t h e preceding lemma we a r e going t o derive
Propodtlon 7.79. W e s e t X : = R x ( R " \ I O I ) . Suppose that v = O . Let N e N , .
Then
the space o f solutions T € B ' ( X ) of ( 7 . 6 4 ) ' which are almost invariant o f degree m and of order 5 N i s ( N + l ) - d i m e n s i o n a l , a basis being given b y the distributions
7.f
347
T h e Heat and t h e Schrodinner Eauation
prooE: by
i n d u c t i o n o n N . Let T E D ' ( X ) be a n invariant s o l u t i o n o f ( 7 . 6 4 ) ' . By
Y
w e d e n o t e t h e d i f f e o m o r p h i s m d e f i n e d by ( 7 . 7 2 ) . S i n c e ( q o Y ) ( r , c )= 1 [ 1 2 p w ( r ) w h e r e p,(r)
: = u r + 1 w e d e d u c e - t a k i n g Lemma 7.78 i n t o a c c o u n t - t h a t
(P,T)o't'=
(pwT')@1~1m'2. 1
C o n s e q u e n t l y , P w T = O if a n d o n l y if p w T = O . S i n c e ( p , ) - ' ( O ) since -l/u
={-l/u}
and
is a zero of p w o f o r d e r 1 t h e e q u a t i o n p w T 1 = 0 is s a t i s f i e d if a n d
o n l y if T ' = C S - ~ ,f~o r s o m e c E @ . S i n c e f r o m ( 7 . 6 9 ) ' o n e deduces t h a t (~rn,Olx)'
= S-l/u
t h e p r o o f o f t h e c a s e N = O is c o m p l e t e . In order to p r o v e t h e i n d u c t i o n s t e p w e f i x N E N a n d a s o l u t i o n T E ~ ' ( Xof ) (7.64)'
w h i c h is a l m o s t i n v a r i a n t o f d e g r e e m a n d of order 5 N . S i n c e t h e n ( d M - m ) N T is a s o l u t i o n of ( 7 . 6 4 ) ' which is invariant of d e g r e e m it is e q u a l to c S
f o r s o m e c E C . S i n c e it f o l l o w s f r o m ( 7 . 7 0 ) t h a t (a,-m) w e conclude t h a t R := T - ( - 1
) Nc
m,NIX
S
N
m.OIx
S l n , N = ( - 1 ) N Sm,o
is a l m o s t invariant of d e g r e e m a n d of
o r d e r 5 N-1. S i n c e R is a s o l u t i o n o f ( 7 . 6 4 ) ' , as w e l l , t h e i n d u c t i o n h y p o t h e s i s i m p l i e s t h a t it is a linear c o m b i n a t i o n o f t h e d i s t r i b u t i o n s S m , j l x , 0.j < N - 1 , i . e . T is a l i n e a r c o m b i n a t i o n of t h e d i s t r i b u t i o n s S,,,,jlx, 0 5 < N . T h a t t h e l a t t e r d i s t r i b u t i o n s a r e linearly i n d e p e n d e n t f o l l o w s f r o m ( 7 . 7 0 ) a n d he f a c t t h a t Sm,-,l
w
is q u a s i h o m o g e n e o u s .
Proof of Theorem 7.77. In view of w h a t w a s s a i d in t h e t e x t p r e c e d i n g Proposit i o n 7.76 w e a l r e a d y k n o w t h a t t h e d i s t r i b u t i o n s m e n t i o n e d in t h e a s s e r t i o n a r e a l m o s t i n v a r i a n t s o l u t i o n s o f ( 7 . 6 4 ) ' . C o n v e r s e l y . l e t TE ~ ' ( RRx" ) be a s o l u t i o n
of ( 7 . 0 4 ) ' which is a l m o s t invariant of degree m a n d of o r d e r < N . T h e n by P r o p o s i t i o n 7.79 w e f i n d a s e q u e n c e o f c o m p l e x n u m b e r s ci , O < i S N , s u c h 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 N
R := T -
ci Sn,,i j=O
is c o n t a i n e d in I R x ( 0 ) . S i n c e t h e s u p p o r t s of T a n d S m , j are c o n t a i n e d in q - ' ( O )
t h i s i m p l i e s t h a t s u p p R C (01 . S i n c e R is a l m o s t invariant a n d solves ( 7 . 6 4 ) ' w e d e d u c e f r o m P r o p o s i t i o n 7.74' t h a t
R v a n i s h e s in c a s e 4 : = - ( m + n + 2 ) / 2 is n o t a
n o n - n e g a t i v e i n t e g e r a n d 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 , i . e . in case
PEN^.
N o w , in t h e l a t t e r c a s e , in view of
( a M- m ) R = 0
348
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
( a , - m ) N + l S m , , = ( - 1 ) ' ( & M - m ) N + l - Js m , o= 0
O<j
1
( s e e ( 7 . 7 0 ) ) w e deduce by Proposition 7.76.(ii) and (7.70) t h a t CN
N
deQ@(-D)S,=(-l) (3M-m)
N+l
T=O
so t h a t c N = O in this case, a s desired.
In view of Proposition 2.19 and ( 2 . 8 ) Theorem 7.77 can b e reformulated as
Theorem7.77'. Suppose that v=O. Then f o r every N 6 N o the space of solutions T E ~ ) ' ( R X R o" f) (7.64) which are almost invariant o f degree m and OF order 5 N is (N+l)-dimensional,a basis being given by the distributions T m * .I := F - ' ( S - m - n - 2 , i ) ,
OSjLN.
in case P := m/B does not belong t o No and by the distributions T o r (see Proposition 7.74) and Trn.i. 0 5 j s N - 1 , in case t is a non-negative integer.
I
From Theorems 7.77' amd 7.72 a n explicit formula f o r T-,,,o is obtained: (7.75)
T-n.onconst (E'-E-)
This s h o w s , in particular, that t h e singular and the analytic singular s u p p o r t of
T - n , o coincide with ( 0 ) x IR". This is valid i n general:
Theorem 7.80. Let TE!S"(RxU?") be an almost invariant solution of the eyudtion ( 7 . 6 4 ) which is not induced by a polynomial function. Then i t s singular and i t s
analytic singular support coincide and are equal to /O1 X U ? " . Proof. " a -singsupp T C /01 XU?"
".
and to assume t h a t T = T , , k .
First we deal with the case "Rem
In view of Theorem 7.77' it suffices t o fix k EN, 0 " . We fix
a ~ ( l , - l ) a, , b , R E I O , + ~ C s u c thh a t a < b , and ~ € C ~ ( C o a , b b l x K ( O , R We ) ) . observe
t h a t by t h e Fourier inversion formula, by ( 7 . 6 8 ) , by Fubini's Theorem and by Lebesgue's Dominated Convergence Theorem w e have
( 2 x ) n ' 1 ( - l ) k < T , , ~ , k , ~ ) = ( -k1 <) S , * , , , , , c p >A"=
3 49
7.f T h e H e a t a n d t h e S c h r o d i n g e r E q u a t i o n
N e x t w e are g o i n g to c h a n g e t h e c o n t o u r o f i n t e g r a t i o n . To t h i s e n d w e have to identify t h e i n t e g r a n d a s a h o l o m o r p h i c f u n c t i o n o n a s u i t a b l e o p e n s u b s e t o f @ . E m p l o y i n g t h a t b r a n c h o f t h e l o g a r i t h m f u n c t i o n which is h o l o m o r p h i c o n t h e r i g h t R e C > O ) a n d r e a l - v a l u e d o n 10,+03Cw e e x t e n d t h e f u n c t i o n
half p l a n e $ : = { C E @ ;
t
to a h o l o m o r p h i c f u n c t i o n h,:.$-C
H t-,-l"k(t)
(7.76)
lh,(<)I
satisfying t h e e s t i m a t e
CC l < i e ~ l + l l o g l < l ~ ~ k ,
1
where C : = n k e x p ( n I I m m I ) and 4 : = - R e m - I . M o r e o v e r , d e n o t i n g by H t h e s u p p o r t i n g f u n c t i o n o f t h e set C o a , o b l x K ( O , R ) w e see t h a t H ( I , - ~ J =) - a l r l + R l u l f o r every r,EIRn a n d f o r e v e r y rEIR s a t i s f y i n g 0 1 5 0 .
S i n c e f o r every < E n : = { z E @ ;R e z > O , G S i g n ( U ) I m Z < o } w e have = 2tRe
oIm<'/u
it f o l l o w s t h a t w i t h t h e a b b r e v i a t i o n A : = 2 a / l u l w e have
C o n s e q u e n t l y , in view o f t h e Paley- W i e n e r - S c h w a r t z t h e o r e m by F ( < ):=
1'
&(<'/u,-<9)d9,
s n-1
a n e n t i r e h o l o m o r p h i c f u n c t i o n is w e l l - d e f i n e d s u c h t h a t f o r every N E N t h e r e is a c o n s t a n t CN s u c h t h a t
(7.77)
I F ( L ) I S C , (l+I
< E n .
fl by s H Ee x p ( - i s a s i g n u ) 1. set c : = ( 1 - i o s i g n u ) / D .
N o w , w e d e f i n e y E : CO,n/41-+
a n d - n o t i n g t h a t y , ( n / 4 ) = E C - o b t a i n by C a u c h y ' s t h e o r e m t h a t 1/E
J' s - m W k ( S ) J'
$(S2/U,
ds - S 8 ) d9 7 =
s n-1
E
1/E
= cJ' h , ( c s ) E
F ( c s ) d s + j ' h , ( < ) F ( < ) d
J'
h,(<)
F(C) d< ,
Yl/E
In view o f (7.76) a n d by t h e f a c t t h a t 4 > - 1 it f o l l o w s t h a t lim € s u p { l h m ( < ) l ;< € $ ,
ICI=E) =O.
E - 0
S i n c e F is b o u n d e d o n K ( 0 . 1 ) t h i s i m p l i e s t h a t
EE10,lC.
350
V I I . 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
C o m b i n i n g ( 7 . 7 6 ) a n d ( 7 . 7 7 ) w e likewise o b t a i n t h a t
M o r e o v e r , t h e e s t i m a t e s ( 7 . 7 6 ) arid ( 7 . 7 7 ) s h o w t h a t t h e f u n c t i o n I O , + ~ C - f l , s H h m ( c s ) F ( c s ) , is a b s o l u t e l y i n t e g r a b l e . C o n s e q u e n t l y , by L e b e s g u e ’ s t h e o r e m ,
again, w e obtain t h a t +cD
= ( - 1 )
k
( 2 ~ t - ” - ’ c fh m ( c s ) F ( c s ) d s . 0
Now w e c a n s h o w t h a t n e a r t h e s u p p o r t o f y
is i n d u c e d b y a r e a l a n a l y t i c
f u n c t i o n . S i n c e f o r a r b i t r a r y s E l O , + ~ l B, E S ” - ’ ,
it f o l l o w s t h a t o n t h e set X , : = ( b l O , t c u C ) x I R r ’
, .
x€!Rn, and t E a l 0 , + m C w e have
bq
+m
f , ( t , x ) : = ( - l ) k ( Z K ) - ~ - ’ c.1
.I
h,,(cs) e x p ( - s ” l t l / l u l + i c s t x . 3 ) ) d a d s
c) s n - 1
a c o n t i n u o u s f u n c t i o n f,:X,--+C
is w e l l - d e f i n e d s a t i s f y i n g ( b y t h e Fubini a n d
t h e Fubini-Tonelli t h e o r e m ) :
< T , , . k , ~ >= J ’ c p ( z ) f ( z ) d z . X O
I t r e m a i n s to b e p r o v e d t h a t F is real a n a l y t i c . To d o t h i s w e f i x
E > 0 .
set p : =
C R l u l / ~a n d observe t h a t f o r ~ E I R \ I - E , E C x, € K ( O , R ) , a n d s E l p , + m l w e have Re(-s’ltl/Iul+ics
<x,B))
c
-qs2
w h e r e q : = L21ul
’
I t f o l l o w s t h a t in t h e d o u b l e i n t e g r a l +m
g,(t,x):=.)’
.I’
h,(cs)exp(-s”ItI/IuI+ics<x,B>)dBds
sn-1
B
w e may i n t e r c h a n g e d i f f e r e n t i a t i o n a n d i n t e g r a t i o n so t h a t
( 3 k, d ,Pg p ) ( t , x ) = +m
= ( - a / ~ u (~i c) )~’ @ J’I P
.
s211+1p1 h , ( c s )
9’exp(
- s2 It I /
I u l + ics < x , B >)
S
sn-1
for a r b i t r a r y kEINo a n d BEN,”.
By (7.7(1) a n d s i n c e w e may a s s u m e w i t h o u t loss
of g e n e r a l i t y t h a t p ? l w e o b t a i n a c o n s t a n t C’ s u c h t h a t I ( a k, a ,Pg p ) ( t , x ) 1 5 C ’ l u l - k
Is”-1I B ( k + I p I + L )
351
7.f T h e Heat a n d t h e S c h r o d i n g e r E q u a t i o n
w h e r e L is t h e s m a l l e s t n a t u r a l n u m b e r s t r i c t l y l a r g e r t h a n
:=s
+m
B(j)
sZi
-r and 2
where
+m
e x p ( - q s 2 ) 2 ~ d s = q - ~ - l J ' r ' e x p ( - r ) d5 r q - ' - ' C ( j )
1
ll
with +m I
CCj) : =
.
J
r' e x p ( - r ) d r = j ! ,
j€lNo.
0
S i n c e ( k + I p l + L ) ! C 2k"""L
[PI!
5 2""lP!
k! ( I p I + L ) ! , s i n c e ( I p I + L ) ! 5 2 ' ( 3 1 + L L [! P I ! , a n d s i n c e
t h i s s h o w s t h a t g, is real a n a l y t i c o n ( I R \ I - E , E [ ) x K ( O , R ) . S i n c e
by i n s e r t i n g t h e T a y l o r e x p a n s i o n o f t h e e x p o n e n t i a l f u n c t i o n o n e sees t h a t t h e f u n k t i o n I k x Rn-@
d e f i n e d by F)
(t.x)
.f .f 0
h,(cs) sn-1
e x p ( - s 2 I t I / l u l + ics t x . 9 ,
)
dads
is real a n a l y t i c , a s w e l l , w e c o n c l u d e t h a t f, is real a n a l y t i c , i n d e e d .
The general c a s e . I t s u f f i c e s to s h o w t h a t f o r a r b i t r a r y
mCC
a n d k € N , w e have
For t h e n t h e g e n e r a l c a s e f o l l o w s f r o m t h e s p e c i a l case p r o v e d a b o v e by i n d u c t i o n . NOW,
from (7.h0)'we deduce t h a t
I < I ~ s , ,is ~e q-u a~l to, ~s r n * , k o n
IRX(R~\(O)).
S i n c e t h e s u p p o r t o f b o t h d i s t r i b u t i o n s is c o n t a i n e d in q - ' ( O ) a n d s i n c e IRX
( 0 ) n q - ' ( O ) = { ( 0 , OI)
w e c o n c l u d e t h a t t h e d i s t r i b u t i o n s d i f f e r by a d i s t r i b u t i o n w i t h s u p p o r t c o n t a i n e d in ( ( 0 . 0 )i . T h i s i m p l i e s t h a t t h e d i s t r i b u t i o n s -ACT,+2,k a n d T,,,
d i f f e r by a
polynomial function. Consequently
M o r e o v e r , t h e e q u a t i o n ( 7 . 6 4 ) m e a n s t h a t w a t T r n + 2 , k = ACT,,+,,,
so t h a t
C o n v e r s e l y , by T h e o r e m 8.6.1 in H o r m a n d e r Clll it f o l l o w s f r o m ( 7 . 7 9 ) t h a t W F A ( T m + 2 , k )C WFA(T,,,,)
u [Char(Ag)n C h a r ( a t ) ]
352
V I I . 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
w h e r e Char(P) d e n o t e s the characteristic set o f P . Since
t h e intersection of b o t h sets is e m p t y so t h a t WFA(T,+2,k)
C W F A ( T m , , k ) . So
t h e proof of ( 7 . 7 8 ) is c o m p l e t e .
"(0)xW"Csingsupp T " . If s i n g s u p p T C ( 0 ) it t h e n f o l l o w s by Proposition 2.23 A
A
-
h
t h a t s i n g s u p p T C ( 0 ) which in view of s u p p T C q '(0)implies t h a t s u p p T C ( 0 ) c o n t r a d i c t i n g t h e a s s u m p t i o n o n T . C o n s e q u e n t l y , in view of t h e inclusion already proven above w e find < E I R n \ ( 0 )
s u c h t h a t ( O , E , ) b e l o n g s t o s i n g s u p p T . Since T
is invariant it f o l l o w s t h a t { ( O , € , ) ; E , E ! R " \ ( O ) }
lies in s i n g s u p p T . Since t h e l a t t e r
is a c l o s e d s u b s e t of RxlR" t h e desired inclusion f o l l o w s .
H
353
Chapter VIII
Extending (Almost) Quasihomogeneous Distributions on X, to the Whole of X
As in C h a p t e r 6 w e a s s u m e t h a t (1.14) h o l d s a n d t h a t X f X , . In s e c t i o n ( a ) t h e
theory of s e c t i o n 4 . ( e ) is carried over to t h e following s e t t i n g : for any fixed
C- f u n c t i o n x : X , + l O , + m C
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 every distri-
bution o n t h e hypersurface S x is e x t e n d e d to a n ( 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 o n X . In p a r t i c u l a r , t h i s leads to a n alternative d e s c r i p t i o n of 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 o n X (see Theorem 8.8 b e l o w ) . In s e c t i o n ( b ) it is s h o w n how t h e m e t h o d of taking q u a s i h o m o g e n e o u s averages a s well as t h e c o n s t r u c t i o n of section ( a ) can b e used to e x t e n d ( a l m o s t ) quasih 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 ( 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 o n t h e w h o l e of X . Employing t h e s e ( 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 s , f o r quasih o m o g e n e o u s C m f u n c t i o n s q:X--+C
w i t h o u t z e r o s in X , o n e c a n easily solve
t h e e q u a t i o n q S = T in t h e set of 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 (see Theorem 8.24 b e l o w ) . While t h e e x t e n s i o n s c o n s t r u c t e d in s e c t i o n ( b ) a r e n o t always unique, in s e c t i o n (c) for every f E Y ’ ( V )
a distinguished e x t e n s i o n of Tf,
to t h e w h o l e of V is con-
s t r u c t e d . Its Fourier t r a n s f o r m is c o m p u t e d in s e c t i o n ( d ) . This leads to a f o r m u l a f o r t h e Fourier t r a n s f o r m of u,,, if u is any t e m p e r a t e d i s t r i b u t i o n with M - t e m perate support.
It s h o u l d b e pointed o u t t h a t t h e p r e s e n t c h a p t e r is c o m p l e t e l y independent of C h a p t e r 7 a n d c a n b e read s u b s e q u e n t to C h a p t e r 6 .
