Weighted Morrey Spaces Related to Certain Nonnegative Potentials and Riesz Transforms

Wang, Hua

doi:https://doi.org/10.1155/2019/7057512

Journal of Function Spaces

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Research Article | Open Access

Volume 2019 | Article ID 7057512 | https://doi.org/10.1155/2019/7057512

Weighted Morrey Spaces Related to Certain Nonnegative Potentials and Riesz Transforms

Hua Wang^1,2

Academic Editor: Maria Alessandra Ragusa

Received31 Dec 2018

Accepted11 Feb 2019

Published24 Apr 2019

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 [1–3].

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, where . For the other term , notice that, for any and , one has . It then follows from Lemma 2 that, for any and any positive integer ,In view of (4) in Lemma 1, we further obtainMoreover, by using Hölder’s inequality and the condition on , we get Hence, Recall that with and ; then there exist two positive numbers such that (19) holds. This allows us to obtain Thus, by choosing large enough so that , we then have Summing up the above estimates for and and letting , we obtain our desired inequality (53). This completes the proof of Theorem 13.

Proof of Theorem 14. We denote by either or . To prove Theorem 14, by definition, it is sufficient to prove that, for any given ball of , there is some such thatholds for any with . Now suppose that for some and for some . We decompose , in the classical way, as Then for any given , by the linearity of , we can write We first give the estimate for the term . By Theorem 6, we get Since with , similar to the proof of (57), we can also show the following estimate as well:In fact, by the definition of , we can deduce that If we choose , then the above expression becomeswhich in turn implies (71). Therefore, where . As for the other term , by using pointwise inequality (62) and Chebyshev’s inequality, we deduce that Moreover, by the condition on , we compute Consequently, Recall that with ; then there exist two positive numbers such that (19) holds. Therefore, By selecting large enough so that , we thus have Let . Here is an appropriate constant. Summing up the above estimates for and and then taking the supremum over all , we obtain our desired inequality (67). This finishes the proof of Theorem 14.

4. Proofs of Theorems 15 and 17

For the results involving commutators, we need the following properties of functions, which are extensions of well-known properties of functions.

Lemma 18. If and with , then there exist positive constants and such that, for every ball in , we havewhere .

Proof. We may assume that with . According to Proposition 8, we can deduce that Making change of variables, then we get which yields the desired inequality if we choose and .

Lemma 19. If with and , then there exist positive constants and such that, for every ball in , we havewhere and and is the constant appearing in Lemma 1.

Proof. Recall the following identity (see Proposition 1.1.4 in [5]): Using this identity and Proposition 8, we obtain where is given by If we take small enough so that , then the conclusion follows immediately.

Lemma 20. If with , then, for any positive integer , there exists a positive constant such that, for every ball in , we have

Proof. For any positive integer , we have Since, for any , the estimate holds trivially, then We obtain the desired result. This completes the proof.

Now, we are in a position to prove our main results in this section.

Proof of Theorem 15. We denote by either or . By definition, we only need to show that, for any given ball of , there is some such thatholds for any with and , whenever belongs to . Suppose that for some , for some , and for some . We decompose as Then, by the linearity of , we write Now we give the estimates for and , respectively. According to Theorem 9, we have Moreover, in view of inequality (57), we get where . On the other hand, by definition (23), we can see that, for any ,So we can divide into two parts: From pointwise estimate (62) and (80) in Lemma 18, it then follows that Following along the same lines as that of Theorem 13, we are able to show that The last inequality is obtained by using (19). For any and any positive integer , similar to the proof of (61) and (62), we can also deduce thatwhere in the last inequality we have used (4) in Lemma 1. Hence, by the above pointwise estimate for , Moreover, for each integer ,By using Hölder’s inequality, the first term of expression (102) is bounded by Since with and , then, by the definition of , it can be easily shown that if and only if , where (see [7, 17, 18] and the references therein). If we denote , then . This fact together with Lemma 18 implies Therefore, the first term of expression (102) can be bounded by a constant times Since with , then, by Lemma 20, Hölder’s inequality, and the condition on , the latter term of expression (102) can be estimated by Consequently,Thus, in view of (107), Notice that with . A further application of (19) yields Combining the above estimates for and , we get By choosing large enough so that , we thus have Finally, collecting the above estimates for and and letting , we obtain the desired result (91). The proof of Theorem 15 is finished.

Proof of Theorem 17. We denote by either or . We are going to prove that, for any given and any given ball of , there is some such that the inequalityholds for those functions such that with some fixed . Now assume that for some and for some . As before, we decompose as Then for any given , by the linearity of , we can write Let us first estimate the term . By using Theorem 10, we get A further application of (39) yields where the last inequality is due to (71). If we denote , thenas desired. Next let us deal with the term . Taking into account (96), we can divide it into two parts, namely, where Since for some , from pointwise inequality (62) and Chebyshev’s inequality, we then have where in the last inequality we have used (80) in Lemma 18. Moreover, it follows directly from the condition that, for each integer , Notice also that triviallyThis fact along with (39) implies that, for each integer , Consequently, Since with , then there exist two positive numbers and such that (19) holds. Therefore, On the other hand, it follows from pointwise inequality (100) and Chebyshev’s inequality thatwhere the last inequality follows from (122). Furthermore, by the definition of , we computeBy using generalized Hölder inequality (37), the first term of expression (127) is bounded by where in the last inequality we have used the fact that which is equivalent to inequality (83) in Lemma 19. By Lemma 20 and (39), the latter term of expression (127) can be estimated by Consequently, Hence, combining the above estimates for and , we have Now can be chosen sufficiently large such that , and hence the above series is convergent. Finally, Fix this and set . Thus, combining the above estimates for and , inequality (112) is proved and then the proof of Theorem 17 is finished.

The higher order commutators formed by a function and the operator and its adjoint are usually defined byLet denote or . Obviously, which is just the linear commutator (23), and By induction on , we are able to show that the conclusions of Theorems 15 and 17 also hold for the higher order commutators with . The details are omitted here.

Theorem 21. Let , , and . If with , then, for any positive integer , the higher order commutators and map into itself, whenever .

Theorem 22. 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 , .

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interests regarding the publication of this article.

Acknowledgments

The author would like to thank Professor L. Tang for providing [7].

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Copyright

Copyright © 2019 Hua Wang. 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.

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