Semigroup Maximal Functions, Riesz Transforms, and Morrey Spaces Associated with Schrödinger Operators on the Heisenberg Groups

Wang, Hua

doi:https://doi.org/10.1155/2020/8839785

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Research Article | Open Access

Volume 2020 | Article ID 8839785 | https://doi.org/10.1155/2020/8839785

Semigroup Maximal Functions, Riesz Transforms, and Morrey Spaces Associated with Schrödinger Operators on the Heisenberg Groups

Hua Wang¹

Academic Editor: Eva A. Gallardo Gutiérrez

Received08 Aug 2020

Accepted28 Oct 2020

Published21 Nov 2020

Abstract

Let be a Schrödinger operator on the Heisenberg group , where is the sub-Laplacian on and the nonnegative potential belongs to the reverse Hölder class with . Here, is the homogeneous dimension of . Assume that is the heat semigroup generated by . The semigroup maximal function related to the Schrödinger operator is defined by . The Riesz transform associated with the operator is defined by , and the dual Riesz transform is defined by , where is the gradient operator on . In this paper, the author first introduces a class of Morrey spaces associated with the Schrödinger operator on . Then, by using some pointwise estimates of the kernels related to the nonnegative potential, the author establishes the boundedness properties of these operators , , and acting on the Morrey spaces. In addition, it is shown that the Riesz transform is of weak-type . It can be shown that the same conclusions are also true for these operators on generalized Morrey spaces.

1. Introduction

1.1. The Heisenberg Group

The Heisenberg group is the most well-known example from the realm of nilpotent Lie groups and plays an important role in several branches of mathematics, such as the representation theory, partial differential equations, several complex variables, and harmonic analysis. It is a remarkable fact that the Heisenberg group, an important example of the simply connected nilpotent Lie group, naturally arises in two fundamental but different settings in modern analysis. On the one hand, it can be naturally identified with the group of translations of the Siegel upper half-space in and plays an important role in our understanding of several problems in the complex function theory of the unit ball. On the other hand, it can also be realized as the group of unitary operators generated by the position and momentum operators in the context of quantum mechanics.

We begin by recalling some notions and well-known results from [1–3]. We write for the set of natural numbers. The sets of real and complex numbers are denoted by and , respectively. Let be a Heisenberg group of dimension , that is, a two-step nilpotent Lie group with underlying manifold . The group operation is given by where , , and .

Under this group operation, becomes a nilpotent unimodular Lie group. It can easily be seen that the inverse element of is , and the identity element of this group is the origin . The corresponding Lie algebra of left-invariant vector fields on is spanned by

All nontrivial commutation relations are given by

Here, is the usual Lie bracket. The sub-Laplacian and the gradient are defined, respectively, by

The Heisenberg group has a natural dilation structure which is consistent with the Lie group structure mentioned above. For each positive number , we define the dilation on by

Observe that is an automorphism of the group . For any given , the homogeneous norm of is given by the following form:

Observe that and

In addition, this norm satisfies the triangle inequality and then leads to a left-invariant distance for any . If and , let be the (open) ball with center and radius . Both left and right Haar measures on coincide with the Lebesgue measure on . For any measurable set , the Lebesgue measure of is denoted by . For , it can be proved that the volume of is where is the homogeneous dimension of and | is the volume of the unit ball in . A simple calculation shows that

Given a ball and , we adopt the notation to denote the ball with the same center and radius . Clearly, by (8), we have

For a radial function on , we have the following integration formula: where is a positive constant which is independent of . For more information about harmonic analysis on the Heisenberg group, the reader is referred to [2, 4, 5] and the references therein.

1.2. The Schrödinger Operator

We recall some standard notation and definitions.

Definition 1. 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 .

In this article, we will always assume that with and . We now consider the Schrödinger operator with the potential on the Heisenberg group (see [3]):

In recent years, there has been a lot of attention paid to the study of various function spaces associated with the Schrödinger operators, which has been an active research topic in harmonic analysis. For the investigation of Schrödinger operators on the Euclidean space with nonnegative potentials that belong to the reverse Hölder class, see, for example, [6–10]. Concerning the weighted case, one can see [11–16] for more details. The extension to the setting of the Heisenberg group has been given by Lin and Liu in [3]. For further details, we refer the reader to [17–19], among others. Regarding the Schrödinger operators in a more general setting (such as the nilpotent Lie group), see, for example, [20, 21]. As in [3, 22], we introduce the following definition.

