The maximal commutators on weighted Morrey spaces

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(School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China)

Abstract： Let M be the maximal function and [b,M](f)(x)=b(x)Mf(x)-M(bf)(x) be the commutator of maximal functions. Let Cb be the maximal commutator. In this paper, we study the estimates for the commutator of maximal functions [b,M] and the maximal commutators Cb on weighted Morrey spaces on spaces of homogeneous type. The lower bound of the maximal commutator Cb is also obtained.

Key words： commutators; maximal function; weighted Morrey spaces

0 Introduction

In their remarkable result[1], Coifman—Rochberg—Weiss showed that the commutator of Riesz transforms is bounded onLp(Rn) if and only if the symbolbis in the BMO space. See also the subsequent result in Refs[2-7]. In Ref.[8], Bastero— Milman—Ruiz characterized the class of functions for which the commutator with the Hardy—Littlewood maximal function and the maximal sharp function are bounded onLp(Rn). Recently, in Ref.[9] Agcayazi, et al also studied the unweighted version of the maximal commutatorCb(f) onRnby using a different approach, and this was extended to a space of homogeneous type by Fu, et al in Ref.[10].

In this paper, we aim to provide a quantitative estimate for the commutator of maximal functions [b,M] and the maximal commutatorCbon weighted Morrey spaces on spaces of homogeneous type. To be more precise, let (X,d,μ) be a space of homogeneous type. The Hardy-Littlewood maximal functionMf(x) onXis defined as

where the supremum is taken over all ballsB⊂X. The commutator of maximal functions [b,M] is defined by [b,M](f)(x)=b(x)Mf(x)-M(bf)(x). The maximal commutatorCbonXwith the symbolb(x) is defined by

|f(y)|dμ(y)，

where

The main result of this paper is as follows.

Throughout the paper, the letter “C” will denote (possibly different) constants that are independent of the essential variables.

1 Definitions and preliminary results

Letμbe a measure onXand letdbe a metric onX. Then we call topological spaceXto be a space of homogeneous type if it satisfies the doubling property, that is, there exists a constantC≥1, such that for all ballsB(x,r)={y∈X:d(y,x)

μ(B(x,2r))≤Cμ(B(x,r))<∞.

For the definition of homogeneous type space, one can see Ref.[11], Chapter 3.

Using the doubling property, we can obtain that there existC,n>0 such that

μ(B(x,λr))≤Cλnμ(B(x,r))

holds for allλ>1. The parameternis a measure of the dimension of the space.

Letwbe a nonnegative locally integrable function onX. For 1

Here the suprema are taken over all ballsB⊂X. The quantity [w]Apis called theApconstant ofw. Next we note that forw∈Apthe measurew(x)dμ(x) is a doubling measure onX.

2 The proof of the main results

In order to prove Theorem 1, we need the following lemma.

ProofofTheorem1

This implies thatb∈BMO(X), and

The proof of Theorem 1 is complete.

In order to prove Theorem 2, we need the following lemma.

|[b,M]f(x)|≤CCb(f)(x).

ProofofTheorem2

Letp∈(1,∞),κ∈(0,1) andw∈Ap(X). From Lemma 2 and Theorem 1, we have