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

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## Abstract

Let Open image in new window , Open image in new window , and Open image in new window denote the Marcinkiewicz integral, the parameterized area integral, and the parameterized Littlewood-Paley Open image in new window function, respectively. In this paper, the authors give a characterization of BMO space by the boundedness of the commutators of Open image in new window , Open image in new window , and Open image in new window on the generalized Morrey space Open image in new window .

## Keywords

Positive Constant Linear Operator Harmonic Analysis High Dimension Lebesgue Measure## 1. Introduction

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

We refer to see [1, 2] for the properties of Open image in new window .

respectively. Open image in new window and Open image in new window play very important roles in harmonic analysis and PDE (e.g., see [3, 4, 5, 6, 7, 8]).

and Open image in new window .

There are some results about the boundedness of the commutators formed by BMO functions with Open image in new window , Open image in new window , and Open image in new window (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 Open image in new window , Open image in new window , and Open image in new window on the generalized Morrey space Open image in new window .

Definition 1.1.

*.*Suppose that Open image in new window be such that Open image in new window is nonincreasing and Open image in new window is nondecreasing. The generalized Morrey space Open image in new window is defined by

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

The main result in this paper is as follows.

Theorem 1.2.

If Open image in new window is bounded on Open image in new window for some Open image in new window , then Open image in new window

Theorem 1.3.

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

Theorem 1.4.

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

Remark 1.5.

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

Remark 1.6.

It is easy to see that the condition (1.15) is weaker than Open image in new window for Open image in new window . 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 Open image in new window is a bounded operator on Open image in new window , we are going to prove that Open image in new window

Now, we consider the Open image in new window norm of Open image in new window in the following two cases.

Case 1 ( Open image in new window ).

Case 2 ( Open image in new window ).

Now, the estimate of Open image in new window is divided into two cases, namely, 1: Open image in new window ; 2: Open image in new window .

Case 1 ( Open image in new window ).

Case 2 ( Open image in new window ).

Then, Open image in new window Theorem 1.2 is proved.

## 3. Proof of Theorem 1.3

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

Suppose that Open image in new window is a bounded operator on Open image in new window , we are going to prove that Open image in new window

Then, Open image in new window Theorem 1.3 is proved.

## Notes

### 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).

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