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

Guliyev, Vagif S.; Deringoz, Fatih

doi:https://doi.org/10.1155/2014/617414

Journal of Function Spaces

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Morrey Spaces and Related Function Spaces

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Volume 2014 | Article ID 617414 | https://doi.org/10.1155/2014/617414

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

Vagif S. Guliyev^1,2and Fatih Deringoz¹

Academic Editor: Yoshihiro Sawano

Received23 Oct 2013

Revised24 Dec 2013

Accepted25 Dec 2013

Published21 Jan 2014

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

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

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

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, and the statement of Lemma 33 follows by (56).

Theorem 34. Let and . Let be a Young function and defined, via its inverse, by setting, for all , , and and . and satisfy the condition where does not depend on and .
Then the operator is bounded from to . Moreover

Proof. The statement of Theorem 34 follows by Lemma 33 and Theorem 31 in the same manner as in the proof of Theorem 14.

If we take , , , , at Theorem 34 we get following corollary which was proved in [40] (see, also [54]).

Corollary 35. Let , , , , and satisfy the condition where does not depend on and . Then is bounded from to .

In the case , from Theorem 34 we get the following Spanne type theorem for the boundedness of the operator on Orlicz-Morrey spaces.

Corollary 36. Let , , , and . Let also be a Young function and defined, via its inverse, by setting, for all , , , , and satisfy the condition where does not depend on . Then is bounded from to .

Remark 37. If we take , , , , at Corollary 36 we get Spanne type boundedness of ; that is, if , , , , and , then for the operator is bounded from to and for , is bounded from to .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their gratitude to the referees for their very valuable comments and suggestions. The research of Vagif S. Guliyev and Fatih Deringoz was partially supported by the Grant of Ahi Evran University Scientific Research Projects (PYO.FEN.4003.13.003) and (PYO.FEN.4003-2.13.007).

