Elias m. Stein - Localization and Summability of Multiple Fourier Series

7/28/2019 Elias m. Stein - Localization and Summability of Multiple Fourier Series

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L O C A L I Z A T IO N A N D S U M M A B I L IT Y O F M U L T I P L EF O U R I E R S E R IE S

BYE L I A S M . S T E I N

Cambridge, ~Mass . , U.S.A .(1)

I n t r o d u c t i o n1 . D e f i n i t i o n s

I n t h i s p a p e r w e s h a ll d e a l w i t h t h e t h e o r y o f " s p h e r i c a l " s u m m a b i l i t y o f m u l -t i p l e F o u r i e r s e r i e s .

L e t / ( x ) = / ( x l , x 2 . . . x k ) b e a L e b e s g u e i n t e g r ab l e f u n c t i o n d e f in e d o n t h e f u n d a -m e n t a l c u b e Q ~, - ~ < x i ~ < g , i = 1 . . . . k , i n E u c l i d e a n k - s p ac e . W e fo r m t h e F o u r i e rs e r ie s o f / ( x )

/ ( x ) = ~ a n e ~ '~ x = ~ a .... . . , k e t( . . . " + " k % ) , ( 1 .1 )w h e r e n = ( n 1 . . . . n k ) Lu a v e c t o r w i t h i n t e g r a l c o m p o n e n t s , n . x = n 1 x 1 4 - n 2 x ~ . . . n ~ x k ,w i t h

a n = ( 2 : ~ ) - k f [ ( x ) e t '~ ': :d x ,Qka n d d x = d x l d X 2 . . . d x k .

W e n e x t f o r m t h e s p h e r i c a l R i c s z m e a n s o f o r d e r ~ o f ] ( x)~- (1 I n 1 2 ~ e , n . z ,I nl< R - - ~ - ) a n ( 1 .2 )

w h e r e I n I = (n ~-t . . . . + n l ) t. U n l e s s s t a t e d t o t h e c o n t r a r y , W e s h a l l a s s u m e t h a t k> ~ 2 .T h e g e n e r a l p r o b l e m o f t h e t h e o r y c o n c e r n s it se lf w i t h t h e v a l i d i t y ( a n d m e a n -

i n g ) o fl i m 8 ~ R ( x , / ) = / ( x ) , (1 .3 )R~-~f o r s o m e a p p r o p r i a t e ~.

( i) Th is resea rch wa s sup ported by the U ni ted S~a te3 Ai r Force und er Con t rac t No. AF 49 (638)-42.mon i t o re d by t he A F O f fi ce o f S c ie n ti fi c R e se a rch o f t he Ai r R e , a rc h a nd D e ve l opme n t C omm a nd .


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94 ELIAS M. STEIN

2 . L o c a l i z a t i o nI n t h e t h e o r y , t h e s o- c a ll e d " c r i ti c a l e x p o n e n t " a ( a = 8 9 k - 1 ) ) , p l a y s a s i g n i-

f i c a n t r o le . I f 0 > ~ , t h e b e h a v i o u r o f t h e R i e s z m e a n s S~R ( x , / ) i s " F e j d r - l i k e " : t h er e l a ti o n s h i p ( 1.3 ) h o l d s a l m o s t e v e r y w h e r e ; t h e c o n v e r g e n c e i s b o u n d e d i f / ( x ) i sl ik e w i s e b o u n d e d , a n d i s u n i f o r m i f /(x) is c o n t i n u o u s ; f i n a l l y t h e v a l i d i t y o f t h er e l a t i o n s h i p ( 1. 3) d e p e n d s o n l y o n t h e v a l u e s o f /(x) i n a n y n e i g h b o r h o o d o f x .W h e n ~t ~< :e, t h e a b o v e i s n o l o n g e r g e n e r a l l y t r u e . I n t h e c l a s s i c al c a s e , k= 1 , a ni m p o r t a n t p r o p e r t y r e m a i n s f o r (~ = ~ . A c c o r d i n g t o t h e l o c a li z a ti o n t h e o r e m o f R i e -m a n n , t h e e x is t e n ce o f ( 1.3 ) ( w h e n k = l , ( ~ = 0 ) d e p e n d s o n l y o n t h e v a l u es o f /(x)i n a n y n e i g h b o r h o o d o f x . I t is n a t u r a l , t h e r e f o r e , t o a s k w h e t h e r t h e l o c al i za t i o np r o p e r t y f o r ( 1. 3) s ti ll h o l d s f o r 5 = ~ w h e n k ~> 2 . T w o r e s u l t s f o r t h e c r it i c al e x -p o n e n t a , w h i c h g i v e a p a r t ia l a n s w e r t o t h e a b o v e q u e st io n , a re d u e t o B o c h n e r [3 ].

F i r s t , t h e r e e x i s t s a n / ( x ) i n t e g r a b l e o v e r Qk , a n d v a n i s h i n g i n a n e i g h b o r h o o do f t h e o r i g i n f o r w h i c h

l i m s u p S ~ ( 0 , / ) = + o ~ .R-~oO

T h u s t h e l o c a l i z a ti o n p r i n c ip l e f a il s t o h o l d u n r e s t r i c t e d l y a t t h e c r i t ic a l e x p o -n e n t , w h e n k ~ > 2. H o w e v e r , b y a n o t h e r r e s u lt o f B o c h n e r , t h e l o c a l i z a t io n p r i n ci p lef o r ( 1 .3 ) , w h e n ( ~ = a , s t i ll h o l d s i f w e r e s t r i c t o u r s e l v e s t o f u n c t i o n s i n L 2 ( Q k ). T h u st h e n a t u r a l q u e s t i o n a r o s e w h e t h e r l o c a l i z a t i o n st il l h o l d s a t t h e c r i t ic a l e x p o n e n t i fw e l i m i t o u r s e l v e s t o f u n c t i o n s o f t h e c l a ss L v (Q k ), 1 < p .

I t w i ll b e o n e o f t h e p u r p o s e s o f t h i s p a p e r t o g i v e a n a//irmative a n s w e r t ot h e a b o v e p r o b l e m . I n f a c t , w e s h a ll s h o w t h a t l o c a l i z a t io n f o r ~ = a s t il l h o l d s i fw e r e s t r i c t o u r s e l v e s t o t h e c l as s o f f u n c t i o n s f o r w h i c h

f I / ( x ) l l o g + I / ( x ) ( 2 . 1 )QkO f c o u r s e , t h e c l as s o f f u n c t i o n s f o r w h i c h (2 .1 ) h o l d s i n c l u d e s e v e r y Lr(Qk)c l a s s , 1 < p .

3 . P o i n t w i s e a n d d o m i n a t e d s u m m a b i l i t yI f w e n o w c o n s id e r t h e r e l a t i o n s h i p ( 1.3 ) in t h e s e n s e o f " a l m o s t e w e r y w h e r e " ,

a n d n o t o f i n d i v i d u a l p o i n ts , w e m a y t h e n o b t a i n r e s u l t s c o n c e r n i n g i ts v a l i d i t y f o r( ~ < ~ , o r 5 = ~ , I n f a c t, f f /(x)ELV(Qk), 1 < p ~ < 2 , w e s h a ll s h o w t h a t ( 1.3 ) w i ll h o lda l m o s t e v e r y w h e r e w h e n e v e r (~ > a ( 2 / p - 1 ).(1 ) ( T h e p o i n t b e in g t h a t a ( 2 / p - 1) < a ,

(1) Fo r p = 2 , th i s re sul t i s know n, see [11] . I t i s a consequence of the g enera l the ory of or tho-no rm al series, as develo ped in KXCZARZ an d ST Et. '~AV S, [9], Ch apt . V .


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L O C A L IZ A T IO N A N D S U M M A B I L I T Y O F M U L T I P L E F O U R I E R S E R I E S 9 5w h e n e v e r 1 < p ~ 2 ). T h u s w h e n e v e r / ( x) E L p (Q ~ ), 1 < p , t h e n ( 1 .3 ) h o l d s a l m o s t e v e r y -w h e r e f o r s o m e d} b e lo w t h e c r i t i c al e x p o n e n t . W e s h a ll a ls o s h o w t h a t t h e r e l a t i o n -s h i p ( 1 .3 ) w i l l h o l d a l m o s t e v e r y w h e r e f o r ~ = :r ( ~ = c r i t i c a l e x p o n e n t = 8 9 k - 1 ) ),w h e n e v e r

f I 1 ( ) 1 ( l o g + I t ( x ) I ) 'd < ( 3 .1 )Qk.T h e s e r e s u lt s w i ll b e c o n se q u e n c e s o f re s u l ts c o n c e r n i n g d o m i n a t e d s u m m a b i l i t y

- - w h i c h r e s u lt s s e e m in t e r e s t i n g o n t h e i r o w n r ig h t . F o r t h is p u r p o s e w e in t r o d u c et h e f o l l o w i n g d e f i n i t i o n

S~. (x) = S~ , (x , [ ) = su p S~R (x , / ) I , (3 .2)r

i f

W e s h a h p r o v e9 I) ~l/p( f ( S ~ . ( x ) ) d x ) < ~ A ~ . ~ ( f / ( x ) l ' d x ) 1 /',

Qk Ok( 3 . 3 )

5 > a ( 2 / p - 1 ), a n d l < p < 2 ; ( a = 8 9 1 ) ) .( 1W e s h a l l a l s o s h o w ,

f S $ ( x ) d x < A f l / ( x ) l ( lo g + ] / ( x ) l )2 d x + B .QIc Ok

(3.4)

A s a f u r t h e r c o n s e q u e n c e o f ( 3. 3) w e s h a ll o b t a i nlira fls~(.,f)-t(x)l~d~=O, fl(x)EL " (Q~), l


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96 ELIAS M, STEIN

F o r 6 > a , ( 4. 1) a b o v e i s a n i m m e d i a t e c o n s e q u e n c e o f r e l a t i o n ( 1.3 ) ( w h i c h , o fc o u r s e , h o l d s a l m o s t e v e r y w h e r e i f ~ > ~ , a n d ! ( x) i s i n t e g r a b le ) . A g a i n , o n l y t h e c a s e~} ~< ~ w i l l in te re s t us .

O u r r e s u l t s a r e t w o - f o l d . F i r s t , i f 1 < p < 2 , a n d ! ( x) E L p (Q k ), t h e n ( 4 .1 ) h o l d sa l m o s t e v e r y w h e r e a s l o n g as ~ > a ( 2 / p - 1) - 1 / p ' , w h e r e 1 / p ' + 1 / p = 1 , ( ~ = 89 (k - 1)).S i n c e f o r 1 < p ~ < 2 , a ( 2 / p - 1 ) - 1 / p ' < a ( 2 / p - 1 ) , t h e r e l a t i o n ( 4. 1) i s n o t im p l i e d b yt h e r e s u lt s m e n t i o n e d i n w 3 .

S e c o n d l y , i f ! ( x ) E L I ( Q k ) i t i s p o ss i b le t o p r o v e a m o r e p r e c i s e r e s u l t : T h er e l a t i o n ( 4. 1) h o l d s a l m o s t e v e r y w h e r e i f 5 = ~ . T h i s is t h e s t r i c t a n a l o g u e o f a t h e o -r e m o f M a r e in k i e w ic z o n t h e s t r o n g - s u m m a b i l i t y o f F o u r i e r se ri es w h e n k = 1 . W es h al l h o w e v e r p o s t p o n e t h e p r o o f o f th i s t o a n o t h e r t i m e , s in c e t h e m e t h o d u s edd i ff e rs i n e s s en c e f r o m t h a t o f t h e r e s t o f t h is p a p e r . I t s h o u l d b e p o i n t e d o u t t h a tB o e h n e r a n d C h a n d r a s e k h a r a n [ 4] h a d s h o w n th a t i f ! ( x ) E L I ( Q k ) t h e r e l a t i o n ( 4 . 1 )w i t h 5 = ~ r e fl e ct s o n l y t h e l o c a l b e h a v i o u r o f ! ( x ) .

5 . S n m m a r y o f r e su l tsF o r t h e s a k e o f c o n v e n i e n c e w e s h a ll b r i ef l y s u m m a r i z e o u r m a i n r e su l ts . T h e y

f a ll i n t o t w o c la ss e s, a n d a r e l i st e d a c c o r d i n g t o s e l f - e x p l a n a t o r y n o t a t i o n :

Results for LP(Qk), l


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LOCALIZATION AND SUMMABIL ITY OF MULTIPL E FOURIER SERIES 9 7R e s u l ts " n e a r " L 1 ( Q , )

A s a l w ay s , a ~ 8 9 k - 1) :( L * ) I / ] ( x ) van i shes i n a ne ighborhood o /Xo , and f 1] ( x )] l o g + [ / ( x ) [ d x i s [ i n i t e , t h e n

Ql im S~ (x0 , [ ) = 0 .R - ~ c r

( D * ) f s ~ ( x , l ) d x < ~ A f I / ( x ) l ( log + I / ( x ) l ) 2 d x + B .Ok Qk

XA.E .*) I / f 1 / (x ) [ ( log + ] [ (x )] )2 d x i s f i n i t e , t h e n l im SR ( , [ ) = [ ( x ) , / o r a lm os t every x .Q k R--~oo(N*) aim f l S ~ ( x , / ) - / ( x ) l d x = O , i /, f l f ( x ) l l o g + [ [ ( x ) ] d x < ~ o .

