Chapter V Vector Valued Inequalities

CHAPTER V VECTOR VALUED INEQUALITIES

The estimates for the operators that we have considered so far can be formulated in the context of Banach space valued functions. The first half of this chapter is devoted to describe the general methods which allow to obtain such estimates. There are at least two reasons which motivate this type of extension lapart from the search of greater generality forits own sake): a) The non-linear operators which appear in Fourier Analysis can be viewed, in almost all cases, as linear operators whose range constists of vector valued functions. This will be the case for the maximal operators and square functions which are studied in sections 4 and 5 respectively.

b) There is an intimate connection between vector valued inequalities and weighted norm inequalities. We begin to explore this connection in the last section, but further insight will be gained with the theory to be developed in Chapter VI.

1 . OPERATORS ACTING ON VECTOR VALUED FUNCTIONS: SOME BASIC FACTS

Let (X,m) be a a-finite measure space. If B is a Banach space, a function F : X B is said to be (strongly) measurable if the following two conditions hold +

i) There is a separab e subspace F ( x ) 6 Bo

for

a.e.

ii) F o r each b ' E B" = dual of R , xt+ is measurable

Bo

of

B

such that

x E X the mapping:

A very casy consequence of this definition is that the positive f u n & is measurable. As in the s c a l n r l y v a l u e d tion IF[B : x+ [F(x)llg

case, we shall identify B-valued functions which coincide m-a.e. For 0 < p < m, we define the Bochner-Lebesgue space L E ( m ) con475

476

1:. VECTOR VALUED INEQUALITIES

F

s i s t i n g o f a l l measurable B-valued f u n c t i o n s

f o r which

Then Li(m) i s a Banach ( w i t h t h e u s u a l m o d i f i c a t i o n when p = m ) . ( p - B a n a c h i f 0 < p < 1 ) s p a c e . I n t h e same way, w e d e f i n e t h e s p a c e L!B(m) = weak - LE(m) which i s “normed“ by

The f o l l o w i n g f a c t s a r e v e r y s i m p l e t o v e r i f y : (1.1)

f 6 Lp

LE,

If.bl

=

lfl,

b E B, IblB.

then Lf

(f.b)(x) = f(x)b 0 < p <

LPB f o r m e d b y a l l f i n i t e l i n e a r c o m b i n a t i o n s of f u n c t i o n s

in

belongs t o

LpPB

t h e subspace

m,

L~B.

f.b

i s dense

F = I: f . . b . € L 1 P B , we c a n d e f i n e i t s i n t e g r a l (which w i l l 3 1 j be a n e l e m e n t o f B ) i n t h e o b v i o u s way: Given

I F ( x ) dm(x) = C ( j ( 1 . 2 ) For e v e r y

Fo 6 L 1 P B ,

I

i

f j ( x ) dm(x))b

I I P o dmlB

5 IFoI

j

1.

T h u s we c a n

LBn t i n u i t y . If e x t e n d t h e m a p p i x FW F dm t o a l l by c o 1 then I F dm i s t h e o n l y e i e m e n t of B w h i c h s a t i s f i e s F E LB, <]F If

dm, b ’ > =

L B1

I


b’)

( b ’ E B*)

F E L; with 1 2 p 5 m, and G 6 L,: t h e n (x) = G(x)> i s t r i v a l l y a n i n t e g r a b l e f u n c t i o n , and we have

T h i s means t h a t

LEi

i s ( w i t h t h e n a t u r a l i d e n t i f i c a t i o n ) isome-

t r i c a l l y c o n t a i n e d i n t o t h e d u a l of

When

dm(x)

B

&

LE,

i s r e f l e x i v e one a c t u a l l y h a s

b u t we s h a l l have no n e e d o f t h i s f a c t .

L:?’

=

L E l C (L!)’.

(LEI*,

1 2 p <

m,

V. 1 . VECTOR VALUED I:U?.CTIONS

477

In this chapter we shall deal with operators on spaces LE. The following is a method of producing such operators which occurs naturally in many instances and which follows the same pattern used in ( 1 . 2 ) to define the integral: Suppose that T is a linear operator which maps L p into L q ; then T can be extended to an o p e rator TB = T @ IdB on Lp @ B by

TB (C f..b.)(x)

(1.4)

If

TB

j

J

C

=

J

j

Tf.(x)b. J J

happens to be bounded, i.e. (fj E Lp,

J 6 B) b. -

then it can be uniquely extended to a bounded operator from LE B into L z which we still denote by T . In this case, we say that T has a B-valued extension, and the following identity holds: (1.5)

< TBF(x),

b'>

=

T(
.)

,b'>) (x)

(this is actually a characterization of TB). with an operator T mapping L p into Lj.

(FELg, b'€B*)

The same can be done

Examples 1.6.(a). Let B = Lr, 1 < r < m. Then a measurable B - v a lued function is just a sequence F(x) = (fj(x))jEa where each f. J is a (real or complex valued) measurable function, and c lfj(x)lr < m a.e. The space Lg LP(L~) consists of all se-

j

quences

(f.) J

such that r l/r

I,

<

Given a linear operator T of strong type (p,q), TB is defined on functions F = (f.). with a finite number of nonvanishing c o p J JEN ponents by

Thus, 1' has a B-valued extension if and only if the following ine qua 1 it y h o 1d s (1.8)

I (? 3

r l/r ITfjI 1

I,

< -

c no ifj( j

r l/r

1

I,

for a l l countable sequences of functions fj 6 Lp. If this is the case, TB is defined by ( 1 .7) over the whole space LP(Lr).

V. VECTOR VALUED INEqUALITIES

178

where (b) As a more general example, take B = Lr(Q,u), is another o-finite measure space. Then L E ( m ) can be idec tified with the mixed norm space LrYP(~xX, ulm) which is formed by all p@m-measurable functions f(w,x) such that (Q,p)

The details of this identification are left to the reader, who is also referred to Benedek and Panzone [ l ] . If an operator T has a B-valued extension TB, this is defined by the analogue of ( 1 . 7 ) , namely (1.9)

TBf(w,x)

=

T(f(w,.))(x)

(c) There are also interesting operators on vector valued functions which do not arise a s extensions of operators in L p . If A and B are Banach spaces, we denote by L ( A , B ) the space of all bounded linear operators from A into B. Given a measurable L(A,B)-valued kernel K(x,y) defined in XxX such that K(x,.) 6 Lp’ f o r a.e. x , then the operator L (A,Bl (1 .lo)

TF(~) = j~(x,y).F(y)dm(y)

is well defined on Li(m), and one may try to investigate its continuity properties. If A = B = Lr, and if K(x,y) is, for each (x,y), a diagonal operator with entries (Kj(x,y)) j 6 N , then T is given by

where

T. is the linear operator defined on

Kj (X,Y)

J

Lp

by the kernel

We shall now prove two elementary results. First of all, we present the simplest case in which the R-valued extension of an operator can be assured to exist THEOREM 1 . 1 2 . L e t 1’ b e a l i n e a r o p e r a t o r iuhicii ?’s p o s i t i v r i i . ~ . g(x) 1. 0 i m p Z i e s Tg(x) 2 0 ) a n d b o u n d e d , with n o r m ITI, as i i r i o p e r u t o r f r o m LP(m) to L q ( m ) jzpsp. ~:(m)). hen, f o r a n y B a n a c h s p a c e B, T h a s a B - v a l u e d e x t e n s i o n T B w h i c h maps ~,:(rn) in~o l,i(mj iresp. L:B(m)) w i t h t h e samc n o r m : ITB I = I l l ,

4 79

V.l. VECTOR VALUED FGNCTIONS

Proof: I t is enough to show that,

for every

n

1 f.(x)b. and let (bi)ksg J' j=1 J of the unit sphere in B* such that lslB Let

F(x)

every

=

s

ITBF(x)

I

=

sup I
= SUP

k

be a countable subset =

{bl,b2,. . . ,bn].

in the span of

IT(
,

F 6 Lp f3 B

sup I
b;(>l,

for

Then, we have

b;(>( =

bi>) (x) I 5 T(IFIB)

(XI

where we have used the identity ( 1 . 5 ) and, f o r the last inequality, the positivity of T. 0 In particular, for a positive linear operator Lp, we have the inequalities

T

which is bounded

in

(when

r

=

this must be understood in the usual way).

m

The second result deals with operators of the type described i n the example 1.6.(c), and it generalizes the fact that convolution with 1 1 n a kernel in L (R") produces a bounded operator in L ( R ) . THEOREM 1 . 1 4 . Let K(x) b e a m e a s u r a b l e Rn and s u c h t h a t

L(A,B)-valued k e r n e l de-

fined i n

(1.15) Then, f o r every

( 1 .16) exists a.e.,

dx 5

IK(x).alB 1 n ), F E LA(R

IalA

the integral I

TF(x)

JK(x-y) .F(y)dy

=

,

5 C lFILl A Fubini's theorem and ( 1 . 1 5 ) give

and t h e o p e r a t o r

T

verifies:

lTF1

LB

Proof:

Given

F E LA(Rn),

and both assertions of the theorem follow immediately.

0

480

\I,

VECTOR VALUED INEQUALITIES

Simple as i t is, this result will nevertheless be useful in proving some Littlewood-Paley inequalities in section 2. Remarks 1.17.

If (1.15)

is replaced by the stronger condition

p W ( A , B )

dx <

m

Li

then the operator T defined by (1 . 1 6 ) is bounded from Lf3 to for all 1 5 p 5 m . More generally, Young's inequalities for convolution hold in this context if K € L:(A,B). However, one cannot obtain from (1 . 1 5 ) Lp boundedness of T if p > 1 , as the following example shows: Given functions k. 6 L 1 (Rn) (j=1,2,. . . ) with lkjil = 1, define A with values in the kernel KA(x) =' (kj(x))jsN L(L1,a) = L", where kx. = k.X The corresponding operator J {x : Ikj(x)l 5 A}' J Tx : L'(L1) L 1 is given by TA((f.))(x) = C k?*f.(x), and i t is I j J J clear that (1.15) holds with C = 1 . Suppose f o r a moment that T A : Lp(L') +. Lp is bounded for some p > 1 uniformly f o r all A > 0. By duality (see (1.3)) and letting h m, this would imply -+

-+

lsyp I k j

* fHpI 5

c

PIpl

which is known to be false if we take for instance where {R.}. 1 ICY tangles in R

(f E: LP'(R")) k. = (R.1' 3 I

1

xR,,

is a "dense" sequence in the family of all reccontaining the origin (see de Guzmgn [2]).

J

As i t is well known, interpolation is a basic tool to deal with operators in Lp spaces, and this turns out to be also the case f o r operators acting on B-valued functions. Here we shall simply state the Marcinkiewicz and Riesz-Thorin theorems in the form that we shall need to use them. No proof will be given since both theorems are proved exactly as in the scalar case. THEOREM 1.18. (Narcinkiewicz). L e t A , B b e Banach s p a c e s , a n d 2 T be a n o p e r a t o r d e f i n e d on L F + Lpl ( 0 < po < p 1 5 m) whose A v a l u e s a r e B-valued f u n c t i o n s . I f T i s sublinear (with the naturaZ d e f i n i t i o n of t h i s ) and of weak t y p e ( p o , p o ) and (pl,pl), i.e. -

V.l.

t h e n , for a22

where

Cp

p

VECTOR VALUED FUNCTIONS

with

po < p < p,,

Co,

depe nds o n l y on

C,

481

we h a v e

&

p.

(Riesz-Thorin). Let T b e a l i n e a r o p e r a t o r w h i c h i s b o u n d e d i n Lp0(Lro) and in Lpl (Lrl), d.

THEOREM 1 . 1 9 .

(i=O,1; ci IF1 Pi ri L i ('I-& e and w i t h 1 5 po,pl,ro,rl 5 m . Lf - = - + P Po P1 -

I TFI LPi(Lri)

with

0 < 8 < 1,

then

<

T

i s boundeu i n

Lp(Lr),

Pi

F E: L

1 = 1-8 r

r.

(1 '1) +

r0

and more

6

'1'

precisely

(F E LP(Lr)) Further generalization o f these results can be found in BerghLofstrom [ l ] . In particular, the Riesz-Thorin theorem remains true r. if T' is bounded from Lpi(Lii) into Lpi (eB1), i=O,l, where A, B are fixed Banach spaces, and

This observation is sometimes useful for the study of certain opera tors which we now describe:

T

I_ zs c a l l e d l i n e a r i z a b l e if t h e r e e x i s t s a l i n e a r o p e r a t o r U d e f i n e d i n LP(m) whose v a l u e s a r e B-va2ued f u n c t i o n s ( f o r some Banach s p a c e BI a n d DEFINITION 1.20. A n o p e r a t o r

defined i n

LP(m)

such t h a t

ITf(x)

I

=

IUf(x)AB

( 5 E. ~"(m))

I t is clear that linear operators are linearizable, but there are some other interesting examples like (1

(1 where

(Tn n6B

is a sequence of linear operators

(we take

B

=

Lm

482

\I.

