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Sharp reversed Hardy–Littlewood–Sobolev inequality with extension kernel

Volume 271 / 2023

Wei Dai, Yunyun Hu, Zhao Liu Studia Mathematica 271 (2023), 1-38 MSC: Primary 42B25; Secondary 35A23, 42B37. DOI: 10.4064/sm220323-26-1 Published online: 20 March 2023


In this paper, we prove the following reversed Hardy–Littlewood–Sobolev inequality with extension kernel: $$\int _{\mathbb R_+^n}\int _{\partial\mathbb R^n_+}\frac{x_n^\beta}{|x-y|^{n-\alpha}}f(y)g(x)\,dy\,dx\geq C_{n,\alpha ,\beta ,p}\|f\|_{L^{p}(\partial\mathbb R_+^n)}\|g\|_{L^{q’}(\mathbb R_+^n)} $$ for any nonnegative functions $f\in L^{p}(\partial\mathbb R_+^n)$ and $g\in L^{q’}(\mathbb R_+^n)$, where $n\geq 2$, $p, q’\in (0,1)$, $\alpha \gt n$, $0\leq \beta \lt \frac{\alpha-n}{n-1}$, $p \gt \frac{n-1}{\alpha -1-(n-1)\beta}$ are such that $\frac{n-1}{n}\frac{1}{p}+\frac{1}{q’}-\frac{\alpha +\beta -1}{n}=1$. We prove the existence of extremal functions for the above inequality. Moreover, in the conformal invariant case, we classify all the extremal functions and hence derive the best constant via the method of moving spheres. It is quite surprising that the extremal functions do not depend on $\beta $. Finally, we derive the sufficient and necessary conditions for existence of positive solutions to the Euler–Lagrange equations by using Pohozaev identities.


  • Wei DaiSchool of Mathematical Sciences
    Beihang University (BUAA)
    100191 Beijing, P.R. China
  • Yunyun HuSchool of Mathematics and Statistics
    Shaanxi Normal University
    710119 Xi’an, Shaanxi, P.R. China
  • Zhao LiuSchool of Mathematics and Computer Science
    Jiangxi Science and Technology Normal University
    330038 Nanchang, P.R. China

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