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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 406540, 17 pages
http://dx.doi.org/10.1155/2012/406540
Research Article

Some Estimates of Rough Bilinear Fractional Integral

1Department of Mathematics, Zhejiang University, Hangzhou 310027, China
2Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China

Received 23 July 2012; Revised 11 September 2012; Accepted 25 September 2012

Academic Editor: Ti-Jun Xiao

Copyright © 2012 Yun Fan and Guilian Gao. 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 boundedness of rough bilinear fractional integral on Morrey spaces and modified Morrey spaces and obtain some sufficient and necessary conditions on the parameters. Furthermore, we consider the boundedness of on generalized central Morrey space . These extend some known results.

1. Introduction

In recent years, multilinear analysis becomes a very active research topic in studying harmonic analysis. As one of the most important operators, the multilinear fractional integral has also attracted much attention. In this note, we will consider the multilinear fractional integral with rough kernel. For fixed distinct and nonzero real numbers , and , the -linear fractional with rough kernel is defined by where ( ) is homogeneous of degree zero on , and denotes the unit sphere of .

When , The boundedness of operator has been well studied in [1, 2]. Recently, Hendar and Idha discussed the boundedness property of on generalized Morrey space in [3].

Here, without loss of generality, we will study the case . More specifically, we will study the rough bilinear fractional integral:

The study of the operators and its related operators with rough kernel recently attracted many attentions. In 2002, Ding and Chin first discussed its boundedness. The following theorem is their main result:

Theorem A (see [4]). Let ,   and , . If there exists a positive constant such that for any , ,
(1) when ,
(2) when ,

Later, when , Chen and Fan in [5] relaxed the conditions of in Theorem A using Hölder inequality. Their main result is as follows.

Theorem B. Let , , and If , then there exists a positive constant such that

We note that when , Hölder inequality is not sufficient in Theorem B. So how to relax the index of is left. In fact, in [6, 7] the authors have obtained the necessary and sufficient conditions on the parameters for the -linear fractional integral operator with rough kernel from to by using the pointwise rearrangement estimate of the -linear convolution.

Theorem C. Let , and be homogeneous of degree zero on , , let be the harmonic mean of , and . Then the condition is necessary and sufficient for the boundedness of from to .

This paper is organized as follows: in the second part of this work we prove some boundedness properties of on Morrey space and extend Theorem C to Morrey spaces; in the third part, we obtain the sufficient and necessary conditions on the parameters for the boundedness of on modified Morrey space; in the last part, we find the sufficient condition on the pair which ensures the boundedness of the operators on the generalized center Morrey space. Since Morrey space, modified Morrey space and central Morrey space all can be seen as generalized space.

2. The Boundedness of on Morrey Space

The classical Morrey spaces were originally introduced by Morrey in [8] to study the local behavior of solutions to second-order elliptic partial differential equations. The reader can find more details in [9].

For and , let denotes the open ball centered at of radius , and is the Lebesgue measure of the ball . When and , Morrey space is defined by where

If , then and . When , . So we only consider the case .

Since Morrey space can be seen as the generalized space, we will be interested in the boundedness of on Morry space . In order to prove our results, we need the following bilinear maximal function:

Lemma 2.1. Let , and . If then there exists a positive constant such that

Proof. In [10], Fefferman and Stein have proved that for every , , there is a constant such that for any measurable functions on and , the following inequality holds, where is the Hardy-LittleWood maximal function. Set be the characteristic function , when , by the above inequality, we can get where .
Taking , , is the characteristic function of a ball in , by simple calculating, that is, . For More details, see [11] about the boundedness of Hardy-Littlewood maximal function on Morrey space.
So when ,   ,   , we have

Theorem 2.2. Suppose , and let be homogeneous of degree zero on , let be the harmonic mean of and , , and . If then there exists a positive constant such that

Proof. Let , , , for and , we can get , . First, is decomposed by
Estimate of is and estimate of is For , we have the following estimates: Combining the above estimates, we have Let , then By computation, we get Taking the supremum of , we have
Hence

Theorem 2.3. Suppose , and let be homogeneous of degree zero on , let be the harmonic mean of and , , , and , , then the condition is necessary and sufficient for the boundedness of from to .

Proof. Sufficiency part of Theorem 2.3 is proved in Theorem 2.2.
Necessity. Let and , . Denote and . Then we have
Since is bounded from to , it is true that where depends only on ,   ,   , and .
If , then in the case , for all , , we have .
If , then in the case , for all , , we have .
Therefore, we get .

