Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 283285, 19 pages

doi:10.1155/2012/283285

Research Article

Weighted Hardy Operators in Complementary Morrey Spaces

Dag Lukkassen,¹ Lars-Erik Persson,² and Stefan Samko³

¹Department of Technology, Narvik University College, P.O. Box 385, 8505 Narvik, Norway

²Department of Mathematics, Luleå University of Technology, SE 921 87 Luleå, Sweden

³Departamento de Matematica, Universidade do Algarve, 6005-139 Faro, Portugal

Received 23 July 2012; Accepted 20 September 2012

Academic Editor: Alois Kufner

Copyright © 2012 Dag Lukkassen et al. 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 the multidimensional weighted Hardy-type operators and with radial type weight , in the generalized complementary Morrey spaces defined by an almost increasing function . We prove a theorem which provides conditions, in terms of some integral inequalities imposed on and , for such a boundedness. These conditions are sufficient in the general case, but we prove that they are also necessary when the function and the weight are power functions. We also prove that the spaces over bounded domains Ω are embedded between weighted Lebesgue space with the weight and such a space with the weight , perturbed by a logarithmic factor. Both the embeddings are sharp.

1. Introduction

Hardy operators and related Hardy inequalities are widely studied in various function spaces, and we refer to the books [1–4] and references therein. They continue to attract attention of researchers both as an interesting mathematical object and a useful tool for many purposes: see for instance the recent papers [5, 6]. Results on weighted estimations of Hardy operators in Lebesgue spaces may be found in the abovementioned books. In the papers [7–9] the weighted boundedness of the Hardy type operators was studied in Morrey spaces.

In this paper we study multi-dimensional weighted Hardy operators

where , in the so called complementary Morrey spaces. The one-dimensional case will include the versions

adjusted for the half-axis , so that in the sequel with may be read either as or .

The classical Morrey spaces , defined by the norm

where , are well known, in particular, because of their usage in the study of regularity properties of solutions to PDE; see for instance the books [10–12] and references therein. There are also known various generalizations of the classical Morrey spaces , and we refer for instance to the surveying paper [13]. One of the direct generalizations is obtained by replacing in (1.3) by a function , usually satisfying some monotonicity type conditions. We also denote it as without danger of confusion. Such spaces appeared in [14, 15] and were widely studied in [16, 17]. The spaces , defined by the norm

where , are known as generalized local Morrey spaces

Hardy type operators (1.1) in the spaces have been studied in [7–9].

The norm in Morrey spaces controls the smallness of the integral over small balls (and also a possible growth of this integral for in the case is unbounded). There are also known spaces , called complementary Morrey spaces, with the norm controlling possible growth, as , of the integral

over exterior of balls. Such spaces have sense in the local setting only. It is introduced in [16, 17] that the space is defined as the space of all functions with the finite norm

where admission of the multiplier vanishing at controls the growth of the norm . Such spaces were also studied in some later papers, and we refer for instance to [18]. We refer also to the paper [19] where variable exponent complementary spaces of such type were introduced.

During the last decades various classical operators, such as maximal, singular, and potential operators were widely investigated both in classical and generalized Morrey spaces, including the complementary Morrey spaces. The mapping properties of Hardy type operators in complementary Morrey spaces were not known. In this paper we obtain conditions for the -boundedness of Hardy type operators in complementary Morrey spaces. Thus we make certain contributions to the known theory of Hardy type inequalities; see, for example, the books [2–4] and references given there.

We also prove a new property for the generalized complementary Morrey spaces; see Theorem 3.1, by showing that the spaces over bounded domains are embedded between the weighted Lebesgue space with the weight , and such a space with the weight , perturbed by a logarithmic factor. In the case where was a power function, this was proved in [19].

The paper is organized as follows. In Section 2 we give definitions and necessary preliminaries, including conditions for radial functions to belong to complementary Morrey spaces. In Section 3 we prove the abovementioned embedding of the generalized Morrey space between Lebesgue weighted spaces. In Section 4, which plays a crucial role in the preparation of the proofs of the main results, we prove pointwise estimate for the Hardy-type constructions via the norm defining the complementary Morrey space. In Section 5 we give the final theorems on the weighted -boundedness of Hardy operators in complementary Morrey spaces. In the appendix we collect various properties of weights from the Bary-Stechkin class which we need when we formulate some sufficient conditions for the boundedness in terms of the Matuszewska-Orlicz indices of the functions and .

2. Complementary Morrey Spaces and Their Properties

2.1. Definitions

Let be an open set in , and , , and . Let also be a continuous function nonnegative on with which may arbitrarily grow as . Let also .

Definition 2.1. Let . The complementary Morrey space , where , is the space of functions such that

In the case of power factor , , we also denote

without danger of confusion.

The space is nontrivial in the case of any locally bounded function with an arbitrary behaviour at infinity, because bounded functions with compact support belong then to this space.

We will also use the following notation for the modular:

The function , defining the complementary Morrey space, will be called definitive function.