Definition 2. Suppose that with . For any given , the critical radius function is defined by where denotes the ball in centered at and with radius .

It should be pointed out that the auxiliary function on the Euclidean space was introduced by Shen in [10]. Later, Li [21] defined it on the (simply connected) nilpotent Lie group. It is well known that the auxiliary function determined by satisfies for any given (see [3, 22]). In particular, with , and with (Hermite operator).

It is easy to check that if , then by the Hölder inequality. Furthermore, it can be shown that the class has a property of self-improvement. More precisely, if , then for some . By this fact, we know that the assumption is equivalent to .

When , we also write

Let us give some elementary properties of the class. Assume that with .

Lemma 3. The measure satisfies the doubling condition; that is, there exists a constant such that for all balls in .

Lemma 4. If , then for any .

Lemma 5. For , we have for any .

For more details, the reader may consult [3, 22].

We also need the following technical lemma concerning the critical radius function (14).

Lemma 6. Let be the auxiliary function determined by . For any and in , there exist constants and such that

Here, and in what follows, is simply denoted by . Lemma 6 has been proved by Lu [22] (see also [3], Lemma 4). In the setting of , this result was first given by Shen in [10] (Lemma 1.4). As a direct consequence of (20), we can see that for each fixed , the following estimate holds for any with and , where is the same as in (20). Let us verify (21). An application of (20) yields which further implies that for each fixed ,

Hence, which is the desired estimate. This estimate will often be used in the sequel.

1.3. Semigroup Maximal Functions and Riesz Transforms

Let be a Schrödinger operator on the Heisenberg group , where is the sub-Laplacian and the nonnegative potential belongs to the reverse Hölder class for , and is the homogeneous dimension of . Since is nonnegative and belongs to , generates a contraction semigroup . Let denote the kernel of the semigroup .

We also denote by the convolution kernel of the heat semigroup . Namely,

For any , it is well known that the heat kernel has the explicit expression:

We consider the heat equation associated with the sub-Laplacian with the initial condition . In fact, the function stated the above exists as a solution to the heat equation. Moreover, by [23] (Theorem 2), we know that the heat kernel satisfies the Gaussian upper bound estimate: where the positive constants and are independent of and . By the Trotter product formula (see [24] for instance) and (29), one has where and are positive constants independent of , , and . Furthermore, by using the estimates of the fundamental solution for the Schrödinger operator on , this estimate (30) can be significantly improved when belongs to the reverse Hölder class for some . The auxiliary function arises naturally in the present situation.

Lemma 7. Let be the auxiliary function determined by . For every positive integer , there exists a positive constant such that, for any and in ,

Remark 8. This estimate of is much better than (30), which was given by Lin and Liu in [3] (Lemma 7). In the setting of , this result can be found in [9] (Proposition 2).

In this article, we investigate the semigroup maximal function related to the Schrödinger operator , which is defined by (see [3])

We shall establish the strong-type and weak-type estimates of the operator . Some other maximal functions will be discussed at the end of Section 3.

Let us also consider the Riesz transforms and the dual Riesz transforms for the Schrödinger operator , which are defined, respectively, by (see [3]) where the are left-invariant vector fields that generate the Lie algebra of . Let

Here, is the gradient operator on . We shall be interested in the behavior of the (vector-valued) operators and associated with the Schrödinger operator on .

For any , the Lebesgue space is defined to be the set of all measurable functions on such that

The weak Lebesgue space consists of all measurable functions on such that

Recently, Lin and Liu ([3], Theorem 6 and Remark 3) established the strong-type and weak-type estimates of the operator on the Lebesgue spaces.