354
V I I I . Extending ( 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
(a) Pulling n a c k Dlslrihullons on S x l o (Almost) Quasihomogeneous IBislr*ihulions on X
We fix a C m function x : X + +
l O , + m [ which is quasihomogeneous of degree 1
( t h i s is possible by Corollary 3.36 and Theorem 3 . 3 9 ) . Recall from sections l . ( f ) and 4 . ( e ) that Sx is a C"submanifo1d
of X and that a)'(Sx) is identified w i t h
t h e dual of CTCSx) via the density w
~-
see( (4.32). ~ In ~analogy with Lemma 6.23
we have
Lemma 8.1. If K i s a compact subse t of X then the s e t S X nM & ' ( K ) is compact. Pr o o f . By Lemma 0.23 we find a compact subset L of X such that the s e t L,
contains M o ' ( K ) . By Remark 6 . 8 the set S x n &
is compact. Since by Lemma 5.46
is a closed subset of S x n q the assertion follows. rn
S"nM,'(K)
The following proposition corresponds to Proposition 0 . 2 4 .
Proposition 8 . 2 . Let v € 9 ' ( S x ) .BJ
:= < v . (0,. cp)lSx>
Q;,v
6 3'(X) i s wel l -de fine d having the following properties:
(i)
Q L v is al mos t quasihomogeneous of degree m and of order
iii)
the support of Q;,,V is contained in
Moisupp v ) n X
_<
a distribution
N(mr);
the latter being a
cl o sed s u bs et of X \ X, and hence c lose d in X:
where
7 is
defined by
(8.1)
A
v :=Q:,,v.
proOf. Let K be a compact subset of X . By Lemma 8.1 L : = SXnM,'(K)
is a
compact subset of Sx. In view of Corollary 5.47 the map C;(K)+%(L), 'p H ( Q m +'p)
Isx
,
is well-defined and, of course, linear and continuous. Conse-
quently, Q A v is a well-defined distribution o n X .
(i):
By Proposition 5.48.(iv) and Proposition 5.45 we deduce for every tElO,+mC
that for N := N ( m X ) we have
355
8.a Pulling Back D i s t r i b u t i o n s on S x
~
N
= t-,*
Qm*(q0M1,,) = (Q,*q)
wj(l/t) (dM-m*)JQm*q. j=O
This implies the assertion.
(ii):
Since S" is a closed subset of X suppu is a closed subset of X , a s well.
By Lemmata 6.7 and O.v.(ii) Mo(suppu) n X is a closed subset of X . Now, if K
n Mo(suppu) = @ then suppu does not
is a compact subset of X such that K
M,'(K)
intersect the s e t
so that in view of Corollary 5.47 we conclude that
< Q , d , , u , v > = O for every (pECT(K).
liii):
rn
t h i s is clear from the definitions.
Next we state the analogue of Propositions 6.15 (for I = I O . + ~ C) and 6.35. To this end we f i x a ( l + k ) - t u p l e w = ( w O , . . . , w k ) of distributions w i ~ a ) ' ( S x )O, < i ~ k .
Propodtlon 8 . 3 . BJ ( 4 . 3 3 ) a distribution v,,, E B ' ( X I is well-defined having the Following properties: k
(i)
supp v,
lii)
v,
c
u (supp v j ) M : i=O
i,
is almost quasihomogeneous of degree m and o f order 5 k + N ( m +
such that j-1
(8.B.aI
( d M - m ) ' v , = (v, ...., vk),,,
+
' i (dM-m)j-i-l
I
QAvi
*
j61Nk
i = o
and
(iiil by IUZ,lml:=
v,,
m € C \ ( - 2 l ( M ) - p ) , ameromorphicfunction S?,,: @+B'(X)
is defined, i t s poles ?,,ing in ( - X ( M ) - p ) ; f o r every m € ( - X ( M ) - p ) aO(m:3)1,,)
equals v,. Note that urn extends the distribution defined i n Proposition 4.24.(i) w i t h
X
replaced by X , . That both distributions are denoted by one and t h e same symbol should therefore not lead to confusion.
proOf. Let K be a compact subset of X . Then by Remark 0 . 8 L : = S X n K M is a compact subset of S". Since in view of Theorem 5.37.(ii) the map C T ( K ) - + a ) ( L ) defined by qHrpm*ISx is well-defined,
urn is a well-defined distribution on S x .
linear and continuous it follows that
356
(i):
VIII. E x t e n d 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 D i s t r i b u t i o n s
S i n c e s u p p u l is a c l o s e d s u b s e t o f X Lemma 6 . 9 . ( i i i ) s h o w s t h a t
K, n s u p p vi
i s e m p t y f o r a n y c o m p a c t s u b s e t K o f X s u c h t h a t K n ( s u p p u i ) M is e m p t y . I n view of T h e o r e m 5.37. ( i i ) t h e a s s e r t i o n f o l l o w s .
(ii):
f r o m (6.31) w e d e d u c e t h a t k
t-m
> = t m * 2 < v i , ( c p o M l / t ) m * , w i l S X> = A + B i=O
where k
I
and
Changing t h e order of summation and substituting l = i - j w e obtain k
k
M o r e o v e r , s u b s t i t u t i n g J = i + l + j 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 leads to
m
=
C tijj(t)
J=O
min(J-l,k)
C
i=O
>.
< U i , ( - a M + m * ) J - i - l Qrn*qISx
C o n s e q u e n t l y , in view o f ( a M - m * ) ' (Q,,*p) = Qmr ( ( a M - m * ) j q ) a n d ( 2 . 5 ) t h e assertion follows. ( i i i ) : t h i s f o l l o w s f r o m t h e c o r r e s p o n d i n g a s s e r t i o n s a b o u t nZ,,c,,i
- see T h e o r e m
S.37.(iv) - because ( c o m p a r e (0.14)) k
(8.3)
= (-1)'
2 < u i , a j ( m * ; a I v , w i )> ,
? c c ~ ( x )j e, z .
rn
i=O
If t h e d i s t r i b u t i o n s ui are i n d u c e d by f u n c t i o n s a n d if R e m > computed:
-p
t h e n urn is e a s i l y
357
8.a Pulling Back Distributions o n S x
k
f IX+ := x
,r “i i=O
0
x gi o p x
(see Notation 1.71)
and
f/,,,+:=O.
proOf. First of all we observe that by Remark 1.73’.(i) f is almost quasihomogeneous of degree m and of order 5 N and that for every k E (0) u N,
the restriction
of its kth order deficiency to S x is equal to g k . Hence, by Lemma 1.90 f is locally integrable. By (4.34) ( T g o , .. . , T,,),
and Tf coincide on X ,
.
Since they are both
almost quasihomogeneous of degree m and since by the assumption on m we have m @ ( - X ( M ) - p ) they coincide everywhere by Corollary 6 . 2 8 .
w
Likewise, if from the outset t h e components of v are given as restrictions to Sy of the deficiencies of a locally integrable function on X then we have
Remark 8.S. Let qo ~ d t I; , , l X ) be almost quasihomogeneous of degree m and of order 5 k . Then d l Tq, Jrn = Tqo . Proof. Note that by Proposition I.Sl.(i) the deficiencies q, , . . . , q k of qo are locally integrable, as well. In view of Theorem 4.25.(i) the support of the distribution c : = d(Tqo),-Tqo
is contained in X \ X +
.
By Proposition 8.3.(i) and by Remark
2.30.(i) and Proposition 2.37.(i) the support of c is also contained i n t h e u n i o n of the sets supp qi , 0 5 i 5 k . Hence, if supp qi C X , for every i E (0) u [N,
then c = 0 .
On the other hand, if suppqi \ X , # @ for some i E ( O ) u Nk then by Proposition 1.01 we have Rem > - p , and c is equal to zero by Corollary 6 . 2 8 . w
The preceding remark is a special case of Theorem 8 . 8 below. This theorem completes the description of almost quasihomogeneous distributions o n X w i t h a
( k + l ) t h order deficiency the support of which is contained i n X \ X , . I t requires special cut-off functions the existence of which is established i n
Lemma 8.6. Let h E N o . One can choose functions G o , . . . , Gk (1.181 f o r
.J= N o ,
E
C z ( X ) satisfying
( 3 . 1 9 ) , and, in addition,
(8.41 Proof. Since the function
WhoX
is almost quasihomogeneous of degree 0 and of
order 5 h it follows from Remark 1.74 and Theorem 3.48 that we can find a f u n c -
358
V I I I . Extending ( A l m o s t ) Q u a s i h o m o g e n e o u s Distributions
satisfying $o
= ( - ~ ) J ( a ~ for a x j E ( 0 ) u N h . In particular, w e have J9 $o ~ 1 Hence, . t h e a s s e r t i o n f o l l o w s by Lemma 4.12.(ii) when we t a k e into a c c o u n t tion $EC;(X+)
that
($o)o,Wk+j = ( - a M )
k drO,Wk+j
= 40,Wj
'
'
Choosing $ o , . . . , $k a s in Lemma 8.6 with h replaced by k + l + h a n d applying Lemm a 6 . 4 7 we obtain
Lemma8.7. Let ,y:X+C
be any C"function
such that s u p p x is a weakly
(M, Cl,+mC)-bounded subset o f X . and let hCN,. (i)
Then we have:
The following conditions are equivalent:
xx
l a ) there esists a function (8.51
( x x / X o ) ~ , m .(=- 1 )
j+ 1
I
6
(Jj+r
C'"(XI satisfj+ng
o~I(xo)+
j
8
f o r J =/O1uNk+,, such that the support o f
x-xX
is a weaklj
( M .Cl.+aCl-boun-
ded subset o f X and, at the same time, an M-bounded subset of (b) XIMo liil
(X)n
E.J,
1x4, :
1.
The following conditions are equivalent:
( a ) there e\ists a function such that the support o f (b)
XI^^)^.^^,^+^
x -xX
x X CC'"(X)
satistving ( 8 . 5 ) f o r
D =( O ) u N , + ) ,
is a weaklj. M - bounded subset o f X ;
e.vtends t o a C'"function g : X + - + @ such that
(-aM)k+h+ig,i.
Note t h a t t h e existence of functions
x satisfying t h e condition ( b ) in
( i i ) is e s t a b -
lished by the assertion ( i ) . Of c o u r s e , if (5.17) holds, i.e. if X = X o t h e n ( b ) is trivially satisfied, and t h e formulation of Lemma 8 . 7 becomes much simpler.
For t h e following theorem we fix a function x x having t h e properties in condition ( a ) of Lemma 8 . 7 . ( i ) f o r h = N ( m ' ) . Moreover, by assuming t h a t to 1 o n a neighbourhood of
x b e identically equal
X n M o ( X ) we achieve t h a t x x has t h e s a m e property.
Theorem 8 . 8 . For every TEB'IXI each o f the (equivalent) conditions l a ) - Icl o f Theorem 6 . 3 7 . ( i l is equivalent t o ( d ) the ( l + k ) - t u p l e d ( T ) o f distributions ( ( d M - m J ' T ) l S x , O S i l k , is well-defined. and T = d(T),,
+
QA ( x x T I .
8.a Pulling
Back Distributions on Sy
359
Proof. ( a ) * ( d ) : Since t h e restriction o f ( d M - r n ) ' T to X , is a l m o s t quasihomogeneous o f degree m a n d of o r d e r 5 k it follows from Theorem 2.42 t h a t t h e distributions ui := (3, - m ) ' TI,,
, 0 C i C k , a r e well-defined and f r o m Theorem 4.25
t h a t urn and T coincide o n X, . Since by Proposition 8 . 3 . ( i i ) T - u r n quasihomogeneous o f degree m a n d since of X \ X, T-urn e q u a l s xx(T-v,,,)
xx
is a l m o s t
is equal to 1 on a neighbourhood
so t h a t by Proposition 6.27 we have
1.
T - urn = Q k ( x , (T-v,)
By Theorem 5.37.(vi) (applied to Qm*'p instead of 9 0 ) and by ( 8 . 5 ) we deduce f o r arbitrary q E C r ( X ) a n d iEN, ( xx
that
Qm* 'P ) m * ,qIs x
xo =
Since by Corollary 5.47 and Remark 5.60 t h e s u p p o r t o f ( X ~ Q , , , * ~ J ) , , , is * , con~~~ tained in Xo we conclude t h a t Q ; , , ( X ~ U , , , ) = O so t h a t ( d ) is proved.
( d l + f a l : t h i s f o l l o w s f r o m Propositions 8 . 3 . ( i i ) and 6 . 2 4 .
Corollary 8.9. Let #*, . . . , #k E C z ( X 1 be as in Lemma
8 . 6 . I f T E B ' f X I satisfies one
(and hence each) of the conditions ( a ) - i d ) o f Theorems 6.37. ( i ) and 8 . 8 then li
(8.6) i=O
(Gi(aM - m l
T)I,l.cdli= d ( T),,,
where d l T l is defined in Theorem 8.8. Proof. Since in view o f ( 8 . 4 ) a n d (8.5) we can t a k e
x,+
t o b e equal to
xx
in
Theorem 6 . 4 2 , we obtain t h e assertion by combining t h e equation ( 6 . 4 3 ) and t h e o n e in condition ( d ) of Theorem 8 . 8 .
Corollary 8.10. Suppose that 16..521 holds. Then ever). distribution d E D ' f X ) with support contained in X \ X, which is almost quasihomogeneous o f degree m is o f the form N
(8.7)
d
=r( d M
- m ) ' Q ; ui
i=o
f o r some choice o f distributions u o , .... u N E D ' ( S " ) where N : = N ( m ' l . proOf. By Corollary 6.51 w e find a distribution T E D ' ( X ) s u c h t h a t ( 3 , - m ) " ' T = d .
360
VIII. Extending ( 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
S i n c e T 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 T h e o r e m 8 . 8 , ( 8 . 2 . b ) , a n d Coroll a r y 6.28 s h o w t h a t t h e a s s e r t i o n is valid f o r vi = ( a , - m ) N - i
Tlsx.
If m E ( - U . ( M ) - p ) t h e n n o t every d i s t r i b u t i o n o n X s a t i s f y i n g t h e c o n d i t i o n ( a ) of T h e o r e m 6.37.(i) is of t h e f o r m v m f o r s o m e vE 3 ' ( S x ) " k .
T h i s is a c o n s e q u e n c e
of t h e f o l l o w i n g r e m a r k which is p r o v 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 . 2 S . ( i i ) to T : = u
mlx;
Remark 8.11. If s u pp
u,
CX
\ X , then
u
= 0 and hence u,,, = 0 .
If k e r M = ( 0 ) t h e n T h e o r e m 6 . 4 5 s h o w s t h a t v,
I
is t e m p e r a t e . W e are g o i n g to
a p p r o x i m a t e its F o u r i e r t r a n s f o r m . To (8.8)
~ , , ~ :(= tw)j ( t ) e - ' t ,
a n d state a p r e p a r a t o r y l e m m a .
a function K i , , i EY;*,-(
V ' x V ) (see Notation 5.34) i s well-defined such that
and
prooF. W e f i r s t observe t h a t t h e f u n c t i o n f
:=
e - i <*
' ' '
> 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<, Mt,,x>)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.
) (<,
*
) .
I .E
Since t h e Laurent coefficients a - j ( r n ; ' 3 1 f , v .
)
belong to Y G * , M ( V * ~ V ) t h i s
J.E
c o m p l e t e s t h e proof of 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 . As for t h e s e c o n d p a r t , if R e m < 0 t h e n f o r every c p E C T ( V * ) Fubini's t h e o r e m
gives +m
= j 't - m $ ( ~ , x )w i ( t ) e -
(cp),,,i,z(x) h
E t
dt =
0
Again by t h e principle o f analytic continuation m E C \ X , ( M ) . This means that
t h i s r e m a i n s valid f o r every
~ J E Q , , ~ , is ~ equal t o t h e m e r o m o r p h i c f u n c t i o n
defined bq
h:@+YM(V)
h ( m ) ( x ): = G ( K ; , i ( - , x ) ) , w h e r e 8 : Y M * ( V * )+@
xEV, mE@\'U,(M),
d e n o t e s t h e c o n t i n u o u s linear o p e r a t o r defined by
CS(f) : = \ ' ( f c p ) ( < ) d < . V*
Since for any fixed X E V t h e function g,:C+
K A , j ( - , x ) , m E C \ ' U , ( M ) , is m e r o m o r p h i c it
Y M * ( v * ) defined bq
g,(m) : =
f o l l o w s by t h e integral f o r m u l a for
t h e Laurent coefficients t h a t a j ( m; h ) ( x ) = CS ( aj ( m ; g,) ) for every j E Z
.
Hence
t h e s e c o n d f o r m u l a in t h e a s s e r t i o n f o l l o w s for m E X , ( M ) , a s w e l l . T h e first f o r m u l a in t h e a s s e r t i o n is obtained by interchanging t h e r o l e s of ( V , M ) a n d ( V * , M * ) a n d taking ( 8 . 9 ) i n t o a c c o u n t .
Ropodtion 8.13. Suppose that ker M = 10) and X = V . Then for ever)' k Urn,€
(El :=
< v i , K&+,i ([,
E
> 0 b)
)Isx>
i=O
a function
E
Yg*(V*) is
lim Tv, E + O
well-defined such that
= F ( v n 3 ) in the topology o f Y ' I V*). .E
F o r t h e proof w e require t h e following c o n s e q u e n c e o f t h e mean value t h e o r e m :
362
V I I I . Extending ( 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
Lemma 8.14. Let 23 be a bounded subset of Y'( V ) . Then (i)
f o r every q
>
~
S
qn+(qz)
converges to
uniformly f o r
+ €23
1
R,,(+):=
Z E Z ~ ,IzISr
qn *(qz)
z EZ"
a s r + +a;
( i i ) R,,(#) converges to J L + ( s ) d x
uniformly f o r + € 2 3 a s q + O .
I
Proof of Proposition 8.13. 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 ( S n - ' ) M =V so t h a t S x = S X n ( S " - ' ) ,
is c o m p a c t by Remark 0.8. Hence w
~ is ,well-defined ~
a n d b e l o n g s to Y ' z * ( V * ) by t h e f i r s t p a r t of Lemma 8.12.
To prove 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 w e o b s e r v e t h a t by Proposition 5 . 4 3 a n d Remark 5 . 4 0 f o r every cpEY'(V*) we have k
k
Now. l e t h b e t h e maximum of t h e o r d e r s of t h e d i s t r i b u t i o n s ui o n t h e ( h e r e c o m p a c t ! ) manifold S". T h e n by T h e o r e m 5.42 t h e set
? 8 : = ( d ~ K ~ + , i ( * , ~X )EcS p" ,; l a l i h , O < i < k } is a b o u n d e d s u b s e t of Y ' ( V * ) . H e n c e by Lemma 8.14 w e have
f o r every 0 5 i 5 k . Combining t h i s w i t h Lemma 8.12 w e o b t a i n
By Lemma 8.14 t h e right-hand s i d e is equal to
Summing over i E ( O ) U N k w e o b t a i 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 .
rn
W e close t h i s s e c t i o n by giving a duality interpretation of t h e f o r m u l a in Theor e m 8.8. I t is a n a l o g o u s to t h e a s s e r t i o n of Remark 4 . 3 0 . Below we s h a l l d e n o t e
c in Proposition 6.59 t h e duality b r a c k e t ' u ~ , k ( a , ( x ) ) x . u " , , , ( a , ( X ) ) ~ defined f o r t h e c a s e G = ( I d v } by ( - .