Remark 2.2. The function in (1.4), defining the Morrey spaces, in some papers is called weight function. We prefer to call both and as definitive functions, keeping the word weight for its natural use in the theory of function spaces, that is, for the cases where the function itself is controlled by a weight.

2.2. On Belongness of Radial Functions to the Space

Lemma 2.3. For a nonnegative radial type functions , , the condition

is sufficient to belong to . (It is also necessary, when or is a ball centered at .)

The proof of Lemma 2.3 is obvious. By means of this lemma we easily obtain the following corollary.

Corollary 2.4. Let be bounded. A power function with belongs to the space if and only if

In the case , the same holds with replaced by . When , the necessary and sufficient conditions are and .

As is known, there may be given sufficient numerical inequalities for the validity of the integral condition in (2.4) in terms of Matuszewska-Orlicz indices and of the function . We give below such sufficient inequalities, but in order not to interrupt the main body of the paper by notions connected with such indices and related Bary-Stechkin function classes , we put these notions in the appendix.

Let now be a nonnegative function in Bary-Stechkin class, namely,

Then , , so that the condition is sufficient for to be in . Note also that (2.6) is equivalent to the inequalities , in terms of the indices, where the second inequality is to be used only in the case ; see (A.17).

3. Complementary Morrey Spaces Are Closely Embedded between Weighted Lebesgue Spaces

The complementary Morrey spaces are close to a certain weighted Lebesgue space as stated in the following theorem, which provides sharp embeddings of independent interest. The notation for the weighted space is taken in the form . The class , used in this theorem, is defined in the appendix; see Definitions A.1-A.2 therein. In the case where is a power function, Theorem 3.1 was proved in [19].

Theorem 3.1. Let be a bounded open set, , , , and let be absolutely continuous and

Then

where , . The left-hand side embedding in (3.2) is strict, when satisfies the condition

(in particular, if , ); that is, there exists a function such that

The right-hand side embedding in (3.2) is strict, when satisfies the doubling condition , (in particular, if , ), there exists a function such that

Proof. Without loss of generality, we may take for simplicity, supposing that .

1⁰. The left-hand side embedding: since the function is almost increasing, we have

so that

. The right-hand side embedding: we have

where

so that

Therefore,

where the last integral converges when since .

3⁰. The strictness of the embeddings: the corresponding counterexamples are

Calculations for the function , which is obviously not in , are easy. Indeed,

where the right-hand side is bounded by (3.3).

In the case of the function we have

for every . However, for small , where , we obtain

Taking into account that is almost increasing and satisfies the doubling condition, we get

which completes the proof of the theorem.

Remark 3.2. Under the assumption (3.3), the upper embedding in (3.2) does not hold with . The corresponding counterexample is which is in , but .

4. Weighted Estimates of Functions in Complementary Morrey Spaces

In the following lemmas we give a pointwise estimate of the “Hardy-type” constructions in terms of the modular with . These estimates are of independent interest and also crucial for our study of the Hardy operators in the complementary Morrey space.

Lemma 4.1. Let , , let , and . Then

for , where does not depend on and and

Proof. We use the dyadic decomposition as follows:

where . Since there exists a such that is almost increasing, we observe that

on . Applying this in (4.3) and making use of the Hölder inequality with the exponent , we obtain

Hence

On the other hand we have

The function is decreasing, and the function is almost decreasing after multiplication by some power functions. Therefore,

Then (4.1) follows from (4.6) and (4.8).

Lemma 4.2. Let , , and , . Let also

Then

where does not depend on and .

Proof. We use the corresponding dyadic decomposition:

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

On the other hand, the integral on the right-hand side of (4.10) can be estimated as follows:

which completes the proof.

Corollary 4.3. Let , let , and . Let also

Then

5. Weighted Hardy Operators in Complementary Morrey Spaces

5.1. Pointwise Estimations

The proof of our main result of this Section given in Theorems 5.3 and 5.6 is prepared by the following Theorems 5.1 and 5.2 on the pointwise estimates of the Hardy-type operators.

Theorem 5.1. Let , let , and . The condition

is sufficient for the Hardy operator to be defined on the space . Under this condition, the pointwise estimate

holds.

Proof. Apply Lemma 4.1 with .

Theorem 5.2. Let and , or and . The condition

is sufficient for the Hardy operator to be defined on the space , and in this case the following pointwise estimate:

holds.

Proof. Apply Lemma 4.2 with .

5.2. Weighted Boundedness of Hardy Operators in Complementary Morrey Spaces

5.2.1. The Case of the Operator

The -boundedness of the multidimensional Hardy operators within the frameworks of Lebesgue spaces (the case in (1.4)) with and is well known: see for instance, [4, page 54]. For Morrey spaces, both local and global, the boundedness of Hardy operators was studied in [7, 8]. We call attention of the reader to the fact that, in contrast to the case of Lebesgue spaces, Hardy-type inequalities in both usual and complementary Morrey spaces different from Lebesgue spaces (i.e., in the case or ) admit the value .

