Journal of Function Spaces

Volume 2014 (2014), Article ID 901267, 11 pages

http://dx.doi.org/10.1155/2014/901267

Research Article

The Boundedness of Marcinkiewicz Integrals Associated with Schrödinger Operator on Morrey Spaces

Dongxiang Chen and Fangting Jin

Jiangxi Normal University, Nanchang 330022, China

Received 22 April 2014; Accepted 18 July 2014; Published 20 August 2014

Academic Editor: Dachun Yang

Copyright © 2014 Dongxiang Chen and Fangting Jin. 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.

Abstract.

Let be a Schrödinger operator, where belongs to some reverse Hölder class. The authors establish the boundedness of Marcinkiewicz integrals associated with Schrödinger operators and their commutators on Morrey spaces.

1. Introduction

In this paper, we consider the Schrödinger operator

in , , where is a nonnegative potential belonging to the reverse Hölder class for some exponent .

A nonnegative locally integrable function on is said to belong to if there exists such that the reverse Hölder inequality,

holds true for every ball . We introduce the definition of the reverse Hölder index of as . It is known that implies for some .

The Marcinkiewicz integral operator is defined by

The above operator was introduced by Stein in [1] as an extension of the notion of Marcinkiewicz function from one dimension to higher dimensions.

Similar to the classical Marcinkiewicz function , we define the Marcinkiewicz function associated with the Schrödinger operator by

where and is the kernel of , . In particular, when , and is the kernel of , . So, is defined by

Lemma 1 (see [2]). For any , there exists such that

where is the auxiliary function as follows.

Gao and Tang [2] have shown that Marcinkiewicz integral is bounded on for and bounded from to weak . Meanwhile they also proved are bounded on and also mapped from to under the assumption that satisfy the condition in Lemma 1.

When satisfies the estimates in Lemma 8 in Section 2 below, Chen and Zou [3] also proved that the Marcinkiewicz integral has the same boundedness.

Now we give the definition of the commutator generalized by and by

and the definition of the commutator generalized by and by

Let be the auxiliary function defined by

Obviously, if . In particular, with and with .

In this paper, we write where and , and denotes the radius of .

The maximal operator is defined by

When , we denote by (the standard Hardy-Littlewood maximal function). It is easy to see that for a.e . This information can be found in [4].

Proposition 2 (see [5]). There exists such that

In particular, if .

Proposition 3 (see [6]). There exists a sequence of points , , in , so that the family , , satisfies the following.

(i).(ii)For every there exist constants and such that .

Tang and Dong [7] first introduced some type Morrey space associated with Schrödinger operators. Meanwhile they obtained the strong and weak boundedness of singular integral, fractional integral, and their commutators in Morrey spaces. Inspired by their work, we give another type of Morrey space, but in fact these two kinds of Morrey spaces are the same when their exponents are restricted in some exceeding. Recently, plenty of famous results on Schrödinger operators have appeared;we refer to [8–13].

We now present the definition of Morrey spaces associated with Schrödinger operators which we needed in this paper.

Definition 4. Let , ; we say (Morrey spaces) provided that

where .

Obviously, when , the space is the class Lebesgue space .

Definition 5. Let , , , and ; we say (Morrey space associated with Schrödinger operators) provided that

where .

In this note we will investigate the boundedness for the commutators of Marcinkiewicz inegral associated with Schrödinger operators on Morrey spaces given in Definition 5.

Our results can be formulated as follows.

Theorem 6. Suppose and .

(i)If , then

 where is independent of .(ii)If , then, for any ,

 holds for all balls , where is independent of , , , and .

For , we define the class of locally integrable functions such that

for all and , where . A norm for , denoted by , is given by the infimum of the constants in the inequalities above. Notice that if we let , we obtain the John-Nirenberg space .

Theorem 7. Suppose , , .

(i) If , then

 where is independent of .(ii) There exists a constant , such that, for all ,

Throughout this paper, denotes the constants that are independent of the main parameters involved but whose value may differ from line to line. By , we mean that there exists a constant such that .

2. Notation and Preliminaries

Shen [5] gave the following kernel estimate that we need.

Lemma 8. If , then one has

(i)for every there exists a constant such that

(ii) for every and there exists a constant such that

 where ,(iii) for every , one has

 where .

Lemma 9 (see [14]). Let and . If , then

for all , with and , where and is the constant appearing in Proposition 2.

Corollary 10 (see [14]). Let , , and ; then

for all , with as in Lemma 9.

From Lemma 9, the author [15] proved the John-Nirenberg inequality for .

Proposition 11 (see [15]). Suppose that is in . There exist positive constants and such that

where and , , and .