*
* J M . Moreover, w e set N : = N ( m * ) .
By applying Proposition 2.45 with ( X , w ) replaced by ( X , , log x ) w e d e c o m p o s e every f E U:*.,k(
D ( x ) ) according to
363
8 . a P u l l i n g Back D i s t r i b u t i o n s o n S x
where
N
k
:=c
WgOX Q=O
P@(f)
and
PD(f)
:=I
Wk+l+p
O x
Pk+i+P(f)
P=O
with
C
p,(f) :=
(aM-m*)i+Pf
( - I ) ~ ~ ) ~ O X
icN0
b e i n g q u a s i h o m o g e n e o u s of d e g r e e m L f o r every 4€lN,.
By P r o p o s i t i o n 2 . 4 5 Px
is a p r o j e c t i o n o n t o t h e s u b s p a c e X a z + , k ( X ) . H e n c e P,
s u b s p a c e im P,.
is a p r o j e c t i o n o n t o t h e
Note t h a t
P k + , + g ( f )= P,( ( a M - m * ) k + if). C o n s e q u e n t l y , in view of P r o p o s i t i o n 5.5') a n d N o t a t i o n 5.55 o n e d e d u c e s t h a t im pD =
(8.11)
Moreover, choosing
{
N (')k+,+@OX
e=o
'pE D ( X )
P,(RI,+)
;
REQ,,,*(~(X))}
s u c h t h a t f = ~ p , , , * , ~ . , ~a n d t a k i n g ( 5 . 0 3 ) i n t o a c c o u n t
w e observe that M =
>=
k
=
X < ( a M - m ) i ~ I S X , ~ , n * , w i I S=x ) i=O
k
=
<
( - I ) ~ - ~(
flsx
>.
a M - m ) ' ~ l s x, ( a M - m * ) k - i
i=O
S i n c e by Lemma 2 . 3 4 , R e m a r k 1 . 7 4 , a n d P r o p o s i t i o n 2.31 w e have t h a t N
( d M - m * ) i PD( f)
I s x eTO[ =
(dk+ 1 + p - i
O x
Pk
+
@
+I(
f )]
Isx
0,
0 5 i 2k ,
w e conclude t h a t < d ( T ) , , , P D ( f ) > 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 <(dM-rn)'TlsX, (dM-m*)k-iPx(f)lsx
>]+
i=O +
In view of T h e o r e m 4.20 a n d Remark 4.30 t h i s e q u a t i o n e x h i b i t s
(
-i)k
< .,
)M. *.
>M
364
VIII.
Extending ( A l m o s t ) Quasihomogeneous
as the direct s u m of the canonical duality bracket
a ) ' ( S x ) ' ' k x a)(Sx)"k--3
C
the spaces Q A ( a ( X ) ) and i m p D induced by
and the duality bracket between
<*, -- >M.
Distributions
In view of Remark 4.30 the first of these duality brackets can
also be looked upon as the bracket between the space x D k , k ( X + ) (see Notation 4.28) and its pre-dual x D z , k ( x + ) .
t 1) B
E x 1end i ng t A I mos 1) Q u a s i honiogtbntwu s I)is1r- i b u 1i on s oII k
l o the Whalc- al' X
The basic result of this section is
Theorem 8.15. Every almost quasihomogeneous distribution
on X + extends t o an
almost quasihomogeneous distribution on X .
In the proof, actually, w e are going to be more precise by keeping track of the almost quasihomogeneous orders of the extensions. Indeed, let k€IN,,
and let
T E a ' ( X + ) be almost quasihomogeneous of degree m and of order 5 k . Then there are essentially two different ways of extending T to X . The first one is based on section ( a ) : by Theorem2.42 we know that (dM-rn)'TI,,
is a well-
defined distribution on S x for every i € { O ) u N k , and from Theorem 4.25.(i) we deduce that the distribution d ( T ) ,
defined in Proposition 8.3 extends T where
d ( T ) denotes the ( l + k ) - t u p l e of distributions ( a M - m ) i T I S x ,O < i < k . The second
way makes use of the functions
.
$ 0 , .. , $k
E
cE(X+
satisfying the assumptions
of Theorem 6.42: by Remark 0.14 for every i € . ( O ) u N k the distribution
$i
(aM-m)' T
- which a priori belongs to a ' ( X + ) - can be identified w i t h a distribution i n
ah(X)
so that by Proposition 6.15 ( $ , ( a M - m ) i T ) , , W k is a well-defined distribution o n
X , and by Proposition 4.13 -applied to ( X + ,k ) instead of ( X , N ) - the distribution k
extends T. (Since o r d M ( T ) may be strictly smaller than k it would be more precise to write Tk instead of T ; however, t h i s ambiguity should cause no confusion).
8.b Extending ( A l m o s t ) Quasihomogeneous Distributions
36s
T h e s e e x t e n s i o n s are a l m o s t q u a s i h o m o g e n e o u s b u t in case m E ( - U ( M ) - p )
not
n e c e s s a r i l y of t h e s a m e order a s T (see P r o p o s i t i o n s 6.35 a n d 6 . 2 4 ) . M o r e o v e r , t h e y d e p e n d o n t h e c h o i c e s of x a n d
Jli.
Different extensions differ from each
.
o t h e r b y 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 c o n t a i n e d in X \ X ,
H e n c e , w e c o n c l u d e f r o m C o r o l l a r y 6.28 t h a t t h e e x t e n s i o n s are u n i q u e e x c e p t when m E ( - % ( M ) - p ) . From Corollary 8.9 w e deduce
Remark 8.16. Suppose that m i ? ( - 2 l ( M ) - p )
or
that ( 8 . 4 ) is satisfied. Then the
distribution T defined by (8.13) is equal t o d ( T ) , .
I
Summarizing t h e discussion above w e obtain
Theorem 8.1s'. Under the preceding assumptions the distribution
f 6 3 ' t X ) defined
by (8.13)extends T t o the whole o f X T is almost quasihomogeneous o f degree m of order 5 k + N + 1 where N : = N ( m ' More precisel>., i f m
).
( - X ( M ) - p ) then T is almost quasihomogeneous o f order 5 k ,
being the unique almost quasihomogeneous e.\tension o f T : i f m 6 ( - X ( M ) - p ) then as an almost quasihomogeneous e-vtension o f T T is unique u p t o an additive term o f the form described in Proposition 6.27. ( i ) .
I
If R e m > - p a n d if T is i n d u c e d by a locally i n t e g r a b l e f u n c t i o n o n X,
t h e n its
e x t e n s i o n T h a s , in f a c t , a l r e a d y b e e n c o m p u t e d in P r o p o s i t i o n 1.91 : I Hxample 8.17. Let qo E .2,= ' ( X , I be almost quasihomogeneous o f degree m and
o f order 5 k . I f Rem > - p then for ever) i E { O J u N k the estension order deficiencj, qi of qo to X defined bj
..
fi
IX,
fi
o f the it*'
x+:= 0 is locall-v integrable on X .
and IdM -ml ' Tq, = Tfi . proOf. T h e f i r s t 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 1.91. S i n c e T f 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 m t h e s e c o n d a s s e r t i o n f o l l o w s in view of P r o p o s i t i o n 2.31 by t h e u n i q u e n e s s p a r t of T h e o r e m 8.15'.
rn
T h e f o l l o w i n g p r o p o s i t i o n c o n t a i n s r u l e s of c o m p u t a t i o n f o r t h e e x t e n s i o n s T
366
VIII. E x t e n d 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 D i s t r i b u t i o n s
Roposltlon 8.18. ( i ) q f = ( q T)' for every quasihomogeneous C -'function q :X + C ; ( i i ) f o A = (ToA)'
for every A E G L ( V , V ) commuting with M provided that
m t ( - Z ( M ) - p ) or IClioA= 4i ( r e s p . x o A = x ) ; (iii)
fm = ( T @ ) '
i f @ satisfies the assumptions o f Remark -7.67.(ii) and i f
m t ! ( - A ( M ) - p ) or each o f the functions
mf. m: t h i s is a n
4i
( r e s p . x ) is G-invariant.
i m m e d i a t e c o n s e q u e n c e of ( 4 . 8 ) (see P r o p o s i t i o n 6.16) a n d
Corollary 2.36.(ii).
( i i ) : Since
A - ' ( X + ) = ( A - ' ( X ) ) + o n e observes t h a t f o A e x t e n d s T . A .
S i n c e by
R e m a r k 2 . 6 7 . ( i ) T o A 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 n d of t h e s a m e o r d e r a s T t h e u n i q u e n e s s p a r t of T h e o r e m 8.15' i m p l i e s t h e d e s i r e d e q u a l i t y pro-
vided t h a t m does n o t b e l o n g to ( - 2 l ( M ) - p ) . To d e a l w i t h t h e e x c e p t i o n a l c a s e , by P r o p o s i t i o n O . l 6 . ( i v ) a n d ( 2 . 3 8 ) o n e d e d u c e s f o r every m E C f r o m ( 8 . 1 3 ) t h a t k
If m E ( - ' U ( M ) - p ) t h e n t h e r i g h t - h a n d side e q u a l s ( T o A ) by t h e a s s u m p t i o n o n t h e functions $ .
(iii):
S i n c e A ( X + ) = X + o n e observes in view of ( 2 . 3 5 ) t h a t (T),
S i n c e by Remark 2 . 6 7 . ( i i ) (T),
e x t e n d s TCV.
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 n d of
order n o t l a r g e r t h a n t h a t of T t h e u n i q u e n e s s p a r t of T h e o r e m 8.15' i m p l i e s t h e
desired e q u a l i t y in c a s e m does n o t b e l o n g to ( - X ( M ) - p ) . To deal w i t h t h e e x c e p t i o n a l case " m E ( - S I ( M ) - y ) " , by P r o p o s i t i o n 6 , 1 6 . ( v ) , by ( 2 . 3 5 ) a n d t h e a s s u m p t i o n on t h e functions
Jli,
a n d by ( 2 . 3 9 ) o n e d e d u c e s f o r every m E C t h a t
k
k
N o t e t h a t in case A E G L ( V , V ) c o m m u t e s w i t h M w e have
(8.14)
(+iOA)o,mj =
so t h a t t h e f u n c t i o n s
JZi
0
($i)O,mj
i E k ,
7
A , 0 C i C k , s a t i s f y t h e c o n d i t i o n s (4.18) a n d ( 4 . 1 9 ) , as
w e l l . M o r e o v e r , in view of
(8 .1 4 )'
((h)G)O,wi=((h)O.wj)G
t h e G-invariant f u n c t i o n s ( $ i ) c , and (4.19).
3
jENO,
O < i C k , c o n t i n u e to s a t i s f y t h e c o n d i t i o n s ( 4 . 1 8 )
367
8.b E x t e n d 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 D i s t r i b u t i o n s
T h e r u l e s f o r t h e d e r i v a t i v e s are m o r e c o m p l i c a t e d ; f o r simplicity w e c o m p u t e t h e e x c e p t i o n a l c a s e s o n l y 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 t h a t M be s e m i - s i m p l e .
Proporitton 8.19. ( i )
a a f = ( d a T ) ' for ever)' U E #
such that m - u M C ( - X ( M ) - ~ ) .
(ii) Suppose that M is semi-simple, and let U C X such that l a l = I ; if aM + m * belongs t o U ( M ) then k
aa f - (aaT)' = Q;
-aM
[,Zaa([+ r=O
proOf. of ~
a:S i n c e a"T
+
,
ox )
G~(aM - m ) r].
a n d (d"T) are b o t h 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 s
aaT t h e a s s e r t i o n f o l l o w s by t h e u n i q u e n e s s p a r t of T h e o r e m 8.15'.
f i i ) : By
x
w e denote t h e function
x,+
s a t i s f y i n g (0.41) fixed in s e c t i o n 6 . ( d ) . In
view of L e m m a 6 . 4 7 w e may a s s u m e t h a t
X \ X + and that s u p p x
C
Xo. W e set h : =
x
is e q u a l to I o n a n e i g h b o u r h o o d of
( ~ l , ~ ) ~ , ~ , ,F~r o. m
( 3 . 8 ) ' , (3.11). (6.41),
a n d (1.38) w e d e d u c e t h a t
S i n c e it f o l l o w s f r o m (3.11) a n d (4.19) t h a t
w e conclude, making u s e o f (4.18), t h a t t h e left-hand
side is e q u a l to t h e
r e s t r i c t i o n of
to ( X O ) , . S i n c e by L e m m a 6 . 4 0 t h e s u m in s q u a r e b r a c k e t s e q u a l s ( - l ) k S O , j + S j , k + l
t h e w h o l e e x p r e s s i o n v a n i s h e s . H e n c e it f o l l o w s by t h e Leibniz r u l e a n d ( 3 . 6 ) a n d by (8.5) t h a t
k
Now, since c o n t a i n e d in
a"+
a n d ( a a T i b o t h e x t e n d aaT t h e s u p p o r t of d : =
a U f -(3"Tj
is
X \ X + . In p a r t i c u l a r , w e have d = x d . S i n c e d is a l m o s t q u a s i h o m o -
368
V111. E x t e n d 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 D i s t r i b u t i o n s
g e n e o u s of degree m - a M P r o p o s i t i o n 6 . 2 7 . ( i ) s h o w s t h a t d =Q;-,,(xd).
Now,
by P r o p o s i t i o n 6 . 1 6 . ( i ) , by L e m m a 2.34, a n d by t h e Leibniz r u l e w e have k
d=
( ~ M - m ) i T ) m , c , , k (- ~ i ( 3 M - m + o c ~ ) ~ a ~ T )= ~ - ~ ~ , ~
[a"(+i i=O
k
=
( a ~ - m ) ~ T ] , - , ~ , ~ ~ .
[(a"$i) i=O
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 t h e e q u a t i o n d = Q ; _ , , ( x d )
w e see
t h a t for every q E C ; ; ) ( X ) w e have < d , q > = < T , @ w >here k
cb
:=
( - 1 1 ~1 (
- I ) ~~ ~ ~ - m * ~ i ~ ~ ~ a + i ~ ~ ~ ~ m ~ + n M q ~ , l ,
i=O
N o t e t h a t by ( 3 . 8 ) ' , Remark S.OO.(ii), a n d C o r o l l a r y 5 . 4 7 w e have @
I x,xo
=0
and
k
@lxo = ( - - I ) ~X ( - 1 1 ~( a M - m * ) i [ c a a + i )
hQm*+,M~)XO)~.
i=O
U s i n g (3.10), ( S . h 3 ) , ( 3 . 0 ) , a n d (8.15) w e d e d u c e t h a t o n t h e s e t ( X o ) + w e have L
S i n c e by Remark 5.60 t h e s u p p o r t o f t h e f u n c t i o n o n t h e l e f t a n d of t h e o n e o n t h e r i g h t is c o n t a i n e d in ( X o ) + t h e s e f u n c t i o n s are e q u a l , a n d t h e c o n d i t i o n ( b )
of T h e o r e m 6 . 3 7 . ( i ) i m p l i e s t h a t k
2 ( - l ) ' < T , ( d M - m * ) ' [ + i a"((.)k+,ox)
=
i=O k
=
Y
<(aM-m)'T,+id,([Jk+,OX)
i=O
Consequently, t h e desired formula follows.
Qm*+,M(P>.
Qm*+,M~]> =
8.b E x t e n d i n g (Almost) 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
369
Since by Lemma 2.34 and by Remark 1.74
t h e right-hand side is equal t o
Combining this with (1.38) and (4.18) w e obtain
Since by Lernma6.40 t h e s u m in square brackets is equal to ( - l ) k S O , j + S J , k + f and since d o L ( w k + l -oj x ) vanishes f o r j = k + l w e conclude by (4.19) t h a t this is equal to
370
VIII. E x t e n d 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 D i s t r i b u t i o n s
Is every almost quasihomogeneous extension of T equal to T for a suitable choice of +o,...,$k? In general the answer is in the negative as the example T = 0 shows. A reformulation of Proposition 0.48 gives a complete description of t h e extensions of T obtained by the method of Theorem 8.1S':
Proposition 8.21. For ever) d €21'1 X ) which is almost quasihomogeneous o f degree m with support contained in X \ X, the Followirig equivalence is valid: There exi s t s another l I + k ) - t u p l e o f Functions G;,..
, ,
,
i,!JL€C;(X)
s a t i s f ) i n g the
assumptions OF Theorem 0.4-3 such that k
f + d =sli,!J; 1~3~-m)'T)~,,,,,,,, ;= 0
if and on/-),i f 18.10)
d = & 1 j; 7'1 Ibr
j; E C M l X )
SOl?l.?
m F . Note first that k
Consequently, applying Theorem (1.37 t o
.f
instead of T we deduce that Q i T I ( x + ~ f
vanishes. Hence, by Proposition 0,27.(i) it follows that d = Q:,,(x,,,d) = Q : , , ( q , S where S : = ? + d . By Proposition 0.48 we have k
(8.18)
s =I(+; (aM-m)is),,,&,k i =0
for some choice of functions I&,,.
..
,+;
+ E C G ; ( X I such I,
if and only if there exists
that Q : , ( x + S ) = Q ; , , ( $ T ) . Since by (8.17) the right-hand side of (8.18) is equal t o k
S0 ( +;(a,
- m ) iT
i=
the proof is complete.
Corollary 8.22. The Following conditions are equivalent: l a ) f o r ever:) e\tensron
T ' E 3 , ' l X I oE
r
which is dlmost quasihomogeneous o f d e -
371
8 . b E x t e n d l n g (Almost) 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
gree rn one can choose t h e f u n c t i o n s 40,. . . , @k in s u c h a war. t h a t T ' is equal to t h e distribution
f
deFined in (8.13):
Ib) every. distribution d € a ' ( X ) with support contained in X \ X , which i s a l m o s t quasihomogeneous o f degree m satisfies ( 8 . 1 6 ) .
I
Next w e note an observation concerning the extension of more general distributions o n X, to the whole of X . The proof reduces the general estension problem t o the extension problem for quasihomogeneous distributions.
Theorem 8.23. Suppose that (6.52) h o l d s . Let R be a distribution o n X , s u c h that I d M - m ) R e s t e n d s t o a distribution d € D ' f X ) . Then R e r t e n d s t o a distribution R E D ' ( X ) s u c h that with support contained
(aM - m ) R - d i s almost quasihomogeneous o f degree rn in x \ x,. R i s unique u p t o d i i additive distribution o f
t h e form described in Proposition 6.27. ( I ) . proOf. By Theorem6.49 we choose S E D ' ( X ) such that ( d , - m ) S = d . distribution T : = R - S
Ix
+
Then the
is quasihomogeneous of degree m , Consequently, R : = S + T
is the desired extension. For the uniqueness assertion one has to look at Propo-
sition 0 . 2 7 . rn
We close t h i s section bj pointing out a slightly different wa? for solving the equation (7 . 1) if ( 0 . 2 8 ) holds (compare Theorems 3.2.4 and 7.1.20 i n Hormander C 1 1 1 ) .
I t turns out that in this case the assumptions ( 7 . 3 ) and ( 7 . 4 ) are not required.
