�9 58 " L'u Shanzhen etai: Weighted Weak Type Estimates

W E I G H T E D WEAK TYPE E S T I M A T E S I N V O L V I N G S P A C E S GENERATED BY BLOCKS*

Lu Shanzhen

( Bei/ing Normal University)

and G u i d o Weiss

( Washington University in st. Louis)

Received Jan. 5, 1993

Abstract

L 1 The problem studied in lhi.t paper LT how to establish weighted estimates o f weak type near for those

operators that are not o f weak type (1,1).

w 1. Introduction

In 1957 E. M. Stein ob t a i ned the fol lowing result[51:

Suppose K ( x , y ) is a kernel in IR n x IR ~ sat isfying

]K(x,Y)I <~ C / Ix - YI"

when x r y a n d

(Tf)(x) = p .v . f R" K ( x , y ) f ( y ) d y

(1.1)

defines a b o u n d e d o p e r a t o r on LP(IR n), ! < P < oo. T h e n T is also a b o u n d e d o p e r a t o r

on the weighted space L e ( I R ~ , [ x l ' d x ) whenever -- n < ~ < n(p -- 1).

The a b o v e res t r ic t ion on ~ is precisely the one tha t g u a r a n t e e s tha t the power weight

w ( x ) -- Ix]" be long to the M u c k e n h o u p t class A e (see [1] for the def in i t ions and p rope r t i e s

of these classes). F. Soria and the s e c o n d a u t h o r of this p a p e r have recent ly ex tended this

result o f Stein in several d i rec t ions [4]:

(i) T h e p o w e r weights in Stein/s result can be replaced by m o r e general weights w ( x )

tha t , in add i t ion to be long ing to the class A e ' satisfy

* The first author is supported by the NSFC (Tian Yuan) and the second is supported by NSF grant DMS-900749 I.

Approx. Ttworg & its Appl , I0:I. Mar. 1994 �9 59 �9

sup w(x) <~ C I inf w ( x ) ( i .2 ) 21 - I ( i z l ~ 2 k + t 2 l - I t; i x l < 2 * + l

where C I is i ndependen t o f k e Z .

.(ii) Stein/s result is ex tended to the case p = 1. M o r e precisely, Soria and Weiss show

that if w satisfies (1.2) and belongs to the M u c k e n h o u p t class A ~, then, whenever T is o f

weak t y p e ( L ~ 1 ,L ), then it is of weak type (L ~ w 'L w)"

(iii) Var iants to the size condi t ion (1.1) were i n t r o d u c e d in [4] tha t were a p p r o p r i a t e

for several appl icat ions. One such condi t ion is

I(T/~x) I < C I/(y) I r~ dy (1.3)

for some P0 > O.

There are, however , m a n y opera tors that are no t o f weak type (L ~ ,L I ), yet t hey d o sat-

isfy o ther condi t ions tha t can be considered as subst i tutes for the boundedness co n d i t i o n ,

type (1,1). Examples are furnished by the maximal B o c h n e r - R i e s z means at the crit ical in-

dex and the mult ipl ier t r ans fo rm in t roduced by Meyer , Ta ib leson and Weiss in [3]. Th u s , it

is o f interest to search for substi tute condi t ions satisfied by such opera to r s tha t lead to

weighted est imates tha t are ana logous to the results ob t a ined by Stein, Sofia and Weiss.

The purpose of this p a p e r is to do this by employing certain spaces genera ted by b locks (see

[2] or [7]) as subst i tutes fo r weak type spaces. Mo reo v e r , some o f ou r results are valid fo r

the ent ire M u c k e n h o u p t class in question; tha t is, they need no t be restricted by cond i t i on

( 1 . 2 ) .

w 2. S tatements o f our results

We need to i n t roduce a few definit ions and establish some no ta t ions in o rde r to s ta te

our theorems. I f w(x) is a non -nega t i ve , locally in tegrable funct ion on R ~ we use the

same symbol w to d e n o t e the measure , it induces on the Lebesgue measurab le sets, E , in

by lett ing w(E) -- f w ( x ) d x . We write [El when w ( x ) = i . Th e space L ~ consis ts o f R n ,

E w

all those measurab le func t ions f on R ~ such that

{S }"" II/llL, " -- I/(x) I'~w(x)dx < oo.

