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Commutators of Littlewood-Paley Operators on the Generalized Morrey Space

  • Yanping ChenEmail author
  • Yong Ding
  • Xinxia Wang
Open Access
Research Article

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

(a) Open image in new window is the homogeneous function of degree zero on Open image in new window , that is,
In 1958, Stein [1] defined the Marcinkiewicz integral of higher dimension Open image in new window as

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

Let Open image in new window and Open image in new window The parameterized area integral Open image in new window and the parameterized Littlewood-Paley Open image in new window function Open image in new window are defined by

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]).

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

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.

Let Open image in new window . 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.

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 (1.8), Open image in new window satisfies (1.1), (1.2), and

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.

If Open image in new window satisfies conditions (1.1), (1.2), and (1.15), let Open image in new window then for Open image in new window , we have

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

We may assume that Open image in new window . We want to prove that, for any Open image in new window and Open image in new window , the inequality
where Open image in new window is the measure on Open image in new window which is induced from the Lebesgue measure on Open image in new window . By the condition (1.15), it is easy to see that
is a closed set. We claim that
Using (2.11), we get Open image in new window Noting that Open image in new window it follows from (2.5), (2.7), (2.8), and Hölder's inequality that
For Open image in new window , by Open image in new window , (2.4), (2.5), (2.6), the Minkowski inequality, and Lemma 2.1, we obtain
Without loss of generality, we may assume that Open image in new window , otherwise, we get the desired result. Since Open image in new window is nonincreasing, it follows that Open image in new window . By (2.13), (2.15), and (2.16), we have
Now, we claim that

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, (2.20) is established. Then, by (2.19) and (2.20), we get
If Open image in new window then Theorem 1.2 is proved. If Open image in new window then
Thus, by (2.28), (2.29), and (2.30), we get, for Open image in new window ,
Similar to the proof of (2.20), we can easily get Open image in new window . Thus, by (2.31), Open image in new window , and Open image in new window when Open image in new window , we have

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 ).

From Cases 1 and 2, we know that there exists a constant Open image in new window such that
So by (2.32), (2.33), and (2.36), we get

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

We may assume that Open image in new window . We want to prove that, for any Open image in new window and Open image in new window , the inequality
where Open image in new window is the measure on Open image in new window which is induced from the Lebesgue measure on Open image in new window . 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:
Then by (3.4), we get Open image in new window . Since Open image in new window and Open image in new window we get Open image in new window and Open image in new window . Thus, by (2.5), (2.7), (2.8), and the Hölder inequality, we get
By (2.5) and (2.6), we have
From (3.9) and (3.10), we get
Without loss of generality, we may assume that Open image in new window , otherwise, we get the desired result. Since Open image in new window is nonincreasing, we have Open image in new window . Then by, (3.6), (3.8), and (3.11), we get
Then, by (2.20) and (3.14), we get
If Open image in new window then Theorem 1.3 is proved. If Open image in new window then
For Open image in new window as above mentioned, we have Open image in new window Since Open image in new window and Open image in new window , it follows the Hölder inequality that
Thus, by (3.17), (3.18), and (3.19), we get, for Open image in new window ,
Thus, by (3.20), Open image in new window , Open image in new window when Open image in new window and the Hölder inequality, we have
As the proof of (2.33) and (2.36), we can get that there exists a constant Open image in new window such that
So, by (3.21) and (3.22), we get

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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© Yanping Chen et al. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of Mathematics and Mechanics, Applied Science SchoolUniversity of Science and Technology BeijingBeijingChina
  2. 2.Laboratory of Mathematics and Complex Systems (BNU), School of Mathematical SciencesBeijing Normal University, Ministry of EducationBeijingChina
  3. 3.The College of Mathematics and System ScienceXinjiang UniversityUrumqiChina

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