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 [813].

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 .
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.
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