#### Abstract

Let be a Schrödinger operator, where is the Laplacian on and the nonnegative potential belongs to the reverse Hölder class for . The Riesz transform associated with the operator is denoted by and the dual Riesz transform is denoted by . In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class for . Then we will establish the mapping properties of the operator and its adjoint on these new spaces. Furthermore, the weighted strong-type estimate and weighted endpoint estimate for the corresponding commutators and are also obtained. The classes of weights, classes of symbol functions, and weighted Morrey spaces discussed in this paper are larger than , , and corresponding to the classical Riesz transforms ().

#### 1. Introduction

##### 1.1. The Critical Radius Function

Let be a positive integer and let be the -dimensional Euclidean space. A nonnegative locally integrable function on is said to belong to the reverse Hölder class for some exponent , if there exists a positive constant such that the reverse Hölder inequality holds for every ball in . For given with , we introduce the critical radius function which is given bywhere denotes the open ball centered at and with radius . It is well known that for any under our assumption (see [1]). We need the following known result concerning the critical radius function.

Lemma 1 (from [1]). If with , then there exist two constants and such thatfor all . As a straightforward consequence of (3), we have that, for all , the estimateis valid for any with and .

##### 1.2. Schrödinger Operators

On , , we consider the Schrödinger operator where for . The Riesz transform associated with the Schrödinger operator is defined byand the associated dual Riesz transform is defined byBoundedness properties of and its adjoint have been obtained by Shen in [1], where he showed that they are all bounded on for any when with . Actually, and its adjoint are standard Calderón-Zygmund operators in such a situation. The operators and have singular kernels that will be denoted by and , respectively. For such kernels, we have the following key estimates, which can be found in [13].

Lemma 2. Let with . For any positive integer , there exists a positive constant such that

##### 1.3. Weights

A weight will always mean a nonnegative function which is locally integrable on . Given a Lebesgue measurable set and a weight , will denote the Lebesgue measure of and Given and , we will write for the -dilate ball, which is the ball with the same center and with radius . In [4] (see also [2, 3]), Bongioanni, Harboure, and Salinas introduced the following classes of weights that are given in terms of critical radius function (2). Following the terminology of [4], for given , we define where is the set of all weights such that holds for every ball with and , where is the dual exponent of such that . For we define where is the set of all weights such that holds for every ball in . For , let us introduce the maximal operator that is given in terms of critical radius function (2). Observe that a weight belongs to the class if and only if there exists a positive number such that , where the constant is independent of . Since for , then, for given with , one has where denotes the classical Muckenhoupt class (see [5, Chapter 7]), and hence . In addition, for some fixed , whenever . Now, we present an important property of the classes of weights in with , which was given by Bongioanni et al. in [4, Lemma 5].

Lemma 3 (from [4]). If with and , then there exist positive constants , , and such thatfor every ball in .

As a direct consequence of Lemma 3, we have the following result.

Lemma 4. If with and , then there exist two positive numbers and such thatfor any measurable subset of a ball , where is a constant which does not depend on and .

For any given ball with and , suppose that ; then by Hölder’s inequality with exponent and (18), we can deduce that This gives (19) with .

Given a weight on , as usual, the weighted Lebesgue space for is defined to be the set of all functions such that We also denote by the weighted weak Lebesgue space consisting of all measurable functions for which

Recently, Bongioanni et al. [4] obtained weighted strong-type and weak-type estimates for the operators and defined in (6) and (7). Their results can be summarized as follows.

Theorem 5 (from [4]). Let and . If with , then the operators and are all bounded on .

Theorem 6 (from [4]). Let and . If with , then the operators and are all bounded from into .

##### 1.4. The Space

We denote by either or . For a locally integrable function on (usually called the symbol), we will also consider the commutator operatorRecently, Bongioanni et al. [3] introduced a new space defined by where for the space is defined to be the set of all locally integrable functions satisfyingfor all and , and denotes the mean value of on ; that is, A norm for , denoted by , is given by the infimum of the constants satisfying (25), or, equivalently, where the supremum is taken over all balls with and . With the above definition in mind, one has for , and hence . Moreover, the classical BMO space [6] is properly contained in (see [2, 3] for some examples). We need the following key result for the space , which was proved by Tang in [7].

Proposition 7 (from [7]). Let with . Then there exist two positive constants and such that, for any given ball in and for any , we havewhere and is the constant appearing in Lemma 1.

As a consequence of Proposition 7 and Lemma 4, we have the following result.

Proposition 8. Let with and with . Then there exist positive constants , and such that, for any given ball in and for any , we havewhere and is the constant appearing in Lemma 1.