R--*.or Qk QkR

(S*) l i ra = ~ [S~ ( x , / ) - ! ( x) d u = O , / o r a lmos t an y x , i / / ( x ) e L 1 ( QD.0

6 . M e th o d s u s e dS i n c e o u r r e s u l t s d e a l w i t h s u m m a b i l i t y o f o r d e r (~, ~ ~ a , w e m u s t i n e a c h c a s e

s u r m o u n t t h e s a m e i n i ti a l d i f f i c u l t y - - w h i c h w e m a y d e s c r ib e a s f o ll o w s .L e t K ~n ( x) d e n o t e t h e f u n c t i o n w h o s e F o u r i e r e x p a n s i o n i s

K ~ ( x ) = Z ' 1 i i " ' " ' ~. ,< R - ~ ) e . ( 6 . 1 )T h u s w e m a y w r i t e

S~R (x , / ) = (2 ~) -k f K ~ ( x - y ) / ( y ) d y . (6.2)Qk

W h e n 5 > ~ , ( o r 5 ~> 0 , w h e n / c = 1 ), w e m a y o b t a i n e s t i m a t e s f o r t h e k e r n e l K ~ (x )w h i c h a r e s a t i s f a c t o r y f o r m o s t p u r p o s e s . (1)

H o w e v e r , w h e n (~ ~< ~ , ]r 2 , e s t i m a t e s f o r t h e k e r n e l K ~ ( x) d e p e n d h e a v i l y o nt h e d i s t r ib u t i o n o f l a t t i c e p o i n t s in I t- s p a c e - - a n d t h is is a v e r y su b t l e m a t t e r . F o rt h i s r e a s o n n o e s t i m a t e s f o r K ~ ( x) w h e n (~ ~ ~ , s a t i s f a c t o r y f o r g e n e r a l p u r p o s e s , h a v eb e e n g i v e n .

A n o v e l a p p r o a c h t o t h e p r o b l e m is t h e r e f o r e n e e d e d . T h e i d e a o f t h i s m e t h o dw a s c o n t a i n e d i n t h e p r o o f o f ( N ), w h i c h a p p e a r e d e a r l i e r ( 2 ) - - a n d th i s re s u l t p r e -s e n t s t h e s i m p l e s t i ll u s t r a t i o n o f t h e m e t h o d u s e d . T h e g e n e r a l i d e a is a s f o l l o w s :

(1) Se e, for exam ple, (10], formula (7).(~) See [13].


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98 ELIAS 5 . STERNT h e e x p r e s s i o n ~ ( x , / ) d e f i n e d in ( 1 .2 ) f o r p o s i t i v e 6 , i s n o w e x t e n d e d t o co m -

p l ex v a l u e s o f (~, t h u s b e c o m i n g a n a n a l y t i c f u n c t i o n in 6. W e t h e n r e s t r i c t o u r a t te n -t i o n t o a s u i t a b l e s t r i p a ~< ~ ((~)~< b . T h e b o u n d a r y l i n e ~ (( ~) = a i s m a d e t o c o r r e -s p o n d t o a n L 2 r e s u l t , a n d t h e l in e ~ ( ( ~ ) - - - b i s m a d e t o c o r r e s p o n d t o a n L 1 r e s u l t .T h e L ~ r e s u l t o n t h e l in e ~ ((~ )= a i s d e d u c e d v i a P a r s e v a l ' s r e l a t i o n , w h i l e f o r t h eL 1 r e s u l t o n ~ ( ( ~ ) = b , r a t h e r s t r a i g h t f o r w a r d e s t i m a t e s a r e s u f fi c ie n t .

W e t h e n u s e a " P h r a g m e n - L i n d e l S f " t y p e a r g u m e n t t o o b t a i n a n L ~ e s t im a t eo n a n i n t e r m e d i a t e l in e o f th e s t ri p . T h i s is d o n e v i a a n i n t e r p o l a t i o n t h e o r e mf o r a n a n a l y t i c f a m i l y o f o p e r a t o r s - - a t h e o r e m w h i c h g e n e r a li z e s M . R i e s z ' s w e ll -k n o w n c o n v e x i t y t h e o r e m ( L e m m a 1 ).

T h e a b o v e i s t h e g e n e r a l p r o c e d u r e f o r p r o v i n g t h e L ~ t h e o r e m s ( A E ) , ( D ), ( N ) ,a n d (S ). T h e l o c a l i z a t i o n r e s u l t , ( L ) , is m o r e d i f f i c u l t s i n c e t h e i n d e x (5 c o n t a i n e d int h e r e s u l t i s a l w a y s f i x e d a t :r H o w e v e r , b y i n t r o d u c i n g " f r a c t i o n a l i n t e g r a t i o n "i n t o t h e p r o b l e m , w e m a y a g a i n o b t a i n a s i t u a t io n f o r w h i c h t h e in t e r p o l a t i o n m e t h o da p p l ie s . T h e s i t u a t i o n i s d e s c ri b e d m o r e f u l l y in w12.

O n c e t h e L p r e s u l t s a r e o b t a i n e d , t h e r e s u l t s " n e a r " L 1 ( i. e (L * ) , ( D * ) , ( A E * )a n d ( ~I *) ) a r e o b t a i n e d b y c e r t a in li m i t i n g a r g u m e n t s f r o m t h e i r c o r r e s p o n d i n g L ~r e s u l t s .

A w o r d s h o u l d b e a d d e d a b o u t a g e n e r a l h e u r i s ti c p r i n c ip l e w h i c h m a k e s t h ec o n v e x i t y p r o p e r t y o f a n a l y t i c f u n c ti o n s a p p l i c a b l e t o o u r s i t u a t io n . I t i s t h i s : I f (~i s c o m p l e x , t h e n t h e b e h a v i o u r o f S ~ ( x , / ) i s e s s e n t ia l l y r e f l ec t e d b y S ~ ( x , / ) , w h e r ea = ~ ( 6 ) .

7 . Genera l remarks ; convent ionW e s h o u l d p o i n t o u t h e r e t h a t , p r e v i o u s l y , r e s u l ts c o n c e r n i n g s u m m a b i l i t y of

o r d e r 6 , 5 ~< ~ , h a d i n g e n e r a l b e e n o b t a i n e d o n l y a t t h e h e a v y p r i c e o f m a k i n g r e -s t r ic t i o n s o n t h e s m o o t h n e s s o f / ( x ) in t h e e n t ir e c u b e Q k. I n s o m e c i r c u m s t a n c e st h e s e r e s t r i c t io n s w e r e i n c o r p o r a t e d i n t o r e s t r ic t i o n s o n t h e o r d e r o f m a g n i t u d e o f t h eF o u r i e r c o e f f ic i e n t s. T o b e s u r e , t h e r e s u l t s t h u s o b t a i n e d h e ld a t i n d i v i d u a l p o i n t s . (1)T h e t h e o r e m s s t a t e d i n w3 - w 5 a b o v e s h o w t h a t i f w e a r e c o n t e n t w i t h b e h a v i o u ralmost everywhere, t h e n w e m a y d e a l w i t h s u m m a b i l i t y o f o r d e r 6 , 5 ~< ~ b y m a k i n gm u c h m i l d e r g l o b a l r e s t r i c t i o n s o n / ( x ) .

W e t h u s h a v e t h e i n t e r es t i n g p h e n o m e n o n t h a t a f u n c t i o n i n L ~, 1 < p h a s aF o u r i e r s e ri es w h i c h i s s u m m a b l e a l m o s t e v e r y w h e r e o f s o m e o r d e r ~, ~ t< ~ , w h i l et h i s s u m m a b i I i t y m a y f ai l a t i n d iv i d u a l p o i n t s w h e r e th e f u n c t i o n is v e r y " s m o o t h " .

1 See , for example , [6 ] , Chap ter V.


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LOCALTZATION AND SUMMABILITY OF MULTIPLE FOURIER SERIES 9 9A n o t h e r p h e n o m e n o n w h i c h s e e m s n o v e l fo r k> ~ 2, i s t h a t i f [ / ( x ) ] ( log + [ / ( x ) [ ) 2

i s i n t e g r a b le , t h e F o u r i e r s e ri es o f ] ( x) is s u m m a b l e a l m o s t e v e r y w h e r e f o r t h e c r it ic a le x p o n e n t . S u c h a re s u l t is u n k n o w n f o r k = 1, a n d i ts p r o o f ( or d i s p r o o f ) w o u l ds e e m t o b e e x t r e m e l y d i f f i c u l t .

C e r t a i n c o n j e c t u r e s s e e m p r o b a b l e , b u t f o r w h i c h w e h a v e n o d e c is i v e ev i d e n ce .(1 ) T h a t t h e r e s u l t ( L * ) c a n n o t b e i m p r o v e d . ( 2 ) T h a t t h e r e s u lt ( A . E . * ) c a n b ee x t e n d e d t o f u n c t i o n s f o r w h i c h ] / ( x ) [ lo g + ] / '( x ) [ i s i n t e g r a b l e .

I t w o u l d b e in t e r e s t in g t o d e c id e w h e t h e r t h e r e s u l ts ( D ), ( A .E . ), a n d ( N ) a r ev a l id f o r a n y r a n g e o f ~ f o r w h i c h ~ < ~ ( 2 / p - 1 ) . W h a t s ee m s t o b e n e e d e d h e r em o s t a r e s o m e g o o d c o u n t e r - e x a m p l e s .

W e w i s h n o w t o m a k e e x p l ic i t a c o n v e n t i o n w h i c h w e s h a ll u s e c o n s i s te n t l y i nt h i s p a p e r .

( i) B o u n d s s u c h a s r A p , B ~, e t c . w i ll b e u s e d r e p e a t e d l y t o s h o w t h a t t h e b o u n d sd e p e n d o n t h e i n d i c a t e d p a r a m e t e r s . T h e s e b o u n d s , h o w e v e r , m a y b e d i f f e r en t i nd i f f e r e n t c o n t e x t s .

( ii) W h e n a n i n e q u a l i t y i s g i v e n w i t h a b o u n d d e p e n d i n g o n a p a r a m e t e r ( e.g .A ~ ) , t h e r a n g e o f t h e p a r a m e t e r w i ll h a v e th e f o ll o w in g m e a n i n g : T h e f u n c t i o n ,4~i s b o u n d e d ( i n d e p e n d e n t l y o f ~ ) f o r ~ i n a n y c l o s e d i n t e r v a l o f t h e r a n g e o f $. F o re x a m p l e , a n i n e q u a l i t y w i t h b o u n d A ~ , f o r ~ > ~0 , w il l m e a n t h a t A ~ i s b o u n d e d i ne v e r y i n t e r v a l 0 ~ ~ ~< a < cr H o w e v e r , a n i n e q u a l i t y w i t h b o u n d A ~ , f o r ~ > 0 , w i l lm e a n t h a t ,4~ m a y b e c o m e i n fi n it e a s ~ - + 0 .

C H A P T E R I

B a s i c L e m m a $T h i s c h a p t e r c o n t a i n s t h e b a s i c t o o l s w h i c h a r e n e e d e d in t h e fo l lo w i n g c h a p t e r s .

8 . I n t e r p o l a t i o n t h e o r e mL e t M a n d N b e t w o g i v e n m e a s u r e s p a c es w i t h m e a s u r e s d /x a n d d v r e s p e c -

t i v e l y . W e s h a ll d e a l w i t h a f a m i l y o f l i n e a r o p e r a t o r s T z ( d e p e n d i n g o n t h e c o m p l e xp a r a m e t e r z ). W e s h a ll a s s u m e t h a t t h e f a m i l y T ~ s a ti s fi e s t h e f o l lo w i n g p r o p e r t i e s :

(i) f o r e a c h z , 0 4 ~ ( z ) < 1 , T z i s a l i n e a r t r a n s f o r m a t i o n o f " s i m p l e " f u n c t i o n so n M t o m e a s u r a b l e f u n c ti o n s o n N .

(ii) I f ~ is a s i m p l e f u n c t i o n o n M , r a s im p l e f u n c t i o n o n N , t h e n


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100 ~ L L , .S M . S T E r ~

r fT~(~) r i s a n a l y t i c i n O < ~ ( z ) < la n d c o n t i n u o u s o n t h e c l o se d s t r i p 0 ~ < ~ ( z) ~ < 1 .

( i i i) S up s u p l og I ( I) ( x + i y ) ] - < ~ a r-~e , a


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L O C A L I Z A T IO N A N D S U M M A B I L IT Y O F M U L T I P L E F O U R I E R S E R I E S 1 0 1W e s h a ll s a y t h a t a n o p e r a t o r T d e f i n e d o n si m p l e f u n c t i o n s o n M t o m e a s u r -

a b l e f u n c t i o n s o n N i s sub - l i near i f(i) I T (~)1 -~- ~/)2) I < I T (~t)i) [ ~- I T (V l) l

w h e n e v e r ~/)1 a n d v22 a r e s i m p l e , a n d( ii ) I T (kw t ) l = I k I I T (Vx)] , fo r e v e r y s c a l a r k .T h e f o l l o w i n g l e m m a h a s b e e n u s e d i n a p a r t i c u l a r e a se b y T i t e h m a r s h [ 14 ].L ~ M M X 2 . Le t T be a sub - l i near opera tor , de / i ncd on s imp le / unc t ion s o / M ,

( / z ( M ) < ~ o ) as above . Suppose t ha tII T ( t) I I 1 ~ < A ( p - 1 ) - ' I 1 / l ip ( 9 . 2 )

]or every p , l < p ~ < 2 , every s imp le ] , and some r , r>~O, wi th t he cons tan t A indep end-e n t o f ] a n d p . T h e n w e m a y c o n c l u d e t h a t

II T ( / ) 1 1 1 < K A [ f l / ( x ) ] ( l o g + I I ( x ) I ) ' d ~ + 1 ] , ( 9 .3 )M/ o r e v e r y s i m p l e / ; A i s t h e b o u n d o / (9 .2 ) and K depends on l y on the t o ta l measureo/ the s l :ace M .

Proo]. W e w r i t e ] (x ) = ~ / n ( x) ,n - 0w h e r e / . ( x) = / (x) , i f 2 " - * ~ < 1 / ( x ) l < 2 " , n > ~ l ;

/ n (x ) = 0 , o t h e r w i s e , n ~> 1 ;l o ( X ) = l ( x ) , if I I ( x ) l < a ;/0 ( x ) = 0 , o t h e r w i s e .

