Weak type inequality for Poisson type integral operators

WEAK TYPE INEQUALITY FOR POISSON TYPE INTEGRAL OPERATORS

Yoo, YOON JAE

ABSTRACT. A condition for a certain maximal operator to be of weak type (P,p) is studied. This operator unifies various maximal operators cited in the literatures.

1. Introduction

Given a functionf in]Rn, we define a functionMf in ]R~+1= {(x, t) : xE ]Rn,t _~ O}

bysetting

Mf(x, t) =^sup{I~I

k

It is well known that this maximal operatorM controls Poisson integral defined by, for x E ]Rn,t _~ 0,

P(f)(x, t) = r f(y)P(x - y, t)dy,

JJR.n

where

is the Poisson kerneL

Received October20, 1997.

1991 Mathematics Subject Classification: 42B25.

Key words and phrases: maximal function, weights, Carleson measure, spaces of homogeneous type.

Given a positive Borel measurev in ]R+.+l, we define a functionNv in]Rn by setting

where the supremum is taken over allcubes Qcontaining x,

Q= ((x,t) ^E R~+l :x ^E Qand 0 ~t ~ sidelength(Q)}.

Also it is well known that

(A) Let N(x) ~ G < 00. For every p with 1 < p < 00, there is a constant Gp such that for every f and everyJL

~ ¹

[1,;+'[Mf(x, t)]Pdv(x,

t)]'

r-

For this, see [6].

(B) For everyp with 1 < p < 00, M is bounded from V(lRn,dx) intoV (]R+.+l,dv)ifand onlyifv satisfies the Carleson condition

v(Q) ~ GIQI

for each cube Q in]Rn. For this, see [1].

(C) For everypwith 1<^p<^00,M is bounded fromV(n;n,v(x )dx) into _V(]R~+l,v) ifthe following condition is satisfied

~~glQfv(Q) ~ cv(x) a.e. x.

For this, see [3].

(D) Recently, Sueiro studied a certain maxiaml operator on spaces of homogeneous type and applied to Poisson-Szego integral on homo- geneous domains.

Inthis paper we generalized some results in (A), (B), (C), and (D) to spaces of homogeneous type with suitable conditions concerning a family of balls.

2. Preliminaries

DEFINITION 2.1. Let X be a topological space. Assume d is a pseudo-distance on X, Le., a nonnegative function defined on X x X satisfying

(1) d(x,x) = 0;d(x,y) > °^{ifx #- y;}

(2) d(x,y) = d(y,x);

(3) d(x,z) ~ K[d(x, y)+d(y,z)], whereK is some fixed constant.

Assume further that

(a) the balls B(x,r) = {y EX: d(x,y) < r} form a basis of open neighborhoods at x E X and that JL is a Borel measure on X such that

(b) 0 <JL(B(x,2r) ~ AJL(B(x,r)) <00,whereA is some fixed constant.

Then we call(X, d,JL) a space of homogeneos type.

REMARK. Properties (3) and (b) will be referred to as the triangle inequality and the doubling property respectively.

Note that the condition (b) is equivalent that for every c >0, there exists a constant Ac^{such that} JL(B(x,cr)) ~AcJL(B(x,r)).

DEFINITION 2.2. Assume for eachx E X we are given a set Ox C

X x [0,00). Let 0 denote the family {Ox: x E X}. For eacht _~ 0 set

O(x,t) =^Oxn(X ^x [t,oo))

and

'R,a(x,t) =^{(Cy,r) E X x [0,00) :O(y,r)(t)nB(x,at)}#- 0},

whereO(y,r)(t) = {z EX: (z, t) EO(y,r)} is the cross-section ofO(y,r) at height t.

DEFINITION 2.3. Assume that we have a family {Ox: x E X}. For f E L~ocCX,dJL) and x E X, t ~0 set

1

1

Mof(x,t) = sup IfldJL.

(y,s)EO(x,t)JL(B(y, s)) B(y,s)

DEFINITION 2.4. Let 1 ::; p < ^00. A pair (v,w) is said to satisfy the conditionCp(n) if there are constantsC = C(K,A,p,a) so that

v(R-a(x,r) ( ) ( )

JL(B(x,r» ::;Cw y , a.e. on B x,r ifp= 1and

v(na(x,r» ( 1 ( w-1/(P-l)dP,)P-l <C,

p,(B(x,r» JL(B(x,r» ^{JB(x,r) -}

ifp>!'

DEFINITION 2.5. An operatorT defined inV(dJL) is said to be of weak type (P,p) ifthere is a constant Cpso that

JL{x: IT(f)(x)1 > A}::; Cp Il~~:

for all A>O.

3. Main Theorems

The following lemma is given in [2]. See. also [8].