VECTOR VALUED INEQII.I?I,ITIES

for the operator M and B = t 2 for G ) . Continuous analogues of the operators defined by ( 1 . 2 1 ) and ( 1 . 2 2 ) are also linearizable, and these include all kinds of maximal operators and g-functions of wide use in Fourier Analysis. COROLLARY 1 . 2 3 . Let T b e a Z i n e a r i z a b l e o p e r a t o r w h i c h i s b o u n d e d i n Lp(m) f o r some 1 p < m. Lf T is p o s i t i v e ( i n t h e s e n s e that: If(x)l 5 g(x) a . e . i m p l i e s ITf(x)i 5 Tg(x) a . e . 1 t h e n , the following i n e q u a l i t i e s hold:

Proof: Observe that ( 1 . 2 4 )

is trivially true if

r

=

p,

and since

it is also true for r = m . Now ( 1 . 2 4 ) would follow for all p 5 r 5 m if we were allowed to use interpolation. But we are, because ITf(x)I = IUf(x)lB for some linear operator U , and thus, ( 1 . 2 4 ) is equivalent to the boundedness of the linear operator:

u

defined by

: LP(P)

c((fj) j 6 B ) (x)

=

-+

LP(Q

(Ufj (x)) j6N. 0

It may be interesting to remark that what we have just proved is a generalization of ( 1 . 1 3 ) (only for some values of r ) to a wider

class o f operators. I t is natural to ask whether ( 1 . 2 4 ) would be true for 1 < r < a. We shall present below some examples in which the answer is affirmative, but, for the general class of operators considered in the Corollary, ( 1 . 2 4 ) is best possible.

2. A THEOREM OF MARCINKIEWICZ AND ZYGMUND

The basic result to be proved in this section it that 12 -valued extensions of linear operators in LP spaces always exist. TO begin with, we need to describe some elementary facts from Probabi lity. DEFINITION 2 . 1 . By a G a u s s i a n s e q u e n c e we s h a l Z mean a s e q u e n c e

(

z

~

Q random v a r i a b l e s

)

~

~

~

( i . e . r e a l valued measurable f u n c t i o n s )

V . 2 . THEORE?? OF MARCI!JKIEWICC.-2YYGMUND

( R , P ) w h i c h a r e i n d e p e n d e n t and i d e n t i c a l l y

i n a p r o b a b i l i t y space

h ( x ) = e-TIx2,

distributed with density function

P ( z j E A)

(2.2)

483

=

f o r every Borel subset

P({w € R : z j ( w ) E A}) = A

0 f

& - TIX 2 e dx

I

(j€:eJ)

A

R.

5

What ( 2 . 2 ) means i s t h a t , f o r e v e r y j N, u n d e r t h e m a p p i n g z . : R + R i s e-TIx d x . J is

t h e image m e a s u r e o f P An e q u i v a l e n t c o n d i t i o n

f o r e v e r y B o r e l f u n c t i o n $ ( x ) on R f o r which t h e i n t e g r a l t o t h e r i g h t e x i s t s . I n p a r t i c u l a r , we s e e t h a t z . € I,'(,) for all J r < m, s i n c e

What makes G a u s s i a n s e q u e n c e s u s e f u l f o r u s i s t h e f o l l o w i n g p r o p e r t y : LEMMA 2 . 4 .

(Xj)jsN r < m,

Let

(zj)jEN

(fi,P),

be a Gaussian sequence i n

6 12. T h e n , t h e sereies X.z converges i n J j and t h e f o l l o w i n g i d e n t i t i h o l d s

( 0 < r < -)

(2.5)

where

br

i s d e f i n e d by ( 2 . 3 1

( t h u s , i t d e p e n d s o n l y on r l

Proof: I t s u f f i c e s t o prove (Z.5) such t h a t

N

1

\hjI2 = 1.

Let

.. z

f o r a f i n i t e sequence =

c 1.z.

J J' o r t h o g o n a l t r a n s f o r m a t i o n i n RN sudh t h a t : = (1,0,. ..,0). Then, f o r e v e r y Borel s u b s e t i=l

. I

P ( z € A) =

I{x€RN: C X J- X J.€A}

j

I

N ( x ~ ) ~ = ,

a n d d e n o t e by

h(xl)h(x2)

=

p(h,

A

an

, x z , . . . , l N )=

of

...

p

R: h(xN)dx =

e ~X€RN:x,€A~

--

dXl

= \ A

Therefore,

and l e t

L r (0) f o r a l l

z

i s a l s o d s t r i b u t e d w t h t h e same d e n s t y f u n c t on :

V. VECTOR VALUEP INEQUALITIES

484

h ( x ) = e-"

2

,

and

1121,

=

br.

0

Remark 2 . 6 . G a u s s i a n s e q u e n c e s d o e x i s t . An e x p l i c i t c o n s t r u c t i o n may b e t h e f o l l o w i n g : T a k e R = 'R = RxR x . . . x R x . . . , a n d d e f i n e P a s t h e p r o d u c t o f c o u n t a b l y many c o p i e s o f t h e m e a s u r e d v ( x ) = =

h(x)dx;

t h e n , a Gaussian sequence

by z.(w) = J

( ' j ) jEN

( 0 = (wl , w 2 , . .

0 .

J

in

(Q,P)

. ' W n , . . .)

Now we a r e a b l e t o p r o v e o u r m a i n r e s u l t . A s a l w a y s ,

THEOREM 2 . 7 .

( M a r c i n k i e w i c z a n d Zygmund)

,

T : Lp

b o unded l i n e a r o p e r a t o r , 0 < p , q m, w i t h norm 2 h a s a n L - v a l u e d e x t e n s i o n , and more p r e c i s e l y :

with

P 5 9 .

C

P19

depending o n l y on

Proof: Consider f i r s t t h e case sequence

(zj)jEN

in

(fi,P),

and t h e t h e o r e m i s p r o v e d w i t h C = 1. P9P For every satisfies

E n)

(X,m)

will

u - f i n i t e m e a s u r e s p a c e , a n d we s h a l l w r i t e s i m p l y LP(X,m).

be a f i x e d instead of

-

i s defined

p

and

q.

IT1

Moreover,

+

.

C

Lq

Lp

be a T

Then,

P 34

= 1

9

q 5 p . I f we t a k e a G a u s s i a n Lemma 2 . 4 i m p l i e s

= b /b In particular P99 P 9' Now we c o n s i d e r t h e c a s e p < q , a n d d e n o t e s = q / p . u(x) 0 w i t h I u ~ , , 5 1 , t h e o p e r a t o r Tuf = u 1 I P . T f

C

485

V . 2 . THEOREM OF I,!.~RCINKIEWICZ-ZYCMUND

and by the case already proved, we have

4 J

Next, we shall establish the analogue of the theorem for weak type operators. The dea is of a function as a supremum of L r -norms, by Kolmogorov condition, which can be stated as LEMMA 2.8. Let

0 < r < q <

m,

n

2 1/2 lfjl 1 1,.

Marcinkiewicz-Zygmund to express the L2-norm means o f the so called follows: f(x)

and for e a c h f u n c t i o n

0,

define

Iflq*

t m(Ix

= SUD

t>o

N

q9r

(f)

=

sup

E

If

: If(x)l

> t}) 119

'Elr m

(-1 =

s

w h e r e t h e r r ~ u p ' ir s t a k e n f o r a l l m e a s u r a b l e s e t s

0 < m(E)

<

m.

E

-1 r

- -) 1

q

with

T h e n we h a v e t h e i n e q u a l i t i e s

Observe that I .Iq* is the usual weak-Lq norm. On the other hand, if instead of XE we allow in the definition of N (f) arbitrary q,r functions @ E Ls, then we get exactly If1 (by the converse of 9 Halder's inequality). Proof: Given t > 0, for an arbitrary set of finite measure, we have

E C {x : If(x)l

> t}

and the first inequality follows. To prove the second inequality, we can assume Iflq* = 1, so that the distribution function of f satisfies

Then, for every set E of finite measure, it is clear that X (t) 5 inf(m(E),t-q), and we get fXE

V . VECTOR VALUED INEQIJALITIES

486

< {r

1,”

m(E)

= ( h r m(E) +

T a k i n g h = m(E) - l l q desired. THEOREM 2 . 9 .

Let

tr-q-l dt}’lr =

rhr-q l/r

1 q-r

(p,q),

1/r

l f X E Q r 5 (&)

we o b t a i n

0 < p,q <

r a t o r of weak t y p e

,,”

tr-’dt + r

and a s s u m e t h a t

a,

m(E)”’

T

as

i s a l i n e a r ope-

i.e.

m({x : I T f ( x ) I > t l ) l / q 5 $l l f l p

Then

T

L 2 - v a l u e d e x t e n s i o n o f weak t y p e

has an

precisely,

there exists

C’

that

P,9

d e p e n d i n g o n l y 03

m({x : (C l T f j ( x ) \ 2 ) 1 ’ 2 > t } )

-

j

P99 t

(p,q). p

and

More q

such

,

1,

P r o o f : W r i t e H = L 2 and F = ( f . ) 6 LE, so that 1 j€N T H F(x) = ( T f . ( x ) ) j c N . Take r q , and u s e t h e p r e v i o u s lemma t o 1 obtain

l T E f l r 5 (&)’Ir

(2.10) where

E

IXEls

lflp

i s an a r b i t r a r y s e t o f f i n i t e measure,

and T E f = (Tf)XE. t o (2.10) gives

( f € LP) l/s = l/r - l/q,

Now, t h e Marcinkiewicz-Zygmund t h e o r e m a p p l i e d

and t h e p r o o f i s ended by i n v o k i n g t h e f i r s t i n e q u a l i t y i n Lemma 2 . 8 . We remark t h a t t h e o r e m s 2 . 7 and 2 . 9 a r e e q u a l l y t r u e f o r a l i n e a r operator

T

from

LP(X,m)

to

Lq(Y,u)

(or

L:(Y,p)),

where

( y , ~ ) i s a n o t h e r measure s p a c e .

The f o l l o w i n g p o s s i b l e g e n e r a l i z a t i o n o f t h e o r e m 2 . 7 a r i s e s n a t u r a l l y : Given a f a m i l y

T

o f l i n e a r o p e r a t o r s u n i f o r m l y bounded i n

wip5 M vnp

(f 6 Lp,

T 6 T)

Lp(m):

487

V. 2. THEOREM OF ?IARCINh'IEWICZ-ZVC?!UND

does the

L 2-valued inequality

(2.11)

I (;J

2 1/2 lTjfjl 1 I],

5.

c I ( Xj l f j l

2 1/2

1

IP

(Tj E 7 )

automatically hold?. The answer is trivally affirmative when but only in this case, as the following examples show.

p = 2,

EXAMPLES 2.12. (a) Let us consider in Lp(R) the translation operators T.f(x) = f(x-j), j=l ,2,3,. . . . If (2.11) is supposed to hold J (j=1,2,. . . ,N) to for some p > 2, we apply it to fj = 1-1 ,1 -jl obtain

i/ c ~ N 1 p , which is impossible for large N. On the i.e., N ~ other hand, if (2. 1 ) holds for some p < 2, we take fl = f 2 = . . . = fN = x so that T.f. = X and we J J [j ,J+11 /2 [ 0 , ! 1 ' < C N obtain ,'Ip which again is absurd for large N.

(b) A much deeper and interesting counterexample is provided by the directional Hilbert transforms in R", n > 1 . These are defined by (Huf)^(c)

= -i

sign([.u)!?(<)

for each unit vector u 6 C n - l = Ix E Rn : 1x1 = 1). If p is a rotation taking u into the vector ( l , O , . . . , 0), one easily shows that HUf(PX)

=

H(fOP(. 'X2,. . . ,xn)) (x,)

where H is the Hilbert transform in R 1 . Thus, each HU i s a bounded operator in Lp(Rn) with norm equal to the norm of H as an operator in Lp(R), 1 < p < -. However, the inequality

is false for every p # 2. This was shown by C. Fefferman [2] in order to prove that the characteristic function of the unit ball is not a multiplier in Lp(Rn). We refer to M. de Guzmrin [2] for this and related results. Here is however an example in which inequality (2.11) does hold. By an interval I in Rn we shall mean the Cartesian product of n

V. VECTOR VALUED INEQUALITIES

488

i n t e r v a l s o f R . The " p a r t i a l sum" o p e r a t o r the multiplier xI:

SI

i s t h e n d e f i n e d by

(SIf)^(S) = m x I ( s )

Let

COROLLARY 2 . 1 3 .

n = 1,

In t h e c a s e

{Ij]

be a r b i t r a r y i n t e r v a l s i n

Then

Rn.

t h e f o l l o w i n g weak t y p e i n e q u a l i t y a l s o h o l d s :

(2.15) Proof: Matters a r e e a s i l y reduced t o considering i n t e r v a l s o f t h e form I ( a ) = [al,m)x[a2,m) a n d we w r i t e

Sa

instead of

(sof)"(SI

x...x

SI(a).

= m x p ( S 1) X P ( S 2 )

( a 6 R")

[an,m) Then

...