W e l e t E , - - s e t w h e r e /~ ( x ): r 0 , /~ ( E , ) i t s m e a s u r e . S i n c e / ( x ) i s s i m p l e , o n l y af i n i t e n u m b e r o f t e r m s a p p e a r i n t h e a b o v e a n d f o l l o w i n g s u m s .N o w , b y p r o p e r t i e s (i) a n d (ii) a b o v e :

IT(/)l


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1 0 2 ~ .L I A S M . S T E I Nw h e r e t h e e x p o n e n t s p . , m a y b e c h o s e n a r b i t r a r i l y , s u b j e c t t o 1 < Pn ~< 2 . W e choo sep . as fol low s : Po = 2, p~ = 1 + 1 / n , n >~ 1 . W e n o t i c e t h a t 1 2 - " / ~ I ~< l , an d / , v an i sh e so u t s i d e E . .

W e t h e r e f o r e h a v e~ n p ( ~ , j , n> ~ l . I

H1o ]12 ~< (/~ (M )) ~- ] (9 .6)Combining (9.6) , (9.5) , and (9.4) gives

t l T ( 1 ) I d ~ _~ A (/~ (M )) t + ~ 2" n ' (# (E~)) "~ (~') .N

(9.7)

O n e a c h t e r m o f t h e i n f i n i t e s e ri es a p p e a r i n g i n (9 .7 ) w e s h a l l a p p l y t h e i n-e q u a l i t y o f Y o u n g : a b 4 a P / p + b q / q , 1 / p + 1 / q = 1 .W e c h o o s e a = 2 " + l n r # ( E , ) n l ( " + l ) , b = 2 - I , p = l + l / n , q = n + l . T h u s

~ ( n + 1 ) ( 1 + 1 / n ) n r ( 1 + 1 / . )2" n ' ( /z (E ' ) )" ' ( "+ 1 ' ~< ~ 1 + 1 / nn - 1 n ~ l #(E ,) + ~ 2-n-~/n -4- 1.n- 1

B u t a s i s e a s i l y v e r i f i ed ,2(, + ) (1+ 1/,),l~r(1+ ln)

~-1 1"4" ~ 1C o m b i n i n g t h e s e e s t i m a t e s w i t h ( 9.7 ), w e o b t a i nf T ( J ) I d v < ~ A K ~ 2 " n r # ( E n ) + A K .9 n - I

N

(9.8)

H o w e v e r , , 1 ~ 2 n n r ju (E , )~ < f [ / ( x ) l ( lo g + I f( x )[ ) r d # . (9.9)MT h u s ( 9 . 9 ) a n d ( 9 . 8 ) t o g e t h e r p r o v e L e m m a 2 .

1 0 . M a x i m a l f u n c t i o nW e i n t r o d u c e t h e s p h e r i c a l m e a n s o f / ( x ) a n d o f ] / ( x ) [ , d e f i n e d a s f o l l o w s :

/ ( x ; 0 = ~ g ' f ! ( x ~ + t ~ 1 , x 2 + t ~ 2 . . . . x ~ + t ~ k ) d E ~ , ( 1 0 . 1 )] ( ~ ; t ) = ~ ; , l f l l ( x , + t ~ , . . . . . x ~ + t ~ ) l d Y . ~ . ( 1 0 . 2 )


11/55

L O C A L IZ A T IO N A N D S U M M A B I L I T Y O F M U L T I P L E F O U R I E R S E R I E S 1 0 3H e r e O k = 2 ( n ) 8 9 (89k ) , a n d E i s t h e u n i t s p h e r e : $ ~ + ~ : ~ . . - + $ ~ = 1 ; d Z e i t s E u c l id -e a n m e a s u r e .

T h e f o l lo w i n g l e m m a is e a s il y d e d u c e d f r o m i ts w e l l - k n o w n " n o n - p e r i o d i c " a n a -l o g u e . ( 1 )

L E M M A 3 .grab le over t he / undamen ta l cube Qk .

L e t/ * ( x ) = s u p N - k o ~ ' k f [ [ ( x + y ) l d y

oo>N>0 ]y[


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10~ ELIAS M. STEIN

B y d e f i n i ti o n o f g ( x ), h o w e v e r ,f l g ( ) l ' d = 2 (10.9)Ek O k

A c o m b i n a t i o n o f ( 1 0 .9 ) , (1 0 . 8 ), a n d ( 1 0 . 5 ) g i v e sli p( f (/* (x ) )~d~ ) . .~A (p / (p -1 ) )( fl /( x ) l~d~ ) ~ , l

1 o f th e b o u n d A ( p / p - l ) ) a p p e a r i n g i n ( 1 0 . 1 0 ) ,

w i ll b e i m p o r t a n t f o r l a t e r p u r p o s e s .

1 1 . R i e s z m e a n s o f c o m p l e x o r d e rL e t ~ a ~ b e a n u m e r i c a l s e ri es . W e sh a ll d e f i n e t h e R i e s z m e a n s o f c o m p l e x

r

order (~ , ~ = a + i v , a s f o l l o w s .L e t a = ~ ( 5 ) > - 1 . D e f i n e S ~ , b y ( ' ;~ = 5~ 1 - ~ , , ( 1 1 . 1 )

w h e r e , o f c o u r se , t h e p r i n c ip a l v a l u e i s t a k e n f o r t h e c o m p l e x e x p o n e n t i a l s a p p e a r i n gi n ( 1 1 . 1 ) . T h u s S ~ R = A ~ / R ~ , w h e r e A ~ = ~ ( R 2 - v ) ~ a , . (1 1.2 )

v< R I

W e n o t e t h a t i f ~ ( ~ ) > - l / p , t h e n S ~R , a s a fu n c t i o n o f R , i s l o c a l ly i n L r .T h e r e l a t io n b e t w e e n S ~ , f o r d i f f e r e n t c o m p l e x 5 's , i s c o n t a i n e d i n t h e f ol lo w i n g .L E M M A 4 . Let f l , 8 , be comp lex num bers , ~ ( f l ) > 0 , ~ ( ( ~ )> - 1 , a n d ~ (f l- t-(~) >0.

T h e nR

A ~,~ 2 1 ' ( 5 + f l + 1 ) t"= F-- (6%-1- F ( f l i . (R2 - t2)t~-I A ~ t d t, (11 .3 )0

the. integ ral con verging absolutely.P r o o / . W e r e c a ll t h a t S R i s l o c a ll y i n L p, a s l o ng a s ~ ( ( ~ ) > - 1 / p . N o w , s i n c e

~ ( ( ~ ) > - 1 , ~ ( f l ) > 0 , a n d ~ ( f l + 5 ) > 0 , w e c a n f in d e x p on e n t s p, a n d q s o t h a t1 / /p + l / q = 1, a n d b o t h

R Rf lA ? . tl t, l l q t0 0

c o n v e r g e . T h u s t h e i n t e g r a l in (1 1 .3 ) c o n v e r g e s a b s o l u t e l y , b y H h l d e r ' s i n e q u a l i t y .


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L O C A L I Z A T I O N A N D S U M MA B I L I T Y O F M U L T I P L E F O U R I E R S E R I E S 105N o w , A ~ = ~ ( t ~ - v ) O a , . T h e re fo re t o v e r i fy t h e i d e n t i t y (11 .3 ) i t i s su f f i c i en t t o

v < t I

v e r i f y t h a t R2 P ( 8 + f l + 1) f( R g - v ) '~ + ~ - F ( ~ + 1 ) F ( f l ) ( R ~ - t ~) ~ - I ( t~ - v ) ~ t d t ,

v

for ~(~)>0, ~(6)>-1, ~(~+8)>0. (11.4)W e s e e f i rs t t h a t t h e i n t e g r a l i n ( 1 1.4 ) c o n v e r g e s a b s o l u t e l y , b y t h e s a m e a r g u -

m e n t u s e d t o e s t a b l is h t h e a b s o l u t e c o n v e r g e n c e o f t h e i n t e g r a l (1 1.3 ). F o r f i x e d fl,t h i s a r g u m e n t a l so s h o w s t h a t t h e c o n v e r g e n c e i s u n i f o r m i n 8 , w h e n e v e r 8 i s r e -s t r i c t e d t o a c l o se d b o u n d e d s e t l y i n g i n ~ ( 6 ) > - 1 , a n d ~ ( f l + ~ t ) > 0 . T h u s f o r f i x e df l, t h e r i g h t s id e o f ( 11 .4 ) i s a n a l y t i c i n 6 ; w h e n 8 > 0 , h o w e v e r , ( 1 1 . 4 ) i s e a s i l yv e r i f ie d b y t h e w e l l - k n o w n e q u a t i o n o f t h e B e t a f u n c t i o n . S in c e t h e l e f t s i de o f (1 1.4 )i s c l e a r l y an a l y t i c i n 8 , (11 .4 ) i s t h e n d e m o n s t r a t ed fo r a l l v a l u e s o f /5 an d 8 i nq u e s t i o n . T h i s c o n c l u d e s t h e p r o o f o f t h e l e m m a .

L e t n o w [ ( x ) = [ ( x l , x 2 . . . . xk ) b e o f p e r i o d 2 ~ i n e ach x~, an d l e t i t b e i n t c -g r a b l e o v e r t h e f u n d a m e n t a l c u b e Q k . W e f o r m t h e F o u r i e r e x p a n s i o n o f ] ( x )

/ ( X ) " ~ . a n e 'n '= , (11.5)w h e r e a n = (2 ~) - k f / (x) e - '~ ~ d x .QkI f 8 i s c o m p l e x , , ~ ( 6 ) > - 1 , w e d e f i n e S ~ n ( x ) b y

S ~ ( x ) = S ~ ( x , / ) = 2 1 - { 2 a , e ' ~ }v< R , Inp -p. . . . a , e ~ . ( 11 .6 )

I n l < n

n 2 -- 2-H e r e I n 12 = t n z + n ~ . W e m a y n o w e x t e n d B o c h n e r ' s r e p r e s e n t a ti o n t h e o r e mt o s u m m a b i l i t y o f c o m p l e x o r d e r .

L X M ~ t A 5 . L e t [ ( x ) b e i n t e g r a b l e o v e r Q ~ , a n d l e t ~ ( 8 ) > 8 9 6 = a + i z . L e tS ~ ( x ) b e a s d e / i n e d a b o v e . T h e n ,

c o

( x ) = cl R * ~-~f /( x; t) t ~-~-1 d~+j~ (tR ) dr,0 (11.7)w h e r e c I = 2 ~ - t k +1 F ( 8 + 1) { F ( 8 9 k ) } - 1 ; t h e i n t e g r a l i n ( 1 1 . 7 ) c o n v e r g e s a b s o l u t e l y .

8 -- 583801. A a a m a t h em a t ic a . 100. Impr im 6 le 25 oc tob re 1958.


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106 ELIAS M. STEIN

P r o o f . W e a s s u m e a s k n o w n t h e e a s e w h e r e (~ i s p o s i t iv e a n d d i > 8 9N o w fo r e a ch f i x ed x a n d R , t h e l e f t s i d e o f (11 .7 ) is c l e a r l y an a l y t i c i n (~. T o p ro v et h e i d e n t i t y (11 .7 ) i t w i l l t h e re fo re b e s u f f i c i en t t o sh o w t h a t t h e r i g h t s i d e o f (11 .7 )i s an a l y t i c i n (~, w h en ~ (5 )> 8 9 k - 1 ). T h u s t h e p r o o f o f t h e l e m m a w i l l b e c o n c l u d e da s s o o n a s w e s h o w :

(i) f o r e a c h f i x e d x a n d R , t h e i n t e g r a l

f f (x; t ) t ~k -6 -1 J~+ t~ ( t R ) d t (11.8)0co n v e rg es ab so l u t e l y an d u n i fo rm l y i n d}, w h en e v e r dt l ie s i n a c l o sed b o u n d e d se tw it h in ~ (d}) > 8 9 k - 1 ) ;

( ii) t h e i n t e g r a l i n (1 1.8 ) a b o v e i s a n a l y t i c i n 5 f o r e a c h f i x e d x , R > 0 a n d t > 0 .F o r t h i s p u r p o s e w e r e c a l l t h e f o l lo w i n g w e l l - k n o w n f a c t s i n t h e t h e o r y o f B e s s e lfu n c t i o n . (2) 1

F ( r 1 8 9 ( l - u 2 ) ~ -t c o s u t d u , ~ ( ~ ) > - ~ . ( 1 1 . 9 )o

[J r t>~l, ~>~0. (11.10)] J ~ , , n ( t ) l < ~ A r t> O , E>~0. (11.11)

B y (11 .9 ) w e see t h a t fo r e ach f ix ed x , R > 0 , a n d t > O , t h e i n t eg ran d i n (1 1 . 8 ) i sa n a l y t i c i n 5 . W e a l s o r e c a l l t h a t

Uf l f ( x ; t ) l t k - l d t < ~ A u ~, i f u > ~ u 0 > 0 . (11 .1 2 )0W e n o w b r e a k u p t h e r a n g e o f i n t e g r a t i o n f o r t h e i n t e g r a l o f (1 1.8 ) i n t o t h e

i n t e r v a l s ( 0 , l / R ) , a n d ( 1 / R , ~ ) . W e f u r t h e r b r e a k u p t h e i n t e r v a l ( I / R , c ~ ) i n t oi n t e r v a l s o f t h e f o r m ( 2 " / R , 2 " + 1 / R ) . T h u s w e w r i t e (1 1 . 8 ) a s

I /R 2n+llRfbI ' ( x ; t ) , ] k 6 ~-1J t , k ( ' R ) d t + ~ .. ..]1 / ( x ; t ) t ' k - ~ J-1 g ~ ( t R ) d , . ( 1 1 . 3 )9 n - o0 2 n l RI f w e r e p l a c e e a c h i n t e g r a n d i n (1 1.1 3) b y i ts a b s o l u t e v a l u e , t h e r e s u l t i n g s u m

m a y b e e s t i m a t e d a s f o l lo w s . T h e f i r s t t e r m i n ( 1 1.1 3) m a y b e e s t i m a t e d b y ( 11 .1 1) ,(1) The proof of this case may be found in [6], Chapter V.(2) See the references in Lernma 8, below.