Theorem 9. Let . Then, the following statements are true: (1)If , then the operator is bounded on (2)If , then the operator is bounded from into

Remark 10. (i)It was also shown by Lin and Liu that this operator is bounded on (ii)On the Euclidean space , this maximal operator was studied by Dziubański et al. [9] (see also [11, 25])

As for the (vector-valued) dual Riesz transform defined above, we have the following estimate given in [21] (see also [19]).

Theorem 11. Let with , and let be a number such that . Then, the (vector-valued) dual Riesz transform is bounded on for .

By duality, we could obtain the following result.

Theorem 12. Let with , and let be a number such that . Then, the (vector-valued) Riesz transform is bounded on for .

Moreover, it will be proved in Section 4 that is of weak-type on the Heisenberg group. The case where is also considered in Section 4.

Remark 13. (i)It can be shown that the range of in the above theorems is optimal (see [3]). In this paper, the authors also proved that the dual Riesz transform is bounded on and gave the Fefferman-Stein-type decomposition of functions with respect to , (ii)It was shown in [26] that when , the operators and are uniformly bounded on with respect to . More specifically, for every , there exists a constant such that for every (iii)Recall that in the setting of , the Riesz transform and its dual form were originally studied by Shen in [10]. It can be proved that the analog of Theorem 11 (also Theorem 12) on the Euclidean space is also true by the same argument (see [10, 27]). For the corresponding estimates for commutators generated by functions, the reader is referred to [11, 12, 28] for more details

The paper is organized as follows. In Section 2, we introduce the Morrey space and weak Morrey space associated with the Schrödinger operator on and state our main results: Theorems 18, 19, 22, 23, and 24. Section 3 is devoted to Proofs of Theorems 18 and 19, which establish the strong-type and weak-type estimates for the semigroup maximal function in the framework of Morrey spaces. The corresponding estimates for some other maximal functions are also proved in this section. Section 4 is devoted to proving the boundedness properties of the Riesz transform and its dual form . In Section 5, we extend the above results to the generalized Morrey spaces.

Throughout this paper, denotes a universal constant which may change from line to line, and a subscript is added when we wish to make clear its dependence on the parameter in the subscript. The notation means that for some positive constant . If and , then we write to denote the equivalence of and . For any , the notation denotes its conjugate number, namely, and .

2. Definitions and Main Theorems

A few historic remarks are in order. The classical Morrey space was originally introduced and studied by Morrey in [29] to deal with the local behavior of solutions to second-order elliptic partial differential equations. Since then, this space was systematically developed by a number of authors. Nowadays, this space has been investigated extensively and widely used in analysis, geometry, mathematical physics, and other related fields. We denote by the Morrey space, which consists of all -locally integrable functions on such that where and . It is known that is an extension of in the sense that . Note that by the Lebesgue differentiation theorem. If or , then , where is the set of all functions equivalent to 0 on . We also denote by the weak Morrey space, which consists of all measurable functions on such that

For the properties of classical Morrey spaces, we refer the readers to [30–34] and the references therein. Moreover, the Morrey spaces were found to have many important applications to the Navier-Stokes equations, the Schrödinger equations, the elliptic equations with discontinuous coefficients, and the potential analysis (one can see [30, 35–38]).

In this section, we introduce some types of Morrey spaces associated with the Schrödinger operator on (see [39]) and then give our main results.

Definition 14. Let be the auxiliary function determined by with . Let and . For each given , the Morrey space is defined to be the set of all -locally integrable functions on such that holds for every ball in , where and denote the center and radius of , respectively. The smallest constant appearing in (40) is called the norm of , which is denoted by . It is a Banach space with respect to the norm . Define

Definition 15. Let be the auxiliary function determined by with . Let and . For each given , the weak Morrey space is defined to be the set of all measurable functions on such that holds for every ball in . The smallest constant appearing in (42) is called the (quasi-)norm of , which is denoted by . It is a (quasi-)Banach space with respect to the (quasi-)norm . Correspondingly, we define

Remark 16. (i)Obviously, if we take or , then this Morrey space (or weak Morrey space ) is just the Morrey space (or weak Morrey space ), which was defined and studied by Guliyev et al. [40](ii)According to the above definitions, one haswhenever . Hence, for all . When , the spaces and reduce to and , respectively. (iii)It follows directly from Chebyshev’s inequality that , and hence,Moreover, the inclusion is strict.