Xp(Sn)

(P =

[O,m))

a n d we c a n w r i t e S o = 2 - " ( I d + i H l ) o ( I d + i H 2 ) O . . . O (Id + iHn)y w h e r e Hk d e n o t e s t h e H i l b e r t t r a n s f o r m i n t h e d i r e c t i o n o f (k) e k = ( O , O , . . . ,O , l , O y...,O). T h e r e f o r e , So i s a bounded o p e r a t o r i n Lp(Rn), 1 < p < m. On t h e o t h e r h a n d

a n d Theorem 2 . 7 g i v e s u s 2 1/2

J

When

n = 1,

the operator

So =

71 ( I d

+ iH)

IP

i s a l s o o f weak t y p e

( l , l ) , a n d t h e same a r g u m e n t c o m b i n e d w i t h Theorem 2 . 9 g i v e s (2.15).

n

T h i s c o r o l l a r y i s v e r y u s e f u l i n L i t t l w e o o d - P a l e y t h e o r y , a s we s h a l l s e e i n s e c t i o n 4 . We c a n a l r e a d y show how i t w o r k s t o p r o d u c e some i n e q u a l i t i e s o f L i t t l e w o o d - P a l e y t y p e w h i c h a r e s i m p l e r b u t l e s s known t h a n t h e u s u a l o n e s .

V . 2 . THEOREM OF E~4L\P,CINI;IEWICZ-ZYI;MUND

489

THEOREM 2 . 1 6 . Let I b e a bounded n - d i m e n s i o n a 2 i n t e r v a l , a n d f o r e a c h l a t t i c e p o i n t k 6 2" consider the trans2ated i n t e r v a l k + I .

f 6 Lp(Rn)

Then, f o r e v e r y

with

2 5 p

t h e foZ2owing i n e -

m,

G a l i t y holds:

1';

Isk+IfI

2 1/2

8,

5 'p

iflp

@ ( x ) i n R", and c o n s i d e r t h e 2 f i n i t e - d i m e n s i o n a l H i l b e r t s p a c e B = 1 ( F ) , where F i s a f i n i t e

P r o o f : Take a Schwartz f u n c t i o n subset of

2".

maps L:(Rn) for q = 2

and f o r

The o p e r a t o r

b o u n d e d l y i n t o Lq(Rn) for all 1 5 q 5 2 . In fact, t h i s i s a consequence o f P l a n c h e r e l ' s theorem, s i n c e

q = 1, K(x)

i t f o l l o w s f r o m Theorem 1 . 1 4 , s i n c e t h e k e r n e l

(e

=

defining the operator

=

where

Q

2nik.x T

@(x))k6F

= L(B,C)

s a t i s f i e s , f o r each

b = (bk)ksF

IblB denotes t h e u n i t cube of

R",

and

C

because

By i n t e r p o l a t i o n we o b t a i n t h e r e s u l t f o r @ 6 S(Rn). The a d j o i n t o p e r a t o r = (Tkf

(where

(Tkf)^(S)

=

f(c)+(c-k))

from Lq'(Rn) i n t o Lj'(Rn) letting F increase t o a l l

1

5 q 5 2.

kEF

i s t h e r e f o r e a bounded o p e r a t o r

w i t h norm i n d e p e n d e n t o f 2" we g e t

F,

so t h a t

490

V. VECTOR VALUED INEQUALITIES

Now, assume that we take @ such that $ ( E ) = 1 for all 5 E I. Then SkcIf = Sk+I(Tkf), and we can invoke Corollary 2.13 (applied to the sequence of functions (Tkf)kEzn) together with the preceding inequality to complete the proof. 0 Remarks 2.17. (a) One can replace in Theorems 2.7 and 2.9 L" by any other (separable) Hilbert space B. In particular, we can obtain a continuous version of both theorems by choosing B = L 2 (R, w(t)dt), where w(t) 2 0. If this is applied in the proof of 2.13, one obtains the corresponding continuous version of that Corollary, namely:

Where ft(x) is a measurable function of (t,x) € RxR", and I(t) are intervals in Rn whose vertices are measurable functions of t. Of course, there i s also the corresponding continuous version of the weak type inequality (2.15)

.

(b) By taking into account the known weighted norm inequalities for the Hilbert transform and the product operators arising from i t (see Chapter IV, (6.2)) it follows immediately that inequality (2.14) holds in Lp(w) provided that w E A* and 1 < p < m . P Similarly, the weighted analogue of (2.15) (with a weight w(x) in R) holds if w E A 1 . (c) When n > 1, we cannot replace the intervals { I j } in Corollary 2.13 by parallelepipeds in arbitrary directions. This fact is actually equivalent to the counterexample 2.12 (b) (see however 7.3 below). (d) The inequality of Theorem 2.16 is false if p < 2. To N -1 see this, define fk in R by 1, = X [k,k+l] and = fk 2nikx k = o (so that f^ = X[o,N~). Then, Sk+If(x) = fk(x) = e f0W,

k = O,l, . . . ,N-1, and Theorem 2.16 implies:

f(x)

=

Nfo(Nx),

so that the inequality of

V.3. VECTOR VALUED SINGULAR INTEGRALS

which, when

N

+ m,

forces

49 1

2 5 p

3. VECTOR VALUED SINGULAR INTEGRALS

So far, we have proved two general results concerning the existence of B-valued extension for a linear operator T which is bounded in LP(m): The extension exists if T is positive (theorem 1 .12) 3 r if B is a Hilbert space (theorem 2.7). This is all that can be said in general. It is the study of particular cases what has to be done now (take an interesting non positive operator T appearing in Fourier Analysis, take some fixed Banach space, such as B = L r with r # 2, and try to know wheter T has a B-valued extension o r not). This is, by far, the more interesting part, and the first natural question would concern the Hilbert transform: Does the inequality

hold? An affirmative answer can be given if r = 2 and 1 < p < m by the theorem of Marcinkiewicz and Zygmund, and this can be slightly improved by interpolating with the obvious case: 1 < p = r < -. However, as early as 1939, Boas and Bochner [l] proved that (3.1) does hold for all 1 < p , r < a. They used a very clever argument involving complex function theory, but now, with the Calder6n-Zygmund machinery available, the proof of (3.1) and its n-dimensional analogues becomes fairly easy. We shall adopt a quite general viewpoint: Let A and B be Banach spaces, and consider a kernel K(x) defined in Rn whose values are bounded linear operators from A to B : K(x) 6 L(A,B). We assume that K(x) is measurable and locally integrable away from the origin, so that the integral (3.2)

is well defined for all F € Li(Rn) with compact support and for all x i! supp(F). To make the Calder6n-Zygmund theory work in this context, some HUrmander type condition must be imposed on K[x), like

V. VECTOR VALUED INEQUALITIES

492

Now, we can state the fundamental result, essentially ta.sn from Benedek, CalderBn and Panzone [ l ] : THEOREM 3.4. Let T b e a bounded l i n e a r o p e r a t o r from Li(Rn) to Li(R") f o r some f i x e d r, 1 < r < m, and a s s u m e t h a t , when F bounded and w i t h compact s u p p o r t , TF(x) i s d e f i n e d b y ( 3 . 2 ) for e v e r y x i? supp(F), g j t h a k e r n e l K(x) s a t i s f y i n g 1 3 . 3 1 . Then, T c a n be e x t e n d e d t o an operdator d e f i n e d i n

that -

ti,

1 5 p <

m,

such

LA

every

A-atom a(x)

The space BMO(B) appearing in (3.7) is defined in a natural way: it consists oE all B-valued functions F(x) for which

Observe that

F € BMO(B)

I (IFllB)IB)lo

implies

5

lFIB € BMO,

and

IFIBhfo(B)

On the other hand, A-atoms are also as expected, i.e., functions m a E: LA supported in a cube Q and such that: -1 Ia(X)8,

5

1Q1

Proof of Theorem 3.4: I t is a straightforward generalization of Theorem 5.7 in Chapter 11. One first proves (3.6) by using a Calder6n-Zygmund decomposition of IFIA, where F € Ll L;, and estimating its "good" part by means of the L*-boundedness of T (a slight modification is required here in the case r = m ) and its "bad" part by means of (3.2) and (3.3). Interpolation (use Theorem 1.18) thengives the estimates (3.5) for 1 < p 5 r.

n

T h e second step is (3.7), whose p r o o f ( a s well as that of (3.8)) is

493

V.3. V E C T O R VALUED SINGULAR INTEGRALS

e x a c t l y a s i n t h e s c a l a r c a s e . Then, t h e s u b l i n e a r o p e r a t o r SF(x)

=

ff

(ITFIB) (x)

satisfies

because t h i s i s t r u e f o r Theorem 1 . 1 8 . Now, s i n c e

..

p = r,

p

=

and we c a n a p p l y a g a i n

m

( s e e C h a p t e r 11, 3 . 6 ) t h e r e m a i n i n g i n e q u a l i t i e s i n ( 3 . 5 ) a r e a l s o proved.

n

As i n the scalar case, the constants

p,

on

M

a n d on t h e

Cp,

(Li,Li)-norm of

1 5 p

5

m,

depend o n l y

T.

T h e r e a r e c e r t a i n r e m a r k s t o be made now which w i l l p a v e t h e way t o some o f t h e a p p l i c a t i o n s o f v e c t o r v a l u e d s i n g u l a r i n t e g r a l s . The f i r s t one i s t h e s t r i k i n g f a c t t h a t t h e v e r y s t a t e m e n t o f Theorem 3 . 4 . l e a d s t o i t s immediate self-improvement i n t h e f o l l o w i n g s e n s e : THEOREM 3 . 9 . Under t h e h y p o t h e s i s of t h e p r e c e d i n g t h e o r e m , t h e

following i n e q u a l i t i e s a r e a l s o v e r i f i e d f o r I{X

: ( TJ :

I T F ~ ( x ) I ~ )l/q

>

All

<

c x -’

- 9

1 < p,q < I(1

m:

I F j ( x ) I A )l / q d x

j

1 < q < m, t h e Banach s p a c e o f a l l a . E A s u c h t h a t I(..)[= 3 J The p r e v i o u s t h e o r e m t e l l s u s t h a t

P r o o f : Denote by l q ( A ) , sequences ( a j ) j c N with =

(z j

l a j l A ) ’Iq <

m.

T : L: -+ L i i s a bounded o p e r a t o r . I t i s t h e r e f o r e t r i v i a l t h a t t h e o p e r a t o r .T : ( F . ) . I-+ (TF.). maps ~9 boundedly i n t o J JcN J JEN lq (A) Lq , and i t i s d e f i n e d by t h e k e r n e l R(x) E L ( l q ( A ) , l q ( B ) ) eq ( B ) g i v e n by k ( x ) . ( ( a j l j c N ) = ( K ( x 1 . aJ. ) J. E N The o p e r a t o r norms o f

K(x)

and

IK(x-y) - K ( x ) l

so t h a t

k

K(x) =

c o i n c i d e , and s i m i l a r l y

IK(x-y) - K(x)[

s a t i s f i e s ( 3 . 3 ) a n d Theorem 3 . 4 a p p l i e s a g a i n p r o v i n g

V. VECTOR VALUED INEQUALITIES

494

0

the desired inequalities.

The second important fact t o be noted is that, if the Banach space A is one-dimensional: A = C , then L(A,B) = B. Moreover, in this case the L 1 -estimate (3.8) has a more interesting meaning due to the atomic decomposition of H 1 (Rn) described in Chapter 1 1 1 . We state explicitly the corresponding results as a Corollary. COROLLARY 3.10. Let

T

: Lr(Rn)

o p e r a t o r f o r some f i x e d

Tf(x) f o r some

=

r,

Li(Rn)

+

1

r 5

(x

B-valued measurable f u n c t i o n

(with I . I

e z t e n d s t o a bounded o p e r a t o r f r o m

1 n H (R )

and assume t h a t

If(y)K(x-y)dy

Harmunder‘s c o n d i t i o n ( 3 . 3 1

from

m,

b e a bounded l i n e a r

1 n LB(R )

and f r o m

K(x)

Lp(R ) 1 n LB(R )

e

supp(f))

which v e r i f i e s

.IB).

I Then T LE(R”), 1 < p < 1 n t o weak-LB(R ) . =

2

a,

1 n ) can be defined in terms of atoms for an arbiOf course, HA(R trary Banach space A , and if this is done, the estimate (3.8) can 1 1 be read as follows: T : HA LB. -+

A final remark, which should have been observed by anyone trying to adapt the proof of Theorem 5 . 7 in Cahpter I1 to the vector valued case, is the following: In the proof of the inequalities (3.6), (3.8) and (3.5) for 1 < p 5 r , one can replace (3.3) by the weaker condition

(for all a 6 A , y E R”). However, (3.7) and the remaining inequalities in (3.5) do not follow from (3.3’). We do not emphasize very much the interest of this weaker condition since, in most of the applications of vector valued singular integrals, (3.3) does hold. Turning to the problems stated at the beginning of this section, we have the first immediate applications of our general result. Let THEOREM 3.11. -

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

(k.)jcN 1ijim

i n Rn with s u p < m. Assume t h a t e a c h k. c o i n c i d e s a u a y 3 J f r om t h e o r i g i n w i t h 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 , and

V . 3 . VECTOR VALUED SINGULAR INTEGRALS

495

T h e n , t h e f o l l o w i n g i n e q u a l i t i e s h o l d , p r o v i d e d t h e r i g h t hand s i d e

is f i n i t g : (3.12)

I { x : (; <

> A l l 5

lkj*fj(x)\r)”r

1

cr

-1

A

r l/r

I(C Ifjl 1 J

(1 < r <

I1

m)

P r o o f : F i r s t o f a l l , we m u s t r e c a l l t h a t , u n d e r t h e h y p o t h e s i s , k.*f 1

1 < p <

f € Lp(Rn),

can be d e f i n e d f o r a l l

r e s u l t i n g o p e r a t o r s a r e u n i f o r m l y hounded i n Now we f i x

r > 1

Lp

and t h e

m,

for a l l

and c o n s i d e r a s Banach s p a c e s :

l
A = B = br.

