Weighted Trudinger inequality associated with rough multilinear fractional type operators
© Feng and Xue; licensee Springer 2012
Received: 7 June 2012
Accepted: 3 August 2012
Published: 23 August 2012
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 .
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.
- Adams D: A sharp inequality of J. Moser for higher order derivatives. Ann. Math. 1988, 128: 385–398. 10.2307/1971445View ArticleMathSciNetMATHGoogle Scholar
- Cerny R, Gurka P, Hencl S: Concentration compactness principle for generalized Trudinger inequalities. Z. Anal. Anwend. 2011, 30(3):355–375.MathSciNetView ArticleMATHGoogle Scholar
- Carleson L, Chang S: On the existence of an extremal function for an inequality of J. Moser. Bull. Sci. Math. (2) 1986, 110(2):113–127.MathSciNetMATHGoogle Scholar
- Chang S, Marshall D: On a sharp inequality concerning the Dirichlet integral. Am. J. Math. 1985, 107: 1015–1033. 10.2307/2374345MathSciNetView ArticleMATHGoogle Scholar
- Chen X, Xue Q: Weighted estimates for a class of multilinear fractional type operators. J. Math. Anal. Appl. 2010, 362(2):355–373. 10.1016/j.jmaa.2009.08.022MathSciNetView ArticleMATHGoogle Scholar
- Ding Y, Lu S:The boundedness for some rough operators. J. Math. Anal. Appl. 1996, 203: 166–186. 10.1006/jmaa.1996.0373MathSciNetView ArticleGoogle Scholar
- Grafakos L: On multilinear fractional integrals. Stud. Math. 1992, 102(1):49–56.MathSciNetMATHGoogle Scholar
- Grafakos, L, Torres, RH: On multilinear singular integrals of Calderón-Zygmund type. In: Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorual). Publ. Mat. Vol. Extra, 57–91 (2002)Google Scholar
- Hedberg LI: On certain convolution inequalities. Proc. Am. Math. Soc. 1972, 36: 505–510. 10.1090/S0002-9939-1972-0312232-4MathSciNetView ArticleMATHGoogle Scholar
- Hempel J, Morris G, Trudinger N: On the sharpness of a limiting case of the Sobolev embedding theorem. Bull. Aust. Math. Soc. 1970, 3: 369–373. 10.1017/S0004972700046074MathSciNetView ArticleMATHGoogle Scholar
- Kenig C, Stein E: Multilinear estimates and fractional integration. Math. Res. Lett. 1999, 6: 1–15.MathSciNetView ArticleMATHGoogle Scholar
- Lerner AK, Ombrosi S, Pérez C, Torres RH, Trujillo-González R: New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory. Adv. Math. 2009, 220(4):1222–1264. 10.1016/j.aim.2008.10.014MathSciNetView ArticleMATHGoogle Scholar
- Moen K: Weighted inequalities for multilinear fractional integral operators. Collect. Math. 2009, 60: 213–238. 10.1007/BF03191210MathSciNetView ArticleMATHGoogle Scholar
- Moser J: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 1971, 20: 1077–1092. 10.1512/iumj.1971.20.20101View ArticleMathSciNetMATHGoogle Scholar
- Muckenhoupt B, Wheeden R: Weighted norm inequalities for fractional integrals. Trans. Am. Math. Soc. 1974, 192: 261–274.MathSciNetView ArticleMATHGoogle Scholar
- Li W, Xue Q, Yabuta K: Multilinear Calderón-Zygmund operators on weighted Hardy spaces. Stud. Math. 2010, 199(1):1–16. 10.4064/sm199-1-1MathSciNetView ArticleMATHGoogle Scholar
- Lu G, Yang Y:Sharp constant and extremal function for the improved Moser-Trudinger inequality involving norm in two dimension. Discrete Contin. Dyn. Syst., Ser. A 2009, 25: 963–979.MathSciNetView ArticleMATHGoogle Scholar
- Ruf B:A sharp Trudinger-Moser type inequality for unbounded domains in . J. Funct. Anal. 2005, 219(2):340–367. 10.1016/j.jfa.2004.06.013MathSciNetView ArticleMATHGoogle Scholar
- Strichartz RS: A note on Trudinger’s extension of Sobolev’s inequalities. Indiana Univ. Math. J. 1972, 21: 841–842. 10.1512/iumj.1972.21.21066MathSciNetView ArticleMATHGoogle Scholar
- Tang L: Endpoint estimates for multilinear fractional integrals. J. Aust. Math. Soc. 2008, 84: 419–429.MathSciNetView ArticleMATHGoogle Scholar
- Tian G, Zhu X: A nonlinear inequality of Moser-Trudinger type. Calc. Var. Partial Differ. Equ. 2000, 10(4):349–354. 10.1007/s005260010349MathSciNetView ArticleMATHGoogle Scholar
- Trudinger N: On imbedding into Orlicz spaces and some applications. J. Math. Mech. 1967, 17: 473–483.MathSciNetMATHGoogle Scholar
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