Chapter II Calderon-Zygmund Theory

CHAPTER I 1 CALDERON-ZYGMUND THEORY

I n 1952, A . P . C a l d e r 6 n and A . Zygmund [l] i n v e n t e d a s i m p l e b u t p o w e r f u l method which h a s become w i d e l y known a s t h e Calder6n-Zygmund decomposition. We aim t o d e s c r i b e h e r e t h i s method t o g e t h e r w i t h some of i t s most immediate and i n t e r e s t i n g a p p l i c a t i o n s . The f i r s t two s e c t i o n s g i v e a d e s c r i p t i o n of t h e method i n c o n n e c t i $ > n w i t h t h e ( v e r y c l o s e l y r e l a t e d ) H a r d y - L i t t l e w o o d maximal o p e r a t o r . Apart from t h e u s u a l e s t i m a t e s f o r t h i s maximal f u n c t i o n , we a l s o o b t a i n some w e i g h t e d i n e q u a l i t i e s which a n t i c i p a t e t h e A t h e o r y t o P be d e v e l o p e d i n C h a p t e r IV, and we s t u d y some v a r i a n t s o f t h e HardyL i t t l e w o o d o p e r a t o r when Lebesgue measure i s r e p l a c e d by a more gene r a l measure. T h i s l e a d s u s i n a n a t u r a l way t o t h e d e f i n i t i o n and s t u d y of C a r l e s o n m e a s u r e s . T h i s i s n o t t h e o n l y maximal o p e r a t o r t o a p p e a r i n t h i s c h a p t e r . The s o - c a l l e d s h a r p maximal f u n c t i o n s h a r e s enough p r o p e r t i e s w i t h t h e H a r d y - L i t t l e w o o d o p e r a t o r , b u t b e h a v e s i n a d i f f e r e n t way i n L m , which i s somehow r e p l a c e d by o u r f r i e n d t h e s p a c e i n t e r e s t i n g r e l a t i o n between w e i g h t s and

BMO

BMO.

There i s an

which w i l l be f u r t h e r

e x p l o i t e d i n C h a p t e r IV. T h i s r e l a t i o n comes t o l i g h t a f t e r p r o v i n g t h e John-Niremberg i n e q u a l i t y f o r BMO f u n c t i o n s , which i s y e t a n o t h e r a p p l i c a t i o n of t h e Calder6n-Zygmund d e c o m p o s i t i o n . The l a s t two s e c t i o n s c o n t a i n a v e r y condensed a c c o u n t on 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 and t h e i r a p p l i c a t i o n t o m u l t i p l i e r t h e o r e m s , s i n c e t h e s e o p e r a t o r s o r i g i n a l l y m o t i v a t e d t h e Calder6n-Zygmund d e c o m p o s i t i o n , and t h e y a r e s t i l l t h e most s t r i k i n g and i m p o r t a n t a p p l i c a t i o n of t h e method.

133

1I.CALDERON-ZYGMUND THEORY

134

1.

THE HARDY-LITTLEWOOD MAXIMAL FUNCTION AND THE CALDERON-ZYGMUND DECOMPOSITION

be a l o c a l l y i n t e g r a b l e f u n c t i o n i n

Let f define

Rn.

For

x

6

Wn

we

where t h e supremum i s t a k e n o v e r a l l c u b e s Q c o n t a i n i n g x ( c u b e w i l l a l w a y s mean a compact cube w i t h s i d e s p a r a l l e l t o t h e a x e s a n d nonempty i n t e r i o r ) , and s t a n d s f o r t h e Lebesgue measure of Q .

IQI

Mf w i l l be c a l l e d t h e ( H a r d y - L i t t l e w o o d ) maximal f u n c t i o n o f f , and t h e o p e r a t o r M s e n d i n g f t o Mf, t h e ( H a r d y - L i t t l e w o o d ) maximal o p e r a t o r . Observe t h a t we o b t a i n t h e same v a l u e M f ( x ) , which c a n b e + m , if we a l l o w i n t h e d e f i n i t i o n o n l y t h o s e c u b e s Q f o r which x i s an i n t e r i o r p o i n t . I t f o l l o w s from t h i s remark t h a t t h e f u n c t i o n Mf i s lower semicontinuous, i . e . , f o r every t > 0 , t h e s e t

i s open.

I n o r d e r t o s t u d y t h e s i z e of Mf, we s h a l l l o o k a t i t s d i s t r i b u t i o n function h ( t ) = I E t l . I t w i l l be i n s t r u c t i v e t o s t a r t with t h e c a s e n = 1 , which i s p a r t i c u l a r l y s i m p l e . L e t f 6 L 1 ( R ) . The open s e t Et i s a d i s j o i n t u n i o n o f open i n t e r v a l s I . : i t s c o n n e c t e d compo3 n e n t s . L e t u s l o o k a t one of t h e I . ' s , and l e t u s c a l l i t I . 3 Take any compact s e t K c I . For e a c h x 6 K , we have some compact i n t e r v a l Q, c o n t a i n i n g x i n i t s i n t e r i o r and s a t i s f y i n g

Since

Q,

c E,,

it follows t h a t

Qx c I .

Since

K

i s c o m p a c t , we

c a n c o v e r i t w i t h t h e i n t e r i o r s of j u s t f i n i t e l y many o f t h e QI's, s a y { Q j I . We c a n even assume t h a t t h i s f i n i t e c o v e r i n g i s minimal i n t h e s e n s e t h a t n o Q j i s s u p e r f l u o u s . Then, no p o i n t i s i n more t h a n two of t h e i n t e r i o r s o f t h e Q j ' s . I t f o l l o w s t h a t :

11.1. HARDY-LITTLEWOOD MAXIMAL FUNCTION

Since this is true for every compact

+I

III

(1.1)

I

KcI, we obtain:

If(Y)IdY

This implies,in particular, that I is bounded. Then, since b c 7 and b 6 Et, we can write: 1 l-q

Finally

(1.1)

1

I

135

If(y)Idy

5 Mf(b1

Let

I

=

(a,b)

.

6 t

implies:

We have obtained the following result: L e t f 6 L'(R) . T h e n , f o r e v e r y t > 0 , t h e s e t R : Mf(x) > t } c a n b e w r i t t e n a s a d i s j o i n t u n i o n of b o u n d e d o p e n i n t e r v a l s I. s u c h t h a t , f o r e v e r y j = 1 , 2 , ... : THEOREM

Et

=

{x

1.2.

6

1'

and. a s a conseauence:

Now we seek an analogue of theorem extension is not straightforward.

1.2

in dimension

n > 1.

The

Let f 6 L1(IRn) , n > 1 , and let t > 0. Instead of looking at the maximal function Mf, we shall try to obtain directly a family of cubes { Q j } such that the average of If I over each is comparable to t in the sense that a relation like ( 1 . 3 ) holds. This is quite easy and it will be done most effectively by considering only

11. CALDERON-ZYGMUND THEORY

136

d y a d i c c u b e s . For k 6 ZL formed by t h o s e p o i n t s of

,

we c o n s i d e r t h e l a t t i c e A k = 2-kZIn IRn whose c o o r d i n a t e s a r e i n t e g r a l mul-

Let Dk b e t h e c o l l e c t i o n of t h e c u b e s d e t e r m i n e d t i p l e s of 2 - k . by Ak, t h a t i s , t h o s e c u b e s w i t h s i d e l e n g t h Z m k and v e r t i c e s 03

The c u b e s b e l o n g i n g t o D =-k Dk a r e c a l l e d d y a d i c c u b e s . in Ak. Observe t h a t i f Q , Q ' 6 D and ( Q ' ( 2 I Q I , then e i t h e r Q ' c Q o r e l s e Q and Q ' do n o t o v e r l a p (by which we mean t h a t t h e i r interiors are disjoint).

Each

Q

6

Dk

i s t h e u n i o n of

2"

non-

We s h a l l c a l l ct t h e f a m i l y overlapping cubes belonging t o D k + l . formed by t h e c u b e s Q 6 D which s a t i s f y t h e c o n d i t i o n : (1.4)

t <

1 -

IQI

Q

If(x)Idx

and a r e maximal among t h o s e which s a t i s f y i t . Every Q c D s a t i s fying ( 1 . 4 ) i s c o n t a i n e d i n some Q ' 6 C t b e c a u s e c o n d i t i o n imposes an u p p e r bound on t h e s i z e of Q ( I Q I 2 t - ' ~ ~ f ~ ~ l ) . (1.4) The c u b e s i n C t a r e , by d e f i n i t i o n , n o n o v e r l a p p i n g . Also, i f Q 6 Dk is in Ct and Q ' i s t h e only cube i n Dk-, containing Q,

we s h a l l h a v e :

but, since

I Q ' I = 2"1Q1,

we g e t :

We have a c h i e v e d o u r p u r p o s e by o b t a i n i n g a f a m i l y cubes such t h a t , f o r every

Ct

= {Qj}

of

j:

N e x t , w e s h a l l i n v e s t i g a t e t h e r e l a t i o n w i t h t h e maximal f u n c t i o n M f . Suppose x 6 IRn i s such t h a t M f (x) > t . There w i l l be some c u b e

R

containing

x

i n i t s i n t e r i o r and s a t i s f y i n g

11.1. HARDY-LITTLEWOOD MAXIMAL FUNCTION

137

We l o o k f o r a d y a d i c c u b e of c o m p a r a b l e s i z e o v e r which t h e a v e r a g e

of

If1

i s comparab y b i g .

Let

T h e r e i s a t most one p o i n t of

k

be t h e only i n t e g e r such t h a t

interior to

Ak

R.

Consequently,

of them, m e e t i n g t h e t h e r e i s some c u b e i n D ~ , and a t most 2" i n t e r i o r of R . I t f o l l o w s t h a t t h e r e i s some Q 6 D k m e e t i n g t h e i n t e r i o r of R a n d s a t i s f y i n g : '

Since

IR1 5

191

we h a v e :

< Zn1RI,

and t h e r e f o r e :

I t follows t h a t

Q c Q.

J

6

C

4-"t

f o r some

I n g e n e r a l f o r any c u b e Q and any t h e cube w i t h t h e same c e n t e r a s Q

j

I

a > 0 , we s h a l l d e n o t e by Q" but with s i d e length a times

I n o u r p a r t i c u l a r s i t u a t i o n , s i n c e R and Q meet t h a t of Q . 3 and I R I 5 I Q I , i t f o l l o w s t h a t R c Q3 c Q j . We c o n c l u d e t h a t i f and t h i s l e a d s t o the estimate c E t c U Q ? t h e n = {Qj}, 4-"t 3 1

We h a v e o b t a i n e d t h e f o l l o w i n g r e s u l t : THEOREM

1.6.

n

Et = {x G IR of c u b e s { Q i }

Let f 6 L1(Rn). Then, f o r every t > 0 , t h e s e t : M f (x) > t } i s c o n t a i n e d i n t h e u n i o n o f a f a m i l y w h i c h r e s u l t f r o m e x p a n d i n g b y a f a c t o r of

nonoverlapping cubes

{Qj}

which s a t i s f y :

3

the

11. CALDERON-ZYGMUND THEORY

138

I t follows t h a t :

where t h e c o n s t a n t

C

d e p e n d s o n l y on t h e d i m e n s i o n

n.

0

We s h a l l d e r i v e some c o n s e q u e n c e s o f t h e b a s i c i n e q u a l i t y ( 1 . 8 ) which i l l u s t r a t e t h e r o l e p l a y e d by t h e maximal o p e r a t o r M . The importance of t h e o p e r a t o r M stems from t h e f a c t t h a t i t c o n t r o l s many o p e r a t o r s a r i s i n g n a t u r a l l y i n A n a l y s i s . A s a n e x a m p l e , we a r e going t o prove a n e x t e n s i o n of Lebesgue's d i f f e r e n t i a t i o n theorem.

.

1.9.

THEOREM

Q(x;r)

=

{y

every

x

G

G

IR":

1 L e t f G L ~ ~ ~ ( I RFX ~ ) x G IR" & a r > 0 , let n : IR ( y - x l , :m a x l y . - x . ) 5 r l . T h e n , f o r a l m o s t j J 1

We may assume

Proof:

f o r every

f

G

L1(IRn).

I t w i l l b e enough t o

show t h a t ,

t > 0, t h e set

h a s z e r o measure. m precisely U Al,j.

I n d e e d , t h e s e t where

(1.10)

does not h o l d , i s

j=1

Given

E

>O,

we c a n w r i t e

compact s u p p o r t and

Therefore

llhl <

f = g + h, E.

For

is continuous w i t h w e c l e a r l y have:

where g

g

MAXIMAL FUNCTION

11.1. HARDY-LITTLEWOOD

But

1

{x

6

R" : M h ( x )

11 5 C I I h l ( l / t < C

> t/2

E

139

/t

and

Thus

i s c o n t a i n e d i n a set of measure 5

At

b e done f o r any

we g e t

> 0,

E

lAtl

The p o i n t s x f o r which ( 1 . 1 0 ) f o r f . We c a n r e p h r a s e t h e o r e m point

x

G

COROLLARY

a)

1

whilst If

x 3

IQ(x;r)

b)

f.

Then, f o r e v e r y Lebesgue a.e.

point

x

G

IR":

f ( y ) dy

In o r d e r t o prove

\ Q ( x ; r )I

Q1

f

f(x) = l i m

Proof:

I

for

1

f G Lloc(Rn). and, therefore, f o r Let

il

0.

h o l d s a r e c a l l e d Lebesgue p o i n t s 1 . 9 by s a y i n g t h a t a l m o s t e v e r y

i s a Lebesgue p o i n t f o r

IRn

1.11.

x

point

=

Since t h i s can

CE/t.

f(y)dy

a ) , just note that

-

f(x)

1

5

IQ(x;r) I

i s a n immediate c o n s e q u e n c e o f

JQ(x;r) If(y) a).

-

f(x)Idy

'0

f and we have a sequence of cubes fl Q . = { X I , t h e n :

i s a Lebesgue p o i n t f o r Q,

3

....

with

1

'

3

f(x) = l i m flyldy Qj j+w

IQj(

Indeed i f

Q. 3

has side length

rj,

we h a v e

Qj c Q(x;rj)

and

11. CALDERON-ZYGMUND THEORY

140

rn = I Q . J J

I

J.

0,

so t h a t

r. J

-+

Therefore

0.

Ct(f) = IQ.1 be the c o l l e c t i o n 3 f o r m e d by t h o s e maximal d y a d i c c u b e s o v e r w h i c h t h e a v e r a g e o f If( i s > t ( c a l l e d Calderh-Zygmund cubes f o r f corresponding t o t ) . Let

f 6 L1(IRn)

and l e t

Ct

I

Let x $ U Q . Then t h e a v e r a g e o f If o v e r any d y a d i c c u b e w i l l be 5 t. "Let {Rk} b e a s e q u e n c e o f d y a d i c c u b e s o f d e c r e a s i n g size such t h a t Rk = { X I . F o r e a c h o f them

'

ft

I f , b e s i d e s , x i s a L e b e s g u e p o i n t f o r f ( a n d h e n c e f o r I f ( ) we g e t , by p a s s i n g t o t h e l i m i t : I f ( x 1 5 t . Thus I f ( x ) 6 t f o r

I

I

a.e. x $ U Q j . 3

IRn i n t o a s u b s e t R made up o f n o n If I i s o v e r l a p p i n g cubes Q j o v e r e a c h of which t h e a v e r a g e of between t and Z n t , a n d a c o m p l e m e n t a r y s u b s e t F where (f(x)1 5 t a.e., i s t h e f i r s t s t e p of t h e s o - c a l l e d Calder6n-Zygmund d e c o m p o s i t i o n w h i c h w i l l b e a t o o l o f c o n s t a n t u s e i n t h e s e q u e l . Let u s r e c o r d t h e f o l l o w i n g : The s p l i t t i n g o f t h e s p a c e

THEOREM

1.12.

Given

non-overlapping cubes

f c L1(Rn) Ct = C t ( f )

&

t > 0,

t h e r e i s a f a m i l y of

c o n s i s t i n g of t h o s e m a x i m a l d y a -

If

d i c c u b e s o v e r w h i c h t h e a v e r a g e of

I

> t.

T h i s famiZy

satisfies: (il

f o r every

(ii)

f o r a.e.

Q

x

Besides, f o r every where

f

ct

: t <

UQ,

t > 0,

ranges over

where

1

1

IQI

Q

Q

I f ( x ) Idx 6 2"t

ranges over

ct,

is

If(x)l 6 t.

E t = { X c R n : M f (x) > t } c U Q 3

c4-nt.

0

11.1. HARDY-LITTLEWOOD MAXIMAL FUNCTION

141

Next we a r e g o i n g t o s t u d y a u s e f u l g e n e r a l i z a t i o n o f t h e maximal f u n c t i o n . Let p b e a p o s i t i v e B o r e 1 m e a s u r e on IR", f i n i t e on compact s e t s and s a t i s f y i n g t h e f o l l o w i n g "doubling" c o n d i t i o n :

f o r e v e r y c u b e Q , w i t h C > 0 i n d e p e n d e n t o f Q.We s h a l l o f t e n s a y s i m p l y t h a t . 1-1 i s a d o u b l i n g m e a s u r e . T h i s i m p l i e s , o f c o u r s e , t h a t f o r e v e r y c1 > 0 , t h e r e i s a c o n s t a n t C = C c1 > 0 , d e p e n d i n g o n l y on c1 , s u c h t h a t p(Q') S Cp(Q) f o r e v e r y cube Q. Since w e a r e i n R n , t h e f i n i t e n e s s o f p on c o m p a c t s u b s e t s i m p l i e s p ( Q ) > 0. t h a t p i s r e g u l a r . Notice t h a t f o r every cube Q , I n d e e d , i f w e h a d p(Q) = 0 f o r some c u b e Q , we w o u l d h a v e k p ( Q ) 2 Ckp(Q) = 0 , f r o m w h i c h p(lRn) = 0 , w h i c h i s e x c l u d e d a s trivial.

Now, f o r

p

as above,

f

G

1

Lloc(p)

and

x

G

IRn

,

define:

w h e r e t h e s u p . i s t a k e n o v e r a l l c u b e s Q c o n t a i n i n g x . As b e f o r e , we o b t a i n t h e same v a l u e Mu f ( X I i f we j u s t t a k e i n t h e d e f i n i t i c q those cubes Q containing x i n t h e i r i n t e r i o r . This is a consequence of t h e r e g u l a r i t y o f p .

Let f E L 1 ( p ] a n d t > 0 . We w a n t t o o b t a i n a CalderBn-Zygmund decomposition f o r f and t r e l a t i v e t o t h e measure p and, a t t h e same t i m e , w e w a n t t o e s t i m a t e t h e p - m e a s u r e o f t h e s e t E t = { x G IRn : Mu f ( x ) > t l , w h i c h i s , o f c c u r s e , o p e n . We a r e g o i n g t o a p p l y t h e same i d e a s t h a t l e d t o t h e o r e m 1 . 1 2 . We n e e d t o make two o b s e r v a t i o n s .

First, we a r e g o i n g t o see t h a t t h e r e i s a c o n s t a n t K > 1 s u c h C it follows t h a t t h a t , e v e r y t i m e we h a v e d y a d i c c u b e s Q ' + Q , p(Q) 2 Kp(Q'). To see t h i s , l e t Q" be a dyadic cube contained i n Q, contiguous t o Q' a n d w i t h t h e same d i a m e t e r . Then Q ' c Q 1 1 3 a n d c o n s e q u e n t l y , f o r C = C 3 we h a v e : i i ( Q ' ) 2 C p ( Q " ) 5 C(p(Q) - p ( Q ' ) ) . T h i s i m p l i e s ( 1 + C ) u ( Q ' ) 6 Cp(Q), which g i v e s p ( Q ) 2 Kp(Q') w i t h K = ( 1 + C ) / C > 1 , and o u r claim is j u s t i f i e d . As a c o n s e q u e n c e , i f w e h a v e a s t r i c t l y i n c r e a s i n g s e q u e n c e

11. CALDERON-ZYGMUND THEORY

142

C

o f d y a d i c c u b e s Qo Q, Q2 k p(Qk) 2 K p(Qo) m as k m. of dyadic cubes is such t h a t t h e -+

...

,

we have t h e i n e q u a l i t y

The c o n c l u s i o n i s t h a t i f a c h a i n p - m e a s u r e o f t h e c u b e s i s bounded

-+

a b o v e , t h e n t h e i r d i a m e t e r i s a l s o bounded a b o v e o r , what i s t h e same, t h e c h a i n t c r m i n a t e s a t a g i v e n c u b e c o n t a i n i n g a l l t h e o t h e r s . The o t h e r o b s e r v a t i o n w e n e e d i s t h e f o l l o w i n g :

f o r every

t h e r e i s B > 0 s u c h t h a t e v e r y time we h a v e c u b e s Q a n d which meet a n d s a t i s f y 1 Q l < A I R I , then they a l s o s a t i s f y p(Q) < B p ( R ) . To s e e t h i s , j u s t n o t e t h a t Q c R' with c1 = 2A1'n t 1, and choose B a c c o r d i n g l y . Now we g o b a c k t o o u r p r o b l e m .

D e n o t e by

t i o n f o r m e d by t h e maximal d y a d i c c u b e s

Since t h i s condition forces

Q

A > 0

R

C = Ct(f;v) the collect satisfying the condition

t o b e bounded by

p(Q)

t - ' lRn ( f ( y ) ( d p ( y ) c m , O U T f i r s t o b s e r v a t i o n i m p l i e s t h a t e v e r y d y a d i c c u b e s a t i s f y i n g o u r c o n d i t i o n i s c o n t a i n e d i n some member o f Ct. Take Q G C t . I f Q G Dk a n d Q ' i s t h e o n l y c u b e i n Dk-, containing

Q,

Thus, f o r e v e r y

we h a v e

Q

6

C t:

Let x G E t , t h a t i s : Mp f (XI > t . Then t h e r e w i l l b e some c u b e R containing x i n i t s i n t e r i o r such t h a t

A s we d i d f o r t h e c a s e when

p = Lebesgue measure,

d y a d i c cube which o v e r l a p s w i t h

R

and s a t i s f i e s

let

Q

be a

I R I 6 IQ I < I R 1 2 "

11.1. HARDY-LITTLEWOOD MAXIMAL FUNCTION

iRnQI I

and

f dv > 2 - " t v ( R )

.

L e t B be t h e c o n s t a n t c o r r e s p o n d i n g t o s e r v a t i o n . Then

jRnQ I

' -'tv

I

f du > B-

2

143

A = 2

n

i n o u r s e c o n d ob-

(Q)

and h e n c e

Q c Qj

I t follows t h a t = I f c i-n - 1 B t

,

f o r some Q j 6 3 then Et c U Q . . j J

'2 - " B - ' t

and

R c Q3 c Q 3j .

We f i n a l l y o b t a i n t h e e s t i m a t e :

This b a s i c estimate

c a n be u s e d t o e x t e n d t h e o r e m

1 . 9 and i t s

corollary, obtaining: THEOREM

With

1.14.

almost every

x

6

li

IR"

as above, l e t

f

( w i t h r e s p e c t t o 111

Then, f o r

:

u

C)

Ct(f,v) = CQ.1, I (with respect t o

In p a r t c u l a r , i f a.e.

1

Lloc(p).

f

x B U Q. j J

we have

v).

If(x)l S t

for

We c a n f i n a l l y s t a t e t h e f o l l o w i n g : THEOREM

1.15.

For

1-1

a s above,

let

t h e r e i s a f a m i l y of n o n - o v e r l a p p i n g

1 f 6 L ( p ) g& t > 0. Then, consistcubes Ct = c t ( f , p ) ,

144

11. CALDERON-ZYGMIJND THEORY

If\

i n g of t h o s e m a x i m a l d y a d i c c u b e s o v e r w h i c h t h e u v e r a g e of relative t o

il

Q

for e v e r y

for a.e. x iil -

wit

fi

p

Here

UQ

@

f

ctIc,

which s a t i s f i e s

: t

V(Q)

Q

where

IR":

M

P

jQI f l d p

5 Ct

ranges over

If ( x )

we h a v e :

pI

Et = { x

ranges over

Ct

G

respect t o

the set

> t,

f (xj>t}

I

6 t.

ct

(and

a.e.

i s

Besides, for every t >

UQ3

is c o n t a i n e e

where

1,

Q

and we h a v e a n e s t i m a t e :

r e p r e s e n t s an a b s o l u t e c o n s t a n t , p o s s i b l y d i f f e r e n t a t each

C

occurrence.

2.

NORM ESTIMATES FOR THE MAXIMAL FUNCTION

THEOREM

t

0.

2.1.

Let

b e a m e a s u r a b l e f u n c t i o n on

IR" a n d l e t

Then we h a v e t h e f o l l o w i n g e s t i m a t e s f o r t h e L e b e s g u e mea-

E t = { x c IR":

s u r e of t h e s e t

with constants

Proof:

f

Write

f,(x) = 0

C

and C '

f = f, + f2,

otherwise.

Then

M f (x) > t}:

w h i c h d o n o t d e p e n d on

f

or

t

where f , ( X I = f ( x ) i f I f ( x ) l > t / Z , a n d M f(x) 5 M f ,

( x ) + M f 2 ( x ) 6 M f l (x)+

11.2. NORM ESTIMATES FOR THE MAXIMAL FUNCTION

since

+ t/2,

which g i v e s

f2 2 t/2

implies t h a t

Mf2 2 t / 2

also.

145

Thus

(2.2).

As f o r (2.3), we may a s s u m e t h a t f 6 L 1 ( R n ) ( o t h e r w i s e we t r u r r c a t e a n d a p p l y a l i m i t i n g p r o c e s s ) . Then we u s e t h e C a l d e r B n Zygmund d e c o m p o s i t i o n f o r f a n d t . We h a v e n o n - o v e r l a p p i n g c u b e s Q j9

such t h a t

f o r every X

f

Qj

j,

and

If(x)I 5 t

implies t h a t

x d UQ.. J J w rite: we c a n

for

Mf(x) > t ,

a.e

Now, s i n c e

1% and

(2.3)

i s proved with

= 2-".

C'

c]

The n e x t r e s u l t i s p r o v e d i n e x a c t l y t h e same way.

THEOREM

Rn

2.4.

Suppose

p

i s a r e g u l a r p o s i t i v e Borel measure i n

s a t i s f y i n g a "doubling" condition l i k e

are constants,

C,C',

(1.13).

Then, t h e r e

such t h a t , f o r any Borel f u n c t i o n

f

and any

t > 0:

we c a n e a s i l y d e r i v e s e v e r a l norm e s t i m a t e s f o r t h e maximal f u n c t i o n .

From t h e o r e m

THEOREM

c

P

> 0

2.1.

p such t h a t , f o r every 2.5.

For e v e r y

with f

G

1 < p <

L p ( R n ):

m,

t h e r e is a c o n s t a n t

11. CALDERON-ZYGMUND THEORY

146

I n e x a c t l y t h e same way we o b t a i n t h e f o l l o w i n g THEOREM

Let

2.6.

