Journal of Function Spaces and ApplicationsVolume 2012 (2012), Article ID 678171, 21 pagesdoi:10.1155/2012/678171

Research Article

Weighted Hardy and Potential Operators in Morrey Spaces

Natasha Samko

Department of Mathematics, Research Center CEAF, Instituto Superior Técnico, 1049-003 Lisbon, Portugal

Received 15 November 2009; Accepted 9 June 2010

Academic Editor: V. Stepanov

Copyright © 2012 Natasha Samko. 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.

Abstract.

We study the weighted -boundedness of Hardy-type operators in Morrey spaces () (or () in the one-dimensional case) for a class of almost monotonic weights. The obtained results are applied to a similar weighted -boundedness of the Riesz potential operator. The conditions on weights, both for the Hardy and potential operators, are necessary and sufficient in the case of power weights. In the case of more general weights, we provide separately necessary and sufficient conditions in terms of Matuszewska-Orlicz indices of weights.

1. Introduction

The well-known Morrey spaces introduced in [1] in relation to the study of partial differential equations and presented in various books, see [2–4], were widely investigated during the last decades, including the study of classical operators of harmonic analysis—maximal, singular, and potential operators—in these spaces; we refer for instance to papers [5–23], where Morrey spaces on metric measure spaces may be also found. Surprisingly, weighted estimates of these classical operators, in fact, were not studied. Just recently, in [24] we proved weighted -estimates in Morrey spaces for Hardy operators on and one-dimensional singular operators (on or on Carleson curves in the complex plane).

In this paper we develop an approach which allows us both to obtain weighted of Hardy-type operators and potential operators and extend them to the multidimensional case, for the Hardy operators (related to integration over balls). Note that, in contrast with the case of Lebesgue spaces, Hardy inequalities in Morrey norms admit the value for ; see Theorem 4.3. The progress in comparison with [24] is based on the pointwise estimation of the Hardy operators we present in Sections 3.2 and 4.1. Such an estimation is helpless in the case of Lebesgue spaces (), but proved to be effective in the case of Morrey spaces (). Roughly speaking, it is based on the simple fact that

The admitted weights are generated by functions from the Bary-Stechkin-type class; they may be characterized as weights continuous and positive for , with possible decay or growth at and , which become almost increasing or almost decreasing after the multiplication by some power. Such weights are oscillating between two powers at the origin and infinity (with different exponents for the origin and infinity).

We also note that the obtained estimates show that the Hardy operators (with admitted weights) act boundedly not only in local and global Morrey spaces (see definitions in Section 3.1), but also from a larger local Morrey space into a more narrow global Morrey space (see Theorems 4.3 and 4.4).

The paper is organized as follows. In Section 2 we give necessary preliminaries on some classes of weight functions. In Section 3 we prove some statements on embedding of Morrey spaces into some weighted -spaces. In Section 4 we prove theorems on the weighted -boundedness of Hardy operators in Morrey spaces. Finally, in Section 5 we apply the results of Section 4 to a similar weighted boundedness of potential operators. The conditions on weights, both for the Hardy and potential operators are necessary and sufficient in the case of power weights. In the case of more general weights, we provide separately necessary and sufficient conditions in terms of Matuszewska-Orlicz indices of weights.

The main results are given in Theorems 4.3, 4.4, 4.5, and 5.3 and Corollary 5.4.

2. Preliminaries on Weight Functions

2.1. Zygmund-Bary-Stechkin (ZBS) Classes and Matuszewska-Orlicz (MO) Type Indices

2.1.1. On Classes of the Type

In the sequel, a nonnegative function on , is called almost increasing (almost decreasing) if there exists a constant (≥1) such that for all (, resp.). Equivalently, a function is almost increasing (almost decreasing) if it is equivalent to an increasing (decreasing, resp.) function , that is, .

Definition 2.1. Let .(1)By one denotes the class of functions continuous and positive on such that there exists the finite limit , and is almost increasing on;(2)by one denotes the class of functions on such that for some .