Remark 17. (i)We can define a norm on the space (see [39]), which makes it into a Banach space. In view of (44), for any given , letNow define the functional by It is easy to check that the functional defined by (49) is indeed a norm on provided ; i.e., it satisfies the following conditions: (a)It is positive definite: , and if and only if (b)It is multiplicative: , for any (c)It satisfies the triangle inequality: , for any (ii)In view of (45), for any given , letSimilarly, we define the functional by It is easily checked that the functional defined by (51) is a (quasi-)norm on for all ; i.e., it satisfies the following conditions: (a)It is positive definite: , and if and only if (b)It is multiplicative: , for any (c)It satisfies the inequality: , for any The space is a (quasi-)Banach space with respect to the (quasi-)norm .

Since Morrey space (or weak Morrey space ) could be viewed as an extension of the Lebesgue space (or the weak Lebesgue space) on (when , or ), it is natural to study the boundedness properties of the operators , , and in the context of Morrey spaces. In this paper, we will extend Theorems 9, 11, and 12 to the Morrey spaces on . Let be the same as before. Now let us formulate our main results as follows.

Theorem 18. Let , , and . If with , then the semigroup maximal function is a bounded sublinear operator on .

Theorem 19. Let , , and . If with , then the semigroup maximal function is a bounded sublinear operator from into .

As an immediate consequence of Theorems 18 and 19 and Remark 17, we have the following results.

Corollary 20. Let and . If with , then the semigroup maximal function is a bounded sublinear operator on .

Corollary 21. Let and . If with , then the semigroup maximal function is a bounded sublinear operator from into .

Theorem 22. Let and . If with , and is a number such that , then the dual Riesz transform is a bounded linear operator on provided that .

Theorem 23. Let . If with , and is a number such that , then the Riesz transform is a bounded linear operator on provided that and with .

It is worth pointing out that we cannot use Theorem 22 to prove Theorem 23 in a direct way by duality, since the predual to is unknown. Motivated by the ideas in [30, 31], it is an interesting and natural problem to investigate the dual theory for the Morrey space through a geometric analysis of the Hausdorff capacity and Choquet integrals, which will be treated in a subsequent paper. In addition, we will prove that the operator is of weak-type . Based on this result, we can further prove the following.

Theorem 24. Let . If with , and is a number such that , then the Riesz transform is a bounded linear operator from into provided that .

As a straightforward consequence of Theorems 22–24 and Remark 17, we obtain the following estimates.

Corollary 25. Let . If with , and is a number such that , then the dual Riesz transform is a bounded linear operator on provided that .

Corollary 26. If with , and is a number such that , then the Riesz transform is a bounded linear operator on provided that and with .

Corollary 27. If with , and is a number such that , then the Riesz transform is a bounded linear operator from into provided that .

3. Boundedness of the Semigroup Maximal Functions

In this section, we will prove the conclusions of Theorems 18 and 19. Let denote the integral kernel of related to the Schrödinger operator (see [3]). Then, we can write as follows:

Proof of Theorem 18. For any given with and , by definition, we only need to show that for any given ball of , the following inequality holds true. By a standard argument, we decompose the function as where denotes the open ball centered at of radius , denotes the characteristic function of the set , and denotes its complement. Then, by the sublinearity of , we write Let us consider the first term . Making use of the first part of Theorem 9, we have Moreover, note that for any fixed , This, combined with (10), yields We now turn to estimate the second term . We first assert that the following inequality holds for any , where and are given as in Lemma 7. Indeed, this can be done by considering the following two cases: and . It follows directly from Lemma 7 that When , then , and hence, On the other hand, we can easily see that When , then . In this case, it is easy to check that for any , From this, it follows immediately that Putting all together produces the required inequality (59). Notice that for any and , one has Thus, That is, . Combining this fact with (59) yields that for any and any positive integer , Furthermore, in view of (21) and (57), we can see that the above expression (67) does not exceed By using the Hölder inequality, we obtain that for each fixed , Substituting the above inequality into formula (68), we conclude that By choosing some sufficiently large number such that , then we have where the last inequality follows from the fact that and . Combining the above estimates for and , we obtain the desired inequality (53). This ends Proof of Theorem 18.