1~

be a r e g u l a r p o s i t i v e Bore1 measure i n

s a t i s f y i n g a "doubZing" c o n d i t i o n l i k e with

p

1

C

there i s a constant

a,

(1.131.

f e ~p(p):

P

> 0

Then, f o r each

Rn p

such t h a t f o r every

We h a v e s e e n t h a t t h e o p e r a t o r bl i s b o u n d e d i n L p ( R n ) f o r 1 < p 5 w. H o wev er , i t i s n o t b o u n d e d i n L 1 ( I R n ) A s a matter of f a c t , f o r f 2 0 , Mf f a i l s t o b e i n L 1 u n l e s s f ( x ) = 0 f o r a . e . x , s i n c e Mf(x) 2 Clxl-" f o r l a r g e x , with C > 0 i f f

.

is not t r i v i a l .

Th e b a s i c e s t i m a t e

(1.8)

L 1- b o u n d e d n e s s , a s w i l l b e s e e n b e l o w .

a c t s a s a substitute €or As regards local integrabi-

l i t y o f t h e m a x i mal f u n c t i o n , we h a v e t h e f o l l o w i n g r e s u l t : THEOREM

Mf

I f ( x ) I l o g + l f ( x ) Idx <

(2.8) Proof:

f b e an i n t e g r a b l e f u n c t i o n s u p p o r t e d i n u b u l l i s i n t e g r a b l e o v e r B i f a n d o n l y if:

Let

2.7.

B c RP.=

If

m.

(2.8) holds, then co

Mf(x)dx = j,l{x

m

f

B : M f ( x ) > t l / d t = Z j I{x e B : Mf(x1 > Z t } / d t 5 0

11.2. NORM ESTIMATES FOR THE MAXIMAL FUNCTION

147

O b s e r v e t h a t f o r t h i s p a r t o f t h e p r o o f we do n o t n e e d t o u s e t h e f a c t t h a t f i s supported i n B. I n d e e d , t h e same p r o o f shows t h a t i f IIRn If ( x ) I l o g + l f ( x ) ldx < a, which we s h a l l i n d i c a t e by s a y i n g f E L l o g L(IRn 1 ,

that

then

Mf

is locally integrable.

Going b a c k t o t h e p r o o f of t h e t h e o r e m , s u p p o s e t h a t IBMf(x)dx < m . I f we d e n o t e by B ' the b a l l concentric with 5 b u t w i t h r a d i u s t w i c e a s b i g , we c a n e a s i l y s e e t h a t I B I M f ( x ) d x < I t is sufficient to realize that for

*

x

6

m.

Mf(x) 5 CMf(x*),

B'W,

i s t h e p o i n t s y m m e t r i c t o x w i t h r e s p e c t t o t h e bounwhere x d a r y o f B. I n t h e complement o f B ' , Mf i s b o u n d e d , s i n c e we

a r e a t l e a s t a f i x e d d i s t a n c e f a r from t h e s u p p o r t of f . I t is a l s o o b v i o u s t h a t Mf(x) 0 as 1x1 a. Thus, we c o n c l u d e t h a t , +

f o r any

+

to > 0

Mf(x)dx < jfx:Mf ( x ) > t O }

m.

to = 1 , t h i s i n t e g r a l i s b i g g e r t h a n o r e q u a l t o

For

/ ; I {X

: Mf(x) > t l l d t 2

Thus

(2.8)

( f(x) ( d x d t =

h o l d s and t h e p r o o f i s c o m p l e t e .

The t h e o r e m e x t e n d s c l e a r l y t o doubling condition.

Mp

0

f o r a measure

!J

satisfying a

T h e r e i s no n e e d t o w r i t e a new s t a t e m e n t .

Suppose now t h a t we hav.e two m e a s u r e s p a c e s w i t h r e s p e c t i v e m e a s u r e s 1-1

and

Lq(v), (2.9)

Then,

V

, and t h a t

that is:

T

i s an o p e r a t o r bounded from

LP(u)

to

11. CALDERON-ZYGMUND THEORY

148

a n d we o b t a i n v({x : ITf(x)

(2.10)

I

> t}) 5

[ c II t II LP f

When T s a t i s f i e s ( 2 . 1 0 ) we s a y t h a t t h e o p e r a t o r T i s o f weak t y p e (p,q) w i t h r e s p e c t t o t h e p a i r o f m e a s u r e s ( v , p ) . F o r example, ( 1 . 8 ) i s r e a d by s a y i n g t h a t M i s o f weak t y p e ( 1 , l ) ( w i t h r e s p e c t t o L e b e s g u e m e a s u r e ) . However, we know t h a t M f a i l s t o b e bounded i n L 1 . I n g e n e r a l , (2.10) may h o l d w h e r e a s ( 2 . 9 ) does n o t hold f o r a given operator T. I t i s c o n v e n i e n t t o see (2.10) a s a s u b s t i t u t e o r a weakening of ( 2 . 9 ) . With t h i s i n m i n d , when ( 2 . 9 ) h o l d s , w e s a y t h a t T i s o f s t r o n g t y p e (p,q) w i t h r e s p e c t t o t h e p a i r of measures ( v , p ) . Sometimes i t i s c o n v e n i e n t t o i n d i c a t e t h a t ( 2 . 1 0 ) h o l d s by s a y i n g t h a t T s e n d s L p ( p ) b o u n d e d l y i n t o L:(v). T h i s means t h a t w e a r e c o n s i d e r i n g t h e s p a c e ( c a l l e d weak-Lq(v))

L:(v)

f o r m e d by t h o s e f u n c t i o n s

g

f o r which

T h i s s p a c e w i t h t h e "norm" n ( f a i l s t o be s u b a d d i t i v e even though 9 we h a v e a s i m i l a r i n e q u a l i t y w i t h a c o n s t a n t i n t h e r i g h t h a n d s i d e ) is bigger (topologically a l s o ) than

Lq(v).

Weak type i n e q u a l i t i e s such a s (2.10) can be used t o o b t a i n s t r o n g t y p e i n e q u a l i t i e s . T h i s i s w h a t we h a v e d o n e t o p r o v e t h e o r e m 2 . 5 . We a r e g o i n g t o p r e s e n t a r e s u l t , w h i c h i s a p a r t i c u l a r c a s e o f t h e M a r c i n k i e w i c z i n t e r p o l a t i o n t h e o r e m a n d i s b a s e d upon t h e same i d e a a s o u r proof of 2.5. 2.11.

THEOREM

measures

S u p p o s e we h a v e t w o m e a s u r e s p a c e s w i t h r e s p e c t i v e

p

L'o(~) + ~ P 1 ( p ) pose t h a t :

ii

T

v

to

.

T

b e an o p e r a t o r s e n d i ' n g f u n c t i o n s i n

+measurable

f u n c t i o n s , 1 6 po < p 1 5

i s subadditive, that i s , f o r

fl,f2

6

m.

L PO ( p ) + L p ' ( p ) ,

%-

11.2. NORM ESTIMATES FOR THE MAXIMAL FUNCTION

ii) T

iii) T p1 < m,

i s of weak t y p e whiZe, i f

def i n it i o n :

IITf

Then, f o r every

(p,p)

(po,po),

i s of weak t y p e

~

p1 =

11 Lm (") p

149

that i s :

(pl,pl) w h i c h means t h e same a s a b o v e i f weak t y p e and s t r o n g t y p e c o i n c i d e by

m,

5 Clllfll

such t h a t

Lrn ( p 1

p,

< p < pl,

T

i s of s t r o n g t y E

/lTflPdv 5 C /Iflpdp. P

that i s :

P r o o f : F i x p w i t h Po < P < P l t a n d l e t f G L p ( u ) c L p o ( p ) + L p l ( p ) . t For e v e r y t > 0 w r i t e f ( x ) = f ( x ) + f t ( x ) where f ( x ) = f ( x ) t if I f ( x ) > t a n d f ( x ) = 0 o t h e r w i s e . Clearly f t 6 LpO(u),

I

I

and s i n c e I f t ( x ) 1 t , f t 6 L p l ( p ) . S u p p o s e p 1 < m. Then, we c a n w r i t e : s i n c e ITf(x) 1 5 IT(ft) (x) 1 + I T ( f t ) (XI

I,

Thus Tf(x) 1 2 P o C o p ~ ; l t P - P " -1

I

> t1)dt 5

L:

If(x

x) =

P-Po

J '

P1-P

J

11. CALDERON-ZYGMUND THEORY

150

For t h e c a s e

where

2-1

=

CL

p, =

=

2

=

c

PO

P

we s i m p l y o b s e r v e t h a t

m y

and,consequently

C;'

co P j

1

Jf(x1

N e x t we s h a l l e s t a b l i s h a g e n e r a l i n e q u a l i t y f o r t h e maximal f u n c t i o n . This i n e q u a l i t y involves a weight f u n c t i o n

@(x).

THEOREM 2 . 1 2 . F o r e v e r y p __ with 1 < p < m y there i s a constant s u c h t h a t , f o r a n y m e a s u r a b l e f u n c t i o n s o n IRn @ 2 0 and f , -

C

P

we h a v e t h e i n e q u a Z i t y :

Proof:

E x c e p t when

trivially,

M@

M$(x) =

m

a.e.,

i n which case

is t h e d e n s i t y of a p o s i t i v e measure

(2.13)

holds

p(du(x) =

i s t h e d e n s i t y of a n o t h e r p o s i t i v e measure ( 2 . 1 3 ) means t h a t M i s a b o u n d e d o p e r a t o r f r o m

M$(x)dx) and @ w(dv(x) = $(x)dx). =

Lp(u) t o LP(v). This i s c l e a r l y t r u e f o r p = m. Indeed, i f M$(x) = 0 f o r some x , t h e n $ ( X I = 0 a . e . a n d L m ( v ) = 0 , s o t h a t nothing needs t o be proved. I f M$(x) > 0 f o r e v e r y x a n d

, we h a v e IIx : [ f ( x ) l > a l l = 0 which

Mf(x) 5

c1

a.e.

1

t If I > c d

M$ = 0

and consequently

o r , w h a t i s t h e same, Thus

I/MflI

5

If(x)I 5 a a.e., c1

from

and, f i n a l l y ,

L"(V)

Having t h e

(m,m)

result,

i f we a r e a b l e t o show t h a t

M

i s of

11.2. NORM ESTIMATES FOR THE MAXIMAL FUNCTION

151

weak t y p e ( 1 , l ) w i t h r e s p e c t t o t h e p a i r ( v , p ) , the interpolation theorem 2 . 1 1 w i l l give (2.13). T h u s , a l l we n e e d i s t o show t h a t :

We c a n o b v i o u s l y assume t h a t f t 0 . We c a n a l s o assume t h a t f G L1 (Wn) I n d e e d , we c a n f i n d i n t e g r a b l e f u n c t i o n s f . such 3' and o b s e r v e t h a t t h a t f l S f 2 5 --- + f a . e .

.

{ X : Mf(x) > t j = U j

{X

: Mf.(x) > t } .

J

f G L1(IRn) a n d f b 0 . Given t > 0 , we know (Theorem t h a t t h e r e i s a f a m i l y of n o n - o v e r l a p p i n g c u b e s { Q j } s u c h

So, l e t

1.6)

t h a t , f o r each

j:

and a l s o {X

: Mf(x) > t } c U Q3 j J

Then

3"*4"

5 t J

f ( x ) ( M @ ) ( x ) d x6

C

Qj

f(x)(M@)(x)dx

.

fl

The t h e o r e m we h a v e j u s t p r o v e d i d e n t i f i e s a whole c l a s s o f w e i g h t f u n c t i o n s @ f o r which t h e o p e r a t o r M i s bounded i n L p ( @ ) f o r every

p

E

(1

,a]

and o f weak t y p e

(1,l)

with respect t o

$,

namely, t h e c l a s s , c u s t o m a r i l y d e n o t e d by A 1 , o f t h o s e @ 2 0 s a t i s f y i n g M@(x) 2 C @ ( x ) a . e . f o r some c o n s t a n t C . For i n s t a n c e , i t i s e a s y t o c h e c k t h a t @ ( x ) = 1x1' i s an A 1 w e i g h t i n IRn provided

-n <

c1 S

0.

We s h a l l 20 back t o w e i g h t s i n c h a p t e r

IV.

11. CALDERON-ZYGMUND THEORY

152

T h e r e i s a n i n t e r e s t i n g e x t e n s i o n of t h e o r e m 2 . 1 2 whose p r o o f i s b u t a r e p e t i t i o n o f t h e a r g u m e n t s which l e d t o 1 . 6 , 2 . 5 and 2.12. I n o r d e r t o p r e s e n t t h i s r e s u l t we make s e v e r a l d e f i n i t i o n s : IR", we d e f i n e a f u n c t i o n , t 2 01 by s e t t i n g

Given a f u n c t i o n f in = { ( x , t ) : x G IRn

I

M f ( x , t ) = sup{--- 1

IQI

If(y)ldy :

Q

Given a p o s i t i v e B o r e 1 m e a s u r e ~ ( p ) in Wn by s e t t i n g

in

1-1

x

Wn+ 1

where t h e s u p i s t a k e n o v e r a l l c u b e s cube

+

Q

Q

G

,

Mf

in

and s i d e l e n g h t ( Q ) t t ) we d e f i n e a f u n c t i o n

containing

x

and f o r a

Q, 'L

Q = {(x,t)

that is,

'L

Q

f

IRn++l : x

G

Q

--

i s t h e cube i n

Rn++l

and

0 5 t 5 side length

having

Q

as a face.

(Q))

,

With t h e

above d e f i n i t i o n s , we c a n s t a t e t h e f o l l o w i n g : THEOREM

C

P

2.15.

p

For e v e r y

such t h a t , f o r every

f

with

1 < p <

and e v e r y

m,

t h e r e is a c o n s t a n t

p:

B e f o r e we s t a r t t h e p r o o f , l e t u s n o t e t h a t t h i s r e s u l t i n c l u d e s t h e p r e v i o u s o n e . J u s t t a k e d p ( x , t ) = Q(x)dxrndG(t) where

Proof:

i s t h e u n i t mass c o n c e n t r a t e d a t t h e o r i g i n i n t h e t a x i s . Then since M f ( x , O ) = Mf(x) a n d , f o r o u r p a r t i c u l a r p , U p ( x ) = M Q ( x ) , (2.16) reduces t o (2.13) i n t h i s case. 6

Now we p r o v e t h e t h e o r e m . A s i n t h e p r o o f o f t h e p r e c e d i n g r e s u l t , i f we e x c l u d e t h e t r i v i a l c a s e when flp(x) = m a . e . , Up(x)dx is a measure and M i s bounded f r o m L a ( N p ( x ) d x ) t o Lm(IRn++' , p ) . A l l we n e e d t o do i s t o p r o v e a weak t y p e ( 1 , l ) & q u a l i t y and t h e n

u s e i n t e r p o l a t i o n . I f we c a l l E~ = { ( x , t ) G IR"++' : M f ( X , t ) > c r ~ , we h a v e t o show t h a t t h e r e i s a c o n s t a n t C s u c h t h a t , f o r e v e r y a > o

153

11.2. NORM ESTIMATES FOR THE MAXIMAL FUNCTION

is integrable. R and s u c h t h a t

Let us s e e t h i s .

W e may assume t h a t

and s u p p o s e t h a t

(x,t)

6

Ea.

x , w i t h s i d e l e n g t h (R) t t

Let

k

f

Fix

Then, t h e r e i s a c u b e

be t h e only i n t e g e r such t h a t

< ]R1 4 2

2-(k+1)n

t h e proof t h a t l e d t o theorem 1.6, t h e r e i s some meets t h e i n t e r i o r o f R and s a t i s f i e s

Q

G

ct >

0

containing

-kn

Dk

.

As i n

which

so t h a t

3 3 I t f o l l o w s t h a t Q c Q j G C4-na f o r some j a n d x G R c Q c Q j . 3 On t h e o t h e r hand t S s i d e l e n g t h ( R ) 5 s i d e l e n g t h ( Q . ) , so t h a t "3 J ( x , t ) G Q j . Thus, we h a v e s e e n t h a t Ea c U where C4-n,= {Qj}. 1 Then

4

I n p a r t i c u l a r , i f t h e measure

f o r every cube ~ ( x S) C and

Q c IRn

(2.16)

i s such t h a t

p

with C independent of Q , then i m p l i e s t h a t f W Mf i s an o p e r a t o r boun-

f o r every p ded from Lp(IRn) t o LP(IRn++l , u ) A c t u a l l y , g i v e n any p w i t h 1 < p < m, (2.17) c i e n t but a l s o necessary f o r LP(Rn+" , P I . Indeed, s i n c e t h e b o u n d e d n e s s of

M

M M(x

with 1 < p < m. i s not only s u f f i -

t o b e bounded f r o m

Q

)(x,t) 2 1

implies t h a t

Lp(IRn) to 21 f o r every ( x , t ) G Q ,

11. CALDERON-ZYGMUND THEORY

154

P(Q plr ) 5 ja(ULXg)(x,t))Pdu(x,tl

5 ClQl

The i m p o r t a n c e of t h e o p e r a t o r M stems f r o m t h e f a c t t h a t c o n t r o l s t h e P o i s s o n i n t e g r a l o f f , P ( f ) , d e f i n e d by:

for

x

and

IR"

6

c

=

t > 0,

where

[IR"

J k

2 t
Actually,

Mf

for

ktl,

1

f 2 0

'L

In p a r t i c u l a r i f f = x Q a n d ( x , t ) G Q , we g e t P(X Q ) ( x , t ) 2 a n > O , w h e r e an d e p e n d s o n l y on t h e d i m e n s i o n n . T h e n , t h e same a r g u m e n t

11.3. SHARP MAXIMAL FUNCTION AND B.M.O.

155

t h a t we u s e d f o r M shows t h a t i f t h e o p e r a t o r f W P ( f ) i s b o unded f r o m L p ( I R n ) t o LP (Rn++l,p), t h e n p s a t i s f i e s ( 2 . 1 7 ) . The m e a s u r e s p s a t i s f y i n g ( 2 . 1 7 ) a r e c a l l e d C a r l e s o n m e a s u r e s . We c a n s t a t e t h e f o l l o w i n g : THEOREM

Let

2.18.

let

Then

i f and o n l y i f

(2.17)

IRn+l and i s a b o u n d e d o p e r a t o r from L P ( R n )

b e a p o s i t i v e B o r e 2 m e a s u r e on

p

1 < p < 03. p n + l ,p) to L (IR+

f W P ( f )

i f and o n l y i f

i s a C a r l e s o n m e a s u r e , that&,

p

h o l d s for some c o n s t a n t

C.

0

O b s e r v e t h a t t h e c o n d i t i o n o b t a i n e d d o e s n o t d e p e n d on p a n d i s a l s o e q u i v a l e n t t o t h e f a c t t h a t f H P ( f ) sends L1(Rn) boundedly i n t o L ,1( I- R + ,p).

3.

THE SHARP MAXIMAL F U N C T I O N AND THE SPACE

For a r e a l l o c a l l y i n t e g r a b l e f u n c t i o n

mal f u n c t i o n

is defined a t

f#

x

G

f IRn

where t h e s u p . i s t a k e n o v e r a l l t h e c u b e s f s t a n d s f o r t h e average of f o v e r Q,

Q

f

1

Q=lQl

Q

in

B.M.O. R",

t h e s h a r p m a xi-

by s e t t i n g

containing t h a t is:

Q

x,

a nd

f(x)dx.

The s h a r p maximal o p e r a t o r

f

H f#

i s a n a n a l o g u e o f t h e H a rdy-

L i t t l e w o o d maximal o p e r a t o r M , b u t i t h a s c e r t a i n a d v a n t a g e s o v e r i t wh ich we s h a l l p r e s e n t l y s e e . O f c o u r s e , f # ( x ) 5 Z M f(x). I t is a l s o c l e a r t h a t i n t h e d e f i n i t i o n o f f#(x) one c a n t a k e o n l y t h o s e cubes

Q

containing

x

in its interior

Jtctual l y

wh e re

i s u s e d t o i n d i c a t e t h a t e a c h s i d e i s bounde d b y t h e o t h e r

11. CALDERON-ZYGMUND THEORY

156

times an a b s o l u t e c o n s t a n t . ( 3 . 1 ) i s 5f'

I t i s c l e a r t h a t t h e r i g h t ha nd s i d e o f

F o r t h e o p p o s i t e i n e q u a l i t y we see t h a t

(x).

I t f o l l o w s t h a t f # ( x ) i s bounde d by t w i c e t h e (3.1) We a l s o n o t e t h a t :

f o r e v e r y a G IR. r i g h t hand s i d e of

.

( I f l ) # ( x ) 2 2f#(x)

(3.2) Indeed

dx 5 f # ( x ) f o r each

x

G

By u s i n g

Q

we g e t

(3.11,

(3.2).

is a function I f f i s such t h a t f # i s bounded, we s a y t h a t f o f bounded mean o s c i a l l a t i o n , a n d w e d e n o t e by t h e i n i t i a l s B . M . O . t h e s p a c e formed by t h e s e f u n c t i o n s . B.M.O.

We w r i t e For

f

G

{f

=

B.M.O.(IRn) B.M.O.

G

T hus

: f #

Lloc(IRn)

G

Lwl

when we n e e d t o s p e c i f y t h e u n d e r l y i n g s p a c e .

we w r i t e x)

Of c o u r s e we g e t a f t e r

-

f

Q

Idx.

(3.1):

Thus, i n o r d e r t o be a b l e t o say t h a t make s u r e t h a t t h e r e e x i s t s

C <

w

aldx 2

IIflI.

f 6 B.M.O., and, f o r each

it s u f f i c e s t o Q, a constant

11.3. SHARP MAXIMAL FUNCTION A N D B.M.O.

(1

157

T h i s i s t h e u s u a l way t o s e e t h a t a c e r t a i n

Then [If * 5 2C. f c B.M.O.

C l e a r l y , f W l l f I I * i s a seminorm a n d IIf I I * = 0 i f a n d o n l y i f f is constant. I t i s n a t u r a l t o consider the quotient space of B . M . O . modulo c o n s t a n t s , which i s a normed s p a c e a n d , a c t u a l l y a Banach s p a c e ( t h e c o m p l e t e n e s s i s l e f t a s a n e x e r c i s e f o r t h e I t c a n b e s e e n i n Neri [l]). reader. T h i s space of e q u i v a l e n c e c l a s s e s modulo c o n s t a n t s w i l l a l s o b e c a l l e d B . M . O . The a m b i g u i t y does n o t c a u s e any problem. Of course

Lm c B . M . O . a n d IIflI. 5 211fllm. However, t h e r e a r e B.M.O. f u n c t i o n s a s we s h a l l s o o n s e e . We s h a l l g i v e

u n bounded

two r e s u l t s w h i c h p r o v i d e many e x a m p l e s o f THEOREM

a.e.,

A,

3.3.

then

IQI

w,

i.e.

e'(x)dx

Q

the smallest

that is

log w = 4 ,

Call

1

functions.

If w i s an A1 w e i g h t , t h a t i s , i f Mw(x) i Cw(x) l o g w i s i n B.M.O. u i t h a norm d e p e n d i n g o n l y on t h e

constant f o r

Proof:

B.M.O.

w

=

C.

e'.

for

5 Ce'")

We h a v e , f o r e v e r y c u b e Q a.e. x c Q

or, equivalently

I ,Q 1

eo(x)dx]

e s s . s u p (e-"')) x f Q implies

But

- 11 IQI Thus,

e'(x)dx

'Q

exp(QQ

-

=

.

e s s . s ~ p . ( e - ' ( ~ ) )5 C x f Q

e x p ( - e s s . i n f . @ ( x ) ) , and J e n s e n ' s i n e q u a l i t y x f Q

2 exp($ )

Q

ess. i n f . '(x))

f i e s , f o r some o t h e r c o n s t a n t QQ

-

e s s . i n f . '(x) x f Q

5 C

C

and, consequently,

independent of

Q,

@

satis-

the property

6 C

We e x p r e s s t h i s by s a y i n g t h a t $ i s o f bounde d l o w e r o s c i l l a t i o n , t h e c l a s s f o r m e d by a l l t h e f u n c t i o n s o f a n d d e n o t e by B . L . O .

11. CALDERON-ZYGMUND THEORY

158

Now, we s e e t h a t

bounded l o w e r o s c i l l a t i o n .

Q

Averaging over

IQI

I,

Indeed

we o b t a i n

[ $ ( X I - OQldx

and t h e i n c l u s i o n

c B.M.O.

B.L.O.

B.L.O.

Observe t h a t t h e c l a s s

-

6 Z($Q

c B.M.0 B.L.O.

ess. i n f . +I

Q

0

follows readily.

i n t r o d u c e d above f a i l s t o b e a vec-

t o r s p a c e e v e n t h o u g h i t i s s t a b l e u n d e r t h e sum a n d t h e p r o d u c t by

.

a n o n - n e g a t i v e n u m b e r . A c t u a l l y B . L . O . n (-B.L.O .) = L Indeed, i f both and -0 a r e i n B . L . O . , w e h a v e a t t h e same t i m e m

+

QQ - e s s . i n f . $

Q

A d d i n g u p we g e t : cube

2 C

ess.sup @

Q

and

-

Theorem

3.3

gives us

B.M.O.

Q

Q

C

f u n c t i o n s from M.

6 C

independent of t h e

i s e s s e n t i a l l y bounded.

$J

p r e s e n t l y see a n i c e way t o p r o d u c e L i t t l e w o o d maximal o p e r a t o r

+ ess. sup. @

e s s . i n f $J 6

This is only possible i f

Q.

-$JQ

A1

weights.

We s h a l l

w e i g h t s by u s i n g t h e H a r d y -

A,

Let p b e a p o s i t i v e B o r e l m e a s u r e on ELn, f i n i t e on c o m p a c t s e t s , and hence r e g u l a r . I t makes s e n s e t o c o n s i d e r t h e m a xim a l f u n c t i o n Mp(x) where t h e

sup

=

sup xfQ

1

~

IQI

Q

dp

is taken over a l l cubes containing

x.

Exactly as

i n t h e c a s e of i n t e g r a b l e f u n c t i o n s , one o b t a i n s f o r measures t h e fundamental e s t i m a t e

with

d e p e n d i n g o n l y on t h e d i m e n s i o n .

C

THEOREM Mp(x) < w(x)

=

3.4. m

Let

for a.e.

p

We c a n s t a t e t h e f o l l o w i n g :

be a gositive Borel measure such t h a t

x

( ~ p ( x ) ) y is a n

G

IR",

and l e t

0 < y < 1.

T h e n t h e function

A, w e i g h t w i t h a c o n s t a n t d e p e n d i n g o n l y

on

11.3. SHARP MAXIMAL FUNCTION AND B.M.O.

y

159

a n d t h e d i m e n s i o n n.

Let

Proof:

1

IQI

with

Q

be a f i x e d cube. w ( x ) d x 5 Cw(x)

independent of

C

for

Q

Q.

We s h a l l s e e t h a t a.e.

x

G

Q

'Q =L Q 3 , t h e 3 - d i l a t e d of Q . We 'L P p , t h e r e s t r i c t i o n o f p t o Q. Q and, s i n c e 0 < y < 1 , a l s o

Let

write p = p, + p2 with p l = Then Mp(x) 5 Mpl(x) + MpZ(x)

( M ~ ( X ) )2 ~ ( M p l ( x ) ) ' + (Mp2(x))'. T h e r e f o r e , i t w i l l b e enough t o s e e t h a t t h e averages of (Mpl(x))' and ( M I . I , ( X ) ) ~ over Q a r e b o t h bounded by Cw(x) f o r any x c Q w i t h C d e p e n d i n g o n l y on y

and t h e d i m e n s i o n .

