Journal of Function Spaces

Volume 2014, Article ID 617414, 11 pages

http://dx.doi.org/10.1155/2014/617414

## On the Riesz Potential and Its Commutators on Generalized Orlicz-Morrey Spaces

^{1}Department of Mathematics, Ahi Evran University, 40200 Kirsehir, Turkey^{2}Institute of Mathematics and Mechanics, 1141 Baku, Azerbaijan

Received 23 October 2013; Revised 24 December 2013; Accepted 25 December 2013; Published 21 January 2014

Academic Editor: Yoshihiro Sawano

Copyright © 2014 Vagif S. Guliyev and Fatih Deringoz. 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 consider generalized Orlicz-Morrey spaces including their weak versions . In these spaces we prove the boundedness of the Riesz potential from to and from to . As applications of those results, the boundedness of the commutators of the Riesz potential on generalized Orlicz-Morrey space is also obtained. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on , which do not assume any assumption on monotonicity of , in .

#### 1. Introduction

The theory of boundedness of classical operators of the real analysis, such as the maximal operator, fractional maximal operator, Riesz potential, and the singular integral operators, and so forth, has been extensively investigated in various function spaces. Results on weak and strong type inequalities for operators of this kind in Lebesgue spaces are classical and can be found for example in [1–3]. This boundedness extended to several function spaces which are generalizations of -spaces, for example, Orlicz spaces, Morrey spaces, Lorentz spaces, Herz spaces, and so forth.

Orlicz spaces, introduced in [4, 5], are generalizations of Lebesgue spaces . 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 [6–8]).

It is well known that the Riesz potential of order () plays an important role in harmonic analysis, PDE, and potential theory (see [2]). Recall that is defined by

The classical result by Hardy-Littlewood-Sobolev states that, if , then the operator is bounded from to if and only if and, for , the operator is bounded from to if and only if . For boundedness of on Morrey spaces , see Peetre (Spanne) [9] and Adams [10].

The boundedness of from Orlicz space to was studied by O’Neil [11] and Torchinsky [12] under some restrictions involving the growths and certain monotonicity properties of and . Moreover Cianchi [6] gave a necessary and sufficient condition for the boundedness of from to and from to weak Orlicz space , which contain results above.

In [13] the authors study the boundedness of the maximal operator and the Calderón-Zygmund operator from one generalized Orlicz-Morrey space to and from to the weak space .

Our definition of Orlicz-Morrey spaces (see Section 3) is different from that of Sawano et al. [14] and Nakai [15, 16].

The main purpose of this paper is to find sufficient conditions on general Young functions and functions , which ensure the boundedness of the Riesz potential from one generalized Orlicz-Morrey spaces to another and from to weak generalized Orlicz-Morrey spaces and the boundedness of the commutator of the Riesz potential from to .

In the next section we recall the definitions of Orlicz and Morrey spaces and give the definition of Orlicz-Morrey and generalized Orlicz-Morrey spaces in Section 3. In Section 4 and Section 5 the results on the boundedness of the Riesz potential and its commutator on generalized Orlicz-Morrey spaces are obtained.

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

#### 2. Some Preliminaries on Orlicz and Morrey Spaces

In the study of local properties of solutions of partial differential equations, together with weighted Lebesgue spaces, Morrey spaces play an important role; see [17]. Introduced by Morrey Jr. [18] in 1938, they are defined by the norm 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 is the volume of the unit ball in .

Note that and . If or , then , where is the set of all functions equivalent to on .

We also denote by the weak Morrey space of all functions for which where denotes the weak -space.

We refer in particular to [19] for the classical Morrey spaces.

We recall the definition of Young functions.

*Definition 1. *A function is called a Young function if is convex and left-continuous, , and .

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

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 2 2 (Orlicz space). *For a Young function , the set
is called Orlicz space. If , , then . If and , then . The space endowed with the natural topology is defined as the set of all functions such that for all balls . We refer to the books [20–22] for the theory of Orlicz spaces.

is a Banach space with respect to the norm
We note that
For a measurable set , a measurable function , and , let
In the case , we shortly denote it by .

*Definition 3. *The weak Orlicz space
is defined by the norm

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 by , if
for some . If , then . A Young function is said to satisfy the -condition, denoted also by , if
for some . The function satisfies the -condition but does not satisfy the -condition. If , then satisfies both the conditions. The function satisfies the -condition but does not satisfy the -condition.

