calderon singular op

7/30/2019 Calderon Singular Op

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COMMUTATORS OF SINGULAR INTEGRAL OPERATORS*BY A . P . C A L D E R O N

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CHICAGOC o m m u n i c a t e d b y A . A d r i a n A l b e r t , M a r c h 2 6 , 1 9 6 5

L e tA f ) =limk(x - y ) f ( y ) d y ,

w h e r e x , y a r e p o i n t s i n n - d i me n s i on a l E uc l i d e a n s p a c e R ' a n d k ( x ) i s a h o m o g e n e o u sf u n c t i o n o f d e g r e e -n w i t h mean v a l u e z e r o o n x | = 1 , a n d l e t B ( f ) = b ( x ) f ( x ) .I t i s w e l l k n o w n ( s e e r e f . 1 ) t h a t i f k a n d b a r e s u f f i c i e n t l y s m o o t h a n d b i s b o u n d e d ,t h e n ( A B - B A ) ( O / c x j ) a n d ( a / a x j ) ( A B - BA) a r e b o u n d e d o p e r a t o r s i n L P ,1 0 ,a n d s u p p o s e t h a t t h e p a r t i a l s o f k ( x ) + k ( - x ) b e l o n g l o c a l l y t o L l o g + L i n x j > 0 .L e t b ( x ) h a v e f i r s t - o r d e r d e r i v a t i v e s i n L r , 1


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V O L . 5 3 , 1 9 6 5 MATHEMATICS: A . P . CALDER 6N 1 0 9 3f u n c t i o n o f t h e s e t s > 0 , I t I < s . T h e n t h e r e e x i s t t w o p o s i t i v e c o n s t a n t s c l a n d c 2d e p e n d i n g o n p o n l y , s u c h t h a t c 4 I | F ( t ) I I p < I I S ( F ) f I j < c 2 1 l F ( t ) I I p , w h e r e F ( t ) = l i mt o - OF ( t + i s ) .T h e n o v e l t y i n t h e p r e c e d i n g s t a t e m e n t i s t h e f i r s t i n e q u a l i t y f o r p < 1 . As i m i l a r r e s u l t f o r t h e f u n c t i o n g o f L i t t l e w o o d a n d P a l e y w h e n F h a s n o z e r o s w a sp r o v e d b y T . M. F l e t t ( r e f . 3 ) , w h o s e m e t h o d w e b o r r o w p a r t i a l l y . A c t u a l l y , o n l yt h e c a s e p ) 1 w i l l b e n e e d e d i n t h i s n o t e , b u t i t s p r o o f i s n o l e s s l a b o r i o u s t h a nt h a t o f t h e g e n e r a l c a s e .

P r o o f o f T h e o r e m 3 : We w i l l a s s u m e f i r s t t h a t F ( t + i s ) i s a n a l y t i c i n s ) 0a n d t h a t F I ( t 2 + 8 2 ) k - 0 a s ( t 2 + s 2 ) X _ c D f o r e v e r y k > 0 . T h e n , o f c o u r s e ,F b e l o n g s t o HP f o r e v e r y p > 0 . We i n t r o d u c e now s o m e n o t a t i o n . F o r a f u n c t i o nG d e f i n e d o n t h e r e a l l i n e we w r i t e~ +o 1 / PM t ( G ) = [ : G P d t P > o .

I f G i s a l s o d e f i n e d i n t h e u p p e r h a l f - p l a n e , we w r i t em ( G ) = s u p X ( t - u , s ) I G ( u , s ) I , S ( G ) = [ f x ( t - u , s ) I g r a d G I 2 d u d s ] ' 1 2 ,U,8

w h e r e X ( t , s ) i s t h e c h a r a c t e r i s t i c f u n c t i o n o f t h e s e t s > 0 , t I < s . By i n t e g r a t i o nw e o b t a i n M 2 2 [ S ( G ) ] = 2 f s I g r a d G I 2 d t d s . Now i f a i s a n y p o s i t i v e n u m b e r , wes e t G = F J ' , t h e n a s i m p l e c a l c u l a t i o n g i v e sA2 ) = 4 1 g r a d G I 2 ( 0 )a n d a n a p p l i c a t i o n o f G r e e n ' s f o r m u l a y i e l d s 4