&[lo

R

=

We c a r r y o u t t h e s e two e s t i m a t e s s e p a r a t e l y :

I,]

m

+

b e s p l i t t h e i n t e g r a l by u s i n g an a r b i t r a r y

R > 0).

Near

0

we

u s e t h e t r i v i a l e s t i m a t e f o r t h e d i s t r i b u t i o n f u n c t i o n , which i s ,

IQI.

clearly, 5 From R t o m we u s e t h e weak t y p e e s t i m a t e , t h e d i s t r i b u t i o n f u n c t i o n i s 5 C t - 1 lip, 1 1 , where l l p l 11 = p 1 ( R n ) Thus

1-Y RIQI

f o r e v e r y x E Q . What we h a v e j u s t done i s t o r e a l i z e t h a t e v e r y in a f i n i t e measure s p a c e , a c t u a l l y o p e r a t o r of weak t y p e ( 1 , l ) t a k e s L 1 boundedly i n t o L p , i f p < 1 . T h i s f a c t i s known a s Kolmogorov's i n e q u a l i t y ( s e e a l s o C h a p t e r V , Lemma 2 . 8 . )

.

11. CALDERON-ZYGMUND THEORY

160

To d e a l w i t h

I t i s enough t o see t h a t ,

i s even s i m p l e r .

Mp2

b e c a u s e o f t h e fact. t h a t p 2 l i v e s f a r f r o m Q ( o u t s i d e a n y two p o i n t s x , y G Q ,we h a v e Mu2(x) 5 CMp2(y), w i t h i s a cube containing so t h a t

absolute constant. I n d e e d i f Q' m e e t i n g R n \ :, then Q c Q I 3 ,

&

'Q'

dP2 5 3n

lQt31

1Q

a), C

x

for an

and

1 3 d p 2 5 3"Mp2(y).

Thus

f o r every

x

For example, g i n i n Rn. x = (xl,

...

G

[I

Q.

take = 6 , t h e Dirac d e l t a o r u n i t mass a t t h e o r i Then M6(x) = Ixlmn where 1x1, = max I x j I f o r 15j5n , x n ) . Thus M6(x)

I t f o l l o w s t h a t f o r any Y w e i g h t , o r , i n o t h e r words

with 1x1'

0 5 Y < 1, Ixl-nY i s an i s a n A1 w e i g h t f o r a n y

A1 a

with

-n < a 5 0 ( o n l y f o r t h e s e a ' s a c t u a l l y , s i n c e w h a s t o b e l o c a l l y i n t e g r a b l e a n d w-' l o c a l l y b o u n d e d ) . However, o u r main c o n c e r n h e r e i s t h e f a c t t h a t l o g 1x1 i s a n e x a m p l e o f a n unboun1 is a c t u a l l y i n B.L.O. ded B.M.O. f u n c t i o n . Note t h a t l o g I n g e n e r a l we have COROLLARY

a)

3.5.

For any p o s i t i v e Bore1 measure

~ p ( x )< m f o r a . e . x G IR", l o g M ~ ( x ) w i t h norm d e p e n d i n g o n l y on t h e d i m e n s i o n . bl

log

1x1

i s in

B.M.O.

!J

such t h a t function

B.M.O.

we know t h a t f G B . M . O . implies B.M.O. is a lattice ( i f f , g G B.M.O., t h e n t h e f u n c t i o n s max ( f , g ) = ( I f - g l + f + g ) / 2 a n d min ( f , g ) = However, we may h a v e ( f + g - If-g ) / 2 w i l l a s o b e i n B . M . O . ) . For example i f : If1 G B . M 0 . w i t h o u t h a v i n g f G B . M . O . 0 f o r 1x1 > 1 if O < X < I f x) = -log X I Since ( I f l ) # ( x ) 6 2f#(x), If1 G B.M.O. Consequently

1

log

XI

i f -1 < x <

o

11.3. SHARP MAXIMAL FUNCTION AND B.M.O.

is in l-$p

it is c l e a r t h a t I f ( x ) ) = max(1og f i s n o t i n B.M.O. Since f is

average

1 2'

I

a -a

0

on e v e r y i n t e r v a l

B.M.O. However, odd a n d , c o n s e q u e n t l y , h a s

[-a,a],

If(x)Idx =

=

-

1

161

we j u s t n e e d t o o b s e r v e t h a t

log a

-+

m

for

a

+

0.

T h e r e i s an i n t i m a t e r e l a t i o n between t h e o p e r a t o r f H f # and t h e H a r d y - L i t t l e w o o d maximal o p e r a t o r M . I t is contained i n the f o l lowing s t a t e m e n t :

po <

0

a,

f

If

3.6.

THEOREM

then f o r every

J Rn(Mf (xDPdx with

C

Proof:

s

i s such t h a t

p

M f

f

such t h a t

L

PO

po

f o r some

po 6 p

with

we h a v e :

a,

2 C

independent of

f.

We may assume t h a t

f 2 0

since

Mf = M(1f

1)

and

(If[)#2

2f#.

First The p r o o f i s b a s e d upon t h e Calder6n-Zygmund d e c o m p o s i t i o n . we s e e t h a t t h i s d e c o m p o s i t i o n c a n b e c a r r i e d o u t f o r o u r f u n c t i o n f . L e t t > 0 and f o r every x 6 Q

suppose

Q

i s a cube such t h a t

fQ > t.

Then,

and t h u s tpo 2

I

1

IQI

Q

P ( M f ( x ) ) Odx 2

,

I t follows t h a t i f Q Q, c u b e s f o r e a c h of which i s 1 -

IQkI

I

Qk

...

1

IQI

P (Mf ( x ) ) Odx =

-!?

IQI

i s an i n c r e a s i n g f a m i l y o f d y a d i c

f(yldy > t,

t h e n t h e f a m i l y i s n e c e s s a r i l y f i n i t e s i n c e IQ,I i s bounded i n d e p e n d e n t l y o f k. T h u s , e a c h d y a d i c c u b e Q s a t i s f y i n g f Q > t w i l l b e c o n t a i n e d i n a maximal o n e . I f 19.1 i s t h e f a m i l y c o n s i s t i n g 3 of t h e s e maximal d y a d i c c u b e s , f o r e a c h o f them w i l l b e

11. CALDERON-ZYGMUND THEORY

162

t < --1 IQj

I

I

Qj

f ( y ) d y 5 2"t.

I n o r d e r t o i n d i c a t e t h e d e p e n d e n c e on t , we s h a l l d e n o t e t h i s For a . e . x f4 U Q t , j i s f ( x ) 2 t . f a m i l y by { Q . I . t,J 1 Observe t h a t i f

t < s,

then each

Qs,j C Qt,k and t a k e Given t > 0 w e f i x Qo = Q 2 - n - 1 two p o s s i b i l i t i e s : e i t h e r t,jo

f o r some k . A > 0. There a r e

I n t h e f i r s t case

I n t h e second case

and,taking

i n t o account t h a t

2"2-'-lt

= t/2,

we can w r i t e

Q05

dyS

Adding up i n a l l t h e p o s s i b l e

I n terms of

CI

Qo's, we g e t :

we h a v e g o t t h e f o l l o w i n g i n e q u a l i t y :

163

11.3. SHARP MAXIMAL FUNCTION AND B.M.O.

a(t) 5 [Ex Now, for

:

f#(x) > t/A}I + 2A-'a(2 -n-1 t)

N > 0, we consider

IN

=

jNpfP-la(t)dt 0 5 jNptP-lB(t)dt 0

since we are assuming

IN s

1

N

Mf

6

Lpo.

2

Also

I

N

ptp-l I{x : f#(x) > t/A) Idt + 2A-

0

ptP-la(2 -n-1 tldt

0

from which: 5 f N ptP-l I{x : f#(x)

I

>

t/A) ldt + CA-lIN

Jo

with a

depending only on

C

n

and

p.

Take

A = 2C

and obtain:

I N 4 ZINptP-ll{x : f#(x) > t/A}ldt. 0

Letting

N

-+

m,

we arrive at

and then

I

m

m

(Mf(x))Pdx

=

0

ptP-'B(t)dt

6 C 1 j0ptP-'a(t/CZ)dt

=

=

11. CALDERQN-ZYGMUND THEORY

164

We have s e e n t h a t t h e maximal f u n c t i o n s Mf and f # a r e c l o s e l y related. We have t h e t r i v i a l p o i n t w i s e e s t i m a t e f '(x) 2 ZMf(x), b u t we a l s o have a n e s t i m a t e g o i n g i n t h e o p p o s i t e d i r e c t i o n , t h i s t i m e an Lp e s t i m a t e . For f s u f f i c e s t o s e e t h a t f # c Lp

"good enough", and 0 < p < w , it t o c o n c l u d e t h a t Mf, and t h e r e f o r e

F o r f qfgood" and f , a r e a l s o i n Lp. Mf c Lp i s e q u i v a l e n t t o s a y i n g t h a t f #

1 i p < m, saying t h a t c L p and t h i s i s a l s o e q u i v a l e n t t o saying t h a t f c Lp. The s i t u a t i o n i s d i f f e r e n t f o r p = m. S a y i n g t h a t M f c L m i s t h e same a s s a y i n g t h a t f E L m . However, t o s a y t h a t f # c L w i s e q u i v a l e n t t o s a y i n g t h a t f 6 B.M.O. The s p a c e B . M . O . i s a n a t u r a l s u b s t i t u t e f o r L m , and t h e o p e r a t o r f b f ' a l l o w s u s t o t r e a t t h e s p a c e s L p and B . M . O . i n a u n i f o r m way. We s h a l l s e e t h a t many c l a s s i c a l o p e r a t o r s c a r r y Lm t o B.M.O. b o u n d e d l y . T h i s i s what makes t h e f o l l o w i n g i n t e r p o l a t i o n theorem i n t e r e s t i n g .

T be a l i n e a r o p e r a t o r bounded i n some w i t h 1 < po < m. Assume a l s o t h a t T Po B.M.O. boundedZy. T h e n , f o r e v e r y p with po < p < bounded i n Lp. THEOREM

3.7.

We c o n s i d e r t h e o p e r a t o r

Proof:

for

LP O

Let

m,

T

This i s a s u b l i n e a r

f'(Tf)#.

o p e r a t o r bounded i n Lpo and a l s o i n Lw. By M a r c i n k i e w i c z ' s i n t e r p o l a t i o n t h e o r e m , i t w i l l a l s o b e bounded i n L p . L e t f G L p .n. Lpo. Then

Tf E

and

I(Tf)'p

L'O

and

M(Tf) E L p o .

S Cllf p .

On t h e o t h e r hand

(Tf)'

G

Lp

The p r e c e d i n g t h e o r e m g i v e s :

and t h i s i n e q u a l i t y e x t e n d s t o e v e r y

The most i m p o r t a n t r e s u l t r e g a r d i n g

B.M.O.

f E Lp.

Ll

i s the following theo-

rem of F . John and L . N i r e n b e r g . THEOREM

Q (3.9)

There e x i s t constants n , s u c h t h a t for e v e r y f and e v e r y t > 0: 3.8.

dimension

)Ix

E

Q : If(x)

- fQI

G

C,,C2 B.M.O.

d e p e n d i n g onZu on t h e = B.M.O.(IRn) , e v e r y

It

-CC,/llfll >

t l l 2 Cle

*

191

11.3. SHARP MAXIMAL FUNCTION AND B.M.O.

165

I t i s a g a i n an a p p l i c a t i o n of t h e Calder6n-Zygmund decompos i t i o n . O b s e r v e , f i r s t of a l l , t h a t we c a n assume IIflI* = 1 , because t h e i n e q u a l i t y ( 3 . 9 ) d o e s n o t change i f we r e p l a c e f by a c o n s t a n t t i m e s i t . We f i x Q and t a k e a > 1 . We know t h a t

Proof:

We make t h e Calder6n-Zygmund d e c o m p o s i t i o n of f

-

f

relative to

Q

o b t a i n i n g cubes

a,

Q1 , j

Q ) f o r e a c h of which:

I

a < -

If(x)

Bes d e s , f o r a . e . x $ U Q ,j f o r each Q 1 , j i s I f Q , j - f Q 5 2"a. A l s o :

-

f

is

Q

Q

f o r the function ( d y a d i c s u b c u b e s of

Idx I Zna

If(x)

-

f

Q

1

I t follows t h a t

5 a.

I

On e a c h Q 1 function f of d y a d i c

I f Q2,k If(x)

-

f f

we make t h e Calder6n-Zygmund d e c o m p o s i t i o n f o r t h e - f r e l a t i v e t o a . Thus w e o b t a i n a f a m i l y Q 2 , k Q 1 s j subcubes of Q . , f o r e a c h of which i s

Q1,j

I

I

<

Zna,

151 and a l s o f o r a . e .

Besides,

I a.

Q1 , j together a l l the families

ilQ2,kl

x

G

Ql,j\

(!

Q2,k)

is

Now we p u t

6 a1 l Q l , j l .

{Q2,k} corresponding t o d i f f e r e n t

and c a l l t h e r e s u l t i n g f a m i l y a l s o Q1 ,j t h e u n i o n of t h e Q 2 , k ' s we h a v e : 's

{Q2,k}.

Then, o u t s i d e

and a l s o

S u b s e q u e n t l y , we o b t a i n f o r e a c h n a t u r a l number o v e r l a p p i n g cubes union i s

If(x)

{QN,j}

- f QI


N,

a f a m i l y o f non-

i n s u c h a way t h a t o u t s i d e of t h e i r and s u c h t h a t

? I Q , , ~ 5I

a - N ~ ~ ~ .

11. CALDERON-ZYGMUND THEORY

166

If

N.2"a

S t c (N

+ 1).2"a

with

N = 1,2,

...,

then

On t h e ( w i t h C 2 = 2-"-l ( l o g a ) / a ) s i n c e t < ( N + l ) Z n a 5 N2""a. o t h e r h a n d , i f t < 2"a, t h e n C 2 t < ( l o g a ) / 2 , and we u s e t h e t r i v i a 1 ma j o r i z a t i o n

Thus, we g e t

f o r every

(3.9)

C1 = / a . Finally, a v a l u e of t h e c o n s t a n t COROLLARY

3.10.

II II* f

I f

Q

by c h o o s i n g

C2

a s above and

c a n be c h o s e n i n o r d e r t o g e t a n o p t i m a l C2 (a = e). f

sup(l

,p

t

IQI

then:

B.M.O.

E

1Q

If(x)

-

fQ/Pdx]'/p

s

CpIlfl),

f , i n s u c h a way t h a t , f o r 1 < p < ff-, I l f l l * , p i s a n o r m e q u i v a l e n t t o f ) - - , l l f l I *on B . M . O .

with P

i n d e p e n d e n t of

C

i i ) For e v e r y

X

1 sup -

Q

IQl

1

Q

0 < X < C 2 / \ \ f \ l * , where

such t h a t

same c o n s t a n t a p p e a r i n g i n

e

(3.91,

I f ( x ) -fQ I

dx <

A f t e r a change of v a r i a b l e s

we h a v e : m

C2

m,

i s the

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

167

1 < p , w e h a v e Ilf 11 * 5 Ilf ( I * , p S CpIIf 11 * , s o t h a t t h e norms a n d II a r e e q u i v a l e n t o v e r B.M.O. Also, i f p > 1 , S t i r l i n g ' s f o r m u l a T ( p ) = 6e - p p P - l I 2 c a n b e u s e d t o c o n c l u d e t h a t C 6 C - p w i t h an a b s o l u t e c o n s t a n t C . P If

II I I *

4.

ll,,p

HARMONIC AND SUBHARMONIC FUNCTIONS IN A HALF-SPACE

I n t h e f i r s t two s e c t i o n s of c h a p t e r I , we s t u d i e d h a r m o n i c and s u b h a r m o n i c f u n c t i o n s i n a d i s k o r i n a e u c l i d e a n b a l l , and s o l v e d t h e D i r i c h l e t p r o b l e m f o r s u c h d o m a i n s . Our p u r p o s e i n t h i s s e c IRn++l = t i o n w i l l be t o extend t h i s s t u d y t o t h e h a l f - s p a c e { ( x , t ) f IR"+1 : t > 01. We a l r e a d y p o s e d t h e D i r i c h l e t p r o b l e m f o r t h i s domain i n t h e f i r s t s e c t i o n o f C h a p t e r I , and s t a r t e d

=

l o o k i n g f o r a s o l u t i o n . We s h a l l c o m p l e t e t h i s t a s k i n t h e p r e s e n t s e c t i o n . The main d i f f e r e n c e between t h e s i t u a t i o n a n a l i z e d i n C h a p t e r I and o u r p r e s e n t s e t t i n g i s t h e unboundedness of t h e domain, which f o r c e s u s t o l o o k c a r e f u l l y a t t h e b e h a v i o u r o f o u r functions a t infinity. We s t a r t by showing t h a t P o i s s o n ' s k e r n e l d o e s i n d e e d s o l v e t h e A t t h e end o f s e c t i o n i n t r o d u c e d t h e P o i s s o n k e r n e l f o r iRn++'

D i r i c h l e t problem.

1

i n Chapter

I

we

11. CALDERON-ZYGMUND THEORY

168

so t h a t

a s c a n b e s e e n by i n t e g r a t i n g i n p o l a r c o o r d i n a t e s . We a l s o h a v e , f o r a n y

I X b 6

I

Pt(x)dx =

because

P1

1x1 > 6 / t

I

J

iii)

Ixl>6

Pt(x)dx

I n o t h e r words,

-

6

+

'n a n-1 a t

T h u s , we h a v e : IR"

,

t >

for all 0

as

t

+

o t > 0

+

0

f o r every

i s harmonic i n

(x,t)iPt(x)

I (t2

t * 0

6 > 0.

i s an approximate i d e n t i t y i n

{Ptlt>O

Also, t h e function P (x) = t

x

2 0,

Pt(x)dx = 1

ii)

as

Pl(x)dx * 0

is integrable. P,(x)

i)

6 > 0

IR".

IRn+l

.

Indeed

1 lx12)(n-l)/2 1

and

is Itthe" r a d i a l harmonic f u n c t i o n i n THEOREM

4. I.

g r a l of

f:

For

f

6

??

p= 1

Lp(IRn)

IRn+l\ C O l .

, we c o n s i d e r t h e P o i s s o n i n t e -

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

Then

u(x,t)

a

a)

f

as

LP

in -

E

If

i s harmonic i n

LP(IR") t

with

169

Rn+' and:

+

I 5 p <

-

u(*,t) * f

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

m,

0.

+

f i s c o n t i n u o u s and bounded, it f o l l o w s t h a t u ( . , t ) * f u n i f o r m l y on c o m p a c t s u b s e t s a s t 0. c ) Lf f i s u n i f o r m l y c o n t i n u o u s and bounded, t h e c o n v e r g e n c e b)

+

as

u(-,t) * f

Lf

d)

weak-*

f

G

t *

2a

Pt*v(x)

, then

l o g y of

M(IRn)

( f i n i t e ) Borel measure i n

u

.

c i t y of

Pt(x). u(x,t)

u(.,t)

+

f

i n the

Lm.

i s harmonic and

The h a r m o n i c i t y o f

Proof:

R".

i s uniform i n

L m ( R n ) , a l l we c a n s a y i s t h a t

t o p o l o g y of

Also if

o

u(x,t)

u(.,t)

Rn * p

u(x,t)=P(v)(X,t)= i n - t h e weak-*

topo-

i s a consequence o f t h e harmoni-

Besides

-

f(x) =

(f(x

-

y)

-

f(x))Pt(y)dy.

T h u s , by u s i n g M i n k o w s k i ' s i n e q u a l i t y f o r i n t e g r a l s , w e g e t :

c ) , given E > 0 , 6 > 0 can be chosen i n such - y ) - flI < E / Z w h e n e v e r IyI < 6 . W i t h t h i s P t h e f i r s t term i n o u r sum o f i n t e g r a l s i s < ~ / 2

In c a s e s a ) and a way t h a t llf( choice of

6,

independently of

-

t.

The s e c o n d

term i n t h a t sum i s

j l y l , 6 P t ( y ) d y * 0 a s t * 0 a s we know f r o m p r o p e r t y i i i ) of t h e P o i s s o n k e r n e l . T h e r e f o r e , t h e s e c o n d term i n t h e 5 211fllp

~ / 2 by t a k i n g t c l o s e t o 0 . This proves sum c a n a l s o b e made a ) a n d c ) . The p r o o f o f b ) is e n t i r e l y analogous. P a r t d)

a n d t h e f i n a l s t a t e m e n t c o n c e r n i n g a B o r e l m e a s u r e p a r e p r o v e d by d u a l i t y e x a c t l y a s i n t h e o r e m 1 . 1 8 i n C h a p t e r I . Now M ( I R n ) is t h e d u a l o f Co(lRn) , t h e s p a c e o f c o n t i n u o u s f u n c t i o n s v a n i s h i n g at

m

w i t h t h e supremum norm, a n d t h e s e f u n c t i o n s a r e u n i f o r m l y

continuous, so t h a t p a r t COROLLARY

4.2.

Fo r

f

c)

applies.

[I

c o n t i n u o u s and bounded i n

Rn,

the func-

u(x,t) defined by u(x,t) = P(f)(x,t) 2 2 t > 0 and u ( x , O ) = f ( x ) , i s c o n t i n u o u s z' n a n d h a r m o n i c i n R n + l . I& with i s t h e r e f o r e a s o l u t i o n o f t h e D i r i c h l e t p r o b l e m i n IRn+l + -

.-t i o n

+

11. CALDERON-ZYGMUND THEORY

170

boundary f u n c t i o n Proof:

rem.

f.

I t f o l l o w s i m m e d i a t e l y from p a r t

0

4.3.

PROPOSITION

I(u(

b)

W i t h t h e same h y p o t h e s i s

.,t)llpsllfllp

f o r euery

of t h e p r e v i o u s t h e o -

of

theorem

we

4.1,

t > 0.

Since t h e function x H u ( x , t ) i s t h e c o n v o l u t i o n of Pt and f ( u ( . , t ) = P t * f ) , t h i s i s a p a r t i c u l a r i n s t a n c e o f Y o u n g ' s i n e q u a l i t y and i t i s e a s i l y o b t a i n e d by w r i t i n g u ( x , t ) = Proof:

=

IRn P t ( r ) f ( x - y ) d y

grals.

and a p p l y i n g M i n k o w s k i ' s i n e q u a l i t y f o r i n t e -

u

Now we s h a l l o b t a i n P o i s s o n ' s r e p r e s e n t a t i o n f o r a c l a s s o f n i c e harmonic f u n c t i o n s . 4.4.

THEOREM

R'+'

monic i n

+

Let

-

u(x,t)

be a c o n t i n u o u s f u n c t i o n i n

and bounded.

Then

u

Rn+l + '

harcoincides necessarily with

x + $ u(x,O),

t h e P o i s s o n i n t e g r a l of t h e b o u n d a r y f u n c t i o n

that

i s , we h a v e t h e P o i s s o n r e p r e s e n t a t i o n :

proof:

Consider t h e f u n c t i o n I

V(X,t)

=

v ( x , t ) g i v e n by:

r

if

t > O

1

T h i s f u n c t i o n i s c o n t i n u o u s i n lRn++l , b o u n d e d , h a r m o n i c i n IRn++' and v a n i s h e s a t e v e r y p o i n t of t h e b o u n d a r y . Theorem 1 . 3 3 i n chapter I t e l l s u s t h a t v is i d e n t i c a l l y 0.

[I

Theorem 4 . 4 . c a n b e r e a d by s a y i n g t h a t t h e D i r i c h l e t p r o b l e m i n Rn + 1 f o r a bounded and c o n t i n u o u s b o u n d a r y f u n c t i o n h a s a u n i q u e bounded s o l u t i o n , b u t , o f c o u r s e , we know t h a t u n i q u e n e s s d i s a p p e a r s a s s o o n a s we a l l o w unbounded s o l u t i o n s .

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

171

I n o r d e r t o e x t e n d t o o u r s i t u a t i o n Theorems 1 . 3 and 1 . 8 from C h a p t e r I , and a l s o f o r o t h e r a p p l i c a t i o n s , we s h a l l make some b a s i c e s t i m a t e s f o r harmonic and s u b h a r m o n i c f u n c t i o n s i n Rn++' . v ( x , t ) be a non-negative subharmonic f u n c t i o n IFP++w h i c h i s u n i f o r m l y i n L p ( l R n ) f o r some 1 5 p < m , &

THEOREM

in -

4.5.

Let

w h i c h we mean t h a t :

~ ( x , t ) ~ d =x

Mp <

m

.

C depending only on p & n, such ( x , t ) G Rn:.' In p a r t i c u l a r v ( x , t ) IC M t - " l P f o r every i s bounded i n each p r o p e r s u b - h a l f - s p a c e I ( x , t ) G lRn++' : t 2 v(x,t) 2 t > 01. A c t u a l l y , t h e f o l l o w i n g s t r o n g e r p r o p e r t y h o l d s : 0 v(x,t) + 0 as ( x , t ) + m i n each proper sub-half-space. Then t h e r e i s a c o n s t a n t

that

Given ( x o , t o ) G Rn++' , l e t Bo = B ( ( x o , t o ) , t o / 2 ) t h e b a l l c e n t e r e d a t (xo, to) with r a d i u s t 0 / 2 . Then

Proof:

1 v ( x 0 , t ) I-

v(x,t)dx d t i

0

I C M

denote

tin/P.

Let u s p r o v e now t h a t t o >0 . S i n c e we know t h a t g i v e n E > 0 , we c a n f i n d t l > t o v ( x , t ) IC M t - n / P , s u c h t h a t v ( x , t ) 5 E p r o v i d e d t 2 t 1 ' We j u s t n e e d t o show t h a t f o r t o It 5 t l , v ( x , t ) c a n b e made s m a l l e r t h a n E by t a k i n g x big. P r o c e e d i n g a s i n t h e f i r s t p a r t o f t h e p r o o f , we see t h a t f o r 1x1 > t l , s a y , a n d f o r t o I t It 1 T h i s p r o v e s t h e f i r s t p a r t of t h e t h e o r e m .

v(x,t)

+

0

as

(x,t)

-+

m,

t 2 t 0 > 0.

Fix

11. CALDERON-ZYGMUND THEORY

172

11

and t h e proof i s completed.

We s h a l l s e e b e l o w t h a t t h e o r e m

I

4.5

s t i l l holds f o r

v(x,t) =

w i t h u h a r m o n i c i n Illn+’ a n d a n y p > 0 . When t h e proof of theorem 4 . 5 cannot be used, s i n c e t h e second i n e q u a l i t y i n t h a t p r o o f i s J e n s e n ’ s i n e q u a l i t y which i s no

= lu(x,t) 0 < p < 1,

longer available f o r LEMMA

4.6.

Let

3. T h e n , i f -_ in

p < 1.

We s h a l l u s e , i n s t e a d , t h e f o l l o w i n g

u be harmonic i n a b a l l B c ( x o , t o ) i s t h e c e n t e r o f B,

and c o n t i n u o u s we h a v e , f o r a n y

p > 0:

where

C

P

i s a c o n s t a n t d e p e n d i n g o n l y on

p

& n.