Definition 2.2. Let .(1)By one denotes the class of functions continuous and positive on which have the finite limit , and is almost increasing on ;(2)by one denotes the class of functions such for some .

2.1.2. ZBS Classes and MO Indices of Weights at the Origin

In this subsection we assume that .

Definition 2.3. One says that a function belongs to the Zygmund class , , if and to the Zygmund class , , if One also denotes the latter class being also known as Bary-Stechkin-Zygmund class [25].

It is known that the property of a function is to be almost increasing or almost decreasing after the multiplication (division) by a power function is closely related to the notion of the so called Matuszewska-Orlicz indices. We refer to [26, 27] [28, page 20], [29–32] for the properties of the indices of such a type. For a function , the numbers are known as the Matuszewska-Orlicz type lower and upper indices of the function . Note that in this definition needs not to be an -function: only its behaviour at the origin is of importance. Observe that and , and the formulas are valid for .

The following statement is known see [26, Theorems 3.1, 3.2, and 3.5]. (In the formulation of Theorem  2.4 in [26], it was supposed that , and . It is evidently true also for and all , in view of formulas (2.5).)

Theorem 2.4. Let and . Then Besides this, and for the inequalities hold with an arbitrarily small and .

2.1.3. ZBS Classes and MO Indices of Weights at Infinity

Following [14, Section 4.1] and [29, Section 2.2], we introduce the following definitions.

Definition 2.5. Let . One puts , where is the class of functions satisfying the condition and is the class of functions satisfying the condition where does not depend on .

The indices and responsible for the behavior of functions at infinity are introduced in the way similar to (2.4):

Properties of functions in the class are easily derived from those of functions in because of the following equivalence: where and . Direct calculation shows that

Making use of (2.16) and (2.17), one can easily reformulate properties of functions of the class near the origin, given in Theorem 2.4 for the case of the corresponding behavior at infinity of functions of the class and obtain that

We say that a function continuous in is in the class if its restriction to belongs to and its restriction to belongs to . For functions in , the notation has an obvious meaning. In the case, where the indices coincide: , we will simply write and similarly for . We also denote

Making use of Theorem 2.4 for and relations (2.17), we easily arrive at the following statement.

Lemma 2.6. Let . Then

2.2. On Classes

Note that we slightly changed the notation of the class introduced in the following definition, in comparison with its notation in [32].

Definition 2.7. Let . By , One denotes the classes of functions nonnegative on , defined by the following conditions: where and .

Lemma 2.8. Functions are almost increasing on , and functions are almost decreasing on .

Proof. Let and . By (2.24) we have . Then . The case is similarly treated.

Corollary 2.9. Functions have non-negative indices , and functions have non-positive indices , the same being also valid with respect to the indices in the case .

Note that the classes being trivial for . We also have which follows from the fact that and (see [24, Subsection 2.3, Remark 2.8]) and property (2.26).

An example of a function which is in with some , but does not belong to the total intersection is given by where and .

The following lemmas (see [24], Lemmas 2.10 and 2.11) show that conditions (2.24) and (2.25) are fulfilled with not only for power functions but also for an essentially larger class of functions (which in particular may oscillate between two power functions with different exponents). Note that the information about this class in Lemmas 2.10 and 2.11 is given in terms of increasing or decreasing functions, without the word “almost”.

Lemma 2.10. Let . Then (i) in the case is increasing and the function is decreasing for some ;(ii) in the case is decreasing and there exists a number such that is increasing.

Lemma 2.11. Let . If there exist and such that for , then . If there exist and such that for , then .

3. On Weighted Integrability of Functions in Morrey Spaces

3.1. Definitions and Belongness of Some Functions to Morrey Spaces

Let be an open set in .

Definition 3.1. The Morrey spaces , are defined as the space of functions such that respectively, where .

Obviously,

The spaces are known under the names of global and local Morrey spaces; see for instance, [9, 10].