References

C. Bennett and R. Sharpley, Interpolation of Operators, vol. 129 of Pure and Applied Mathematics, Academic Press, Boston, Mass, USA, 1988.
View at: MathSciNet
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, vol. 30 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1970.
View at: MathSciNet
A. Torchinsky, Real-Variable Methods in Harmonic Analysis, vol. 123 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1986.
View at: MathSciNet
W. Orlicz, “Über eine gewisse Klasse von Räumen vom Typus B,” Bulletin International de l'Académie Polonaise A, no. 8-9, pp. 207–220, 1932, reprinted in Collected Papers, PWN, Warszawa, Poland, pp. 217–230, 1988.
View at: Google Scholar
W. Orlicz, “Über Räume $(L^{M})$ ,” Bulletin International de l'Académie Polonaise A, pp. 93–107, 1936, reprinted in Collected Papers, PWN, Warszawa, Poland, pp. 345–359, 1988.
View at: Google Scholar
A. Cianchi, “Strong and weak type inequalities for some classical operators in Orlicz spaces,” Journal of the London Mathematical Society, vol. 60, no. 1, pp. 187–202, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H.-o. Kita, “On maximal functions in Orlicz spaces,” Proceedings of the American Mathematical Society, vol. 124, no. 10, pp. 3019–3025, 1996.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. Kita, “On Hardy-Littlewood maximal functions in Orlicz spaces,” Mathematische Nachrichten, vol. 183, pp. 135–155, 1997.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. Peetre, “On the theory of $M_{p, λ}$ spaces,” Journal of Functional Analysis, vol. 4, no. 1, pp. 71–87, 1969.
View at: Publisher Site | Google Scholar
D. R. Adams, “A note on Riesz potentials,” Duke Mathematical Journal, vol. 42, no. 4, pp. 765–778, 1975.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. O'Neil, “Fractional integration in Orlicz spaces. I,” Transactions of the American Mathematical Society, vol. 115, pp. 300–328, 1965.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. Torchinsky, “Interpolation of operations and Orlicz classes,” Studia Mathematica, vol. 59, no. 2, pp. 177–207, 1976.
View at: Google Scholar | MathSciNet
F. Deringoz, V. S. Guliyev, and S. G. Samko, “Boundedness of maximal and singular operators on generalized Orlicz-Morrey spaces,” in Operator Theory, Operator Algebras and Applications, vol. 242 of Operator Theory: Advances and Applications, pp. 1–24, 2014.
View at: Google Scholar
Y. Sawano, S. Sugano, and H. Tanaka, “Orlicz-Morrey spaces and fractional operators,” Potential Analysis, vol. 36, no. 4, pp. 517–556, 2012.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
E. Nakai, “Generalized fractional integrals on Orlicz-Morrey spaces,” in Banach and Function Spaces, pp. 323–333, Yokohama Publishers, Yokohama, Japan, 2004.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
E. Nakai, “Orlicz-Morrey spaces and the Hardy-Littlewood maximal function,” Studia Mathematica, vol. 188, no. 3, pp. 193–221, 2008.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, vol. 105 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, USA, 1983.
View at: MathSciNet
C. B. Morrey Jr., “On the solutions of quasi-linear elliptic partial differential equations,” Transactions of the American Mathematical Society, vol. 43, no. 1, pp. 126–166, 1938.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Kufner, O. John, and S. Fučík, Function Spaces, Noordhoff International, Leyden, Mass, USA, Publishing House of the Czechoslovak Academy Of Sciences, Prague, Czech Republic, 1977.
View at: MathSciNet
V. Kokilashvili and M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces, World Scientific, Singapore, 1991.
View at: Publisher Site | MathSciNet
M. A. Krasnoselskii and Ja. B. Rutickii, Convex Functions and Orlicz Spaces, P. Noordhoff, Groningen, The Netherlands, 1961.
View at: MathSciNet
M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, vol. 146 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1991.
View at: MathSciNet
G. Weiss, “A note on Orlicz spaces,” Portugaliae Mathematica, vol. 15, pp. 35–47, 1956.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
P. Liu and M. Wang, “Weak Orlicz spaces: some basic properties and their applications to harmonic analysis,” Science China, vol. 56, no. 4, pp. 789–802, 2013.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Y. Liang, E. Nakai, D. Yang, and J. Zhang, “Boundedness of intrinsic Littlewood-Paley functions on Musielak-Orlicz Morrey and Campanato spaces,” Banach Journal of Mathematical Analysis, vol. 8, no. 1, pp. 221–268, 2014.
View at: Google Scholar
Y. Liang and D. Yang, “Musielak-Orlicz Campanato spaces and applications,” Journal of Mathematical Analysis and Applications, vol. 406, no. 1, pp. 307–322, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
Q. Lu and X. Tao, “Characterization of maximal operators in Orlicz-Morrey spaces of homogeneous type,” Applied Mathematics, vol. 21, no. 1, pp. 52–58, 2006.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
F.-Y. Maeda, Y. Mizuta, T. Ohno, and T. Shimomura, “Trudinger's inequality and continuity of potentials on Musielak-Orlicz-Morrey spaces,” Potential Analysis, vol. 38, no. 2, pp. 515–535, 2013.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