P r o o f : O f c o u r s e , we a l r e a d y know t h a t t h e lemma i s t r u e when p 2 1 w i t h C = 1 . Let u s l o o k now a t t h e c a s e 0 < p < 1 . We P may a s s u m e t h a t B i s t h e u n i t b a l l c e n t e r e d a t t h e o r i g i n a n d a l s o t h a t j B l u ( x , t ) l P d x d t = IB . Write

r o ) lPdo do

YP

b e i n g L e b e s g u e m e a s u r e on t h e u n i t s p h e r e , a n d

mm(r) 2 1 f o r e v e r y r w i t h 0 < r < 1 , s i n c e o t h e r w i s e t h e maximum p r i n c i p l e w o u l d i m m e d i a t e l y g i v e u s t h e required inequality

We may a l s o a s s u m e t h a t

From t h e r e p r e s e n t a t i o n o f u a s a P o i s s o n i n t e g r a l o v e r t h e s p h e r e of r a d i u s r (see t h e formula r i g h t b e f o r e theorem 1.30 i n Chapter I ) , we o b t a i n , f o r 0 < s < r : m m ( s ) S 2 ( 1 t a k e s = r a w i t h a > 1 . We h a v e

- s r - l ) - nm , ( r ) .

Now

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

173

By t a k i n g l o g a r i t h m s and i n t e g r a t i n g we g e t :

l o g m ( r ) d- r P r The l a s t i n t e g r a l i s bounded by a c o n s t a n t d e p e n d i n g Dnlyon p n,

and

s i n c e o u r normalizing assumption i s e q u i v a l e n t t o (n

t

1)

lo 1

mp(r)P

rn d r

=

1.

T h e r e f o r e , a f t e r a c h a n g e of v a r i a b l e s i n t h e i n t e g r a l on t h e l e f t hand s i d e , we c a n w r i t e :

/2

But a > 1 mm(r) 2 1,

implies

(1/2)a

< 1/2

log mm(r)

dr r

-

a n d , s i n c e we h a v e assumed

it f o l l o w s t h a t 2 log mm(r) drr

I

r)

dr

.

Thus log mm(r) Now w e c h o o s e a 1 < a < (l-p)-'.

dr

2 Ca + ( 1

so close t o 1 t h a t Then we o b t a i n :

dr log mm(r) - 5 C

- PI l / a > 1-p,

dr ,log m m ( r ) r t h a t i s , we t a k e

P

ro > 0 ( ( / 2 ) a 5 r o 2 l ) , mm(r ) S C 0 P' a c o n s t a n t d i f f e r e n t from t h e p r e v i o u s o n e , b u t s t i l l d e p e n d i n g o n l y on p and n . Then t h e maximum p r i n c p l e g i v e s u s I 5 cP a s we wanted t o p r o v e .

T h i s i m p l i e s t h a t f o r some

IU(O,O)

Now we c a n g i v e t h e p r o m i s e d e x t e n s i o n o f t h e o r e m

4.5.

11. CALDERON-ZYGMUND THEORY

174 THEOREM

Let

4.7.

i s uniformly i n

u(x,t)

Lp(lRn)

Then t h e r e i s a c o n s t a n t

that

l

u(x,t)

b e a harmonic f u n c t i o n i n 0 < p <

f o r some

C

Rn+l which

+

that i s :

m,

p

d e p e n d i n g o n l y on

u ~ x f o r r e v e r y ( x , t ) f IR"++~ i s bounded i n e a c h p r o p e r s u b - h a l f - s p a c e

and

.

n,

-

such

In particular ( x , t ) G IRn+tl : t t

2 t > 01. A c t u a l l y , t h e f o l l o w i n g s t r o n g e r p r o p e r t y h o l d s 0 u(x,t) + 0 (x,t) m i n each proper sub-half-space. -+

I t i s e x a c t l y t h e same a s t h e p r o o f o f t h e o r e m ing with t h e inequality:

Proof:

with

Bo

=

4.5

start-

B ( ( x o , t o ) , t o / Z ) ) , w h i c h was e s t a b l i s h e d i n lemma 4 . 6 .

We c a n now p r e s e n t t h e a n a l o g u e s o f t h e o r e m s

1.3

and

1.8

in

Chapter I , providing c h a r a c t e r i z a t i o n s of t h e Poisson i n t e g r a l s o f LP f u n c t i o n s , p > I , o r m e a s u r e s i n IR"

.

THEOREM

4.8.

the function a) u bl u

Let

u(x,t)

1

p S

Then t h e f o l l o w i n g t u o c o n d i t i o n s on

m,

defined i n

IRnt+'

are equivalent:

i s harmonic i n and i s u n i f o r m l q i n Lp(IRn) i s t h e P o i s s o n i n t e q r a l o f some f u n c t i o n f G L p ( I R n )

and 4 . 3 , that (b) implies ( a ) i m p l i e s ( b ) . Assuming (a) holds, consider t h e f u n c t i o n s f . ( x ) = u ( x , t . ) for a 3 J s e q u e n c e t . + O . From ( a ) , (fj) i s a bounded sequence i n 1 Lp(IRn) = LP'(JRn)*. A s i n t h e proof of theorem 1.3 i n Chapter

Proof:

(a). that

We a l r e a d y know, f r o m

.

4.1

Let u s p r o v e t h a t , c o n v e r s e l y ,

( f j ) h a s a s u b s e q u e n c e c o n v e r g i n g i n t h e weak-* t o p o l o g y o f L p . I, We may a s s u m e t h a t ( f j ) i t s e l f c o n v e r g e s i n t h e weak-* t o p o l o g y t o a c e r t a i n f G Lp(lRn) , t h a t i s , f o r e v e r y g G L p ' ( l R n )

j, the function (x,t)w u(x,t t t . ) is continuous J it i n IR"++l , h a r m o n i c i n IRn+l a n d , a c c o r d i n g t o t h e o r e m 4 . 7 , i s a l s o bounded. I t f o l l o w s from theorem 4.4 t h a t t h i s f u n c t i o n

NOW, f o r each

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

175

i s t h e Poisson i n t e g r a l of i t s boundary f u n c t i o n , t h a t i s , f o r every j u(x,t + t . ) = J Letting

J IR" P t ( x

go t o

j

since the function

-

y ) u ( y , t . ) d y = jJRnPt(x J

m,

we g e t

y

H Pt(x

As in the torus,

the case p similar p r o o f , t h e following THEOREM

a) b)

in -

y)

=

1

belongs

to

y ) f .J ( y ) d y .

Lp'(IRn)

. fl

i s d i f f e r e n t , a n d we g e t , w i t h a

T h e f o l l o w i n g t w o c o n d i t i o n s on t h e f u n c t i o n

4.9.

an+'

defined i n

-

-

u(x,t)

are equivalent:

u i s h a r r n z n i c i n Rn;l and i s u n i f o r m l y i n L1(IRn) u i s t h e P o i s s o n i n t e g r a l o f some ( f i n i t e ) B o r e 1 m e a s u r e IR", that i s u(x,t) = P(u)(x,t) = P (X - Y ) d U ( y ) . JR"

B a s e d u p o n t h e same i d e a s i s t h e f o l l o w i n g r e s u l t o n h a r m o n i c m a j o r i z a t i o n , which w i l l be a v e r y u s e f u l tool i n t h e s e q u e l

1

Let

4.10.

THEOREM

v

be a n o n n e g a t i v e subharmonic f u n c t i o n i n

w h i c h is u n i f o r m Z y i n

ha: a l e a s t h a r m o n i c m a j o r a n t

Lp(JRn) f o r some 1 6 p IRn++' . M o r e o v e r , i f

iz

m. Then v p > 1 , & I@

h a r m o n i c m a j o r a n t is t h e P o i s s o n i n t e g r a l o f some f u n c t i o n i n Lp(IRn)

a n d if

Bore1 measure on Proof:

p = 1, IR" .

i t i s t h e P o i s s o n i n t e g r a l of some ( f i n i t e )

As before, consider the functions

We may a s s u m e t h a t f . sequence t . G O . J J t h e weak-* t o p o l o g y o f Lp(IRn) if p > logy of M(IRn) if p = 1. Just t o f i x Then f . J f o r every

+

f

j

E

Lp(IRn) t h e r e is

f.(x) = v(x,tj) for a J converges as j m in 1 o r i n t h e weak-* t o p o the notation, let p > 1.

i n t h e weak-* t o p o l o g y .

+

Now w e c l a i m t h a t

The way t o s e e t h i s i s t o o b s e r v e t h a t b o t h f u n c t i o n s

176

11. CALDERON-ZYGMUND THEORY

( x , t ) H v(x,t + t . )

and

1

tend t o

as

0

(x,t) R > 0

h a v e t o make

in

+ m

IRn+l

.

Thus, given

E

we j u s t

> 0,

b i g enough s o t h a t

+ t . ) - P(f.)(x,t)

V(X,t

P(f.)(x,t) 3

3

1

i n t h e boundary of t h e r e g i o n

5E

KR =

I ( x , t ) c JRn++l : I x I 2 + t 2

6

R2 1 .

Then we c a n a p p l y c o r o l l a r y 2 . 6 f r o m C h a p t e r I t o c o n c l u d e t h a t v ( x , t + t . ) - P ( f . 1 ( x , t ) 5 E f o r every ( x , t ) c KR. Since t h i s is 3 1 t r u e f o r a l l l a r g e enough R ' s , we h a v e a c t u a l l y v ( x , t + t . ) 3 - ~ ( f . ) ( x , t )5 E f o r e v e r y ( x , t ) c IR"++' . N O W , s i n c e E was 1 a r b i t r a r y , we g e t ( 4 . 1 1 ) . That t h e function ( x , t ) W v ( x , t + t . ) 3 t e n d s t o 0 a s ( x , t ) + m i n IRn+l f o l l o w s f r o m t h e o r e m 4 . 5 . As f o r t h e f u n c t i o n P ( f . ) , we j u s t nLed t o o b s e r v e t h a t f j c Co(lRn) , 3 t h e s p a c e o f c o n t i n u o u s f u n c t i o n s v a n i s h i n g a t m, a n d t h i s a l r e a d y guarantees t h a t let

h

Co(JRn).

G

By t a k i n g

R

Take

E

+

0

> 0.

as

(x,t) *

in

m

IRn++'

-+

f o r every

.

Indeed,

Then

b i g e n o u g h , t h e s e c o n d i n t e g r a l c a n b e made

absolute value. t h a t P,(x) 0 (4.11)

P(fj)(x,t)

< ~ / 2 in

F o r t h e f i r s t i n t e g r a l w e make t h e o b s e r v a t i o n as (x,t) + i n IR"++' . T h u s , we h a v e e s t a b l i s h e l j

and every

(x,t) c

.

Letting

j

+ m,

we

have:

Thus case

u ( x , t ) = P ( f ) ( x , t ) i s a harmonic majorant of p = 1 we g e t a h a r m o n i c m a j o r a n t u = P(p)

v . For t h e with G M(IRnI

That u i s indeed t h e l e a s t harmonic m a j o r a n t of v f o l l o w s from t h e f a c t t h a t P ( f i ) ( x , t ) i s t h e l e a s t harmonic m a j o r a n t of ( x , t ) l - v ( x , t + t . ) b e c a u s e , i f h i s a h a r m o n i c m a j o r a n t o f v , we 1 would h a v e h ( x , t + t . ) 2 v ( x , t + t . ) a n d , c o n s e q u e n t l y h ( x , t + t . ) t 1 3 1 2 P ( f j ) ( x , t ) . Letting j +m, we g e t : h ( x , t ) 2 u ( x , t ) . We s h a l l n e x t s t u d y t h e p o i n t w i s e c o n v e r g e n c e o f P o i s s o n i n t e g r a l s .

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

177

The a p p r o a c h w i l l be t h e same a s t h a t we h a v e a l r e a d y u s e d t o p r o v e t h e L e b e s g u e t h e o r e m ( c o r o l l a r y 1 . 1 1 ) . Given f G U L p ( I R n ) , 1 o end o f s e c t i o n 2 we p r o v e d t h a t IPt*f(x) S CMf(x,t) with +

1

M f ( x , t ) = s u p I- 1

1QI

'Q

If1 : x E Q ,

s i d e l e n g t h (Q) 2 t )

Of course, it follows t h a t

* P f(x) 5 C

Mf(x,t)

SUP

t>O

CMf(x)

=

T h u s , P * i s a bounded o p e r a t o r i n L p € o r a n y p > 1 a n d i t i s a l s o o f weak t y p e ( 1 , l ) . T h i s i s a l l we n e e d t o p r o v e t h e f o l l o w i ng

f o r some p I t w i l l b e enough t o show t h a t f o r e a c h

Proof:

Assume f i r s t t h a t

1 5 p <

w.

A~

h a s measure true that

=

{x 0.

G

IR"

f

G

LP(IRn)

Indeed, t h e s e t of p o i n t s

Pt*f(x)

+

f(x)

as

t

+

According t o theorem Consequently,

-

4.1,

f(x)

I

part

5 lirn

c),

x

G

>

s

5

0, t h e s e t

S }

IRn

is p r e c i s e l y

0,

Given E > 0 , we c a n w r i t e f = g + h , w i t h compact s u p p o r t a n d J R n l h l P < E .

l i m sup IPt*f(x) t+O

- f(x) I

: l i m sup IPt*f(x) t + O

such t h a t

where i t i s n o t

.

m

U Al,j j=l

where g i s c o n t i n u o u s We h a v e , f o r e v e r y x a n d t :

Pt*g(x)

sup IPt*h(x) t + O

I

+

g(x)

+ Ih(x)

as

1

5

t

+

0.

11. CALDERON-ZYGMUND THEORY

178

I t follows that

a n d we g e t :

Since t h i s is t r u e f o r every

we a c t u a l l y have

> 0,

E

lAsl

=

0.

Thus, t h e theorem i s proved i n t h i s c a s e . The c a s e

f

s e e how.

Take

as t

f o r almost every

+

0

G

Lm(IRn) can be reduced t o t h e previous one. Let u s B = B ( O , R ) , R > 0. We s h a l l p r o v e t h a t P t * f ( x ) + f ( x ) x

G

T h i s w i l l b e enough, s i n c e

B.

R

i s a r b i t r a r y . Let B ' = B ( 0 , R + 1 ) . Write f = f l + f 2 w i t h fl(x) = f(x) if x G B' a n d f , ( x ) = 0 e l s e w h e r e . Then f,

G

L1(Rn)

and, consequently

Pt*fl(x)

-+

fl(x)

a . e . x G IR". T h u s , we j u s t n e e d t o p r o v e t h a t f o r a . e . x G B. But f o r x f B :

as

t

-+

0

for

Pt * f 2 ( x ) + f 2 ( x ) = 0

and

In general, i f

@

G

L1(IRn)

t h e approximate i d e n t i t y =

t-"@(t-'x).

I,n

t > 0,

$ ( x ) d x = 1 , w e may c o n s i d e r o b t a i n e d by m a k i n g

@t(x) =

T h e n , i f we now s e t

a l l p a r t s of theorem 11,

at,

and

4.1,

with t h e exception of t h e harmonicity of

s t i l l h o l d , s i n c e t h e y depend fsnly on t h e s e t h r e e f a c t s :

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

@(xldx= 1

ii)

t

t > 0,

I@(x)

For e v e r y 6 > 0 :

iii)

as

f o r every

179

and Idx

+

0

0.

+

Theorem 4 . 1 2 , h o w e v e r , d e p e n d s upon t h e i n e q u a l i t y P " f ( x ) 6 5 CMf(x). I f we want t o e x t e n d i t t o o t h e r a p p r o x i m a t e i d e n t i t i e s * @,, we s h a l l h a v e t o s t u d y t h e maximal f u n c t i o n @ ( f ) ( x ) =

I.

sup I@,*f(x) I f we g e t a n i n e q u a l i t y @ * ( f ) ( x ) S CMf(x), t>O e x t e n d s immediately t o t h e a p p r o x i m a t e i d e n t i t y {@t}. theorem 4 . 1 2 I f @ i s r a d i a l and d e c r e a s i n g , t h e i n e q u a l i t y i s v e r y e a s y t o =

obtain. THEOREM

$(x) t 0

Let

4.13.

IRn . Then f G u LP(IR") a n d e v e r y 15 p 9 n.

dimension -

Let

Proof:

Call

be a r a d i a Z , d e c r e a s i n g , i n t e g r a b Z e

$ * ( f ) (x) 5 C l l + l l ,Mf(x) f o r e v e r y x G I R ~ , with c d e p e n d i n g o n l y on t h e

function i n

Then

B. = B(0,r.). J J

I

IR"

@(x)dx=

and f o r e v e r y

f

G

m

1

a.IB.1 <

j=1

1

Lp(iRn)

1

,

1 5 p <=

By a p p l y i n g t h i s t o t h e f u n c t i o n

11

IRn f ( x

-

m

f(x

y)@(y)dyl 5 C[jRn

Now o b s e r v e t h a t , f o r

@

m,

-

a )

we obtain

@]Mf(x).

a s above and

t > 0,

@t i s a f u n c t i o n of

11. CALDERON-ZYGMUND THEORY

180

11+, 111

t h e same t y p e a n d

=

llCplll.

Therefore

i s r a d i a l , d e c r e a s i n g a n d i n t e g r a b l e , we c a n f i n d a s e q u e n c e o f f u n c t i o n s $ ( J ) , e a c h o f w h i c h i s a f i n i t e l i n e a r comb i n a t i o n of c h a r a c t e r i s t i c f u n c t i o n s of b a l l s c e n t e r e d a t 0 w i t h

Now i f

J,

x,

c o e f f i c i e n t s 2 0, such t h a t , f o r every monotonically t o $ (x) Then

.

Since, f o r each

Cp(j)(x)

increases

j

we f i n a l l y g e t :

CII

J , * ( f ) ( x ) S CII$IIIMf(x). COROLLARY

4.14.

that the function

i s called every

R".

f

+

$(XI

j I R n $ ( x ) d x = 1 and s u c h i s i n t e g r a b l e ($

= s u p { ) @ ( y ) I :IyI t 1x11

t h e l e a s t r a d i a l d e c r e a s i n g m a j o r a n t of

u

LP (R") 15p5m

G

,

Cpt*f(x)

-t

f(x),

as

is i d e n t i c a l t o that of theorem

It

Proof:

with

Cp G L 1 ( l R n )

Let

t

+

0,

4.12.

4).

Then, f o r

f o r almost every

The f a c t t h a t

i s i n t e g r a b l e a n d J,nCp = 1 guarantees t h e uniform convergence f o r f c o n t i n u o u s w i t h c o m p a c t s u p p o r t . T h e n , we j u s t h a v e t o

observe t h a t

+ * ( f ) ( x ) 2 $ * ( l f l ) ( x ) S CllJ,ll,Mf(x).

A c t u a l l y , t h e p o i n t w i s e convergence o f a P o i s s o n i n t e g r a l a t a bound a r y p o i n t h o l d s i n a s t r o n g e r f o r m t h a n t h e o n e we h a v e d i s c u s s e d . We c a n a l l o w a n y n o n - t a n g e n t i a a p p r o a c h . L e t u s g i v e p r e c i s e definitions. F o r a p o i n t x 6 IRn a n d a n y r e a l number N > 0 , we s h a l l c a l l TN(x) t o t h e o p e n v e r t i c a l c o n e w i t h v e r t e x x a n d aperture

N , t h a t is

rN(x) = { ( y , t )

G

n + l .. y IR+

XI

< Ntj.

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

181

,

Given a f u n c t i o n u d e f i n e d on JRn+l and a p o i n t x G IRn we s h a l l s a y t h a t u ( y , t ) c o n v e r g e s t o II a s ( y , t ) t e n d s t o x ( a c t u a l l y t o (x,O)) n o n t a n g e n t i a l l y , and we s h a l l w r i t e u ( y , t )

+

II

a s ( v , t ) N;T*x, i f and o n l v i f , f o r e v e r y N > 0 , u ( y , t ) c o n v e r ges t o R a s ( y , t ) tends t o x remaining always i n TN(x). This i s t h e n a t u r a l e x t e n s i o n t o o u r c o n t e x t of t h e n o t i o n of n o n - t a n g e n t i a l convergence i n t r o d u c e d f o r t h e d i s k i n Chapter I ( i n theorem 1 . 2 0 ) . To p r o v e t h e n o n - t a n g e n t i a l c o n v e r g e n c e of P o i s s o n i n t e g r a l s we s h a l l l o o k a t t h e n o n t a n g e n t i a l P o i s s o n maximal f u n c t i o n o f a p e r t u r e N > 0 , d e f i n e d f o r any f G U L P ( R n ) as: 1< p 6 m * PV,Nf(X) = sup IPt*f (Y) I (y,t i G r N ( X ) We know t h a t I(Pt*f)(y)l 5 C M f ( y , t ) w i t h C d e p e n d i n g o n l v on n . However, i f ly < Nt, we c a n e a s i l y s e e t h a t M f ( y , t ) 5 S CNMf ( x ) , where CN i s a c o n s t a n t dependinr! on N . To p r o v e t h i s i n e q u a l i t y we j u s t n e e d t o o b s e r v e t h a t i f Q i s a c u b e c o n t a i n i n g y and h a v i n g s i d e l e n g t h 2 t, then x G Q Z N + l , so t h a t w e have

XI

and, consequently.

Thus we c a n w r i t e : (4.15)

P

*

v ,Nf ( x )


T h i s i n e q u a l i t y i s a l l we n e e d t o p r o v e t h e f o l l o w i n g :

(y,t)N;T'x Proof:

f

U Lp 1
4.16.

THEOREM

f

(IR") , x

6

then

Pt*f(y)

+

f(x)

IR'.

The p r o o f i s e n t i r e l y s i m i l a r t o t h a t of t h e o r e m

Suppose f i r s t t h a t

f

G

LP(Rn)

and

1 5 p <

a.

4.12.

Then, a f t e r f i x * P

with the operator ing N > 0, we proceed a s i n 4 . 1 2 , * i n s t e a d of P I n t h i s way, we show t h a t t h e s e t

.

as

v ,N

11. CALDERON-ZYGMUND THEORY

182

h a s measure pt * f ( y )

I f we c a l l

0.

f(x)

+

as

fl

E =

(y,t)N;T.x

we s h a l l h a v e x E.

EN,

4

N=l f o r every

IEl = 0

and

A s f o r t h e c a s e f G Lw(lRn) , we r e a l i z e t h a t t h e a r g u m e n t g i v e n i n 4.12 extends t o our c a s e , s i n c e , as can be e a s i l y s e e n , Pt * f 2 ( y )

+

as

0

( y , t ) N;T.x

if

x

6

n

B.

We s h a l l b e a b l e t o g i v e a more p r e c i s e r e s u l t by u s i n g a more d i r e c t a p p r o a c h b a s e d upon t h e i n e q u a l i t y :

v a l i d f o r every

x

lRn

G

and e v e r y

6

lRnttl

satisfying

w i t h a c o n s t a n t C N t h a t d e p e n d s o n l y on N we may a s s u m e t = 1 . n . To p r o v e ( 4 . 1 7 ) ,

IyI < N t , dimension

n e e d t o show t h a t , f o r e v e r y

lxlprovided

(y,t)

XI2

+ 1

y12

IyI < N .

t

1

x

G

and t h e T h e n , we

IRn,

S CN

This is very simple, since f o r

1x1 > 2 N ,

we c a n

make

and f o r

1x1 5 2 N

we s i m p l y u s e

+

Ix

-

yI2 + 1

5

1x1' + 1 5 4 N 2 + 1 .

The more p r e c i s e v e r s i o n o f t h e n o n t a n g e n t i a l c o n v e r g e n c e i s as f o l lows.

Proof:

Let x b e a L e b e s g u e p o i n t f o r f . G i v e n F > 0 , by d e f i n i t i o n of L e b e s g u e p o i n t , t h e r e i s a 6 > 0 s u c h t h a t f o r e v e r y

O < r < 6

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS Consider t h e f u n c t i o n

XI

g

d e f i n e d by

Iy < 6 and g(y) = 0 otherwise. t e e s t h a t Mg(x) 5 C E . Now, i f Iy -

-

g(y) = f ( y )

XI

183

f(x)

if

The i n e q u a l i t y a b o v e g u a r a n < Nt,

B u t , s i n c e y - z = x - z - ( x - y ) a n d Ix - y I < N t , inequality ( 4 . 1 7 ) a p p l i e s t o y i e l d P t ( y - z ) 5 CNPt(x - z ) Hence

.

The f i r s t i n t e g r a l i n t h i s sum i s Pt * l g l ( x ) 5 CMg(x) 6 C E . Now we c l a i m t h a t t h e s e c o n d i n t e g r a l i n t h e sum c a n a l s o b e made < E by m a k i n g t c l o s e e n o u g h t o 0 . I n d e e d , t h i s s e c o n d i n t e g r a l c a n b e b o u n d e d by

a n d we j u s t n e e d t o o b s e r v e t h a t f o r every

[ I I ~ I > ~ P ~ ( Z ) +~ 0~ Za s] ' t/ ~ 0 +

This is a very simple computation

q 2 1.

A s a c o n s e q u e n c e , we o b t a i n a n e x t e n s i o n t o o u r c o n t e x t o f F a t o u ' s

c l a s s i c a l theorem ( s t a t e d a f t e r c o r o l l a r y COROLLARY

IRn+l

& I

X E

IRn

Proof:

1

.

4.19.

Then __

1.21

i n Chapter

I.)

u ( x , t ) b e harmonic and bounded has non-tangential limits a t almost every p o i n t

Let the function

u

t h e b o u n d a r y of

IRn+l

According t o theorem

.

4.8,

u(x,t)

=

P, * f ( x )

f o r some

11. CALDERON-ZYGMUND THEORY

184 f G L~(R")

.

ing theorems

Then t h e c o n c l u s i o n o f t h e c o r o l l a r y f o l l o w s by a p p l y 4.16 o r 4.18.

We s h a l l p r e s e n t now a l o c a l v e r s i o n o f Fatou's t h e o r e m .

This w i l l

imply, i n p a r t i c u l a r , t h a t , f o r a harmonic f u n c t i o n u ( x , t ) in IRn+ 1 t h e e x i s t e n c e o f n o n t a n g e n t i a l l i m i t s a t a l m o s t e v e r y boun+ ' dary p o i n t is e q u i v a l e n t t o t h e n o n - t a n g e n t i a l boundedness almost F i r s t o f a l l , we make p r e c i s e t h e n o t i o n o f n o n - t a n g e n everywhere. t i a l boundedness. F o r a p o i n t x G IRn a n d r e a l numbers N > 0 , h h > 0 , we s h a l l c a l l T N ( x ) t o t h e open v e r t i c a l c o n e w i t h v e r t e x x

and a p e r t u r e

N , truncated a t height

h rN(x) = { ( y , t )

G

Rn++' : [ y

-

h,

that is

XI<

Nt

and

0 < t < h}.

Then we make t h e f o l l o w i n g : DEFINITION

4-20.

non-tangentially N > O

&

h

u

A function

x G IRn i s bounded i n

bounded a t %he p o i n t

> o

such t h a t

u

JRn++l i s s a i d t o b e

defined i n

_ .

i f t h e r e a r e numbers

Tk(x),

that i s ,

such t h a t

sup{

(Y

O b s e r v e t h a t , whi e t h e n o n - t a n g e n t i a l b o u n d e d n e s s o f u a t x h r e q u i r e s o n l y t h e e x i s t e n c e o f a t r u n c a t e d c o n e r N ( x ) on w h i c h u i s bounded, f o r n o n - t a n g e n t i a l convergence one h a s t o c o n s i d e r a l l possible

apertures.