Theorem 8.24. Let q E C ' ( X ) be yudsilioiriogeneous of degree in X ,
(6
C having no zeros
Let T E D a ' ( Xbe ) dlinost qud,sihornogeiieous oEdegree ni such thdt t h e support
of i t s ( l + h )t" order d e h c i e n q is contained in X \ X ,
has a solution S € D ' ( X ) which
m f . Since TI,+
is
Then t h e equation (7I )
almost quasihomogeneous of degree ni - P
is almost quasihomogeneous of degree m and of order 5 k the
1 (TI
) is almost quasihomogeneous of degree m - 4 and of orq x+ der 5 k . By Proposition t % . l t % . ( i ) the support o f d : = T - q R is contained i n X \ X , .
distribution R : =
Since d is almost quasihomogeneous of degree m it follows by Proposition b . 3 3 that the equation q c = d has a solution c ~ a ' ( X which ) is almost quasihomogeneous of degree m - 4 . Hence S : = R + c is the desired solution of ( 7 . 1) .
rn
372
VIII. E x t e n d 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 D i s t r i b u t i o n s
(c) E x t e n d l n g Trmqolc Lo t h e Whole ol' X
In t h i s s e c t i o n w e s u p p o s e t h a t X satisfies (5.17). Let f E C r n " ( X )
(see Definition
1.18. ( i ) with V, = G M ( o + ) a n d V,= k e r M ) satisfy 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 .
Then
fm.wk
is a well-defined function belonging to CmVo( X + ) . Moreover. fm,b,k
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 n d of o r d e r 5 k + N ( m ) + l with ( k + l ) t h
o r d e r deficiency ( - l ) k + i Q , ( f ) ( , + .
In particular, it 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 n d of o r d e r 5 k provided t h a t m Z l l ( M ) . I f , in addition, m d o e s n o t
b e l o n g to
(
-%( M ) - p ) t h e n by T h e o r e m 8.15' T f m,r,,k p o s s e s s e s a 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 o r d e r 5 k to t h e w h o l e of X , namely T,
m,wk'
If
m E % ( M ) t h e n by Theorem 8.15' TF,,,,,~ uniquely e x t e n d s to a n 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 with ( k + l ) t h o r d e r deficiency ( - l ) k + i T Q m ( f ) . Finally, if m E ( - X ( M ) - p ) t h e n by Theorem 8.15' Tfm,cdk h a s a n a l m o s t q u a s i h o m o g e n e o u s extension which, however, is n o t unique. In t h e p r e s e n t s e c t i o n w e s h o w t h a t t h e r e is a canonical choice for t h e e x t e n s i o n which d o e s n o t d e p e n d o n x or
To find
LJJ.
it w e employ meromorphic f u n c t i o n s again. We require
Lemma8.25. Let w b e a s i n Chapter3. Suppose that m E I - Z ( M ) - p ) .L e t g E C O ( X + ) be such that /g/,,,, , w ,is well-defined bj, ( 3 , i ) 'and continuous. Then bj < Q ~ l ( T g w , , ) , p :>= , / ' g ( . ~ w )( x ( s ) )( Q m r p ) ( s ld s ,
(8.19)
~ E C F ( X ) ~
X+
a distribution Q L ( Tg
ox
)
D ' t X ) is well-defined; i t is equal t o
N
1( - t ) i
(dM-m)i~ril(~I,i)
i=o
where hi :=g,-
wcJi/sx
and N : = N ( m 4 X .Moreover, we have N
& ( T ~ l o ) =( d~M - m ) i Q , ; l ( r g g ( W W I ) D X ) . i=O
-f.
Let v € C z ( X ) , a n d set K : = s u p p q and
qi:=(3M-m*)iQm*rplX
+
.
Then
f o r every i E N 0 by Corollary 5.47 t h e s u p p o r t of q i is c o n t a i n e d in M i l ( K ) so t h a t by Lemma 8.1 t h e set s u p p q i nSX is c o m p a c t , i.e. (4.30)is s a t i s f i e d f o r g i n s t e a d of f . C o n s e q u e n t l y , t h e a s s e r t i o n s f o l l o w by Lemma 4.23 a n d Proposition 5.48.(iii).
373
8.c Extending f,,,k
Note t h a t by Propositions 5.11 a n d 5.13 t h e function g = 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 Lemma 8.25 f o r w = oj , j E l N 0 .
Theorem 0.26. By 31f(ml: = jf,,,,k , tion %?,:C+d)'(X)
rn
€ ~ K ( M ) u ( - X ( M ) - ~a) .rnerornorphic func-
i s defined, i t s p o l e s lying in t h e s e t X ( M ) u ( - 2 K ( M ) - p ) .
Moreover, ( i l i f rn62KtMI then with N : = N ( r n l we have
and
l i i ) if r n € ( - 2 l ( M ) - q ) then with N : = N ( r n ' ) w e have
and
k
Since by T h e o r e m 5 . 3 7 . ( i v ) f o r every i E ( O ) ~ n \ lby~ C \ X ( M ) 3 m H f , , , , , k - i
C \ ( - X ( M )- p ) 3 m
H
)
(resp.
a meromorphic function o n C is defined with poles
lying in t h e set X ( M ) ( r e s p .
(
-'U(M) - p ) ) t h e first assertion is easily deduced
f r o m ( 8 . 2 4 ) . Note t h a t by T h e o r e m S . 3 7 . ( i i ) and since S x is a weakly M-bounded subset of X the support of every j e E o n e o b t a i n s t h a t
'pm,.c,,ilSX
is a compact s u b s e t o f S x . Moreover, f o r
374
(j):
VIII. E x t e n d i n g (Almost) 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
In this case cMq,,i
is holomorphic a t m* so t h a t (8.25) and Theorem 5.37.(iv)
imply t h a t f o r every j C N 0 w e have k
a 7 .
= s j o i ~= O- eC= m a xc ( - 1 ) k + t +)ii':( (k-i+l ,j I
IQ-k+i-l.Q-j.i
where by H we abbreviate t h e right-hand side of ( 8 . 2 4 ) and where
l e , j , i : = J q e ( 8 ) ai(m*;3n,,,i)(8)
dG(8)
S X
with
( a M - m ) e Q m f . Since by PropositionsS.11 and 5.13 and by (1.38)
qe:=(-l)'
we have = ( - l ) J V m * ,',,w . = ( - 1 )
aj(m*;gn,,,i)
1
i
j
(ii j ) Vmm,cJi+j,
jEINo,
we deduce by Lemma 4.23 and (1.38) t h a t
Applying this to
tp-k+i-l,g-i,i,
substituting S = s + e + i -k - 1 and inserting t h e
r e s u l t into t h e equation a t the beginning of t h e proof w e conclude in view of ( 8 . 2 41 t h a t
where the distributions R i , s
E
% ' ( X ) are defined by
: = J[ q s W s + k + l - j O X ] ( X ) q ( x ) d x
x+ Changing t h e order of summation according to N
m
c
e = m a x l k - i +1.1
I S = P + i - k-1
...
N
S+k+l-i
S =0 Stj+i-k-1
e = m a x l k - i + 1,j I
= x
c
and k i=O
N
c
m i n l k . S +k + 1 - j I
N
S=O Szj+i-k-l
... =
c
c
S= max(0.j-k-1)
i=O
and substituting L = 4+i-k-1 we obtain N
(8.26)' where
a - j ( m ; X f )-
Sj0
d(Tfm,L,k)m =
c
S=maxlO,j-k-11
bi,S R i . S
8.c
Extendlng f m , , k
375
Changing t h e o r d e r of summation we see t h a t
Applying Lemma1.76 to ( L + k + l - j , L , j - l ) instead of ( t , k , j ) we deduce t h a t t h e s u m in square b r a c k e t s vanishes in c a s e jENk and equals
(jjiL,)= ( j i l ) in c a s e
j ? k + l . Since in t h e l a t t e r c a s e we have S
S+k+l-j
L = j-k-1
e=o
In o r d e r t o c o m p u t e b 0 . s we observe t h a t f o r j = O t h e t e r m in s q u a r e b r a c k e t s e q u a l s
To c o m p u t e ( 8 . 2 8 ) we derive a variant of
1 . 8 6 ) . Differentiating t h e equation
L+k+l
'1 t L + k - i
i=O
L times with r e s p e c t to t and evaluating t h e r e s u l t a t t = l we obtain k
(k-i)!
i=O
so t h a t ( 8 . 2 8 ) is equal t o
Hence we conclude t h a t
S
(8.29)
bo,s=
C
(-1)
L=O
k
S+k+l
S+k+l
(L+k+l) =
(-1) e=k+l
e S+k+l
(
)=
t h e l a s t equality being valid by formula ( 4 . 2 0 ) . Combining t h i s with t h e o t h e r equations we see t h a t t h e proof of ( i ) is complete.
(iil:
T h e proof is analogous to t h a t of ( i ) . Under t h e p r e s e n t a s s u m p t i o n s t h e
function 3nf,wk-i is holomorphic a t m so t h a t it f o l l o w s by ( 8 . 2 5 ) and by (5.04) (applied to
'p
instead of f ) t h a t
376
VIII.
Extending ( 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
j+Lbl+l
w h e r e , a g a i n , by H w e d e n o t e t h e r i g h t - h a n d side of (8.24) a n d w h e r e JL,j,i :=
.f
aL(rn;IJnf,,i)(B) [ ( a , - m * ) j
Q,*cp](B)d;(B).
S X
S i n c e by P r o p o s i t i o n 5.11 a n d 5.13 a n d by (1.38) w e have
w e d e d u c e by P r o p o s i t i o n 5 . 4 8 . ( i i i ) ( a p p l i e d to P ( x , d ) = d , - m * ) ,
by L e m m a 8 . 2 5 ,
a n d by (1.38) a n d ( 2 . 5 ) t h a t
S u b s t i t u t i n g i = k-I a n d S = s + j + L - 1 - 1 w e o b t a i n
w h e r e t h i s t i m e w e set N : = N ( m ' ) a n d
Ri,s := ( - I
)k'J ( d M
-m
)'
Qh(Tf,,,,
+
k+ -
X )
.
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 a n d s u b s t i t u t i n g k'= L + j w e c o n c l u d e i n view of ( 8 . 2 4 ) t h a t ( 8 . 2 6 ) h o l d s f o r t h e p r e s e n t c h o i c e of R i , s , as w e l l . I t f o l l o w s t h a t ( 8 . 2 6 ) ' is valid. H e n c e , in view of ( 8 . 2 7 ) a n d (8.20) t h e a s s e r t i o n is p r o v e d .
Notation 8.27. f m , f i , k: = a o ( m ; S f ) ; a n d , in p a r t i c u l a r . f, : = f m , + .
Theorem 8.28. f,,, ,,,, is an estension o f Tf,,,,,,,
t o the whole of X having the
following properties: (i)
(a,,,,-m)'
im,uk = 1-11'
.. F
~
l i i ) i f m t a ( M ) u ( - Z K ( M ) - p ) then m and o f order
5
k
, f~o r every ~ - i ~c N k ; frIl,~,,k
is almost quasihomogeneous of degree
;
is almost quasihomogeneous o f degree rn with l l + k )
(iii) i f r n € X ( M ) then order deficiency ( - 1 )
l+k
TQ,
;
( i v ) i f m E ( - X ( M ) - p ) then f m e wis k almost quasihomogeneous o f degree m with ( l + k ) t h order deficiency (-ilk Q ; , , ( T f ) ( s e e ( 8 . 1 9 ) ) .
377
8.c Extending k,,,f
m.By T h e o r e m s 8.15' a n d 8.26 w e k n o w t h a t
% F ( r n ) e x t e n d s T, m.Uk f o r every
m € Q : = C \ ( U ( M ) u ( - U ( M ) - p ) ) . By t h e integral f o r m u l a f o r t h e Laurent coeffi-
cients o f t h e meromorphic functions 3 ' 2, a n d 9Ef,wk(see T h e o r e m 5 . 3 7 . ( i v ) ) t h i s implies 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 .
fi):
Let i € . ( O )uNk . By hi :
c d a ' ( x )w e
d e n o t e t h e m e r o m o r p h i c f u n c t i o n de-
fined by Q 3 m H ( d ~ - m ) ~ % f ( m By ) . (5.63) a n d by t h e a s s e r t i o n already proved above w e have h i ( m ) l x + = ( - l ) ' % f , k - i ( m ) l x + , m e n , w h e r e by t h e m e r o m o r p h i c f u n c t i o n defining f m , w k - i
. By t h e uniqueness
%f,k-i
we d e n o t e
part of T h e o r e m 8.15'
w e c o n c l u d e t h a t hi a n d ( - l ) ' % f , k - i coincide o n fl a n d , hence, are e q u a l as meromorphic f u n c t i o n s . C o n s e q u e n t l y , in view of (8.21) a n d ( 8 . 2 3 ) for j C I N k , t h e a s s e r tion f o l l o w s by Remark 2.54 (applied to j = 0).
f i i ) :this fiii):
f o l l o w s f r o m Theorem 8.15'.
In view of ( 8 . 2 1 ) f o r j E [ N k a n d P r o p o s i t i o n 2 . 5 3 t h i s f o l l o w s f r o m ( i i ) a n d
(8.21) f o r j = l + k . An a l t e r n a t i v e a r g u m e n t is based o n t h e uniqueness p a r t of T h e o r e m 8.15': by ( 5 . 6 3 ) t h e d i s t r i b u t i o n
(aM - m ) k + l Em,ok
and t h e distribution
induced by ( - l ) l ' k Q m f b o t h e x t e n d ( d M - m ) k + l f m , c , , kand a r e a l m o s t quasihomog e n e o u s a n d c o n s e q u e n t l y coincide on t h e w h o l e of X .
fiv):
In view o f ( 8 . 2 3 ) f o r j C N k a n d P r o p o s i t i o n 2 . 5 3 t h i s f o l l o w s f r o m ( i i ) a n d
( 8 . 2 3 ) f o r j = l + k . rn If Re m > -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.<,k
(-I)'(
j=O
and N
(q>)m+p.uk-- T (
(8.311'
j=O
z)
qjim.uk+,
where q j : = ( d M - P J J q . ( i i i ) frn,WkoA= ( f oA)r,l.l,lk for ever)' A EGL( V . V ) commuting with M ;
livl
( f r r , , ~ ~ l k ) C ~=
(fca)rn,l,lk i f @ satisfies the assumptions OF Remark -3.67.(iil
proOf. We first observe that as functions of m
: = @ \ ( 2 L ( M ) u( - X ( M ) - p ) ) all
the terms i n the equations of the proposition are holomorphic and define meromorphic functions o n the whole of @ . Hence, by making use of the integral formula (2.10) for t h e Laurent coefficients one easily reduces the proof to the case "m
~n".
(iiil:By Theorem 5.37.(vii) we have Tf
o A
mswk
= Ttf
A),
.Wk
. Consequently, for
every m e n we deduce the desired equality from Proposition 8.18.(ii).
(iv): -
B y Theorem 5.37.(viii) we have
( T f m , c J k ) BT(fB) =
m ,Wk
.
Hence, for every
m E n the desired equation is obtained by Proposition 8.18.(iii).
( i ) : Here a
proof by direct computation does not seem to work. However, in view
of (5.34)and (5.34)' ( s e e Theorem 5.37.(v)) the equations (8.30) and (8.30)' are valid for the restrictions to
X,. Since the distributions on both sides of these
equations areare almost quasihomogeneous of degree m + P the uniqueness part
8 . d T h e Fourier Transform o f
Gm.t,,L
379
of Theorem 8.15' implies t h a t ( 8 . 3 0 ) and ( 8 . 3 0 ) ' a r e valid f o r every m e n .
fii): This
is a special c a s e of ( i ) .
ad) l ' h e Fouric?r 'I'ranst'orm o f '
+n,.ok
il' cp Ilelongs to YPCVI
In this section we s u p p o s e t h a t X = V . and we fix a function qeY'P(V)
Theorem 8.31. Gm,cJk is a temperate distribution satis6ving Ff+",,cJk) = (-l)k f
$ )m*.wk.
A consequence is t h a t t h e Fourier transform has quasihomogeneous eigenvectors:
Corollary 8.32. Let A
= Ap . a n d s e t mo := - $. - Then S : = pmo ,~',k
E C be such that
is a l m o s t quasihomogeneous of degree mo a n d of order i k satisKving
i =f - l ) k A s .
I
For t h e proof of Theorem 8.31 t h e following lemma is required
Lemma 8.33. S u p p o s e t h a t
Then t h e r e is a c o n s t a n t C such t h a t for a r b i t r a r j 9 .x E2?'I(V)nde"l VI t h e function
f:IO.+~CxV+C,
It.y)
H
t-m-'p(Mt\)Xfv).
is absolute/-v integrable
satisfjing t he estimate
llflLl
c ( l l P l ~ llXILL,, l llPILL,> llxlb, ) . +
m f . Let 4 E C b e such t h a t R e 4 > 0 . Since X E Z'(V) and since
'p
bounded it follows by t h e Fubini-Tonelli theorem t h a t t h e function f ~ : l O , l l ~ V (~t , @ , t'-'q(Mtx)x(~), x) H belongs to L'( 1 0 , l l x V ) satisfying
is essentially
380
VIII. E x t e n d 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 D i s t r i b u t i o n s
1
llf,Il'&
5 j'tRe,-l d t
II [P IIym II x (ILl
.
0
S i n c e t h e f u n c t i o n F:Il,+~CxV-+I0,1IxV, ( s , y ) H ( l / s , M , y ) , is a d i f f e o m o r p h i s m s a t i s f y i n g det D F ( s , y ) = -s-'det s h o w s t h a t t h e function
fpOF
M,=
- s"-'
t h e c h a n g e of v a r i a b l e s f o r m u l a
I d e t D F 1 , m a p p i n g ( s , y ) to sCL-'-lV ( Y ) x ( M , Y ) ,
x V ) satisfying b e l o n g s to Z'( lI,+a[
N o w , if 4 = - m t h e n f, is e q u a l to t h e r e s t r i c t i o n of f to 10,ll x V ; if 4 = m + p a n d if in t h e d e f i n i t i o n of f p t h e roles of
'p
and
x
a r e interchanged then t h e func-
c o i n c i d e s w i t h t h e r e s t r i c t i o n of f to Il,+mlxV. C o m b i n i n g
tion f p o F IdetDFI
this with t h e preceding information o n e obtains t h e assertion with
Proof to
o f Theorem 8.31. By P r o p o s i t i o n 5.41 t h e f u n c t i o n s v ) ~ , ~ ,, 0~5 i- 5~k , b e l o n g
(see N o t a t i o n 5.39). W e c h o o s e $0,. . . , +k a s in C o r o l l a r y 6 . 3 9 . By R e m a r k
5.40 t h i s i m p l i e s t h a t t h e f u n c t i o n k
b e l o n g s to
Y(V), its
s u p p o r t b e i n g M - t e m p e r a t e . C o n s e q u e n t l y , by P r o p o s i t i o n 6.22
(T+)m,t,,kis t e m p e r a t e f o r every m C C . S i n c e in view of (4.10) w e h a v e k
=i =CO (J,i ( a , - m ) i ~ y m , W k ) m . w k= ( T + ) ~ . ' , ~
~ , ( m )
f o r a r b i t r a r y m E n : = @ \ ( ' u ( M ) u ( - ' u ( M ) - I . I ) )i t f o l l o w s t h a t X , ( m )
is temperate
for every m e n . M o r e o v e r , s i n c e t h e r e is a c o m p a c t s u b i n t e r v a l J of I O , + m C s u c h t h a t for a r b i t r a r y m e n a n d x € Y ( V ) w e h a v e
w e c o n c l u d e by P r o p o s i t i o n O . 2 2 t h a t
X,
is m e r o m o r p h i c w i t h v a l u e s in
Y'(V).