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 . Note that if and only if . It is known that

Note that Young functions satisfy the properties

The following analogue of the Hölder inequality is known; see [23].

Theorem 4 (see [23]). *For a Young function and its complementary function , the following inequality is valid:
*

*The following lemma is valid.*

*Lemma 5 (see [1, 24]). Let be a Young function and a set in with finite Lebesgue measure. Then
*

*In the next sections where we prove our main estimates, we use the following lemma, which follows from Theorem 4, Lemma 5, and (16).*

*Lemma 6. For a Young function and , the following inequality is valid:
*

*3. Orlicz-Morrey and Generalized Orlicz-Morrey Spaces*

*3. Orlicz-Morrey and Generalized Orlicz-Morrey Spaces**Definition 7 (Orlicz-Morrey space). *For a Young function and , one denotes by the Orlicz-Morrey space, the space of all functions with finite quasinorm

*Note that and if , then .*

*We also denote by the weak Orlicz-Morrey space of all functions for which
where denotes the weak -space of measurable functions for which
*

*Definition 8 (generalized Orlicz-Morrey space). *Let be a positive measurable function on and any Young function. One denotes by the generalized Orlicz-Morrey space, the space of all functions with finite quasinorm
Also by one denotes the weak generalized Orlicz-Morrey space of all functions for which

According to this definition, we recover the spaces and under the choice :

According to this definition, we recover the generalized Morrey spaces and weak generalized Morrey spaces under the choice , :

*Remark 9. *There are different kinds of Orlicz-Morrey spaces in the literature. We want to make some comment about these spaces.

Let be a function and a Young function. (1)For a cube , define -average over by
and define its -average over by
(2)Define
The function space is defined to be the Orlicz-Morrey space of the first kind as the set of all measurable functions for which the norm is finite.(3)Define
The function space is defined to be the Orlicz-Morrey space of the second kind as the set of all measurable functions for which the norm is finite.

According to our best knowledge, it seems that is more investigated than . The space is investigated in [15, 16, 25–34] and the space is investigated in [14, 35–37].

*4. Boundedness of the Riesz Potential in Generalized Orlicz-Morrey Spaces*

*4. Boundedness of the Riesz Potential in Generalized Orlicz-Morrey Spaces**In this section sufficient conditions on the pairs and for the boundedness of from one generalized Orlicz-Morrey spaces to another and from to the weak space have been obtained.*

*Necessary and sufficient conditions on for the boundedness of from to and to have been obtained in [6, Theorem 2]. In the statement of the theorem, is the Young function associated with the Young function and whose Young conjugate is given by
where
and , the Holder conjugate of , equals either or 1, according to whether or and denotes the Young function defined by
where
*

*Recall that, if and are functions from into , then is said to dominate globally if a positive constant exists such that for all .*

*Theorem 10 (see [6]). Let . Let and Young functions and let and be the Young functions defined as in (34) and (32), respectively. Then(i)the Riesz potential is bounded from to if and only if
(ii)The Riesz potential is bounded from to if and only if
*

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

*The following theorem was proved in [38] (see, also [13]).*

*Theorem 11. Let , , and be weights on and bounded outside a neighborhood of the origin. The inequality
holds for some for all nonnegative and nondecreasing on if and only if
Moreover, the value is the best constant for (39).*

*Lemma 12. Let and Young functions and , , Young function defined as in (34). If and dominates globally, then
*

*Proof. *If , then
For the proof of this claim see [39, page 50].

If dominates globally, then a positive constant exists such that
Indeed,
Thus, (41) follows from (42), (43), and (16).

*The following lemma is valid.*

*Lemma 13. Let , and Young functions, , and . If satisfy the conditions (37), then
and if satisfy the conditions (36), then
*

*Proof. *Suppose that the conditions (37) are satisfied. For arbitrary , set for the ball centered at and of radius , . We represent as
and have

Since , , and from the boundedness of from to (see Theorem 10) it follows that
where constant is independent of .

It is clear that , implies . We get
By Fubini’s theorem we have
By Lemmas 6 and 12 for we get

Moreover,
is valid. Thus

On the other hand, using the property of Young function as it is mentioned in (16)
and we get

Thus

Suppose that the conditions (37) are satisfied. From the boundedness of from to (see Theorem 10) and (56) it follows that
Then by (53) and (58) we get the inequality (46).

