Φ-Admissible Sublinear Singular Operators and Generalized Orlicz-Morrey Spaces

Abstract

We study the boundedness of -admissible sublinear singular operators on Orlicz-Morrey spaces . These conditions are satisfied by most of the operators in harmonic analysis, such as the Hardy-Littlewood maximal operator and Calderón-Zygmund singular integral operator.

1. Introduction

As it is well known that Morrey [1] introduced the classical Morrey spaces to investigate the local behavior of solutions to second-order elliptic partial differential equations (PDE), we recall its definition as where ,  . Here and everywhere in the sequel stands for the ball in of radius centered at . Let be the Lebesgue measure of the ball and , where . was an expansion of in the sense that . We also denote by the weak Morrey space of all functions for which where denotes the weak -space (for see Definition 4). Morrey found that many properties of solutions to PDE can be attributed to the boundedness of some operators on Morrey spaces. Maximal functions and singular integrals play a key role in harmonic analysis since maximal functions could control crucial quantitative information concerning the given functions, despite their larger size, while singular integrals, Hilbert transform as its prototype, nowadays intimately connected with PDE, operator theory and other fields.

Let . The Hardy-Littlewood (H-L) maximal function of is defined by

The Calderón-Zygmund (C-Z) singular integral operator is defined by and bounded on , where is a “standard singular kernel,” that is, a continuous function defined on and satisfying the estimates

It is well known that the maximal and singular integral operators play an important role in harmonic analysis (see [2, 3]).

Orlicz spaces, introduced in [4, 5], are generalizations of Lebesgue spaces (see also, [6–8]). They are useful tools in harmonic analysis and its applications. For example, the Hardy-Littlewood maximal operator is bounded on for , but not on . Using Orlicz spaces, we can investigate the boundedness of the maximal operator near more precisely (see [9–11]).

We find it convenient to define the generalized Orlicz-Morrey spaces in the following form (see Definition 3 for the notion of Young functions).

Definition 1. Let be a positive measurable function on and a Young function. We define the generalized Orlicz-Morrey space as the space of all functions with finite quasinorm

Remark 2. The Calderón-Zygmund (C-Z) singular integral operators are bounded and expressed as (4) for all , with standard kernel . Then, one can prove that is of weak type and type , , for , and then is uniquely extended to an -bounded operator by the density of in . On the other hand, is not dense in Morrey spaces in general. Therefore, we need to give a precise definition of for the function in Morrey spaces; for example, for some ball which contains , with proving the absolutely convergence of the integral in the second term and the independence of the choice of the ball (see [12, 13] for example). Also, is dense in Orlicz spaces if and only if satisfies the condition.

The main purpose of this paper is to find sufficient conditions on general Young function and the functions , ensuring that the sublinear operators generated by singular integral operators are of weak or strong type from generalized Orlicz-Morrey spaces into . Note that the Orlicz-Morrey spaces were introduced and studied by Nakai in [12, 14]. Also the boundedness of the operators of harmonic analysis on Orlicz-Morrey spaces see also, [9, 10, 13–17].

By , we mean that with some positive constant independent of appropriate quantities. If and , we write and say that and are equivalent.

2. Preliminaries

Definition 3. A function is called a Young function if is convex, left-continuous, and .

From the convexity and , it follows that any Young function is increasing. If there exists such that , then for .

We say that if, for any , there exists a constant such that for all .

Recall that a function is said to be quasiconvex if there exist a convex function and a constant such that

Let be the set of all Young functions such that If , then is absolutely continuous on every closed interval in and bijective from to itself.

Definition 4 (Orlicz space). For a Young function , the set is called Orlicz space. The space endowed with the natural topology is defined as the set of all functions such that for all balls .

Note that is a Banach space with respect to the norm see, for example, [18, Section 3, Theorem 10], so that

Definition 5. The weak Orlicz space is defined by the norm where .

For Young functions and , we write if there exists a constant such that If , then with equivalent norms.

For a Young function and , let If , then is the usual inverse function of . We note that

A Young function is said to satisfy the -condition, denoted also by , if for some .

For a Young function , the complementary function is defined by The complementary function is also a Young function and . If , then for and for . If , , and , then . If , then .

It is known that

Let be a sublinear operator; that is, .

Definition 6 (-admissible sublinear singular operator). Let be any Young function. A sublinear operator will be called -admissible singular operator, if(1) satisfies the size condition of the form for and ;(2) is bounded in .

In the case the -admissible singular operator will be called the -admissible singular operator.