M 2 2 ( G ) = 4 f s j g r a d G I 2 d t d s = 2 M 2 2 [ S ( G ) ] ( 1 )On a c c o u n t o f t h e d e f i n i t i o n o f G a n d t h e a n a l y t i c i t y o f F , we h a v e t h e f o l l o w i n gw e l l - k n o w n i n e q u a l i t y

M . [ m ( G ) ] < c M . ( G ) , 0 < p < a . ( 2 )No w l e t p ) 1 , t h e n

S ( G P ) 2= f x ( t - u , s ) p G P - I g r a d G I 2 d u d s < p 2 m ( G ) 2 P - 2 S ( G ) 2 ,t h a t i s ,S ( G P ) < p m ( C ) P - ' S ( C ) , 1


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1 0 9 4 MATHEMATICS: A . P . CALDER6N P R O C . N . A . S .L e t u s a s s u m e no w t h a t we h a v e t h e i n e q u a l i t y

c M r ( G ) 2 M r [ S ( G ) ] ( 5 )f o r s o m e r , r > 0 . L e t 0 < q < r a n d p = r / q . T h e n ( 3 ) a p p l i e d t o G 1 / P g i v e s

S ( G ) < p m n ( G L I P ) P - l S ( G l I P ) = p m ( G ) ( P - 1 ) ' P S ( G " 1 )w h e n c e , a p p l y i n g H o l d e r ' s i n e q u a l i t y , we g e tM q I [ S ( G ) ] < p q M j [ m ( G ) q ( P - l ) l P S ( G l I P ) q ] < p " ' A I r / q [ S ( G l / P ) Q ] M r l ( r - . ) [ m ( G ) q ( P - 1 ) I P ]

= p Q M r ' [ S ( G 1 I P ) ] M s , ( P - l ) I P [ m ( G ) ]a n d f r o m t h e l a s t e x p r e s s i o n , ( 2 ) , a n d ( 5 ) a p p l i e d t o G 1 / P i t f o l l o w s t h a t

Mq [ S ( G ) ] < c p P M r ' I [ G l P ] M q A q ( P - l ) I P ( G ) = c p q M l q ' P ( G ) M j q ( P - l ) / P ( G )o r M q [ S ( G ) ] < c , M , ( G ) . ( 6 )On a c c o u n t o f ( 1 ) , ( 5 ) h o l d s w i t h r = 2 . H e n c e t h e p r e c e d i n g i n e q u a l i t y h o l d sf o r 0 < q < 2 .No w we w i l l s h o w t h a t ( 6 ) h o l d s f o r 0 < q < a . S i n c e ( 5 ) i m p l i e s ( 6 ) w i t hq < r , i t i s e n o u g h t o s h o w t h a t ( 6 ) h o l d s f o r q > 4 . L e t h ( t ) > 0 b e a n y b o u n d e df u n c t i o n w i t h c o m p a c t s u p p o r t . T h e n

.+ C 4 o + c oS f ( G ) 2 h d t = h ( t ) f x ( t - u , s ) g r a d G I 2 d u d s d t- f | g r a d G I 2 f w h ( t ) x ( t - u , s ) d t d u d s .

No w we o b s e r v e t h a t i f P ( t , s ) d e n o t e s t h e P o i s s o n k e r n e l f o r t h e h a l f - p l a n e , t h e nX ( t s ) < c s P ( t , s ) a n d c o n s e q u e n t l yf ' h ( t ) x ( t - u , s ) d t < c h ( t ) s P ( t - u , s ) d t < c s H ( u , s ) ,

w h e r e H ( t , s ) i s t h e P o i s s o n i n t e g r a l o f h ( t ) . T h u s ,w+ c of , 0 s ( G ) 2 h d t < c f g r a d G | 2 s H ( t , s ) d t d s .

N o w , f r o m ( 0 ) w e h a v eA ( G 2 H ) = HAG2 + 2 ( g r a d G 2 ) - ( g r a d H )

= 4 H | g r a d G I 2 + 2 G ( g r a d G ) * ( g r a d H )) 4 H I g r a d G 2- 2 G g r a d G I I g r a d H f

a n dJ S ( G ) 2 h d t < f s A ( G 2 H ) d t d s + 2 f s G | g r a d G I I g r a d H I d t d s


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V O L . 5 3 , 1 9 6 5 MATHEMATICS: A . P . CALDER6N 1 0 9 5a n d a p p l y i n g G r e e n ' s f o r m u l a t o t h e f i r s t t e r m o n t h e r i g h t 4r+cocf S ( G ) 2 h d t < f G 2 h d t

