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

令(M^n,g)为n维无边紧黎曼流形,0<αn/n?α,该文研究了下列Hardy-Littlewood-Sobolev(HLS)不等式||Iαf||L^q(M^n)≤C||f||L^p(M^n),Iαf(x)=∫M^nf(y)/|x?y|g^n?αdVy,p≥nq/n+αq的极值问题.首先,利用算子Iα:L^p(M^n)→L^q(M^n)在次临界情形(即p>nq/n+αq)时的紧致性,证明p>nq/n+αq时极值函数fp∈Lp(Mn)的存在性;进而证明函数列{fp}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{fp}在Lnq/n+αq(Mn)中的一致有界性,利用文献32]建立的集中列紧原理证明{fp}在Lnq/n+αq(M^n)中存在收敛子列,从而给出临界情形(即p=nq/n+αq)时极值函数的存在性.

Extremal Problems of Hardy-Littlewood-Sobolev Inequalities on Compact Riemannian Manifolds:the Approximation Method from Subcritical to Critical

Let(M^n,g)be a n-dimensional compact Riemannian manifolds,0<αn/n?α.This paper is mainly devoted to study the extremal problems of the following HLS inequalities:||Iαf||L^q(M^n)≤C||f||L^p(M^n),with Iαf(x)=∫M^nf(y)/|x?y|g^n?αdVy,p≥nq/n+αq.Firstly,we prove that Iα:L^p(M^n)→L^q(M^n)with p>nq/n+αq is compact and then get the existence of extremal functions fp,p>nq/n+αq.Secondly,we find that the function sequence{fp}is a maximizing sequence for the sharp constant of HLS inequality with p=nq/n+αq.Finally,by the Concentration-Compactness principle established in32],we can prove that there exists a convergence subsequence of{fp}and then give the existence of extremal function for critical case.