H e n c e t h e f i r s t part of t h e a s s e r t i o n f o l l o w s . In order to d e d u c e t h e f o r m u l a for t h e Fourier t r a n s f o r m w e f i r s t o b s e r v e t h a t it s u f f i c e s to p r o v e it if ( 8 . 3 2 ) h o l d s . For by t h e p r i n c i p l e of a n a l y t i c c o n t i n u -
8.d
381
The Fourier TransForm of 'pm,,k
a t i o n it is t h e n valid for every m E n , a n d to derive it f o r m E X ( M ) u ( - X ( M ) - p ) w e c h o o s e E > 0 so s m a l l t h a t
Now s u p p o s e t h a t ( 8 . 3 2 ) h o l d s . By Remark 8 . 2 0 and by t h e definition of T,,,,',,~ w e then conclude that
< Gm,,+, x > = .I'$ 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,-,,
s u c h t h a t f o r s o m e c p E C g ( X ) s a t i s f y i n g cp(xo) # O t h e r e
is a n e i g h b o u r h o o d W o f 50 in V'\(O)
s u c h t h a t for every N E N
s u p { I Y ( ~ ~ T ) ( M , s S) IE; W 1 =
o ( T - ~ )
as
T + a .
385
9.a The Wave Front Set WFMI(T)
In order to s h o w t h a t in Definition 9.1 o n e may t a k e a n y test f u n c t i o n
'p
whose
s u p p o r t i s c o n c e n t r a t e d n e a r xo w e p r o v e a f e w s t a n d a r d l e m m a t a in a p r e c i s e f o r m s u i t a b l e f o r l a t e r a p p l i c a t i o n s , as w e l l . F o r t h i s w e have to t o u c h u p o n a f e w t e c h n i c a l p o i n t s . First of a l l w e r e c a l l t h a t t h e Fourier t r a n s f o r m o f a n y V E & ' ( V ) e x t e n d s to a h o l o m o r p h i c f u n c t i o n ( a g a i n d e n o t e d by $ ) o n t h e c o m p l e x i f i c a t i o n
V*xV* o f V*; n a m e l y , w e have A
v ( ~ , u ) : =< u , e x p ( - i ( 5 , . > + < u : > ) > ,
( < , u )E V * X V + .
D e n o t i n g t h e r e a l - c o m p l e x c o o r d i n a t e s of < ( r e s p . u ) E V * by ( y , z ) ( r e s p . ( v , w ) ) w e s h a l l w r i t e ( c o m p a r e N o t a t i o n 1.2S.A) d
d
(9.3)
17 ( z j + i w j ) y j ( L j + i w js.)
(yj+ivi)'j
([,u)'l:= j=l
o:= ( P , y , 6 )
J ,
EX.
j=1
M o r e o v e r , w e r e c a l l t h a t if K is a c o m p a c t s u b s e t o f V its supporting function
H K :V*'R
is d e f i n e d by H , ( I J ) : =m a x { < u , 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
d e n o t e t h e c a n o n i c a l p r o j e c t i o n o f V* o n t o
GM(h.).
Lemma 9.2. Let K be a compact subset OF V , let q E C f l M ,
E
Then there are constants a , b , C €10,+wC such that for every
0 , and k,PcLVo.
x EC;(KI and for
arbitrary constants A , N 6 to,+at and B E C l , +at the Following implication h o l d s : i f for every p E X satisFying
(9.51
la a+Px 1 5 A B R
I @ / (k + P+l the estimates
e a ~ ,
a E X , ReaM 5 N ,
386
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 Sets
are valid then f o r arbitrary y € Z , s E [ l , + ~ [ l, E V 4 X \ K ( 0 , z ) . and u E V ' w e have
Proof.F i r s t
of a l l w e fix p E E M n C O , q l s u c h t h a t
N e x t w e f i x ~ E V * \ K ( O , E a) n d r . E l l , + c o l . T h e n w e f i r s t c h o o s e X E a s u c h t h a t
lE,Xl?l(I/&
a n d t h e n fix j E I N n s u c h t h a t I ( M , < A ) ~ I LI M , < , l / f i
(here. for a
m o m e n t , w e w o r k w i t h t h e real c o o r d i n a t e s w i t h r e s p e c t to t h e c a n o n i c a l IR-basis a s s o c i a t e d w i t h t h e r e a l - c o m p l e x b a s i s Cr* of V* d e f i n e d in C o n v e n t i o n 1.24';
in
t h i s way w e a c h i e v e t h a t t h e e s t i m a t e s b e l o w a r e valid f o r t h e case L J # O , as w e l l ) . Finally, w e f i x mEIN, s u c h t h a t N - R e X + p < m ( R e X - p ) < N . S i n c e w e may a s s u m e that
E
5fi
w e d e d u c e f r o m ( 0 . 2 ) ' - a p p l i e d to p i n s t e a d of E ) of
N
E
-
that
N
N o w , for every PEU s a t i s f y i n g P 5 w w e fix a , p E ' U s u c h t h a t w - P = a + p . l a l < m , N
I f i I S k + t + l , a n d R e a M = l a l R e A . Applying ( 9 . 5 ) t o
estimates m
instead of p and taking t h e
a n d (9.7) i n t o a c c o u n t w e d e d u c e
ia"-pXt 5 A B I a l R e X 5 A ~ " ' n )
N
387
9.a The Wave F r o n t Set W F M ( T )
Moreover, s e t t i n g C : = max{ 1 + 1x1 ; X E K }
we e s t i m a t e Ixy-'
I o n s u p p x by C I y ' .
Finally, k+P+m+l
2k+t?+l N/(Xmin-B)
2
p sw
Choosing a c o n s t a n t d E l O . + a C s u c h t h a t
and combining a l l t h e e s t i m a t e s w e o b t a i n t h e a s s e r t i o n w i t h b : = ( 4 n / ( c p s ) )1 /
(
Xmin-p)
and
a : = d ( i l . n / ( c , E ) ) e + i 4kJ',ds.
Lemma 9.3. Let K be a compact s u b s e t o f V, and l e t q E P M and
E ,
r , s €10,+ m y ,
Then there e s i s t c o n s t a n t s b , c €10,+wC such that for arbitrary A , N . W E I O . +a[ and B E C l . + a C t h e f o l l o w i n g implication i s valid: i f vE&'(VI s a t i s f i e s
and i f ,y EC;;'(K) f u l f i l l s ( 9 . 5 ) for ever) /3 62l satisfying t h e e s t i m a t e I F I S k + P + l for k : = s + n + l and s o m e
P-'
( s A m a x + r ) / A m j n then
388
1X. Quasihomogeneous W a v e Front Sets
--c.-J'(I+IMK(S-T)I)-"-'d(MK<) .=11(1+1 -
where
IIL,(V*) < + a .
V* N
By t h e Paley-Wiener t h e o r e m o n e f i n d s a c o n s t a n t d o n l y d e p e n d i n g o n K s u c h that
I ?((, u ) I 5
d" A ( 1 + I ( I
J'
'T
K(Eo
e x p ( H,(u) ) , (, u EV*. C o n s e q u e n t l y ,
l g ( M K ( < - ( ) , u ) l d (5
.2 L )
Since w e may a s s u m e t h a t
v
vc
lg((,u)\d( 5 d"A?exp(H,(u)).
i s so s m a l l t h a t s(kmax+q)-k?(krni,,-q) 5 - r
the
N
assertion follows with c : = ? m a x ( aCS, d } , m
Lemma9.4. Let K be a compact subse t o f X . and let F be a closed subset o f S,b
such that K x F n W F M ( T )= 0
K in X and
F"
w
Then there evist open neighbourhoods K o f
o f F in V x \ { O ) , a sequence ( C N ) , , ,
o f positive constants and
another constant ko such that sup { 5
* IFFcpT) ( M , t,u l l e\p(
w
- H , l u ) ) : 5 E Cl,+wC, < E F ,
II E
V *}
5
5 CN s u p { l p ( a ' l \ ) / ; x ~ X ReuM-C , N+k,}
for arbitrary N E N and p E C r ( 2 ) . Pro o f . -
S t e p 1 : W e fix ( X ~ , E , ~E) K x F . T h e n w e c a n c h o o s e + E C g ( X ) a n d
E
> 0
such that + ( x o ) = 2 and
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 U of xo s u c h t h a t
I+[2
1 on
U . Applying
L e m m a 0 . 3 ( t o v = + T a n d B = I ) a n d c o m b i n i n g t h e r e s u l t w i t h (9.10) w e f i n d a c o n s t a n t koEN a n d a s e q u e n c e of c o n s t a n t s C , E l O , + ~ C , N E N , s u c h t h a t r E C l , + c o C , E , E K ( C 0 , ~ ) ,I J E V * } 5 s u p ( r N 17(~+T)(M.E,,i~)Iexp(-H,(u));
5 C , sup{
IIa'xIlL,;
RePM c N + k o }
f o r a r b i t r a r y x E C g ( U ) a n d N E N . N o w , let r p ~ C g ( U ) S. i n c e I + [ - > 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~<) P ) ;, x E X } , where the Bp are constants not depending o n any
we s e t
'p
BE?[,
we deduce the assertion for
~EC~(E).
We can now prove the following standard fact.
Propodtlon 9.S.
The projection o f W F M ( T ) t o the first variable is equal t o
singsupp T .
mf. If xo E X \ sing supp T
t h e n by the discussion preceding Definition 9.1 and
by Proposition 1.72.(ii) the condition ( 9 . 1 ) ' holds for some ' p E C T ( X ) satisfying cp(xo) # 0 so that W F M ( T ) does not intersect the set ( x o ) x S " - ' . I f , on the other
hand, the last condition is valid then there is a compact neighbourhood U of xg such that W F M ( T ) n U x S V + = 0 .Hence by Lemma 0.4 the condition (9.1)' holds
. x g does not belong t o singsupp T . for every c p ~ C g ( U )Consequently,
The following remark is concerned w i t h elementary symmetry properties of W F M ( T ) .
Remark 9.6.
(I)
I f T is real valued then WFM(TI is invariant under the trans-
-€). " (ii) I f T = T then W F M ( T )is invariant under the map
formation
(.\.[) H ( \ ,
(\,€)#(-A,[).
(iii) Let W be another C-vector space, and let N E L ( W t W * ) and A E G L ( W , V ) be such that NoA*=A*oM. Then W F N ( T ~ A=){ 0 , A*€); (AS ,€) I n particular, i f W = V . i f A'
6
WFM I T ) }
and M commute, and if T is A-invariant then
W F M ( T ) is invariant under the transformation I,\.€)
H
,A * € ) .
390
IX. Ouaslhornocceneous Wave F r o n t Sets
proof. li). This follows from the fact t h a t <€
(iii). The assumption on
A implies that ( A * ) - ' o N , = M,o(A*)-',
V*.
r E l 0 , t c o C . Con-
sequently, making u s e of (2.8) w e deduce that
I d e t A1 7(rpT OA ) ON, A* = T ( ( p 0 A - I T ) o ( A * )-'ON, oA*= 0
This implies t h e assertion. rn As f o r t h e behaviour of quasihomogeneous wave front sets under linear partial
differential operators t h e usual results remain valid:
proOf. (i): this follows immediately from Lemma 9 . 4 .
(ii):
Let u € U be s u c h t h a t l a l = l . Then by the Leibniz-rule and i n view of
(see Remark 1.30) we have
Consequently, since s u p p
C s u p p 'p t h e assertion follows from Lemma 9 . 4 .
(p',)
rn
The following result deals with t h e converse of Proposition 9.7 f o r differential operators with quasihomogeneous principal part. We fix m € R e ( U ( M ) ) , and let P=P(x,D)=
(9.12)
2
(where D":=
a,(x)D'
(-ia),)
ReUMSrn
b e a differential operator of (quasihomogeneous) order m e R e ( ' U ( M ) ) with coefficients a,
E
C - ( X ) . Its principal part P,
:XxV'-+C
is defined by
( x , < )E x x v * .
391
9 . a The Wave Front S e t W F M ( T )
N o t e t h a t even if M i s semi-simple
P,(X;)
need n o t b e q u a s i h o m o g e n e o u s .
However, if M is s e m i - s i m p l e a n d i f , in addition, o ( M ) C R t h e n P,(x,
* )
is quasi-
homogeneous of degree m .
Theorem 9.8. Under the preceding assumption we suppose that for some $ E X ( M ) P,,,(x,
.)is
quasihomogeneous of degree
I%
for every S E X . Then for every T c D ' ( X )
we have (9.13)
WFM ( T ) C WFM ( P T ) u P L ' I O )
proOf. T h e proof of T h e o r e m 8.3.1 in H o r m a n d e r Clll is to b e suitably modified. Let ~ X ~ , < ~ ) E Xb eX sSu c~h *t h a t it d o e s n o t belong to t h e r i g h t - h a n d s i d e of (9.13). T h e n w e can c h o o s e c o m p a c t neighbourhoods K of x o in X a n d W of < g
in V*\(O), a s e q u e n c e o f c o n s t a n t s C,
(9.16)
inf{ JP,(z)l
;
,
and a distribution v E & ' ( X ) s u c h t h a t
ZE KxW} > 0
N o t e t h a t (9.14) a m o u n t s to
equal to 1 o n a fixed neighbourhood U of x g a n d
W e fix a f u n c t i o n XEC;(K)
set u : = x T . In o r d e r to prove t h a t
d o e s n o t belong to t h e l e f t - h a n d side
of (9.13) w e have to verify t h a t ';(M,E,) = O ( T - ~uniformly ) for €, near To t h i s e n d we f i x N 2 m ,
1 2
E0
as
T+m.
N a n d €,E W . T h e Leibniz r u l e s h o w s t h a t for every
w E C ~ ( K w) e have (9.17)
'P( w e x p ( - i < M , < ,
-
> ) /Pm(
*
, M,E,)) = ( w -
1
R,w)
exp(-i<M,<,
a > )
PEBm
where d , , , : = { p € R e ( X ( M ) ) ; O < p < r n } and
w i t h ~ ~ , , : = { ( a , B , y , g , f ) € X ' ;R e a M 5 m , p + y + S + f = a , R e f M = m - p } . N o t e t h a t R,
is a differential o p e r a t o r of quasihomogeneous o r d e r 5 p ( f o r Re fM = m - p
implies R e B M = R e a M - R e ( y + 8 ) M + p - m 5 8 ) . For any w E l O , + ~ [defining differential o p e r a t o r s
392
IX. Quasihomogeneous W a v e F r o n t Sets
s,
:=I+ k E N pl+.
c
. .+pkSv
RP,...RPk
and
E, : =
c
k E N p i + . . .+ p k > 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,<) =
(9.19)
T
-
+
B(e,T)(M,<)
a n d e , : = E , ( x ) a n d w h e r e m € X ( M ) is c h o s e n in s u c h
where +,:=S,(x)/P,(-,€,) a way t h a t P,
7 ~( + , v ) ( M , < )
is q u a s i h o m o g e n e o u s of d e g r e e % . S i n c e Re % = m w e d e d u c e t h a t
M o r e o v e r , m a k i n g u s e of ( 9 . 2 ) ' f o r every qE(fM w e find a c o n s t a n t C, s u c h t h a t
Employing t h e l a s t t w o c o n d i t i o n s w e d e d u c e t h a t f o r a r b i t r a r y p, , , . . , pk E Bm,
EX, a n d q < min!Bm t h e r e is a c o n s t a n t C s u c h t h a t IaYRp
...R X I
5 C~-~i-.'.-'k+' <_C,
rE
pk
1
Cl,+cOC,
so t h a t by L e m m a 9 . 3 a n d (9.1s) w e c o n c l u d e t h a t F ( + N ~ ) ( M T =< )O ( r C N ) unif o r m l y for < E W a s r + ~ M . o r e o v e r , w e d e d u c e t h a t f o r every ~ E t hUe r e is a c o n s t a n t C, s u c h t h a t
I a y e NI
5
c,
T€Cl.+cOC.
T-N+",
Now let q be t h e ( q u a s i h o m o g e n e o u s ) o r d e r of T o n K . T h e n f o r s o m e c o n s t a n t
C,
( w h i c h i s i n d e p e n d e n t of e N a n d r ) w e have
(9.22)
I 7 ( e N T )( M , < ) I 5 C, s u p { 11 a p ( e N e x p ( - i < M , t ,
*
> ) I1 1 ~ R~ e ;p M 5 q ) .
By t h e Leibniz r u l e w e o b t a i n a'(.NeXp(-i<M,<,
- > ) I=
exp(-i<M,<,
>)
2 (!)
(M,<)'-'aye
N
Y SP
C o m b i n i n g t h i s w i t h (9.21) w e o b t a i n a c o n s t a n t C' s u c h t h a t ( 9.23 )
I a'
( e N exp ( -i
<M,<,
-
>)
II
5 c ' r q + ' - , s u p { I ayeN I ; y 5 p
1,
Re p M 5 q .
Y.a
393
T h e Wave F r o n t Set WFML(T)
Consequently,
Y ( eNT)( M,<)= O ( r - N +
’
+ q)
u n i f o r m l y f o r < € W as
T+m.
Combining
t h i s w i t h t h e e s t i m a t e for ~ Y ( + N V ) ( M , S ) aI n d w i t h ( 0 . 1 Y ) w e c o n c l u d e t h a t ( x ~ , < ~ )
does n o t b e l o n g to W F M ( T ) , i n d e e d .
W e n o w c o m e to t h e g e n e r a l i z a t i o n o f T h e o r e m 4.20 a l l u d e d to in s e c t i o n 4 . ( d ) . Here we require
Notation9.9. ( i ) By X M w e d e n o t e t h e f u n c t i o n x + : V * \ ( O ) + l O , + ~ C
defined
in P r o p o s i t i o n 1.70 f o r X = V * . ( i i ) By
w e d e n o t e t h e C r n r e t r a c t i o n d e f i n e d in N o t a t i o n
p M : V‘ \lO1+Sv*
1.71.(i) if X = V * \ / O l a n d x = x M .
Remark 9.10. Let N : V -
V be a non-zero linear map, and suppose that X is
quasihomogeneous o f type N . Moreover. let I be a closed subset of 3 0 . + m y . and let Z be a closed subset of X x S V r which is contained in L x S v * f o r some ( N . I ) bounded subset L o f X . Then the set
is a closed subset of X x S p
-
Proof. Let (y,,u,)
M cZN , , n X x S , . . W e c h o o s e s e q u e n c e s ( t j ) j E Nin I a n d
( x ~ , < in~ Z ) s u~c h~ t h~a t
By c h o o s i n g a s u b s e q u e n c e if n e c e s s a r y w e achieve t h a t (ti) c o n v e r g e s to a p o i n t <-ESP
as j+m. Since K : = ( y , ) ~ ( N l / ~ ~ j x€ !~N;} is a c o m p a c t s u b s e t o f X it
f o l l o w s by t h e a s s u m p t i o n o n L t h a t t h e t j s t a y in a c o m p a c t s u b i n t e r v a l o f 10, + a [ . H e n c e w e may a s s u m e t h a t ( t j ) c o n v e r g e s to s o m e t , E I we conclude that
as j+a. By c o n t i n u i t y
394
1X. 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
Since Z is closed in X x S v * t h e pair ( x r n , E r n ) belongs to Z so t h a t ( y , , \ ~ , ) M
lies in Z N , I .