Thus, even though t h e e x i s t e n c e of a non-tan-

g e n t i a l l i m i t of u a t x c l e a r l y i m p l i e s t h e n o n - t a n g e n t i a l b o u n d e d n e s s o f u a t x , t h e c o n v e r s e i s by n o means o b v i o u s . A c t u a l l y t h e converse i s f a l s e i f one l o o k s a t an i n d i v i d u a l p o i n t x , b u t it i s t r u e i f one l o o k s a t a whole s e t of p o i n t s and d i s r e g a r d s a s e t of measure 0 . T h i s i s t h e c o n t e n t of t h e f o l l o w i n g r e s u l t , which c a n b e v i e w e d a s a l o c a l v e r s i o n o f F a t o u ' s t h e o r e m .

Let u

Rn++l. Let E c IRn and s u p p o s e t h a t u i s n o n - t a n g e n t i a l l y b o u n d e d a t e v e r y x G E . Then u h a s n o n - t a n g e n t i a l l i m i t s a t a l m o s t e v e r y p o i n t x c E . THEOREM

Proof:

4.21.

be a harmonic f u n c t i o n i n

A l l we n e e d t o p r o v e i s t h a t i f

u

is non-tangentially

bounded a t e v e r y p o i n t o f a s e t o f p o s i t i v e m e a s u r e , t h e n t h e r e i s some s u b s e t o f p o s i t i v e m e a s u r e s u c h t h a t

u

has non-tangential

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

185

l i m i t s a t every point of t h i s subset.

Indeed, once t h i s i s proved, i f u i s n o n - t a n g e n t i a l l y bounded a t e v e r y p o i n t x G E , then the s e t { x G E : u d o e s n o t have a n o n - t a n g e n t i a l l i m i t a t XI

n e c e s s a r i l y must have m e a s u r e

0.

Assume t h a t u i s n o n - t a n g e n t i a l l y bounded a t e v e r y x 6 E w i t h [ E l > 0 and l e t u s t r y t o f i n d F c E w i t h IF1 > 0 s u c h t h a t u has non-tangential limits a t every

x

F.

G

We s t a r t by making s e v -

e r a l r e d u c t i o n s f o r which t h e h a r m o n i c i t y of u d o e s n o t p l a y any r o l e . F o r t h e s e p r e p a r a t o r y s t e p s we c o u l d assume t h a t u i s merely continuous.

F i r s t of a l l , E c a n b e t a k e n t o b e compact. A l s o , s i n c e E = U E N,h,M) where E(N,h,M) = Cx G E : l u ( y , t ) I S M f o r e v e r y h ( y , t ) c T N ( x ) l and t h e u n i o n i s t a k e n o v e r a l l N , h , M p o s i t i v e and r a t o n a l ; we may p e r f e c t l y w e l l assume t h a t E c o i n c i d e s w i t h a g i v e n E(N,h,M) o r , i n o t h e r words, t h a t t h e r e a r e N > 0 , h > 0 , M > 0, such t h a t f o r every x c E : sup I l u ( y , t ) I : ( y , t ) G r N h (x)}S 5 M. The n e x t s t e p w i l l b e t o show t h a t , f o r e v e r y p o s i t i v e i n t e g e r

k > N,

t h e r e i s a set Ek c E w i t h l E k l > 0 and such t h a t f o r every x G E k , u i s bounded o v e r Tkk ( x ) w i t h a bound Mk independent of x . T h i s p r o o f w i l l be b a s e d upon L e b e s g u e ' s d i f f e r e n t i a t i o n t h e o r e m . We know t h a t i f x G E i s a Lebesgue p o i n t f o r X E , the c h a r a c t e r i s t i c f u n c t i o n of E , then

C o n s e q u e n t l y , t h i s c o n v e r g e n c e h o l d s f o r a l m o s t e v e r y x G E . The p o i n t s x G E f o r which t h i s h o l d s a r e c a l l e d d e n s i t y p o i n t s of E . Take 0 < q < 1 . A f u r t h e r r e s t r i c t i o n on r- w i l l b e imposed b e l o w . If

x

G

E

i s a d e n s i t y p o i n t , we s h a l l h a v e

Now, i f we t a k e with

IF\ > 0

6 > 0

small enough, w e s h a l l have a s u b s e t

such t h a t , f o r every

x

G

F

and e v e r y

F c E

0 < r < 6

is

11. CALDERON-ZYGMUND THEORY

186

I B ( x , r ) n E 1 , i-. S u p p o s e we h a v e a f i x e d i n t e g e r k > N . We c l a i m IB(x,r) t h a t if q i s t a k e n c l o s e e n o u g h t o 1 , d e p e n d i n g on k , a n d i f

1

s > 0

(4.22)

i s t a k e n s m a l l e n o u g h , a l s o d e p e n d i n g on h ziEI'N(~) f o r every

Ti(x) c

x

G

k,

we have:

F.

L e t u s show t h i s . T a k e x G F a n d ( y , t ) G T z ( x ) , t h a t i s : Iy < kt, t < s . We must p r o v e t h a t t h e r e i s some z G E s u c h that Iy - z l < N t (we may a s s u m e , t o s t a r t w i t h , t h a t s < h ) . If

XI

z,

t h e r e e x i s t e d no such

we w o u l d h a v e

B(y,Nt)=

k:I z - y ) < N t j c I E n \ E

i n s u c h a way a n d , c o n s e q u e n t l y , B ( y , N t ) fl B ( x , k t ) c B ( x , k t ) \ E , t h a t t h e set B(x,kt)\E c o n t a i n s a t least a b a l l of diameter N t . Thus :

if q h a s b e e n s o c h o s e n . Now, i f s i s taken so small t h a t k s < 6 , we would h a v e a c o n t r a d i c t i o n w i t h t h e s e l e c t i o n o f 6 d e p e n d i n g on o u r

F

s,

(4.22)

q.

I t follows t h a t , with our choice of

h o l d s and hence, f o r e v e r y

I

x

G

F

and e v e r y

and and

F

(y,t)

G

TE(x), l u ( y , t ) S M. Suppose t h a t F c B ( 0 , R ) a n d t a k e Mk2 M 2 a n d a l s o M 2 s u p { l u ( y , t ) I : IyI 2 R + k , s 5 t I k l ( s i n c e u i s k c o n t i n u o u s , i t w i l l b e b o u n d e d on t h e c o m p a c t s e t E ( o , R + k 2 ) x x [s,k] c IRn++l 1 . Then f o r e v e r y x G F a n d e v e r y ( y , t ) G Tkk ( x ) , s o t h a t , b y s e t t i n g Ek = F , we h a v e t h e s e t we l u ( y , t ) I 5 Mk, were l o o k i n g f o r .

G

O b s e r v e t h a t , g i v e n k > N a n d E > 0 , t h e s e t Ek c a n b e t a k e n such t h a t IE\EkI < E. I n d e e d , o n c e i- h a s b e e n s e l e c t e d , we r e a l i z e t h a t t h e set m

U A. j=1 J

where

Since

(A.)

Eo

of d e n s i t y p o i n t s o f

IA~I

E

c a n be w r i t t e n a s

i s an i n c r e a s i n g f a m i l y , J E = ~J E~ ~ a s j -t a. I Thus by t a k i n g 6 > 0 small e n o u g h , s a y 6 = l / j , F = A . can be 1 t a k e n w i t h m e a s u r e a s c l o s c t o /El a s w e w i s h . -f

Our f i n a l p r e p a r a t o r y s t e p w i l l b e t o r e a l i z e t h a t t h e r e i s a :>et

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS G c E with c o n e r kk ( x ) ,

187

IGI > 0 such t h a t f o r every x G G and every truncated u i s b o u n d e d on r kk ( x ) w i t h a bound Mk d e p e n d i n g

on k , b u t n o t on x G G f o r f i x e d k . A c t u a l l y , g i v e n E > 0 , G c a n b e t a k e n i n s u c h a way t h a t I E \ G I < E . A l l w e have t o do i s t o consider f o r each integer k > N , a s e t Ek c E s a t i s f y i n g , a s i n t h e p r e v i o u s p a r a g r a p h t h a t u i s bounded on U { Tkk ( x ) : x G E k I and a l s o IE\ E k l < 2 - k ~ . Then, w e t a k e G = kfNEk so t h a t

k, u

and,for every

i s b o u n d e d on

U { rkk ( x ) : x

G

GI.

Once t h e s e p r e p a r a t o r y s t e p s h a v e b e e n t a k e n , o u r t a s k h a s b e e n r e d u c e d t o t h e f o l l o w i n g : We h a v e a c o m p a c t s e t E c J R n , and t h e h r e g i o n R = u T N ( x ) f o r some N > 0 a n d h > 0 f i x e d . We know XG E lu(x,t) I S 1 t h a t u i s a harmonic f u n c t i o n i n R n t l such t h a t f o r every (x,t) G R. Then we m u s t show t h a t f o r a l m o s t e v e r y X G E , there exists l i m u(y,t) as ( y , t ) t e n d s t o (x,O) r e m a i n i n g always i n s i d e of j , define wise. For write @ j( x )

R.

Let u s show t h i s .

For every p o s i t i v e i n t e g e r

if ( x , l / j ) G R a n d @ j ( x ) = 0 o t h e r @j(x) = u(x,l/j) ( x , t ) G lRnttl we d e f i n e Q j ( x , t ) = P t * @ j ( x ) , a n d we

u(x,t t l/j) = $.(x,t) t $.(x,t). The s e q u e n c e o f f u n c t i o n s 1 1 i s bounded i n Lm(IRn) a n d , c o n s e q u e n t l y , it h a s a sub-

sequence

converging i n t h e weak-* topology of

(@j

to a

Lm

we d e f i n e @ ( x , t ) = f u n c t i o n @ G L m ( R n ) . F o r ( x , t ) G IRnt+' P t * @ ( x ) . The weak-* c o n v e r g e n c e i m p l i e s t h a t @ j l ( x , t )- + @ ( x , t )

=

u(x,t) and f o r e v e r y ( x , t ) G IRnt+' . S i n c e u ( x , t + l / j ) ~ + ~ , ( x , +t )@ ( x , t ) , i t f o l l o w s t h a t , a l s o $ . l ( x , t ) $(x,t) for 1 e v e r y ( x , t ) G IRnttl, a n d w e h a v e : u ( x , t ) = @ ( x , t ) t $ ( x , t ) . Now, -f

+

since

I$(x,t)

=

Pt

* @ ( x ) and

@

G

Lm(IRn)

,

the ordinary

Fatou

implies t h a t @ ( y , t ) has non-tangential theorem ( c o r o l l a r y 4.19) l i m i t s a t a l m o s t e v e r y p o i n t x G IR" . We s h a l l p r e s e n t l y s e e t h a t f o r almost every x f E , $ ( y , t ) converges t o 0 as ( y , t ) (x,O) remaining i n s i d e of $2. This w i l l f i n i s h t h e proof of t h e theorem. Write t h e b o u n d a r y o f R a s a R = E U B w h e r e B = I ( x , t ) f a R : t > O I . +

We s h a l l c o n s t r u c t a f u n c t i o n

a)

H

b)

H 2 0 i n IR"" H t 2 i n B , and For almost e v e r y x tangentially.

c) d)

i s harmonic i n

H(x,t)

with the following properties:

IRn+tl

6

E

is

H(y,t)

+

0

as

(y,t)

+

(x,O)

non-

11. CALDERON-ZYGMUND THEORY

188

Assuming f o r a moment t h e e x i s t e n c e o f

we s h a l l s e e t h a t l $ ( x , t ) I 5 H ( x , t ) f o r e v e r y ( x , t ) G R , f r o m which t h e c o n v e r folgence $ ( y , t ) + 0 a s ( y , t ) + (x,O) remaining i n s i d e of R , lows i m m e d i a t e l y f o r a l m o s t e v e r y x G E . Now, t o p r o v e t h a t H,

I $ ( x , t ) ( 5 H ( x , t ) f o r e v e r y ( x , t ) G R , we j u s t n e e d t o s e e t h a t , f o r a g i v e n ( x , t ) 6 R , and f o r an a r b i t r a r y E > 0, w e have Ibj(x,t)

I

I H(x,t) +

E

for

b i g enough.

j

If

l / j < h,

call

where B . = I ( x , ~ )G an. : t > 0 1 . J J

we h a v e , f o r j b i g e n o u g h , ( x , t ) G R . a n d J so t h a t l u ( x , t + l / j ) I 5 1 . Also, 1 , and hence I J l j ( x , t ) I 5 2 , and t h i s h o l d s a l s o f o r We know t h a t 2 5 H ( x , t ) f o r ( x , t ) G B . T h u s , by g i v e n E > 0 , we s h a l l have 2 S H ( x , t ) + E f o r e v e r y p r o v i d e d j i s b i g enough. Thus, f o r j b i g enough H(x,t) + E f o r every ( x , t ) c B . . Also, since

Given ( x , t ) G R , ( x , t + l / j ) G ll,

I

I ~ $ ~ ( x , t )6 (x,t) G B j* continuity, (x,t) G B. IQj(x,t)

I

1 5

J

w i t h u n i f o r m c o n v e r g e n c e on c o m p a c t s e t s , i t t u r n s o u t t h a t

I

6 E for ( x , t ) G R and t s m a l l enough. Then, t h e f a c t I$j(x,t) t h a t H 2 0 i m p l i e s I $ j ( x , t ) 6 H ( x , t ) + E. Thus, given E > 0, we know t h a t f o r j b i g e n o u g h a n d f o r t o > 0 small e n o u g h , t h e

I

inequality I q j [ x , t ) 1 5 H ( x , t ) + E h o l d s i n t h e boundary of t h e s e t R t. 0 = I ( x , t ) G R . : t > t o } . But i s a subharmonic funcJ 1 t i o n a n d H + E i s h a r m o n i c . C o n s e q u e n t l y we h a v e I q j ( x , t ) l 6 6 H ( x , t ) + E a l l through Rto and, since t h i s i s v a l i d f o r a l l j s u f f i c i e n t l y small t o > 0 , w e a c t u a l l y h a v e I q j ( x , t ) I 5 H ( x , t ) + E f o r every ( x , t ) c R . . We h a v e s u c c e e d e d i n p r o v i n g w h a t w e w a n t e d : J g i v e n ( x , t ) c Q a n d g i v e n E > 0 , we h a v e l $ j ( x , t J I 5 H ( x , t ) + E f o r b i g enough j . From t h i s w e i m m e d i a t e l y g e t I$(x,t) I 6 H(x,t) f o r e v e r y ( x , t ) G R. I t only remains t o c o n s t r u c t t h e function

H.

Let

X

be t h e charac-

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

t e r i s t i c f u n c t i o n o f t h e complement o f

E,

189

t h a t i s X = XIRn,E.

T h e n , we c l a i m t h a t , f o r a n a p p r o p r i a t e c o n s t a n t

C,

the function

H ( x , t ) = C{(Pt * X ) ( x ) + t } s a t i s f i e s t h e f o u r p r o p e r t i e s r e q u i r e d . In f a c t , a ) , b ) and d) a r e c l e a r and o n l y c ) needs t o be v e r i f i e d . For t h o s e Ch 2 2 .

(x,t)

with

B

6

The p o i n t s

(x,t)

we have

t = h, G

B

with

0 < t < h

s e t { ( x , t ) G Rn++l : d i s t ( x , E ) = N t l , 0 < t h , we h a v e B ( x , N t ) c IRn\ E

T h u s , w e j u s t n e e d t o make H(x,t) 2 2

also

f o r those

H(x,t) 5 2

i f w e make

are included i n the

so t h a t , i f (x,t) and, consequently

6

B

and

C appropriately big t o guarantee t h a t

(x,t)

with

B

6

0 < t < h.

we s e e t h a t , i f a function u(x,t) i s harmonic i n Rn+' and it i s uniformly i n Lp(IRn) f o r some 1 < p < m, then the functions ut(x) = u ( x , t ) i n t h e Lp norm a s converge t o a g i v e n f u n c t i o n f G Lp(Rn) t + 0. B e s i d e s , theorem 4 . 1 2 tells us t h a t there is pointwise c o n v e r g e n c e a l m o s t e v e r y w h e r e , i n s u c h a way t h a t , f o r a l m o s t e v e r y x G R~ is f(x) = l i m u(x,t). By c o m b i n i n g t h e o r e m

4.8

and theorem

4.l(a)

t+O

I n case u ( x , t ) with p = m o r 4.9

i s h a r m o n i c i n IRn" and i s uniformly i n p = 1 , we c a n s t i l l s e e by u s i n g t h e o r e m

and t h e n p a r t

d)

i n theorem

4.1

o r the closing sentence i n

t h i s same t h e o r e m , t h a t t h e f u n c t i o n s f

G

ut o r t o a c e r t a i n Bore1 measure

Lm(IRn)

c o r r e s p o n d i n g weak-* Thus, f o r

u

topology a s

t

harmonic, t h e f a c t t h a t

+

Lp(Rn) 4.8 or

converge t o a c e r t a i n p G M(IRn) in the

0.

u

i s uniformly i n

Lp(IRn)

11. CALDERON-ZYGMUND THEORY

190

with 1 5 p S m implies t h e convergence of t h e family ut a s t + O i n t h e s e n s e o f t e m p e r e d d i s t r i b u t i o n s ( a l l t h e k n o w l e d g e a b o u t temp e r e d d i s t r i b u t i o n s t h a t we s h a l l n e e d i s c o n t a i n e d i n S t e i n - W e i s s [Z], Chapter I , s e c t i o n 3 ) . A l s o , t h e boundary d i s t r i b u t i o n ( f o r p a s t h e c a s e may b e ) u n i q u e l y d e t e r m i n e s u . Now s u p p o s e t h a t u i s s t i l l h a r m o n i c i n IRn++l a n d u i s u n i f o r m l y i n Lp(IRn) but Then t h e o r e m 4 . 7 t e l l s u s t h a t each ut i s a bounded f u n c t i o n , h e n c e a t e m p e r e d d i s t r i b u t i o n . The q u e s t i o n a r i s e s n a t u r a l l y : Does l i m u t e x i s t i n t h e sense of tempered d i s t r i b u t i o n s ? t+O The a n s w e r 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 : 0 < p < 1.

THEOREM

Let u ( x , t )

4.23.

suppose t h a t

u

IR n tl

be a harmonic f u n c t i o n i n

Lp(IRn)

i s uniformly i n

f o r some

p

and

t

-

0.

l i m u t = f e x i s t s i n t h e s e n s e of t e m p e r e d d i s t r i b u t i o n s , a n d t h i s t+O l i m i t f uniquely determines u.

we assume 0 < p < 1 . F o r ( x , t ) 6 IRn+tl a n d 6 > 0 , we d e f i n e u 6 ( x , t ) = = u ( x , t + & ) . Each u 6 i s h a r m o n i c i n IRn++l. From t h e o r e m 4 . 7 we We c a n u s e t h i s e s t i m a t e t o o b t a i n l u ( x , t ) ) 5 CMt-n'P. know t h a t Proof:

S i n c e we know t h e t h e o r e m t o h o l d f o r

This implies,in particular,that

I t follows t h a t , f o r each

f o r each

6 >O,

u (x,t)

g r a l o f some f i n i t e B o r e 1 m e a s u r e .

6

6 > 0

1 5 p 5

a,

is

w i l l be t h e Poisson i n t e -

A c t u a l l y , s i n c e t h i s measure has

t o b e t h e weak-* l i m i t o f t h e f u n c t i o n s

x n u6(x,t)

as

t

+

0,

we

see t h a t it coincides with t h e i n t e g r a b l e f u n c t i o n u,(x) = u ( x , 6 ) . I f q > 6 > 0 , w e have u ( x ) = u ( x , q ) = u ( x , n - 6 ) = P * u,(x). rl 6 ri-6 T a k i n g F o u r i e r t r a n s f o r m s we g e t :

This allows us t o d e f i n e a function 6 > 0.

j,

$(c)

= ~ ( ~ ) e * n ~ 6c G~ I 6 R " ,,

is a well defined continuous function because t h e r i g h t

11.4. HARMONIC AND SUBHARMONIC FUNCTIONS

hand side does not depend on

we obtain the estimate 1$(5)e-Z'151tl where we have made

N

i

lu(x,t) Idx 2 CMt-n((l'P)-l)= IRn n( (l/p) - 1 ) > 0. Thus 2

=

6 > 0. Now, since for any

191

t > 0,

Ct-N

This estimate implies that $ is a slowly increasing function and, consequently, a tempered distribution. We know (see Stein-Weiss [Z]) that the Fourier transform establishes an isomorphism from the topological vector space S ' , formed by the tempered distributions, onto itself. Therefore, there is a unique f G S' such that $ = f. Now it is easy to see that ut + f in S' as t + 0. We have to see that, for every function @ in the Schwartz class s h

as t

+

0. But, by using the Fourier transform

Observe that u.

f

=

0

forces

u = 0, so that

f

uniquely determines

1_1

It is important to note that the estimate obtained for $ in the above proof involves M, and it implies Il 2 C(@)M. We shall have to appeal to this estimate in the next chapter, when we shall d e a l w i t h t h e r e a l v a r i a b l e t h e o r y o f Hardy s p a c e s .

11. CALDERON-ZYGMUND THEORY

192

5.

SINGULAR INTEGRAL OPERATORS

A l l t h r o u g h t h i s c h a p t e r we have s e e n t h e a c t i o n o f t h e Calder6n-Zyg-

mund d e c o m p o s i t i o n i n s e v e r a l s i t u a t i o n s . We s h a l l now a p p l y t h i s p o w e r f u l d e v i c e t o t h e p r o b l e m f o r w h i c h i t was o r i g i n a l l y c r e a t e d ( i n A . C a l d e r 6 n a n d A . Zygmund [ l ] ) , n a m e l y , t h e e s t i m a t i o n o f o p e r a t o r s of t h e form Tf(x) = where

P.V.

I R n Q ( y ) lyl-"f(x

-

y)dy

i s homogeneous o f d e g r e e z e r o a n d i t s i n t e g r a l o v e r t h e

Q

I t t u r n s o u t t h a t t h e same m e t h o d w o r k s f o r a unit sphere is n u l l . w i d e r c l a s s o f o p e r a t o r s w h i c h we s h a l l now d e s c r i b e : DEFINITION

Given a tempered d i s t r i b u t i o n

5.1.

K, t h e c o n v o l u t i o n

opera t o r

is c a l l e d a s i n g u l a r i n t e g r a l o p e r a t o r i f t h e f o l l o w i n g t w o c o n d i tions are satisfied:

function

(a)

i

(b)

K

K(x)

f

L"(IR") coincides i n

Eln\

{Ol

with a l o c a l l y integrabZe

s a t i s f y i n g Hbrmander ' s c o n d i t i o n :

The f i r s t c o n d i t i o n g i v e s , by P l a n c h e r e l ' s t h e o r e m , t h e i n e q u a l i t y

and t h e r e f o r e , h

T

e x t e n d s t o a bounded o p e r a t o r i n

L2(IRn)

with

Under t h e a s s u m p t i o n ( a ) o n l y , n o t h i n g more c a n b e norm IIKII,. s a i d a b o u t T , b u t i f we a d d ( b ) , t h e n o t h e r e s t i m a t e s c a n b e p r o v e d , a s we s h a l l s e e b e l o w . Let u s now l o o k a t t h e e f f e c t w h i c h H o r m a n d e r ' s c o n d i t i o n i s d e s i g n d t o produce: I f K G L:oc(IRn \ I O l ) and a ( x ) i s a n i n t e g r a b l e f u n c t i o n s u p p o r t e d i n a cube Q c e n t e r e d a t t h e o r i g i n and w i t h

1 1 . 5 . S I N G U L A R INTEGRAL

193

mean v a l u e z e r o , t h e n t h e c o n v o l u t i o n K * a(x)

= IQK(x =

makes s e n s e f o r Writing

8

=

%

and x 6 Q ,

Q

a.e. x

[K(x 6

y)a(y)dy =

-

y)

-

K(x1)

and i s l o c a l l y i n t e g r a b l e away from

Q

it i s obvious t h a t

QZfi,

and c o n d i t i o n

I

-

5.l(b)

1x1 > 21yl

6

Q

.

i n an a r b i t r a r y c u b e .

If

5.2.

in a c u b e

y

then implies

By t r a n s l a t i o n i n v a r i a n c e , t h e same i s t r u e i f

LEMMA

whenever

Q.

Q

K

a(x)

is supported

T h u s , we have p r o v e d

satisfies

with

S.l(b)

a(y)dy = 0,

&

a

f

L 1 ( R n ) is supported

then (with

'L Q

= QZ f i )

Before proving t h e e s t i m a t e s f o r 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 , l e t u s l o o k f o r i n t e r e s t i n g e x a m p l e s t o which t h e t h e o r y c a n b e a p p l i e d . More a p p l i c a t i o n s w i l l be g i v e n i n t h e n e x t s e c t i o n . ~EXAMPLES:

( a ) We a r e g o i n g t o d e s c r i b e t h e 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 d e f i n e d by c o n v o l u t i o n w i t h p r i n c i p a l v a l u e d i s t r i b u t i o n s .

These d i s t r i b u t i o n s K ( @ ) = p.v. where

k G Lioc(Rn growth c o n d i t i o n

K = p.v.k(x)

J

k(x)r$(x)dx = l i m

\ {O))

E+O

J

IXI>E

k ( x ) r$ ( x ) dx

i s a f u n c t i o n t o which we impose t h e

( k ( x ) Idx 5 C 1

(5.3)

a c t i n t h e f o l l o w i n g manner:

(r > 0 )

(which i s n a t u r a l i n view of t h e b a s i c example m e n t i o n e d a t t h e b e g i n n i n g of t h i s s e c t i o n ) . Assuming ( 5 . 3 1 , i t i s easy t o check that

K($)

i s w e l l d e f i n e d f o r a l l Schwartz f u n c t i o n s

r$

and

K

is

11. CALDERON-ZYGMUND THEORY

194

a tempered d i s t r i b u t i o n i f

Indeed, then

if

+

(5.3)

I

and

hold,

(5.4)

with

L

=

p.v.j

Ixl
k(x)dx,

k(x)@(x)dx= !L$(O) + I1 + I2 1x121

where t h e i n t e g r a l s

I,

and

I2

a r e a b s o l u t e l y c o n v e r g e n t , and

moreover

and

A l l these principal value d i s t r i b u t i o n s give r i s e t o singular i n t e -

g r a l o p e r a t o r s provided t h a t Harmander's c o n d i t i o n i s s a t i s f i e d : PROPOSITION

5.5.

and

a n d if

15.41,

Lf

k

f

1

Lloc(Rn\

K = p.v.

k(x),

T f ( x ) = I( * f ( x ) = l i m E*O

J / yI

s a t i s f i e s ( 5 . lib)), ( 5 . 3 1

{Ol)

then >E

k(x

-

y)f(y)dy

(f

G

S(Rn))

i s a singular integral operator. I n o r d e r n o t t o d e l a y t o o much t h e s t u d y o f t h e p r o p e r t i e s o f 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 , t h e p r o o f s of t h i s and t h e n e x t p r o p o s i t i o n

195

11.5. SINGULAR INTEGRAL OPERATORS

a r e p o s t p o n e d t o t h e end of t h e s e c t i o n . ( b ) L e t u s c o n s i d e r now t h e p a r t i c u l a r c a s e i n which i s homogeneous of d e g r e e - n , i.e.

k(x)

w i t h R homogeneous o f d e g r e e 0 ( w e s h a l l s y s t e m a t i c a l l y u s e t h e notation: x ' = x/lxl). The o p e r a t o r T o b t a i n e d by c o n v o l u t i o n w i t h k w i l l t h e n b e , wherever i t may b e d e f i n e d , d i l a t i o n i n v a r i ant: T(f6) = (TfI6 Now, c o n d i t i o n s

(5.3)

(with and

(5.4)

f 6 (x) = f ( 6 x ) ,

6 > 0)

a r e respectively equivalent t o

and R(x')da(x') = 0 where da d e n o t e s t h e u n i t sphere.

