Weighted Trudinger inequality associated with rough multilinear fractional type operators
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- Feng, H. & Xue, Q. J Inequal Appl (2012) 2012: 179. doi:10.1186/1029-242X-2012-179
Let be the multilinear fractional type operator defined by . In this paper, we study the weighted estimates for the Trudinger inequality associated to with rough homogeneous kernels, which improve some known results significantly. A similar Trudinger inequality holds for another type of fractional integral defined by , where .
KeywordsRiesz potential multilinear fractional integral weights weights Trudinger inequality
Variant forms of the Trudinger inequality as a generalization of the classical results, especially in the literature associated with multilinear Riesz potential or multilinear fractional integral, have been studied in recently years (see, for example, [2, 3, 6, 7, 10, 14, 16–18, 20, 21]). This kind of inequality plays an important role in Harmonic analysis and other fields, such as PDE.
Multilinear fractional integral can be looked at as a natural generalization of the classical fractional integral, which has a very profound background of partial differential equations and is a very important operator in Harmonic analysis. In fact, if we take , , and , then is just the well-known classical fractional integral operator studied by Muckenhoupt and Wheeden in . We denote it by . If , we simply denote . In recent years, the study of the Trudinger inequality associated to multilinear type operators has received increasing attention. Among them, it is well known that Grafakos considered the boundedness of a family of related fractional integrals in . After that, in , Y. Ding and S. Lu gave the following Trudinger inequality with rough kernels.
Theorem A ()
The definition of multiple weights was given in  and  independently, including some weighted estimates for a class of multilinear fractional type operators. These results together with  answered an open problem in , namely the existence of the multiple weights.
In 2010, W. Li, Q. Xue, and K. Yabuta  obtained the weighted estimates for the Trudinger inequality associated to as follows.
Theorem B ()
where , , .
where is a linear combination of s and x depending on the matrix A. They showed that was of strong type and weak type . When , we denote this multilinear fractional type operator by . In 2008, L. Tang  obtained the estimation of the exponential integrability of the above operator , which is quite similar to Theorem B.
Inspired by the works above, in this paper, we study the Trudinger inequality associated to multilinear fractional integral operators and with rough homogeneous kernels. Precisely, we obtain the following theorems, which give a positive answer to the above questions.
where , , .
Remark 1.1 If we take , then Theorem 1.1 coincides with Theorem B. If for , then Theorem 1.1 is just Theorem A that appeared in . We give an example of as follows: Let ( for each j), then satisfy the conditions of the above Theorem 1.1.
Remark 1.2 Assume , . If , Trudinger  proved exponential integrability of , and Strichartz  for other α. In 1972, Hedberg  gave a simpler proof for all α. In 1970, Hempel-Morris-Trudinger  showed that if , for the inequality in Theorem 1.1 cannot hold, and later Adams  obtained the same conclusion for all α; meanwhile, in the endpoint case , it is true. In 1985, Chang and Marshall  proved a similar sharp exponential inequality concerning the Dirichlet integral. Assume , , then the result was obtained by Grafakos  as we have already mentioned above.
for some constant C depending only on q on n on α and on the ’s.
Remark 1.3 If we take , for , then Theorem 1.3 is just as Theorem 1.3 appeared in . But there is something that needs to be changed in the proof of Theorem 1.3 in . In the case , one cannot obtain the conclusion that . Thus, our proof gives an alternative correction of Theorem 1.3 in .
for some constant C depending only on q on n on α.
Corollary 1.2 and Corollary 1.4 follow since exponential integrability of implies integrability to any power q.
On the other hand, we shall study the boundedness of the multilinear fractional maximal operator with a weighted norm. It follows the following theorem.
where , .
2 The proof of Theorem 1.1
In this section, we will prove Theorem 1.1.
Here, in the above third inequality, we have used the well-known weighted result of Hardy-Littlewood maximal function.
3 The proof of Theorem 1.5
In this section, we will prove Theorem 1.5.
where , holds, also. □
4 The proof of Theorem 1.3
In this section, we will prove Theorem 1.3.
where denotes as .
For , if satisfies , then for some , . Without losing the generalization, we set .
where , are constants depending only on n, m, α, p, and the . □
Feng’s current address: Department of Mathematical and Statistical Sciences, University of Alberta, Canada.
The second author was supported partly by NSFC (Grant No. 10701010), NSFC (Key program Grant No. 10931001), Beijing Natural Science Foundation (Grant: 1102023), Program for Changjiang Scholars and Innovative Research Team in University.
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