_ c o 4 co+ f q d t f x ( t - u , s ) G I g r a d G | g r a d H d u d s

< G 2 h d t + - m ( G ) S ( G ) S ( H ) d t .Now we s e t p = q / ( q - 1 ) a n d a p p l y t h e t h r e e - t e r m H o l d e r i n e q u a l i t y w i t h e x -p o n e n t s 2 q , 2 q , p t o t h e p r e c e d i n g i n t e g r a l s a n d g e t

r + 04 fS(G) 2 h d t < c M 2 , 2 ( G ) M , ( h ) + c M 2 q [ m ( G ) ] M 2 Q [ S ( G ) ] M , [ S ( H ) ] . ( 7 )S i n c e H i s h a r m o n i c a n d 1 < p < a ) , we h a v e M , [ S ( H ) . c p M p ( h ) , a n d s i n c e4 < q < a , we a l s o h a v e M 2 q [ m ( G ) I < c M 2 , ( G ) . S u b s t i t u t i n g i n t h e p r e c e d i n gi n e q u a l i t y , s e t t i n g M , ( h ) = 1 , a n d t a k i n g t h e s u p r e m u m o f t h e l e f t - h a n d s i d e o v e ra l l s u c h h , we f i n d t h a t M . [ S ( G ) 2 ] = M 2 C 2 [ S ( G ) ] < c M 2 C ( G ) [ M 2 q ( G ) + M 2 g S ( G ) ] ,a n d t h i s i m p l i e s t h a t M 2 q [ S ( G ) ] < C ' M 2 , ( G ) p r o v i d e d t h a t M 2 , [ S ( G ) I < a 0 . T os e e t h a t t h i s i s t h e c a s e we o b s e r v e t h a t s i n c e m ( G ) i s b o u n d e d , ( 7 ) h o l d s w i t hM. f [ m ( G ) ] r e p l a c i n g M 2 Q [ m ( G ) J a n d M , [ S ( G ) ] r e p l a c i n g M 2 q [ S ( G ) ] a n d f r o m t h i s ,a r g u i n g a s a b o v e , we o b t a i n

M 2 Q 2 [ S ( G ) ] < c M 2 a 2 ( G ) + c M. [ m ( G ) ] M Q [ S ( G ) ] .S i n c e t h e r i g h t - h a n d s i d e i s f i n i t e f o r q = 2 , i t f o l l o w s b y i n d u c t i o n t h a t t h e l e f t -h a n d s i d e i s f i n i t e f o r a r b i t r a r i l y l a r g e q a n d h e n c e f o r a l l q > 2 . T h u s ( 6 ) i se s t a b l i s h e d f o r 0 < q < a .No w we p r o v e t h e c o n v e r s e i n e q u a l i t y . L e t q > 0 . T h e n ( 1 ) a n d ( 4 ) g i v e

M q g ( G ) = M 2 2 ( G 9 l 2 ) = 2 M 2 2 S ( G " 1 2 ) < c M l [ S ( G 12 ) 2 a S ( G f q l 2 ) 2 ( - 1 ) ]w h e r e a = 2 q / ( q + 2 ) , , 8 = 2 / q , a = ( q + 2 ) / 2 ( q + 1 ) , 1 - = q / 2 ( q + 1 ) .A p p l y i n g H 6 l d e r ' s i n e q u a l i t y t o t h e r i g h t - h a n d s i d e we g e t

MCq ( G ) < C M ( q + l ) I q [ S ( G I 1 I 2 ) 2 f ] M q + 1 [ S ( G ) 2 ( l - ) ] .B u t

M ( q + 1 ) 1 C [ S ( G 1 1 2 ) 2 a ] = M f ( q + 2 ) / q 2 J [ S ( G a q l 2 ) IM q + i[ S ( G ) 2 ( 1 - 0 ) ] = M q 2 ( 1 - u ) [ S ( G ) ] .A p p l y i n g ( 6 ) t o t h e r i g h t - h a n d s i d e o f t h e f i r s t o f t h e p r e c e d i n g i d e n t i t i e s , a n do b s e r v i n g t h a t M ( q + 2 ) 1 q [ G a 1 2 ] - M q a U / 2 ( G ) , s u b s t i t u t i o n i n t h e p r e c e d i n g i n e q u a l i t yy i e l d s