The weighted Morrey space is defined as

Remark 3.2. As is well known, the space as defined above is not necessarily embedded into , in the case when is unbounded. A typical counterexample in the case is Indeed, we have which is bounded (when , take into account that , and when , make use of the inclusion ).

Lemma 3.3. Let , , and . The condition is sufficient for the function to belong to . In the case , the inclusion with holds if , the latter condition being necessary, when is an inner point of or and .

Proof. We have where . Then by (2.12). If , we choose and then the right-hand side of the last inequality is bounded. So let . We distinguish the cases (1) and (2) . In the case (1), . Therefore, which is bounded under the choice . In the case (2), we observe that and then the same estimate follows.In the case , the proof of the “if” part follows the same lines as above with . To prove the “only if” part, it suffices to observe that

Corollary 3.4. If and there exists an such that is almost increasing, then . To derive this corollary from Lemma 3.3, it suffices to refer to formula (2.10).

3.2. Some Weighted Estimates of Functions in Morrey Spaces

Lemma 3.5. Let , and . Then where does not depend on and and under the assumption that the last integral converges.

Proof. We have where . Making use of the fact that there exists a such that is almost increasing, we observe that Applying this in (3.13) and making use of the Hölder inequality with the exponent , we obtain Hence, It remains to prove that We have Making use of the fact that is almost increasing with some , we easily obtain that which proves (3.17).

Corollary 3.6. Let , and . Then

Lemma 3.7. Let , and . Then where does not depend on and and

Proof. The proof is similar to that of Lemma 3.5. We have where . Since there exists a such that is almost increasing, we obtain where may depend on , but does not depend on and . Applying the Hölder inequality with the exponent , we get It remains to prove that We have which completes the proof.

Remark 3.8. The analysis of the proof shows that estimate (3.21) remains in force, if the assumption is replaced by the condition that and satisfies the doubling condition .

Corollary 3.9. Let , and . Then

4. On Weighted Hardy Operators in Morrey Spaces

4.1. Pointwise Estimations

We consider the generalized Hardy operators

In the sequel with may be read either as or with the operators interpreted as

In the case is a power function, we also use the notation and their one-dimensional versions adjusted for the half-axis .

Lemma 4.1. Let and . (I)Let . Then the Hardy operator is defined on the space or on the space , if and only if and in this case (II)Let or and . Then the Hardy operator is defined on the space or on the space , if and only if for every and in this case

Proof. (I) The “If” Part. The sufficiency of condition (4.5) and estimate (4.6) follow from (3.12) under the choice and .The “Only If” Part. We choose a function equal to in a neighborhood of the origin and zero beyond this neighborhood. Then by Lemma 3.3. For this function , the existence of the integral is equivalent to condition (4.5).(II) The “If” part. The sufficiency of condition (4.7) and estimate (4.8) follow from (3.21) under the choice and .The “Only If” Part. We choose a function equal to in a neighborhood of infinity and zero beyond this neighborhood. Then by Remark 3.2. For this function , the existence of the integral is nothing else but condition (4.7).

Corollary 4.2. s (I) The Hardy operator is defined on the space or on the space , if and only if , and in this case (II) The Hardy operator is defined on the space or on the space , if and only if , and in this case

4.2. Weighted -Estimates for Hardy Operators in Morrey Spaces

The statements of Theorem 4.3 are well known in the case of Lebesgue space when ; see, for instance, [33, p. 6, 54]. As can be seen from the results below, inequalities for the Hardy operators in Morrey spaces admit the case when .

4.2.1. The Case of Power Weights

Theorem 4.3. Let , and . The operator (, resp.) is bounded from or to , where , if and only if , resp.).

Proof. The “only if” part follows from Corollary 4.2 and the “if part” from (4.9) and (4.10), since ; see Remark 3.2.

4.2.2. The Case of General Weights

We first deal with Hardy operators on a ball of a finite radius .