Definition 7 (weak -admissible sublinear singular operator). Let be any Young function. A sublinear operator will be called the weak -admissible singular operator, if(1) satisfies the size condition (21);(2) is bounded from to the weak .

In the case the weak -admissible singular operator will be called weak -admissible singular operator.

Necessary and sufficient conditions on for the boundedness of in Orlicz spaces have been obtained in [19, Theorem 2.1] and [20, Theorem ]. Note that if and only if . Also, if is a Young function, then if and only if is quasiconvex for some (see, e.g., [20, page 15]). With this remark taken into account, the known boundedness statement runs as follows.

The following theorem was in fact proved in [11].

Theorem 8 (see [11]). Let be any Young function. Then the maximal operator is bounded from to and for bounded in .

Sufficient conditions on for the boundedness of the singular integral operator in Orlicz spaces are known; see [20, Theorem ] and [17, Theorem 3.3]. The following theorem was proved in [21].

Theorem 9 (see [21]). Let be a Young function and let be a singular integral operator. If , then the operator is bounded on and if , then the operator is bounded from to .

Remark 10. Note that, from Theorems 8 and 9 we get the any Young function the maximal operator and for the Young function the singular integral operator are the weak -admissible singular operator. Also for the Young function the maximal operator and for the Young function the singular integral operator are the -admissible singular operator.

Definition 11 (generalized Orlicz-Morrey space). Let be a positive measurable function on and be any Young function. We denote by the generalized Orlicz-Morrey space, the space of all functions with finite quasinorm Also, by we denote the weak generalized Orlicz-Morrey space of all functions for which
According to this definition, we recover the Orlicz space and weak Orlicz space under the choice : Also according to this definition, we recover the generalized Morrey space and weak generalized Morrey space under the choice :

The following statement, containing Guliyev results obtained in [22–24], was proved in [25] (see also [26]).

Theorem 12. Let , and satisfies the condition where does not depend on and . Then, for a -admissible sublinear singular operator is bounded from to and for a weak -admissible sublinear singular operator is bounded from to .

We will use the following statement on the boundedness of the weighted Hardy operator: where is a weight.

The following theorem was proved in [27] (see also, [28]).

Theorem 13. Let , and be weights on and be bounded outside a neighborhood of the origin. The inequality holds for some for all non-negative and non-decreasing on if and only if Moreover, the value is the best constant for (28).

Remark 14. In (28) and (29) it is assumed that and .

3. -Admissible Sublinear Singular Operator in the Spaces

In this section, sufficient conditions on for the boundedness of the -admissible sublinear singular operator in generalized Orlicz-Morrey spaces are obtained.

The following lemma was generalizations of the Guliyev lemma for Orlicz spaces [22–24].

Lemma 15. Let be any Young function and , , and . Then for the -admissible sublinear singular operator the following inequality is valid and for the weak -admissible sublinear singular operator the following inequality is valid Here defined in (16).

Proof. Let be any Young function and the operator be a -admissible sublinear singular operator. With the notation , we represent as and then Since , by the boundedness of in , it follows that
Next, observe that , imply . We get By Fubini's theorem, we have Applying the following Hölder's inequality we get
Moreover, Thus,
On the other hand, by (20) we get and then Thus,
Let the operator be a weak -admissible sublinear singular operator. Then, by the weak boundedness of on Orlicz space and (39) it follows that
Then by (39) and (44), we get inequality (31).

Corollary 16 (see [15]). Let be any Young function and , , , and . Then, for the singular integral operator the following inequalities are valid: if and if .

Corollary 17 (see [22–24]). Let and , , , and . Then, for the -admissible sublinear singular operator the following inequality is valid: and for the weak -admissible sublinear singular operator the following inequality is valid

Theorem 18. Let be any Young function and , and satisfy the condition where does not depend on and . Then, a -admissible sublinear singular operator is bounded from to and a weak -admissible sublinear singular operator is bounded from to .

Proof. By Lemma 15 and Theorem 13 with , , , , we have

Note that, from Theorems 8, 9, and 18 we get the following corollaries, which are proven in [15].

Corollary 19. Let be any Young function and ,  , and satisfy condition (49). Then the maximal operator is bounded from to and for , the operator is bounded from to .

Corollary 20. Let be any Young function and ,  , and satisfy the condition (49). Then, the singular integral operator is bounded from to for and from to for .

Conflict of Interests

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

Acknowledgments

The author would like to express their gratitude to the referee for his very valuable comments and suggestions. Thanks to referee we added the Remark 2.