M q g f ( G )


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1 0 9 6 MATHEMATICS: A . P . C A L D E R 6 N P R O C . N . A . S .- T o o b t a i n ( 6 ) a n d ( 8 ) f o r F w e s e t G = I F I a n d o b s e r v e t h a t I g r a d G I = I F P I .F i n a l l y , we m u s t r e m o v e t h e c o n d i t i o n s w e i m p o s e d o n F a t t h e b e g i n n i n g o f t h ep r o o f . I f F ( z ) , z = t + i s , i s a n a l y t i c i n t h e u p p e r h a l f - p l a n e a n d b e l o n g s t o H P ,t h e n F ( z + i / n ) = F n ( z ) i s b o u n d e d t h e r e . L e t no w e m ( z ) = e x p ( - z a m ) , w h e r eo < a < 1 / 4 a n d a r g ( z a ) i s b e t w e e n 0 a n d 7 r / 4 . T h e n a s i m p l e c a l c u l a t i o n s h o w st h a t

j f 8 | I e m ' ( t + j s ) 2 d t d s < C 2 a ,8>0w h e r e c i s i n d e p e n d e n t o f m . C o n s e q u e n t l y , S ( e m ) 2 < c 2 a . N o w , t h e f o l l o w i n gi n e q u a l i t i e s c a n b e r e a d i l y v e r i f i e d :

S ( F . e m ) 2 < 2 [ S ( F ) 2 + m ( F n ) 2 S ( e m ) 2 ] < 2 [ S ( F n ) 2 + C 2 m ( F n ) 2 a ]S ( F n e m ) P < 2 P [ S ( F n ) P + c P m ( F n ) P a P / 2 ] .

I n t e g r a t i n g we g e tM / P [ S ( F n e m ) ] < 2 P [ M r P [ S ( F n ) ] + C P a P 1 2 M P [ m ( F n ) ] ] .

S i n c e M p P ( F n ) = l i m M p P ( F n e m ) a n d b y ( 8 ) , M p P ( F n e m ) < c p P M p P [ S ( F n e m ) ] f r o mmt h e i n e q u a l i t y a b o v e we o b t a i nM p P ( F n ) < c p P 2 P [ M p P [ S ( F n ) ] + c P a P / 2 M P [ m ( F n ) ] ] ,

a n d l e t t i n g a t e n d t o z e r oM p ( F n ) < c p 2 M p [ S ( F n ) ]

F i n a l l y , a s n t e n d s t o i n f i n i t y , M p ( F n ) c o n v e r g e s t o M p ( F ) a n d S ( F n ) i n c r e a s e s a n dc o n v e r g e s t o S ( F ) . T h u s we c a n p a s s t o t h e l i m i t i n t h e p r e c e d i n g i n e q u a l i t y a n do b t a i n h a l f o f t h e d e s i r e d r e s u l t . To o b t a i n t h e o t h e r h a l f we o b s e r v e t h a t , s i n c e( F n e m ) ' c o n v e r g e s t o F n ' , we h a v e S ( F n ) = l i m m i n f S ( F n e m ) . T h u s f r o m ( 6 )a p p l i e d t o F n e m a n d F a t o u ' s l e m m a we g e t

Mp [ S ( F n ) I < c p M F n ) ,a n d a p a s s a g e t o t h e l i m i t c o m p l e t e s t h e p r o o f o f t h e t h e o r e m .

P r o o f o f T h e o r e m 2 : We b e g i n w i t h t h e o n e - d i m e n s i o n a l c a s e . H e r e h ( x )b e c o m e s s i m p l y x - 2 , a n d t h e p r o o f r e d u c e s t o e s t i m a t ef ' C e ( f ) g d x = ( x - y ) 2 [ b ( x ) -b ( y ) ] g ( x ) f ( y ) d x d y- c o I - Y 1 > E

i n t e r m s o f t h e n o r m s o f f , g , a n d b ' . F o r t h i s p u r p o s e t h e r e i s n o l o s s o f g e n e r a l i t yi n a s s u m i n g t h a t t h e s e f u n c t i o n s a r e i n f i n i t e l y d i f f e r e n t i a b l e a n d h a v e c o m p a c ts u p p o r t . L e t e ( x ) b e t h e c h a r a c t e r i s t i c f u n c t i o n o f x > 0 a n d x ( x ) t h a t o f x I > E .T h e n r + cb ( x ) = e(x - t ) b ' ( t ) d t ,a n d s u b s t i t u t i n g , t h e i n t e g r a l a b o v e b e c o m e s

r+ cJ b ' ( t ) f ( x - y ) - 2 x ( 1 x - y I ) [ e ( x -t ) -e ( y -t ) ] g ( x ) f ( y ) d x d y d t- , . .