Theorem 5.3. Let and let

The operator is bounded from to if , where

and is the same as in (5.1). Under this condition,

Proof. From the estimate (5.2) of Theorem 5.1 we have

Hence, calculating , passing to polar coordinates, we obtain (5.6).

Remark 5.4. Note that

in (5.6) if we suppose that .

The following corollary gives sufficient conditions for the boundedness of the operator in terms of the Matuszewska-Orlicz indices of the function and the weight (we refer to the appendix for these notions).

Corollary 5.5. Let (with admitted in the case ) and the conditions (5.5) be satisfied. Suppose also that

in the case . The operator is bounded from to if

The condition , is guaranteed by the inequalities

In the power case , , and , the conditions (5.10)-(5.11) reduce to

conditions (5.13) are also necessary for the operator to be bounded from to .

Proof. We have to check that the condition of Theorem 5.3 holds under the assumptions (5.10)-(5.11). From the inequality it follows that

We represent as and by (5.10) obtain

which is bounded in view of the second assumption in (5.11), by the property .

The sufficiency of the conditions (5.13) in the case of power functions is then obvious since and in this case.

Let us show the necessity of these conditions. From the boundedness with , and , by standard homogeneity arguments with the use of the dilation operator it is easily derived that the conditions , necessarily hold, via the relation

(We take into account that we excluded in this theorem.)

The necessity of the remaining condition in (5.13) follows from the fact that by Corollary 2.4, so that this condition is necessary for the operator with to be defined on the space .

5.2.2. The Case of the Operator

Let

where is the same as in (5.3).

Theorem 5.6. Let , and

The operator is bounded from to if

and then

Proof. From the estimate (5.4) we obtain

from which (5.20) follows.

As above, we provide also sufficient conditions for the boundedness of the operator in terms of the Matuszewska-Orlicz indices.

Corollary 5.7. Let (with admitted in the case and and satisfy the conditions (5.18). The operator is bounded from to if

and the condition (5.10) holds; the assumption is guaranteed by the conditions

In the case , and , the conditions (5.22) and (5.10) reduce to

conditions (5.24) are also necessary for the operator to be bounded from to .

Proof. We have to find, in terms of the Matuszewska-Orlicz indices, conditions sufficient for the validity of (5.19). For the latter we have

by the assumption . With and the condition (5.10) we arrive at , where the boundedness of the right-hand side is guaranteed by the condition .

The proof for the case of power functions is similar to that in Corollary 5.5.

Appendix

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

In the sequel, a nonnegative function on , , is called almost increasing (almost decreasing) if there exists a constant () 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 A.1. Let :

(1)by we denote the class of continuous and positive functions on such that there exists finite or infinite limit ;(2)by we denote the class of almost increasing functions on ;(3)by we denote the class of functions such that for some ;(4)by we denote the class of functions such that is almost decreasing for some .

Definition A.2. Let :

(1)by we denote the class of functions which are continuous and positive and almost increasing on and which have the finite or infinite limit ,(2)by we denote the class of functions such for some .

By we denote the set of functions on whose restrictions onto are in and restrictions onto are in . Similarly, the set is defined.

A.1. ZBS Classes and MO Indices of Weights at the Origin

In this subsection we assume that .

We say that a function belongs to a Zygmund class , , if and , , and to a Zygmund class , , if and , . We also denote

the latter class being also known as Bary-Stechkin-Zygmund class [20].

For a function , the numbers

are known as the Matuszewska-Orlicz type lower and upper indices of the function . The property of functions to be almost increasing or almost decreasing after the multiplication (division) by a power function is closely connected with these indices. We refer to [22–28] for such a property and these indices.

Note that in this definition need not to be an -function: only its behaviour at the origin is of importance. Observe that for , and , and the following formulas are valid:

for .

The proof of the following statement may be found in [21], Theorems 3.1, 3.2, and 3.5. (In the formulation of Theorems in [21] it was supposed that , and . It is evidently true also for and all , in view of formulas (A.3)).

Theorem A.3. Let and . Then and . Besides this, , and , and for the inequalities

hold with an arbitrarily small and , .

We define the following subclass in :

A.2. ZBS Classes and MO Indices of Weights at Infinity

Following Section 2.2 in [29], we use the following notation.

Let . We put , 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 (A.2):

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

By (A.10) and (A.11), one can easily reformulate properties of functions of the class near the origin, given in Theorem A.3 for the case of the corresponding behavior at infinity of functions of the class and obtain that

We say that a continuous function 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 (note that in (A.13) we use and , not and ). In the case where the indices coincide, that is, , we will simply write and similarly for . We also denote

Making use of Theorem A.3 for and relations (A.11), one easily arrives at the following statement.

Lemma A.4. Let . Then

Acknowledgment

The third author thanks Luleå University of Technology for financial support for research visits in June of 2012.

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