The

operator TF(x) = ( k . * f . ( x ) ) . J 3 JCN

Li,

i s t h e n bounded i n TF(x) where

=

K(x) 6 L ( B )

(I:

(f.). 6 L;) 1 JGN

=

and i t i s c l e a r t h a t

[ K(x-y) .F(y)dy

(x

J

e

supp(F1)

i s t h e d i a g o n a l o p e r a t o r w hose e n t r i e s a r e

kj ( x ) : K(x) . b = ( k . ( x ) b . ) . 1 1 JCN

(b

=

( b . ) 6 B) 1

Since

I K x - y ) - I < ( x ) \ ~ ( =~ )s u p I k . ( x - y ) - k 1. ( x ) l , (3.3) is 1 1 v e r i f i e d , a n d Th eo r em 3 . 4 a p p l i e s g i v i n g t h e d e s i r e d v e c t o r v a l u e d i n e q u a 1i t e s .

Cl

l h e h y p o t h e s i s o f t h l s t h e o r e m are fulfillcd i n t h c f o l l o w i n g s p e cially interesting cases:

k (XI

a ) When e a c h

lip)I 5 b ) IVhen

is

J

C,

k (x) J

=

C’

o u t s i d e t h e o r i g i n and

IVkJ(x)l 5 C h(x)

f o r a11

Ix~-~-’ J ,

where

( 1 c‘ 8 )

c

6 I,*(R”)

,ind

496

V. VECTOR VALUED INEQUALITIES

The second case can also be obtained as an application of Theorem 3.9 which A = B = C (complex numbers). What this case means is that every singular integral operator has a bounded Lr-valued extension if 1 < r < m. In particular, we can take k(x) = p.v.-TI1X in R , proving (3.1) for all 1 < p , r < m . An application of this inequality is the following improvement of Corollary 2.13 (without modifying its proof) : COROLLARY 3.14. Let (2.15)

&!

(c I

.Ir+'

r

be f i x e d ,

1

r <

m.

The i n e q u a l i t i e s (2.14)

r e m a i n t r u e if we s u b s t i t u t e e v e r y w h e r e

(C

r.

j

1.1

2) 1/2

j

Not surprisingly, weighted estimates can also be obtained for vector singular integral operators, provided that we assume more regularity for the kernel K(x), namely

We invite the reader to look for a complete analogy with the results for regular singular integral operators which, in the scalar case, were described in Chapter 11, Theorem 5.20, and in section IV.3. Here, we merely state the result which we shall need to apply in the next section. THEOREM 3.16. S u p p o s e t h a t T i s a s i n T h e o r e m 3 . 4 w i t a k e r n e l K(x) s a t i s f y i n g , i n s t e a d of Hsrmander ' s c o n d i t i o n , t h e s t r o n g e r a s s u m p t i o n ( 3 . 1 5 ) . S u p p o s e a l s o t h a t , f o r some f i x e d r, 1 < r < a n d for a l l w e i g h t s w E A1, w e h a v e

w i t h C(w)

(3.17) for every

(3.18)

depending o n l y on t h e

w(Ix

: \TF(x)lB

w E A1,

> tl)

A1-constant of

5 C(w) t-l ]lF(x)II$

and a l s o , €or every

jITF(x)lgw(x)dx

w.

w € A

with

P -

T h e n , we

m,

have

w(x)dx 1 < p <

m,

5 Cp(w) IIF(x)Og w(x)dx.

The proof of (3.17) is again a n exact repetition (except for a minor change in the estimation of "good part" when r = m ) of the one given in Chapter IV, Theorem 3.5, for scalar functions (observe that the hypothesis IK(x) 1 5 Clxl-" was not really needed there).

V.4. SOME MAXIMAL INEQUALITIES

497

Once the weak type result is proved, (3.18) follows by the extrapolation theorem of Chapter IV, ( 5 . 2 0 ) . 0

4. APPLICATIONS: SOME MAXIMAL INEOUALITIES

Almost every nonlinear operator appearing in Fourier Analysis is linearizable (in the sense of Definition 1.20). It is sometimes useful to look at such operators as linear operators taking complex valued functions into B-valued functions, and if the kernel happens to satisfy the appropriate condition, the theory developed in the preceding section can he applied. In particular, we shall see that the Hardy-Littlewood maximal operator and some of its generalizations can he viewed as vector valued singular integrals. Given

6 L1(Rn),

@

M f(x) @

=

we define

sup 6O.

he maximal operator

lf*@&(x)

Denoting B = L m ( R + ) , it is which is certainly bounded in Lm(Rn). equivalent to consider the linear operator T : Lm + Li defined (as since in Corollary 3.10) by the B-valued kernel K ( x ) = ($6(x))6,0, we have: M@f(x) = ITf(x)lB. Now, HGrmander's condition for the kernel K(x) amounts to the following:

and we are led to THEOREM 4.2. L e t il M weak t y p e

@

1

n

6 L (R )

be a function satisfying ( 4 . 1 ) .

i s an o p e r a t u r bounded i n

@ (1 ,1) E d m a p p i n g

or

iii

every

f

c

u

H 1 (R")

1 ip
Lp(Rn), 1 < p c L1 (R").

m,

Then: Q

into

LP(R")

I

iiij The f o l l o w i n g v e c t o r valued inequa2itie.s hold f o r a l l 1 < p,q <

m:

498

V. VECTOR VALUED INEQUALITIES

Proof: Since fx@&(x) depends continuously on 6 , it suffices to obtain in (i) and (iii) uniform estimates for the operators I

where

F

is an arbitrary finite subset o f KF(X)

=

(@6(x))6cF

E

R,

and

Lm(F)

Now, we are under the hypothesis of Theorem r = m, A = C , B = L m ( F ) ) , which give (i) Finally, (ii) follows in the usual way (see instance) since it is trivially verified by compact support. 0

3.4 and 3.9 (with and (iii) respectively. Chapter 11, (1 .9), for a l l continuous f with

A trivial but important observation is that, if

is such that ( $ 1 5 1 4 1 , the whole theorem, except for the H’ + L1 result, holds also for the operator M$. In particular, we can take a s $ the characteristic function of the unit cube centered at the origin (thus, M+ = Hardy-Littlewood maximal operator) and as $ a nonnegative Schwartz function such that $(x) 1 when x E Q. Then, we have $

COROLLARY 4.3. T h e H a r d y - L i t t l e w o o d m a x i m a l o p e r a t o r v e r i f i e s t h e v e c t o r v a l u e d i n e q u a l i t i e s l i i i l of t h e p r e v i o u s t h e o r e m .

Next, we observe that, since Lm(w) = Lm(Rn) for every w 6 A 1 , the operator M$ satisfies the estimate required in Theorem 3.1b with r = m , and we can therefore obtain weighted estimates for this kind of maximal operators: THEOREM 4.4. S u p p o s e t h a t I$(X-Y)

-

L1(Rn)

$ 6 $(XI

I

5

satisfies

C l y l 1x1 - n - l

I I

(1x1 > 2 Y > O )

T h e n , t h e m a x i m a l o p e r a t o r M$ i s bounded i n L P ( ~ ) 1 < p < m and w E A and i t i s a l s o of weak t y p (1,l) P’ r e s p e c t t o the m e a s u r e w(x)dx Q w E A, _ .

with -

Proof: I t suffices to apply 3.16 taking into account that the condition imposed on $ is dilation invariant, so that the same

499

V.4. SOME MAXIElAL INEQUALITIES

inequality is verified by sup l$6(x-Y) -

6>0

Q6

for all

6 > 0 , i.e.

5 c IYI

@,(XI1

(1x1 >21yl

and this is exactly the condition (3.15) for the K(x) = ($,(X)),>0-

'

0)

Lm-valued kernel

One can consider more generally maximal operators defined by a i family Q = ( @ )iEI, namely sup * f(x)l iEI where, either I is countable o r I = R+ and $ I ~ ( X ) depends continuously on i. If (4.1) is verified by the family ( @ ' ) , and if M@ happens to be bounded in Lq(Rn) for some q (this is certainly i true with q = m when s u p I @ 1 , < m ) , then, parts (i) and (iii) of Theorem 4.2 hold foriE' Mo. Mof(x)

=

In particular, we can consider approximations of the identity (G6 ) corresponding to a lacunary sequence ( 6 k ) of dilations, and we k have THEOREM 4.5. S u p p o s e t h a t (4.6)

(4.7)

@

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

/$(x)I log(2+lxl)dx

Jl

w,(tI

dt

7

<

m,

where

<

m

w,(t)

=

sup

IhlZt

T h e n , t h e c o n c l u s i o n s of T h e o r e m 4 . 2

i

I$(x+h)-@(x)ldx

h o l d for t h e d y a d i c m a x i m &

o p e r a to r

irepZacing

i n part

(iil

lirn C+O

Ky

lim).

k+m

Proof: Instead of proving (4.1) for the sequence obtain the stronger inequality

(2-k)ym,

wc shall

V. VECTOR VALUED INEQUALITIES

500

Observe that the left hand side of this inequality remains unchanged 1 if we substitute y by 2y, so that we can assume: 5 IyI 5 1 . Then, we study the sums corresponding to k 2 0 and to k < 0 separately: m

m

1 k=o

,

dx

11,

=

=

< -

< 2 f due to ( 4 . 6 ) .

For the remaining sum we have to use (4.7):

dt

=

C2 <

m

and this completes the proof. Examples 4 . 9 . (a) Given a function n ( x ) in Rn which is Dositive, homogeneous of degree 0 and integrable over the unit sphere, we consider the following nonisotropic maximal function, which was suggested by E.M. Stein and studied by R. Fefferman [l]:

This is the maximal operator associated to $(x) = Q(x')x,(x), where B is the unit ball. Since @ 6 L 1 (R n ) and has compact support, (4.6) is verified. Now, as we did in Chapter I 1 for the homogeneous kernels of singular integrals, we define o,(R,t)

=

I

sup

Ihlzt

/x'I=l

IQ(x'+h) - n ( x '

and we try to see the relationship between modulus of continuity of @ , w,(@,t). If

m,(R,

t

rj

rn-1dr

w1

Ih

and the L 1 < t 5 1, we have

n,t)

V . 4 . SOME MAXIMAL INEQlJALITIES

501

and t h e r e f o r e

Thus, we can s t a t e our r e s u l t :

I n f a c t , t h i s i s t r u e f o r t h e o p e r a t o r N Q d e f i n e d by t a k i n g t h e k supremum o v e r a l l r = 2 , k 6 Z ( d u e t o 4 . 5 ) , b u t , s i n c e k @,(x) 5 Z n @ ,(x) when Z k - ' < r < 2 , we h a v e MQ f(x) 5 ZnN Rf(x) 2 f o r a l l p o s i t i v e f , a n d t h e r e s u l t f o r M, follows. (b) For every

consider the kernel

a > 0,

OC1(Xj = - max(1

r la)

Then

, 01~-

h a s compact s u p p o r t and i t s

Oa € L 1 ( R " ) ,

tinuity satisfies (4.7)

2 1x1

-

5 Cata,

wl(Qcl,t)

1 L -modulus o f con-

so t h a t t h e Dini condi ion

i s v e r i f i e d . T h e r e f o r e , t h e max i ma l o p e r a t r NNf(x)

i s bounded i n

Lp(R"j,

1

sup k€Z

=

1 < p

5

f(~-2-~)o"(y)dy, m,

a n d o f weak t y p e ( 1 , l )

An e a s y

c o m p u t a t i o n s h o ws t h a t

or'

lim

=

( i n the d i s t r i b u t i o n sense)

0

01+0

where

0

(Lebesgue measure i n t h e u n i t s p h e r e ) i s i d e n t i f i e d w i t h

a s i n g u l a r Bore1 measure i n

Rn

supported i n

{x € Rn

Therefore, the limiting oaerator (corresponding t o

a

: 1x1 = 1 1 . =

0)

is given

by Nf(x)

=

Nof(x)

=

sup k6Z

I

l/y'l=l

f ( ~ - 2 - ~ y ' ) d o ( yI ' )

H o w e v e r , t h e c o n s t a n t s o b t a i n e d by a p p l i c a t i o n o f The ore m 4 . 5 t o

N"

b l o w u p when

operator

N

c1

+

0,

a n d n o t h i n g c a n b e s a i d a b o u t t h e m a xim a l

by t h e m e t h o d s d e v e l o p n e d s o f a r . E s t i m a t e s f o r

Nf

w i l l be o b t a i n e d i n 5 . 1 9 below.