( n - 1 ) - d i m e n s i o n a l Lebesgue measure on t h e

F i n a l l y , some c o n t i n u i t y c o n d i t i o n must b e imposed on R i n o r d e r t o e n s u r e t h a t S . l ( b ) h o l d s . To t h i s e n d , we i n t r o d u c e t h e f o l l o w i n g L 1 -modulus o f c o n t i n u i t y :

PROPOSITION 5 . 6 . S u p p o s e that R i s homogeneous of d e g r e e integrable over t h e u n i t sphere w i t h i n t e g r a l equal t o zero.

L 1-Dini c o n d i t i o n

holds,

then

0 If

the

11. CALDERON-ZYGMUND THEORY

196

is a s i n g u l a r i n t e g r a l o p e r a t o r . M o r e o v e r , i f 52 is o d d , t h e n t h e e x t e n s i o n o f T 2 L2(lRn) i s g i v e n , i n terms o f t h e Fourier tranof o r m , by ( T f I 1 ( 5 ) = i ( S ) m ( S ) , w h e r e

Observe t h a t

m(S) = ( P . v . Q ( x ) l x l - " ) ' ( C )

i s homogeneous o f d e g r e e

z e r o , s o m e t h i n g w h i c h c o u l d h a v e b e e n p r e d i c t e d d u e t o t h e homog e n e i t y o f 52(x) and t h e behaviour o f t h e F o u r i e r t r a n s f o r m w i t h r e s p e c t t o d i l a t i o n s . An e x p l i c i t f o r m u l a f o r m ( S ) when R i s n o t n e c e s s a r i l y odd c a n b e f o u n d i n S t e i n [l] o r S t e i n - W e i s s [Zl. We a l s o p o i n t o u t t h a t t h e L 1 - D i n i c o n d i t i o n i s r a t h e r w e a k ; i n p a r -

XI-^

t i c u l a r , it i s v e r i f i e d i f

IR(x)

-

R(y)

I

2 C

[ X I

-

f o r some

y'Ia

a > 0. c ) I n IR1 , t h e r e i s b a s i c a l l y o n e homogeneous s i n g u l a r i n t e g r a l o p e r a t o r , c o r r e s p o n d i n g t o Q ( t ) = rr1 s i g n ( t ) ( o r k ( t ) = 1

x),

namely

This i s t h e Hilbert transform,

(Hf)-(E)

= -i

sign

which i s a l s o d e f i n e d i n

L2(IR)

by

([)f(E)

In f a c t , t h e formula of Proposition

5.6

g i v e s i n t h i s case

I n IRn , we h a v e a f a m i l y o f a n a l o g u e s o f t h e H i l b e r t t r a n s f o r m c o r r e s p o n d i n g t o R h ( x l ) = c n ( x ' - h ) , where h i s a n y f i x e d v e c t o r i n IRn , a n d t h e n o r m a l i z i n g c o n s t a n t c n i s c h o s e n ( f o r a r e a s o n w h i c h w i l l become c l e a r i n a moment) s o t h a t

f o r e v e r y u n i t v e c t o r u ( a n Advanced C a l c u l u s c o m p u t a t i o n shows t h a t c n = TI - ( n + 1 ) ' 2 r ( y ) ) . S i n c e Qh d e p e n d s l i n e a r l y on h , it s u f f i c e s t o c o n s i d e r t h e o p e r a t o r s corresponding t o t h e v e c t o r s of t h e s t a n d a r d b a s i s : h = e l , e 2 , ..., e n , n a m e l y R.f(x) = 1

P.V.

cn

J+

f ( x -y)dy

Cj = 1 , 2 ,

...,n )

11.5. SINGULAR INTEGRAL OPERATORS

which a r e c a l l e d t h e R i e s z t r a n s f o r m s . The a c t i o n of c i a l l y s i m p l e i n t e r m s of t h e F o u r i e r t r a n s f o r m :

197

R

j

i s spe-

7

(f

T h i s i s a consequence of

(5.6)

G

L'

; j = I , 2,

..., n )

and t h e f o l l o w i n g i d e n t i t y

whose p r o o f i s now, w i t h o u r c h o i c e of c n , a l m o s t t r i v i a l : We can f i x 5 with 151 = 1 , and t h e n , t h e l e f t hand s i d e i s a l i n e a r f u n c t i o n of

h,

say

such t h a t

!L(h),

lL(h)

I

5

lhl

and

L(5)

= 1;

L(h) = E - h .

t.his n e c e s s a r i l y im plies

We c a n now s t a t e t h e main r e s u l t c o n c e r n i n g s i n g u l a r i n t e g r a l o p e r a tors : THEOREM

5. 7 .

Every 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 t i s f i e s t h e i n -

equalities

where

~

C

P'

1 2 p 2

d e p e n d s o n l y on

m,

~ & k BK ~ ~of tmh e

p,n

and on t h e c o n s t a n t s

kernel.

Thus, we can c o n s i d e r e v e r y s i n g u l a r i n t e g r a l o p e r a t o r e x t e n d e d a s a bounded o p e r a t o r from

Lp

t o i t s e l f , from

L r = I L m f u n c t i o n s w i t h compact s u p p o r t )

L 1 t o weak-L1 t o BMO.

and from

The p l a n of t h e p r o o f i s a s f o l l o w s : We o b t a i n f i r s t t h e b a s i c e s t i mate ( 5 . 9 ) . I n t e r p o l a t i n g by M a r c i n k i e w i c z ' theorem 2 . 1 1 between t h i s and t h e L 2 -boundedness o f T , the inequalities (5.8) follow f o r 1 < p 5 2 . We t h e n p r o v e a t e c h n i c a l lemma (5.11) which w i l l a l s o be u s e d l a t e r , and from such a r e s u l t , t h e Lm-BMO e s t i m a t e F i n a l l y , we u s e ( 5 . 1 0 ) and t h e i n t e r p o l a t o o b t a i n t h e remaining L P - i n e q u a l i t i e s :

w i l l e a s i l y be d e r i v e d .

t i o n theorem 2 < p
3.7

11. CALDERON-ZYGMUND THEORY

198

Take

f

L1

n

L2

and, given

t > 0,

l e t {Qj} b e t h e c u b e s o f t h e Calder6n-Zygmund d e c o m p o s i t i o n c o r r e s p o n d i n g t o If I a t h e i g h t t . (see 1 . 1 2 ) . T h i s decomposition of t h e underl y i n g s p a c e i n t o c u b e s a l l o w s t o d e f i n e what i s c a l l e d t h e CalderBnZygmund d e c o m p o s i t i o n , f ( x ) = g ( x ) + b ( x ) , of t h e f u n c t i o n f i n t o i t s "good p a r t " g a n d i t s "bad p a r t " b , w h i c h a r e g i v e n a s P r o o f of

(5.9):

G

follows:

IRn\ R with

fi = U Q . 3 1

(observe t h a t

-

b(x) = f ( x ) where

b. J

g(x) =

i s supported i n b.(x) = {f(x) 3

Ifi

I

6 Ct- 1

(XI

\ I f 11 1 )

1 b.(x)

j

Qj

J

a n d h a s mean v a l u e z e r o ; p r e c i s e l y :

- -

To e s t i m a t e Tg we u s e t h e f a c t t h a t T i s bounded i n L' gether with the observation t h a t \g(x) \ = (f(x) \ 5 t f o r a, a n d I g ( x ) I 5 2"t f o r a l l x G R ; t h u s a.e. x

to-

4

N e x t , we o b s e r v e t h a t b j (x)

Lemma

can be a p p l i e d t o each p i e c e

5.2

o f t h e bad p a r t , b e c a u s e

4

Tb. (x) = K * b . (XI f o r a . e . x Qj 3 1 b . by t e s t f u n c t i o n s s u p p o r t e d J

( t h i s c a n b e s e e n by a p p r o x i m a t i n g 2& in Qj). Thus, if Q . = Q . J J QJ

On t h e o t h e r h a n d , s i n c e 2

b

G

L2(IRn)

c o n v e r g e i n t h e L -norm t o b lar, ITb(x)\ 5 L)Tb.(x)) a.e., J J

, t h e series

a n d Tb, and

b. 5.1 respectively.

and

LTb.

J

.J

In p a r t i c u -

11.5. SINGULAR INTEGRAL OPERATORS

199

B e f o r e s t a t i n g t h e a u x i l i a r y lemma, we i n t r o d u c e a v a r i a n t of t h e H a r d y - L i t t l e w o o d maximal o p e r a t o r which w i l l a l s o a p p e a r i n t h e sequel : NOTATION:

LEMMA

For

5.11.

1 5 p <

Let

e x i s t s f o r every

f

E

Cp

L:(Rn)

> 0.

ITf(x)

E-n

with

f

w e denote

m,

i n d e p e n d e n t of

,

M f = M(lflP)l'P,

P

so that the integral

1 < p <

Then, f o r a l l

-

E

i.e.

we h a v e

m,

IEldx 5 C M f(0) + P P

and

f

S i n c e M f i n c r e a s e s a s p i n c r e a s e s , we c a n assume t h a t P and i n t h i s c a s e , we have a l r e a d y p r o v e d t h a t T i s bounded i n L p . Set Proof:

1 < p 5 2,

f

1

= fX

tx:

IXl
'

f2 =

-

1

Then 1Tf(x)

-

IEI 5 ITfl(x)I +

and w e o n l y have t o o b s e r v e t h a t , i f of

P,

[K(x p ' = -2P-1

-

y)

-

K(-y)]f(y)dy(

i s t h e d u a l exponent

11. CALDERON-ZYGMUND

200

The f i n a l s t e p t o e s t a b l i s h of

Proof

p = 2,

with all

An

f

(5.10):

because

G

Theorem

THEORY

is:

5.7

We o n l y n e e d t o u s e t h e p r e v i o u s lemma w i t h ( s a y ) M2f (0) 6 ( I f and

(Im

d e p e n d i n g o n l y on t h e d i m e n s i o n of t h e s p a c e .

,

Lr(IRn) (Tf)

#

Thus, f o r

we o b t a i n

(0) 5 A n ( C 2 + B K I

Ilfllm

and s i n c e e v e r y t h i n g i s t r a n s l a t i o n i n v a r i a n t , we a l s o have

which i s ( 5 . 1 0 ) . I f one w i s h e s t o g e t t h e same i n e q u a l i t y f o r a l l (even though t h i s i s a r a t h e r i r r e l e v a n t q u e s t i o n ) , f E L 2 fl La j u s t d e f i n e f . = fX I x : s o t h a t T f = l i m T f . ( i n L 2 ) and J j+a J

~

Il~fllBMo6 l i m

~

SUP

l

<

~

I l ~ f j l I B M o2

~

cIlfllm *

The e s t i m a t e s of Theorem 5 . 7 seem s p e c i a l l y s a t i s f a c t o r y i n L p w i t h 1 < p < m . I t i s n o t t r u e , however, t h a t 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 map L 1 ( o r L”) t o i t s e l f , and t h i s i s what makes t h e s u b s t i t u t e r e s u l t s (5.9) and ( 5 . 1 0 ) i n t e r e s t i n g . We c a n e a s i l y s e e t h i s i n t h e c a s e of t h e H i l b e r t t r a n s f o r m a p p l i e d t o t h e f u n c tion f = 6 L~ n L”:

xLalbI

I t is clear that also follows t h a t

Hf Hf

4 L”, 4

and s i n c e (but

L1

Hf

Hf(x) 6

Q

-,b-a

(weak-Ll)fl

1x1

+

m,

it

BMO).

I n o r d e r t o g e t a b e t t e r u n d e r s t a n d i n g of t h e b e h a v i o u r of o u r o p e r a t o r s i n L 1 and L m , we p o s e o u r s e l v e s two q u e s t i o n s : (Ql) Are t h e r e f u n c t i o n s f G L 1 f o r which we c a n a s s e r t

that

Tf

G

L1

f o r a l l 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 T?

11.5. SINGULAR INTEGRAL OPERATORS

(4.2)

201

T i s a s i n g u l a r i n t e g r a l o p e r a t o r , i s i t poss i b l e t o d e f i n e " i n a n a t u r a l way" Tf f o r a l l f G L m , s o t h a t T : Lm + BMO?.

If

Looking f o r an answer t o t h e f i r s t q u e s t i o n , we i m m e d i a t e l y f i n d a n e c e s s a r y c o n d i t i o n : I f f G L 1 i s s u c h t h a t R . f G L 1 f o r some J we must h a v e Riesz transform R . 1'

and s i n c e t h e l e f t hand s i d e i s c o n t i n u o u s , s o must be t h e r i g h t g ( 0 ) = I f = 0.

hand s i d e , which f o r c e s :

T h i s s u g g e s t s t h e p o s s i b i l i t y of u s i n g we a l s o assume t h a t

Lemma

i s supported i n a cube

f

5.2, Q.

provided t h a t In f a c t

and t h u s , a u n i f o r m e s t i m a t e w i l l be o b t a i n e d , f o r i n s t a n c e , i f llfll,

5

Let 1.

IQI

u s c o l l e c t t h e c o n d i t i o n s imposed s o f a r and g i v e

a name t o t h e f u n c t i o n s s a t i s f y i n g them: DEFINITION

cube

Q

5.12.

'

i s a bounded f u n c t i o n supported i n a Ilallm 5 l a ( x ) d x = 0.

An a t o m ( * )

and such t h a t

IQI

Now, what t h e p r e c e d i n g r e m a r k s p r o v e i s t h a t

f o r e v e r y atom C

a(x)

d e p e n d i n g o n l y on

k e r n e l of

and every s i n g u l a r i n t e g r a l o p e r a t o r n

a n d on t h e c o n s t a n t s

BK

and

I)iII

T,

with of t h e

T.

One may a r g u e t h a t

(5.13)

i s n o t a good a n s w e r t o

(Ql)

due t o

t h e v e r y s p e c i a l c o n d i t i o n s imposed t o a ( x ) . A t r i v i a l ( b u t n e v e r t h e l e s s i n t e r e s t i n g ) e x t e n s i o n c o n s i s t s i n d e f i n i n g t h e Banach (*)

This w i l l be c a l l e d a

(1,m)-atom i n

C h a p t e r 111.

11. CALDERON-ZYGMUND THEORY

202

1 H,~(IR~I

space

f f o r some

(atomic iff

6

(Aj)

G

II'

The s e r i e s I f = 0 f o r 5e v' jearj y

H

1

I as f o l l o w s :

f(x) =

and atoms

1j A J.

a.(x> J

a j (x)

,

and

converges i n L 1 , so t h a t f G H a1 t . Now, we have

H a1t

COROLLARY 5 . 1 4 : Every s i n g u l a r i n t e g r a l o p e r a t o r H a1 t ( l R n ) boundedly i n t o L1(IRn) ,

G.

c L1,

and c l e a r l y

T

More i n s i g h t i n t o HAt w i l l b e g a i n e d i n C h a p t e r 1 1 1 . For t h e time b e i n g , l e t u s merely observe t h a t a Schwartz f u n c t i o n f 1 b e l o n g s t o Hat i f and o n l y i f I f = 0 . Turning t o

(QZ),

l e t us f i r s t observe t h a t i t i s n o t a t r i v i a l

q u e s t i o n . I n f a c t , t h e p r o o f of Theorem 5 . 7 shows t h a t we c a n a c t u a l l y d e f i n e T : Lp fl L m + BMO f o r a r b i t r a r y p my but s i n c e Lp fl Lm i s n o t d e n s e i n L m , t h i s d o e s n o t a l l o w t o e x t e n d T by c o n t i n u i t y t o a l l bounded f u n c t i o n s . F o l l o w i n g t h e argument i n t h e proof of (5.10), one would b e t e m p t e d t o d e f i n e Tf = l i m T ( f X B j ) , where B . = I x : 1x1 2 j } . Such a l i m i t , however, may n o t e x i s t i n 1

But, any r e a s o n a b l e s e n s e ( s e e t h e example below P r o p o s i t i o n 5 . 1 5 ) . s i n c e w e want t o c o n s i d e r Tf and T(fXBj) a s e l e m e n t s of BMO, we may t r y t o add a c o n s t a n t t o e a c h t e r m T(fXBj) s o a s t o make t h e c o r r e c t e d sequence converge. PROPOSITION

nel

K,

5.15.

f o r each

This, w e can achieve:

G i v e n a s i n g u l a r i n t e g r a l opera-

f s Lm

a.e.,

with ker-

L1 and a l s o p o i n t satisfies

The s e q u e n c e t o t h e r i g h t c o n v e r g e s l o c a l l y i n wise

T

we d e f i n e

and t h e e x t e n d e d o p e r a g

T

11.5. SINGULAR INTEGRAL OPERATORS

is t h e same c o n s t a n t of

Cm

where

Observe t h a t , i f

f E L,:

203

(5.10).

t h e o l d and new d e f i n i t i o n s of

Tf(x)

d i f f e r only i n a constant: I l y l > , K ( - y ) f ( y ) d y ; thus they agree a s e l e m e n t s of BMO. I n g e n e r a l , i f f E L m , we d e n o t e by Tf t h e c l a s s i n BMO r e p r e s e n t e d by t h e f u n c t i o n d e f i n e d by ( 5 . 1 6 ) . In t h i s way, we e x t e n d t h e o p e r a t o r T : L: T : Lm + BMO w i t h t h e same norm.

.+

t o an o p e r a t o r

BMO

L e t u s d e n o t e by

g . ( x ) t h e j - t h term of t h e J (5.16). Given a compact s e t F , s e q u e n c e i n t h e r i g h t hand s i d e of l e t R b e t h e f i r s t n a t u r a l number s u c h t h a t R 2 2 s u p l x l . Then, XGF f o r a l l x E F and j > II: Proof

of

(5.15):

The i n t e g r a l t o t h e r i g h t i s u n i f o r m l y bounded (when

x E F)

[Im,

by

and c o n v e r g e s when j * m t o t h e same i n t e g r a l e x t e n d e d BKIIf over a l l ( y I > R. Thus, g j ( x ) c o n v e r g e s p o i n t w i s e a n d i n L 1 ( F ) . A s a c o n s e q u e n c e , we h a v e IITflIBMOI l i m s u p j+co

where we have u s e d f e r in a constant.

I

and t h e f a c t t h a t

u

(5.101

gj

and

T(fXB.) 3

dif-

The n e c e s s i t y o f s u b s t r a c t i n g t h e c o n s t a n t t e r m s i n ( 5 . 1 6 ) can be s e e n i n t h e f o l l o w i n g i n s t r u c t i v e example: Compute t h e H i l b e r t transform of have

f(x) = sign(x).

1 g.(x) = 7 {log 1

and l e t t i n g

=

$

j *

m

1-1X -X J

l o g 1x1 ?

Since

-

we f i n d :

1

log

fXB. = 3

rr2

I

+

2

-

Xc-j,ol,

we

=

+ -

X

2 .2 l o g Ix -1

Hf ( x ) =

X[o,ji

log j

l o g 1x1.

Our n e x t o b j e c t i v e i s t o g i v e more p r e c i s e e s t i m a t e s f o r some s i n g u -

11. CALDERON-ZYGMUND THEORY

204

l a r i n t e g r a l o p e r a t o r s f o r w h i c h more r e g u l a r i t y o f t h e k e r n e l i s assumed . DEFINITION

A singular integral operator

5.17.

Tf = K * f

c a l l e d r e g u l a r i f i t s k e r n e l s a t i s f i e s t h e f o l l o w i n g two c o n d i t i o n s :

Observe t h a t

(5.19)

t r i v i a l l y i m p l i e s Hbrmander's c o n d i t i o n

Ix~-~-'

I

S.l(b),

IVK(x) 6 C o n s t . and it i s a u t o m a t i c a l l y v e r i f i e d i f away f r o m t h e o r i g i n . F o r homogeneous o p e r a t o r s a s t h o s e d e s c r i b e d i n Proposition 5.6, a s u f f i c i e n t condition t o be a regular singular integral operator is:

R c C ' (IR" \ { o I ) .

The g r o w t h l i m i t a t i o n

(5.18)

enables us t o define the truncated

operators

d i r e c t l y f o r L p f u n c t i o n s f ( x ) , 1 5 p < m, a n d o n e n a t u r a l l y w i s h e s t o h a v e a more e x p l i c i t d e f i n i t i o n o f Tf a s a l i m i t o f T E f . T h u s , we a r e i n t e r e s t e d i n u n i f o r m e s t i m a t e s f o r ( T E ) E , O , a n d we a r e g o i n g t o s e e t h a t w e c a n a c t u a l l y e s t i m a t e t h e maximal

If

T

THEOREM

5.20.

f c Lp,

1 S p <

(i)

# (Tf) (x) 5 C M f ( x ) 9 9

(ii)

T*f(x) 5 C M f ( x )

m

9

-

i s a regular singular integral operator and

then the following inequalities are v e r i f i e d :

9 4

t

( 1 < 9) CM(Tf)(x)

( 1 < 9)

By t r a n s l a t i o n i n v a r i a n c e , we o n l y n e e d t o p r o v e t h e p o i n t wise e s t i m a t e s ( i ) a n d ( i i ) a t x = 0 . The b a s i c p o i n t i s t h a t Proof:

we c a n u s e Lemma

5.11

( w h i c h h o l d s now f o r e v e r y

f

G

Lp,

since

11.5. SINGULAR INTEGRAL OPERATORS

205

was o n l y imposed t o e n s u r e t h e e x i s t e n c e o f t h e a s s u m p t i o n f 6 L: w i t h a n i m p r o v e d e s t i m a t e f o r t h e l a s t term, n a m e l y IE)

Since

Mf 2 M f 9

which i m p l i e s IE = TEf(0),

f o r every

we o b t a i n

q > 1,

( i ) ( a t x = 0 ) . On t h e o t h e r h a n d , w e o b s e r v e t h a t and it a l s o follows t h a t

a n d t a k i n g t h e supremum o v e r a l l

E

> 0 ,we g e t

(ii).

which i s a consequence of t h e s t r o n g e r L e t u s now p r o v e ( 5 . 2 1 ) , In f a c t , r e g u l a r i t y p r o v i d e d by ( 5 . 1 9 ) . m

r

(

5

5

Finally,

(iii)

is a t r i v i a l consequence of

(ii) and t h e

Lp

i n e q u a l i t i e s f o r t h e o p e r a t o r s M and T , and t h e proof of (iv) i s p o s t p o n e d t o C h a p t e r V , where i t w i l l a p p e a r i n a n a t u r a l context.

0

COROLLARY

(5.4)

,

5.22.

(5.18)

Tf = ( p . v . k ( x ) ) * f 5.5)

satisfies

Let k

L~oc(IRn\.03) be a f u n c t i o n s a t i s f y i n p (5.19). Then, the singular integral operator G

( w h o s e e x i s t e n c e i s g u a r a n t e e d by P r o p o s i t i o n

11. CALDERON-ZYGMUND THEORY

206

F o r a l l f G S(IRn) , we know t h a t l i m T , f ( x ) = T f ( x ) a t e v e r y x G IR". S i n c e S(IRn) i s d e n s e inE+' L P ( R n ) , t h e r e s u l t f o l l o w s f r o m 5 . 2 0 ( i i i ) a n d ( i v ) by a s t a n d a r d t e c h n i q u e a l r e a d y

Proof:

used t o d e r i v e t h e d i f f e r e n t i a t i o n theorem f o r t h e H a r d y - L i t t l e w o o d maximal o p e r a t o r .

1.9

from t h e e s t i m a t e s

To c o n c l u d e t h i s s e c t i o n , we remember t h a t t h e r e s u l t s n e e d e d t o e s t a b l i s h some e x a m p l e s of 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 were s t a t e d w i t h o u t p r o o f s . Here we p r o v i d e t h e p r o o f s :

K = p.v.k(x).

We must o n l y show t h a t 1? G L m ( l R n ) Let u s d e f i n e t h e t r u n c a t e d k e r n e l s

Since

R kE = K

P r o o f of

15.51:

lim

E+O, R+m

,

w he re

( i n t h e s e n s e of t e m p e r e d d i s t r i b u t i o n s ) ,

i t s u f f i c e s t o p r o v e t h a t t h e F o u r i e r t r a n s f o r m s of { k:lo < < a r e u n i f o r m l y bounded. We b e g i n by o b s e r v i n g t h a t H o r m a n d e r ' s condition t r a n s f e r s t o the truncated kernels i n the following sense:

and t h e r e f o r e ,

Now, we f i x

5

G

IRn

a n d decompose

11.5. SINGULAR INTEGRAL OPERATORS

by t h e p r e c e d i n g o b s e r v a t i o n .

On t h e o t h e r h a n d ,

( t h i s t y p e of i n e q u a l i t y i s a c t u a l l y e q u i v a l e n t t o

Proof of show t h a t

(5.6):

(5.3)

IxI-~

implies

(5.3))

The f i r s t a s s e r t i o n w i l l f o l l o w f r o m

k(x) = Q(x)

it does, because

207

and thus

(5.5)

s a t i s f i e s Hormander's c o n d i t i o n .

if we But

11. CALDERON-ZYGMUND THEORY

208

m(6)

F o r t h e s e c o n d a s s e r t i o n , we h a v e t o p r o v e t h a t

ier transform of

K = p.v.(R(x)

[XI-").

Since

k

i s t h e Four-

i s odd,

The i n n e r i n t e g r a l i s a l w a y s b o u n d e d i n a b s o l u t e v a l u e by

i t s l i m i t when E + 0 , desired expression f o r

R

+

m,

is

7I Ts .i g n ( x ' . ~ ) .

n,

and

This gives the

m(S).

MULTIPLIERS

6.

I n t h e p r e c e d i n g s e c t i o n we o b t a i n e d

Lp

i n e q u a l i t i e s f o r convolu-

t i o n o p e r a t o r s whose k e r n e l s a r e n o t i n t e g r a b l e . k

G

L1(IRn)

,

then

Tf = k * f

Of c o u r s e , i f is also a singular integral operator,

since

(Bk

denoting t h e c o n s t a n t i n Hormander's condition

t h i s c a s e , Young's i n e q u a l i t y g i v e s

1 6 p 6

m,

and theorem

Lp

S.l(b)).

In

estimates for a l l

namely

5.7

seems t o be o f no

i n t e r e s t here.

T h e r e i s , how-

e v e r , o n e p o i n t i n t h a t t h e o r e m w h i c h ma ke s i t u s e f u l e v e n f o r t h i s simple s i t u a t i o n , and it i s t h e f a c t t h a t , f o r a f i x e d 1 < p <

11c11,

m,

and

p

with

t h e e s t i m a t e of I l k * f l l p d e p e n d s o n l y on t h e c o n s t a n t s Bk, w h i c h may b e c o n s i d e r a b l y s m a l l e r t h a n IIklIl.

