# Commutators of Littlewood-Paley Operators on the Generalized Morrey Space

- Yanping Chen
^{1}Email author, - Yong Ding
^{2}and - Xinxia Wang
^{3}

**2010**:961502

https://doi.org/10.1155/2010/961502

© Yanping Chen et al. 2010

**Received: **6 May 2010

**Accepted: **11 July 2010

**Published: **28 July 2010

## Abstract

## Keywords

## 1. Introduction

Let be the unit sphere in equipped with the Lebesgue measure . Suppose that satisfies the following conditions.

We refer to see [1, 2] for the properties of .

respectively. and play very important roles in harmonic analysis and PDE (e.g., see [3–8]).

There are some results about the boundedness of the commutators formed by BMO functions with , , and (see [7, 9, 10]).

Many important operators gave a characterization of BMO space. In 1976, Coifman et al. [11] gave a characterization of BMO space by the commutator of Riesz transform; in 1982, Chanillo [12] studied the commutator formed by Riesz potential and BMO and gave another characterization of BMO space.

The purpose of this paper is to give a characterization of BMO space by the boundedness of the commutators of , , and on the generalized Morrey space .

Definition 1.1.

*.*Suppose that be such that is nonincreasing and is nondecreasing. The generalized Morrey space is defined by

We refer to see [13, 14] for the known results of the generalized Morrey space for some suitable . Noting that , we get the Lebesque space . For , coincides with the Morrey space .

The main result in this paper is as follows.

Theorem 1.2.

If is bounded on for some , then

Theorem 1.3.

Let and . Assume that is nonincreasing and is nondecreasing. Suppose that is defined as (1.9), satisfies (1.1), (1.2), and (1.15). If is a bounded operator on for some , then

Theorem 1.4.

Let , , and . Assume that is nonincreasing and is nondecreasing. Suppose that is defined as (1.10), satisfies (1.1), (1.2), and (1.15). If is on for some , then

Remark 1.5.

It is easy to check that (see, e.g., the proof of ( ) in [15, page 89]), we therefore give only the proofs of Theorem 1.2 for and Theorem 1.3 for .

Remark 1.6.

It is easy to see that the condition (1.15) is weaker than for . In the proof of Theorems 1.2 and 1.3, we will use some ideas in [16]. However, because Marcinkiewicz integral and the parameterized Littlewood-Paley operators are neither the convolution operator nor the linear operators, hence, we need new ideas and nontrivial estimates in the proof.

## 2. Proof of Theorem 1.2

Let us begin with recalling some known conclusion.

Similar to the proof of [17], we can easily get the following.

Lemma 2.1.

Now let us return to the proof of Theorem 1.2. Suppose that is a bounded operator on , we are going to prove that

Now, we consider the norm of in the following two cases.

Now, the estimate of is divided into two cases, namely, 1: ; 2: .

## 3. Proof of Theorem 1.3

Similar to the proof of Theorem 1.2, we only give the outline.

Suppose that is a bounded operator on , we are going to prove that

## Declarations

### Acknowledgments

The authors wish to express their gratitude to the referee for his/her valuable comments and suggestions. The research was supported by NSF of China (Grant nos.: 10901017, 10931001), SRFDP of China (Grant no.: 20090003110018), and NSF of Zhenjiang (Grant no.: Y7080325).

## Authors’ Affiliations

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