*Theorem 14. Let and the functions and satisfy the condition
where does not depend on and . Then for the conditions (37), is bounded from to and for the conditions (36), is bounded from to .*

*Proof. *By Lemma 13 and Theorem 11 we get
if (37) is satisfied and
if (36) is satisfied.

*Remark 15. *Recall that, for ,
hence Theorem 14 implies the boundedness of the fractional maximal operator from to and from to .

If we take , , , , at Theorem 14 we get following corollary which was proved in [40] and containing results obtained in [41–45].

*Corollary 16. Let , , and satisfy the condition
where does not depend on and . Then is bounded from to for and from to for .*

*In the case , from Theorem 14 we get the following Spanne type theorem for the boundedness of the Riesz potential on Orlicz-Morrey spaces.*

*Corollary 17. Let , and Young functions, , and satisfy the condition
where does not depend on . Then for the conditions (37), is bounded from to and for the conditions (36), is bounded from to .*

*Remark 18. *If we take , , , , at Corollary 17 we get Spanne type boundedness of ; that is, if , , , , and , then for the Riesz potential is bounded from to and for , is bounded from to .

*5. Commutators of Riesz Potential in the Spaces *

*5. Commutators of Riesz Potential in the Spaces*

*For a function , let be the corresponding multiplication operator defined by for measurable function . Let be the classical Calderón-Zygmund singular integral operator; then the commutator between and is denoted by . A famous theorem of Coifman et al. [46] gave a characterization of -boundedness of when are the Riesz transforms . Using this characterization, the authors of [46] got a decomposition theorem of the real Hardy spaces. The boundedness result was generalized to other contexts and important applications to some nonlinear PDEs were given by Coifman et al. [47].*

*We recall the definition of the space of .*

*Definition 19. *Suppose that ; let
where
Define

Modulo constants, the space is a Banach space with respect to the norm .

*Remark 20. *(1) The John-Nirenberg inequality: there are constants , , such that for all and

(2) The John-Nirenberg inequality implies that
for .

(3) Let . Then there is a constant such that
where is independent of , , , and .

*Definition 21. *A Young function is said to be of upper type p (resp., lower type ) for some , if there exists a positive constant such that, for all (resp., ) and ,

*Remark 22. *We know that if is lower type and upper type with , then . Conversely if , then is lower type and upper type with (see [20]).

*Lemma 23 (see [48]). Let be a Young function which is lower type and upper type with . Let be a positive constant. Then there exists a positive constant such that for any ball of and
implies that .*

*In the following lemma we provide a generalization of the property (69) from -norms to Orlicz norms.*

*Lemma 24. Let and a Young function. Let is lower type and upper type with ; then
*

*Proof. *By Hölder’s inequality, we have

Now we show that
Without loss of generality, we may assume that ; otherwise, we replace by . By the fact that is lower type and upper type and (12) it follows that
By Lemma 23 we get the desired result.

*Remark 25. *Note that statements of type of Lemma 24 are known in a more general case of rearrangement invariant spaces and also variable exponent Lebesgue spaces , see [49, 50], but we gave a short proof of Lemma 24 for completeness of presentation.

*Definition 26. *Let be a Young function. Let

*Remark 27. *It is known that if and only if (see [21]).

*Remark 28. *Remarks 27 and 22 show us that a Young function is lower type and upper type with if and only if .

*The characterization of boundedness of the commutator between and was given by Chanillo [51].*

*Theorem 29 (see [51]). Let , , and . Then is a bounded operator from to if and only if .*

*The boundedness of the commutator was given by Fu et al. [52].*

*Theorem 30 (see [52]). Let and . Let be a Young function and defined, via its inverse, by setting, for all , . If , and then 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 [53].*

*Theorem 31. Let , , and be weights on and bounded outside a neighborhood of the origin. The inequality
holds for some for all nonnegative and nondecreasing on if and only if
Moreover, the value is the best constant for (79).*

*Remark 32. *In (79) and (80) it is assumed that and .

*The following lemma is valid.*

*Lemma 33. Let and . Let be a Young function and defined, via its inverse, by setting, for all , , and and ; then the inequality
holds for any ball and for all .*

*Proof. *For arbitrary , set for the ball centered at and of radius . Write with and . Hence
From the boundedness of from to (see Theorem 30) it follows that

For we have

Then
Let us estimate :

Applying Hölder’s inequality, by Lemma 24 and (70), we getIn order to estimate note that
By Lemma 24, we get
Thus, by (52)
Summing and we get
Finally,