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L O C A L I Z A T IO N A N D S U M M A B I L I T Y O F M U L T I P L E F O U R I E R S E R I E S 107a n d e a c h t e r m i n t h e i n f in i t e s e ri e s m a y b e e s t i m a t e d b y ( 1 1 .1 0 ). C o m b i n i n g t h e s ee s t im a t e s , w e o b t a i n a s a n e s t i m a t e f o r t h e a b s o l u t e c o n v e r g e n c e o f ( 11 .8 ) t h e f o l-l o w i n g :

l IR 2" ~ IR

A ; e V * l R " + ~ [ l ( x ; t ) I t k - ~ d t - ~ - A " d ' " ' t R - t ~ 2 - ( t k ' t + " ) " I I ( x ; t ) l t ~ - ~ d t , (11 .14)n-- 00 2n/R

w he re 5 = a + i ~ .B y ( 1 1. 12 ) t h e i n f i n i t e s u m a p p e a r i n g i n ( 1 1. 1 4) m a y b e e s t i m a t e d a s f o l lo w s :

A'ae't~l R -k+ t ~ 2 - (~- 8 9 (11 .15). = 0

T h i s l a s t s e ri e s c o n v e r g e s w h e n a > 8 9 k - 1 ); t h a t i s, w h en ~ ( 6) > 89 k - 1 ). Th ere -f o re t h e i n t e g r a l (1 1.8 ) c o nv e r ge s a b s o l u te l y w h e n ~ ( 6 ) > 8 9 k - I ) , a n d b y t h e a b o v ee s t i m a t e s t h e c o n v e r g e n c e i s u n i f o r m i n a n y c l os e d b o u n d e d s e t w i t h i n ~ ( 6 ) > 89 k - 1 ) .T h i s c o n c l u d e s t h e p r o o f o f t h e l e m m a .

C H A P T E R I I

L o c a l i z a t i o n1 2 . Ou t l in e o f me t h o d

L e t ] ( x ) = [ ( x x . . . . xk) b e a p e r i o d i c f u n c t i o n , i n t e g r a b l e o v e r t h e f u n d a m e n t a lc u b e Qk. A s s u m e t h a t / ( x ) v a n i s h e s i n t h e e - s p h e r e ,

(Ixl =x + . . .W e c o n s i d er t h e s p h e r ic a l R i e s z m e a n s o f o r d e r 8 9 k - 1 ) o f t h e F o u r i e r e x p a n s i o n

o f / ( x ) , e v a l u a t e d a t t h e o r ig i n:k ~ R ( k -1 ) ( 0 ) = 8 ~ C k - 1 ) ( 0 ; f ) =

w h er e a , = ( 2 z ) - k f / ( x ) e - i " X d x .Qk

~Z 1 - c ,~ , ( 1 2 . 1 )I n l < n

T h e c r u x o f t h e p r o o f o f t h e l o c a l i z a t i o n t h e o r e m f o r L p, 1 < p , ( T h e o r e m ( L }i n w 5 ) w i ll c o n s i s t i n t h e p r o o f o f t h e f o l lo w i n g i n e q u a l i t y :

sup l < p . (12.2>


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108 ELIAS M. STEINH e r e / ( x ) i s a n y f u n c t i o n i n L"(Q~, ) w h i ch v an i sh es i n t h e e - sp h e re , Ix I ~ 0s u p { U ~ ( O ) { < C , . , { l l { { , i f ~ ( X ) - - - a > 0 . (1 2.5 )R>~0

B , . x a n d C ,. z w i l l b e a p p r o p r i a t e b o u n d s ; t h e i r e s t i m a t e s w i ll b e o f i m p o r t a n c e l a t e r .B a s i c t o t h e c o n s i d e r a t i o n o f t h e a b o v e i s t h e f o l l o w i n g " k e r n e l " :

H(~ ) ([ x ]) = f e - ' x v (1 - ]y ]u)t(k-x '~a ly ]a dy 1 d y z . . , d y k . (12.6)Ivl 0 . T h e d e d u c t i o n o f ( 1 2 . 4 )w i l l b e m o r e su b t le .S i n c e i t i n c l u d e s t h e r e s u l t f o r ) t = 0 , i t m a y b e v i e w e d a s a v a r i a n t o f t h e lo c a li z a-t i o n r e s u l t f o r L 2. O n c e (1 2.4 ) a n d ( 12 .5 ) h a v e b e e n p r o v e d , th e n ( 1 2 . 2 ) c a n b ed e d u c e d b y t h e c o n v e x i t y - i n t e r p o l a t i o n a r g u m e n t m e n t i o n e d e a r l i e r ( L e m m a 1 ) .

I n t h i s c h a p t e r w e s h a l l a d o p t t h e f o l lo w i n g p r o c e d u r e . I n w 1 3 w e sh a l l o b -t a i n a n a s y m p t o t i c e s t i m a t e f o r t h e k e r n e l H ~~ )({ x{ ), ( ; t = a + i v ) , f o r la r g e v a lu e s o fIx ] , a l l v a l u e s o f v , an d - 8 9 1 8 9 In w 1 4 w e sh a l l d e r i v e t h e L 2 r e su l t (12 .4 ).: N ex t, i n w 1 5 w e sh a l l p ro v e t h e L I r e su l t (12 .5 ). W e t h en o b t a i n t h e g en e ra ll o c a l i z a ti o n t h e o r e m s i n w 1 6.

1 3 . A y m p t o t i e f o r m u l a f o r H x(~)( u )W e co n s i d e r t h e fu n c t i o n H(~ ) (u) d e f i n ed b y

H i ~ ' ( I x l ) - - f ( 1 - l y l 2 ) ~ ( ~ - l ) + ~ l Y l ~ e - ' X ~ d y , ; t = a + i v , (13.1)[Yl


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1 0 9T h u s w e m a y

L O C A L I Z A T I O N A N D S U M M A B I L I T Y O F M U L T I P L E F O U R I E R S E R I E S

w r i t t e n a s a n a p p r o p r i a t e F o u r i e r - B e s s e l t r a n s f o r m , ( se e [ 5] , p . 6 9 .)w r i t e ( 1 3 . 1 ) a s ]

H ~k) (u) = (2 ~) tk u- 89 f (1 - t 2 ) 8 9 t t k +a Jt ( k 2) (u t ) d t . (13 .2 )0

W e s h a l l p r o v e t h e f o l l o w i n g . .T H E O a E M 1. L e t 2 = a + i v , 8 9 1 89 u > ~ l . T h e n

H i k ) (u ) = A ( ~ ) u- ~ - ~ + A ( ~ 2 )u - k -~ c o s u + A ( ~ 3 )u - k ~ s i n u + R ( ) , u ) u -~ -~ ~, (13 .3 )I A(~')I 0 . The s a me re ma rks c a n be ma de fo r mo s ts imi la r e s t ima tes in th i s paper .


18/55

1 1 0 ELIAS M. STEINr

0

H e r e , A ~ ( 4 ' ) = A ( h " - ' ( r etc.C o n c l u s i o n : [ ~ ( u ) [ < w k ( 2 n ) - i ~ M u - k -2 , i / u > ~l .

a n d

t h e n

H e r e , e ok i s th e ( k - 1 ) d i m e n s i o n a l v o l u m e o ] t h e u n i t s p h er e .P r o o ] o / t h e l e m m a . W e s h a ll m a k e u s e o f t h e fo l J o w i n g f a c t ( q u o t e d a b o v e ) : L e t,

/ ( x ~ . . . . . x ~ ) = r ( q ) , o 2 = x ~ + . . . + ~,F ( y 1 . . . . y k ) = ( 2 Z t ) - 8 9 f e '~ U ( x 1 . . . . x k ) d x ,

B k

F (Yl . . . y~) = ( I) (u) , z ~U = Y l - f- . - . + y k -I t i s a l s o w e l l k n o w n t h a t A ( r i s t h e s t a n d a r d k - d i m e n s i o n a l l a p l a c e a n o f / ( xx . . . . x ~).H e n c e

1 ~ 1 2 ~ 1 6 2 - ~ f ~ " ~ A o ( t ) d x IEk

< ( 2 ~ ) - ~ f I B koo

0= eo k ( 2 ~ ) - t k M .

T h e r e f o r e , I ( b ( u ) ] ~< o )k ( 2 ~ ) - 89 M u -2q ~< wk M (2 ~ ) - tk u k-~ ,i f u> ~ 1 . T h i s c o n c l u d e s t h e p r o o f o f t h e l e m m a .

L E M M A 7 . L e t ~ = ~ + i z l , ~ > ~ - 8 9 a n d u > ~ l . T h e nf e - ~ t t ~ J ~ ( k - z ) ( t u ) d t = B ; . u - l k - c - 1 + R ~ o ( ~, u ) u -8 90

( 1 3 . 4 )

w h e r e ] B r 8 9 a n d ] R ( x ) ( $ , u ) l < A ~ e " l' ' .P r o o ] o / t h e l e m m a . W e m a k e u s e o f t h e f o l lo w i n g k n o w n i d e nt i t ie s :

f e - t t " J m ( t u ) d t = ( l + u 2) t("~x) F ( m + n + l ) p , m ( ( l + u 2 ) - t ) ,0

( 1 3 . 5 )a n d


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L O C A L I Z A T I O N A N D S U M M A B I L I T Y O F M U L T I P L E F O U R I E R S E R I E S 11 12 0p ~ m ( co s 0 ) F ( m + 8 9 ( 27 e) ~ ( si n 0 )-m [ c o s ( n + 89 [ cos ~ - cos O] m-~ ' dv2 , (13.6)

0 fo r 0 < 0 < ~ .T h e f i r s t i d e n t i t y m a y b e f o u n d i n [2 ], p . 29 , f o r m u l a 6 ; t h e s e c o n d m a y b e

found i n [1 ] , p . 159 , fo rm u l a 27 .I n t h e a b o v e f o r m u l a w e s h a ll l e t n = 8 9 a n d m = 8 9 ( k - 2 ) . N o w d e fi ne

B e b yl ) t ( k - ~ ) ( 13 . 7 )r ( k + ~ ) , ~ + ~ (0).

B y c h oo s in g 0 = 89 in (13 .6) we eas i ly see th a t[Br t"tnt, $ = ~ + i ~ . ( 1 3 . 8 )

B y f u r t h e r i n s p e c t i o n i n ( 1 3 . 6 ) , w e m a y s e e t h a tp~(~- ~)I F ( k + $ ) t ~ + r ' (x)[


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1 1 2 E L I A S M . S T E I NL ~ M M . ~ 8 . L e t ~ = ~ + i ~ , ~ > ~ - 8 9 u > ~ l . T h e n

1f ( 1 - t2) ~(~-~ t ~ J t (k -2 ) ( tU ) dt0

= ( C ~ s in u + D ~ c os u ) u - t k - ~ 1 + R ( 2 ) ( ~ , u ) u - 89 -~ ~, ( 1 3 . 1 4 )w h e r e ] C r < A ~ e t ' l n l , [D e[ < A ~ e t'l 'l ,a n d ]R (~) ($ , u ) [ ~< A s e"l ' j .

P r o o / o / t h e l e m m a . W e u s e t h e i d e n t i t y1u n - m J m ( u ) 2 m - n - 1 Y ( m - n ) = [ ( 1 - ~ ) ~ - n - 1 J n (~o u ) ~n+l d Q ,

0( m - n ) > 0 , ( n ) > 0 , ( 1 3 . 1 5 )

w h i c h m a y b e f o u n d i n [ 2] , p . 2 6. W e a ls o u s e th e a s y m p t o t i c e x p a n s i o n

J , + t , ( u ) = u - t c o s u - ~ - ( # + z r ) ~ + R ( 3 ) ( # + i r , u ) , ( 1 3 . 1 6 )

w h e r e [ R ( S ~ ( # + i v , u ) ] < ~ A , e 'l V l u - t , i f u > ~ l .T h i s a s y m p t o t i c f o r m u l a m a y b e f o u n d i n [1 ], p . 8 5 . W e t h e n t a k e m = k - ~ + ~ ,

n = 8 9 # = k - ~ + ~ , a n d ~ = ~1 . A s t ra i g h t - f o r w a r d c o m b i n a t i o n o f ( 1 3. 15 ) a n d( 13 .1 6 ) le a d s d i r e c t l y t o ( 1 3. 14 ) a n d t h e p r o o f o f t h e l e m m a .

P r o o [ o / T h e o r e m 1 . C o n s i d e r t h e i n t e g r a l[1f (1 - t2) t(k-1)+~ t | k+~ Jt (k -e) ( tU) d t ,0

~t = a + i t . ( 1 3 . 1 7 )

T h e m a i n c o n t r i b u t io n s t o i ts a s y m p t o t i c e x p a n s i o n w i ll b e d u e t o t h e "s i n -g u l a r i t i e s " o f t h e e x p r e s s i o n ( 1 - t ~ )l (k - 1 )+ a t t~ + ~ , a t t = 0 , a n d t = 1 . F o r t h i s r e a s o nw e s e p a r a t e t h e t w o c o n t r i b u t i o n s a s f o ll o w s .