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V O L . 5 3 , 1 9 6 5 MATHEMATICS: A . P . CALDER6N 1 0 9 7a n d t h e p r o b l e m r e d u c e s t o s t u d y i n g t h e c l a s s o f t h e f u n c t i o n r e p r e s e n t e d b y t h ei n n e r i n t e g r a l . F o r t h i s p u r p o s e w e l e t z b e a c o m p l e x v a r i a b l e a n d s e t

1 o + c o 1f , ( z ) = f f ( x ) d x , j = 1 i f I m ( z ) > 0 , j = 2 i f I m ( z ) < 0 ,2 7 r co_x - za n d d e f i n e s i m i l a r l y g j ( z ) . T h e n we h a v e f ( x ) = f i ( x ) - f 2 ( x ) a n d s i m i l a r l y f o r g .F u r t h e r m o r e , t h e f j b e l o n g t o H P , 1 < p < c , i n t h e c o r r e s p o n d i n g h a l f - p l a n e sa n d , w i t h t he n ot at i o n o f t h e p r e c e d i n g p r o o f , w e h a v e

M P ( f j ) : < c p M Y f ) , I t we h a v e

K j ( x , y , t ) f i ( y ) d y = ( x y - i e ) - 2 f 1 ( y ) d y- _f [ ( t + i s ) - ( x - i e ) > - 2 f ( t + i s ) d ( i s ) .0

A s r e a d i l y s e e n , f o r x < t t h e i n t e g r a l o n t h e l e f t a b o v e i s a l s o g i v e n b y t h i s l a s te x p r e s s i o n . T h u s , r + c o +k i ( t ) = g ( x ) J [ ( t + i s ) - ( x - i e ) ] - 2 f 1 ( t + i s ) d ( i s ) ,a n d i n t e r c h a n g i n g t h e o r d e r o f i n t e g r a t i o n w e g e t

r++0ck i ( t ) = - M f i ( t + i s ) [ ( t + i s ) - ( X -i E ) ] 2 g ( x ) d x d ( i s ) .8 =0 JcoS i n c e g ( x ) = g ( x ) - 9 2 ( x ) a n d g 2 ( z ) i s a n a l y t i c i n I m ( z ) < 0 , i t s c o n t r i b u t i o n t ot h e i n n e r i n t e g r a l a b o v e i s z e r o a n d t h e v a l u e o f t h i s r e d u c e s t o 2 7 r i g l ' ( t + i s + i e ) .T h u s we h a v e

r + ck 1 ( t ) = - 2 7 r i , ff(t + i s ) g i ' ( t + i s + i e ) d ( i s ) .L0L e t u s i n t r o d u c e no w


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1 0 9 8 MATHEMATICS: A . P . CALDER6N P R O C . N . A . S .+F ( z ) = -2rif f 1 ( z + i s ) g l ' ( z + i s + i e ) d ( i s ) .8=0

T h e n w e h a v e k i ( t ) = F ( t ) . F u r t h e r m o r e , s i n c e f i a n d g l ' a r e b o u n d e d a n d O ( z - 1 )a n d O ( z - 2 ) , r e s p e c t i v e l y , F ( z ) b e l o n g s t o H P , p > 1 , a n d w i t h t h e n o t a t i o n o f t h ep r e c e d i n g p r o o f we h a v e

( 2 7 r ) - 1 S ( F ) . m ( f i ) S ( g , ( z + i e ) ) < m ( f i ) S ( g i )a n d i f q - 1 = p - 1 + r - ' , 1 < p , q < a ) , r < a , t h e n b y T h e o r e m 3 a n d ( 9 ) w e h a v eM r / t - i ( k i ) = M r l r - l ( F ) . c M r r - 1 [ S ( F ) ] . c M p [ m ( f J ) ] M q , 1 , 1 [ S ( g i ) ]