Theonm 9.11. Let X, N , and I be as in Remark 0.10, and suppose that N * and M commute. Moreover, suppose that X is locallj N-bounded. Let u €.9'(X) be such that supp u is an ( N , 1)-bounded subset o f X . And let w : I 0 , +a[+
C be a locallj
integrable function with support contained in 1 . I f we denote by u,,,,,
the distri-
bution on X defined bv ( 4 . 3 ) with M replaced bj. N then
= p M ( N I P ) . Interchanging t h e roles of k' and t; and replacing l/s b j s we s e e
t h a t t h e converse implication i s valid, a s well. Hence
M
so t h a t by t h e definition of Z N , I t h e equivalence of ( 0 . 2 4 ) and (0.24.)' f o l l o w s
By Proposition 3.12 we c h o o s e an open neighbourhood W of s u p p u such t h a t W is an ( N , 1)-bounded s u b s e t of X . Since K : = K ( X ~ . E is ) a compact s u b s e t of X it follows t h a t J : = { t E l / l ; N , ( W ) n K # @ } is a compact s u b s e t o f l O , + , = l . Let 1 'p EC;f( K ) . Hence, using (4.4),s u b s t i t u t i n g s = - , and s e t t i n g u : = t r N we deduce t
<E
V'.
9 . b The Wave Front Set w i t h R e s p e c t to
395
C'sL
By continuity for every s € J we find an open neighbourhood I, o f s in J s u c h t h a t
(9.25)
( a ) N:(K(F,,,E))
( b ) N1,,(K(xo,
C N:(K(F,O,~E)), E))
and tEI,.
K(xo, ZE)),
C N,/,(
Since J is c o m p a c t we find a finite s u b s e t S of J s u c h t h a t J = U S E SI,. In view of ( 9 . 2 4 ) ' we find c o m p a c t neighbourhoods K, of N l / s ( K ( ~ 0 , 2 ~ )in) X and F, of ( ( I ~ O N : ) ( ~ ( ~ , ~in, ~S,* E ) )s u c h t h a t ( 9 . 2 6)
K,XFsnWF~(ti)=@,
Now we c h o o s e cE Cl.+mC such t h a t
S€S
c1 5 x M ( N : < )
for a r b i t r a r j t E J and
Since N : ( M , < ) = M r x M ( N ; c , p M ( N : < ) it follows in view of
((J
<EK(~;,,E)
25.a) t h a t f o r e v e r j
N E N we have
where CN : = c (9.28)
N
.I,. t R e m + v - l I w ( l / t ) l d t . Since i n
s u p p q j o N N , =N , / , ( s u p p q j )
C
view of (0.2S.b) we have
N 1 / , ( K ( i o , 2 ~ ) )C K,.
tEI,.
SES.
and since b j ( 1 . 4 9 ) - a p p l i e d t o P = P, ( s e e Remark 1.43) and P L , i : = ( d N * - a N ) 'P we have
an application of Lemma 0 . 4 t o t h e s e t s ( K , . F,)
-
t h e assumption being s a t i s -
fied by ( 9 . 2 6 ) - s h o w s t h a t t h e right-hand side and hence t h e left-hand side of ( 9 . 2 7 ) is bounded f o r every N E I N . This means t h a t
(
xo,
d o e s not belong to
WFM(U,,~,,).
Let L : LO, and
+io[+
Cl.+~x~C be a n increasing continuous function s u c h t h a t L ( O ) = I
396
I X . Q u a s i h o m o g e n e o u s W a v e Front Sets
(a) L(t) 2 l+t ,
(9.30)
and
( b ) L ( t + s )5 As+' L ( t ) ,
t,SE
co,+mc,
tE
co, +ac.
for s o m e c o n s t a n t A . Note t h a t (9.30.b) implies L(t) 5 At+',
(9.31)
S o m e t i m e s , in a d d i t i o n w e p o s t u l a t e :
Vd>l 3 C d > I VtcCO,+aC: L(dt) s C , L ( ~ ) ~ .
( 9.32 )
Example 9.12. The following choices f o r L satisfy the preceding assumptions: (i)
L = A where A ( t ) : = / + t :
(ii)
L ( t l = ( t + t ) " f o r some p E l l . + w l ;
(iii) L t t ) = e a t f o r some a 6 1 0 . + m L . Moreover, i f L satisfies the preceding assumptions then so does Ld f o r ever) d>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
it suffices to prove the assertion for the special case M = Id".
-
Let 2 3 = ( V u ) u E u be an open covering of K such that V, C U for every U E U . Then we choose
E
> 0 such that
K + k ( 0 , 2 E )c
U v,,
and
V,+K(O.~E)
LIEU
cu
for every U E U .
By Theorem 1.4.5 in Hormander C 111 we choose a Cm partition of unity on the compact subset K + i i ( 0 , 2 ~subordinated ) to the covering 93, consisting of f u n c t i o n s w i t h values in CO, 11. Applying Theorem 1.4.2 i n Hormander C 11 1 to the sequence
defined by d. :=
E / ~ N if j E N N E
2-i
otherwise
we obtain functions GNEC:(K(O,2~)) such that Then
laa G N If- ( B N ) ' O L 1if
J a I5 N where
C : = I / I K ( O , E ) It 1/SGNdx
IaacpN I
with values in C0,ll equal t o 1 o n K ( O , c )
so that
B is a constant not depending on N . q ~ : = @ ~ / j ' ? ~satisfies d x
f- C (BN)'"' if la1 5 N
After these preparations we can prove
Propodtlon 9.19. Suppose that (9.3-7) is satisfied in case M is not semi-simple. ) some open neighbourLet u E & ' ( X ) and x O E X . Then uILI belongs to C M S L I U for hood U o f xo i f and only i f there are an open neighbourhood U o f so and a bounded sequence o f distributions u N , N E N , in &'(XI coinciding on U with u such that for a suitable family o f constants C , ( 9 .3 6 )
/GN I M , ~ ) / _
for every t E S + .
.
q6
e M ,we
L ~ ) ' + v / ~ ) ~ ,
have rEcl,+mC, V € @ # U ,
399
9.b The Wave Front Set with Respect to CMvL
Proof.
"j" W: e fix
E > 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
<
F(u~+~)(<)
400
I X . 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
for some constant
D, n o t d e p e n d i n g o n N a n d a . S i n c e
( u N ) N ~ Nis
a bounded
s e q u e n c e i n & ' ( V ) t h e f u n c t i o n s I y ( u N + k ) l are b o u n d e d o n K ( 0 , l ) b y a c o n s t a n t n o t depending o n
N.Consequently,
t h e f u n c t i o n d e f i n e d by
H
<"
f?(tiN+k)(c)
b e l o n g s to xi(V*),its L l - n o r m b e i n g b o u n d e d by E p ( E p ( L N + k ) l + p) N + k w h e r e
E, is a c o n s t a n t n o t d e p e n d i n g o n N a n d a . H e n c e ,
aaUN+k=
(2rc ) - " y(<" y ( ~ ~ + ~ ) )
is a c o n t i n u o u s f u n c t i o n o n V . S i n c e t h i s is so f o r e v e r y N a n d e v e r y a it f o l l o w s t h a t u is a C - f u n c t i o n o n U s a t i s f y i n g l+p N + k
Ia"ul 2 I I F ( d a U N + k ) l l L l 5 E p ( E p ( L N + k )
)
.
Now, by (9.31) w e h a v e (LN+k
- A(k
+
) (1+ p )( k +1)
5 A
( N + k +1 ) ( l + p )( k + 1 )
2 ) ( 1 + p )( k + 1 ) A ( N - 1 ) ( l + t ,
)
=
( k +1 )
M o r e o v e r , by ( 9 . 3 7 ) a n d ( 9 . 3 2 ) w e h a v e
A ( 2 k + 3 ) ) ( 1 + p ) ] 1 + P . In view of o u r c h o i c e o f
where F : = [E,(C1+,
t h a t t h e r e s t i c t i o n o f u to U is i n d u c e d by a ChSL1'" f u n c t i o n .
Q
this shows
rn
Definition 9.20. L e t T E T l ' ( X ) . By W F M , L ( T w ) e d e n o t e t h e c o m p l e m e n t i n X xSV* o f t h e set o f a l l ( X ~ , ~ ~ ) E X s X u c hS ~t h U a t t h e r e exist a b o u n d e d s e q u e n c e ( u N ) N E N in & ' ( X ) , o p e n n e i g h b o u r h o o d s U o f x g in X a n d W o f
E0
in V * \ ( O )
a n d a f a m i l y o f p o s i t i v e c o n s t a n t s C, , qEC!M , s u c h t h a t f o r e v e r y N E N u N is e q u a l to T o n U a n d s u c h t h a t ( 9 . 3 6 ) is satisfied f o r e v e r y EEW.
Remark 9.21. In the definition of (9.36) f o r r 2 I L , )
'+'.
W F M . L i t suffices to postulate the estimate in
9.b The Wave Front Set w i t h Respect
to
401
C M s L
proOf. Since l u ~ is ) a bounded sequence in &'(X)we can find c o n s t a n t s W and s ( F ( u N )5I W
such t h a t
1.1'
on V * \ K ( O , I ) .
Hence, choosing k € N such t h a t
k > s X m a x we deduce f r o m ( 9 . 2 ) ' t h a t f o r any c o m p a c t neighbourhood K of svb we find a c o n s t a n t C s u c h t h a t lM,
f o r arbitrary r € [ t , ( L N ) ' + ' ]
r k 5 W C ( L , ) " + ~ ) ~5 W C A ' " ' ) " ' ~ ) ~
and < E K .
The analogue of Lemma 9 . 4 is valid:
Lemma9.22. Let K be a compact s u b s e t of X . a n d l e t F be a closed s u b s e t of = (3. Then t h e r e e \ i s t neighbourhoods
S v * such that W F , , , ( T ) n K x F in X a nd
F"
of F in V' \ (0)s u c h t h a t for an!
c o n s t a n t s A,. P € X , a n d for an!
of K
q € e Mfor , an) famil) of positive
r€lNo there is a c o n s t a n t a s u c h t h a t t h e fol-
lowing implication holds: N
If
(
~ i s a) n y sequence ~ ~ in ~
x
(9.39)
/a""xN/
C
A, ( A p ( L N )
C;;'(K)
satistjing
l+q h(ml
I
.
cr,p € 2 1 . h l n ) < N .
then
I F ( x N T ) ( M ,E.u)l
5
a e\p(H
XN
(1~))
r - r (a ILN)'+q/r)
w
for a r b i t r a r j r E t l . + w C , E 6 F . u E V X , a n d N E N .
mf. Of
c o u r s e , we may a s s u m e t h a t ( A @ ) is increasing ( i . e . Ag < A ,
if
Before coming to t h e actual proof of t h e lemma we observe t h a t if a n o t h e r increasing family of positive c o n s t a n t s and if
('pN)
is
is any sequence in
C m ( X ) such t h a t ('1.39) is satisfied f o r ( q N I K , A k ) instead of ( y N , A g ) then t h e functions
'pN
xN
satisfy (9.39) with Ap replaced by 2 ' p " 1 max{ Ag , A;, A g A;
1.
Indeed, given u , B E2I such t h a t h ( u ) 5 N then f o r every y 5 u+B w e write y = y r r + y g and a + l j - y = y,+Ta
T,+re
where y n , y g , T r r , Tp belong t o X satisfying t h e equations
a and y g + T g = B and obtain by t h e Leibniz rule t h a t
402
I X . Q u a s i h o m o g e n e o u s Wave Front Sets
S t e p f : W e fix x o E K a n d c o C F . T h e n t h e r e e x i s t a n o p e n n e i g h b o u r h o o d U of
xo in X , a c o n s t a n t E > 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
laa+'xNI
UN : = x N T
E ' U , la1 i 3 N .
form a bounded sequence in E ' ( X ) , and i n order
~ ) not belong t o the left-hand side of ( 9 . 4 0 ) we have to prove that ( x ~ , < does
to verify ( 9 . 3 6 ) for
5
near
50. To
this end we fix EEW, N
2
m , q € e Mand ,
T 2
N'*q.
From the arguments in the proof of Theorem 9.8 we conclude that ( 9 . 1 0 ) is valid
w i t h x replaced by (9.43)
xN
and for v : = ( l + q ) N , i.e.
Y ( u N ) ( M ~ E=)
F ( J ~ N V N ) ( M +~
T-'I'
where t h i s time we set + N : = S V ( ~ N ) / P m ( * ,and € , ) e N : = E , ( x N ) . The proof w i l l be complete once we have verified ( 9 . 3 6 ) for both summands on the right-hand side separately.
Estimating b"R,, .LRRPk~NL I n view of the meaning of m we observe that RPl.
'
. RPkXN
( 0 . 44 )
is a s u m of terms o f the form T
-khi (
M,E)tl+ ...+
tk
b, Dpl ( b , . . . D"-'
(b, D P
XN) ... )
405
9 . b T h e W a v e F r o n t Set w i t h R e s p e c t t o C M * L
where t h e
are s u c h t h a t R e ( E i ) M ' m - Q i ,
E~
w h e r e t h e bj b e l o n g t o a s e t
.B
of
real a n a l y t i c f u n c t i o n s which c a n be e x t e n d e d to u n i f o r m l y b o u n d e d h o l o m o r p h i c f u n c t i o n s o n a c o m m o n c o m p l e x n e i g h b o u r h o o d o f K , a n d w h e r e t h e P i ~ Us a t i s f y t h e e s t i m a t e R e ( p j ) M5 pi, i . e .
In order to e s t i m a t e ( 0 . 4 4 ) we fix q'ECLM s u c h t h a t ( m - P ) ( l + q ) q ' L p q f o r every Q
b e l o n g i n g to t h e set Bn1d e f i n e d in t h e tekt f o l l o w i n g ( 0 . 1 7 ) , i.e.
<EW}
w h e r e c : = max{
. N e x t w e observe t h a t t h e c o n c l u s i o n o f Lemma 8 . 0 . 3
in H o r m a n d e r [ I l l r e m a i n s valid if t h e d e r i v a t i v e s
j C N n , a r e replaced b j
( - i a ) " . a€SL1.H e n c e , w r i t i n g t h e derivatives of a n y f u n c t i o n g a p p e a r i n g in ( 0 . 4 4 ) in t h e f o r m D p g = D L 1 ( i D L 2 .( .1. D L P g .). . 1 ) . where 4 :=
I and
...
L ~ , ,tP
b e l o n g to
XI,
we c a n apply H o r m a n d e r ' s L e m m a a n d
o b t a i n a c o n s t a n t E n o t d e p e n d i n g o n N , ~ @ , [aJn d~ b j E $ s u c h t h a t I D p o ( b I D B 1 ( b , . . . D " k - l ( b k D " X ~ ) . . . ) ) C~ E k ' 3 " ( 3 N ) ' p o + ' ' ' + " k '
. .+Pkl
provided t h a t
R,,
BE%,,,.
C
3 N . Let B be t h e m a x i m u m n u m b e r o f s u m m a n d s in
Then combining this with t h e estimates above and using (9.45) w e
obtain t h a t
I DmR p l . . . R ~ , x N I
5
) k 3N+1 +
(
C r n C;n
)k (
I
I+VI
+ ,
.,
Pk
- ( v1 +
. . . + pk ) / ( I +
)
if T E [ I . + ~ [ a n d if (9.46)
+ p1+ . . . +pk 5 3 N .
S i n c e pi 2 X m i n 2 1 t h e e s t i m a t e ( 9 . 4 6 ) i m p l i e s k 5 3 N so t h a t w e find a n o t h e r c o n -
406
I X . Quaslhornogeneous W a v e Front S e t s
s t a n t C not depending on N,r,E, and pi such t h a t (9.47)
I D'+R p , . . . R p k x ~I -
.
la I + p l + . . +Pk T - ( p , +
CN+l
...+ P k ) / ( l + n )
provided t h a t (9.46) holds.
Estimating F ( @ N ~ N E M T [Let L . Q b e t h e number of elements of % m . Since pi 2 X m l n 2 1 t h e inequality p2 + . . , + pk 5 v implies t h a t k 5 u + l . Hence t h e number
of summands in ,$ ,,
is not larger than
cv1+1
(9.48)
2 ok 5
( v + 2 ) o V + '5 ( e Q ) " + ' = e O ( ( e e ) l + q ) N
k=O
Employing t h e Leibniz rule, taking (9.16) into account, and combining ( 9 . 4 8 ) with (9.47) we obtain t h a t
N'"'
lD"JINl 2 C"'
if la1 5 2 N
where C is another constant not depending on N , r and
<.
Moreover, since t h e
sequence ( v N ) is bounded we find constants W , s E l O , + m C not depending on N such t h a t every v N satisfies ( Y . 8 ) . We s e t t > sXma,/hmin
and
E
'0 such t h a t
k : = s + n + l and choose numbers
K(<,,~E)C
W . Then by (9.41) and by Lem-
ma 9.3 w e conclude f o r N 2 k + t + l that
1 7 ( $ N ~ N ) ( M K < )51 C N + l ,,,k+@+l
5 C; C C(L"'+"/, ~
( W (bN'+'/r
)N +
C, (C,( L N ) l + q / r ) N )5
lN
for every €,€K(<,,E) where Ch is a constant not depending o n N , r and < . Here
f o r the proof of t h e last inequality we made use of ( 9 . 3 0 . a ) . Estimating FleN_TILMKe. Let q b e t h e (quasihomogeneous) order of T on K .
