# American Institute of Mathematical Sciences

February  2022, 21(2): 727-747. doi: 10.3934/cpaa.2021196

## Necessary and sufficient conditions on weighted multilinear fractional integral inequality

 1 University of Chinese Academy of Sciences, No.19(A) Yuquan Road, Shijingshan District, Beijing, China 100049 2 Zhejiang University of Science and Technology, No.318 Liuhe Road, Hangzhou, Zhejiang, China 310023

* Corresponding author

Received  June 2021 Revised  October 2021 Published  February 2022 Early access  December 2021

Fund Project: This paper is supported by the National Science Foundation of Zhejiang Province of China (Grant No. LQ18A010002), National Natural Foundation of China (Grant No. 12001488) and in part by the National Natural Foundation of China (Grant No. 11871452 and 12071052)

We consider certain kinds of weighted multi-linear fractional integral inequalities which can be regarded as extensions of the Hardy-Littlewood-Sobolev inequality. For a particular case, we characterize the sufficient and necessary conditions which ensure that the corresponding inequality holds. For the general case, we give some sufficient conditions and necessary conditions, respectively.

Citation: Yongliang Zhou, Yangkendi Deng, Di Wu, Dunyan Yan. Necessary and sufficient conditions on weighted multilinear fractional integral inequality. Communications on Pure & Applied Analysis, 2022, 21 (2) : 727-747. doi: 10.3934/cpaa.2021196
##### References:
 [1] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar [2] W. Rudin, Real and Complex Analysis, 3$^{nd}$ edition, McGraw-Hill Education, New York, 1987.  Google Scholar [3] Z. Shi, D. Wu and D. Yan, Necessary and sufficient conditions of doubly weighted Hardy-Littlewood-Sobolev inequality, Anal. Theor. Appl., 30 (2014), 193-204.  doi: 10.4208/ata.2014.v30.n2.5.  Google Scholar [4] Z. Shi, D. Wu and D. Yan, On the Multi-linear Fractional Integral Operators with Correlation Kernels, J. Fourier Anal. Appl., 25 (2019), 538-587.  doi: 10.1007/s00041-017-9591-1.  Google Scholar [5] E. M. Stein and G. Weiss, Fractional integrals on $n$-dimensional euclidean space, J. Math. Mech., 7 (1958), 503-514.  doi: 10.1512/iumj.1958.7.57030.  Google Scholar [6] D. Wu, Z. Shi and D. Yan, Sharp constants in the doubly weighted Hardy-Littlewood-Sobolev inequality, Sci. Chi. Math., 57 (2014), 963-970.  doi: 10.1007/s11425-013-4681-2.  Google Scholar [7] D. Wu and D. Yan, Sharp constants for a class of multi-linear integral operators and some applications, Sci. Chi. Math., 59 (2016), 907-920.  doi: 10.1007/s11425-015-5120-3.  Google Scholar

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##### References:
 [1] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar [2] W. Rudin, Real and Complex Analysis, 3$^{nd}$ edition, McGraw-Hill Education, New York, 1987.  Google Scholar [3] Z. Shi, D. Wu and D. Yan, Necessary and sufficient conditions of doubly weighted Hardy-Littlewood-Sobolev inequality, Anal. Theor. Appl., 30 (2014), 193-204.  doi: 10.4208/ata.2014.v30.n2.5.  Google Scholar [4] Z. Shi, D. Wu and D. Yan, On the Multi-linear Fractional Integral Operators with Correlation Kernels, J. Fourier Anal. Appl., 25 (2019), 538-587.  doi: 10.1007/s00041-017-9591-1.  Google Scholar [5] E. M. Stein and G. Weiss, Fractional integrals on $n$-dimensional euclidean space, J. Math. Mech., 7 (1958), 503-514.  doi: 10.1512/iumj.1958.7.57030.  Google Scholar [6] D. Wu, Z. Shi and D. Yan, Sharp constants in the doubly weighted Hardy-Littlewood-Sobolev inequality, Sci. Chi. Math., 57 (2014), 963-970.  doi: 10.1007/s11425-013-4681-2.  Google Scholar [7] D. Wu and D. Yan, Sharp constants for a class of multi-linear integral operators and some applications, Sci. Chi. Math., 59 (2016), 907-920.  doi: 10.1007/s11425-015-5120-3.  Google Scholar
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