Multilinear fractional integral operators on generalized weighted Morrey spaces
© Hu and Wang; licensee Springer. 2014
Received: 30 April 2014
Accepted: 28 July 2014
Published: 22 August 2014
Let be multilinear fractional integral operator and let . In this paper, the estimates of , the m-linear commutators and the iterated commutators on the generalized weighted Morrey spaces are established.
1 Introduction and results
The classical Morrey spaces were introduced by Morrey  in 1938, have been studied intensively by various authors, and together with weighted Lebesgue spaces play an important role in the theory of partial differential equations; they appeared to be quite useful in the study of local behavior of the solutions of elliptic differential equations and describe local regularity more precisely than Lebesgue spaces. See [2–4] for details. Moreover, various Morrey spaces have been defined in the process of this study. Mizuhara  introduced the generalized Morrey space ; Komori and Shirai  defined the weighted Morrey spaces ; Guliyev  gave the concept of generalized weighted Morrey space , which could be viewed as an extension of both and . The boundedness of some operators on these Morrey spaces can be seen in [5–9].
As is well known, multilinear fractional integral operator was first studied by Grafakos , subsequently, by Kenig and Stein , Grafakos and Kalton . In 2009, Moen  introduced weight function and gave weighted inequalities for multilinear fractional integral operators; In 2013, Chen and Wu  obtained the weighted norm inequalities for the multilinear commutators and . More results of the weighted inequalities for multilinear fractional integral and its commutators can be found in [15–17].
The aim of the present paper is to investigate the boundedness of multilinear fractional integral operator and its commutator on the generalized weighted Morrey spaces. Our results can be formulated as follows.
2 Definitions and preliminaries
A weight ω is a nonnegative, locally integrable function on . Let denote the ball with the center and radius . For any ball B and , λB denotes the ball concentric with B whose radius is λ times as long. For a given weight function ω and a measurable set E, we also denote the Lebesgue measure of E by and set weighted measure .
When , is understood as .
Lemma 2.2 
If and with , then is the classical Morrey space.
If , then is the weighted Morrey space.
If , then is the two weighted Morrey space.
If , then is the generalized Morrey space.
If , then .
Lemma 2.3 (John-Nirenberg inequality; see )
By Lemma 2.3, it is easy to get the following.
Lemma 2.5 
where is independent of f, , , and .
By Lemma 2.4 and Lemma 2.5, it is easily to prove the following results.
We also need the following result.
Lemma 2.7 
At the end of this section, we list some known results about weighted norm inequalities for the multilinear fractional integrals and their commutators.
Lemma 2.8 
Lemma 2.9 
3 Proof of Theorem 1.1
We first prove the following conclusions.
From (3.11) and (3.12) we know and are not greater than (3.10) for , .
Combining the above estimates, the proof of Theorem 3.1 is completed. □
This completes the proof of first part of Theorem 1.1.
This completes the proof of second part of Theorem 1.1.
4 Proof of Theorem 1.2
Proof We will give the proof for because the proof for is very similar but easier. Moreover, for simplicity of the expansion, we only present the case .
Summing up the above estimates, (4.2) is proved for . □
The authors would like to thank the referees and the Editors for carefully reading the manuscript and making several useful suggestions.
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