Corollary 2.4. Let , be homogeneous of degree zero on , be the harmonic mean of and , , , and . If then there exists a positive constant such that

Proof. By Hölder inequality, it is easy to know when , we have , through the given condition, . Applying Theorem 2.2, we get
From the inequality and Theorem 2.2, we obtain an Olsen inequality involving a multiplication operator.

Corollary 2.5. Suppose , and let be homogeneous of degree zero on , let be the harmonic mean of and , , , , and . If One has

3. The Boundedness of on Modified Morrey Space

After studying Morrey spaces in detail, people are led to considering the local and global counterpart. There are many famous work by V. I. Burenkov, H. V. Guliyev and V. S. Guliyev, and so forth and (see [1220]). Recently, Guliyev et al. have considered the following modified Morrey spaces in [21].

Definition 3.1. Let , and . is defined as the set of all functions , with the finite norms Note that and if or , then .
In [21], the authors discussed the boundedness of maximal function in modified Morrey spaces and obtained the following generalized Hardy-Littlewood-Sobolev inequalities in modified Morrey spaces.

Theorem D. Let and . If , then condition is necessary and sufficient for the boundedness of the operator from to .

We also can extend Theorem D to the multilinear case.

Lemma 3.2. Let , and . If then there exists a positive constant such that

Proof. When , the following inequality: holds, where is the Hardy-littlewood maximal function and .
Taking , . Using the method in [21], we get .
Hence, with the same arguments in Lemma 2.1, we complete the proof of Lemma 3.2.

Theorem 3.3. Suppose , and let be homogeneous of degree zero on , let be the harmonic mean of and , , , and , . Then the condition is necessary and sufficient for the boundedness of from to .

Proof. (1) Sufficiency. Let , , , since and , we can get , and .
Do the same decomposition of in the proof of Theorem 2.2, then we only need to estimate . We can easily obtain
For , we get
While , we obtain
Thus, we obtain Set we have
Hence, by the boundedness of in Lemma 3.2, we prove that is bounded from to .
(2) Necessity. Let and , . Denote , , and . Then from [21], we have
By the boundedness of , we have
If , then in the case , for all , , we have .
If , then in the case , for all , , we have .
Therefor .

4. The Boundedness of on Generalized Center Morrey Space

Definition 4.1. Let be a positive measurable function on and . We denote by the generalized central Morrey space, the space of all functions with finite quasinorm where denotes a ball centered at with side length and is the Lebesgue measure of the ball .
According to this definition, we recover the spaces under the choice . About the space, the readers can refer to [22], In fact, we can easily check that is a Banach space, reduce to when , and .
There are many papers that discussed the conditions on to obtain the boundedness of fractional integral on the generalized Morrey spaces, see [23, 24]. In [25] the following condition was imposed on the pair : for the fractional integral , where and ( ) does not depend on .

Theorem E (see [26]). The inequality holds for all nonnegative and nonincreasing g on if and only if and , where the is the Hardy operator

In this section we are going to discuss the boundedness of on generalized central Morrey space.

Lemma 4.2. Suppose , , , and , then for , the inequality holds for any ball and for all and .

Proof. Let , , and . For any , set , we write
Hence
Since is bounded from to , we have where the constant is independent of and .
To estimate , it follows that So
By the same estimating, we also can obtain
To estimate , we get
Combining the above estimates, we end the proof of Lemma 4.2.

Theorem 4.3. Suppose , , ,   , and . If satisfies the condition and satisfies the condition where the constant does not depend on r. Let , then is bounded from to .

Proof. By Theorem E and Lemma 4.2, we have

Corollary 4.4. Suppose , , , , , , , and , then is bounded from to .

Remark 4.5. Although we worked on the bilinear case. Applying same ideas in the argument, we may obtain similar extension of .

Acknowledgments

The authors thank the referees for useful comments which improve the presentation of this paper. This research is supported by NSFZJ (Grant no. Y604563) and NSFC (Grant no. 11271330).

