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Improved Hardy–Littlewood–Sobolev Inequality on \({\mathbb{S}^n}\) under Constraints

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Abstract

In this paper, we establish an improved Hardy–Littlewood–Sobolev inequality on \({\mathbb{S}^n}\) under higher-order moments constraint. Moreover, by constructing precise test functions, using improved Hardy–Littlewood–Sobolev inequality on \({\mathbb{S}^n}\), we show such inequality is almost optimal in critical case. As an application, we give a simpler proof of the existence of the maximizer for conformal Hardy–Littlewood–Sobolev inequality.

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Acknowledgements The authors would like to thank the referee for his/her careful reading of the manuscript and valuable suggestions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shaanxi Normal University, **’an, 710119, P. R. China

    Yun Yun Hu & **g Bo Dou

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  1. Yun Yun Hu
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  2. **g Bo Dou
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Correspondence to **g Bo Dou.

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Conflict of Interest The authors declare no conflict of interest.

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Supported by the National Science Foundation of China (Grant Nos. 12101380, 12071269), China Postdoctoral Science Foundation (Grant No. 2021M700086), Youth Innovation Team of Shaanxi Universities and the Fundamental Research Funds for the Central Universities (Grant Nos. GK202307001, GK202202007)

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Hu, Y.Y., Dou, J.B. Improved Hardy–Littlewood–Sobolev Inequality on \({\mathbb{S}^n}\) under Constraints. Acta. Math. Sin.-English Ser. 39, 2149–2163 (2023). https://doi.org/10.1007/s10114-023-2630-8

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  • DOI: https://doi.org/10.1007/s10114-023-2630-8

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