Now, f o l l o w i n g F e f f e r m a n a n d S t e i n [ l ] ,

we s h a l l u s e t h e v e c t o r

v a l u e d i n e q u a l i t i e s f o r t h e H a r d y - L i t t l e w o o d m a xim a l o p e r a t o r i n

V. VECTOR VALUED INEQUALITIES

502

order to obtain some estimates for Marcinkiewicz integrals. Given a closed set F whose complement has finite Lebesgue measure, the Marcinkiewicz integral of order X > 0 corresponding to F is the function:

H X (F;x) where

=

H (x)

=

6 (Y)

Ix-y(n+X + 6(X) n+X dy

JR\F

denotes the distance o f the point y from

6(y)

F.

On the other hand, given a family of disjoint b a l l s (or cubes) {Bj}, the Marcinkiewicz integral of order A corresponding to this family is defined as: d?" SA(X) = c j (x-c. + dn+'

' J

1

where

c. J

denotes the center of

R. J

and

d. J

its diameter.

x Then, t h e X > 0 , we Z e t q = 1 + 6. i n te g r a Is HA & r g Sx d e f i n e d a b o v e s a t i s f y :

COROLLARY 4.10. G i v e n Marcinki ewi c z

Moreover, p = :;

we h a v e t h s c o r a r e s p o n d i n g weak t y p e i n e q u a l i t i e s f o r

l{x :

fIX(X)

I{x

SX(x) > t)I

:

> t}l

5 CX t - ' / q 5 C X t-"'

IR\F/

; lBjl 3

Proof: Let us consider first S X . If B denotes the unit b a l l centered a t the origin, i t is very easy t o see that: M ( X B ) ( x ) > > Const.(l + lxln)-'. (Ilere M stands for the Hardy-Littlewood maximal operator). By translation and dilation invnriance of M , we have dn

and there fore

V.4.

SOME MAXIMAL INEQUALITIES

503

The inequalities for Sx result now from an application of Corollary 4.3 with f . = X B . . It is clear that this argument works equally J well for a familyJ of cubes, instead of balls. In order to prove the results for H A , we first observe the dilaX tion invariance of H A in the following sense: Hx(tF;x) = Hx(F;y). This a l l o w s us to assume IRn\Fl = 1 . Now, taking into account that 6(Y) 5

where

+

Rn\F

=

Ix-YI

u

3 p. 1 6 ) ,

Stein [ l ] , and diameter

dj)

is a Whitney decomposition of

Qj

{Qjl

i.e., and

-

x !6

and, since

( R n\ F l

=

1,

I:

c. J

( x 6 Qj)

I

for some absolute constant A. Therefore, letting Marcinkiewixz integral corresponding to the family

On the other hand, for

(see

are disjoint cubes (with center

A-'d. < 6(x) < Ad. J -

Rn\F

SA

be the

{Q41

we can make the Trivial estimate

thc results for

of the corresponding inequalities for

S

A'

HA

n

are now a consequence

We conclude this section by showing how vector valued singulor i n tegrals can be used to prove the w e a k type (1 ,1) for 'i-*f(xi

=

sup I >o

11,

y , >k

K (y) f ( x - y ) d y

1

where K is the kernel of 11 regulrlr singular integral oper;itor. This fact was stated without proof in Chapter 1 1 , Theorem S . 2 0 , hut we did prove there that 'I,* is Ihouncled in 1.' for a l l 1 i p c a'.

v.

504

VECTOR VALUED INEQUALITIES

where h ( t ) i s a C1 f u n c t i o n s u c h t h a t 0 5 h ( t ) 5 1 , h ( t ) = 0 when 0 5 t < 1 a n d h ( t ) = 1 when 2 5 t < a. The f i r s t t h i n g t o notice is that IT*f(x) - T*f(x) < sup c -

I 2

IY

Thus, e s t i m a t e s f o r vation is t h a t

E>O

'E<\y\<2E

IK(Y)f(X-Y) Idy

I f ( x - ~ ) l d5~ C Mf(x)

E-n

E>O

sup

I12E

T"

or

7"

a r e e q u i v a l e n t . The s e c o n d o b s e r -

I h ( l x - y l ) - h ( l x l ) l 5 CIyI/IxI

1x1 > 2 1 ~ 1 . T h e r e f o r e , i f we h a v e

KE(x)

=

whenever

K(x)h(lxl/E)

and

~ K ' ( X - Y ) - K E ( x ) I < I K ( x - y ) - K ( x ) l h I x( - Y 7l ) -

1x1 > 2 1 ~ 1 ,

+

with C ' independent of E . This implies t h e i n e q u a l i t y ( 4 . 1 ) f o r t h e f a m i l y ( K E ) E > O , a n d s i n c e T* i s b o u n d e d i n L p , t h e remark p r e c e d i n g 4 . 5 a p p l i e s a n d t h e weak t y p e ( 1 , l ) f o r M o r e o v e r , s i n c e we h a v e r e a l l y p r o v e d t h a t t h e

?*

follows.

Lm-valued kernel

(KE) E > O s a t i s f i e s ( 3 . 1 5 ) , a n d we know t h a t T* ( a n d t h e r e f o r e a l s o i s bounded i n Lp(w) i f w € A , and 1 < p < m ( s e e Chapter IV, 3 . 6 ) we c a n a p p e a l t o Theorem 3 . 1 6 a n d s t a t e t h e f o l l o w i n g

t*)

COROLLARY 4 . 1 1 . __ Let let

T"

T

b e a 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 , and

be t h e a s s o c i a t e d maximal o p e r a t o r d e f i n e d a s above. Then,

f o r every weight

w E A1

w({x : T * f ( x ) > t l ) 5 C w t

5 . APPLICATIONS: SOME LITTLEWOOD-PALEY THEORY

I n t h i s s e c t i o n we c o n t i n u e t h e a p p l i c a t i o n o f t h e t h e o r e m s f o r v e c t o r v a l u e d s i n g u l a r i n t e g r a l s t o some i m p o r t a n t l i n e a r i z a b l e o p e r a t o r s . T h i s p r o g r a m was c a r r i e d o u t i n s e c t i o n 4 f o r t m - v a l u e d k e r n e l s , and h e r e w e s h a l l u n d e r t a k e t h e c a s e o f H i l b e r t s p a c e valued kernels. This w i l l lead us t o various kinds of square

V.5.

505

LITTLEWOOD-PALEY THEORY

functions, one of the typical ingredients of what is called Littlewood-Paley theory. We emphasize, however, that this theory is much deeper and richer than what we can afford to present here, which must be considered only as an introduction to some of its basic aspects. Suppose we are given a sequence of kernels form the square function m

Gf(x)

(

=

1

j=-m

Ikj

*

f(x)l

kj(x)

in

Rn

and we

2 1/2

1

Then Gf(x) = ITf(x)lB, where B = 1’ and T is the linear operam tor taking f(x) into the B-valued function (kj*f(x))j7-m. By Plancherel’s theorem, G (equivalently T) is bounded in L2(Rn) if and only if (5.1)

If this is verified, Corollary 3.10 can be applied provided that we also have

Looking for some concrete examples satisfying ( 5 . 1 ) find out the following: THEOREM 5.3. Let

@(x)

&

$(O) and

an i n t e g r a b l e f u n c t i o n in

\@(x)dx

=

a s s u m e t h a t , f o r some (5.4)

I@(X)l

(5.5)

/l@(x+h)

=

c 1x1

Rn

suck t h a t

0

it verifies

> 0,

5

and ( 5 . 2 ) we

-n-a

(x 6 Rn)

and __ - @(x)ldx

5 C lhla

(h

€

Rn)

V. VECTOR VALUED INEQUALITIES

506

a r e bounded i n

LP(R"),

1 < p

a r e a l s o b o u n d e d o p e r a t o r s from

<

m,

a n d of weak t y p e

H1(Rn)

2

1

n

(1 ,I).

hey

L (R ) .

As usual, we have denoted by @t the dilation of I$ : @,(x) = = t - n I$(:). Observe that A f is the continuous analogue of Gf S (this is more evident if we make the change of variables t = 2 the definition of nf).

in

Proof of the Theorem: The results for Gf will follow from Corollary 3.10 once we show that the kernels verify (5.1) and (5.2). Similarly, for A f we must verify the conditions:

(@2j)y=_,

and

The properties ( 5 . 1 ) and (5.1') are derived from two different estimates for the Fourier transform of @ : For small 151 we have

?)

with B = inf(?;1 z 0. The estimate for large 151 is based on 2nih.S - 1 ) is the Fourier transform of the fact that @ ( < ) ( e @(x+h)-@(x), and choosing h = [ / ( 2 1 , 1 2 ) we get

2 1 ? ( 5 1 1 5 ~l@(x+h)-@(xlldx 5 C 151-' Since the left hand side of (5.1') is invariant under dilations of 5 , it can be assumed that 151 = 1 , and then

The proof of (5.1) is quite similar, and it is left to the reader. Now, we observe that (5.4) and (5.5) are respectively stronger than the hypothesis of Theorem 4.5, under which, the inequality

(which is stronger than ( 5 . 2 ) 1 was obtnined. Thus, only (5.2 I ) remains to be proved, and again, by dilation invariance, we can assume that

V . 5 . LITTLEWOOD-PALEY THEORY

507

Iy1 = 1 . Then, Schwarz i n e q u a l i t y and F u b i n i ' s theorem a r e u s e d t o m a j o r i z e t h e l e f t h a n d s i d e by

On t h e o t h e r h a n d , i f we u s e ( 5 . 5 ) a n d t h e f a c t t h a t

@(x)

is

bounded : I ( t ) 5 C [ / Q ( x-

$1 - @ ( x ) ( d x<

C t-'

and combining b o t h e s t i m a t e s , w e f i n a l l y o b t a i n t

Remarks 5 . 6 .

'

-

t2a-1

iy

dt +

0

t - l -n/2dtl

<

;".

0

( a ) Th e h y p o t h e s i s ( 5 . 4 ) a n d ( 5 . 5 ) c a n b e s l i g h t l y r e -

l a x e d , b u t t h e y a r e g e n e r a l enough t o c o v e r a l l c a s e s o f i n t e r e s t . I n p a r t i c u l a r , b o t h o f them a r e t r i v i a l l y v e r i f i e d i f llowever, t h e h y p o t h e s i s

&(O)

=

@ 6 S(Rn). is absolutely necessary.

0

(b) I t o c c u r s q u i t e o f t e n t h a t one has t h e e q u i v a l e n c e lGflp

"

lflp,

IGF],

and n o t o n l y t h e i n e q u a l i t y

f a c t , i f B i s a H i l b e r t s p a c e and t o r such t h a t

T : L2

+

5 Cp l f l , .

In

Li- i s a l i n e a r o p e r n (f E L2)

t h e n , a n i n e q u a l i t y o f t h e form

ITfl,n

5

C

LB

some p 2 1 , a u t omn t i c a 1 1y imp 1i e s IfUPl s e e t h i s , we a p p l y p o l a r i z a t i o n t o t h e a b o v e

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

g

such t h a t

If!,

(f 6 L ' n

Idp)

for

V. VECTOR VALUED INEQUALITIES

508

(c) Vector valued inequalities for the operator C; are obtained by an application of Theorem 3 9. In particular, if this is done for q = 2 , we obtain

provided that

@

satisfies the hypothesis of Theorem 5.3.

Our first application of Theorem 5 . 3 will he one of the classical results obtained (in the periodic setting) by Littlewood and Paley in their fundamental series of papers. In order to state it, same notation must be introduced: The dyadic intervals in R are those of the form:

I. I

=

[ 2 j-1 , 2 j ) ;

and they form a partition of R \ { O } . The family of dyadic intervals in Rn, denoted by A = A(Rn), consists of all n-dimensional intervals which are the Cartesian product of n 1-dimensional dyadic intervals. I t is plain that the intervals in A(Rn) are pairwise disjoint, and their union covers almost all Rn. Thus, Plancherel's theorem implies I\f(x)12dx

(5.7)

where

SI

IS

=

p ) I 2dx

stands for the partial sum operator:

THEOREM 5 . 8 . T h e r e e x i s t c o n s t a n t s

c

such that

(5.9) When

n

cp 1,

=

I I X

( f E:

: (

lf~,

5 I(

1

I€A

C

P'

P

> 0

(SIf)^

=

(1 < p <

I,5 cPlfpp

2 1/2 I S ~1 ~ I

2 n L (R 1 )

:XI. a)

(f t:

LP(R"))

(f t:

H~ (R))

we a L s o h a v e t h e weak t y p e r e s u l t :

1

I €A

l s I f ~ x l 1 2 ) 1 ~'2 A l l 5

c x-l

lflHl

Proof: By the observation made in 5.6.(b), and taking into account ( 5 . 7 1 , i t is enough to prove the second inequality in ( 5 . 9 ) . We shall consider first the case n = 1. Take @ € S(R) such that $ ( O ) = 0 and G ( 5 ) = 1 for all 5 6 I o . Then

rsI.(@2-j* f 1 1 ~ 5 )= 1

i.e.,

S

Ij

f

=

S I . ( $ ~ ~* -f)~ 1

.