Since t h e number of summands i n e N is majorized by t h e left-hand side of ( 9 . 4 8 ) it follows by ( 9 . 4 8 ) and ( 9 . 4 7 ) t h a t lDYeNl 5 e O ( ( e Q ) ' ' q ) N B"'
NIYl+"+"'
r-N
i f ReyM 5 q
where B is a constant not depending on < , y , r and N . Note that Nlyl+u+m
(.q
+m
+
( l + q ) ( q + l )) N ( N l + q ) N - q - l
Consequently, taking (9.22)and (9.23) into account we conclude t h a t
I 7 ( e N T ) ( M K < ) 5I
A ( A N ' + , / T )N-q-'
9.c Wave F r o n t Sets of Almost 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
407
where A is s t i l l another constant not depending o n €, T and N . Combining this with LN 2 N and (9.30.b) and with t h e estimate f o r 1 7 ( $ N ~ N ) ( M r c ) 1 we deduce from (9.43) in view of Remark 9.21 t h a t ( x ~ , € ,d~o e) s n o t belong to W F M , L ( T ) , indeed. rn
Theorem 9.28. The assertion o f Theorem 9.11 remains valid i f WFM is replaced by WFM, L . proOf. If W F M ( u ) is replaced by WF,,.(u)
then t h e proof of Theorem 9.11 remains
valid up to t h e estimates (9.27). Moreover, t h e conditions ( 9 . 2 8 ) and ( 9 . 2 9 ) hold. By Lemma9.18 we find functions x ~ E C ~ ( K ( X ~equal , E ) )to 1 o n K ( x ~ , E / ~and ) c o n s t a n t s Ap such t h a t (9.39) is satisfied. By (9.29) we can change t h e c o n s t a n t s Ap in such a way t h a t
xNoNt
satisfies (0.39) f o r every t E J . Then in view of (Y.28)
an application of Lemma 9.22 to the s e t s ( K s , F s ) , s C S , shows t h a t t h e right-hand side of (9.27) with
'p
replaced by
X N is
bounded by C , ( C , ( L N ) l * q ) N where C,
is a c o n s t a n t not depending on N . This means t h a t ( x o
,cob C WFM,L(U,,,).
rn
(c) W a v e Front S e l s of' Almosl Quasihomogcnc?c,us Distribulions
In order to formulate a simple, b u t important inclusion for t h e quasihomogeneous wave f r o n t sets we require
Notation 9.29. If Y is a subset of V'
we define its quasihomogeneous hull ( o f
type M ) at infinity a s t h e set Y M , m consisting of a l l F , E S V * such t h a t t h e r e
exists a sequence (ujIiENin Y and a sequence ( r j ) in C t , + m l converging to
+m
such t h a t (9.49)
F, = lim Ml,Tiuj.
Obviously, YM,-
j-+m
is a closed s u b s e t of Sv+. Moreover, if Y is closed and quasi-
homogeneous of type M then YM,given in
= Y n S v * . For special Y a useful inclusion is
408
IX. Q u a s i h o m o g e n e o u s
Remark 9.30. Let m € 2 1 ( M ) , let P,: V * +
Wave F r o n t S e t s
C be a polynomial function which is
quasihomogeneous of degree m and of t-vpe M , and let P : V * -
C be a pol)-
nomial function such that Q p ( P - P , r , ) = 0 for ever) t E X ( M ) satisfying ReP I m . Then ( P - ' ( O ) ) M , ~C~ (, P , ) - ' ( O ) .
proof. Let quence
(Ti)
R : = P,-P
< E ( P - ' ( o ) ) M , ~and , choose a sequence ( u i ) j e N i n P-'(O) and a sein
Il,+mC
converging to
+m
such that (0.49) is satisfied. Then setting
and N : = m a x { o r d M ( Q p R ) k: ' € A ( R ) } and making use of Proposition
1.34.(ii.a) and of (1.41) we deduce
P m ( M l / T i ~ ~= iT~-"' ) P,,l(i~i) =
R ( M T iM 1 / T i ~ ~=i )
N
=
d
'T r p - m ~ ) k ( ~ R j ), , k ( M l / 'i I J , ) .
3
tEA(R) h=O
In view of ( 0 . 4 0 ) and since by the assumption we have Re(k'-m) < O for every P E A ( R ) the right-hand side converges to 0 as j + m . A s the left-hand side tends to P,(€,)
a s j+cj t h i s shows that P,(€,)=O.
w
A
kopositlon9.31. WFM,,IT)
af. Let U
C
V ~ ( s u p p T T ) ~ for , ~ , ,ever) T E Y ' ( V ) .
and W be bounded open subsets of V satisfying
maO.18 we choose constants D . A B and a sequence of functions
fi C W .
B y Lem-
xN E C T ( W )
equal
to 1 on U such that for arbitrary N E N and P E ' U the condition (9.5) is satisfied h
Finally, we choose a function c p E C C O ( V * ) equal to 1 on V * \ K ( O , Z E ) and vanishing on K ( O , E )and for every r C I R , + ~ set I y T : = OM^,^. Then for every x E C F C V ) one deduces from ( 0 . S O ) : if I J E Y and < E K ( E , E ) then A
x ( M ~ < - ~ =J () V ~ : ) ( M , < - L J )
so that by the Fourier inversion formula we conclude that
9.c W a v e F r o n t S e t 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
109
Distributions
h
C o n s e q u e n t l y , s i n c e T is t e m p e r a t e o n e f i n d s c o n s t a n t s E a n d m E N s u c h t h a t
where
S ( X ; K :, =~ S) U ~ { ( ~ + ~ M , I aJa ~( ' p" ,'; ) ( M , ( < - i J ) ) I ;
5 m , I J E V ' } . Note
t h a t ~ K ( M K ( < - i ~ ) ) = ' p ( ~ - i f~ I) J=EOK ( < , E so ) t h a t it s u f f i c e s t o t a k e t h e s u p r e mum over all ~ J E V * \ K ( < , E )o. n l j . By t h e Leibniz r u l e a n d by ( l . 4 ( j ) ' ( a p p l i e d to
P ( 3 ) = a e ) w e have
(for t h e d e f i n i t i o n of t h e polynomial f u n c t i o n s P p , j see R e m a r k 1.30 w i t h M r e p l a c e d by M* ) . Now w e f i x 4 E C?M so s m a l l t h a t
in
4
1 . By Lemma 0 . 2 w e f i n d c o n s t a n t s
a and b such that
1 ay N; ( M
(c -
IJ
))
I
,< - IJ) I )
5 a (1+ IM (
'I1
T -m ~
f o r a r b i t r a r y N E N , K E C ~ , + ~
Iy ( 1m .
Since t h e derivatives of
'p
N'"l/r
)
satisfying
are bounded and since
w h e r e C, is a c o n s t a n t o n l y d e p e n d i n g o n t h a t 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
(b
UN : =
n.
In view of R e m a r k 0 . 2 1 w e c o n c l u d e
xNT h a s t h e p r o p e r t i e s which i m p l y t h a t
Ux(€,)does n o t i n t e r s e c t W F M , , ( T ) . N e x t w e p r o v e a version of P r o p o s i t i o n 9.31 f o r d i s t r i b u t i o n s ( n o t n e c e s s a r i l y t e m p e r a t e ) given by a Fourier r e p r e s e n t a t i o n f o r m u l a . Let v be a c o m p l e x Radon m e a s u r e o n V * x V * 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 of X t h e r e is a c o n s t a n t m K such that
410
IX.
Quasihomogeneous Wave F r o n t Sets
a distribution T, E D ' ( X ) is defined a s is immediately derived from t h e PaleyWiener estimates for
$
when
'p
ECg(K).
Propodtion 9.32. Under the preceding hypotheses we have
WFM,*(TV)C X X ( K R ( S U PvP ) where rR : V * x V*+
mf. Let
) ~ , ~
V * denotes the projection defined by
Re [
U and W be relatively compact open s u b s e t s of X satisfying
-
fi C W.
We set K : = W . We choose A D , D and XN a s in t h e proof of Proposition 9.31 . From 7 ( X N T,) ( M,
<) =
< T,
, XN exp ( - i < M, 5 ,
>)
>=
cN(M, < -
du(<)
V*XVf
we deduce t h a t
IF,0-Ml/,pI
t 2~
Lemma 9.2 shows t h a t f o r any
if r ~ C R , + a land p E s c l R ( s u p p u ) . there are c o n s t a n t s a,,b,
I C / ( X ~ T , ) ( M , < )5I CK,, ( b n N 1 + " / r I N ,
> 0 such t h a t
r E C R , + a C , [ E K(<,,
E),
NEN,
where CK,, is a constant o n l y depending on K and q . This s h o w s t h a t U X { E , ~ I does not intersect WFM.,(T,).
We now come to t h e behaviour of t h e quasihomogeneous wave f r o n t sets of almost quasihomogeneous distributions. A preparatory remark is in order:
9.c
Wave F r o n t Sets of A l m o s t
Quasihomogeneous D i s t r i b u t i o n s
411
Remark 9.33. Suppose that X is quasihomogeneous o f type M*. Let T6DD'(XIbe almost quasihomogeneous o f type M*. Then f o r arbitrary x E X and ( E S we ~ ha ve : (x,c)€WFM(TI
{ C M E X , ~ ~t €) I; O , + m C } C WFM(TI I ( p M * ( , ~ l , ( I E W F M ( T I
( f o r the definition o f pM+ see Notation 9 . 9 ) . The assertion remains valid i f WFM is replaced by WFM,..
mf.L e t
m EC be such t h a t
T is a l m o s t quasihomogeneous of d e g r e e m . Then
f o r arbitrary r , t E l O , + m C , c p E C z ( X ) , and qESV* we have F ( q T ) ( M , v ) = t-"< ToM:/, , [c pe xp(-i<M.rl,
- t-m-v -
cn
->)]OM,*,,> =
<
o i ( l / t ) (aMf-rn)'T,cpoMT/,exp(-i<M,/,rl, ->)> = i=O m
-
t-m-cl
2 (-i)icdi(t)
~ ( c p o ~ ? (; a~ M , +-mf
T)(M=/,~).
i=O
Since by Proposition 9.7 w e have WFM( ( d M * - m ) ' T ) C W F M ( T ) and since by Proposition 9.26 t h e same is valid with WFM replaced by WFM,,
t h e a s s e r t i o n fol-
lows.
The following theorem is a s t r a i g h t - f o r w a r d generalization of t h e corresponding r e s u l t s f o r homogeneous distributions in Hormander C111 (see Theorems 8.1.8 and 8.4.18).
Proof o f ( i ) . Step 1 : We f i r s t s u p p o s e t h a t T is a l m o s t quasihomogeneous of deg r e e m . By Remark 9.33 and Proposition 2 . 4 0 . ( i ) we may a s s u m e t h a t x lies in S ,
412
and
IX. Q u a s i h o m o g e n e o u s Wave F r o n t
5
S et s
i n S , * . In view of Proposition 2 . 4 0 . ( i ) and the Fourier inversion formula it
suffices to prove one of the implications, only. Suppose that ( x , t ; ) @WF,(T).
Then we find compact neighbourhoods K of x
in V and F of €, in V ' \ I O I such that K x F n W F M ( T )= @ . We fix E > 0 such that K ( x , ~ EC ) k and choose y , € C g ( K ) equal to 1 on K ( x , 2 ~ ) .Moreover, let U be an open neighbourhood of t; i n V x such that
u
and choose x E C F ( F ) equal
C
to 1 on U . Note that by the Fourier inversion formula for arbitrary
j
E V and
t E I O , + a C we have
Since T i s almost quasihomogeneous of degree m we deduce ,-I,
( 'J
T ( x ? ) ( - M:y)
.S4)
= t"'+"
( % + (< t )T i , r y ( ; o M f ) > ,
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 <*
XOM,,,)
' ,1
1' F F q T i ) ( i ~e)i < " ' y >Y ( M , , ~ ~ d(M,,,u) J)
>=
=
V'
= j ' y ( q T i ) ( M t i ~e)x p ( i < M , u , y > )X ( 1 J ) dlJ. V*
Since suppx C F it follows t h a t
(9.55)
/ < q T i , ~ , ( ; o M t ) >Sl d ' l x ( u ) l d u sup{ l F ( q T i ) ( M t i ~ )~l J; E F } , t i O . F
Since by Proposition 0.7 we have WFM(Ti) C WFM(T) we deduce by means of LemniaY.4 that this is equal t o O ( t - N ) as t+'Jj for every N . Consequently, i n view h
of ( 9 . 5 4 ) we can conclude that ( < . - x ) 4 WFM*(T) once we have proved that
sup{ I < ( l - c p ) T i , r , ( ; o M : ) ) I ; for every N E N
.
Y E K ( x , E ) }= O ( t - N ) as
t+co
Since T and hence the Ti are temperate it suffices to show that
for every N the s e t ! B N : = ( t N ( l - q )T,(;oM:);
y € K ( x , ~ ) t,E C l , + a C }
is a bounded subset of Y ( V ) . N o w , since 1-y, vanishes o n K ( x , 2 ~ we ) have
413
9.c W a v e F r o n t Sets of Almost 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
N o w , by ( 1 . 4 9 ) w e have No
( f o r t h e d e f i n i t i o n o f t h e Pp,j see R e m a r k 1.30). H e n c e , by L e m m a O . 2 f o r a r b i t r a r y
PE'U a n d S E Nt h e r e is a c o n s t a n t C ( ( 3 , s )s u c h t h a t I a B ( $ o M f ) ( z - y ) l 5 C ( p , s ) ( l + l M : ( ~ - y ) ( ) - t~- N f o r a r b i t r a r y ~ E ~ ( x , E z)e ,V \ K ( y , ~ )a n d t E C l , + a C . S i n c e t H IM:(z)l
is increa-
s i n g w e have
so t h a t t h e desired a s s e r t i o n a b o u t 2 3 ~f o l l o w s by t h e Leibniz r u l e . S t e p . ? ; Now w e s u p p o s e t h a t T 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 . Again by t h e F o u r i e r inversion f o r m u l a w e may a s s u m e t h a t T i s a l m o s t q u a s i h o m o g e n e o u s on
+ = V \ C O l . T h e n by T h e o r e m 8.15 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 d i s t r i b u t i o n S E ~ ' ( V )s u c h t h a t t h e s u p p o r t o f d : = T-S is c o n t a i n e d in (01. In p a r t i c u l a r , A
S is t e m p e r a t e , as w e l l . I t f o l l o w s t h a t d is i n d u c e d by a polynomial f u n c t i o n . Hence ( 9.50 )
Applying S t e p 1 to S i n s t e a d of T w e d e d u c e t h e desired a s s e r t i o n f o r T , a s w e l l .
Proof of (iil. S i n c e t h e s e c o n d s t e p of t h e p r o o f of t h e a s s e r t i o n ( i ) r e m a i n s valid if W F M is r e p l a c e d by W F M , L w e may a s s u m e t h a t T 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 d e g r e e m . A s b e f o r e , w e may a l s o a s s u m e t h a t Y E S v a n d t; E S v i , a n d it s u f f i c e s to p r o v e o n e o f t h e t w o i m p l i c a t i o n s , o n l y .
W e s u p p o s e t h a t ( x , € , ) 4 W F M , L ( T ) a n d c h o o s e K , F , s , a n d W as in S t e p 1 of t h e p r o o f o f ( i ) a f t e r having r e p l a c e d W F M by WFM.,.
By L e n i m a O . l 8 w e c h o o s e
c o n s t a n t s C a n d AD a n d s e q u e n c e s ( [ P N ) N € N in C g ( K ) a n d ( x N ) N € N in C T ( F ) s u c h t h a t f o r every N E I N q N - l o n K ( x . 2 ~ a)n d x N - 1 both
'pN
and
xN
o n W a n d f o r every 8641
s a t i s f y ( 9 . 5 ) with ( A , B ) r e p l a c e d by ( A p , C N ) . W e f i x i E N 0 .
T h e n by Lemma 9 . 2 2 f o r ever) q C V M w e o b t a i n a c o n s t a n t a q s u c h t h a t
I F ( ~ ~ T ~ ) ( M5 a,, ~ ct -) ' IL - R e m - '( a , ( L N ) " q / t ) N ,
tECI.+a[,<EF,NEN.
414
I X . Q u a s i h o m o g e n e o u s Wave F r o n t Sets
By (9.55) -applied to
( ~ N , x instead ~ )
of ( q , -~ )this implies
where a; is another c o n s t a n t depending only on q . Moreover, fixing k E N , making u s e of (9.57) and applying Lemrna0.2 to some P > ( 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,)(<)l
5
n
5 llM1/Tll
2 j=1
Since
~ ' ( l + l < l I:(C)l )k
v*
a n d diJ' b e l o n g to
condition (9.64)' follows.
Y(V * )
I~ldC~up{(l+lul)~l(a,J,)(iJ)I; i,EV*}.
a n d s i n c e by R e m a r k 1.8.( i ) lim
I1 M1,, I1 = 0
,+a
H
~
the
417
9.d T h e He a t a n d t h e Schrodinger Equation
(d) Quasihomogeneous Wave F r o n l Sels 01' l h e Slandard Fundamental Solutions ol' l h e H e a t and of' the Schrodinger Eyuatlon
Let q be a s in section ( f ) of C h a p t e r 7 . We fix rCCl,+mC, s e t p : = ( r , l , . _ .I ,) , and a s s u m e t h a t M is of t h e form ( 1 . l . a ) . Instead of WFM and C M V Lwe shall write WF, and C P V L .In t h e p r e s e n t section t h e Cm and C P y L quasihomogeneous wave f r o n t sets ( w i t h respect to p ) of t h e a l m o s t invariant (see Definition 7 . 7 0 ) fundamental s o l u t i o n s of q ( D) a r e determined. The s t a n d a r d homogeneous wave f r o n t sets, of c o u r s e , correspond t o r = l , so t h e t r e a t m e n t includes C"
and ana-
L
lytic wave f r o n t sets a s well a s t h o s e with respect t o C . In t h e c o u r s e of t h e proof we give s o m e r a t h e r precise e s t i m a t e s which might be interesting f o r their own s a k e . We a l s o refer t o t h e discussion following t h e proof of Theorem 0.35.B. Most of t h e material is taken f r o m 1 7 1 . We recall t h a t f o r any o p e n s u b s e t LI of R1+" t h e space C " . L ( f l ) c o n s i s t s of all Cm functions f : n - C
such t h a t f o r ever] compact s u b s e t K of fl t h e r e is a
c o n s t a n t C satisfying
where L a : = L ( r a l + a 2 +...+a,,, ) . Note t h a t C P ' A ( f l ) is t h e s t a n d a r d inhomogeneous Gevrey c l a s s TP(RI of type p which f o r r = l coincides with t h e space of all real analytic functions on fl. From Proposition 9.7 and Theorem 9.27 we see t h a t for any EE3'(lR1'")
t h e fol-
lowing inclusions hold: W F , ( q ( D ) E ) C WF,(E) C WF,,,L(E) C WF,,,,(E)
C W F , , , A ( ~ ( D ) EU) R ' + " X q , ~ p ( 0 )
where qW,, d e n o t e s t h e quasihomogeneous (of type p ) principal p a r t of q which is given by
if r > 2 (9.60)
if r < 2 If E is a n a l m o s t invariant fundamental solution of q ( D ) then in view of (7.61) and by Theorem 7.80 E is real analytic outside ( O l x l R " so t h a t in view of t h e equalities WF,(S,)=(Ol x S "
WF,,,,(S,)
t h e preceding inclusions become
418
IX. Quaelhomogeneous Wave F r o n t
(9.67)
(0)xS"C
WF,(E) C WF,,=(E) C WF,,,(E)
C
~
O
~
~
S
"
U
Sets
~
~
O
The equations (9.66) s u g g e s t t h a t when determining the wave f r o n t sets of E o n e has to deal with three different cases. The first t w o of them can b e handled together, they are easier than t h e third. In order to avoid repetitions, from now on we denote by E one of t h e distributions defined in Notation 7.71, i.e. in case v # O E is t h e unique invariant fundamental solution of q ( D ) whereas in case v = O E is one of t h e t w o invariant fundamental solutions E' o r E - o f q ( D ) .