LEMMA 3.1. Let E be a bounded subsetofX i.e. E iscontained in some ball. Let r(x) bea positivenumberfor each x E E. Then there is a (finite orinfinite)sequence ofdisjoint balls B(xi,r(xi»' Xi E E, suchthat the balls B(xi,4.Kr(xi» cover E, where K istheconstantin the triangle inequality. Furthermore, every x E E iscontained insome ball B(xi,4.Kr(xi» satisfyingr(x) ::; 2r(xi)'

THEOREM 3.1. Suppose that if(y,rl) E n(x,t) and rl < .r2 then (y, r2) E n(x,t). IfMo beofweak type (P,p) with respect tothe pair (v,w), p~ 1,then (v,w) satisfies the condition Cp(n).

Conversely, if(v,w) satisfies the condition Cp(n), then Mo is of weaktype (P,p) with respect to thepair (v,w).

Proof. Suppose that Mn be of weak type (p,p) with respect to the pair(v, w). Let f ~ 0 without loss of generality. Let (xo ,t) E1?o(x, r).

Then O(xo,t)(r) n B(x,ar) =1= 0. So we can choose y E O(xo,t)(r) n B(x,ar). By the triangle inequality,

(1) B(x,r) C B(y,K(o.+l)r) C B(x, (K²a+Ko.+K²)r).

Thus

fB(y,K(o+!)r) :s: Mn(fxB(y,K(o+!)r))(xo ,t), where

fB = p,(~)

is

Choose A so that 0< A< fB(y,K(o+!)r). Ifwe denote EA = {Mn(fxB(y,K(o+!)r)) >A},

then 1?o(x, r) C EA, and so

v(1?o(x,r)):s:'; ( ^fPwdp,.

JB(y,K(o+!)r) Henceifwe replace f by fXB(X, r), then

(2)

In particualr, if 8 C B(x,r), then by the doubling property ofp, we have

Iff = 1, then (3) can be written as

(4) ( p,(8) )P ¹

p,(B(x, r)) v(1?o(x, r)) :s:^{C w(8),}

wherew(S) = Is^wdp,.

Hp= 1, then

(5) v('Ra(x, r)) <C_1_ f wdp, p,(B(x,r)) - p,(S)Js holds for every measurable set Se B(y,r). Thus

v('Ra(x, r)) . ( ) ( )

(6) p,(B(x, r)) ::; Cess.mfyEB{x,r)W Y ::; Cw Y

a.e. yEB(x,r).

Next letp> 1. Choose / = W-1/(p-l) so that / = /Pw on B(x,r).

Then from (3), we have

( 1 f w-1/{P-l)dP,)Pv('Ra(x, r)) ::; C f w-1/(p-l)dp,.

p,(B(x,r))Js Js

for allmeasurable set Se B(x,r).

Thus

(

1 f ^{-l/{p-l) )P-l}v('Ra(x, r)) (

p,(B(x,r)) Js w dp, p,(B(x,r))::; C K,A,a:,p).

and so by lettingSt ^B(x,r), we obtain (7)

(

1 f ^-l/{p-l) )P-lv('Ra(x,r)) (

p,(B(x,r)) JB{x,r) W dp, p,(B(x,r))::; C K,A,a:,p).

To prove the converse, suppose that (v, w) satisfies the condition Cp(O),p ~ 1. We follow the idea of Sueiro[8]. For each,X>0, define

EA = {(x,t) EX ^x [0,00) :Mo/(x,t) >^,X}

and E~^{= {xEX: sup}_{r>O P,}^(Bt_{x, r}⁾⁾ _JB{x,r)^{f I/Idp, >,X}}

Alsofor each x E E~, if we put

r(x) = sup{r>0: (Bt )) ( IfldJL> A}, JL x, r J^B(x,r)

thenr(x) >0 and

JL(B(X~r(x))) l(x,r(x» IfldJL 2 A.

Assume for a moment that E~ is bounded. Then by the covering lemma, there exists a sequence of balls {B(xi,r(xi))} so that _E~ C

UiB(Xi,4Kri)' Now we want to verify

(8) E>. C U{Ro.(Xi,4Kri/ai) Infact, if(x,t) E E>., then

JL(Bty,r)) l(y,r) IfldJL >A

for some (y,r) E O(x,t). Thus Y E _E~ and t :::; r :::; r(y). By the last part of the Covering lemma, y E B(Xi,4Kri) for some i such that r(y) :::; 2ri. Here we may assume that a < 2K. Consequently, t :::; r :::; r(y) :::; 2ri < (4K/a)ri and so (y,4Kri/a) E O(x,t). But yE B(xi,a(4K/a)ri), so we get

yEO(x,t)(4Kri/a)n B(Xi, a(4K/a)ri), and hence (x,t) E'Ro.(Xi,4Kri/a). Thus (8) holds.

Since 4K/a>1, we have

(9) B(xi,ri) C B(xi,4Kri/a).