F.-Y. Maeda, Y. Mizuta, T. Ohno, and T. Shimomura, “Boundedness of maximal operators and Sobolev's inequality on Musielak-Orlicz-Morrey spaces,” Bulletin des Sciences Mathématiques, vol. 137, no. 1, pp. 76–96, 2013.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Y. Mizuta, E. Nakai, T. Ohno, and T. Shimomura, “Boundedness of fractional integral operators on Morrey spaces and Sobolev embeddings for generalized Riesz potentials,” Journal of the Mathematical Society of Japan, vol. 62, no. 3, pp. 707–744, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Y. Mizuta, E. Nakai, T. Ohno, and T. Shimomura, “Maximal functions, Riesz potentials and Sobolev embeddings on Musielak-Orlicz-Morrey spaces of variable exponent in $ℝ^{n}$ ,” Revista Matemática Complutense, vol. 25, no. 2, pp. 413–434, 2012.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
E. Nakai, “Calderón-Zygmund operators on Orlicz-Morrey spaces and modular inequalities,” in Banach and Function Spaces II, pp. 393–410, Yokohama Publishers, Yokohama, Japan, 2008.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
E. Nakai, “Orlicz-Morrey spaces and their preduals,” in Banach and Function Spaces III, pp. 187–205, Yokohama Publishers, Yokohama, Japan, 2011.
View at: Google Scholar | MathSciNet
Y. Sawano, T. Sobukawa, and H. Tanaka, “Limiting case of the boundedness of fractional integral operators on nonhomogeneous space,” Journal of Inequalities and Applications, vol. 2006, Article ID 92470, 16 pages, 2006.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Gala, Y. Sawano, and H. Tanaka, “A new Beale-Kato-Majda criteria for the 3D magneto-micropolar fluid equations in the Orlicz-Morrey space,” Mathematical Methods in the Applied Sciences, vol. 35, no. 11, pp. 1321–1334, 2012.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Gala, Y. Sawano, and H. Tanaka, “On the uniqueness of weak solutions of the 3D MHD equations in the Orlicz-Morrey space,” Applicable Analysis, vol. 92, no. 4, pp. 776–783, 2013.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
T. Iida, E. Sato, Y. Sawano, and H. Tanaka, “Multilinear fractional integrals on Morrey spaces,” Acta Mathematica Sinica, vol. 28, no. 7, pp. 1375–1384, 2012.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
V. S. Guliyev, “Generalized local Morrey spaces and fractional integral operators with rough kernel,” Journal of Mathematical Sciences, vol. 193, no. 2, pp. 211–227, 2013.
View at: Publisher Site | Google Scholar
A. Cianchi, “A sharp embedding theorem for Orlicz-Sobolev spaces,” Indiana University Mathematics Journal, vol. 45, no. 1, pp. 39–65, 1996.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
V. S. Guliyev, S. S. Aliyev, T. Karaman, and P. S. Shukurov, “Boundedness of sublinear operators and commutators on generalized Morrey spaces,” Integral Equations and Operator Theory, vol. 71, no. 3, pp. 327–355, 2011.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
V. S. Guliyev, Integral operators on function spaces on the homogeneous groups and on domains in ℝⁿ [Ph.D. dissertation], Steklov Mathematical Institute, Moscow, Russia, 1994, (Russian).
V. S. Guliyev, Function Spaces, Integral Operators and Two Weighted Inequalities on Homogeneous Groups. Some Applications, Casioglu, Baku, Azerbaijan, 1999, (Russian).
V. S. Guliyev, “Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces,” Journal of Inequalities and Applications, vol. 2009, Article ID 503948, 20 pages, 2009.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
T. Mizuhara, “Boundedness of some classical operators on generalized Morrey spaces,” in ICM-90 Satellite Conference Proceedings: Harmonic Analysis, pp. 183–189, Springer, Tokyo, Japan, 1991.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
E. Nakai, “Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces,” Mathematische Nachrichten, vol. 166, pp. 95–103, 1994.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. R. Coifman, R. Rochberg, and G. Weiss, “Factorization theorems for Hardy spaces in several variables,” Annals of Mathematics, vol. 103, no. 3, pp. 611–635, 1976.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes, “Compensated compactness and Hardy spaces,” Journal de Mathématiques Pures et Appliquées, vol. 72, no. 3, pp. 247–286, 1993.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
L. D. Ky, “New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators,” Integral Equations and Operator Theory, 2013.
View at: Publisher Site | Google Scholar
K.-P. Ho, “Characterization of BMO in terms of rearrangement-invariant Banach function spaces,” Expositiones Mathematicae, vol. 27, no. 4, pp. 363–372, 2009.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. Izuki and Y. Sawano, “Variable Lebesgue norm estimates for BMO functions,” Czechoslovak Mathematical Journal, vol. 62, no. 3, pp. 717–727, 2012.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Chanillo, “A note on commutators,” Indiana University Mathematics Journal, vol. 31, no. 1, pp. 7–16, 1982.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
X. Fu, D. Yang, and W. Yuan, “Generalized fractional integrals and their commutators over non-homogeneous metric measure spaces,” http://arxiv.org/abs/1308.5877.
View at: Google Scholar
V. S. Guliyev, “Generalized weighted Morrey spaces and higher order commutators of sublinear operators,” Eurasian Mathematical Journal, vol. 3, no. 3, pp. 33–61, 2012.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
V. S. Guliyev and P. S. Shukurov, “On the boundedness of the fractional maximal operator, Riesz potential and their commutators in generalized Morrey spaces,” in Advances in Harmonic Analysis and Operator Theory, vol. 229 of Operator Theory: Advances and Applications, pp. 175–199, 2013.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

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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.

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