To p u t i t i n a more p r e c i s e f o r m : I f w e a r e g i v e n a f a m i l y o f k N 6 L 1 , a n d we w s h t o o b t a i n u n i f o r m e s t i m a t e s f o r

kernels

Ilp

IlkN*f by means o f Young s i n e q u a l i t y , we a r e f o r c e d t o i m p o s e the strong condition s u p I k N I I l < m . H ow e ve r, t h e o r e m 5 . 7 h a s N

11.6. MULTIPLIERS

209

t h e f o l l o w i n g c o n s e q u e n c e w h i c h w i l l b e u s e d f o r o u r main r e s u l t b e low:

Let

6.1.

LEMMA

(kd

be a sequence o f k e r n e l s i n

suppose t h a t t h e r e e x i s t s

C > 0

lkN(x-y)

b)

-

kN(x) ldx S C

&

L1(IRn), N = 1,2,

such t h a t , f o r a l l

(y

...

IR")

f

.. .

Then, T f = kN*f, N = 1 , 2 , , are uniformly bounded operators from N Lp t o i t s e l f ( 1 < p < a ) , from L 1 t o w e a k - L 1 a n d f r o m Lm to

BMO

.

This suggests a d i f f e r e n t (though equivalent) approach t o 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 d e f i n e d by p r i n c i p a l v a l u e d i s t r i b u t i o n s . Indeed, under t h e hypothesis of Proposition 5.5, the kernels k = N s a t i s f y ( a ) a n d ( b ) of t h e p r e c e d i n g lemma. = k X ~ xI :/ N S ~ S N I Therefore IIkN

with limit

C

*

flip

' CplIfllp

independent of

P

Tf

=

(f

N = 1,2

,..., (in

l i m kN * f

6

LP; 1 < p <

and f o r a l l

f

G

w)

Lp,

the

LP-norm)

N+W

exists (because it obviously e x i s t s f o r Schwartz f u n c t i o n s ) d e f i n i n g 1 < p < w, and which c e r an o p e r a t o r T which i s bounded i n L p , t a i n l y a g r e e s w i t h t h e u n i q u e bounded e x t e n s i o n t o Lp of t h e ft+ (P.v. k(x))*f, i n i t i a l l y d e f i n e d i n convolution operator: S(IR").

T h i s i s n o t , h o w e v e r , o u r r e a s o n f o r s t a t i n g lemma 6 . 1 . h e r e . We aim a t a d i f f e r e n t a p p l i c a t i o n o f t h e Calder6n-Zygmund method w h i c h gives sufficient conditions f o r Fourier multipliers in DEFINITION

6.2.

Let

1 5 p <

a.

Given

i s a ( F o u r i e r ) m u l t i p l i e r f o r Lp by t h e r e l a t i o n d e f i n e d i n L2(IRn)

m

m

f

Lm(IRn)

if t h e o p e r a t o r

Lp.

, we s a y t h a t Tm,

initiaZZy

11. CALDERON-ZYGMUND THEORY

210

satisfies the inequality

(f E L2

llTmflIp 5 C l l f l l P Tm

In t h i s ease,

n

LP)

Lp

h a s a u n i q u e bounded e x t e n s i o n t o

w h i c h we

Tm.

s t i l l denote by

Some g e n e r a l f a c t s a b o u t m u l t i p l i e r s a r e c o n t a i n e d i n 7 . 1 5 a n d a t t h e

m

By P l a n c h e r e l ' s t h e o r e m , e v e r y

end of t h i s s e c t i o n .

6

Lm

is a

L 2 , a n d t h e n o r m o f T~ a s a n o p e r a t o r i n L' is The f a c t t h a t t h e R i e s z t r a n s f o r m s a r e b o u n d e d Lp, c a n b e r e s t a t e d by s a y i n g t h a t m j ( t ) = [ . / I F 1

multiplier for e q u a l t o Ilrnll,. operators in (1 = 1 , 2

,...,n )

are multipliers for

Lp

if

1 < p <

-.

1

The Horman

d e r - M i h l i n m u l t i p l i e r t h e o r e m p r o v i d e s a g e n e r a l i z a t i o n of t h i s :

If

Let

6.3.

THEOREM

n 7 .

m

G

{R-"

(6.4)

+ 1

be t h e f i r s t i n t e g e r g r e a t e r than

ca ~outside the

o r i g i n and s a t i s f i e s

Ill< I x I < 2 R

for every multi-index

LP, 1

p l i e r for

[y]

a =

L - ( R ~ ) is of c l a s s

< p

c1

-.

such t h a t

10.1

5 a,

then

m

is _a m ti- ~ _u l _

F o r many a p p l i c a t i o n s , o n e m e r e l y v e r i f i e s t h e c o n d i t i o n

which i s s t r o n g e r t h a n

m([) (with

When

of c l a s s

Ca

b E IR)

b = 0

,

(6.4).

6 - n f ($1

.

ib

i.e.

(which i s t h e most i n t e r e s t i n g c a s e ) ,

is dilation invariant, i.e., =

T h i s i s s a t i s f i e d by e v e r y f u n c t i o n

o u t s i d e t h e o r i g i n a n d hom oge ne ous o f d e g r e e

'Tm(fg) = ( T m f ) 6 ,

the operator

where

f6(x)

=

We w i s h t o p r o v e t h e t h e o r e m by r e d u c i n g i t t o a n a p p l i c a t i o n O K

Lemma

6.1.

Two s t a n d a r d t e c h n i q u e s a r e n e e d e d f o r t h i s r e d u c t i o n .

m

11.6. MULTIPLIERS

211

The f i r s t o n e i s t h e smooth c u t t i n g o f t h e m u l t i p l i e r i n t o d y a d i c p i e c e s , which i s b a s e d on t h e f o l l o w i n g LEMMA

T h e r e i s a non n e g a t i v e f u n c t i o n

6.5.

15

i n the spherical shell

Take

Proof:

@ 6

@(El

such t h a t

Cm

> 0

:

4 < 151 < 2 1

supported i n when

1 6

J2

$

supported

such t h a t

4 < 151 < 2 ,

151 S J Z .

c"(Rn)

6

non n e g a t i v e a n d

Then,

i s a Cm f u n c t i o n s t r i c t l y p o s i t i v e a t e v e r y 5 f 0 a n d s u c h t h a t 00(25) = O o ( < ) . Thus, $ ( 5 ) = @ ( 5 ) / @ o ( 5 ) s a t i s f i e s t h e r e q u i r e d

0

conditions.

The s e c o n d p r e v i o u s r e s u l t i s a n e x p l i c i t f o r m u l a t i o n o f t h e w e l l known f a c t t h a t t h e r e g u l a r i t y o f t h e m u l t i p l i e r i s t r a n s l a t e d i n t o c o n t r o l of t h e s i z e of t h e k e r n e l : LEMMA

Let

6.6.

(so that

s

a = 21. I-f

6

n + 1 + 1 , and l e t s b e s u c h t h a t a = s 2 k 6 L2(Rn) i s such t h a t i s of c l a s s C a ,

];[

then, -

Proof:

Since

Ixla 6 C(lxl

la+ ...

+Ixnla),

i s (-2niy a Da^ .k(t) 1 1 Hausdorff-Young's i n e q u a l i t y t o g e t t r a n s f o r m of

xak(x)

(with

and s i n c e t h e F o u r i e r

D . = 9/85.), 1 1

we u s e

We c a n now p r o c e e d t o p r o v e t h e m u l t i p l i e r t h e o r e m .

Proof of function

6.3: $

of

W e u s e t h e p a r t i t i o n o f t h e u n i t y d e f i n e d by t h e 6.5

i n o r d e r t o decompose t h e m u l t i p l i e r a s

11. CALDERON-ZYGMUND THEORY

212

s u p p ( m j ) c 1 5 : Z j - ’ < l c l < Z j + l } , and t h a t we c a n c o n t r o l t h e s i z e o f t h e d e r i v a t v e s o f m. a s f o l l o w s : J

Observe t h a t

llDa m j / I s

(6.7)

6 C Zj(-I’

I n d e e d , by t h e L e i b n i t z r u l e o f d i f f e r e n t i a t i o n , i f

and i t i s now t h a t t h e h y p o t h e s i s

(6.4)

la1 5 a

is used, together with

Holder’s inequality, t o obtain

from which

(6.7)

follows.

Now

o u r p l a n i s t o a p p l y lemma = k . ( x ) , with -N 3

$

k.(x) = i . ( - x ) = 3 J Since

Llc.(S)

J

J

=

Im(S)

I

t o the kernels

6.1

kN ( x ) =

m j (SIdS B

Lm(Rn)

,

we o n l y have t o p r o v e

I f we a r e a b l e t o d o s o , t h e o p e r a t o r s

TNf ( x ) w i l l be u n i f o r m l y bounded i n

( N = 1, 2,

Lp,

1 < p <

m,

and s i n c e

1;. J

=

...) m.

1’

11.6. MULTIPLIERS

TN

i s t h e o p e r a t o r d e f i n e d by t h e m u l t i p l i e r N m (5) =

N

1

j=-N

which c o n v e r g e s t o TNf

213

Tmf

+

(in

l i m inf N - t m

L2)

N

I

m. ( 5 ) = m ( 6 )

j=-N

J

m(5)

when

N

f o r every

IT N f ( x )

-

+

w.

f

f

L

I

=

0

Tmf(x)

2

$(2-j~)

,

By P l a n c h e r e l ' s t h e o r e m , and i n p a r t i c u l a r a.e.

Then, F a t o u ' s lemma g i v e s , f o r e v e r y

f

Thus, m a t t e r s a r e reduced t o p r o v i n g

(6.8).

G

L 2 fl L p

For a f i x e d

y c IR",

we s h a l l s e p a r a t e t h e sum i n t o two p a r t s : W

Ik.(x-y) I

J=-

-

k.(x)\dx= 1

1

jGI

+

1

jcII

I = { j : Z1lyl 2 1 ) and I 1 = { j : 2 J 1 y l < 1 ) . For t h e t e r m s i n t h e f i r s t sum, w e u s e t h e t r i v i a l m a j o r i z a t i o n

where

(x-y)

which i s b a s e d on

-

k . ( x ) Idx 5 3

and

(6.6)

(6.7)

(with

Therefore

For t h e terms corresponding t o k . (x J

-

y) 2.

-

k . (x) 3

j

G

must p l a y a r o l e .

k.(x) = k.(x-y) I 3

-

11,

s

such t h a t

a-fls

1 2

= -).

t h e c a n c e l l a t i o n of

Define

kj(x)

Our t a s k c o n s i s t s now i n o b t a i n i n g a n e s t i m a t e f o r t h e d e r i v a t i v e s 2. of m . which i s good e n o u g h , namely 1

11. CALDERON-ZYGMUND THEORY

214

j ( 1 - a +n/s) sup \\Da%.\\ 5 C ) y \ 2 la =a I S

(6.9)

(l5s52; 2’1y)
I

5

To p r o v e t h i s , we f i r s t o b s e r v e t h a t , f o r a l l

I Dl ( e - 2 n i y .

5 -1)

( t h i s i s o b v i o u s when

cly12J(1-l~l)

y = 0

because

IDy(e-2niY‘S 1

we h a v e

IyI > 0

I

I

=

‘L

G

supp(m.)

1

(j

Iy.[I

5 2j”

f

Iy

I ( 2 . r r ~ ) ~2 I Clyl’).

11)

I,

a n d when By t h e L e i b -

nitz rule sup

lal=a

ID

c42’

m.(c) 1

1

5

c IY

and i n s e r t i n g t h e e s t i m a es b t a i n e d i n ( 6 . 7 ) f o r D’m. we h a v e 3’ proved ( 6 . 9 ) . Now, we a r e i n p o s i t i o n t o i n v o k e a g a i n ( 6 . 6 ) , which g i v e s

and t h e r e f o r e ,

Under t h e h y p o t h e s i s o f type

(l,l),

6.3,

the operator

Tm

i s a l s o o f weak

and s a t i s f i e s

I t i s n a t u r a l t o a s k whether t h e l a s t i n e q u a l i t y c a n be improved t o o b t a i n t h e same k i n d o f e s t i m a t e t h a t r e g u l a r 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 s a t i s f y (see

5.20(i)).

We s h a l l now p r o v e , w i t h a l m o s t

n o a d d i t i o n a l e f f o r t , a more p r e c i s e f o r m o f h i g h e r smoothness of t h e m u l t i p l i e r of t h e o p e r a t o r

Tm.

6.3

w h i c h shows how

implies higher regularity

III

In p a r t i c u l a r , the operators described in the

l a s t theorem always s a t i s f y : ( T m f ) # 5 C M Z f , and t h e y behave e x a c t l y a s r e g u l a r 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 i f we a s s u m e t h e d e c a y ( 6 . 4 ) f o r a l l t h e d e r i v a t i v e s of o r d e r 5 n

THEOREM

6.10.

__ Let

a

&an

of

m(t).

i n t e g e r such t h a t

n < a 5 n,

arid

__

11.6. MULTIPLIERS

m(S)

suppose that the multiplier

la1 2 a .

Then, for e v e r y

q >

n

--,

215

satisfies the operator

(6.4)

Tm

f o r all satisfies

We u s e t h e same n o t a t i o n a s i n t h e p r o o f o f 6 . 3 . f i c e s t o o b t a i n t h e p o i n t w i s e estimate f o r e v e r y o p e r a t o r

Proof:

N T f ( x ) = kN * f ( x ) =

N

1

j=-N

I t suf-

k. * f(x) J

(with C independent of N) and t h i s w i l l b e a n e a s y consequence 9 of t h e c r u c i a l i n e q u a l i t y f o r t h e k e r n e l s kN:

where denote (6.11)

E >

0

w i l l b e f i x e d below.

t . = 2 J Iy

1 that

1,

In fact, given

and f o r an a r b i t r a r y

f,

y E IR",

we

it follows from

$ 7

j=1

5

cly

m

1 t ;-E - n / q t n 'q3M

j=1

9

f ( 0 ) 5 C'Mqf(0)

This i s a s u b s t i t u t e , f o r our p r e s e n t problem of t h e i n e q u a l i t y (5.21),

a n d t h e r e s t of t h e p r o o f i s a s i n t h e o r e m

5.2O(i).

Now, i n o r d e r t o e s t a b l i s h ( 6 . 1 1 1 , g i v e n q , we t a k e s 5 2 s u c h n s c q . We c a n a l s o a s s u m e t h a t E = a - < 1 , and . t h i s that S i s o u r c h o i c e of E . The v a r i a n t o f lemma 6 . 6 w h i c h w e n e e d h e r e is as follows: I f k G L2 ( I R n ) and i(5) i s of c l a s s C a , then

11. CALDERON-ZYGMUND THEORY

216

The p r o o f i s e n t i r e l y s i m i l a r , b u t H o l d e r ' s i n e q u a l i t y i s a p p l i e d i n

t o order

\Ir

ll-l\ql

5 11. 1 1 - ( I s l , w i t h -r = -s - -q ' remain t o h o l d f o r t h e d e r i v a t i v e s of m

t h e form: and (6.9)

a,

if

t 2

IyI

, we

j

have

T h e r e f o r e , t h e l e f t hand s i d e of

Since (6.7) 'L and m . up 1

(6.11)

i s m a j o r i z e d by

ii

and t h e proof is ended.

To c o n c l u d e t h i s s e c t i o n ,we l i s t some e l e m e n t a r y p r o p e r t i e s o f multipliers:

1 < p <

is a muZtipZier f o r

Lp

if and

only if it is a multiplier f o r L p ' , and t h e norms of o p e r a t o x L p and on L p ' are identical.

Tm

as an

Let

(6.22).

m

m.

T h i s f o l l o w s from t h e i d e n t i t y jTmf(xl (f

6

L2

n

Lp,

n i t i o n of

Tm

g(x) dx

1

-_.

f ( x ) T m g ( x ) dx

w h i c h i s o b t a i n e d by u s i n g t h e d e f i g f L2 n L p ' ) , and P l a n c h e r e l ' s theorem.

m

(6.13).

If

then

is a multiplier=

m

=

The c o n d i t i o n on

is a muZtiplier for

q

Lp,

and if

I-1

~ 9 .

means t h a t , e i t h e r

p 6 q 5 p'

9

-

or

- 11

6

2

I-1

P

- - 11

2

p' 5 q S p.

The r e s u l t i s a c o n s e q u e n c e o f t h e p r e c e d i n g s t a t e m e n t a n d

11.6. MULTIPLIERS

217

Marcinkiewicz' i n t e r p o l a t i o n theorem,

2 , l l . . I f one u s e s i n s t e a d t h e (see 7 . 5 b e l o w ) , i t f o l l o w s t h a t i s n o t g r e a t e r t h a n t h e norm i n L p . In

Riesz-Thorin i n t e r p o l a t i o n theorem t h e norm o f

Tm

in

particular, taking

Lq

q = 2,

we a l w a y s h a v e

The m u l t i p l i e r s f o r

(6.14).

Lp

Ilmll,.

IITmllp,p 2

Lm(Rn)

f o r m a s u b a l g e b r a of

w h i c h i s i n v a r i a n t u n d e r t r a n s l a t i o n s , r o t a t i o n s and d i l a t i o n s .

f = T

( T f ) . The ml m2 i n v a r i a n c e p r o p e r t i e s f o l l o w from t h e b e h a v i o u r of t h e F o u r i e r t r a n s form w i t h r e s p e c t t o a f f i n e t r a n s f o r m a t i o n s : A(x) = h + L ( x )

The f i r s t a s s e r t i o n i s o b v i o u s , b e c a u s e

(L l i n e a r ) ,

(with

"v

L = (L*)-').

(6.15).

all 1 -

which i s very simple

c

The f o l l o w i n g f u n c t i o n s a r e m u l t i p l i e r s f o r p c m:

m.(S) = s i g n ( S . 1 3 J

where

P

,

j = l,Z,

i s a (bounded o r unbounded) polyhedron i n

LP(IRn)

for

...,n IRn

, h., F

i s t h e i n t e r s e c t i o n of a f i n i t e number of h a l f - s p a c e s

In f a c t , the multiplier

m, c o r r e s p o n d s t o t h e H i l b e r t t r a n s f o r m a c t i n g on t h e f i r s t v a r i a b l e : Tm f ( x 1 , x 2 , 1

... '

xn ) = i H ( f ( . , x 2 ,

which i s bounded i n L p ( I R n ) , 1 < p and F u b i n i ' s t h e o r e m . S i m i l a r l y f o r

m,

m. 1'

..., x n ) ) ( x l ) by t h e b o u n d e d n e s s of 2 5 j 5 n.

H

C o n c e r n i n g XF, i t s u f f i c e s t o p r o v e t h e r e s u l t f o r t h e c h a r a c t e r i s t i c f u n c i t o n of a h a l f - s p a c e (because t h e product of m u l t i p l i e r s is a g a i n a m u l t i p l i e r ) , a n d by t r a n s l a t i o n and d i l a t i o n i n v a r i a n c e , we

11. CALDERON-ZYGMUND THEORY

218

o n l y have t o prove it f o r

s = s eo

1

But

7.

7.1

Xs

1

= z(l

= { c ~ I R ": t , = ~ * e ~ s 0 1

+ ml)

.

NOTES AND FURTHER RESULTS

.-

The d e f i n i t i o n o f t h e maximal f u n c t i o n i n

IR"

i s i n Wie-

n e r [l]. I t i s t h e n - d i m e n s i o n a l a n a l o g u e of t h e maximal f u n c t i o n d e f i n e d by Hardy a n d L i t t l e w o o d [l] f o r f u n c t i o n s i n t h e t o r u s T . The c l a s s L l o g L was i n t r o d u c e d by Zygmund t o g i v e a s u f f i c i e n t c o n d i t i o n f o r t h e l o c a l i n t e g r a b i l i t y of i s a c t u a l l y necessary (theorem 2.7) 7.2.-

Mf.

That t h i s condition

was o b s e r v e d by S t e i n [4].

T h e r e a r e many v a r i a n t s o f t h e maximal f u n c t i o n i n

IR".

It

i s e q u i v a l e n t , f o r i n s t a n c e , t o consider b a l l s i n s t e a d of cubes. More g e n e r a l l y , g i v e n a f a m i l y o f s e t s F w h i c h c o v e r s IRn i n t h e

belongs t o s e t s " n a r r o w s e n s e " ( i . e . , e v e r y x c IRn a r b i t r a r i l y small d i a m e t e r ) , we can d e f i n e MFf(x) =

F c P

of

SUP

I f every set

F E F c a n b e p a c k e d w i t h i n two c u b e s : Q1 c F c Q 2 C = fixed constant, i n s u c h a way t h a t d i a m (Q,) S C d i a m ( Q 2 ) , t h e n MF a n d M a r e e q u i v a l e n t , i n t h e s e n s e t h a t C-"Mf(x)

5 M,f(x)

5 CnMf(x)

This i s not the case f o r the family

i?

o f a l l bounded n - d i m e n s i o n a l

MRf i s c a l l e d t h e i n t e r v a l s : R = [ a l , b l ] x [aZ,b2] x . . . x [an,bI,]. s t r o n g m a x i m a l function a n d c o n t r o l s t h e d i f f e r e n t i a t i o n w i t h r e s p e c t t o a r b i t r a r y i n t e r v a l s i n IRn , a s w e l l a s t h e u n r e s t r i c t e 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 ( s e e Zygmund [I], Chap. X V I I ) . I t was f i r s t s t u d i e d by J e s s e n , M a r c i n k i e w i c z a n d Zygmund [l], who proved t h a t M i s a bounded o p e r a t o r irom Lp(IRn) t o i t s e l f , K 1 < p < -, f r o m L ( 1 o g L ) " t o L' a n d f r o m L ( l o g L ) " - ' t o weak L 1 . A s a c o r o l l a r y of t h e l a s t r e s u l t :

11.7.

1i m XER diam (R)+O f o r every

IRI I R

NOTES AND FURTHER RESULTS

f(y)dy = f(x)

219

a.e.

w h i c h i s l o c a l l y i n L(1og L ) ~ - ' ( R ~ )The . estimates f o r

f

M R a r e b a s e d o n i t s m a j o r i z a t i o n by t h e n - f o l d a p p l i c a t i o n o f o n e d i m e n s i o n a l m a xi mal f u n c t i o n s ( s e e s e c t i o n I V . 6 ) . The b o u n d s o b t a i n e d i n t h i s c h a p t e r f o r t h e m a xim a l f u n c t i o n

7.3.-

in

Lp(IRn)

large

n

, for a fixed [with

a > 1).

operator over centered b a l l s i n M(")f(x)

B

(with

=

=

sup

O
are of t h e o r d e r of an f o r M[n) d e n o t e s t h e m a xim a l

p > 1,

Ho wev er , i f

IBlrn

rB

,

IRn

namely

If ( x + y ) Idy

I x E IRn : 1x1 < 1 1 1 ,

then, the following inequalities

hold

with Stein

independent of

C

$1, [a].

n.

T h i s s t r i k i n g r e s u l t is due t o

A s a n a p p l i c a t i o n , o n e c a n o b t a i n p o i n t w i s e summa-

b i l i t y o f F o u r i e r s e r i e s o f i n f i n i t e l y many v a r i a b l e s . We m e n t i o n two o p e n q u e s t i o n s i n t h i s d i r e c t i o n (Ql)

I s t h e r e a weak t y p e a n a l o g u e o f

(Q2)

I s ( * ) t r u e i f we r e p l a c e t r i c , o p e n s e t i n IR"?

Concerning

(Ql),

B

for

(*)

p = l?

by a c o n v e x , c e n t r a l l y symme-

a weak t y p e e s t i m a t e w i t h a f a c t o r

o b t a i n e d by S t e i n a n d S t r o m b e r g [l].

As for

(Q2),

n

h a s been

B o u r g a i n [3]

h a s r e c e n t l y g i v e n a n a f f i r m a t i v e a n s w e r f o r p 2 2 when n B = { x E IR : IIxII 5 1 1 , b e i n g a n u n c o n d i t i o n a l norm : T h i s i n c l u d e s t h e maxi= ( x , , x2,. ,xn) ( 2x, , 2 x 2 , . ,+ x n )

11

..

11

1 1 . 11 11

..

11.

mal f u n c t i o n o v e r c e n t e r e d c u b e s .

7.4.-

C a l d e r h a n d Zygmund [l].

i n t r o d u c e d t h e d e c o m p o s i t i o n which

b e a r s t h e i r name i n o r d e r t o s t u d y t h e e x i s t e n c e a n d b o u n d e d n e s s o f singular integrals.

The a p p r o a c h f o l l o w e d h e r e t o d e r i v e t h e e s t i -

mates f o r t h e max i mal f u n c t i o n f r o m s u c h a d e c o m p o s i t i o n i s a l s o

11. CALDERON-ZYGMUND THEORY

220

s u g g e s t e d i n t h e same p a p e r . The more u s u a l a p p r o a c h t o s u c h e s t i mates i s v i a c o v e r i n g lemmas, a p a r t i c u l a r example of which i s B e s i c o v i t c h C o v e r i n g Lemma:

each

*en

A c IR”,

a set

suppose t h a t f o r

Q(x) c e n t e r e d & x . Then, f r o m t h e , f a m i Z y { Q ( x )l x G A i t i s p o s s i b l e t o s e l e c t a s u b f a m i l y of c u b e s { Q j } which s t i l l c o v e r A and such t h a t i x Q j ( x ) 5 4 n ( i . e . , no more t h a n 4” c u b e s Q j c a n o v e r l a p ) . x

G

A

we h a v e a c u b e

For t h i s , s i m i l a r r e s u l t s and t h e i r a p p l i c a t i o n t o e s t i m a t e s of maximal f u n c t i o n s , we r e f e r t o t h e monographs b y d e G u z m h [l], The B e s i c o v i t c h lemma c a n b e u s e d S t e i n [lJ, S t e i n a n d Weiss [2]. t o g i v e an improved v e r s i o n of theorems

2.4

and

2.6:

“Let p a r e g u l a r p o s i t i v e B o r e 1 m e a s u r e i n EXn, and l e t b e t h e m a x i m a l o p e r a t o r d e f i n e s in s e c t i o n I , b u t w i t h c e n t e r e d

cubes

only.

Then

G,,

bounded i n

Lp(p)

a n d of weak t y p e

(l,l)I!

Observe t h a t no d o u b l i n g c o n d i t i o n i s imposed. For t h e (non-cent e r e d ) o p e r a t o r M p , t h e r e s u l t i s s t i l l t r u e i n R1 , b u t f a l s e in

IR”

for

n > 1

( S j o g r e n [I]).

C o v e r i n g lemmas f o r f a m i l i e s F s u c h a s t h o s e c o n s i d e r e d i n 7 . 2 a r e s o m e t i m e s much h a r d e r t o o b t a i n . One s u c h lemma f o r t h e f a m i l y R h a s b e e n o b t a i n e d by A . CBrdoba a n d R . F e f f e r m a n yielding a d i f f e r e n t ( g e o m e t r i c ) p r o o f o f t h e Jessen-Marcinkiewicz-Zygmund t h e @ rem.

131,

7.5.-

The M a r c i n k i e w i c z i n t e r p o l a t i o n t h e o r e m

(2.11)

can be

s t a t e d more g e n e r a l l y f o r o p e r a t o r s o f weak t y p e ( p o , q o ) and ( p l , q l ) w i t h po 5 q o , p 1 2 4,. A companion o f t h i s i n t e r p o l a t i o n t h e o r e m whose p r o o f ( b a s e d on t h e maximum m o d u l u s p r i n c i p l e f o r a n a l y t i c functions) is completely d i f f e r e n t i n s p i r i t , is: Riesz-Thorin -(p,,qo) and

tMPe

Theorem:

(Po,q0)9

a n d we h a v e

(pl,ql),

wherv

If T -

with

i s a l i n e a r o p e r a t o r of s t r o n g t y p e s

1 ,< p i , q i

5

a,

then

T

i s of s t r o n g

NOTES AND FURTHER RESULTS

11.7.

