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

• Yanping Chen
• Yong Ding
• Xinxia Wang
Open Access
Research Article

## 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
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 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, 4, 5, 6, 7, 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
(1.10)
Let . It is said that if
(1.11)
where denotes the ball in centered at and with radius ,
(1.12)

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
(1.13)
where
(1.14)

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
(1.15)

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
(2.10)
is a closed set. We claim that
(2.11)
In fact, since note that we can get Taking , let
(2.12)
For , we have
(2.13)
For noting that if , then for . Thus, we have
(2.14)
Using (2.11), we get Noting that it follows from (2.5), (2.7), (2.8), and Hölder's inequality that
(2.15)
For , by , (2.4), (2.5), (2.6), the Minkowski inequality, and Lemma 2.1, we obtain
(2.16)
Let
(2.17)
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
(2.18)
Thus,
(2.19)
Now, we claim that
(2.20)
where is independent of . In fact,
(2.21)

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

Case 1 ().

Since is nondecreasing in , then
(2.22)
Thus,
(2.23)

Case 2 ().

Since is nonincreasing in , then
(2.24)
Thus,
(2.25)
Now, (2.20) is established. Then, by (2.19) and (2.20), we get
(2.26)
If then Theorem 1.2 is proved. If then
(2.27)
Let . For , we have
(2.28)
Noting that if and , we get . Applying (2.11), we have . Since when and , it follows that
(2.29)
By ,   when and and the Minkowski inequality, we have
(2.30)
Thus, by (2.28), (2.29), and (2.30), we get, for ,
(2.31)
Similar to the proof of (2.20), we can easily get . Thus, by (2.31), , and when , we have
(2.32)
We first estimate Since for we have
(2.33)

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
(2.34)

Case 2 ().

Since the function is decreasing for and for , by (2.27), we have
(2.35)
From Cases 1 and 2, we know that there exists a constant such that
(2.36)
So by (2.32), (2.33), and (2.36), we get
(2.37)

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
(3.10)
From (3.9) and (3.10), we get
(3.11)
Let
(3.12)
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
(3.13)
Thus,
(3.14)
Then, by (2.20) and (3.14), we get
(3.15)
If then Theorem 1.3 is proved. If then
(3.16)
Let . For , we have
(3.17)
For as above mentioned, we have Since and , it follows the Hölder inequality that
(3.18)
By , the Minkowski inequality, and for and , we get
(3.19)
Thus, by (3.17), (3.18), and (3.19), we get, for ,
(3.20)
Thus, by (3.20), , when and the Hölder inequality, we have
(3.21)
As the proof of (2.33) and (2.36), we can get that there exists a constant such that
(3.22)
So, by (3.21) and (3.22), we get
(3.23)

Then, 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).

## References

1. 1.
Stein EM: On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz. Transactions of the American Mathematical Society 1958, 88: 430–466. 10.1090/S0002-9947-1958-0112932-2
2. 2.
Ding Y, Fan D, Pan Y: Weighted boundedness for a class of rough Marcinkiewicz integrals. Indiana University Mathematics Journal 1999, 48(3):1037–1055.
3. 3.
Chang S-YA, Wilson JM, Wolff TH: Some weighted norm inequalities concerning the Schrödinger operators. Commentarii Mathematici Helvetici 1985, 60(2):217–246.
4. 4.
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.
5. 5.
Xue Q, Ding Y: Weighted boundedness for parametrized Littlewood-Paley operators. Taiwanese Journal of Mathematics 2007, 11(4):1143–1165.
6. 6.
Sakamoto M, Yabuta K: Boundedness of Marcinkiewicz functions. Studia Mathematica 1999, 135(2):103–142.
7. 7.
Torchinsky A, Wang SL: A note on the Marcinkiewicz integral. Colloquium Mathematicum 1990, 60–61(1):235–243.
8. 8.
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/S0273-0979-1982-15040-6
9. 9.
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)00230-5
10. 10.
Ding Y, Xue Q: Endpoint estimates for commutators of a class of Littlewood-Paley operators. Hokkaido Mathematical Journal 2007, 36(2):245–282.
11. 11.
Coifman R, Rochberg R, Weiss G: Factorization theorems for Hardy spaces in several variables. The Annals of Mathematics 1976, 103(3):611–635. 10.2307/1970954
12. 12.
Chanillo S: A note on commutators. Indiana University Mathematics Journal 1982, 31(1):7–16. 10.1512/iumj.1982.31.31002
13. 13.
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
14. 14.
Komori Y, Mizuhara T: Factorization of functions in and generalized Morrey spaces. Mathematische Nachrichten 2006, 279(5–6):619–624. 10.1002/mana.200310381
15. 15.
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
16. 16.
Uchiyama A: On the compactness of operators of Hankel type. Tôhoku Mathematical Journal 1978, 30(1):163–171.
17. 17.
Ding Y: A note on end properties of Marcinkiewicz integral. Journal of the Korean Mathematical Society 2005, 42(5):1087–1100.

© Yanping Chen et al. 2010

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## 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