Proof of Theorem 19. Let with and . Fix . Our aim is to prove, by definition, that for each given ball of , the following estimate holds true. To this end, we decompose the function as Then, for any fixed , we can write We first give the estimate for the term . By the second part of Theorem 9, we get Therefore, in view of (57) and (10), we have To estimate the second term , by using the pointwise inequality (68) and Chebyshev’s inequality, we can deduce that Moreover, for each fixed , we compute Consequently, substituting this inequality into formula (77), By selecting some large enough such that , we thus have where the last step is again due to the fact that and . Summing up the above estimates for and , and then taking the supremum over all , we obtain our desired result (72). This completes Proof of Theorem 19.

We also consider the maximal function with respect to the Poisson semigroup , which is defined by

We now begin to prove that under the conditions of Theorems 18 and 19, the same results also hold for the operator related to . Recall the subordination formula (see [41], p. 6)

Thus, from the heat semigroup, we can define the Poisson semigroup by

From this, it follows immediately that for all ,

Hence, as an immediate consequence of Theorems 18 and 19 and Corollaries 20 and 21, we have the following results. Let be the same as in (14).

Theorem 28. Let , , and . If with , then the operator is bounded on and hence bounded on .

Theorem 29. Let , , and . If with , then the operator is bounded from into and hence bounded from into .

Remark 30. (i)A slightly more general point of view is as follows. Motivated by the work in [9, 25], we introduce the semigroup nontangential maximal function related to , which is given as follows:It is interesting to investigate the boundedness of the operator related to . Following the same arguments as in Proofs of Theorems 4 and 6 in [3], we are able to prove that the operator is bounded on for all and bounded from into . Based on this result, we can further prove that the corresponding estimates for the operator remain valid in the context of Morrey spaces. The proof needs appropriate but minor modifications, and we leave this to the interested reader. (ii)In view of the above results, we consider here the nontangential maximal function with respect to the Poisson semigroup , which is defined byFor the same reason as above, it can be shown that this new maximal operator is dominated by in some sense. Therefore, all the results mentioned above hold as well for the maximal operator in this more general situation.

4. Boundedness of the Riesz Transforms

This section is concerned with Proofs of Theorems 22, 23, and 24. Recall that the operators and have singular kernels with values in that will be denoted by and , respectively (see [3]).

Obviously,

The next lemma plays a crucial role in our Proofs of Theorems 22–24.

Lemma 31. Let with , and let be the auxiliary function determined by . For every positive integer , there exists a positive constant such that, for any and in, where denotes the (vector-valued) kernel of the operator . Moreover, the above inequality also holds with replaced by .

Lemma 31 was proved by Li [21] in a more general setting (connected nilpotent Lie group) (see also [19] for ). For such kernels, we also give the following result, which establishes the Lipschitz regularity of .

Lemma 32. Let with , and let be the auxiliary function determined by . For every positive integer , there exists a positive constant such that, for any and in , and for some fixed , is given as in (16), whenever . Moreover, the above inequality also holds with replaced by .

The above kernel estimate in Lemma 32 was obtained by Pengtao and Lizhong in [19], which will be used to prove that the operator is of weak-type . Recall that in the Euclidean setting, the kernel estimates (89) and (90) were proved in [10, 42].

We are now in a position to give the proofs of our main theorems.