L e t ~ ( t ) E C oo (0 , 1) , w i t h ~o (0 = 1 , i f 0 ~< t ~< ~ , a n d ~0 ( t ) = 0 , i f ~ ~< t ~< 1 . T h e n w r i te1f (1 - t 2 ) ~ ( k - 1 ) + j l t ~k+~ Ji (*-~) ( tu ) d t = I 1 + 12, ( 1 3 . 1 8 )

0

!w h e r e 11 = f ( 1 - t~) 89 t i~+a ~o (t) J t(~ -2) (t u) d t, ( 1 3 . 1 9 )0


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LOCALIZATION AND SUMMABIL1TY OF MULTII~LE FOURIER SERIES 1131

a n d 12 = f (1 - t2) t(~-1)+~ t ~k+~ [1 - ~v (t)] J t ( k - 2 ) ( t U ) d t, (13.20)t

T h e i n t e g r a n d i n (1 3.1 9) h a s n o w o n l y o n e " s i n g u l a r i t y " , a t t = O . I n o r d e r t oo b t a i n a n a s y m p t o t i c e x p a n s i o n f o r i t w e s h a l l c o m p a r e i t w i t h t h e i n t e g r a l i n ( 13 .4 )w h i ch d i sp l ay s t h e s am e s i n g u l a r i t y . S i m i l a r l y , ( 13 .2 0 ) w i ll b e co m p are d w i t h (13 .1 4 ).Cons ider (13 .19) f i r s t .

L e t q b e t h e s m a l l e s t i n te g e r s o t h a t 2 q~> k + 2 . D e f i n e a p o l y n o m i a l P ( t) o fd e g r e e 2 q b y t h e f o l lo w i n g p r o p e r t ie s :

2 qP (t) = 1 + ~ aj tJ;i -4

i f w e s e t ( t ) = e - t P (t) - (1 - t~) (k-1)+ a ~p (t), (13.2 1)t h e n 6(n~ ( 0 ) = 0 , O < ~ n < ~ 2 q . (13.22)

I t i s c l e a r t h a t t h e co n d i t i o n s (1 3.2 2 ) d e t e rm i n e t h e co e f f i c i en t s a j co m p l e t e l y .B e c a u s e t h e s e c o n d i t i o n s i n v o l v e t h e d e r i v a t i v e s u p t o o r d e r 2 q o f W ( t ) ( 1 - t~) ~(~-l~+a,t h e n

l a~[~< A (1 + 12 I~q). (13.2 3)B y T a y l o r ' s t h e o r e m w i t h t h e r e m a i n d e r , (1 3.2 2) , a n d (1 3.2 3) it a l s o f o l lo w s t h a t

I O ," '(t)l


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1 1 4

N o w

:ELIAS M. STEIN

[ A x ( u ) l ~ < A ' ( l + 1 2 [ 4 q ) u - 8 9 u > ~ l . ( 1 3 . 2 7 )

] R ~4 ) ( 2 , u ) I ~< A ( 1 + [ 2 1 4 q ) e"~TI~< A e~"1.1,[ Ba[


23/55

L O C A L I Z A T I O N A N D S U M M A B I L I T Y O F M U L T I P L E F O U R I E R S E R I E S 1 1 5W e a l s o r e c a l l t h a t 1 - ~p ( t) = 0 , f o r 0 ~< t ~< ~ .

T h u s ( $ ~ ( t ) = 5 2 ( t ) = Q ( 1 - t 2 ) , f o r 0 ~ < t ~ < ] ,

a n d t h e re f o r e f [ A q ( ( ~ ( ~ ) ) I ~ k 1 d ~) ~< A ' (1 + 1412q). (13 .34 )0

N o w c o n s i d e r A 2 ( u) d e f i n e d b y1

A 2 (u ) = I a - f Q (1 - t ~) (1 - t~) i (~ 1)+~J t ( k - 2 ) ( t u ) d t .0

( 1 3 . 3 5 )

B e c a u s e o f t h e d e f i n i t i o n s ( 13 .2 0 ) a n d ( 1 3. 3 0) w e h a v e1 o9

- A 2 ( u ) = f 5 2 ( t ) t ~ k j i ( ~ _ 2 ) ( t u ) d t = f 5 ~ ' ( t ) t t k j t ( ~ _ ~ ) ( t u ) d t .0 0

( 1 3 . 3 6 )

W e s e e b y ( 1 3 .3 3 ) a n d ( 1 3 .3 4 ) ( a n d t h e f a c t t h a t ~ ( t) = 0 f o r t> ~ 1 ) t h a t ($~' t) s a t i s f ie st h e c o n d i t io n s o f L e m m a 6 , w i t h M = A ( 1 + 1 41 2~ T h u s ,

[ A 2 ( u ) I < A ' ( 1 + ] 4 [ ~ ) u - ~ " 3 , u> ~ 1 . ( 1 3 . 3 7 ): N o w

1f Q (1 - t 2) (1 - t2) t (k 1)+4 t i k j t ( k _ 2 ) ( t u ) d t0

1 2q 1= f (1 - t2) 89 ~~ t I k J t ( k - 2 ) ( t u ) d t + ~ b j f (1 - t 2 ) | ( k - 1 ) + 2 + ! t t k J i ( k - 2 ) ( t u ) d t .0 t -1 0

W e m a y n o w a p p l y L e m m a 8 , w i t h ~ = 4 , ~ t+ 1 . . . . 4 + 2 q t o t h e a b o v e . Ac o m b i n a t i o n o f t h i s a n d ( 1 3 .3 7 ) g i v e s u s a n e s t i m a t e f o r (1 3 .3 5 ) . I t i s

1 2 = ( C ~ s i n u + D ~ c o s u ) u - t ~ - ~ - ~ + R (s~ ( 4 , u ) u - t ~ - ~ - a , ( 1 3 . 3 8 )w h e r e I C a ] < A e~ '~ l , I Da [ < A e~ M , [ R ~ , ( 4 , u ) I < A e~ ' ' j ,

4 = ( r + i % - 8 9 1 8 9 a n d u ~ l .I f w e c o m b i n e ( 13 .3 8 ) w i t h t h e a s y m p t o t i c e x p a n s i o n f o r 11 i n ( 1 3 . 2 9 ) w e o b t a i n

t h e a s y m p t o t i c e x p a n s i o n f o r ( 1 3 . 17 ) . W e a l s o n o t i c e t h a t t h e f u n c t i o n H (~ ) ( u) ( d e-f i n e d i n ( 13 .2 )) d i f fe r s f r o m ( 1 3. 17 ) o n l y b y a f a c t o r ( 2 ~ ) ~ k u - i ~ + l ; t h u s t h e p r o o f o fT h e o r e m 1 i s c o m p l e t e .


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1 1 6 m x a s M . S TE IN

9 1 4 . T h e L 2 e s t i m a t e

L e t ] ( x) b e i n t e g r a b l e o v e r t h e f u n d a m e n t a l c u b e Q = Q k, a n d l e t i t b e p e r i o d i c .D e f i n e U ~a ( x ; / ) b y

U ~ ( x ; / ) = E a , 1 - - I ~ l ~ , ~ ~ , 2 = a + i v , - - 8 9 1 8 9 ( 1 4. 1)O < l n l < n - R ~ - ]T h e a . a r e t h e F o u r i e r c o e f f ic i e n ts ; a . ~ ( 2 z~ ) - k f / ( x ) e - ~ ' x d x . T h e m a i n r e s u l t

oo f th i s s e c t i o n w i l l b e t h e f o l l o w i n g :T ~ ~ 0 R ~ ~ 2 . L e t 1 >1 e >~ O, t / ix ed . L e t / ( x ) 6 L 2 ( Q ) , a n d a s s u m e t h a t / ( x )

v a n i s h e s i n t h e s p h e r e I x IO

W e a l s o h a v e t h e e s t i m a t eI B ~ . ,l


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LOCALIZATION AND SUMMABI LITY OF MULTIPLE FOUR IER SERIES 117T h e r e f o re b y t h e A b e l - s u m m a b i l i ty o f t h e F o u r i e r i n v er s io n w e h a v e th a t t h e

f o l l o w i n g l im ( 2 ~ ) - k f H ~ ( I x [ ) e ' ~ U e - n l ~ ' d x (14 .5 )~--~0 F~k

c o n v e r g e s u n i f o r m l y i n y , i f lul lS>0; m o r e o v e r , t h i s l i m i t i s ( 1 - ] y ] U ) t ( ~ - i ) + a l y l a ,i f l y ] 4 l , z e r o o t h e rw i s e . B y a c h a n g e o f v a r i a b l e w e t h e n h a v e , f o r e a c h f i x e d y ,l u l>0 ,

l i r a (2 z ) -k " 8~ +a f H ~ (8 ]x [) e xu e - " lz l d x (14 .6 )v--*0 Ek

c o n v e r g i n g u n i f o r m l y i n 8, O < a < ~ s < b < oo , t h e l i m i t b e i n g ( 1 - 1 y [ 2 ~ 2 ) ]( k- 1) +~ l l y ] jl,i f [ y] ~< 8 , z e r o o t h e r w i s e . N o w i n t h e a b o v e , l e t I x ] = ! , a n d y = n , ( w h e r e n i s av e c t o r w i t h i n t eg r a l c o m p o n e n t s , I n ! , 0 ) . B e c a u s e o f t h e u n i fo r m c o n v e rg e n c e i n( 14 .6 ) w e m a y i n t e g r a t e t h e e x p r e s s io n i n a < . s < ~ b , a f t e r m u l t i p l y i n g b y ~ p (s ), a n di n t e r c h a n g e l i m i t s. T h u s w e o b t a i n ( 1 4 .4 ) i n t h e c a s e ! ( x ) = e ~ ~ , ] n ] * 0 . A f in i t el i n e a r c o m b i n a t i o n o f s u c h m o n o m i a l s w i ll c o m p l e t e t h e p r o o f o f t h e l e m m a .

L ] ~ M ~ A 1 0 . G i v e n a ! i x e d e , l ~ > e > 0 ; a ss um e th at ! ( x ) e L a ( Q ) a n d ! ( x ) = O i tI x l


26/55

1 1 8 ~ L ~ . S M . S T E IN

W e s h a l l f i r s t s h o w t h a t i f / ( x ) i s a n e x p o n e n t i a l p o l y n o m i a l a n d f / ( x ) d x = 0 , t h enQ

i f R ) I ,I U * n ~ ( O , / ) l ~ o ~ l , ( 2 ~ t ) - ~ ' ] f H ' ~ ( R , t ) [ ( O , t ) t k - l d t + A ~ e3 "IT I[ I[ II, , (14.11)0

w i t h A , , i n d e p e n d e n t o f R . F o r t h i s p u rp o s e , w e b r e a k u p t h e r a n g e o f i n t e g r a t i o ni n (14 .1 0) i n t o t h e i n t e rv a l s (0, e ) an d ( e, o o ). I t i s t h e re fo re su f f ic i en t, t o sh o w t h a t

ooI f ( R , t) / (O , t ) d r I ~ e, 2 - - a + i v , a n d w i t h A ~ ) d e p e n d i n g o n R , b u t I A ~ 4 ) I ~ A e ~t~L w i th Ai n d e p e n d e n t o f R .

I n f a c t , a c c o r d i n g t o T h e o r e m 1 , t h e f i r s t t h r e e t e r m s o f t h e a s y m p t o t i c ex -p a n s i o n o f H ~k) (u) ar e

A(~t) u- k - ) . + A(~) u -k - ) , co s u + A ( ~ ) u k - ~ . s in u .A p p l y i n g f o r m u l a ( 14 .9 ) t o t h e f i r s t t e r m a b o v e , w e o b t a i n

1

0( R + s ) k + ) . ~ b ( s ) t - k ~ ( R + s ) k ) . d s

1 8 ~ k ~ 1 ~ 2 ) .= A ' ) t " f ( i + R ) ~ b , s ) d s .0

T h i s l a s t i s t h e t e r m A(x4)t k-~" w h i ch ap p ea r s i n (1 4 . 1 3 ) , w i t h1 8 ' ~ k - - 1 + 2 2A (4)= A~ ) , [ ( 1 + ~ ) ~ ( s ) d s .

0

G o i ng o v e r t o t h e s e c o nd t e r m o f t h e a s y m p t o t i c e x p a n s io n , A~e)u '~ ).t h en t h e co n t r i b u t i o n i n (1 4 . 9 ) i s

1 8 ) k - 1 + 2 ) .A ~ ) t - k - ) . f 1 + ~ ( R + s ) ~ + ~ ' ~ ( s ) ( R + s ) - k ~ ) . c o s [ t ( R + s ) ] d s .0

C O S U ,


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LOCALIZATION AND SUMMABILITY OF MULTIPLE FOU RIE R SERIE S 11 9

M a k i n g u s e o f t h e f a c t t h a t ~ ( s ) E C 1 , w e i n t e g r at e t h e a b o v e i n t e g ra l b y p a r t sa n d o b t a i n t h a t i t i s 0 ( t 1 ) u n i f o r m l y i n R , R > ~ 1 .

T h u s t h e e n t i r e c o n t r i b u t i o n o f A ~ ) u - k -a c o s u is i n c o r p o r a t e d i n th e r ig h t - h a n ds i de o f (14 .13) .

A s i m i l a r a r g u m e n t i s a p p l ie d t o t h e t e r m A ~ ) u - k - a s in u . F i n a l l y , t h e r e m a i n d e rt e r m , R 0 ,, u ) u - k= a =l , o f t h e a s y m p t o t i c e x p a n s i o n i s a l so d i r e c t l y i n c o r p o r a t e d i n t h er i g h t - h a n d s i d e o f { 1 4 . 1 3 ) .