When ! < q < m we say tha t a funct ion b ~ L q is a (q,w)--block if it is suppor t ed in a ball w

B and satisfies the inequal i ty

Ilbll L' ~< [w(B) ]( i / , )-~ (2.1)

�9 60 �9 Lu Shanzhen etal: Weigh:ed Weak Type Estimates

The weighted Block Space B q,l*, consists of all functions f o f the form

f-- ~ , . ,b , . k a Z

(2.2)

where each b k is a (q ,w) -b lock and the numerical sequence m = {m k } satisfies

N ( m ) ~ ~ l m k l { l + l ~ Imkl} < ,,~

A topology is in t roduced on the space B by letting the " no rm " of f be defined by put- q,W

t ing II/]l j , , . - inf{N(m): f--- ~ k m k b # } . Wherr w~-- i the space Bq., is the space Bq intro-

duced by Taibleson and Weiss in [7]. Finally, let Lq'= deno te the " weak q - t y p e " Loren tz w

space associated with the measure w (see [4] for the def ini t ions and properties o f these

spaces ). We are now able to state the results tha t we establish in this paper. We begin with

our principal theorem:

Theorem. Suppose a sublinear operator T satisfies inequality (1.3) for some P o e(0,1]

and it maps L : ( R ~) boundedly into L : ' | ~) for a f ixed w e A I and q(~(1,oo),

�9 | ,o~ then T maps B ( R ") into L (R "); moreover,

q , w

w( {x: I(T/) (x) I> A} ) < CIIJql a.. / ~l, (2.3)

where C is independent o f f and ;~ > O.

Corollary. Suppose w r A i satisfies condition (1.2). I ra sublinear operator T satisfies

(!.3) for some p0~(0,1] and maps L q (R ") boundedly into L q : ~ ~) for q~(1,oc.), then T

],o0 n maps B . , ( I R ' ) into L , ( R ) andinequality (2.3) holds.

w 3. Proof s o f the results

In order to establish the above theorem it suffices to show tha t if b is a (q,w)--block

then

C w( {x: I (Tb) (x) I> ;~ > 0} ) < T ' (3.1)

where C is a cons tan t tha t i~ independent of b(x) and ,1.. This is a consequence o f the St~in

- N. Weiss L e m m a (see L e m m a (1.3) in [2] for the complete a rgumen t ) that shows bow

such uni form w e a k - t y p e est imates can be added when one forms linear combina t ions as in

(2.2) satisfying the finite en t ropy condit ion N(m) < oo. Let us then suppose that b(x) is a

(q ,w)-b lock suppor ted on a ball B such that

~< CwfB) ]o/~)-I llb II L'. (3.2)

Approx. Theory & its Appi., I0:1, Mar. 1994 �9 61 �9

Let x b be the center and r ( B ) t h e radius

showing (3.1) into three cases.

Case 1. 2 > l / w ( B ) .

Since T maps L q boundedly into L q'| , w w

of B. We break up the a r g u m e n t for

w( {x: I ( rb) (x) l > 2} ) g C a - q J " I b ( x ) I ' w ( x ) d x <<. �9 B

ca - ' [w(n) 11-, .. ca - ' [aw(n) 11-, < Ca- i

Case 2. 2 <<. I / w(n) and Ix - xbl > 2r(B).

Using condi t ion (1.3) and the fact tha t Po < q, we have

[(Tb)(x),<~ C/{f Ib__.~_)['-- dy} i/'* <~ C n / , o

s Ix - y l " Ix - x b I

C 1/q

Ix -x . l

{f alb(y)l'* dy} 1/,. <~

L e t w ( B ) ---- ess. inf .w(x). Using (2.1) and the fact tha t w~A X E 8

if and only if !

w(n) In--i- "< C_wfn), (3.3)

we see tha t the last expression does not exceed

Ci , /p * {w---~} 1/q IBI(~/#, I-c~/q) - - II b II L' I x - x b ~

C IBI 1/,* fae(n)/Inll i/o C IBI 1/~ .:,0 w(n---Y-'- ~t~7~ "< I':" w(n) "< Ix-x~l ix--xb

(we are using the c o m m o n convent ion tha t the cons tan t C varies f rom place to place; nev-

ertheless, it is clear f rom the context tha t it is independent o f the " essential " variables).

This shows

r(B) }'/'* 1 I(Tb)(x)l <~ C Ix - Xbl w(n)" (3.4)

It follows, therefore, tha t

w ( { x : Ix --Xbl > 2r(B) and I(Tb)(x)l > a}) ~<

w({x: ,x- %, < {aw~ } "/'r(n)}). (3.5)

�9 62 �9 Lu Shanzhen etal: Weighted Weak Type Estimates

W e can suppose C > 1 and, since we are assuming tha t 2w(B)<~ 1, we -*n assert that

2w(B---~ > 1. Thus, B ~ ' B ----{x:lx--xb]< r(B)}----~B; where aB denotes

the ball with same center as B and radius equal to ct -- r(B). W e also observe

that weA = A ; thus (see L e m m a (2.2) in Chapte r IV o f [1]), 1 1 / Pe

( I B I / I B " I) ~/'~ ~ C w ( B ) / w ( B ' ) ,

which, together w'.'th (3.5), gives us

w({x: Ix --Xsl > 2r(B) and I(Tb)(x)l > 2}) ~< w(B ~ )<~

C{ IB � 9 }1/,, w(B)-- C~t "/p. w(B)--- C / 2.

Case 3.

In this case

2 <~ 1 / w(B) and IX--Xbl <~ 2r(B).

w({x: Ix --xbl <~ 2r(B) and IT(b)(x)l > 2}) ~<

w((x: Ix -- xbl ~< 2r(B)}) = w(2B) <~ Cw(B) <~ C / 2,

the second to last inequal i ty being an immediate consequence o f L e m m a (2.2) in Chap te r

r r o f [11.