? ( E ~ Z - ~ C ) X ~ . ( C =) ? ( O X I. ( 5 ) I

I

Now we use inequality (2.14) and

V.S.

LITTLEWOOD-PALEY THEORY

509

Theorem 5 . 3 t o g e t

.

a n d t h e same i n e q u a l i t y h o l d s f o r t h e i n t e r v a l s (5.9)

{(-I.)}? Thus, 3 J=-Y i s p r o v e d i n t h i s c a s e . The weak t y p e r e s u l t f o l l o w s i n t h e

same way b y u s i n g now ( 2 . 1 5 ) , i n s t e a d o f ( 2 . 1 4 ) :

(c

({x : where

I S I , f ( x ) 1 2 ) 1 / 2 > All 5 C A - 1 J

J

Gf(x)

lGfll

5

C'

A-1

i s t h e s q u a r e f u n c t i o n d e f i n e d i n Theorem 5 . 3 .

The p r o o f f o r

Rn

w i l l b e by i n d u c t i o n on

t h a t (5.9) holds f o r

Rn-l

.

n,

s o t h a t we a s s u m e x € Rn a s

We s h a l l w r i t e a p o i n t

x = (xl,x') w i t h x 1 € R a n d x ' € R n - l . Let A a n d A ' denote r e s p e c t i v e l y t h e f a m i l i e s o f d y a d i c i n t e r v a l s i n Rn a n d i n R n - l ,

so that

A

= A+

u

A _ , where

and i t s u f f i c e s t o p r o v e t h e s e c o n d i n e q u a l i t y i n ( 5 . 9 ) f o r t h e family

Let

A+.

@ E

S(R)

b e d e f i n e d as i n t h e f i r s t p a r t o f t h e

p r o o f , a n d d e f i n e t h e o p e r a t o r T . i n Lp(Rn) a s c o n v o l u t i o n w i t h J in the first variable, letting the other variables fixed, i.e. 92 - j ( T j f l A ( C l ,<'I = ? ( C l , < ' ) ? ( 2 - j E 1 ) By F u b i n i ' s t h e o r e m , t h e r e m a r k 5 . 6 . ( c ) operators

Ti

c a n be a p p l i e d t o t h e

t o yield

and t h e induction hypothesis i m p l i e s

Combining b o t h i n e q u a l i t i e s , we g e t

one immediatly v e r i f i e s ( b y c o m p u t i n g F o u r i e r t r a n s f o r m s ) t h a t : S I f = S I o T j OSRxI f . T h e n , t h e t r u n c a t i o n a r g u m e n t ( b a s e d on C o r o l l a r y 2 . 1 3 ) a l r e a d y Finally, f o r every interval

I

=

1 . ~ 1 6' A + , J

V. VECTOR VALUED INEQUALITIES

510

used in the case o f

R

applies again, and this completes the proof.

0

One would like to have the Littlewood-Paley inequalities (5.9) for other different decompositions of Rn. For instance, we can consider the decomposition into lacunary spherical shells:

Now, the proof of Theorem 5.8 presents basically two steps. First, the result is proved for a smooth modification of the desired decom position, by using Theorem 5.3. This works in many other situations and, in particular, in the one we have in mind. Thus, we canstate: COROLLARY 5.10. Let m(t) R, = ( 0 , ~ ) s u c h t h a t

in

be a

m(t)

Cm =

1

f u n c t i o n w i t h compact s u p p o r t f o r a22

t 6 [1/2,1]

and

m

1

For e a c h

f 6 L~

n

m ( ~ ~ t =) 2

(0 < t <

k=-m

LP(R")

a)

we d e f i n e t h e s q u a r e f u n c t i o n

The second step in the theorem just proved is the truncation of the smooth decomposition by means of Corollary 2.13. This part is more specific of n-dimensional intervals (though it also holds in othcr cases where the proof is harder, see 7.3 below). In particular, we should need in o u r case the inequality

?xD),

and this is certainly false for every p # 2, since the characteristic function of a spherical1 shell is not a multiplier for I,P(R") i f n > 1 and p # 2 (c. Fefferman [?.I). ( (SDf)^ =

*

A

I n the study of multiplier transformations: (T,f) = fm, the equivalence of Lp-norms given by (5.9) can be used as a scissor theorem [we have borrowed this expression from A. C6rdoba), since i t allows to cut the multiplier into its dyadic pieces. To be precise, we have COROLLARY 5 . 1 1 . L e t

m 6 I."(Rn),

and Z e t ~-

u s d e c o m p o s e -it us

V.5.

51 1

LITTLEWOOD-PALEY THEORY

1 mI, w h e r e , f o r e a c h d y a d i c i n t e r v a l I, we d e n o t e I€A mI = mXI. G i v e n 1 < p < m, m i s a n Lp m u l t i p l i e r , i .e. i f a n d o n l y if t h e i n e q u a Z i t y ITmflp 5 Cp lfl,, m

=

fI € L 2

hoZds for a r b i t r a r y f u n c t i o n s

n

Lp(Rn).

then ( 5 . 1 2 ) follows from Proof: If Tm is bounded in Lp(Rn), the consecutive application of Corollary 2 . 1 3 and the MarcinkiewiczZygmund theorem. On the other hand, if ( 5 . 1 2 ) holds, we take an arbitrary f 6 L 2 n Lp and apply Theorem 5.8: -1

ITmflp 5 cp =

-1

cp

l'f

I

'E

2 1/2

lSITmfl 1 2 1/2

lTmlSIfl 1

I,

5.

I,

c, I

=

(I:I P I f l 2 1 1 / 2 I,

5

c; lfl,.

n

The corollary remains true if we use instead of any other decomposition of Rn for which the conclusions of Theorem 5.8 hold; in particular we can take = {IxR"-~ / I 6 A(Rk)}. We shall now see how this works to produce significant multiplier theorems. For the sake of simplicity, we shall consider only the case of R . The operators St defined by (Stf)^ = ^fX [t,-) are uniformly bounded in L p ( R ) , 1 < p < m , and so is eve y "linear combination" of them

(X € L

T

But

is the operator corresponding to the multiplier

and every bounded C 1 Function m(<) whose derivative is integrable can be written in this form. Thus, we have established a simple result: 1'1f m c L~(R) i s o p c l a s s c 1 , a n d ml c L'(R), t h e n ITmfl, 5 C, Ifl,, 1 < p < -'I. The Littlewood-Paley theory provides the following improvement of this result: (Marcinkiewicz multiplier theorem). Let m € Lm(R) C1 o u t s i d e t h e o r i g i n and s u c h t h a t Im'(C) Id6 5 B f o r e v e r y d y a d i c i n t e r v u Z I. T h e n , t h e o p e r a t o r

COROLLARY 5 . 1 3 .

1,

be

Q

function of class

51 2

Tm d e f i n e d by

V. VECTOR VALUED INEQUALITIES

(Tmf)^

=

?m

i s bounded i n

P r o o f : F o r every Schwartz function

Lp(R),

1 < p <

f , one easily verifies by

1,

taking Fourier transforms that

T f(x) = m(2j-')SIf(x) + StSIf(x).m'(t)dt mI where I is the dyadic interval: I = [Zj-', Z j ) , and S t defined as above. Therefore, the hypothesis on m imply ITm f(x)I 5 C {ISIf(x)I I By Corollary 2.13, we have

+

-.

(1,

is

IstSIf(x)12~m'(t)ldt)1/2}

n

I and ft = f I , where, given t € R, we define J(t) = [t,-) if I is the dyadic interval to which t belongs. The second term can be estimated by means of the continuous version of Corollary 2.13 stated in 2.17.(a) so that it is majorized by

Thus, we have proved that (5.12) applies. El

holds, and the preceding corollary

The hypothesis of this corollary can be slightly weakened: It suffices to assume that m E L m ( R ) and that m has uniformly bounded variation on each dyadic interval. As the reader may guess, there is a corresponding version of the Marcinkiewicz multiplier theorem for Rn. F o r this we refer to Stein [ l ] , Ch. IV. We shall now turn to one of the early applications of quadratic expressions, namely, the majorization of maximal functions in order to establish results of pointwise convergence. The idea is roughly as follows: We need to obtain Lp estimates for the maximal operator sup ITkf(x) I k and we take an appropriate Schwartz function t rivi a1 major i zation T*f(x)

=

$(x)

to make the

V.5. LITTLEWOOD-PALEY THEORY

T*f(x) 5 sup /Tkf(x) k 5

(i ITkf(x)

-

I

@ -k*f(x)

2

- @ -k*f(x) I 2)1/2 2

+

I@

sup k

+

51 3

2

-k*f(x)

I

CMf(x)

where M denotes the Hardy-Littlewood maximal operator. Thus, matters are reduced to estimate the Lp norm of 2 1/2 Qf = (1 lTkf - @ - k x f ( I k 2 When p = 2 , this is easily made by means of the Fourier transform, and the method dates back to Kolmogorov [ l ] , who proved the a.e. convergence of lacunary partial sums of Fourier series for L2 periodic functions. Now, a basic feature of Littlewood-Paley theory is that it allows to extend orthogonality arguments (which are simple in L2) to Lp, 1 < p < m . Not surprisingly, therefore, it was up to Littlewood and Paley to extend Kolmogorov's result to Lp functions, 1 < p < m , as we shall now see. THEOREM 5.14. a)

Let

m 6 Lm(Rn)

be a f u n c t i o n o f c l a s s m(0)

n e i g h b o u r h o o d of t h e o r i g i n , w i t h f o r some -

E

L2(Rn),

> 0.

=

1

C1

& Im(c) I 5

a

CIcl-'

define the multiplier transfor-

we

mat i o n s

(Tkf) ( 5 ) A

=

Then, t h e maxima2 o p e r a t z 2 n 1,

(R )

^f ( 5 ) m(

2-

T*f(x)

=

kc)

(k 6 2 )

s u p ITkf(x)

@

k

I

i s bounded i n

2 n lim Tkf(x) = f(x) a.e. ( f 6 L (R 1 ) k-tm b) Trike m = X I i n p a r t ( a ) , w h e r e I i s b o u n d e d o p e n i n t e r -~ v a l i n Rn c o n t a i n i n g t h e o r i g i n . T h e n , Tkf = S f, and t h e m a x i m a l o p e r a t o r S*f(x) = sup I S f(x) I i s i n ' t h i s c a s e b o u n d e d k 2 1 i n Lp(Rn) f o r a 2 1 1 < p < m. A s a c o n s e c u e n c i a , LIB h a v e __ (5.15)

(5.16)

1

lim f(c)e 2nix*Sdg = f(x) k-tm 2 k I

c) I n t h e c a s e

A

n

= 1,

(5.16)

I{x : S*f(X) > All 5

a.e.

holds f o r every

c x-l

If$

(f6

u

Lp(Rn)) 1
f 6 H1(R),

and

(f 6 H 1 ( R ) I

Proof: We shall limit ourselves to prove the maximal inequalities.

v.

514

VECTOR VALUED INEQUALITIES

The c o r r e s p o n d i n g p o i n t w i s e c o n v e r g e n c e r e s u l t s a r e i m m e d i a t e l y d e r i v e d from them, s i n c e l i m T k f ( x ) = f ( x ) t o t h e f a c t t h a t m(0) = 1 ) . a ) Take

$ € S(Rn)

such t h a t

G(0)

=

for all 1.

f € S(Rn) (due

Then

Im(S) 5 CIS1 a n d Im(S) - ? ( 5 ) 1 5 C I S 1 - E , a n d b o t h i n e q u a l i t i e s t o g e t h e r i m p l y ( a s i n t h e p r o o f o f Theorem 5 . 3 ) t h a t

I f we d e f i n e that

Qf

and t h i s p r o v e s :

6

as above, i t f o l l o w s from P l a n c h e r e l ' s theorem

lT*f12 5 Const. l f I 2 .

b ) Take $ E S ( R n ) s u c h t h a t $ ( O ) = 0 a n d $ ( E ) i n a neighbourhood o f a I . Then, ( 1 - $ ) x I 6 C:(Rn),

e x i s t s @ € S(Rn) s u c h t h a t $ = ( 1 - $ ) x I . Qf a s a b o v e , a n d o b s e r v e t h a t

With t h i s

=

1 for all and t h e r e

9,

we define

S k f - @ * f = S (+ * f ) 2 1 2-k 2% 2-k T h e n , t h e same a r g u m e n t o f Theorem 5 . 8 Theorem 5 . 3 ) p r o v e s t h a t t o be proved.