H e n c e ( 1 4 . 1 3 ) i s d e m o n s t r a t e d .N o w i t i s e a s y t o s e e t h a t

o o

f t - k - ~ - ' l / ( O , t ) I t k - i d t < - . . A ~ l l / l l l i f - 8 9 s a y .$

T h u s i n o r d e r t o c o n c l u d e th e e s t i m a t e ( 1 4 .1 2 ), w e m u s t e s t i m a t e t h e q u a n t i t yoOA~4 .) e - n t t - k - k ] ( 0 , t ) t k 1 d t . (14.14)e

B y c h a n g in g b a c k t o t h e c a r t e s i a n c o o r d in a t es x = ( x , . . . . . x~ ) i n E u c l i d e a n s p a c ek - s p a c e , w e m a y w r i t e ( 1 4 . 1 4 ) a sA i 4, ( o k ) - ' f e - 'l zl [ x [ - k - a / ( x ) d x . (14 .15)

W e r e c al l t h a t / ( x ) i s a n e x p o n e n t i a l p o l3 r n om i a l , p e r i o d ic o v e r t h e f u n d a m e n t a lc ub e Q , - z e < x ~ < g , i - - 1 . . . . . k ; a n d t h a t I ] ( x ) d x = 0 "

L e t n o w Q ~ d e n o t e t h e t r a n s l a t i o n o f t h e c u b e Q b y t h e v e c t o r 2z e n , w h e r en = ( n , . . . . . n ~ ) , n , a r e i n t e g e r s . T h u s Q ~ = Q + 2 ~ n , a n d E k = U ~ Q " , w h e r e t h e u n i o nr a n g e s o v e r al l i n t e g r a l c o m p o n e n t v e c t o r s . T h u s e x c e p t f o r t h e c o n s t a n t A ~4) (oJk) 1,w e m a y r e w r i t e ( 1 4 . 1 5 ) a s

f e - ~ l ~ [ x [ k ~ ' / ( x ) d x + Y Y , , f e - " : ~ [ x [ ' ~ a / ( x ) d x , (14.16){[x[>~e}nQ Qnw h e r e Z ' i n d i c a te s t h a t w e s u m o v e r a ll n , w i t h I n [ ~ : 0 . S i n c e f / ( x ) d x = O , t h e nQn

/ [ , I (14 .17)Qn Qn


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12 0 E L I A S M . S T E I NN o w i t i s a n e a s y m a t t e r t o v e r i f y t h a t i f x E Q " t h e n

I ~ , ~ ~ I ~ 1 - ~ - ~ - ~ - , ~ I n l ~ ~ 1 ~ < A [ 1 + I ~ 1 ] I n l - ~ - ~ - ~0 < ~ < 1 , 2 = ( ~ + i v , - 89

T h e r e f o r e b y ( 14 .1 7 )I f ~ - " ' " I ~ 1 - ~ - ~ / ( ~ ) a x [ ~ < A I ~ l ' l n l - ~ - ' ~ f I / ( ~ ) 1 a - ,Qn Q i f I ~ 1 " 0 ,

s i n ce f l / ( x ) l a ~ = f l / ( ~ ) l a ~ . T h u s t h e i n f i n i t e su m a p p e a r i n g i n ( 1 4 .1 6 ) is e s t i-Qnm a t e d b y A I ~I ( X ' I n l - k- l- ~) f l f l d x .QS i n c e - 89 t h e n c e r t a i n l y Z ' [ n [ - k - 1 - ~ T h e f ir s t m e m b e r o f ( 1 4 . 1 6 ) i sc l e a r ly e s ti m a t e d b y A , f I t ( x ) I d x . C o m b i n in g t h e s e t w o e s t i m a t e s , w e o b t a i n a s a n

Qe s t i m a t e f o r ( 1 4 . 1 6 )

A . ( I + I ~ I ) f l t ( x ) l d x .0

W e t h u s o b t a i n t h e e s t i m a t e f o r ( 1 4 . 1 5 ) , a n d t h e n v i a ( 1 4 . 1 4 ) w e a r r i v e a t t h ee s t i m a t e ( 1 4. 1 2) . ( H e r e w e u s e d (1 + ] 3 ] ) e2~1"1~< es= l' l. ) He nc e t he pr oo f fo r (14 .11) i s co m-p l e t e d , w h e n / ( x ) i s a n e x p o n e n t i a l p o l y n o m i a l , a n d f / ( x ) d x - - 0 . A s i m p l e l i m i t i n g

O

a r g u m e n t ( k e e p i n g R f i x e d ) s h o w s t h a t ( 1 4 .1 1 ) st il l h o l d s if / ( x ) f i L l ( Q ) , a n df / ( x ) d x = 0 . I f w e n o w a s su m e t h a t / ( x ) v a n i s h es i f [x [~ < e , t h e n / ( 0 , t ) = 0 i fO0 ~ < t~ < e . T h e r e f o r e ( 14 . 11 ) b e c o m e s

I u * " ( 0 , 1 ) 1 . < A . ~ 3 ~ r I1 1 1 11 (14.18)w i t h A , i n d e p e n d e n t o f R , a n d / ( x ) a s s u m e d t o v a n i s h f o r I x [ ~ e . T h e a b o v e c o m -p l e t e s t h e p r o o f o f t h e l e m m a .

C O R O L L A R Y . T h e c o n c l u s i o n e l L e m m a 10 s t i l l ho lds i / we drop the assu mp t iontha t f / (x) d x = O.

Q

P r o o / . C h o o s e g ( x) a s a f i x e d p e r i o d i c f u n c t i o n o f c la s s C ~ w i t h p r o p e r t i e sg ( x ) = O , f o r I x l< l , a n d fg (x )dx=l . A p p l y L e m m a 10 t o th e f u n c t i o n

0

h ( x ) = / ( ~ ) - g ( x ) f / ( x ) d ~ .0


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LOCALIZATION AND SUMMAB ILITY OF MULTIPL E FOURIE R SERIES 12 1T h e n c l e ar l y f f x ( x ) d x = 0 , a n d i t x ( x ) = 0 i f [ x[ ~< e .QM o r e o v e r , U ~ ~ ( 0 , i t 1 ) = U ? r ( 0 , i t ) - ( f i t ( x ) d ~ ) V * ~ , -~ - , g ) -Q

rr*~ t oI t i s e a s y t o v e r i f y t h a t ] ~ n ~ , g ) [ ~< A , b y t h e a b s o l u t e c o n v e r g e n c e o f t h e F o u r i e re x p a n s i o n o f g ( x ) . T h u s

] U ~ (0, jt)[ < A~ ea 'l' l ][ it [[1 + A II it Ih,a n d t h e c o r o l l a r y i s p r o v e d .

Proof oi t Theorem 2 . W e f i x t h e f u n c t i o n ~ (s ) a p p e a r i n g i n (1 4.7 ) o n c e a n d f o ra l l, a s fo l lo w s : L e t ~0 ( s) b e t h e p o l y n o m i a l o f d eg ree 2 k - 1 w h i ch s a t i s fi e s :

1 1f c f ( s ) d s = l , f q ~ ( s )s 'd s = O , l < i < 2 k - 1 . ( 1 4 . 1 9 )0 0

W i t h ~ (s ) so d e f i n e d , t h e p r o o f o f T h e o r e m 2 w i l l b e c o n c l u d e d a s s o o n a s w es h o w t h a t

I UaR (0, /) - U*n (0, i t )[< A ea" l'l [[ it , , R >/2, (~4.20)w h e r e it i s a n y f u n c t i o n i n L 2 ( Q) , a n d A i s i n d e p e n d e n t o f R .

W rit e / (x ) , .. Z an e"~'z. T h e nZ ] a . [ ' = (2 . ) -~ f I / ( ~ ) I ' ~ -Q

W e r e c a l l t h a t t h e n u m b e r o f l a t t ic e p o i n t s i n t h e s p h e r i c a l s h e l l c o n t a i n e d b e -t w e e n s p h e re s o f r a d i u s R - 1 , a n d R + 1 , i s 0 (R ~ - I) . T h u s a n a p p l ic a t io n o f S c h w a r z ' si n e q u a l i t y y i e l d s :

i sb e ' w r i t t en a s

1( R ~ - n ~ ) J ( ~ - a ) + a l n ] a a . - f { ~ ( ( R + s ) 2 - n 2 ) t ' ~ - x ) + a [ n [ a a . } c p ( s ) d s .l~


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122 ~rLr~S M. STEINW e sha l l w r i t e (14 .22) (o r 14.23 ) i n t he fo rm S 1 + S z , w he re S z i nvo l ves a l l t e rm s

w i t h R - 1 < i n I < R + 1 , a n d S 1 t h e r e m a i n i n g t e r m s o f t h e s u m s . N o w i f R - 1 < I n ],t h e n c l ea r ly ( R 2 - n 9 ) ~< 2 R ; a n d s i m i la r l y ( R + s ) ~ - n z ~ < 4 R . T h u s f o r S 2 w e h a v et h e f o l l o w i n g e s t i m a t e

[S~ [ ~< A R 89

Because of (14 .21) , t h i s becomesI ~ l ~ l ~ l . (14.24)R - I ~ < [ n l < R ~ - 1

(14.25)

( R 9" - n Z ) t ( k - 1 ) + ; t - ((R + s) 2 n 2 ) 8 9 = c I 8 ~ - c 2 8 2 . . . C 2 k - 1 8 2 k 1 0 2 k ,

I o ~ [ < ~ [ s u p [ (( R + ~ ) ~ - ~ ) ~ ( ~ -1 , + , .] (~ , [ .h e r e 0 < ~ s < lT h e n i t is a n e a s y m a t t e r t o v e r if y , i f I n [ < R - 1 , t h a t

Io~I< A[I+ IvI~],~ -~-~+~", a=a+ iv , - 89

(14.26)

N o w by T a y l o r ' s expan s i on , i f 0 ~< s~ < 1 ,(14.27)

S u b s t i t u t i n g ( 14 .2 7) i n ( 14 .2 6) , a n d u s i n g t h e o r t h o g o n a l i t y r e l a ti o n ( 1 4 . 1 9 ) w e o b t a i nI s l l - < A [ l + l ~ l ~ ] 5 I ~ ] t ~ + 2 ~ 9l ~ < ] n l < R - 1

I f w e n o w u s e S c h w a r z ' s i n e q u a l i t y , a n d t h e f a c t t h a t - 8 9 < a ~< 0, we o bt ai nIS 1 [~


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L O C A L IZ A T IO N A N D S U M M A B IL IT Y O F M U L T IP L E F O U R IE R S E R IE S 123

s u p I v ~ ( 0 , / ) l ~ < A . e ~ * ' I I / 1 1 ~ + A e ~ ' ' ' I I/ 1 1 ~ ~ < A : e ~ ' * ~ 1 1 /1 1 2.R~>2S i n c e t h e a b o v e i n e q u a l i t y i s tr i v i a l if 0 < R < 2 , w e h a v e t h u s c o n c l u d e d t h e p r o o fo f T h e o r e m 2 .

1 5 . T h e L 1 e s t i m a t e

L E M M A 1 1. L e t / ( x ) e L l ( Q k ) , a n d a s s u m e t h at 8 9 t hen

U~ (x, [ ) = cok (2 7t) - k R ~ ~ ~ H I (R t ) [ (x; t ) t k- x d t , (15.1)0

t h e i n t e g r a l c o n v e r g i n g a b s o l u t e l y . T h e q u a n t i t y U ~ ( x , [ ) i s d e [i n e d i n (14 .1) , a n d t h ek e r n e l H ~ ( u ) i s d e [ i n e d i n (13.1) .

P r o o f . U s i n g t h e a s y m p t o t i c e s t i m a t e o f T h e o r e m 1 , w e s ee t h a t f o r f i x e d R ,t h e k e r n e l R k + ~ H ~ ( R I x I) i s a b s o l u t e l y i n t e g r a b l e o v e r E k, w h e n e v e r R ( ; t ) > 0 . T h u sw e m a y c o n v o l v e R k w i t h a n a r b i t r a r y p e r i o d i c i n t e g r a b l e f u n c t i o n / ( x ) ,a n d t h e u s u a l m u l t i p l ic a t i o n f o r m u l a f o r t h e F o u r i e r c o e f fi c ie n t s h o l d s. T h e F o u r i e rt r a n s f o r m o f R ~ i s i m m e d i a t e l y d e d u c e d f r o m ( 1 3.1 ), a n d f r o m t h i s t h ep r o o f o f L e m m a 11 is c o n c l u d e d .

T H E O R E M 3. L e t e be [ i xed , e > 0 . A s s u m e th at [ ( x ) E L I ( Q ) a n d th at / ( x ) = O ,i / I x [ ~ ~tl (,~) > O, ,~ = a + i 3. T h e n

2~ r , 1su p I V~ (0, l ) l ~< A , . e " - " I I / I h (15 .2)R~>O O"

P r o o / . I f w e m a k e u s e o f ( 1 5 .1 ) a n d t h e f a c t [ ( O ; t ) = O , i f O ~


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124 ~ .L~S M. STEm

a n d

T h u s ( 1 5 . 4 ) b e c o m e s

a n d t h e r e f o r e

n + lf I / ( o ; t ) l d t < A I I / I h , n = l , 2 . . . . .n

[U~(O,/I OP ro o / . I t i s c l e a r l y s u f f i c i e n t t o p r o v e ( 1 6 .1 ) f o r S ~ ( ~- 1) ( 0) i n p l a c e o f StR k 1 ) ( 0 ),

w h e r eS~ (k-1) (0) = ~ a , e ' z 1 -0


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L O C A L I Z A T I O N A N D S U M MA B I L I T Y O F M U L T I P L E F O U R I E R S E R I E S 125N , w e p i c k a n a r b i t r a r y p o s i t i v e R o , a n d w e id e n t i f y N w i t h t h e

T h e f a m i l yT o d e f i n e

i n t e r v a l [ 0 , R o] , g i v i n g t h e s p a c e N t h e s t a n d a r d L e b e s g u e m e a s u r e .T z ( ' ) is n o w d e f in e d b yTz ( / ) (R ) = U ~ (0 , f ) , (16 .2 )

w h e r e 2 = 2 ( z) = ( p - 1 ) / 2 - z . p / 4 . (16.3)I n d e f in i ng U ~ ( 0 , / ) w e h a v e s et / = 0 , f o r I x l < e . F o l lo w i n g th e n o t a ti o n o fL e m m a 1 , w e sh a ll le t P 1 = 1 , T 2 = 2 , q t = q 2 = q = ~ 1 7 6 t w ill b e t h e p a r a m e t e r s o t h a t0 < t < l , a n d 1 / p = l - t + t / 2 .