Proof of Theorem 22. Let with and , where . Fix . By definition, we only need to show that for any given ball of , the following inequality holds true. Using the standard technique, we decompose the function as Then, by using the linearity of , we write Let us consider the first term . Making use of (10), (57), and Theorem 11, we have Now let us turn to estimate the second term . By using Lemma 31, we obtain that for any , where Hence, can be written as follows: Arguing as in Proof of Theorem 18, we can also obtain We only have to deal with the term . As mentioned in the previous proof, one has whenever and . Thus, for any positive integer , It is not difficult to check that when and , one has , which implies where stands for the fractional integral operator of order one defined by For this operator, a classical result of Folland and Stein [43] states that is bounded from into for and (see also [39, 44]). Namely, there exists a constant such that for any , Since , we can choose a number such that . A combination of the Hölder inequality and (103) gives where the last inequality holds by our assumption . We now claim that the following inequality holds. For any ( is the doubling constant in Lemma 3), there exists a constant such that for any and , Taking this claim momentarily for granted, then we have where in the last step we have invoked (57) and (10). In addition, it follows immediately from (21) and (57) that A trivial computation shows that Therefore, in view of (108) and (107), we conclude that Consequently, By choosing some sufficiently large number such that , then we have where the last inequality follows again from the fact that and . Combining the above estimates for , , and produces the desired inequality (91).

Finally, let us verify (105). Suppose that with . Using the same method as in Proof of Lemma 1 in [42], for any and , there must exist an integer such that . Two cases are considered below.

Case 1. . In this case, one has . This fact together with Lemmas 3 and 4 yields Since , it is easy to see that

Case 2. . In this case, one has . This fact, along with Lemmas 5 and 4, implies that

Thus, in both cases, (105) holds. This completes Proof of Theorem 22.

In order to prove Theorem 24, let us first set up the following result, which is based on a version of the Calderón-Zygmund decomposition on and Lemma 32.

Theorem 33. Let with . Then, the Riesz transform is bounded from into .

Proof. For any given and , making use of the Calderón-Zygmund decomposition of at height (see [43]), we have the decomposition with such that (1), for almost every and(2)each is supported in the ball , and we denote the center and the radius of by and , respectively(3)the sets are finitely overlapping andFrom this construction, we have that for any fixed , The part of the argument involving the function proceeds as follows. By using the boundedness of (see Theorem 12 with ), we obtain Setting , we split into two parts as follows: It is obvious that Therefore, in order to complete our proof, we need only to show that An application of Chebyshev’s inequality yields We observe that whenever and . Then, we apply Lemma 32, (88), and the cancelation condition of to get Obviously, the first term on the right-hand side of (125) is bounded by Using this estimate together with (11), we can deduce that On the other hand, the latter term on the right-hand side of (125) is controlled by When , by the triangle inequality, one has From this, it follows that the above expression is bounded by Consequently, where is the fractional integral operator of order one given in (102). A combination of the Hölder inequality and (103) implies that for each fixed , where the last inequality is obtained by the hypothesis . Moreover, in view of (10), (105), and (57), we can see that the above expression (132) is bounded by where the last step is due to the fact that . Therefore, by selecting some large enough such that , we thus have Collecting all these estimates and then taking the supremum over all , we conclude the proof of Theorem 33.

Proof of Theorem 24. Let be a positive number such that . To prove Theorem 24, it is enough to prove that for each given ball of , the following estimate holds true for any given with some and . Using the standard technique, we decompose the function as Then, for any fixed , we can write Let us estimate the first term . By Theorem 33, (57), and (10), we get As for the second term , from (88) and Lemma 31, it follows that for any , where Thus, by (139) and Chebyshev’s inequality, can be written as follows: We now proceed exactly as we did in Proof of Theorem 19 and have the following estimate as well: Let us analyze the latter term . In order to do this, we first observe that whenever and . Hence, for any positive integer , It is easy to verify that when and , one has . This fact together with (107) implies that for any , Consequently, Applying the Hölder inequality along with (103), we can compute the above integral as follows: where the last inequality holds since . Moreover, by (105), (57), and (10), we can see that the above expression is bounded by Taking into account the fact that , then we have where a large enough is chosen satisfying . By the choice of , it guarantees that the exponent of the last summation is negative, and hence, it is convergent. Therefore, we conclude that Combining these estimates for , , and , and then taking the supremum over all , we get the desired estimate (135). This finishes Proof of Theorem 24.