The dyadic maximal operator is defined by

where is a dyadic cube.

The dyadic sharp maximal operator is defined by

where denotes dyadic cubes and .

A variant of dyadic maximal operator and dyadic sharp maximal operator is defined as following:

In our paper, we need the following proposition when .

Proposition 12 (see [15]). Let and suppose that . If , then the equality

Further, let , , if and only if

A function is said to be a Young function if it is continuous, convex, and increasing satisfying and as . We defined the -average of a function over a cube by means of the following Luxemburg norm:

and the maximal operator associated to by . Then the following generalized Hölder inequality holds:

where is the complementary Young function of .

We define the corresponding maximal function

In [16], one has given a general result that can be applied to prove the boundedness of the localized classical operators. One considers a covering of balls such that the family of a fixed dilation of them, , has bounded overlapping (e.g. a covering associated to like in Proposition 3).

An operator is defined by

Proposition 13 (see [16]). Let , and a weight on with the following property: for each , , admits an extension to such that

bound with a constant independent of . Then,

continuously. If , the assumption of is changed by weak type , and the corresponding weak type can be concluded for .

In our paper, set and . As we know, is bounded from to . So, Proposition 13 also holds for and .

From [16], we have the following result.

Lemma 14. Let . Assume that . Then there exists a constant such that

Furthermore, when , there exists a constant such that, for all ,

Proof. Let , with and as in Proposition 2. Let be the family given by Proposition 3 and set .

Clearly, we have

First, for , Minkowski’s inequality says that

where we have used when and .

Therefore, for ,

Hence from Proposition 13 we have done it .

For the case , using the estimate of again, we have, for each ,

Besides, from Proposition 13,

Therefore, summing over we have the weak type .

Lemma 15. Let , and . Set . Let . Then,

holds for any .

Proof. We fix and let (dyadic cube). We consider two cases about ; that is, and .

Case  1. When , decomposing , where , where . Let a constant be fixed along the proof. Since , we then have

where .

We start with . For any , note that for any and ; by Lemma 9, we obtain

where .

For , note that for any and ; by Kolmogorov’s inequality, Proposition 11, and Lemma 14, we have

To deal with , we first fix the value of by taking with ; we have

For , since and , using Minkowski’s inequality and Proposition 11, we obtain

where the integer satisfies .

Similarly,

For , since and , by Minkowski’s inequality and Proposition 11, we have

where the integer is the same as above.

For , since and , using Minkowski’s inequality and Proposition 11, we obtain

where the integer is the same as above, and we know it is finite.

Similarly,

So,

Case  2. When , decomposing , where , where . Since , so and ; then,

where .

To deal with , for any , note that ; by Lemma 9, we then have

where .

For , using Kolmogorov’s inequality, Proposition 11, and Lemma 14, we have

Finally, for , notice that for any ; then .

Hence the proof is finished.

The following information can be founded in [4]. Define the following maximal functions:

and their commutators:

where .

Obviously, we have

where and .

Lemma 16 (see [1]). Let , and , , and . Let ; then,

holds for any .

In the following Lemma, we set .

Lemma 17 (see [4]). Let , , and . Then there exists a constant such that for all

Lemma 18 (see [15]). Let and let be locally integral. Then there exist positive constants and independent of and such that

3. Proof of the Main Theorems

Proof of Theorem 6. (i) Without loss of generality, we may assume that . Pick any and , and write

where and , for . Hence, we have

By the boundedness of , we obtain

By Proposition 2, Lemma 8, and Minkowski’s inequality, we have

Let , and we obtain

(ii) When and noting that   are bounded from to weak , we have

We have finished the proof of Theorem 6.

Proof of Theorem 7. (i) Pick any and , as in Theorem 6; we write

and, by the boundedness of , we obtain

Set .

We write

When , by Lemmas 8 and 9, Proposition 2, Corollary 10, and Minkowski’s inequality, we have

where . Choosing large enough such that , we obtain

(ii) We adapt a similar argument to that of Theorem 1.2 in [5].

Define

We start with . Using Lemma 8 with , we have

Therefore, the estimates for follow from those for by Lemmas 16 and 18.

To deal with we write

where is defined in Lemma 15 and

For , by Lemma 8 with ,

For , using Lemma 8 again, we have

Using Lemmas 15, 16, 17, and 18 and Proposition 12, by adapting an argument in [2], we can obtain the desired result.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their hearty thanks to the referee’s comments. This paper is supported by the National Natural Science Foundation of China (10961015, 11261023, and 11326092) and the Jiangxi Natural Science Foundation of China (20122BAB201011), GJJ12203.

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