( W e s h o u l d p o i n t o u t h e r e t h a t i t w i ll b e i m p o r t a n t t h a t e s t i m a t e s t h a t f o l lo wa r e m a d e i n d e p e n d e n t l y o f R 0. A t t h e c o n c l u s i o n o f t h e p r o o f w e s h a l l l e t R o - -> c ~ . )

I t is a n e a s y m a t t e r t o v e r i f y t h a t t h e f a m i l y o f o p e r a t o r s T z i s a n a n a l y t i cfa m i l y i n t h e se n se o f ( i) , ( if ), a n d ( ii i) o f w 8 .

W e s h a l l a l s o u s e t h e f o l l o w i n g n o t a t i o n , w h i c h s h o u l d n o t l e a d t o c o n f u s i o n :

M Qfl{Izl> ~}

w h e r ea n d

W e t h e n c l a i m t h e f o l l o w i n g b o u n d s o n T z (]):I I T,~ ( / ) I I , ~ < A o ( y ) I I f l I .

I I T I ( f ) I1 ~ < A 1 ( y ) l i f ] l ~ ,A o ( y ) < . ( A ~ / ( p - 1)) e"r~l,

A 1 (y) ~


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126 ELLIS M. ST E ~S i n c e 1 / p = 1 - t / 2 , t h e n ~ ( t) = 0 . T h u s

T t ( / ) (R )= U ~ (0 , / ) = S ~ (k ~) (0 ) .T h e r e f o r e ( 1 6 . 8 ) b e c o m e s

s u p I S ~+(k-l) ( o ) 1 < A ~ I I / I I . "O


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L O C AL IZ A TIO N A N D S U M M A B I L IT Y O F M U L T I P L E F O U R I E R S E R I E S 127s u p I ( O ) I B , f l / ( x ) I l o g + I ( x ) l d + B , .R>~0 Q (16 .13)

P r o o / . I f w e f ix R , t h e n T h e o r e m 4 i m p li e s t h a t f o r e v e r y s im p l e [, v a n i sh i n g( 0 ) [ < (AJ(v- 1 ) ) . 1 < v < 2 . ( 1 6 .1 4 )

A ~ i s i n d e p e n d e n t o f R , o f co u r s e .N o w a p p l y L e m m a 2 t o t h e a b o v e , w h e r e T ( / ) = S ~ ( k - 1 ) ( O ) ; t h e s p a c e N c o n -

s i st s o f t h e s i n g l e p o i n t w i t h m e a s u r e 1 ; a n d r = 1 . T h u s w e o b t a i nI S ~ k - l ) ( 0 ) ] < B ~ ( f I [ (x ) l l ~ + I [ ( x ) l d x + 1) , (16 .15)O

w h e r e B e i s i n d e p e n d e n t o f R . T h i s c o n c l u d e s t h e p r o o f o f t h e t h e o r e m i n t h e c a s ew h e n / ( x ) i s s im p l e . T h e g e n e r a l c a se t h e n f o ll o w s b y a s t a n d a r d l i m i ti n g a r g u m e n t .

W e a r e n o w i n a p o s i t i o n t o p r o v e o u r m a i n r e s u l t .T H E O R ~ . r a 6 . L e t / ( x ) e L l og + L ( Q ) , a n d a s s u m e t h a t [ ( z ) v a n i s h e s i n a n e i g h -

bor hood o f the po in t x o . Th en l ir a S in (k -a ) (Xo , ] ) ex i s t s an d i s z er o .R..-~ oO

P r o o / . W e m a y , a f t e r t r a n s l a t i o n , a s s u m e t h a t x o = 0 , a n d t h a t o u r n e i g h b o r h o o dco n t a i n s t h e s p h e r e I x [ ~ < e , f o r s o m e e , 0 < e ~< 1 .I t w i ll b e s u f fi c ie n t t o s h o w t h a t g i v e n a n y ~ / > 0 , t h e r e e x i s t s a n R o = R 0 (r/),

s o t h a t[S ~R (k 1 ) ( 0 , / ) [ < 7 ' w h e n e v e r R > R o ( r/) .

N o w f o r a n y ~ , 0 < $ , ( 16 .3 ) m a y b e r e w r i t t e n a sn>~oSUpS ~ (k -' )( O )I ~ < B ~ f l / ( x ) l l o g t [ - ~ ! d x + ~ e B ~ .

Q(16.16)

T h i s f o l l o w s b y w r i t i n g f ( x ) / ~ i n s t e a d o f / ( x ) i n ( 1 6. 3) . N o w c h o o s e ~ n o s m a l lt h a t ~ B , < ~ // 3, a n d k e e p ~ f ix e d .

N e x t w r i t e / (x ) = [1 (x) + / 2 (x) wh ere [x (x) E C ~ an d [1 (x) an d /2 (x) v an i she s int h e e - n e i g h b o r h o o d o f 0 ; a n d

B 6 f [ / 2 ( x ) [ l og + ( [ / - ~ x - ) - [ ) d x < ~ / 3 .Q

N O R ' S ~ ( k 1 ) ( 0 , l ) = S ~ ( k 1 ) ( 0 , 1 1) + S ~ ( k 1 ) ( 0 , / 2 ) "


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128 ELIAS M. STERNB e c a u s e f x i s s u f f i c i e n t l y s m o o t h a n d / ( 0 ) = 0 , t h e n S ~ ( k - l ' (0 , / 1 )- - > 0 , a s R - - > o o . T h u s

I s ' k i f R > R 0 ( ~ ] ) .H o w e v e r , b y a p p l y i n g ( 1 6 .1 6 ) t o ] 2 ( x) w e o b t a i n I S le k 1 ) ( 0 ,/ 2 )1 < 2 ~ ] /3 . C o m b i n i n gt h e s e t w o , w e g e t : ] S ~ k -~ ) ( 0, ] ) [ < ~], w h e n e v e r R > R 0 0 ] ). T h i s c o n c l u d e s t h e p r o o fo f T h e o r e m 6 .

C H A P T E R I I ID o m i n a t e d S n m m a b i l i t y

1 7 . A n L 1 e s t i m a t e f o r d o m i n a t e d s n m m a b i l i t yW i t h S a , ( x , / ) = ~ n ( 1 - L n R ~ ' ) ~ a , ,e ' n ~ ,

w h e r e a , = ( 2 ~ ) - k I f ( x ) e - ~ a ' ~ d x ,

w e d ef in e S~ . (x , { ) b y S~ . (x , f ) - - su p I S~n (x , f ) I - (17 .1}R >OT h e r e s u l t o f t h i s s e c t io n i s c o n t a i n e d i n t h e f o l l o w i n g :L ~ M ~ A 1 2 . Le t [ (x ) E L 1 (Q), an d le t ~ () > 89 k - 1 ) . A l so l e t ]* (x ) be the m ax i m al

/ u n d i o u d e /i n ed i n L e m m a 3 . T h e n( a ) ~ , ( x , l ) < A o e " ~1 ( a - 89 k - 1)) -1 /* ( x ) ,

Ok Ok

w h e r e A , i s in d e p e n d e n t o / R a n d / , a n d A , r e m a i n s b o u n d e d a s a - > 8 9P r o o ] . A c c o r d i n g t o L e m m a 5

oof T $ ( X , , / ) = C U ~ k - r f / ( z ; ~ ) ~ 8 9 J a + t k ( t R ) d t

0w i t h c l = 2a t k + 1 F ( 5 + 1 ) / F (89k ) .

W e b r e a k u p t h e a b o v e i n t eg r a l i n t o t w o i n t e g ra l s c o r r e s p o n d i n g r e s p e c t iv e l y t ot h e i n t e r v a l s ( 0 , l / R ) , u n d ( l / R , o o ) . U s i n g t h e e s t i m a t e ( 11 .1 1) fo r t h e f i r s t i n t e g r a l ,a n d ( 11 .1 0 ) f o r t h e s e c o n d i n t e g r a l , w e t h e n e a s i l y o b t a i n

(I) See footnote p. 109.


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L O C A L I Z A T I O N A N D S U M M A B I L I T Y O F M U L T I P L E F O U R I E R S E R I E S 129I/ R ooI ~ R ( ~ , / ) I < A o e ~ M { R ~ f l / ( ~ ; t ) lt ~ - Z d t + R t ( k - l ' - ~ f l / ( ~ ; t ) l t i ( k - x ' - ~ ~d t } 9 (1 7 .2 )

0 I l R( W e n o t e h e r e t h a t A ~ r e m a i n s b o u n d e d a s l o n g a s a > / 0, a n d a i s r e s t r i c te d to s o m ef i n i t e i n t e r v a l . )

N o w b y d e f i n i t i o n o f /* ( x) g i v e n i n ( 10 . 3) , w e h a v e1///

R ~ f I / ( x ; 0 I ~ ' - ~ d t < / * (x ) k - *.0M o r e o v e r , a s i m p l e a r g u m e n t o f i n t e g r a t i o n b y p a r t s g i v e s

oo

R ~ ' k - " - ~ f l / ( ~ ; t ) l t ~ ( ~ x ' - ~ - * d t < A [ ( 7 - 8 9 f o r a > 8 9I f w e c o m b i n e t h e s e t w o e s t i m a t e s i n ( 1 7 . 2 ) w e o b t a i n

S~, ( x , / ) = s u p I S ~ ( x , / ) l < A o e~l~l [a - 89 /r - 1)] -* /* (x ),Rw i t h A , b o u n d e d a s a - -> 8 9 / c - 1 ) . T h i s p r o v e s p a r t ( a) .

I n o r d e r t o p r o v e p a r t ( b) , w e m a y r e w r i t e ( 17 .2 ) i n t e r m s o f a c o n v o l u t i o nw i t h a n i n t e g r a b l e k e r n e l .

I n e f f e c t w e h a v eIS % (x , / )l < A . e = '~ ' R k f ] / ( x - y ) l r (17.3)

E k

w h e re ~ ( u ) = l , i f 0 4 u < l , a n d ~ ( u ) = u - " - t k ~ l , i f l ~< u .I n t e g r a t i n g ( 17 .3 ) w i t h r e s p e c t t o x a n d i n v e r t i n g t h e o r d e r o f i n t e g r a t io n , w e

o b t a i nf I s ~ ( ~ , / ) l d x < A o ~ " " { f I / (~ ) 1 d ~ } ~ f r

Q k O k E koo

H o w e v e r , R k f r 1 6 2 1 8 9E k 0

T h u s p a r t (b ) i s a l so p r o v e d , a n d t h e p r o o f o f t h e l e m m a i s c o m p l e t e .18 . A n L" es t ima te

T h e r e s u l t o f t h i s s e c t i o n is c o n t a i n e d i n t h e f o l l o w i n g t h e o r e m .T H E O R ] ~ M 7 : L et / (x ) E L 2 ( Q ) ; a n d ~ = a + i v . T h en

( a ) ( f l ~ , ( = , / ) l " d = ) ' < . A o ~ " " ' ( f l / ( = ) l ' d = ) ' / o , ~ > 0 ,Q Q( b ) ( f l s % ( ~ , / ) l ~ d x ) ' < . ( f l / ( ~ ) r d ~ ) ~ , / o r ~ > ~ o .Q Q


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130 ELIAS M. STEl~P r o o / . W e i n t r o d u c e t h e f o l lo w i n g t w o a u x i l i a r y f u n c t i o n s :

D ~(x' )=( f l~(x ' l ) -~E ' (x ' l)P dR )0

a n d A ~ ( x ' l ) = s u p { l I [ s ~ ; l( x ' (18 .2)0

(18 .1)

T h e p r o o f o f t h e t h e o r e m w i ll b e a c o n s e q u e n c e o f t h e f o l lo w i n g l e m m a .L ~ M ~ A 1 3 . L e t / ( x ) E L 2 ( Q ) , ~ ( 5 ) > ~ - , ( 5 = a + i ~ ) . T h e n D ~ ( x , l ) a n d A o ( x , l )

are ] in i t e a lmo s t everywhere ; mo reover( f E a ~ ( z , l ) l ~ x ) ' . < A o ( f i i ( ~ ) l ~ d x ) ' ( 1 8 .3 )Q Q

a n d ( . I [ A ~ ( x , 1 ) ] ~ d x ) t ~ A ( , d 1 ' ' ( f I 1 ( ~ )1 ~ d ~ ) ~ . ( 1 8 . 4 )Q Q

P r o o / o / t h e l em m a : W e c o n s i d e r ~ f i r st . S i n c e[ ~ ( x , / ) ] 2 = ~ 1 s % ( x , / ) - 8 % 1 (~ , / ) IR - d R ,

0

W e i n t e g r a t e w i t h r e s p e c t t o x a n d i n t e r c h a n g e t h e o r d e r o f i n t e g r a t i o n . T h u sor

Q o Q(18.5)

W e n o w e v a l u a t e t h e i n n e r i n t e g r a l b y P a r s e v a l ' s f o r m u l a .

S 6 R ( Z , / ) - - S ~ R I ( x , f ) = nl~


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L O C A L IZ A T I O N A N D S U M M A B I L I T Y O F M U L T I P L E F O U R I E R S E R I E S 131S u b s t i t u t i n g t h i s i n (1 8 .5 ) w e t h e n h a v e

f [ f 2 o ( x , / ) ] 2 doo { . l l ' l a n l }0

Inl< A Z l a ~l 2.T h i s g i v e s f [ f~ o ( x , / ) ] 2 d x < A f l / ( x ) l ~ d x , i f a > ~ l ,

Q O

a n d t h e r e f o r e ( 1 8. 3 ) i s p r o v e d i n t h i s c a s e .A s s u m e n o w a < 1 . T h e n1 - - ~ - ) < A ,

U s i n g ( 1 8 . 6 ) w e o b t a i n( 2 ~ )- ~ f i s ~ ( ~ ' / ) _ ~ , - 1 ( ~ , / ) [ ~ d ~

Q~ A ~ " I n l 4

Inl~


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132 ~.~As M. sTEn~B u t i t i s e a s y t o c h e c k t h a t

21hif R ' - ~ ( R - ] n l ) 2 ~ 1 8 9 if o > 89I n l

} { Z , } aR 8 9 (18.8)ence9 -R -o o

A c o m b i n a t i o n o f (1 8.7 ) a n d ( 18 .8 ) p r o v e s ( 1 8 . 3 ) w h e n 8 9 T h u s ( 1 8 . 3 ) i sc o m p l e t e l y p r o v e d . T h e f i n i te n e s s a l m o s t e v e r y w h e r e o f f ~ ( x , [ ) fo l lo w s f r o m ( 18 .3 ),o f co u r se .