221

The M a r c i n k i e w i c z a n d R i e s z - T h o r i n t h e o r e m s a r e t h e p r o t o t y p e s o f t h e r e a l and complex methods of i n t e r p o l a t i o n , r e s p e c t i v e l y .

These

a r e d e s c r i b e d i n B e r g h a n d L o f s t r o m [l]. Thorin theorem due t o S t e i n with t h e exponents

If TZ a r e l i n e a r o p e r a t o r s depending 0 5 Re(z) 5 1 , the hypothesis

p,q:

z,

a n a l y t i c a l l y on

An e x t e n s i o n o f t h e R i e s z allows t h e o p e r a t o r T t o change

[S]

provided t h a t

7:;

a m o d e r a t e g r o a t h when

-t

and

Ml(t)

have

An a p p l i c a t i o n o f t h i s t h e o r e m w i l l

m.

be i n d i c a t e d i n C h a p t e r V , 5.19

Mo(t)

and

7.7.

Further generalizations

h a v e b e e n o b t a i n e d by C o i f m a n , C w i c k e l , R o c h b e r g , S a g h e r a n d Weiss

C1I -

The i n e q u a l i t y

7.6.-

i s d u e t o C . F e f f e r m a n a n d S t e i n El].

(2.13)

The i d e a o f r e l a t i n g i n e q u a l i t i e s o f t h i s t y p e w i t h C a r l e s o n measu r e s i s t a k e n f r o m t h e same p a p e r . The s p a c e

BMO

was i n t r o d u c e d i n J o h n a n d N i r e m b e r g [l],

p r o v e d t h e t h e o r e m b e a r i n g t h e i r name, S t e i n [Z] equivalence BMO

(theorems

7.7.-

lip

3.6

The c o n d i t i o n

Zygmund

C. Fefferman and

(3.8).

d e f i n e d t h e s h a r p maximal f u n c t i o n

IIMf

c11,

%

llf#llp and

where t h e y

f#

and obtained t h e

Lp

and t h e i n t e r p o l a t i o n between

and

3.7).

S.l(b)

was a l r e a d y i m p l i c i t i n C a l d e r B n a n d

b u t i t was Hormander

[l]

who f i r s t f o r m u l a t e d i t e x p l i -

c i t l y i n t h i s form and p o i n t e d o u t t h a t o t h e r o p e r a t o r s a p a r t from t h e homogeneous s i n g u l a r i n t e g r a l s c o u l d b e d e a l t w i t h b y t h e same method. 6.3,

He u s e d i t , i n p a r t i c u l a r , t o p r o v e t h e m u l t i p l i e r t h e o r e m

.

i m p r o v i n g a p r e v i o u s r e s u l t o f M i h l i n [l]

One o f t h e e a r l y

r e a s o n s t o d e v e l o p t h e t h e o r y o f s i n g u l a r i n t e g r a l s was i t s a p p l i c a b i l i t y t o problems i n l i n e a r p a r t i a l d i f f e r e n t i a l equations.

Just

t o show t h i s c o n n e c t i o n , we w r i t e a s i m p l e consequence o f theorem 6 . 3 :

Let

P(D)

of degree

b e a n e l l i p t i c c o n s t a n t c o e f f i c i e n t o p e r a t o r homogeneous k.

IIDaflIp<

Then, t h e a p r i o r i i n e q u a l i t i e s

CP 11 P ( D ) f l / p

(la1

=

k, 1 < p <

m)

11. CALDERON-ZYGMUND THEORY

222

hold f o r every =

T (P(D)f) m

f

Ck

E

w i t h compact s u p p o r t ( o b s e r v e t h a t

with the multiplier

m(E,) = E,'/P(t)

DQf =

satisfying

(6.4l)).

The t h e o r y o f 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 h a s b e e n g e n e r a l i z e d a l o n g many d i f f e r e n t l i n e s . five notes.

o r i n p r o d u c t domains) 7.8.-

by

J u s t a few of them a r e d e s c r i b e d i n t h e n e x t

Others (singular i n t e g r a l s f o r vector valued functions w i l l be considered i n Chapters

One c a n c o n s i d e r " n o n - i s o t r o p i c " A x H 6(x) = 6 x

,

dilations in

V

and

,

IRn

IV.

defined

6 > 0

i s a p o s i t i v e - d e f i n i t e n x n m a t r i x (When A = i d e n t i t y , we g e t t h e u s u a l " i s o t r o p i c " dilations). If d = t r a c e (A) is where

A

t h e homogeneous d i m e n s i o n o f

IRn with respect t o these d i l a t i o n s , then t h e analogue of Proposition 5.6 is true f o r kernels k(x) S e e d e Guzman w i t h t h e r i g h t homogeneity: k(6(x)) = 6-dk(x).

p],

Chapter 1 1 , and t h e r e f e r e n c e s c i t e d t h e r e . T h i s n o t i o n o f d i l a t i o n c a n b e c a r r i e d o v e r t o t h e more g e n e r a l s e t t i n g o f l o c a l l y c o m p a c t g r o u p s ( R i v i e r e Cl]).

These i d e a s have been

s p e c i a l l y f r u i t f u l when a p p l i e d t o some n i l p o t e n t Lie g r o u p s ( l i k e t h e Heisenberg group) i n c o n n e c t i o n w i t h s o l v a b i l i t y and r e g u l a r i t y p r o p e r t i e s of n o n - e l l i p t i c p a r t i a l d i f f e r e n t i a l o p e r a t o r s ( s e e F o l l a n d a n d S t e i n [l]). The H i l b e r t t r a n s f o r m a l o n g a f i x e d d i r e c t i o n 7.9.IuI = 1 , i s d e f i n e d a s f o l l o w s f

f

u

G

IR" ,

.

L2(lRn)

T h i s i s t h e s i m p l e s t e x a m p l e o f a s i n g u l a r i n t e g r a l whose k e r n e l i s supported in a submanifold of

the line

IRn :

Itu : t

G

R1.

In

t h i s c a s e , t h e r o t a t i o n i n v a r i a n c e and F u b i n i ' s theorem immediately

HU i s b o u n d ed i n L p ( I R n ) a n d o f weak t y p e ( 1 , l ) . A e x t e n s i o n c o n s i s t s i n r e p l a c i n g t h e l i n e by a c u r v e l i k e

prove t h a t non-trivial y(t)

=

$1,

ta2

,...)t a n )

The H i l b e r t t r a n s f o r m a l o n g

y

, is

tsIR

1 1 . 7 . NOTES AND FURTHER RESULTS

I t i s known t h a t i n e q u a l i t y for

1 < p < m, IIHyfllp 5 CplIflIp, p = 1 i s s t i l l an open q u e s t i o n .

a n d W a i n g e r Cl]

b u t t h e weak t y p e We r e f e r t o S t e i n

f o r t h i s and r e l a t e d r e s u l t s .

The f a c t t h a t

7.10.-

223

IIHUflIp 6 C p I l f l l p

s u g g e s t s t h a t t h e same

i n e q u a l i t y w i l l b e t r u e f o r any "convex combination'! o f d i r e c t i o n a l Hilbert transforms:

HUf ( x ) R ( u ) d o ( u )

Tnf(x) = lul=l bith

R

L

G

1

When

i s odd, i n t e g r a t i o n i n p o l a r coordi-

R

n a t e s shows t h a t

which a r e t h e o p e r a t o r s c o n s i d e r e d i n P r o p o s i t i o n l a r i t y b e i n g now a s s u m e d o n

R.

Thus, f o r

R

5.6,

no regu-

odd

C = Norm i n L p ( R ) of t h e H i l b e r t transform. T h i s simple P a n d e l e g a n t m e t h o d , c a l l e d t h e m e t h o d of r o t a t i o n s , i s a l s o d u e t o

with

[z],

C a l d e r 6 n a n d Zygmund

r e s u l t is obtained f o r

i t s m a j o r d r a w b a c k b e i n g t h a t n o weak t y p e

.

p = 1

w o r k s p r o v i d e d t h a t we a s s u m e I f we a p p l y

IIRjlll

independent of P following:

C

for a l l

n

G

N

When R i s n o t o d d , t h e m e t h o d s t i l l G L log L(Cn-l).

t o the Riesz transforms

(*)

turns out t h a t

R

and

n.

=

1,

and t h e r e f o r e ,

R . j = 1 , 2 ,...,n , it 7' I I R j f l / P 5 Cpllfllp with

T h i s h a s been improved by S t e i n

1 < p <

m.

[8]

t o the

F o r a p r o o f o f t h i s r e s u l t b a s e d on

t h e method o f r o t a t i o n s , see Duoandikoetxea and Rubio d e F r a n c i a 7.11.-

[l].

The C a l der 6 n - Z y g mu n d m e t h o d c a n a l s o b e a p p l i e d t o o p e r a t o r s

which a r e n o t t r a n s l a t i o n i n v a r i a n t . C o if m a n a n d Weiss [l]

A g e n e r a l s e t t i n g p r o v i d e d by

is the following:

11. CALDERON-ZYGMUND THEORY

224

We c o n s i d e r

a s p a c e of h o m o g e n e o u s t y p e X , w h i c h means t h a t X i s endowed w i t h a p s e u d o - m e t r i c p and a "doubling" measure dm(x), and we t r y t o s t u d y o p e r a t o r s o f t h e form

lx

Tf(x) =

K(x,y)f(y)dm(y)

Now, H o r m a n d e r ' s c o n d i t i o n s p l i t s i n t o two p a r t s :

I

(HI

P(x,Yo)>kP(Y ,Yo)

f o r some f i x e d

k, C > 0

is the condition i)

LP,

IK(x,y)

-

and f o r a l l

y,yo

for the kernel:

(H)

K(x,yJIdm(x) G

X,

11

i i ) IITf

)I2 2

2 < p <

Cllf m,

)I2

plus and from

( H I )

Lm

to

and

imply t h a t BMO.

which

(HI)

K'(x,y) = K(y,x).

llTf ( 1 6 C I I f p l u s (H) imply t h a t 1 < p < 2 , a n d o f weak t y p e ( 1 , l )

Lp,

5 C

Then

i s bounded i n

T T

i s bounded i n

Examples o f o p e r a t o r s i n c l u d e d i n t h i s g e n e r a l framework a r e t h o s e considered i n

7.8

a n d s i n g u l a r i n t e g r a l s i n some c o m p a c t g r o u p s o r

i n homogeneous s p a c e s u n d e r t h e a c t i o n o f s u c h g r o u p s , l i k e t h e u n i t sphere

of

IR",

t h e u n i t b a l l Bn

of

En,

etc.

When X = IRn , t h e c o n d i t i o n s a n a l o g o u s t o t h o s e o f r e g u l a r 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 (see 5.17) are

(it i s obvious t h a t

i m p l i e s (H) a n d ( H I ) ) . Operators bounded i n L 2 ( I R n ) a n d d e f i n e d b y a k e r n e l s a t i s f y i n g (CZ.l) and ( C Z . 2 ) are sometimes c a l l e d Calderdn-Zygmund operators. See C o i f man a n d Meyer

(CZ.2)

111, J o u r n 6

[l].

is not a convolution operator, the Fourier transf o r m i s no l o n g e r a n e f f e c t i v e t o o l f o r p r o v i n g L 2 - b o u n d e d n e s s , a n d t h i s i s u s u a l l y t h e d i f f i c u l t p a r t when a n a l y s i n g w h e t h e r T i s a Calder6n-Zygmund o p e r a t o r . A n i c e necessary and s u f f i c i e n t condit i o n h a s b e e n f o u n d by D a v i d a n d Journt? El]. For o p e r a t o r s w i t h 7.12.-

When

T

225

1 1 . 7 . NOTES AND FURTHER RESULTS

antisymmetric kernels, t h i s c r i t e r i o n takes t h e following s p e c i a l l y s i m p l e form:

-

If

Theorem":

T

i s g i v e n by a kerneZ s a t i s f y i n g

K(x,y) = - K ( y , x ) , T ( 1 ) E BMO.

(CZ.2) and onZy i f

then

T

is b o u n d e d i n

A s a n a p p l i c a t i o n , we c o n s i d e r t h e commutator of o r d e r

(CZ.l),

if

L2(rd)

k:

where A i s a L i p s c h i t z f u n c t i o n , i . e . A ' G Lm ( s u c h c o m m u t a t o r s a p p e a r n a t u r a l l y when t r y i n g t o c o n s t r u c t a n a l g e b r a o f p s e u d o Now, s i n c e To i s d i f f e r e n t i a l o p e r a t o r s , see A.P. C a l d e r 6 n [4]). t h e 'IT1 t h e o r e m " p l u s t h e H i l b e r t transform and Tk(l) = T k - l ( A ' ) , induction gives

This i s t h e theorem of

A.P. C a l d e r B n [3],

which c a n a l s o b e formu-

l a t e d i n terms o f a n o t h e r i m p o r t a n t o p e r a t o r : along a Lipschitz curve

where

11,

z(t)

=

< 1,

r

The Cauchy i n t e g r a l

i n t h e complex p l a n e , d e f i n e d by

t + iA(t) i s t h e p a r a m e t r i z a t i o n o f r . Now, i f we c a n e x p a n d t h e i n t e g r a n d i n t o a g e o m e t r i c s e r i e s

and, w r i t i n g

The e s t i m a t e s

E

= 1/C,

(*)

(*)

is equivalent t o

have been improved by Coifman, McIntosh a n d

a l l o w i n g t o remove t h e r e s t r i c t i o n llA'\I, < E i n ( * * ) . Meyer [l], We r e f e r t o Y.hleyer T h i s h a s a l s o b e e n o b t a i n e d by D a v i d [l]. [Z. f o r a n e x c e l l e n t e x p o s i t i o n o f t h e s e a n d r e l a t e d r e s u l t s .

D],

7.13.-

Lemma

4.6

i s d u e t o H a r d y a n d L i t t l e w o o d , b u t we h a v e p r e -

11. CALDERON-ZYGMUND THEORY

226

The a r g u m e n t u s e d s e n t e d h e r e t h e p r o o f o f F e f f e r m a n a n d S t e i n [2]. t o p r o v e 4 . 1 4 comes f r o m C a l d e r d n a n d Zygmund [l]. By u s i n g S t e i n ' s s p h e r i c a l m e a n s , which w e s h a l l d i s c u s s i n C h a p t e r V , 7 . 7 , t h i s r e s u l t c a n b e p a r t i a l l y i m p r o v e d : F o r J, r a d i a l a n d i n t e g r a b l e i n Rn , t h e o p e r a t o r f ( x ) t - + ; y g l $ t * f ( x ) i s b o u n d e d i n Lp(IRn) , p > L A d i f f e r e n t r e s u l t f o r approximations of t h e i d e n t i t y i s n-1' Zo's theorem (see s e c t i o n V . 4 ) .

1

i s d u e t o A.P. C a l d e r d n [l] who s u c c e e d e d i n 7 . 1 4 . - Theorem 4 . 2 1 e x t e n d i n g t o h a r m o n i c f u n c t i o n s i n IRn++' a p r e v i o u s theorem of P r i v a l o v [I] v a l i d f o r h a r m o n i c f u n c t i o n s i n t h e u n i t d i s k D . Pri-

v a l o v ' s t h e o r e m was p r o v e d o r i g i n a l l y v i a c o n f o r m a l m a p p i n g . We s h a l l g i v e a n o u t l i n e o f t h i s p r o o f b e l o w . The method i s n o l o n g e r a v a i l a b l e i n h i g h e r d i m e n s i o n s . T h i s i s w h a t makes C a l d e r 6 n ' s c o n t r i b u t i o n a b a s i c one. In o r d e r t o given a n account of P r i v a l o v ' s p r o o f , l e t as assume, w i t h o u t l o s s of g e n e r a l i t y , t h a t u i s a harmonic f u n c t i o n i n D such t h a t f o r e v e r y t f o r which ei t 6 E ,

a s u b s e t of T = boundary of D w i t h I E l > 0 , (u(z)I is bounded by a f i x e d number n on t h e S t o l z domain r a ( t ) , where c1 is also fixed. (The S t o l z d o m a i n s a r e d e f i n e d i n p a g e 1 0 9 ) . We may a l s o a s s u m e t h a t E i s c o m p a c t . We c o n s i d e r t h e s e t U = U T a ( t ) where e i t r a n g e s o v e r E . Then U i s a n open s e t bounded by a r e c t i f i a b l e Jordan curve.

We know t h a t

I

lu(z)

2 n

on

U.

Now t h e

U and look a t t h e idea is t o use a conformal equivalence $ : D f u n c t i o n u ( $ ( z ) ) w h i c h i s h o l o m o r p h i c a n d bounded i n D . We c a n -+

a p p l y F a t o u ' s theorem t o c o n c l u d e t h a t u ( $ ( z ) ) c o n v e r g e s nont a n g e n t i a l l y a t a l m o s t e v e r y p o i n t i n t h e b o u n d a r y o f D . We know (corollary 4.7 i n c h a p t e r I ) t h a t , e x c e p t f o r t h e p o i n t s where t h e boundary of

U

h a s n o t a n g e n t , which f o r m a s e t o f a r c l e n g t h

t h e c o n f o r m a l mapping r e m a i n s c o n f o r m a l a t t h e b o u n d a r y p o i n t s . We a l s o know ( s e e a f t e r c o r o l l a r y 3 . 1 3 i n c h a p t e r I ) t h a t t h e 0,

homeomorphism i n d u c e d b e t w e e n t h e b o u n d a r i e s o f

sets of length

0

t o s e t s of Lebesgue measure

U 0.

and

D

carries

T h u s , we h a v e

t h a t f o r almost every t and f o r every 6 , $ ( r g ( t ) i s a r e g i o n non-tangential a t $(eit) w i t h r e s p e c t t o U, a n d we f i n a l l y g e t t h e e x i s t e n c e of n o n - t a n g e n t i a l limits f o r u a t a most e v e r y p o i n t o f E , w h i c h i s p a r t o f t h e b o u n d a r y of U. Note t h a t i n E a r c l e n g t h c o i n c i d e s w i t h Lebesgue measure. The method works e v e n f o r a f u n c t i o n meromorphic i n D . a f t e r u n i f o r m i z i n g t h e s i t u a t i o n , so t h a t w e c a n assume

Indeed, c1 and n

1 1 . 7 . NOTES AND FURTHER RESULTS

227

I

f i x e d , we r e a l i z e t h a t t h e f a c t t h a t lu(z) I n f o r each z s ra(t) it with e G E, i m p l i e s t h a t t h e s i n g u l a r i t i e s of u i n U a r e a l l i n s i d e a d i s k D(o,r) with r < 1 and, consequently, t h e r e is o n l y a f i n i t e number of them. M u l t i p l y i n g by a n a p p r o p r i a t e p o l y n o m i a l , we g e t a h o l o m o r p h i c f u n c t i o n on U, t i a l b o u n d a r y v a l u e s e x i s t a t a . e . p o i n t of

f o r which n o n - t a n g e n E. Obviously, t h e

same t h i n g h a p p e n s f o r t h e o r i g i n a l f u n c t i o n

u.

Note t h a t t h e same t e c h n i q u e o f c h a n g i n g t h e domain c o n f o r m a l l y , c a n b e u s e d t o p r o v e t h e f o l l o w i n g t h e o r e m , a l s o due t o P r i v a l o v [l]: u(z)

If a f u n c t i o n

meromorphic i n

T

a t e a c h p o i n t of a s u b s e t of identically

D

has non-tangential

limit

of p o s i t i v e m e a s u r e , t h e n

0

u

D.

0

By u s i n g t h e two m e n t i o n e d t h e o r e m s , o n e c a n e a s i l y o b t a i n t h e f o l lowing result, due t o Plessmer

Let u(z)

111 ( s e e

b e an a n a l y t i c f u n c t i o n i n

p o i n t s of L e b e s g u e m e a s u r e

0,

a l s o Zygmund D.

to

eit

o r e l s e , t h e image

u(r,(t))

XIV).

Then, e x c e p t f o r a s e t of

a t every point

e i t h e r the function has a f i n i t e l i m i t a s

c ] , Chap.

z

e it

of t h e b o u n d a r y ,

tends non-tangentially

i s dense i n t h e plane f o r

a.

every

A c t u a l l y A . P . CalderBn [l] e x t e n d s P r i v a l o v ' s theorem t o p l u r i h a r monic f u n c t i o n s i n a C a r t e s i a n p r o d u c t o f e u c l i d e a n h a l f - s p a c e s and o b t a i n s non-tangential convergence a t a . e . p o i n t of t h e d i s t i n guished boundary.

By u s i n g t h i s r e s u l t , h e i s a b l e t o e x t e n d

P l e s s n e r ' s t h e o r e m t o a n a l y t i c f u n c t i o n s o f s e v e r a l complex v a r i a b l e s . T h e r e i s a c o n v e n i e n t way t o r e f l e c t a n a l y t i c a l l y t h e f a c t t h a t a harmonic f u n c t i o n u h a s n o n - t a n g e n t i a l l i m i t s a t a . e . p o i n t of a s u b s e t E of t h e boundary w i t h IEl > 0. This i s achieved through the area f u n c t i o n

For

u

harmonic i n

I f we j u s t f i x

A(u) D,

i n t r o d u c e d by

Lusin.

we d e f i n e

a, we c a n s i m p l y w r i t e

r(t)

and

A(u)(t).

11. CALDERON-ZYGMUND THEORY

228

F = u + iv

Observe t h a t i f

is analytic i n

D,

then

is the t h e J a c o b i a n o f t h e mapping F . T h u s , we s e e t h a t A , ( u ) ( t ) a r e a ( p o i n t s c o u n t e d a c c o r d i n g t o t h e i r m u l t i p l i c i t y ) of F ( T , ( t ) ) . T h i s e x p l a i n s t h e name " a r e a f u n c t i o n " . The f o l l o w i n g r e s u l t h o l d s :

Let u

D with a.e. x

f o r a.e.

D and l e t E b e a s u b s e t o f t h e boundary o f [ E l > 0. Then, f o r u t o have a non-tangential l i m i t a t E, i t i s n e c e s s a r y and s u f f i c i e n t t h a t A(u)(x) i s f i n i t e

b e harmonic i n 6

x

6

E.

The n e c e s s i t y was p r o v e d by M a r c i n k i e w i c z a n d Zygmund [2] s u f f i c i e n c y by S p e n c e r [IJ

.

Note t h a t i f

is analytic in

F = u + iV

by t h e Cauchy-Riemann e q u a t i o n s . u

has a non-tangential

v

same i s t r u e f o r

For a f u n c t i o n

A(u)

l i m i t f o r a.e.

then:

=

A(v)

point of

a n d we g e t : E,

then the

and c o n v e r s e l y .

u(x,t)

harmonic i n t h e upper h a l f - s p a c e

(generalized) area function

A.P. C a l d e r B n [7]

Thus

D

and t h e

A(u)

Rn+'

,

the

is defined a s follows:

e x t e n d e d t h e t h e o r e m o f M a r c i n k i e w i c a a n d Zygmund

t o t h i s higher dimensional s e t t i n g and t h e extension of Spencer's The g e n e r a l i z e d a r e a f u n c r e s u l t was c a r r i e d o u t by E.M. S t e i n t i o n was p r e v i o u s l y c o n s i d e r e d i n c o n n e c t i o n w i t h o t h e r r e l a t e d

@I.

11.7.

o p e r a t o r s i n E.M. S t e i n In s e c t i o n

N O T E S AND FURTHER R E S U L T S

GO].

of c h a p t e r

8

229

I11

we s h a l l i n d i c a t e some r e s u l t s on t h e

a r e a f u n c t i o n which a r e c o n n e c t e d w i t h t h e t h e o r y of Hardy s p a c e s . L 2 , the Fourier multipliers f o r Lp a r e c o m p l e t e l y c h a r a c t e r i z e d o n l y when p = 1 : t h e c l a s s of L’ m u l t i p l i e r s c o i n c i d e s w i t h t h e F o u r i e r - S t i e l t j e s t r a n s f o r m s of f i n i t e Bore1 measures i n lRn. Sufficient conditions f o r m(S) to be an Lp m u l t i p l i e r f o r a l l 1 < p < = a r e g i v e n by t h e HBrmander7.15.-

A p a r t from t h e t r i v i a l c a s e of

M i h l i n theorem, 6 . 3 , a n d by a n o t h e r c l a s s i c a l r e s u l t : t h e Marcink i e w i c z m u l t i p l i e r theorem, whose 1 - d i m e n s i o n a l v e r s i o n w i l l be There i s an e x t e n s i v e l i t e r a t u r e c o n proved i n c h a p t e r V , 5 . 1 3 . c e r n i n g more s p e c i f i c m u l t i p l i e r s f o r d i f f e r e n t r a n g e s of p ’ s . We s h a l l b r i e f l y d i s c u s s h e r e t h o s e o n e s r e l a t e d t o s u m m a b i l i t y of Fourier series o r i n t e g r a l s . Given

a 20,

f o r every f t h e r a n g e of

i s it t r u e t h a t

G

Lp(Rn) ? The q u e s t i o n i s e q u i v a l e n t t o t h e s t u d y of p ’ s corresponding t o t h e Bochner-Riesz multipliers:

Our p r e s e n t knowledge a b o u t t h i s c h a l l e n g i n g problem i s t h e f o l l o w i n g :

i) kernel

( c r i t i c a l index).

Ka

Then

ma(S)

= i?,(C)

which c a n b e e x p l i c i t l y computed and b e l o n g s t o

Thus, m_U Weiss [ZJ.

i s an

Lp

multiplier for a l l

1 5 p < =.

with a L1(Rn)

.

See S t e i n and

ii) a = 0 . For n = 1 ’ mo = X[-1,1] i s an LP m u l t i p l i e r 1 < p < m ( t h i s i s e q u i v a l e n t t o M. R i e s z t h e o r e m ) , w h i l e f o r i s n o t an LP m u l t i p l i e r f o r any p 2 n 2 2 , m 0 - X[]l’l51’ ( C Fef f erman n- 1 iii) 0 < a S The c o n j e c t u r e i s t h a t ma 1 s an Lp multi2 p l i e r i f and o n l y i f

for

+

.

-.

Pa

=

2n n + I + 2a < P < n

-

2n 1

-

2a = P A

11. CALDERON-ZYGMUND THEORY

230

The n e c e s s i t y o f t h i s c o n d i t i o n i s e a s i l y s e e n , b e c a u s e Ka $ L p a For n = 2 , t h e c o n j e c t u r e i s t r u e , and i t was f i r s t p r o v e d by

n];

.

d i f f e r e n t p r o o f s h a v e b e e n g i v e n by C . F e f For n > 2 , t h e c o n j e c t u r e h a s been ferman [4] and C6rdoba [lJ. n- 1 p r o v e d when a T h i s f o l l o w s f r o m r e s u l t s o f S t e i n a n d Tomas on r e s t r i c t i o n o f F o u r i e r t r a n s f o r m s ( s e e P. Tomas [l]); it can a l s o C a r l e s o n and S j a l i n

=.

be p r o v e d by u s i n g t h e a p p r o a c h i n C6rdoba [l].