References

  1. L. Grafakos, “On multilinear fractional integrals,” Studia Mathematica, vol. 102, no. 1, pp. 49–56, 1992. View at Zentralblatt MATH
  2. C. E. Kenig and E. M. Stein, “Multilinear estimates and fractional integration,” Mathematical Research Letters, vol. 6, no. 1, pp. 1–15, 1999. View at Zentralblatt MATH
  3. G. Hendar and S. Idha, “Multilinear maximal functions and fractional integrals on generalized Morrey spaces,” http://personal.fmipa.itb.ac.id/hgunawan/files/2007/11/multilinear-maximal-functions-n-fractional-integrals-ver-3.pdf.
  4. Y. Ding and C.-L. Lin, “Rough bilinear fractional integrals,” Mathematische Nachrichten, vol. 246-247, pp. 47–52, 2002. View at Zentralblatt MATH
  5. J. Chen and D. Fan, “Rough bilinear fractional integrals with variable kernels,” Frontiers of Mathematics in China, vol. 5, no. 3, pp. 369–378, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. V. S. Guliev and Sh. A. Nazirova, “Rearrangement estimates for generalized multilinear fractional integrals,” Siberian Mathematical Journal, vol. 48, no. 3, pp. 463–470, 2007. View at Publisher · View at Google Scholar
  7. V. S. Guliyev and Sh. A. Nazirova, “O'Neil inequality for multilinear convolutions and some applications,” Integral Equations and Operator Theory, vol. 60, no. 4, pp. 485–497, 2008. View at Publisher · View at Google Scholar
  8. 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 · View at Google Scholar · View at Zentralblatt MATH
  9. J. Peetre, “On the theory of Mp,λ spaces,” Journal of Functional Analysis, vol. 4, pp. 71–87, 1969. View at Zentralblatt MATH
  10. C. Fefferman and E. M. Stein, “Some maximal inequalities,” American Journal of Mathematics, vol. 93, pp. 107–115, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. F. Chiarenza and M. Frasca, “Morrey spaces and Hardy-Littlewood maximal function,” Rendiconti di Matematica e delle sue Applicazioni, vol. 7, no. 3-4, pp. 273–279, 1987. View at Zentralblatt MATH
  12. V. I. Burenkov and H. V. Guliyev, “Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces,” Studia Mathematica, vol. 163, no. 2, pp. 157–176, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. V. I. Burenkov, V. S. Guliyev, A. Serbetci, and T. V. Tararykova, “Necessary and sufficient conditions for the boundedness of genuine singular integral operators in local Morrey-type spaces,” Eurasian Mathematical Journal, vol. 1, no. 1, pp. 32–53, 2010.
  14. V. I. Burenkov, H. V. Guliyev, and V. S. Guliyev, “Necessary and sufficient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces,” Journal of Computational and Applied Mathematics, vol. 208, no. 1, pp. 280–301, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. V. I. Burenkov, H. V. Guliyev, and V. S. Guliyev, “On boundedness of the fractional maxiaml operator from complementary Morrey type spaces,” in The Interaction of Analysis and Geometry, vol. 424 of Contemporary Mathematics, pp. 17–32, American Mathematical Society, Providence, RI, USA, 2007.
  16. V. I. Burenkov and V. S. Guliyev, “Necessary and sufficient conditions for the boundedness of the Riesz potential in local Morrey-type spaces,” Potential Analysis, vol. 30, no. 3, pp. 211–249, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. V. S. Guliyev, Integral operators on function spaces on the homogeneous groups and on domains in ℝn [Ph.D. thesis], Steklov Mathematical Institute, Moscow, Russia, 1994.
  18. V. S. Guliyev, Fuction Spaces, Integral Operators and Two Weighted Inequalities on Homogeneous Groups, Some Applications, Casioglu, Baku, Azerbaijan, 1999.
  19. V. S. Guliev and R. Ch. Mustafaev, “Integral operators of potential type in spaces of homogeneous type,” Doklady Ross. Skad. Nauk. Matematika, vol. 354, no. 6, pp. 730–732, 1997 (Russian). View at Zentralblatt MATH
  20. V. S. Guliev and R. Ch. Mustafaev, “Fractional integrals in spaces of functions defined on spaces of homogeneous type,” Analysis Mathematica, vol. 24, no. 3, pp. 181–200, 1998. View at Publisher · View at Google Scholar
  21. V. S. Guliyev, J. J. Hasanov, and Y. Zeren, “Necessary and sufficient conditions for the boundedness of the Riesz potential in modified Morrey spaces,” Journal of Mathematical Inequalities, vol. 5, no. 4, pp. 491–506, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. J. Alvarez, J. Lakey, and M. Guzmán-Partida, “Spaces of bounded λ-central mean oscillation, Morrey spaces, and λ-central Carleson measures,” Collectanea Mathematica, vol. 51, no. 1, pp. 1–47, 2000.
  23. 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 · View at Google Scholar · View at Zentralblatt MATH
  24. 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 · View at Google Scholar · View at Zentralblatt MATH
  25. 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 · View at Google Scholar · View at Zentralblatt MATH
  26. M. Carro, L. Pick, J. Soria, and V. D. Stepanov, “On embeddings between classical Lorentz spaces,” Mathematical Inequalities & Applications, vol. 4, no. 3, pp. 397–428, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH