- Research Article
- Open Access

# 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

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 .

## Keywords

- Positive Constant
- Linear Operator
- Harmonic Analysis
- High Dimension
- Lebesgue Measure

## 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]).

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.

*.*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.

Case 1 ( ).

Case 2 ( ).

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

Case 1 ( ).

Case 2 ( ).

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

Then, Theorem 1.3 is proved.

## 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

## References

- Stein EM: On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz.
*Transactions of the American Mathematical Society*1958, 88: 430–466. 10.1090/S0002-9947-1958-0112932-2MathSciNetView ArticleMATHGoogle Scholar - Ding Y, Fan D, Pan Y: Weighted boundedness for a class of rough Marcinkiewicz integrals.
*Indiana University Mathematics Journal*1999, 48(3):1037–1055.MathSciNetView ArticleMATHGoogle Scholar - Chang S-YA, Wilson JM, Wolff TH: Some weighted norm inequalities concerning the Schrödinger operators.
*Commentarii Mathematici Helvetici*1985, 60(2):217–246.MathSciNetView ArticleMATHGoogle Scholar - Kenig CE:
*Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Regional Conference Series in Mathematics*.*Volume 83*. American Mathematical Society, Washington, DC, USA; 1994:xii+146.View ArticleGoogle Scholar - Xue Q, Ding Y: Weighted boundedness for parametrized Littlewood-Paley operators.
*Taiwanese Journal of Mathematics*2007, 11(4):1143–1165.MathSciNetMATHGoogle Scholar - Sakamoto M, Yabuta K: Boundedness of Marcinkiewicz functions.
*Studia Mathematica*1999, 135(2):103–142.MathSciNetMATHGoogle Scholar - Torchinsky A, Wang SL: A note on the Marcinkiewicz integral.
*Colloquium Mathematicum*1990, 60–61(1):235–243.MathSciNetMATHGoogle Scholar - Stein EM: The development of square functions in the work of A. Zygmund.
*Bulletin of the American Mathematical Society*1982, 7(2):359–376. 10.1090/S0273-0979-1982-15040-6View ArticleMathSciNetMATHGoogle Scholar - Ding Y, Lu S, Yabuta K: On commutators of Marcinkiewicz integrals with rough kernel.
*Journal of Mathematical Analysis and Applications*2002, 275(1):60–68. 10.1016/S0022-247X(02)00230-5MathSciNetView ArticleMATHGoogle Scholar - Ding Y, Xue Q: Endpoint estimates for commutators of a class of Littlewood-Paley operators.
*Hokkaido Mathematical Journal*2007, 36(2):245–282.MathSciNetView ArticleMATHGoogle Scholar - Coifman R, Rochberg R, Weiss G: Factorization theorems for Hardy spaces in several variables.
*The Annals of Mathematics*1976, 103(3):611–635. 10.2307/1970954MathSciNetView ArticleMATHGoogle Scholar - Chanillo S: A note on commutators.
*Indiana University Mathematics Journal*1982, 31(1):7–16. 10.1512/iumj.1982.31.31002MathSciNetView ArticleMATHGoogle Scholar - Mizuhara T: Commutators of singular integral operators on Morrey spaces with general growth functions. Sūrikaisekikenkyūsho Kōkyūroku 1999, (1102):49–63. Proceedings of the Coference on Harmonic Analysis and Nonlinear Partial Differential Equations, Kyoto, Japan, 1998Google Scholar
- Komori Y, Mizuhara T: Factorization of functions in and generalized Morrey spaces.
*Mathematische Nachrichten*2006, 279(5–6):619–624. 10.1002/mana.200310381MathSciNetView ArticleMATHGoogle Scholar - Stein EM:
*Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, no. 30*. Princeton University Press, Princeton, NJ, USA; 1970:xiv+290.Google Scholar - Uchiyama A: On the compactness of operators of Hankel type.
*Tôhoku Mathematical Journal*1978, 30(1):163–171.MathSciNetView ArticleMATHGoogle Scholar - Ding Y: A note on end properties of Marcinkiewicz integral.
*Journal of the Korean Mathematical Society*2005, 42(5):1087–1100.MathSciNetView ArticleMATHGoogle Scholar

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