< C M p ( J ' i ) M q i , ( g i ) < C M p @ f ) M ' q i q ( g ) . ( 1 0 )No w w e e s t i m a t e k 2 . We h a v er+co+f K 2 ( x y t ) I f ( y ) dy e [ ( x - t ) 2 + E 2 ] - 1 s u pP2f [ ( y - t ) 2+ 6 2 ] - 3 / 2 I f ( y ) I d y < c E [ ( x - t ) 2 + e 2 ] - f ( t ) ,w h e r e f i s t h e m a x i m a l f u n c t i o n o f H a r d y a n d L i t t l e w o o d a s s o c i a t e d w i t h I j f .C o n s e q u e n t l y , r + ODk 2 ( t ) | < c f ( t ) s u p e f [ ( x - t ) 2 + E 2 ] - 1 g ( x ) I d X < c f ( t ) g ( t ) .

M r / r , 1 ( k 2 ) . C M p ( ) M q l q - j ( g ) < C M p ( f ) M/l ( g ) .T h i s c o m b i n e d w i t h ( 1 0 ) s h o w s t h a t M I , - 1 ( k o ) < c M p ( f ) M q l f q I ( g ) w h e r e c d e p e n d so n p a n d r b u t n o t o n e . A s r e a d i l y s e e n , t h i s i m p l i e s t h a t Mq [ C ( f ) ] < c M r ( b ' )M P ( f ) .We no w p a s s t o d i s c u s s t h e n - d i m e n s i o n a l c a s e . A s b e f o r e , we a s s u m e t h a t fa n d t h e p a r t i a l d e r i v a t i v e s b j o f b a r e i n f i n i t e l y d i f f e r e n t i a b l e a n d h a v e c o m p a c ts u p p o r t . We d e n o t e b y P a u n i t v e c t o r i n R n a n d b y E i t s o r t h o g o n a l c o m p l e m e n ta n d f i x E , E > 0 . L e t s b e a r e a l v a r i a b l e a n d

k ( x , v ) = f h ( v ) s - 2 [ b ( x ) - b ( x + v s ) ] f ( x + v s ) d s .1 8 > eT h e n s e t t i n g y = x + v s , i n t e g r a t i o n i n p o l a r c o o r d i n a t e s s h o w s t h a tC E f ) = 1 2 f k ( x , v ) d v , ( 1 1 )w h e r e d v d e n o t e s t h e s u r f a c e a r e a e l e m e n t o f t h e u n i t s p h e r e i n R n . We now f i x va n d s e t x = z + v t , w h e r e z e E . T h e n f r o m t h e i n e q u a l i t y f o r t h e o n e - d i m e n s i o n a lc a s e e s t a b l i s h e d a b o v e we g e t

k ( z + P v t v ) d t < c1 { g r a d b ( z + v t , P ) l r d tX [ f ( z + v t v ) I P d t ] I h ( v ) j .

I n t e g r a t i n g w i t h r e s p e c t t o z o v e r E a n d a p p l y i n g H 6 l d e r ' s i n e q u a l i t y t o t h e r i g h t -h a n d s i d e , we o b t a i n


8/8

V O L . 5 3 , 1 9 6 5 MATHEMATICS: A . P . CALDERON 1 0 9 9[ f I k ( x , p ) e d x p l I q < c [ J I g r a d b j d x ] 1 ' ` J [ l f ( x ) | P d x ] I P I h ( v ) |

From t h i s a n d M i n k o w s k i ' s i n t e g r a l i n e q u a l i t y a p p l i e d t o ( 1 1 ) we o b t a i nI C J j f ) I g . : < c 1 1 g r a d b | j r 1 1 f l j p f h ( v ) I d p ,w h e r e c d e p e n d s o n p , q , a n d r b u t n o t o n e .

C o n c e r n i n g t he c o nv e r g e n c e o f C C f ) a s e t e n d s t o z e r o w e m e r e l y o b s e r v e t h a to u r a s s e r t i o n o b v i o u s l y h o l d s i f f a n d t h e b j a r e a s s u m e d t o b e i n f i n i t e l y d i f f e r e n t i a b l ea n d h a v e c o m p a c t s u p p o r t , w h e n c e 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 i n e q u a l i t ya b o v e b y a p p r o x i m a t i o n .

