Convolution and Equations

PART I1 CONVOLUTION AND

a

EQUATIONS

INTRODUCTION

T h e p u r p o s e of Part I1 is t o study s o m e m a i n b a s i c p a r t i a l differ e n t i a l e q u a t i o n s , whose s o l u t i o n s w e r e a l r e a d y known i n t h e f i n i t e d i m e n s i o n a l c a s e . In the infinite d i m e n s i o n a l c a s e , t h e i r s o l u t i o n s a r e m a i n l y obtained by extending c l a s s i c a l m e t h o d s , but t h e s e e x t e n s i o n s a r e in g e n e ral m u c h m o r e t e c h n i c a l t h a n t h e p r o o f s i n t h e f i n i t e d i m e n s i o n a l c a s e a n d

r e q u i r e t o u s e new t o o l s s u c h a s f o r i n s t a n c e n u c l e a r i t y of t h e s p a c e s , t o pological t e n s o r p r o d u c t s , i n t e g r a t i o n t h e o r y i n infinite d i m e n s i o n , t h e t h e o r y of m a p p i n g s of type

QP

...

a s well a s new m e t h o d s t h a t have no

analogues o r a r e t r i v i a l i n t h e finite dimensional c a s e a n d that a r e added t o the c l a s s i c a l m e t h o d s t o y i e l d a s o l u t i o n . In c h a p t e r 9 a n d 10 we obtain

v e r y g e n e r a l r e s u l t s of e x i s t e n c e

of s o l u t i o n s of convolution e q u a t i o n s i n s p a c e s of p o l y n o m i a l s a n d of e n t i r e functions of exponential t y p e . I n c h a p t e r 11, f r o m a v e r s i o n of t h e W e i e r s t r a s s p r e p a r a t i o n t h e o r e m , we obtain a g e n e r a l r e s u l t of d i v i s i o n of a d i s t r i b u t i o n by infinite d i m e n s i o n a l non z e r o h o l o m o r p h i c f u n c t i o n s , which g i v e s a new proof of a n e x i s t e n c e r e s u l t of c h a p t e r 10 a n d t h e n we s t u d y t h e d i v i s i o n by r e a l a n a l y t i c f u n c t i o n s . I n c h a p t e r 12 we study t h e convolution e q u a t i o n s in v a r i o u s s p a c e s of h o l o m o r p h i c f u n c t i o n s on n o r m e d a n d l o c a l l y convex s p a c e s , and in c h a p t e r 13 we obtain e x i s t e n c e a n d a p p r o x i m a t i o n of s o l u t i o n s f o r f i n i t e - d i f f e r e n c e 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 i n s p a c e s of

Coo

func-

t i o n s on n o r m e d a n d l o c a l l y convex s p a c e s . T h e e n d of P a r t I1 t r e a t s of t h e s o m e w h a t d i f f e r e n t , but a l s o basic,, p r o b l e m of t h e r e s o l u t i o n of t h e

6

e q u a t i o n , i n pseudo-convex open s u b -

s e t s of l o c a l l y convex s p a c e s . In c h a p t e r 14 we e x p o s e m a i n b a s i c r e s u l t s o n pseudo-convexity in infinite d i m e n s i o n , a n d s o m e a p p r o x i m a t i o n r e s u l t s 206

Convolutionand a equations

201

in pseudo-convex d o m a i n s . In c h a p t e r 15 we p r o v e t h e r e s o l u t i o n of t h e b

e q u a t i o n in D F N s p a c e s , a n d m o r e g e n e r a l l y i n n u c l e a r s p a c e s , (with

s o m e a s s u m p t i o n on the given s e c o n d m e m b e r in t h i s l a t t e r c a s e ) . C h a p t e r 16 is c o n c e r n e d with a p p l i c a t i o n s t o t h e f i r s t C o u s i n p r o b l e m a n d t o s o l u t i o n s of s o m e homogeneous convolution e q u a t i o n s in s p a c e s of e n t i r e f u n c t i o n s of exponential t y p e . T h e r e s u l t s i n t h i s p a r t show t h e d i f f i c u l t i e s , but a l s o t h e r i c h n e s s , of the t h e o r y of p a r t i a l d i f f e r e n t i a l e q u a t i o n s i n i n f i n i t e d i m e n s i o n .