Boundedness for some Schrödinger Type Operators on Weighted Morrey Spaces

Abstract

We establish the boundedness of some Schrödinger type operators on weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.

1. Introduction

In this paper, we consider the Schrödinger differential operator where is a nonnegative potential belonging to the reverse Hölder class for .

A nonnegative locally integrable function on is said to belong to if there exists such that the reverse Hölder inequality holds for every ball in ; see [1].

For , the function is defined by Let ,   and . For and , we say (weighted Morrey spaces related to the potential ) provided that where denotes a ball with centered at and radius , and the weight functions (see Section 2). The space could be viewed as an extension of weighted Lebesgue spaces (i.e., when , ). In particular, when or ,   and , the space is the classic Morrey space (see [2]). When or and , was first introduced in [3] (see also [4]), where (Muckenhoupt weights class). It is easy to see that for , and for . In addition, when , the has been studied in [5].

From [1, 6], we know some Schrödinger type operators, such as with , with , with , with and , and with is a nonnegative polynomial, are standard Calderón-Zygmund operators; see [7]. In particular, the kernels of operators above all satisfy for any . Hence, in the rest of this paper, we always assume that denotes the above operators.

Recently, Bongioanni et al. [8] proved boundedness for commutators of Riesz transforms associated with Schrödinger operator with functions which include the classic BMO function, and they [9] established the weighted boundedness for Riesz transforms, fractional integrals, and Littlewood-Paley functions associated with Schrödinger operator with weight class which includes the Muckenhoupt weight class. Very recently, the author [10, 11] established the weighted norm inequalities for some Schrödinger type operators, which include commutators of Riesz transforms, fractional integrals, and Littlewood-Paley functions with functions; see also [12, 13].

The aim of this paper is to study the boundedness properties of some Schrödinger type operators on the weighted Morrey spaces . Our main results in this paper are formulated as follows.

Theorem 1. Suppose and . (i)If and , then where is independent of .(ii)If and , then for any , holds for all balls , where is independent of , , , and .

Let (see its definition in Section 2); we define the commutator of by

Theorem 2. Suppose , and . (i)If and , then where is independent of .(ii)If and , then, for any , holds for all balls , where is independent of , , , and .

Next, we discuss the Littlewood-Paley function related to Schrödinger operators defined by and the commutator of with is defined by

Theorem 3. Suppose , , and . (i)If and , then where is independent of .(ii)If and , then, for any , holds for all balls , where is independent of , , , and .

Theorem 4. Suppose , , , and . (i)If and , then where is independent of .(ii)If and , then, for any , holds for all balls , where is independent of , , , and .

Finally, we consider the boundedness of fractional integrals related to Schrödinger operators.

Let with for and its associated semigroup:

The -fractional integral operator is defined by

Theorem 5. Suppose , and . (i)If ,  ,  ,  , and , where , then where is independent of .(ii)If ,  ,  , and , then, for any , holds for all balls , where is independent of , and .

Let ; we define the commutator of by

Theorem 6. Let ,  ,  , and . (i)If  ,  ,  ,  , and , then where is independent of .(ii)If  ,  ,  , and , then, for any , holds for all balls , where and , and is independent of , and .

We remark that even if in the case, Theorems 2, 3, 4, and 6 are also new; see [5].

Throughout this paper, is a positive constant which is independent of the main parameters and not necessary the same at each occurrence.

2. Some Notation and Basic Results

We first recall some notation. Given and , we will write for the -dilate ball, which is the ball with the same center and with radius . Given a Lebesgue measurable set and a weight , will denote the Lebesgue measure of and . will denote for .

Lemma 7 (see [1]). There exists and such that In particular, if .

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

A weight will always mean a nonnegative function which is locally integrable. As in [9], we say that a weight belongs to the class for , if there is a constant such that for all ball We also say that a nonnegative function satisfies the condition if there exists a constant for all balls where Since , obviously, for , where denote the classical Muckenhoupt weights; see [7]. We will see that for in some cases. In fact, let and ; it is easy to check that and is not a doubling measure, but provided that and .

For convenience, we always assume that denotes , , and .

Lemma 8 (see [10]). Let ; then (i)if , then .(ii) if and only if , where .(iii)If for , then there exists a constant such that for any

Lemma 9 (see [12]). Let , . If , then there exists positive constants , and such that for all ball .

As a consequence of Lemma 9, we have the following result.