For the case r = ? we note that
Proof in case v Z 0 . In this case we have q - ' ( O ) n S " =
0.Hence
f o r r = 2 t h e asser-
tion follows from ( 0 . 0 6 ) and ( 0 . 6 7 ) . This means t h a t t h e restriction T of E to IRl+ n
\CO)
belongs to t h e Gevrey class ~ ' 2 " ' ~ " ' 1 ) ( I R 1 + n \ ( O l ) . Since t h e latter is
contained in I'p(IR1'"\(0))if r > 2 this implies t h a t T belongs to P ' ( " ' + ' ' \ ( O ) i.e. WF,,,(T)=@
),
for r > 2 , a s well. rn
When v = O the main s t e p in the proof of TheoremO.35.A is the following lemma which we a r e going t o prove in subsection (d.1) below by microlocally cutting off a suitable real analytic function.
Lemma9.36. Suppose that v = O . Let [ o < S r l - ' . Then W F p , A ( E ) n f ? x A = Q ) where A : = { ( r , f ) E l R x I R " ; s i g n ( o u ) <[,to> ~ 0 and ) where
Proof o f Theorem 9.3S.A
if v = O . Let xoEIRn\(OI and <EIR"\(sign(ciu) C O , + ~ l x o ) .
Then t h e compact convex segment K:= C - s i g n ( o u ) < , x o l in IR" does not contain
~
~
9.d
419
T h e H e a t and t h e Schr6dlnger Equation
t h e origin. Hence by t h e Hahn-Banach Separation Theorem w e find > O foreveryxEK,inparticular:
C < ~ , X >~ O>
and s i g n ( a u ) < < , < o > < O .
Hence Lemma 9.36 implies t h a t ( ( 0 , x o ) , ( r , < ) ) 4 W F , , A ( E ) f o r every T E R . Consequently, by (9.67) we have
Since t h e analytic singular support of E equals ( 0 ) x l R " t h e left-hand side of ( 9 . 7 0 ) is non-empty, and since in view of ( 9 . 0 6 ) and (9.68) for r 2 2 t h e set on t h e right-
hand side consists of one element only we conclude t h a t equality holds. rn
We now c o m e to t h e more difficult case " r < 2". In this case it follows from (0.67) and (9.06)
- or
from ( 9 . 7 0 ) - t h a t f o r every xo
E
IR"\(O)
we have
As here t h e right-hand side consists of t w o elements, equality d o e s not automatically follow from t h e non-emptiness of t h e left-hand side. In fact, it depends o n L whether equality holds o r not:
"heorem 9.3S.B. Suppose that r < 2 , and set S : = s u p { k " L L ' ; (i)
I f S < + w then WF,(E) = WF,,,(E) = i 0 l x S " u W
w={
kgV\I}.
where
iE v f O @
(10)x (W" \ / O l ) x i ( - s i g n u , ~ ) l i f v = o
( i i ) I f S = + w then WF,,=(E) = WF,,,(E) = 1 0 l x S " u ~ ~ 0 l x R " ~ x { ~ l , 0 ~ , ~ - 1 , 0 ~
The proof is based on t h e following more precise result establishing a microlocal decomposition of E . I t will b e derived in subsections ( d . 2 ) and ( d . 3 ) below. Note C W F A ( E ) is a l s o a consequence of t h a t t h e inclusion ((O)xlR")x{(l,O),(-l,O)}
Theorem 8.5.0 in Hormander C111.
Theorem9.37. Suppose &hat r < 2 . Let v € i + l , - l I such that v = s i g n u i f v = O . Let co6S"-'. and define R and X a s in ( 9 . 6 9 ) . Then E l , has a decomposition E l , = E , + E- where E , and E- are distributions on R having the following properties :
420
IX. Q u a s i h o m o g e n e o u s Wave Front Sets
Proof of Theorem 9.35.B.fi). N o t e t h a t t h e c o n d i t i o n " S < +a''i m p l i e s t h a t t h e
r'
Gevrey c l a s s
2s1*.
. . s l
)
(n) is
c o n t a i n e d in C p S L ( f l ) C . onsequently, t h e function
f in T h e o r e m 9.37 s a t i s f i e s : W F p , ~ ( T f=) @ , so t h a t WF,,,L(EI,) = W F l , , L ( E - ) , a n d WF,,L(EI,)
is c o n t a i n e d in ( { O ) x X ) x { ( - w , O ) } . 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 u c a n be r e p l a c e d by - u . a n d for every
X
~
XE t h e l e f t - h a n d side
of (9.71) i s e m p t y ; a n d ( b ) if v = O t h e n f o r every x o E X t h e l e f t - h a n d side of
{ ( - s i g n u , O ) } a n d - s i n c e it is n o n - e m p t y - e q u a l to
(9.71) is c o n t a i n e d in
{ ( - s i g n u , O ) } . S i n c e for every x o E I R n \ ( 0 ) t h e r e is < o € S n - ' s u c h t h a t ( O . x o ) ~ f l t h e p r o o f is c o m p l e t e . rn
ProoF
OF
Theorem 9 . 3 , 5 . B . f i i ) . Let x,EIR"\(O),
a n d set < o : = x n / I x o I . T h e hypo-
t h e s i s " S = +a'' i m p l i e s t h e e x i s t e n c e of a s t r i c t l y i n c r e a s i n g s e q u e n c e ( j i ) i E N
in
N such that :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,<)); (t,c)EZ,, In view of WF,,,(F)
C WF,,,,(E')
u WF,,,(T,)
sER} C
WF,(F).
t h e inverse inclusion is valid w i t h
WF,
replaced by WF,,,
by T h e o r e m s 9 . 3 5 . A a n d 9 . 3 9 .
(ii).
W e first s u p p o s e t h a t t h e number S in Theorem 9.35.B is finite. Then it fol-
l o w s f r o m T h e o r e m s 9.35.B and 9.39 t h a t
Since q ( D ) is n o t hypoelliptic t h e s i n g u l a r s u p p o r t of F i n t e r s e c t s { O ) x ( l R " \ { O ) ) . Since F is invariant u n d e r t h e action of O ( n ) on t h e s p a c e s variables x it f o l l o w s t h a t (01x ( l R " \ ( O l ) is contained in s i n g s u p p F . C o n s e q u e n t l y , equality h o l d s everyw h e r e in ( 9 . 7 6 ) Now w e s u p p o s e t h a t S = +a.Since by Theorem 9.39 W F p , L ( T , ) d o e s n o t intersect ( R x R " ) x { ( u , O ) } i t f o l l o w s f r o m T h e o r e m 9.35.B.(ii) t h a t
424
IX. Quasihomogeneous Wave Front Sets
WF,,L(F) n ( I R x I R " \ ( O l ) x { ( v , O ) } = = WF,,,(E')
Since WF,(F)
n (IRxlR"\(O))x{(v,O)} = ((0)xRn\~O))x{(v,O)},
is contained in WF,,L(F)
t h e first part of the proof s h o w s t h a t
WF,,L(F) contains ( ( 0 ~ ~ I R " \ ( O ~ ) x ( ( - u , 0Hence ) } . t h e assertion follows. m
(d.1) IBroor ol' Lemma 9.30
The first s t e p relies on a suitable Fourier representation formula for E which is valid f o r t h e case v f O , a s well.
Proporitlon 9.40. For every
s €10.+a>[
we have
The proof requires a deformation of t h e contour of integration:
Lemma9.41. Let Z be an open subset o f C " , let F:Z-C
be a holomorphic
function, and let g:R"x~O,lI---irIR" be a C' function such that { € + i g ( f , t )E;E R " , ~ E [ O , I I }c Z . Suppose that there are constants C and L such that
Moreover, let
n be
an open subset of IR" such that f or ever) compact subset K
of D there are constants L K and C K satisfiing t E to,11.
9.d.l
425
Proof of Lemma9.36
all coincide where O t : W n -----;,C" is defined by
Proof. -
By P:
5 H (=+ig(<, t)
w e d e n o t e t h e p o l y d i s c in R" of r a d i u s r > 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 <1212+ v 2 s 2
+ v2T2 +
u2 s2 + 2 I t; 1' ~v I s , i . e .
D e n o t i n g by 9,t h e p a r t i a l F o u r i e r t r a n s f o r m w i t h r e s p e c t to t h e t i m e v a r i a b l e t
426
I X . Q u a s i h o m o g e n e o u s Wave Front Sets
h
we see t h a t E = 7,E" where
E"
is t h e partial Fourier transform of E with respect
to t h e space variables x . Since
E"
is induced by t h e function (7.60) it follows
f o r every J , E Y ' ( [ R X [ R ~ ) that
By t h e definition of 7,and by Fubini's theorem t h e integral wit.. respect to t is equal to j - f ( t . r ) d t J,(r,t;)dr R O
where f ( t . r ) : = 'We x p [ - t ( E + i 0 r t ~ 1 < 1 ~ ) 1X=e x p [ - t aw q ( r - i E O , < ) ] =
I
0
= - q(r-iEo,S)a , e x p [ - t w q ( r - i ~ b . < ) l .
Note t h a t (9.81) s h o w s t h a t q ( r - i e a , [ ) # O , indeed: more precisely, we have
lq(r-iEcs,<)l 2 I W I E
>o. .m
The main theorem of calculus implies t h a t
f ( t , r ) d t = l / q ( r - i s o , < ) so t h a t
Now we assume t h a t J, E 7Cg([RxlRn) so t h a t J, is well-defined on @ x @ " a s a n entire holornorphic function. Since by (9.81), again, we have Iq(r-ia(E+s).c)I> l w l ~ > O ,
S€CO,+aJE,
one can deform t h e contour of integration by applying Lemma 9.41 to g ( K , < , t ): = ( T - i o ( t s + ( l - t ) E , O ) so t h a t t h e double integral on t h e right-hand side of (9.82) is equal to
f o r every s ~ l O , + a CBy . (9.81) we a l s o have I q ( r - i o ( E + s ) , < ) I2 I w I s f o r arbitrary
9.d.l
427
Proof of Lernrna9.36
s , ~ ~ l O , + c o CHence, . in view of t h e Paley-Wiener e s t i m a t e s f o r J, o n e concludes
by Lebesgue's Dominated Convergence Theorem t h a t t h e limit a s E + O c a n be taken under t h e integral sign. This gives
(9.83)
<E^,$)
=s IR"
:(r-ios,<)
drdc.
IR
An application of t h e Fourier inversion formula leads to t h e desired formula.
Proof of Lemma 9 . 3 6 . Suppose t h a t v = 0 ,and let
co
rn
and A be a s in t h e s t a t e m e n t
o f Lemma 9.36. The f i r s t s t e p in t h e proof c o n s i s t s in deforming t h e c o n t o u r of integration in t h e formula of Proposition 0 . 4 0 . This is based o n t h e following estimate
I Im q ( r - i a s
which follows f r o m
, < + i p < o ) I = I uI s + 2p (
- s i g n ( a u ) < <,c0> ) .
We now fix a f u n c t i o n ~ E C ' ( I R X I R " ) with values in 1 0 , 1 1 and with s u p p o r t contained in A satisfying ( 0.85)
f o r s o m e N E IR t h e functions ( 1 + I
- 1 ) - N di x , j E IN,,
, a r e bounded.
Then in view of ( 9 . 8 4 ) t h e assumptions of Lemma 9.41 a r e satisfied f o r F = l / q and g ( < , t )= ( - o , t x ( < ) I
Hence, s e t t i n g
: = ( r - i o , [ + i x ( < ) l
< = ( T , < ) E IR x IR",
one obtains
where J ( < ): d e t (I)'(<). Now we fix C , E> 0 and set
and A , : = IRxIR"\A2
where A , : = A c , , \ I R x K ( O , l ) .
such way t h a t , in addition,
x = 1 on A, and t h a t x
We s u p p o s e t h a t
x
is chosen in
is quasihomogeneous of d e g r e e 0
and of type p o n IRxIR"\ K ( 0 , l ) . Then t h e derivatives of
x a r e even bounded so
t h a t (9.85)holds f o r N = O . For j E ( l , Z l we define E j E 3 ' ( f l ) by
428
Then
IX. Q u a a i h o m o g e n e o u s Wave F r o n t Sets
Eln= E,+E2.
Since A,-c
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 t h e set
A l \ ( ! R ~ R n \ A c , c ) = A C , = \ A 2 = AC,= nlRxK(O,l) is c o n t a i n e d in t h e b o u n d e d set C - C , C l x K ( O , l ) it f o l l o w s by Proposition 9.32 t h a t
WF,,,(E,)
(9.86)
n f l ~ A , -=~@ .
N e x t w e are going to s h o w t h a t a function g : f l - C
is well-defined by
g ( z ) : = , \ ' l ( z , < dC ) A2
where
(9.87)
To t h i s e n d w e have to e s t i m a t e I . Note t h a t for < = ( r , < ) E A 2 w e have x C < ) = l so t h a t @ ( < I
( T - i o , < + i l < l < o ) and
and
Hence
IJ I
5
dy o n
A2,
a n d - in view of ( 9 . 8 4 )-
Since I is measureable a n d since e x p ( -
I*I
< < o , x >) 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(<)d < ,
c,-cn,.
r+ Since T-\(IRxR"\T)
= T n I-R,RCxlR" is c o n t a i n e d in t h e b o u n d e d set
I - R , R E x K(O,CR'/')
it f o l l o w s by P r o p o s i t i o n 9.32 t h a t
In order to s t u d y E, w e a r e g o i n g to s h o w t h a t a Cc' f u n c t i o n f : f l + @
is w e l l -
d e f i n e d by
1I ( z , < )d<
f ( z ) :=
I-+
w h e r e I is d e f i n e d a s in ( 9 . 8 7 ) . To t h i s e n d w e have to e s t i m a t e I . F o r w e have by ( 9 . 9 8 ) : x ( c ) = l a n d x ( < ) = O so t h a t
c = ( r , c )E r ,
@ ( c ) =(r,[+iap(<)Co) and
as well a s
Since I is m e a s u r a b l e a n d s i n c e e x p ( - a < s 0 , x > p ) is a b s o l u t e l y i n t e g r a b l e o n
r+
w e c o n c l u d e t h a t f is w e l l - d e f i n e d . S i n c e for 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 . 9 0 ) is p o s i t i v e , Fubini's t h e o r e m s h o w s t h a t E,
=Tf.
In order to see t h a t f is d i f f e r e n t i a b l e w e fix o ! = ( j , ~ ) E [ N , x b J , " a n d c = ( ~ , < ) E r + . Since
151
Cp(<)2 and
151 5 p ( < ) w e d e d u c e t h a t
432
IX. Quasihomogeneous Wave Front Sets
S e t t i n g c : = a < c 0 , x > / 2 and making use of ( Y . Y l ) and (9.101) we conclude t h a t
I D Z I ( z , c ) l 5 co-' 2''' B , e x p ( - c p ( < ) ) .
Z € n ,
where B , : = s u p { p ( 5 ) i i ' a ' e x p ( - c ~ ( < ) ) ;< € r + }I t. f o l l o w s by (9.92) t h a t
Since (9.102)
J'
J'e-cp 5
r+
exp[-p(c2r,c<)]dt~d< =
c-I'-?
IR X I R K '
J'
e-p,
IRXR"
combining t h e preceding e s t i m a t e s we see t h a t f is a C" function such t h a t
where C is a c o n s t a n t not depending o n z and a . In particular, this s h o w s t h a t
f belongs to
r(2.'.....1)(n).
To obtain information o n W F , , , ( E + ) ( s e e ( 9 . 9 7 ) ) is contained in h(z) : =
f
r
we let E > O be any c o n s t a n t s u c h t h a t
and define h:n-C
rE
by
I(z,<)d<.
r+\rlBy t h e s a m e a r g u m e n t s a s with f we see t h a t h is a well-defined Cm function such t h a t (9.103) holds with ( f , T + ) replaced by ( h , r + \ r ' ) . To find more precise e s t i m a t e s f o r t h e derivatives of h we observe t h a t
if < = ( r , < ) e r ' + \ r r then
l r l < ( ~ < I / E ) " <_ (p(<)/E)r so t h a t
1 = ~ r ~ j l ( c + i a p ( < ) < ~~ E) '-lr J ( " a ) ' ' ' p ( ~ ) " + ' ' '
433
9.d.2 Proof of T h e o r e m 9 . 3 7 . Part 1
C o m b i n i n g t h e e s t i m a t e s a b o v e w i t h (9.102) w e find a c o n s t a n t C n o t d e p e n d i n g o n L and a such that
I h'"'(z)(
cl-l+l
1)
+
''I
,
In p a r t i c u l a r , t h i s s h o w s t h a t h b e l o n g s to C P V A ( n ) ,i . e . WF,.,(Th)
z=(t.X)Efl.
= @ and
WF,,A(E+) = WF,,*(Tf-h). Since
" $((I)(<)) h
< T f - h , 'p > = ( 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
<=
(5, ,El) EIRX[R"-'
IR; s u c h t h a t to= ( l , O ) , w r i t e
and suppose that
r={(r,f)€IRxIR": U T > O . max{l
r
to t h e set
rE d e f i n e d
in ( 9 . 9 7 ) s i n c e it m a k e s t h e
a p p l i c a t i o n o f Fubini's t h e o r e m e a s i e r ) . Let x E R " s u c h t h a t x1 > 0 . M o r e o v e r , let a=(j,p)EN,xN,".
We write p=(k,y)ElN,xN,"-'.
(9.100) o n e o b t a i n s
Combining (9.103). ( 9 . 8 7 ) and
436
1X. 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
By o u r c h o i c e of
r
and
r,
a n d by Fubini's t h e o r e m t h i s is e q u a l to
+m
(9.112)
.I' ("TI'
R
1'
I o ( T , < * , x l () < @ ) ' e x p < i < ' , x ' >d t ' d r
K'(0,srd)
where
w h e r e w e w r i t e < = (T,
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'
(<')Y ( i W r + 1
K ' ( 0 . r ~ ~ )
~ * 1 ~ ) ( ~ - e~ x) p/[ <~ i < ' , x ' >- x l
(iWr+
1c'12)"2]
dc'
Y.d.3
P r o o f of Theorem 9 . 3 7 . P a r t 2
439
Introducing polar coordinates we find
M(r,x)=
9'L(r,S,x) d 9
sn-2
where
,4 ~ ( r , ~ ,: =x f)
2 (k-l)/2 p l ~ l + (~i w - r~+ p )
2 1/2 )
e x p [ i p < S , x ' >- 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.
where y : = x ' / x l and A ( s , t ) : = X ( s ) - i & t , and 2 12cl
L(r,a,x)=
1
T~
-1
,I
(1~1+n-2)/2 '
~ ~ X ( s ) ~ - ' e x p [ - J ; x , A ( s( H , ,y>)]ds
0
where r n : = ( l y l + n - 3 ) / 2 . Substituting +m
rj
j' 0 2 j + l p l + n - 1
m
R
where g ( a ) : = ~
~
6
i.e. $ d r = o d o , leads to
g(0)
+m
L ( r , a , x )d r =
2.=02,
1
'
sm
X ( s ) k - l e x p [ - o x l A ( s , < 9 , y >) ] d s d o
0
By ~Fubini's ~ -theorem ~ . t h i s is equal to
Note, moreover, t h a t by t h e choice of R we have T > 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
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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.
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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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