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

*Journal of Inequalities and Applications*
**volume 2010**, Article number: 961502 (2010)

## Abstract

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

## 1. Introduction

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

(a) is the homogeneous function of degree zero on , that is,

(b) has mean zero on , that is,

(c), that is,

In 1958, Stein [1] defined the Marcinkiewicz integral of higher dimension as

where

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

Let and The parameterized area integral and the parameterized Littlewood-Paley function are defined by

where and

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

Before stating our result, let us recall some definitions. For the commutator formed by and the Marcinkiewicz integral are defined by

Let and The commutator of and the commutator of are defined, respectively, by

Let . It is said that if

where denotes the ball in centered at and with radius ,

and .

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.

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

where

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.

Assume that is nonincreasing and is nondecreasing. Suppose that is defined as (1.8), satisfies (1.1), (1.2), and

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.

If satisfies conditions (1.1), (1.2), and (1.15), let then for , we have

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

We may assume that . We want to prove that, for any and , the inequality

holds, where Since we may assume that Let

where Since , we can easily get Then, has the following properties:

In this proof for , is a positive constant depending only on , , , and . Since satisfies (1.2), then there exists an such that and

where is the measure on which is induced from the Lebesgue measure on . By the condition (1.15), it is easy to see that

is a closed set. We claim that

In fact, since note that we can get Taking , let

For , we have

For noting that if , then for . Thus, we have

Using (2.11), we get Noting that it follows from (2.5), (2.7), (2.8), and Hölder's inequality that

For , by , (2.4), (2.5), (2.6), the Minkowski inequality, and Lemma 2.1, we obtain

Let

Without loss of generality, we may assume that , otherwise, we get the desired result. Since is nonincreasing, it follows that . By (2.13), (2.15), and (2.16), we have

Thus,

Now, we claim that

where is independent of . In fact,

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

Case 1 ().

Since is nondecreasing in , then

Thus,

Case 2 ().

Since is nonincreasing in , then

Thus,

Now, (2.20) is established. Then, by (2.19) and (2.20), we get

If then Theorem 1.2 is proved. If then

Let . For , we have

Noting that if and , we get . Applying (2.11), we have . Since when and , it follows that

By , when and and the Minkowski inequality, we have

Thus, by (2.28), (2.29), and (2.30), we get, for ,

Similar to the proof of (2.20), we can easily get . Thus, by (2.31), , and when , we have

We first estimate Since for we have

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

Case 1 ().

Since the function is decreasing for and for by (2.27), we get

Case 2 ().

Since the function is decreasing for and for , by (2.27), we have

From Cases 1 and 2, we know that there exists a constant such that

So by (2.32), (2.33), and (2.36), we get

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

We may assume that . We want to prove that, for any and , the inequality

holds, where Since we may assume that Let be as (2.3), then (2.4)–(2.8) hold. In this proof for , is a positive constant depending only on , , , and . Since satisfies (1.2), then there exists a such that and

where is the measure on which is induced from the Lebesgue measure on . By the condition (1.15), it is easy to see that

is a closed set. As the proof of (2.11), we can get the following:

Taking , let

For , we have

For noting that if , , and then we get

Then by (3.4), we get . Since and we get and . Thus, by (2.5), (2.7), (2.8), and the Hölder inequality, we get

By (2.5) and (2.6), we have

In we have and . In we get and It is easy to see that Now, we estimate by , the Minkowski inequality, Lemma 2.1 for , and (2.4), we get

From (3.9) and (3.10), we get

Let

Without loss of generality, we may assume that , otherwise, we get the desired result. Since is nonincreasing, we have . Then by, (3.6), (3.8), and (3.11), we get

Thus,

Then, by (2.20) and (3.14), we get

If then Theorem 1.3 is proved. If then

Let . For , we have

For as above mentioned, we have Since and , it follows the Hölder inequality that

By , the Minkowski inequality, and for and , we get

Thus, by (3.17), (3.18), and (3.19), we get, for ,

Thus, by (3.20), , when and the Hölder inequality, we have

As the proof of (2.33) and (2.36), we can get that there exists a constant such that

So, by (3.21) and (3.22), we get

Then, Theorem 1.3 is proved.

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## 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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Chen, Y., Ding, Y. & Wang, X. Commutators of Littlewood-Paley Operators on the Generalized Morrey Space.
*J Inequal Appl* **2010**, 961502 (2010). https://doi.org/10.1155/2010/961502

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