Proof of Theorem 23. Since the proof is similar to that of Theorem 22, we shall only indicate the necessary modifications. As before, it is enough for us to show that for an arbitrary fixed ball in , the following estimate holds true for any given with some , , and , where and . To this end, we split through and . Then, the left-hand side of (151) will be divided into two parts given below. Since the Riesz transform is bounded on for (see Theorem 12), we can deal with in the same manner as in Proof of Theorem 22 and obtain On the other hand, from (139), it follows that Here We follow the same arguments as in Proof of Theorem 18 and obtain the following estimate as well: It remains to check that (151) holds for the last term . As it was shown in Theorem 24, it holds that for any , Moreover, by using the Hölder inequality, we obtain that for each fixed , which in turn gives For the latter integral, we use the Hölder inequality with exponent and (103) to derive where the last inequality is obtained by the fact that . Furthermore, in view of (105), (57), and (10), we can see that the above expression is controlled by A trivial computation leads to Taking into account this fact, then we have where a large enough is chosen such that . By the hypothesis, we know that the exponent in the last summation is negative, and hence, it is convergent. Therefore, we conclude that as desired. This finishes Proof of Theorem 23.

If one has a slightly stronger assumption on the potential , then we have the following improved estimates for the kernels and .

Lemma 34. Let with , and let be the auxiliary function determined by . For every positive integer , there exists a positive constant such that, for any and in ,

We remark that in the Euclidean case, this lemma was already obtained by Shen in [10]. Moreover, Shen [10] actually showed that and its dual form are standard Calderón-Zygmund singular integral operators in , and hence, these two operators and are all bounded on for and are of weak-type , when with . We adapt the arguments used in [10, 42] (see also [19, 21]) to our present situation and prove Lemma 34 similarly. Furthermore, by adopting the same method given in [3, 10], we can also prove that the operators and are bounded on for all and are bounded from into in such a situation. Repeating the arguments above, we are able to show that under the same assumptions as in Theorems 18 and 19, the corresponding results also hold for the operators and on .

Theorem 35. Let and . If with , then the operators and are bounded linear operators on for all and hence bounded on .

Theorem 36. Let and . If with , then the operators and are bounded linear operators from into and hence bounded from into .

We recall the relation . Since tends to as , so we have the following: tends to 1, and tends to 1 with . Hence, the above theorems can be regarded as the limiting case of the results of Theorems 22, 23, and 24.

5. Generalized Morrey Spaces

In the last section, let us give the definitions of the generalized Morrey spaces related to the nonnegative potential on . Let , , be a growth function, that is, a positive increasing function on , and satisfy the following doubling condition: for all , where is a doubling constant independent of .

Definition 37. Let be the auxiliary function determined by with . Let and be a growth function. For any given , the generalized Morrey space is defined to be the set of all -locally integrable functions on such that holds for every ball in , and we denote the smallest constant satisfying (167) by . It is easy to see that the functional is a norm on the linear space that makes it into a Banach space under this norm. Define

Definition 38. Let be the auxiliary function determined by with . Let and be a growth function. For any given , the generalized weak Morrey space is defined to be the set of all measurable functions on such that holds for every ball in , and we denote the smallest constant satisfying (169) by . Correspondingly, we define

Remark 39. (i)As in Section 2 (Remark 17), we can also define a norm and a (quasi-)norm on the linear spaces and , respectively(ii)According to this definition, we recover the spaces and under the choice , for all (iii)In the Euclidean setting, when or , the classes and reduce to the classes and , which were introduced and studied by Mizuhara in [45]. We refer the reader to [46, 47] for further details

By using the same procedure as in the proofs of our main results, we have the following theorems. We leave the details to the reader. Let be as in (14).

Theorem 40. Let , , and . If with , then the operators and are bounded on and hence bounded on .

Theorem 41. Let , , and . If with , then the operators and are bounded from into and hence bounded from into .

We point out that the same conclusions also hold for and other maximal functions (, , and ) discussed in Section 3.

Theorem 42. Let and . If with , and is a number such that , then the operator is bounded on and hence bounded on provided that .

Theorem 43. Let . If with , and is a number such that , then the operator is bounded on and hence bounded on provided that and with .

Theorem 44. Let . If with , and is a number such that , then the operator is bounded from into and hence bounded from into provided that .

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there is no conflict of interest regarding the publication of this paper.

References

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