P r o o f o f T h e o r e m 1 : S i n c e ( b ) c a n r e a d i l y b e o b t a i n e d f r o m ( a ) b y d u a l i t y ,we s h a l l o n l y p r o v e t h e l a t t e r . L e t u s c o n s i d e r f i r s t t h e c a s e w h e n k ( x ) i s a n o d df u n c t i o n . T h e r e w i l l b e n o l o s s i n g e n e r a l i t y i n a s s u m i n g t h a t k ( x ) i s i n f i n i t e l yd i f f e r e n t i a b l e i n x | > 0 a n d t h a t f a n d t h e b j a r e i n f i n i t e l y d i f f e r e n t i a b l e a n d h a v ec o m p a c t s u p p o r t . L e t f j , b j , a n d k , d e n o t e t h e j t h p a r t i a l d e r i v a t i v e s o f f , b , a n dk , r e s p e c t i v e l y . T h e n i n t e g r a t i o n b y p a r t s y i e l d s

f k l-/k(xy ) [ b ( x ) - b ( y ) ] f , ( y ) d y = k ( x - y ) b j ( y ) f ( y ) d - y+ k j ( x - y ) [ b ( x ) - b ( y ) ] f ( y ) d y

- f n k ( v e ) [ b ( x ) - b ( x + v e ) ] f ( x + v e ) v j e ' - d P ,w h e r e v j d e n o t e s t h e j t h c o m p o n e n t o f t h e u n i t v e c t o r v a n d d v d e n o t e s t h e s u r f a c ea r e a e l e m e n t o f t h e u n i t s p h e r e i n R ' . N o w , t h e f i r s t t e r m o n t h e r i g h t r e p r e s e n t sa a o r d i n a r y t r u n c a t e d s i n g u l a r i n t e g r a l a n d i t s norm i n B c a n b e e s t i m a t e d i nt e r m s o f t h e n o r m s o f b j a n d f . T o e s t i m a t e t h e n o r m o f t h e s e c o n d t e r m we u s eT h e o r e m 2 , a n d i n t h e l a s t t e r m we r e p l a c e b ( x ) - b ( x + v e ) b y

2 b j ( x + t P e ) v j e d ta n d a p p l y M i n k o w s k i ' s i n t e g r a l i n e q u a l i t y t o t h e r e s u l t i n g i n t e g r a l . C o l l e c t i n gr e s u l t s a n d l e t t i n g e t e n d t o z e r o , ( a ) f o l l o w s .

I n t h e c a s e w h e n k ( x ) i s e v e n , t h e o p e r a t o r A c a n b e r e p r e s e n t e d a s a f i n i t e sumo f o p e r a t o r s o f t h e f o r m A 1 A 2 w h e r e A l a n d A 2 h a v e o d d k e r n e l s a n d s a t i s f y t h eh y p o t h e s i s o f t h e t h e o r e m ( s e e r e f . 2 ) . S i n c e 0 / O x 1 c o m m u t e s w i t h A 2 , we h a v e

( A 1 A 2 B - B A 1 A 2 ) = A(A2B-BA2) + ( A 1 B - B A 1 ) - A 2 ,1 x j C 1 x j 4 9 x js i n c e A , a n d A 2 a r e b o u n d e d i n L P f o r e v e r y p , 1 < p < c , t h e d e s i r e d r e s u l t f o l l o w s .

* T h i s r e s e a r c h w a s p a r t l y s u p p o r t e d b y t h e NSF g r a n t G P - 3 9 8 4 .1 C a l d e r 6 n , A . P . , a n d A . Z y g m u n d , " S i n g u l a r i n t e g r a l o p e r a t o r s a n d d i f f e r e n t i a l e q u a t i o n s , "Am. J . M a t h . , 7 9 , 9 0 1 - 9 2 1 ( 1 9 5 7 ) .2 I b i d . , "On s i n g u l a r i n t e g r a l s , " 7 8 , 2 8 9 - 3 0 9 ( 1 9 5 6 ) .3 F l e t t , T . M . , "On s o m e t h e o r e m s o f L i t t l e w o o d a n d P a l e y , " J . L o n d o n M a t h . S o c . , 3 1 , 3 3 6 - 3 4 4( 1 9 5 6 ) .4 To o b t a i n G r e e n ' s f o r m u l a f o r t h e h a l f - p l a n e u n d e r o u r a s s u m p t i o n s w e a p p l y i t t o n c o s ( n - ' t )s i n ( n - 1 s ) a n d t h e f u n c t i o n G 2 o r G2H o v e r t h e s q u a r e -n < t < n, 0

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