Theorem 4.4. Let , and and . Then the weighted Hardy operators and are bounded from or to , where , if respectively, or, equivalently, The conditions are necessary for the boundedness of the operators and , respectively.

Proof. By (2.10) and (2.11), the function is almost increasing, while is almost decreasing for every . Consequently, for and then supposing that . From the right-hand side inequality in (4.16) and Theorem 4.3, we obtain that the operator is bounded if , which is satisfied under the choice of sufficiently small, the latter being possible by (4.12). It remains to recall that condition (4.12) is equivalent to the assumption by Theorem 2.4. The necessity of the condition follows from the left-hand side inequality in (4.16). The case of the operator is similarly treated.

In the case of the whole space (), we admit that the weight may have an “oscillation between power functions” different at the origin and infinity. Correspondingly, the behavior at the origin and infinity is characterized by different indices and , as described in Section 2.1.3.

Theorem 4.5. Let , and and . Then the weighted Hardy operators and are bounded from or to , if respectively, or, equivalently,         The conditions are necessary for the boundedness of the operators and , respectively.

Proof. The restriction of to is covered by Theorem 4.4, so that it suffices to estimate . For we have where . By Lemma 3.5 we have , where the integral converges since with an arbitrarily small and . Then by (2.18). Here , since for sufficiently small ; see Remark 3.2.To deal with the second term in (4.20), it suffices to observe that for we have inequality (4.15) with replaced by and then the proof follows the same lines as in Theorem (4.4) after formula (4.16).The operator is considered in a similar way.

5. Application to Potential Operators

We consider the potential operator and in Theorem 5.3 show that its weighted boundedness in Morrey spaces—in the case of weights with —is a consequence of the nonweighted boundedness due to Adams [5] and the weighted boundedness of Hardy operators provided by Theorem 4.5.

The necessity of the boundedness of the Hardy operators for that of potential operators is a consequence of the following simple fact, where and are arbitrary Banach function spaces in the sense of Luxemburg (cf., e.g., [34]).

Lemma 5.1. Let be any weight function. For the boundedness of the weighted potential operator from to , it is necessary that the Hardy operators and are bounded from to , where .

The proof of the sufficiency of the obtained conditions is based on the pointwise estimate of the following lemma.

Lemma 5.2. Let with be a weight and a non-negative function. Then the following pointwise estimate holds:

Proof. We have We first consider the case . For with in this case, by the definition of the classes , we have which yield with and prove (5.2).Let now . We denote , where and stands for the fractional part of . Now The procedure is similar to the previous case; we can first manage with the fractional part , treating as a function in like in the previous case, and then repeat a similar procedure times treating as a function in .For definiteness we consider the case where ; the case of is similarly treated. By the definition of the class , we have (this step should be omitted when is an integer), that is, where and . We only have to take care about the kernel . We have We make use of the fact that and obtain where again only the first term must be studied. We repeat the same procedure times more and finally arrive at the kernel which is the kernel of the Hardy operator .

We are ready for the following statement, where notation (2.22) is used.

Theorem 5.3. Let , , and . (i)Let with . Then the condition or equivalently is sufficient for the boundedness of the potential operator (5.1) from the weighted space to the space , where .(ii)Let . Then the condition is necessary for the boundedness of the potential operator (5.1) from to .

Proof. The necessity part (ii) follows from Lemma 5.1 and Theorem 4.5.Part (i). We have to prove the boundedness of the operator from to . Since the non-weighted -boundedness of the potential operator is known [5], it suffices to show the boundedness of the operator . For that it remains to make use of Theorem 4.5. This completes the proof.

Corollary 5.4. Let , and . Then the potential operator (5.1) is bounded from into , , if and only if

Remark 5.5. As can be seen from the proof of Theorem 5.3, its statement remains valid under the condition more general than (5.13). Correspondingly, condition (5.14) may be written in a more general form: (Recall that in the case and in the case ; see Corollary 2.9.)

Acknowledgment

This work was supported by Research Grant SFRH/BPD/34258/2006, FCT, Portugal.

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