Now letp = ^1. Then by the doubling property and (9), we have v(E>.):::; ~V('Ro.(xi,4Kri/a»)

i

:::;~ JL(B(Xi, 4Kri/a))ess.infyEB(Xi,4Kri/o.)w(y)

i

:::; C~ f.L(B(Xi, ri»ess.infyEB(xi,4Krdo.)w(y)

i

Next, letp> 1. By Holder's inequality we have (10)

J-L(B(Xi, ri)) :::; ~J^{(B(Zi,Ti)fdJ-L}

1 (l ) l / P ( l ) (p-l)/p

:::; - fPwdJ-L w-1/(p-l)dJ-L

A ^{B(Zi,Ti) B(Zi,Ti)}

Therefore by (9) and (10), we obtain

v(E_{A) :::;} L ^V(Ra(xi, 4Krifa))

i

:::; [J-L(B(Xi, 4Kri/a))]P [ ( w-1/(P-l)dJ-L]l-p

JB(Zi,4cKTi/a)

:::; [J-L(B(Xi,ri))]P[ ( w-1/(P-l)dJ-L]1-P

J^B(Zi,Ti)

:::;

1

ITE~ is not bounded, :fixa E X and R > o. Now we replace E~

by _E~nB(a,R) to apply the previous argument. Letting R ~ ^00,

then the weak type estimate is obtained as before. Thiscompletes the

~~ 0

REMARK 1. Let M is the original maximal operator given in the introduction. Following Ruiz[4], a measure v on _]R~+l satisfies the condtionC_{p (}w) if, for 1 :::;P< 00,

v(Q)· ( 1

1 )

sup - - - - w-1/(p-l) < 00

Q IQI IQI ^{Q '}

where the supremum is taken overallcubes QC ]Rn and, for p = 1 s~p1QI :::;v(Q) Cw(x), a.e.

With this in mind, we have then the following:

COROLLARY 3.1 (RUIZ[4]). Let1 ~p< 00, then v E Cp(w) ifand onlyifthemaximal operatorM isof weak type (P, p) with respect to the pair (v,w).

THOREM3.2. Assume that 0 satisfies the conditions (1) oftheorem 3.1. If(v,w) satisfies the condition C₁(0), then Mo isbounded from V(X,wdp,) intoV(X x [O,oo),dv).

Proof. It suffices to show that Mo is of strong type (00,00). Sup- pose that w(x) =I 0 for all x and A> IIfIlLoo(wdp.). (There is no loss of generalityindoing so for ifw(x) = ^{0 for somex, thenv(na(y,r))} = ⁰

for any y E X and r > O. This says that v(X x [0,00), so there is nothing to prove.) We have then J{lfl>>'}wdp, = 0 and consequently p,{lfl > O} = O. Hence If(x)1 ~ A, a.e., from which we haveMn ~ A and so IIfIILoo(wdp.). This proves that Mo is of strong type (00,00).0

REMARK 3.1. Ifdv = udp,®d~o, where d~o is the Dirac measure concentrated on t = 0, then the Cp(O)-condition is reduced to the condition obtained by Wenjie[9]. In fact, Cp(O)- condition can be simplifies as

1

1

(1

p,(B(x, r))p ^{B(x,r) B(x,r) - ,}

ifp > 1,and

(Bt )) [ udp, ~^{Cw(y), a.e. ony E B(x, r),}

p, x, r J^B(x,r)

ifp= 1. In fact, our condition is slightly improved than that ofWenjie.

References

[1] L. Carleson,Interpolation by bounded analytic junctions and the corona prob- lem, Ann. Math. 76 (1962), 547-559.

[2] R. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur cer- tain espaces homogenes, Lecture Notes in Math. 242 (1971), Springer-Verlag, Berlin.

[3] C. Feff'erman C. and E. M. Stein, Some maximal inequalities, Amer. J. Math.

93 (1971), 107-115.

[4] F.J. Ruiz, A Unified approach to Carleson measures and Ap weights, Pacific J. Math. 117 (1985).

[5] F.J.Ruiz andJ. L. Torrea,A Unified approach to Carleson measures and Ap

weights Il, Pacific J. Math. 120 (1985).

[6] J.Garcia-Cuerva andJ.L. Rubio de Francia, Weighted Norm Inequalities and Related Topic, Elsevier Science, North-Holland, 1985.

[7] E. M. Stein, Harmonic Analysis: Real variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, N. J., 1993.

[8] J. Sueiro, On ma:timal junctions and Poisson-Szego integrals, '!'rans. Amer.

Math. Soc. 298 (1986),653-669.

[9] P. Wenjie, Weighted Norm Inequaitiesfor Certain Ma:timal Operator with Ap- proach Region, Lecture NotesinMath. 1494 (1988), Springer Verlag, Berlin, 169-175.

Department of Mathematical Education Kyungpook National University

Taegu 702-701, Korea

E-mail: [email protected]