W e n o w c o n s i d e r A ~ . L e t v = [89 ( k - 1 ) ] = g r e a t e s t i n t e g e r i n 8 9 1 ). T h e n b yM i n k o w s k i ' s i n e q u a l i t y ,

R

0 R R{ ~ , ~ ' ( x , , ) - ~ + " ( x , , ) I ' a ~ , } ' + { l f l ~ ' : " ( ~ , , ) I~ a u } '0 O

= 11 + 12. (18.9)S in ce a = R ( 6 ) > 8 9 t h e n .R(6+~)>89 a n d w e m a y a p p l y L e m m a 1 2 , p a r t

(a ) t o S~+" (x, [) . Th is giv esI , s ' ; " ( x , 1 ) 1 A , , ~ * ' l * ( x ) .

S u b s t i t u t i n g t h i s e s t i m a t e i n t h e s e c o n d t e r m o f ( 1 8 . 9 ) g i v e s12 ~< A a e* 1 1 " ( x ) . ( 1 8 . 1 0 )

M o r e o v e r ,R

, ,

0oo

0

< ~ ,~ ( x , 1 ) + ~ + 1 ( x , 1 ) + " " + ~ , + , ( x , I ) . (18.11)W h e n w e co m b i n e (1 8 .9 ), ( 18 .1 0) a n d (18 .1 1) w e o b t a i n

A~ (x , / ) ~< ~0 (x , [ ) + . - . + D~+, (x , f ) + A , e~l 'l [* (x). (18 .12)


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~ , ~ + 1 . . . . 5 + ~. T o t h e t e r m c o n t a i n i n g ]* (x ) w e a p p l y L e m m a 3 . W e t h e r e f o r e o b t a i n( f A , ( x , 1 ) ] ' - d . ) ' < A ~ e * ' ( f I 1 ( . ) I * d ~ ) ' .Q Q

T h i s c o m p l e t e s t h e p r o o f o f L e m m a 1 3 .P r o o / o / t h e o r e m 7 . W e c o n s i d e r ( a ) f i r s t . W e s h a l l m a k e u s e o f L e m m a 4 ( o f

w r e c a l li n g t h a t R ~ E ~ a = A ~ . F o r ~ , f l a p p e a r i n g i n t h e s t a t e m e n t o f L e m m a 4 ,w e s u b s t i t u t e 89 a - 1 ), 8 9 a + 1 ) + i v , r e s p e c ti v e ly . W e t h u s h a v e :

R2 F ( a + i T + I ) f t ~ ) t ( , _ x ) + ~ p t ~ ,R U S ~ n = F ( 8 9 1 8 9 ) . ( R 2 - S ~ (~ (18.13)0

I f a > 0 , t h e f a c t o r i n v o l v i n g t h e r f u n c t i o n s i s c e r t a i n l y b o u n d e d b y A o e " 1" 1.T h u s f r o m { 1 8 .1 3 ) w e o b t a i n

RI ~ ( ~ , l ) l < - A . e ' l " t R - ~ ' f ( R 2 - t 2 ) t ( ~ - ~ ) P ' l S ~ ( ~ I ) 1 a t .0

A p p l y i n g n o w S c h w a r z ' s i n e q u a l it y t n t h e a b o v e w e o b t a i nRI ~ '~ ( ~ , / ) l ~ A , e * ' R - ~l f ( R ~ - t 2 ) ~ t { f l s ~ ( ~ '

0 0

T h e r e f o r e , [S~R(x , l ) ] < ~ A o e * l 'l " B ( R ) " A t ( , + 1 ) (x , 1 ) ,w h e r e

W e t h u s h a v e

RB ( R ) = R - ~ ' + t { f ( R e - t 2 ) " - x t " dr } t

01= { f ( 1 - t ' ) " - ' t ~ d z } t

0< ~ A / a = A a , i f a > 0 .

S~. (x, 1) = su p I~R (x , t ) 1< A o e * ' A t (, ~1)(x, 1).RF i n a l l y t h e nf IS , ( x , / ) ] 2 d x ~ A ~ e ~ i ' l f [A 8 9 A ~ e2 . 1 f I t ( x ) p d x ,Q Q Q

(18.14)

b y a p p l y i n g (1 8.4 ) o f L e m m a 13 ( to t h e c a se 8 9 s in c e ] ( a + l ) > 8 9 w h e na > O ) .

10- 583801. Acta mathemat&~a. 100. Imprim~t le 25 octobre 1958.


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134 ELLIS M. STEINT h i s p r o v e s p a r t ( a) o f T h e o r e m 7 .P a r t ( b) is p r o v e d b y o b s e r v i n g w h e n ~ ( 0 ) ~ > 0 , t h a t S ~ n ( x , /) m u l t i p l ie s t h eF o u r i e r c o e f f i c i e n t s o f ] ( x ) b y c o n s t a n t s o f a b s o l u t e v a l u e n o t e x c e e d i n g o ne .

1 9 . P r o o f o f d o m i n a t e d , p o i n t w i s e , a n d n o r m s u m m a b i l it y( a ) P o o l o f T h e o r e m ( D ) , ( se e w f o r s t a t e m e n t ) .T h e i d e a o f t h e p r o o f is a s fo l lo w s . W e n o t i c e t h a t t h e c a s e w h i c h c o r r e s p o n d s

t o p = 2 h a s a l r e a d y b e e n d i s p o s e d o f i n T h e o r e m 7 , p a r t ( a) , o f w1 8 . W e s h o u l d l i k et o h a v e a n a n a l o g o u s r e s u l t f o r p = 1 , a n d t h e n i n t e r p o l a t e b e t w e e n i n d i c es p = 2,a n d p = 1 . H o w e v e r , T h e o r e m ( D ) f ai ls w h e n p = . l , s o t h a t w e m u s t c o n t e n t o u r -s e lv e s w i t h a w e a k e r s u b s t i t u t e . S u c h a s u b s t i t u t e r e s u lt , s a ti s f a c t o r y f o r o u r p u r -p o s e s , is c o n t a i n e d i n L e m m a 1 2, p a r t ( a ), o f w 1 7.

N o w t o t h e p r o o f . L e t p a n d 0 b e t h e i n d ic e s g i v e n in T h e o r e m ( D ). A s s u m et h a t 1 < p < 2 , s i n ce t h e e a s e p = 2 i s c o n t a i n e d i n T h e o r e m 7 . L e t P l b e a n e x p o n e n t( t o b e d e t e r m i n e d l a te r ) w h i c h s a ti sf i es 1 < p l < p . T h u s w e m a y w r i t e

1 1, 0


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l _ ~ + ~ l t + e 0 ( 1 - t )k - 1 E ' ~ -E "= ( - - ~ - ) ( ~ - 1 ) + E ( ~) ) + ( e l) - (e 0).

H o w e v e r , b y a s s u m p ti o n , 8 > ( 8 9 T h u s w e m a y f in d a n ~1 s m a lle n o u g h s o t h a t s ti ll 5 > ( 8 9 W e f ix s u c h a n ~ . T h i s d e t e r m i n e sP l ( b y ( 19 . 1) ), a n d t , ( b y ( 1 9. 2) ). H o w e v e r , 0 < t < 1 , t h u s w e m a y f i n d ~0, a n d e l s o t h a t

5 ( t ) = ~ - ( 2 - l ) + E OT) + E ' ( ~ ) + E " (eo).T h i s p r o v e s ( 1 9 .4 ) , a n d w e p r o c eed w i t h t h e p a r a m e t e r s ~7, e0, ~1 f i x ed i n t h i s

m a n n e r .L e t R ( x ) b e a m e a s u r a b l e f u n c t i o n d e f i n e d o n Qk s u b j e c t o n l y to t h e c o n d i t io n s

t h a t : O < ~ R ( x ) ~ < R o < ~ . ( 1 9 .5 )W i t h t h e a i d o f (~ ( t) a n d R ( x ) w e n o w d e f in e a n a n a l y t i c f a m i l y o f t r a n s f o r m a -

t i o n s , T z ( " ) , a s f o l l o w s :T~ ( / ) (x ) = r (x, f ) , (19.6),,x)

a n d w e v e r i f y t h a t t h e f a m i l y s at is fi es t h e c o n d i t i o n s o f L e m m a 1 o f w8 . T h a t t h eco n d i t i o n s { i ) , ( if ) an d ( iii) a r e s a t i s f i ed f o l l o w s ea s i l y w h en o n e m ak es u s e o f t h er e s t r i c t i o n ( 1 9 . 5 ) . W e n e x t c l a i m t h a t T ~ ( - ) s a t i s f i e s t h e f o l l o w i n g b o u n d s

I[ T , y ( / )I 1 ~ , ~ < a 0 ( y ) I I ! l ip . , / (19 .7)l J . , y ( /) ] l~, < A , ( y )I I f l [ , ,,w h e r e A ~ ( y ) 0 . W e m a y t h e r ef o r e a p p l y T h e o r e m 7 , p a r t (a ) a n d o b t a i n

I }T ,~( /) [ [~.=I I T ,y ( f ) J [ 2


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1 3 6 E L I A S M . S T E I ~

w h e r e a o = ~ [ ~ ( k - 1 ) + e 1 --C ol.( T h e c o n s t a n t A o c l e a r ly d o e s n o t d e p e n d o n R(x) o r R o . ) N e x t ,

(8 (1 + i y) ) = - to (i y) + (89 ( k - 1) + e, ) (1 + i y) .H en ce , ~ (~ (1 + iy)) = ~ (k - 1) + ex > 89 k - 1).W e m a y t h u s u s e L e m m a 12, p a r t ( a ), a n d o b t a i n

I I T,+,~ I ) I I , , < I I s ~ + ' ~ ) ( / ) I1 ~ , < A ~ o , , ,, I I l * I 1 , , ,w h e r e a , = ~ g I 8 9 ( ] g - - 1 ) - ~- ~ 1 - - ~ 0 [ "S i n c e 1 < P l, w e m a y a p p l y L e m m a 3 o f w1 0 . T h e r e f o r e ,

II T I + ' y ( t ) I k < A 1 e a ' i Y [ I l l l l ~ , . (19 .11)( T h e c o n s t a n t A 1 a g a i n c l e a r ly d o e s n o t d e p e n d o n R(x) a n d R e . ) T h i s e s t a b l i s h e s( 1 9 . 7 ) an d ( 1 9 . 8 ) .

U s i ng t h e i n t e r p o l a t io n l e m m a , w e t h u s o b t a i nI I T , (I ) I I ~< A , I I 1 1 1 ~ - ( 1 9 . 1 2 )

N o w b y ( 8.5 ) o f L e m m a 1 t h e c o n s t a n t A I a p p e a r i n g i n ( 19 .1 2 ) d e p e n d s o n l yo n t h e Aj(y) o f ( 1 9 . 7 ) . S i n c e t h e s e l a t t e r a r e i n d e p e n d e n t o f R(x) a n d R 0, t h e s a m eh o l d s f o r A v H o w e v e r , ( 19 .1 2 ) m a y b e r e w r i t t e n a s

p xllv( f l .~`' , .. (~, (~,I ) d~) - ~A , ( f l / ( ~ ) l ' d x ) ' " ( 1 9 . 1 3 )Ok Ok

a n d b y (1 9.4 ) 5 ( 0 = 5 .9 w e h a ve ( f l ~ . , . , ( ~ , l ) l ' d ~ ) l " < ~ A , ( f

Qk Okw i t h R (x ) s u b j e c t o n l y t o t h e c o n d i t i o n 0 ~ R ( x)of R(x) a n d R o. B y a n a p p r o p r i a t e c h o i ce o f R(x)

\ l i p ~ < ~( f ( o 2 2 g ~ . l ~ ( x , l ) l ) ' e x 2 ~ A ,N o w s in c e th e i n t e g r a n d o f t h e l e f t - h a n d s i d e

I t ( ~ ) l ~ d x ) ' ' ~R 0 < ~ , a n d w i t h At i n d e p e n d e n tw e d e d u c e

( f I I ( x ) [ ' ) " .Qki n cr e a s e s w i t h R o, w e o b t a i n

o < ~ < = f I / ( x ) l ~ d x ) l ' ~! ( s u p I ( x , l ) , )T h i s c o n c l u d e s t h e p r o o f o f T h e o r e m ( D) .


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( b ) P r o o f o ! T h e o r e m ( D * ) ( se e w f o r s t at e m e n t) .Theo rem (D*) w i l l be a consequence o f t he fo l l owing l emma .L E M M A 1 4. L e t / ( x ) E L ~(Q k ), l < p ~ < 2 . T h e n ,

_ p xl / p( f t s l ( k - ' ( f l / ( ) l , ( 1 9 . 1 4 )Ok Ok

where A does no t depend on p or / .P r o o / . Thi s l emm a i s a l r eady con t a ined imp l i c it l y in t he p roo f o f Theo rem (D)

a b o v e . I n fa c t, fo r o u r g iv e n p , l < p ~ < 2 , f ix t h e in d e x Pl, l < p ~ < p , b y1 - - - = - 1 1 ( 1 _ 1 ) . (19.15)p~ 2

T h u s i f 1 / p = ( 1 - t ) / p 0 + t / p o, ( p 0 = 2 ) , t h e nt = 2 - p , a n d 1 - t = p - 1 .

N ow def ine J (z ) by

As i s eas i ly ver

Elias m. Stein - Localization and Summability of Multiple Fourier Series

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Transcript of Elias m. Stein - Localization and Summability of Multiple Fourier Series