Put t ing together the inequalit ies ob ta ined in Cases 2. and 3., we see that w({x: IT(b)(x)l > 2}) is the sum o f w({x: Ix -- Xb[ > 2r(B) and IT(b)(x)l > 2}) and w({x: Ix - xs l ~< 2r(B)

and IT(b)(x)[ > ;t}) when 2 ~< 1 / w(B). Since we have shown that each o f these s u m m a n d s

does no t exceed C / 2 when 2 is so restricted, Case 1. comple tes the es t imate (3'.1) tha t we

wan ted to establish and the theorem is proved.

Let us now turn our a t ten t ion to the p roof of the Corol lary . It is clear from the theo-

rem we just establ ished that , under the assumptions we are mak ing in the s ta tement o f the

Corol lary , it suffices to show that T maps L q ( R ") into L"| "). In order to do this we

use a techniqu~ in t roduced in [4] by letting

~--- 2 1 " + 1 I k {x: 2 *-1 ~< Ixl < 2 k} and I~ {x: 2 k-2 ~< Ixl < }

and, then, put t ing fk,0 = fX,~ 'fk.l -- f - - fk.0 so that

I(T/)(x)l ~< ~ I(Tfk,0)(x)lzt, + ~ I(Z/'k.t)(x)lz,, = (Zo] ) (x ) + (T,])(x). k t Z k f Z

Let m k = inf{w(x): xe l~ }. Then , making use o f ( ! .2),

Approx. Theory & its Appl., I0:!, Mar. 1994 �9 63 �9

w({x: I(Toj0(x)l > 2}) = ~ W({X~Ik: I(Tfho)(X)l > 2}) = J

I t , s t

{ x ~ l j. : I(T[,,.e Xx)! > 2} I ~'~z "

c c ,.,,Era" I/k.0(x)l d x / f f <~ C C , ';

C I l ~ x ) l q w ( x ) d x / 2 q .

Final ly , we n o w es t imate the o p e r a t o r T i and begin with the o b s e r v a t i o n tha t , if

x ~ 1 k and y ~ ( I k )c , then Ix -- YI" "" Ixl" + lYI ' . Us ing (1.3), we then h a v e

w({x: I(Ta/)(x)l > / l } ) = ~ W({X~Ik : I(Tfk,a)(X)l > 2}) <~ k ~ Z

~ w ( { x e l j : I I/(r)l'* dr>Ca'*})~< Ewf(~I,:t.(I/l'~

where the o p e r a t o r L is defined by

(rg)(x)-- f -gfY)- dr. dlxl" + lYI"

It is shown in [4] tha t L is a b o u n d e d o p e r a t o r o n each of the spaces L P ( I R ' ) , 1 < p < oo,

whenever v ~ A . I f we apply this result when v ---- w e A i c A p and p = q / P0 > 1 we see P

tha t

w({x: I(f~J~(x)l > ~.}) < ~ w ( { x ~ I k : t ( I / l ' * )(x) > C: . '* }) k G Z

= w({x: L(I/]'* )(x) > CA p* }) ~< C~ll(x)lCw(x)dx / 2 r .

References

[1] Garcia--Cuerva, I. and Rubio de Francia, J. L., Weighted Norm Inequalities and Related Topica,

North l-iollland Math. Studies, I 16, Elsevier Science Publishers, Amsterdam (1985) pp. 1-604.

[2] Lu, S. Z., Taibleson, M. H. and Weiss, Guido, Spaces Generated by Blocks, Math. Series, Beijing

Normal Univ. Press, Beijing.(1989).

[3] Meyer, Y, Taibleson, M. H. and Weiss, G,aido, Some Functional Analytic Properties of the spaces Bq

Generated by Blocks, Ind. U. Math J., 34 (1985) pp. 493-515.

[4] Soria, F. and Weiss, Guido, A Remark on Singular Integrals and Power Weights, Preprint.

[5] Stein, E. M., Note on Singular lntegxals, Proc. A. M. S. 8(1957), pp.250-4.

[6] Stein, E. M. and Weiss, Guido, Introduction to Fourier Analysis on Euclidean Spaces, Princeton

�9 " 64 �9 Lu Shanzhen etal: Weighted Weak Type Estimates

[7]

Univ. Press, Princeton (197 I), pp. 1-294.

Taibleson, M. H. and Weiss, Guido, Certain Spaces Associated with a. e. Convergence of Fourier Se-

ries, Vol. l, Univ. of Chicago Conf. in Honor of A. Zygmund, Wardsworth Publ. (1983) pp.95-113.

Department of Mathemat i c s

Beijing N o r m a l Univers i ty

Beijing, ! 00875

P.R. china

and

D e p a r t m e n t of Ma themat i c s

W a s h i n g t o n Univers i ty in St. Louis

M O 63130

U.S.A.