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Weighted Trudinger inequality associated with rough multilinear fractional type operators
Journal of Inequalities and Applications volume 2012, Article number: 179 (2012)
Abstract
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 .
1 Introduction
The Trudinger inequality (also sometimes called the Moser-Trudinger inequality) is named after N. Trudinger who first put forward this inequality in [22]. Later, J. Moser [14] gave a sharp form of this Trudinger inequality. It provides an inequality between a certain Sobolev space norm and an Orlicz space norm of a function. In [14], J. Moser gave the largest positive number , such that if , normalized and supported in a domain D with finite measure in , such that , then there is a constant depending only on n such that for all , where is the area of the surface of the unit n-ball. The following inequality holds:
In 1971, D. Adams [1] considered the similar inequality of J. Moser for higher order derivatives. The key, for him, was to write the function u as a potential (see the definition below) and prove the analogue of (1.1) as follows:
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.
We begin by introducing a class of multilinear maximal function and multilinear fractional integral operators. Suppose that , , Ω is homogeneous of degree zero, and (), where denotes the unit sphere of . The multilinear maximal function and multilinear fractional integral is defined by
and the fractional maximal operator defined by
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 [15]. 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 [7]. After that, in [6], Y. Ding and S. Lu gave the following Trudinger inequality with rough kernels.
Theorem A ([6])
Let , , , , , . Denote B as a ball with a radius R in . If , , and , then for any , there is a constant C, independent of n, α, , γ, such that
where , , and
The definition of multiple weights was given in [5] and [13] independently, including some weighted estimates for a class of multilinear fractional type operators. These results together with [12] answered an open problem in [8], namely the existence of the multiple weights.
In 2010, W. Li, Q. Xue, and K. Yabuta [16] obtained the weighted estimates for the Trudinger inequality associated to as follows.
Theorem B ([16])
Let , , , , , and , , , . Denote B as a ball with the radius R in , if , , , then for any , there is a constant C, independent of n, α, , γ, such that
where , , .
On the other hand, in 1999, Kenig and Stein [11] considered another more general type of multilinear fractional integral which was defined by
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 [20] obtained the estimation of the exponential integrability of the above operator , which is quite similar to Theorem B.
Thus, it is natural to ask whether Theorem B is true or not for with rough kernels. Moreover, one may ask if Theorem B still holds or not for the operator with rough kernels defined by
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.
Theorem 1.1 Let , , , , , . Denote B as a ball with radius R in ; if , (), , and , where , . Then for any , there is a constant C, independent of n, α, , γ, such that
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 [6]. 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 [20] proved exponential integrability of , and Strichartz [19] for other α. In 1972, Hedberg [9] gave a simpler proof for all α. In 1970, Hempel-Morris-Trudinger [10] showed that if , for the inequality in Theorem 1.1 cannot hold, and later Adams [1] obtained the same conclusion for all α; meanwhile, in the endpoint case , it is true. In 1985, Chang and Marshall [4] proved a similar sharp exponential inequality concerning the Dirichlet integral. Assume , , then the result was obtained by Grafakos [7] as we have already mentioned above.
Corollary 1.2 Let B, , , s, and be the same as in Theorem 1.1, then is in for every , that is,
for some constant C depending only on q on n on α and on the ’s.
Theorem 1.3 Let , , with for . Let B be a ball with radius R in and let be supported in B, and if is homogeneous of degree zero, and , where denotes the sphere of , and , where and , . Then there exist constants , depending only on n, m, α, p, and the such that
Remark 1.3 If we take , for , then Theorem 1.3 is just as Theorem 1.3 appeared in [20]. But there is something that needs to be changed in the proof of Theorem 1.3 in [20]. In the case , one cannot obtain the conclusion that . Thus, our proof gives an alternative correction of Theorem 1.3 in [20].
Corollary 1.4 Let B, , , s, and be the same as in Theorem 1.3. Then is in for every , that is,
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.
Theorem 1.5 If , , , , , , , then there is a constant C, independent , such that
where , .
2 The proof of Theorem 1.1
In this section, we will prove Theorem 1.1.
Proof For any ,
Set . For any , denote as a ball with radius R in , then for any , when , for . Therefore, . So,
According to the relationship between s and : , from the Hölder’s inequality and , it follows that
Hence, we obtain that
Set , then
Now we put , , thus
By the fact that
Therefore, we get
Here, in the above third inequality, we have used the well-known weighted result of Hardy-Littlewood maximal function.
From (), we get
Hence,
On the other hand,
From the above all, we obtain that
□
3 The proof of Theorem 1.5
In this section, we will prove Theorem 1.5.
Proof By the well-known Hölder’s inequality, we get
Hence,
In addition, from the condition , it follows that
According to the above, we obtain that
It is easy to see that
where , holds, also. □
4 The proof of Theorem 1.3
In this section, we will prove Theorem 1.3.
Proof For any and ,
For , let with for . Then
where denotes as .
For , if satisfies , then for some , . Without losing the generalization, we set .
Thus,
Define that and for . By the condition of , we have
where . In the case that , by the fact that
we have
Consider the case where exactly l of the are ∞ for some . Without losing the generalization, we only give the argument for , , then
Combining the above cases, we obtain
Thus, by the estimates for , , we have
In particular, we chose for all , then
Now, we set
where .
Then
Hence,
Let and , then
On the other hand,
Combining the above results, we obtain
where , are constants depending only on n, m, α, p, and the . □
Authors’ information
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H.
Feng’s current address: Department of Mathematical and Statistical Sciences, University of Alberta, Canada.
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Acknowledgement
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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Feng, H., Xue, Q. Weighted Trudinger inequality associated with rough multilinear fractional type operators. J Inequal Appl 2012, 179 (2012). https://doi.org/10.1186/1029-242X-2012-179
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DOI: https://doi.org/10.1186/1029-242X-2012-179