紧黎曼流形上Hardy-Littlewood-Sobolev不等式的极值问题:次临界逼近法

摘要/Abstract

摘要：

令（Mⁿ，g）为n维无边紧黎曼流形， $0<\alpha<n,\ q>\frac{n}{n-\alpha}$，该文研究了下列Hardy-Littlewood-Sobolev（HLS）不等式

$\|I_\alpha f\|_{L^q(M^n)}\leq C\|f\|_{L^p(M^n)},\quad I_\alpha f(x)= \int_{M^n}\frac{f(y)}{|x-y|_g^{n-\alpha}}{\rm d}V_y,\ p\geq\frac{nq}{n+\alpha q} $

的极值问题.首先,利用算子$I_\alpha: L^p(M^n)\rightarrow L^q(M^n)$在次临界情形(即$p>\frac{nq}{n+\alpha q}$)时的紧致性,证明$p>\frac{nq}{n+\alpha q}$时极值函数$f_p\in L^p(M^n)$的存在性;进而证明函数列$\{f_p\}$为临界情形时HLS不等式的最佳常数的极值列;最后,结合极值列$\{f_p\}$在$L^{\frac{nq}{n+\alpha q}}(M^n)$中的一致有界性,利用文献[32]建立的集中列紧原理证明$\{f_p\}$在$L^{\frac{nq}{n+\alpha q}}(M^n)$中存在收敛子列,从而给出临界情形(即$p=\frac{nq}{n+\alpha q}$)时极值函数的存在性.

关键词: Hardy-Littlewood-Sobolev不等式, 紧黎曼流形, 极值问题

Abstract:

Let $(M^n,g)$ be a $n$-dimensional compact Riemannian manifolds, $0<\alpha<n$ and $q>\frac{n}{n-\alpha}$. This paper is mainly devoted to study the extremal problems of the following HLS inequalities:

$\|I_\alpha f\|_{L^q(M^n)}\leq C\|f\|_{L^p(M^n)},\quad\mbox{with}\quad I_\alpha f(x)=\int_{M^n}\frac{f(y)}{|x-y|_g^{n-\alpha}}{\rm d}V_y,\ p\geq\frac{nq}{n+\alpha q}.$

Firstly, we prove that $I_\alpha: L^p(M^n)\rightarrow L^q(M^n)$ with $p>\frac{nq}{n+\alpha q}$ is compact and then get the existence of extremal functions $f_p, p>\frac{nq}{n+\alpha q}$. Secondly, we find that the function sequence $\{f_p\}$ is a maximizing sequence for the sharp constant of HLS inequality with $p=\frac{nq}{n+\alpha q}$. Finally, by the Concentration-Compactness principle established in [32], we can prove that there exists a convergence subsequence of $\{f_p\}$ and then give the existence of extremal function for critical case.

Key words: Hardy-Littlewood-Sobolev intqualities, Compact Riemannian manifolds, Extremal problems

中图分类号:

O175.2

张书陶,韩亚洲. 紧黎曼流形上Hardy-Littlewood-Sobolev不等式的极值问题:次临界逼近法[J]. 数学物理学报, 2020, 40(1): 63-71.

Shutao Zhang,Yazhou Han. Extremal Problems of Hardy-Littlewood-Sobolev Inequalities on Compact Riemannian Manifolds: the Approximation Method from Subcritical to Critical[J]. Acta mathematica scientia,Series A, 2020, 40(1): 63-71.

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