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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 406540, 17 pages
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.
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.
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 .
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 ). Let , and , . If
there exists a positive constant such that for any , ,
(1) when ,
(2) when ,
Later, when , Chen and Fan in  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  to study the local behavior of solutions to second-order elliptic partial differential equations. The reader can find more details in .
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 , 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
Taking , , is the characteristic function of a ball in , by simple calculating, that is, . For More details, see  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
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 [12–20]). Recently, Guliyev et al. have considered the following modified Morrey spaces in .
Definition 3.1. Let , and . is defined as the set of all functions , with the finite norms
and if or , then .
In , 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 , 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 , 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 .
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 , 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  the following condition was imposed on the pair : for the fractional integral , where and () does not depend on .
Theorem E (see ). 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
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 .
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).
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