(i.e.,

l Q f l p 5 Cp l f l p ,

c ) I t s u f f i c e s t o o b s e r v e t h a t , when

i n e q u a l i t y (2.14) p l u s

w h i c h was t h e o n l y p o i n t

n = 1,

the operator

d e f i n e d i n p a r t ( b ) maps H'(R) b o u n d e d l y i n t o weak-L1(R) p r o o f i s a g a i n a s i n Theorem 5 . 8 ) .

Q

(the

We h a v e s t a t e d p a r t ( b ) b e c a u s e o f t h e s i m p l i c i t y o f i t s p r o o f . A much d e e p e r r e s u l t i s t r u e , h o w e v e r : I f then (5.17) n = 1,

1 < p <

m,

I

?(S)e2nix*Sdg = f ( x ) a.e. tI t h i s i s t h e t h e o r e m o f C a r l e s o n [3] a n d Hunt [ l ] lim

t-tm

For

f € Lp(Rn),

.

The

r e s u l t f o r Rn c a n b e e a s i l y r e d u c e d t o t h e o n e - d i m e n s i o n a l c a s e ( s e e C . F e f f e r m a n [3]). We m u s t m e n t i o n , h o w e v e r , t h a t p a r t ( c ) i s s h a r p i n two s e n s e s : F i r s t o f a l l , ( 5 . 1 7 ) , d o e s n o t h o l d f o r e v e r y f E H1(R) ( o n e c a n a c t u a l l y h a v e l i m s u p I S t I f ( x ) I = +m a . e . for such

f);

t

+

s e c o n d l y , one c a n n o t r e p l a c e

m

H1(R)

by

L1(R)

in the

V.5. LITTLEWOOD-PALEY THEORY

51 5

statement of 5.14.(c). Proofs of both facts, in the periodic setting, will be found in Zygmund [l], Ch. VIII. A n application of Theorem 5.14(a)

denotes the unit ball in

with gives:

R",

COROLLARY 5.18. For e v e r y

m(5)

=

XB([)

where

B

f E L2(Rn)

The corresponding result for Lp(Rn) is false if p < 2 and n > 1. This is a consequence of the negative result of C. Fefferman [ Z ] for the ball multiplier together with some general principles to be studied in the next chapter (see VI.2.8(c)). Concerning L2 functions, it is not known (and this is a beautiful open question) wheter dyadic partial sums can be replaced by arbitrary partial sums or not, i.e.: Does (5.17) hold for every f 6 L 2 (Rn) after replacing I by the unit ball B?. Another application of Theorem 5.14(a) is the followi.ng: Remember the kernels introduced in Example 4.9(b) Qa(x)

=

2

max(1 - \ x 1 2 , 0la-l

which we shall consider now defined €or each complex number a with Re(a) > 0. By the usual formula for computing Fourier transforms of radial functions (see Stein-Weiss [l], Ch. IV) we have

*,

where, for each 6 > - 1 Jg(t) denotes the corresponding Bessel function. The definition of ma makes sense whenever Re(a) > and for every such a we introduce the maximal operator N"f(x)

=

sup

kEZ

/j~([)ma(2-k~)e2nix.FdEl

(f E

S(Rn))

which coincides with the one defined in 4.9(b) when a 2 0 (for a = 0, we identified 0' with Lebesgue measure in the unit sphere, 0 , and it turns out that ;(<) = mO(c)]. We can now state. The maximal o p e r a t o r ( d e f i n e d a p r i o r i i n COROLLARY 5.19. -

Nf(x)

=

sup k62

jly' \=I

f(x-Z-ky') do(y')

I

S(Rn))

516

V. VECTOR VALUED INEQUALITIES

Lp(Rn)

is b o u n d e d in

Proof: In 4.9(b)

Q

n 2 2

1 < p <

m.

we have obtained

""flp

lfl,

5 ca,p

(1 < p <

On the other hand, since t-'JB(t) is a Cm (whose value at t = 0 is Z-B/r(l+B)) and

m;

Re(a)

> 0)

function in F O , m ) JB(t) 5 Cgt' 1 2

(t m ) , the multiplier ma satisfies (except f o r a multiplicative constant depending on a ) the hypothesis of Theorem 5.14(a), and we -f

get lNUf12 5

ca

(f E S(Rn),

If12

Re(a)

>

'1 - n

We shall omit the technical de ails needed to conclude the proof, but the reader who is familiar with the interpolation of analytic families of operators (see Ste n-We s s [ l ] , Ch. V) should have no difficulty in interpolating both estimates for the operators N " , after a suitable linearization, to obtain lNaflp

I cayp

for ID 1 - TI 1 < 7 7 Re(a) and what we wanted to prove. 0

lfl,

(f E S(R"))

1 - n < Re(a) 5 0.

+

The case

a = 0

is

To finish this section, we shall describe some results corresponding to continuous type square functions. We begin by the so called Littlewood-Paley g-function, which was first studied by complex methods, in the periodic setting and for n = 1 , as a previous step to the inequalities of Theorem 5.8 (This approach is described in Zygmund [l] , Ch. XIV). F o r a function f(x) in Rn, we define

(1

m

g(f)(x)

=

tlVu(xyt)12dt)"2

0

where u(x,t) we have

=

Pt*f(x)

denotes the Poisson integral of

f. Then,

COROLLARY 5.20. Let 1 < p < m. T h e n f € Lp(Rn) if a n d o n Z y if g(f) E Lp(Rn), a n d t h e r e exists c o n s t a n t s c C > 0 such t h a t P'

Proof: Let us first l o o k at what happens f o r

P

p = 2:

V . 5 . LITTLEWOOD-PALEY THEORY

51 7

T h u s , we a r e i n c o n d i t i o n s o f u s i n g t h e o b s e r v a t i o n 5 . 6 ( b ) , a n d i t w i l l be enough t o p r o v e

I g ( f ) l p 5 Cp b e made s e p a r a t e l y f o r t h e o p e r a t o r s

Take

a

@(x) =

Pt(x)

I t=T.

1 < p <

-.

This w i l l

A s i m p l e c o m p u t a t i o n shows t h a t

I4(x)I 5 C(1 IV@(X)I 5

ifi,,

c

+

1x1

IxI(1

+

2 -(n+1)/2

1

1x121-(n+3)'2

The f i r s t i n e q u a l i t y shows t h a t ( 5 . 4 ) h o l d s , w h i l e t h e s e c o n d e a s i l y implies (5.5).

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

+(c)

=

2 1 ~ 1 5 1 e - ~ ' 1 ~ 1 s, o t h a t

a n d w e c a n a p p l y Theorem 5 . 3 t o t h e o p e r a t o r G(0) = 0 , b y means o f 4 , w h i c h c o i n c i d e s w i t h t h e o p e r a t o r g o ,

ga

u(x,t) =

(3 a Pt)*f(x)

=

A

defined because

t - l a t.* f ( x )

The i n e q u a l i t i e s f o r g k ( f ) , 1 < k < n, f o l l o w i n t h e same way b y a n a p p l i c a t i o n o f Theorem 5 . 3 t o t h e f u n c t i o n ~ ( x )= a P, ( x ) . 0 axk The n e x t a p p l i c a t i o n i s c o n c e r n e d w i t h a n o p e r a t o r i n t r o d u c e d by Marcinkiewicz. For a f u n c t i o n F i n R , we d e f i n e u(F) (x) =

(1

t - 3 l F ( x + t ) + F ( x - t ) - 2 F ( x )1 2 d t ) " 2

m

0

The f u n c t i o n

p(F)

i s c l o s e l y connected with t h e d i f f e r e n t i a b i l i t y

p r o p e r t i e s o f F(x) : A t a l m o s t e v e r y p o i n t x a t which F ' ( x ) e x i s t s , one h a s p ( F ) ( x ) < ( s e e Zygmund [ l ] ) a n d t h e c o n v e r s e i s t r u e i f t h e d e r i v a t i v e i s t a k e n t o e x i s t i n t h e L 2 s e n s e . The ~0

r e s u l t t o be proved h e r e is: COROLLARY 5 . 2 1 . G i v e n integral of

f , i.e.

f E: L 1l o c ( R ) , F(x) =

/:

f.

u e d e n o t e b2

F

the indefinite

Then, t h e i n e q u a l i t i e s

V. VECTOR VALUED INEQUALITIES

518

Proof: Define (It*f(x)

I$ =

t

=

X [-1 , o ] -

-'1

t

X[O,l]

Then

*

[f(x+y)-f(x-y)]dy

=

t-l [F(x+t)+F(x-t)

-ZF(x)]

0

Thus, with this choice of 4 , p(F) equals the function nf o f Theorem 5.3. It is clear that I$ satisfies (5.4) and ( 5 . 5 ) , and ; ( O ) = 0, so that: lp(F)pp 5 Cp lfl,, 1 < p < m. On the other hand :

I

m

I$(tC)l2

0

$

=

Im 1 1

1,-2

0

- cos 2ntlzt-3

=

c

<

w

(5

+

0)

and this proves the reverse inequality by a new appeal to 5.6(b).

0

Finally, we shall mention the continuous analogue of the dyadic maximal operators considered in Theorem 5.14. Given a multiplier m E Lm(Rn), we define the maximal operator (5.22)

sup

R>o

(A iR

(Tsf(x) (2ds)1'2

0

where (TSf)^(C;) = ?(()m(sC). As we did for the dyadic case, this maximal operator is trivially majorized by m

c Eff(x)

+

(J 0

where

@

E S(Rn),

ITsf(x) - aS*f(x)

'ds) - 1/2 S

and we can state:

COROLLARY 5 . 2 3 . _The r e s u l t s of Theorem 5 . 1 4 r e m a i n t o hoZd a f t e r T*f by t h e maximal o p e r a t o r d e f i n e d b y (5.w.

repZacing

6. VECTOR VALUED INEQUALITIES AND A_ WEIGHTS r

A new application o f weighted norm inequalities will be given now:

They can be used to obtain inequalities like (1.8) for a (not necessarily linear) operator. This fact was actually at the very source o f the A theory, since the inequality (see Chapter 1 1 , P 2.12)

V . 6 . VECTOR INEQUALITIES AND An WEIGHTS

519

which i s t h e c l o s e s t a n t e c e d e n t o f

A w e i g h t s , was p r o v e d b y P F e f f e r m a n a n d S t e i n a s a t o o l t o o b t a i n t h e r e s u l t we s t a t e d a s Corollary 4.3. This approach t o v e c t o r valued i n e q u a l i t i e s i s not t h e o n e we h a v e f o l l o w e d h e r e , b u t i t i s n e v e r t h e l e s s i n t e r e s t i n g and w e a r e g o i n g t o d e s c r i b e i t .

THEOREM 6 . 1 . Let p > 0

and

s 1. 1

(Tj)

be f i x e d , and l e t

s e q u e n c e of s u b l i n e a r o p e r a t o r s i n Lp(Rn) such t h a t , t o each u c L;(R") we c a n a s s o c i a t e u c L:(R") with 1 ~ 1 , 5 I u I s and satisfying

(6.2) f o r a21

l l T j f ( x ) I p u ( x ) d x 5 Cp I I f ( x ) J p U(x)dx

f € Lp(U)

E d every

j.

(Tj)

Then, t h e o p e r a t o r s

Lq(Rn), where q = p s ' , following vector valued i n e q u a l i t y holds u n i f o r m l y bounded i n

and m o r e o v e r ,

the

P r o o f : We s h a l l v e r i f y t h e s e c o n d a s s e r t i o n , w h i c h c o n s i s t s i n a v e r y s i m p l e a p p l i c a t i o n o f H B l d e r ' s i n e q u a l i t y . Given a s e q u e n c e (fj)

in

Lq(Rn),

there exists

u(x)

2

0

such t h a t

1.1,

= 1

and

I n t h e c a s e o f 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

(T. = M f o r 1 j) we c a n a p p l y t h e t h e o r e m w i t h a r b i t r a r y p > 1 a n d s > 1 by c h o o s n g U(x) = cs Mu(x), w h e r e c s d e n o t e s t h e i n v e r s e o f t h e norm o f M i n L s ( R n ) . T h u s , we o b t a i n all

w h i c h i s t h e more d i f f i c u l t h a l f o f t h e s t r o n g t y p e i n e q u a l i t i e s c o n t a i n e d i n 4 . 3 . More g e n e r a l l y , we c a n s t a t e t h e f o l l o w i n g THEOREM 6 . 4 . Let

( T j ) be a s e q u e n c e of Z i n e a r i z a b Z e o p e r a t o r s ( i n t h e s e n s e of 1 . 2 0 ) and s u p p o s e t h a t , for some f i x e d r > 1, these o p e r a t o r s a r e u n i f o r m l y b o u n d e d i n Lr(w) for e v e r y w e i g h t

V. VECTOR VALUED INEQUALITIES

520

w 6 Ar,

3 JITjf(x)Irw(x)dx

with C,(w)

5 c,(w)

(w E Ar)

!lf(x)/'w(x)dx

d e p e n d i n g onZy on t h e A,

c o n s t a n t of 1 < p,

v e c t o r v a l u e d i n e q u a l i t 2 ( 6 . 3 ) h o l d s f o r alZ

c = cP991 .

w(x). q <

Then, t h e

(with

oJ

Proof: It suffices to prove (6.3) for p = r and 1 < q < m , since the extrapolation theorem for A weights proved in Chapter IV P shows that the hypothesis of the theorem are actually verified for every r > 1. If r < q and s = (q/r)I, we take, for instance, s+l o = 7 ,s o that 1 < u s, and observe that, given u E Ls(Rn), the function w(x) = {M(u5) (x) }l/u satisfies: i) u(x) 5 w(x)

ii) ,1.1 5 cs IUI, iii) w 6 A I C A, with

A1

constant depending only on

Therefore, the previous theorem can be applied with and the proof of this case is complete. Now, we consider the case are linearizable, i.e.

=

Ci'w(x),

We recall that the operators T. J

r > q. ITjf(x)l

U(x)

5.

=

lujf(X)IB

for some linear operators U which are uniformly bounded from j Lr(w) t o Li(w) whenever w E Ar. Here B is a Banach space which, for notational simplicity, will be assumed to be the same for all j. Given a sequence (fj) E Lq(lr), there exists a sequence (gj) E E Lq'(tr') with unit norm and such that

=

sup

1

1

kgj(x).U.f.(x), J J

Gj(x)>

dx m

where the supremum is taken over all G . 6 LB* with IGj(x)IB* 5 1 J a.e. (see 1 . 3 ) . Now, if U! denotes the adjoint of U (which will 1 j map B*-valued functions into complex valued functions):

V.6. VECTOR INEQUALITIES AND A

P

WEIGHTS

521

defines a and matters are reduced to proving that the sequence ( U ! ) 1 bounded operator from Lq' (Lf;:) into Lq' (lr'). But, since r ' < q', this in turn will be a consequence of the part already proved (which works equally well for vector valued functions) once the following weighted norm inequalities are obtained: IIU;F(x)l"w(x)dx

5 Crl(w) IlF(x)lii

w(x)dx

(W 6

Art)

(with Crl(w) depending only on the A r l constant of w). Given -r/r w E A r l , we denote v = w E Ar (observe that the A, constant of v is equal to the A r , constant of w raised to the power r F), and take an arbitrary F € Li:(w). There is some g 6 Lr(Rn) with lgl, = 1 such that (jlUiF(x) Ir'w(x)dx) '/" =

<

kuj (iI I f;:

(wl'r'g) (x), F(x)>

F (x)

w (x) dx) 'Ir '

IU;F(x)

=

w(x)l/''g(x)dx

v(x)l"w(x)l/rldx

(1I

=

<

T (v-llrg) (x) I rv(x) dx) ' I r

and the last factor is bounded by theorem.

n

Cr(v)

by the hypothesis of the

As an application of this theorem, we obtain again the vector valued inequalities

IxI-~

where [ c j l m5 C, Ikj(x)l 5 C which were proved in section 3.

and

lvkj(x)l

5 Clxl - n -1 ,

Remarks 6.5. We mention here some extensions of Theorem 6.4: a) The conclusion of the theorem can be strenghtened as follows: If 1 c p,q < m and w E A then the weighted vector valued ine9' quality

V . VECTOR VALUED INEQUALITIES

522

ho ds. This can be easily proved by using the relationsh p be tween weights of different A classes, as in the proof o f the extrapolaP tion theorem. A direct proof of (6.6) for the Hardy-Littlewood maximal operator and f o r singular integrals has been given by Andersen and John [ l ]

.

b) The class Ar can be replaced in the statement of the (This relaxes the hypothesis, since theorem by the class .:A In this case, the weighted vector valued inequalities of A:c A,). the previous remark still hold, with weights w E A" 9'

A final comment about Theorem 6.1 is in order. What this theorem (which can be stated for an arbitrary measure space (X,m) instead of (Rn,dx)) shows is that vector valued inequalities for a family of operators can be obtained if one has enough information about the weights associated to these operators. In 6.4, this process was efectively carried out by using our information about A weights. P What is perhaps more surprising, is that the process can be reversed, and some information about the weighted norm inequalities which an operator verifies can be obtained if one knows that certain vector valued inequalities hold (and the results in sections 1 , 2 and 3 can be used to this end). Bringing to light the general principles on which this reversed process is based is the principal aim of the next chapter.

7 . NOTES AND FURTHER RESULTS

The only linear operators which admit B-valued extensions for every Banach space B are essentially the positive operators. To be precise: Let T : Lp + Lq be a linear operator having a bounded 1 1 -valued extension; then, there exists a bounded linear operator T+ : Lp + Lq which is positive and verifies: lTf(x) 1 5 T+(lfJ) (x) for every f € Lp. The same happens if t 1 is replaced by 1". See Virot [ I ] . 7.1.-

Theorem 2 . 7 was first proved by Paley [l] in the case q = p and with a bigger constant C The version presented here P,P' follows Marcinkiewicz and Zygmund [ l ] rather closely. F o r a general operator of weak o r strong type ( p , q ) , the values of r for which 7.2.-

V.7. NOTES AND FURTHER RESULTS

523

a bounded Lr-valued extensions exists are described in Rubio de Francia and Torrea [ l ] . An interesting extension of theorem 2.7 which is based on Grothedieck's fundamental theorem is the following: Given Banach lattices A , B and a bounded linear operator T : A -+ B, then, for arbitrary fl,f2,..., fn € A:

where KG is the so called Grothendieckls universal constant, whose exact value is still unknown (however, 1 < KG < 2). See Krivine [l]. 7.3.- The following result is of interest in relation with the negative result of C. Fefferman described in 2.12(b) : Take in R' a (for instance, lacunary sequence of directions u . € E l J Then, the L 2-valued u. = exp(2-jni)). inequality .7

3

2) 1 /2

I,

5

cp I ' f

2 1/2 lfjl 1 1,

holds. A related result is that the maximal operator associated to the family of all rectangles having one of their sides parallel to some u. is bounded in Lp(R2), 1 < p m. For a geometric proof 3 of this fact when p > 2, see Stromberg [ l ] and CBrdoba and R. Fefferman [2]. The general case, p > 1 , was proved by Nagel, Stein and Wainger [ l ] . 7.4. The more usual form of the Marcinkiewicz integral corresponding to the closed set F is

x

J (XI =

j s(YP

Ix-Yl

-n-X dY

(x E F)

This is the natural extensions to Rn of Marcinkiewicz' original definition for the case n = 1 , which he successfully applied to several problems of convergence and summability of Fourier series. However, JA(x) is infinite for all x E F , and the modified definition, H A , has the advantage o f being defined over all R", while HX(x) = JX(x) for every x 6 F. On the other hand, the Marcinkiewicz integral S , corresponding to a family of intervals in R has been used by Carleson [3] in proving the pointwise convergence of Fourier series. We refer to the expository paper by Zygmund [2] for more details o n this.

V . VECTOR VALUED I N E Q U A L I T I E S

524

The i n t e g r a l s

Jx and

HA

have been extended t o t h e p a r a b o l i c

s e t t i n g ( d e s c r i b e d i n 1 1 . 7 ) b y M . W a l i a s [ l ] and C. Calderdn [2] . The r e s u l t s o b t a i n e d a r e e x a c t l y a s i n 4 . 1 0 , w i t h n r e p l a c e d by t h e

homogeneous d i m e n s i o n . 7 . 5 . - The s t a t e m e n t s ( i ) a n d ( i i ) o f Theorem 4 . 2 a r e d u e t o Zo [ l ] ,

w h i l e 4 . 3 was p r o v e d by C . F e f f e r m a n a n d S t e i n [ l ] b y t h e m e t h o d d e s c r i b e d i n s e c t i o n 6 . The u n i f i e d a n d s i m p l e r a p p r o a c h f o l l o w e d h e r e o r i g i n a t e s i n Rubio d e F r a n c i a , R u i z a n d T o r r e a [ l ] . Given a Banach s p a c e

B,

i f the Hilbert transform has a

bounded B-valued e x t e n s i o n

HB

t o L i ( R ) f o r some q , t h e n HB 1 < p < a, a n d o f weak t y p e ( 1 , l ) .

7.6.-

i s a l s o bounded i n a l l ,L{(R), T h i s f o l l o w s f r o m Theorem 3 . 4 . The good s p a c e s

f o r which t h i s

B

h a p p e n s a r e c h a r a c t e r i z e d by a c o n d i t i o n c a l l e d c - c o n v e x i t y : B

c - e o n v e x if t h e r e e x i s t s

is

<

: BxB + R

symmetric, convex

i n e a c h v a r i a b l e and s a t i s f y i n g :

<(o,o)

'

See Burkholder [ Z ]

c(x,y) 5 Ix+YIB

0;

5 IlYlB.

lxlB 5

when

f o r t h e s u f f i c i e n c y and Bourgain [l] f o r t h e

necessity. The R i e s z t r a n s f o r m c a n b e d e f i n e d i n 1 n < - c o n v e x , a n d t h e n , t h e s p a c e HB(R ) coincides with

1 HB

=

1

I f 6 LB

such t h a t

LE,

1,

p

when

B

d e f i n e d i n terms o f

B 1 R . f € LB, J

j=1,2,

is B-atoms

. . . ,n l

A l s o , i f B i s a <-convex Banach l a t t i c e of s e q u e n c e s , t h e n e v e r y 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 h a s a bounded B - v a l u e d e x t e n s i o n t o L E ( R n ) , 1 < p < m, and t h e f o l l o w i n g e x t e n s i o n o f t h e Fefferman-Stein i n e q u a l i t i e s holds: jI(Mfj(x)IjsrlBP

(1 < p <

m).

d x 5 C P ll(fj(x))jsNIi

S e e B o u r g a i n [Z]

dx

.

7 . 7 . - The o p e r a t o r N = N o d e f i n e d i n 4 . 9 ( b ) a n d 5 . 1 9 i s t h e d y a d i c v e r s i o n o f S t e i n ' s maximal s p h e r i c a l mean:

Mf(x)

=

sup

I

f(x-ry

' 1 do ( y ' 1 I

(f E S(Rn))

V.7. NOTES AND FURTFIER RESULTS

525

For Re(") > 1 - n are defines also Ma, whose dyadic version N" was considered in 5.19 and (when a > 0) in 4.9(b). Stein's theorem is the following:

and this is best possible when n 2 3 (see Stein [ Z ] ) , while the results for R 2 remain unknown. The proof of this theorem is not far avay from the techniques developped in section 5, and we can sketch it: One applies analytic interpolation to the two estimates: a) lM"flp Ic",p

lfl,

(Re(")

= 1;

1 < p < ")

When Re(") = 1, I Oa(x) I = C,XB(x), and M a is essentially the Hardy-Littlewood maximal operator. For the second estimate, one first proves the following inequality which is based on the properties of Bessel functions sup (E 1 5 Ca-B R>o

A4"f(x) A

:1

ITEf(x)[2ds)"2

(a-B > -) 1

2

* B

(where (T'f) = fm ; see the discussion previous to 5.19), and observes that the maximal operator to the right is bounded in 1-n L2(Rn) if B > -z (by Corollary 5 . 2 3 ) . We refer to Stein and Wainger [l] for details. The fact that the dyadic operator N is bounded in all the Lp spaces (i.e., 5.19) was observed by C. CalderBn [ l ] , and also by R. Coifman and G. Weiss. 7.8.- Let Snf f 6 L'(T).

of

denote the n-th partial sum of the Fourier series If 1 < p,q < and (1 lfj]q)l'q 6 Lp, then j

This is a vector valued version of the Carleson-Hunt theorem which follows very easily from Theorem 6.4 and the weighted norm inequalities for S*f = sup lSnf] described in Chapter IV, (see Rubio de n Francia [ l ] ) . 7.9. Vector valued inequalities can be used to obtain mixed norm estimates, i.e., estimates involving the Benedek and Panzone

526

V. VECTOR VALUED INEQUALITIES

Lq2p-norms defined in 1.6(b): "Let S be a linear operator in Rn+m = RnxRm which is translation invariant and satisfies

for some fixed p , q 2 1. Then S is a bounded operator in L ~ ~ P ( R " X R ~AS ) ~ a~ .corollary, every classical singular integral o p e rator in Rn is bounded in Lp1'p2'""pn(Rn) for arbitrary p1,p2,pn > 1, and every multiplier operator in LP(Rn+m) is bounded in L2,P(RnxRm). Details and generalizations can be found in Benedek, Calder6n and Panzone [ l ] , Herz and Rivi6re [ l ] , Rubio de Here we can sketch a proof in the compact Francia and Torrea [l]. case ( T " + ~ instead of Rn+m): First of all, it is not difficult to replace lq by Lq(Tn) in the vector valued inequality, so that:

Then, given g 6 L q y p , we define fu(x,y) = g(u+x,y), and the result is proved because Lebesgue measure and the operator S are translation invariant. 7.10.- A generalization of theorem 2.16 has been recently found by J.L. Rubio de Francia (unpublished). For an arbitrary sequence { I . ) 1 of disjoint intervals in R, the inequality

holds. The p r o o f is a combination of the ideas in 2.16 and 3.8.