##### 1.5. Orlicz Spaces

In this subsection, let us give the definition of and some basic facts about Orlicz spaces. For more information on this subject, the reader may consult book [8]. Recall that a function is called a Young function if it is continuous, convex, and strictly increasing with An important example of Young functions is with some . Given a Young function and a function defined on a ball , we consider the -average of a function given by the following Luxemburg norm: Associated with each Young function , one can define its complementary function as follows: Such a function is also a Young function. It is well known that the following generalized Hölder inequality in Orlicz spaces holds for any given ball : In particular, for the Young function , the Luxemburg norm will be denoted by . A simple computation shows that the complementary Young function of is (see [9, 10] for instance). The corresponding Luxemburg norm will be denoted by . In this situation, we haveWe next define the weighted -average of a function over a ball . Given a Young function and a weight function , let (see [8] for instance) When , we denote , and when , we denote . Also, the complementary Young function of is given by with the corresponding Luxemburg norm denoted by . Given a weight on , we can also show the weighted version of (35). That is, the generalized Hölder inequality in the weighted setting (see [11] for instance)holds for every ball in . It is a simple but important observation that, for any ball in , This is because for all . So we have

In [2], Bongioanni et al. obtained weighted strong , , and weak estimates for the commutators of the Riesz transform and its adjoint associated with the Schrödinger operator , where satisfies some reverse Hölder inequality. Their results can be summarized as follows.

Theorem 9 (from [2]). Let and . If with , then the commutator operators and are all bounded on , whenever belongs to .

Theorem 10 (from [2]). Let and . If with and , then, for any given , there exists a positive constant such that, for those functions such that , where and ; that is,

In this paper, firstly, we will define some kinds of weighted Morrey spaces related to certain nonnegative potentials. Secondly, we prove that the Riesz transform and its adjoint are both bounded operators on these new spaces. Finally, we also obtain the weighted estimates for the commutators and defined in (23).

Throughout this paper denotes a positive constant not necessarily the same at each occurrence, and a subscript is added when we wish to make clear its dependence on the parameter in the subscript. We also use to denote the equivalence of and ; that is, there exist two positive constants and independent of such that .

#### 2. Our Main Results

In this section, we introduce some types of weighted Morrey spaces related to the potential and then give our main results.

Definition 11. Let , let , and let be a weight. For given , the weighted Morrey space is defined to be the set of all locally integrable functions on for whichfor every ball in . A norm for , denoted by , is given by the infimum of the constants in (42), or, equivalently, where the supremum is taken over all balls in and and denote the center and radius of , respectively. Define

Note that this definition does not coincide with the one given in [12] (see also [13] for the unweighted case), but in view of the space defined above it is more natural in our setting. Obviously, if we take or , then this new space is just the weighted Morrey space , which was first defined by Komori and Shirai in [14] (see also [15]).

Definition 12. Let , let , and let be a weight. For given , the weighted weak Morrey space is defined to be the set of all measurable functions on for which for every ball in , or, equivalently, Correspondingly, we define

Clearly, if we take or , then this space is just the weighted weak Morrey space (see [16]). According to the above definitions, one has for . Hence for and for .

The space (or ) could be viewed as an extension of the weighted (or weak) Lebesgue space (when ). Naturally, one may ask the question whether the above conclusions (i.e., Theorems 5 and 6 as well as Theorems 9 and 10) still hold if we replace the weighted Lebesgue spaces by the weighted Morrey spaces. In this work, we give a positive answer to this question. Our main results in this work are presented as follows.

Theorem 13. Let , , and . If with , then the operators and map into itself.

Theorem 14. Let , , and . If with , then the operators and map into .

Theorem 15. Let , , and . If with , then the commutator operators and map into itself, whenever .

To deal with the commutators in the endpoint case, we need to consider a new kind of weighted Morrey spaces of the type.

Definition 16. Let , let , and let be a weight. For given , the weighted Morrey space is defined to be the set of all locally integrable functions on for which for every ball in , or, equivalently,

Concerning the mapping properties of and in the weighted Morrey spaces of the type, we have the following.

Theorem 17. Let , , and . If with and , then, for any given and any given ball of , there exist some constants and such that the inequalities hold for those functions such that with some fixed , where .

If we denote then Theorem 17 now tells us that the commutators and map into , when is in .

#### 3. Proofs of Theorems 13 and 14

In this section, we will prove the conclusions of Theorems 13 and 14.

Proof of Theorem 13. We denote by either or . By definition, we only have to show that, for any given ball of , there is some such thatholds for any with and . Suppose that for some and for some . We decompose , in the classical way, as where is the ball centered at and radius and is the characteristic function of . Then by the linearity of , we write We now analyze each term separately. By Theorem 5, we get Since with and , then we know that the inequalityis valid. In fact, for , by Hölder’s inequality and the definition of , we have If we take , then the above expression becomeswhich in turn implies (57). Therefore,