Corollary 10 (see [12]). Let , . If , then there exist positive constants , and such that for any measurable subset of a ball .

Bongioanni et al. [8] introduce a new space defined by where and and .

In particular, Bongioanni et al. [8] proved the following result for .

Proposition 11. Let and . If , then for all , with and , where and is defined in Lemma 7 and is a constant depending only on .

Obviously, the classical is properly contained in ; for more examples, see [8]. For convenience, we let .

Tang [10] gave the following result, which is equivalent to Proposition 11.

Proposition 12. Suppose that . There exist positive constants and such that for any ball

Applying Corollary 10 and Proposition 11, we can obtain the following result.

Proposition 13. If and   , then there exist positive constants , and such that for every ball and every , one has where , and .

Proof. We adapt the same argument of pages 145-146 in [7]. We first assume . We apply Chebysheff’s inequality and Proposition 11; we obtain for , . From this and by Corollary 10, there exist constants and such that for , .
If , we take . Then where . However, if , then , and in that range of . Altogether then, if we drop the normalization on by replacing by , we can obtain the conclusion by taking and .

From Proposition 13, it is easy to see the following.

Corollary 14. If and , then there exist positive constants and such that for every ball , one has where .

3. Proof of Theorems 1–4

Proof of Theorem 1. Without loss of generality, we may assume that and . Pick any ball and write where for . Hence, we have By the boundedness of    (see [10]), we obtain By Lemma 8 and Corollary 10, as well as (5), there exist some positive constants and such that From (41), (42), and (43) with , we obtain As for the case , the proof can be given by replacing (42) with the corresponding weak estimate.

Proof of Theorem 2. Without loss of generality, we may assume that . Pick any ball , as in the proof of Theorem 1 we write Set for . By the boundedness of   (see [10]), we get Set When , by Lemmas 7 and 9 and Corollaries 10 and 14, there exist some positive constants and such that If we take in (48), then we obtain It remains to consider the case . Set . From [10], we know that for any From this, we have Set When , by Lemma 7, Corollary 10, and Proposition 12, note that , then there exist some positive constants and such that by taking and , where in the sixth inequality, we used the following facts (see [2]); the generalized Hölder inequality Combing (51) and (53), we obtain that holds for all balls , where is independent of , and .
Thus, Theorem 2 is proved.

Finally, we give some sketch proof of Theorems 3 and 4.

Proof of Theorems 3 and 4. Let us denote the set of measurable functions with norm , the set of measurable functions defined on valued in , and the set of Bochner-measurable functions . The space is the set of with finite norm We simply name the space as when .
Thus, we can redefine the as follows: which has associated kernel It is easy to see that for any .
Thus, we can adapt the same argument of Theorems 1 and 2 to prove Theorems 3 and 4; we omit the details here.

4. Proof of Theorems 5 and 6

We first need the following lemma.

Lemma 15 (see [5]). Let be as in (17). For any , there exists a such that for all and .

Proof of Theorem 5. Without loss of generality, we may assume that and . Pick any ball , as the proof of Theorem 1, we write Hence, we have Let . By the boundedness of (see [10]), we get From (61), by Corollary 10, then there exists a constant such that where .
Note that . So
As for the case , the proof is similar.

Proof of Theorem 6. Without loss of generality, we may assume that and . Pick any ball , as in the proof of Theorem 1, we write For . Set . Let . By the boundedness of (see [10]), we get Set When , by (61) and Corollary 10, then there exists some positive constant such that by taking .
Then, It remains to consider the case and . Set . From [10], we know where and .
From this, we have
Set When , by (61) and Corollary 10 and Proposition 12, note that , then there exist some positive constants and such that by taking and .
Hence, By (73) and (76), we obtain the desired result. This completes the proof.

5. The Calderón-Zygmund Inequality

For the open set ,    and , we say if In this section, we consider the behavior of the solution of the following Schrödinger equation: where , ,  ,  , and .

Theorem 16. Let be an open set in and . If , then there exists a function , where , , , , and , such that Furthermore, where , , and ; where , , and ; where , ,  , , and .

Proof. From the proof of Theorem 5, we have From the proof of Theorem 1, we obtain Thus, (80) and (82) hold.
From page 543 in [1], we know where is the fundamental solution for .
Using (85) and adapting the argument for Theorem 5, we then have Thus, (81) holds. Hence, Theorem 16 is proved.

Conflict of Interests

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