Hardy–Littlewood–Sobolev Inequalities with the Fractional Poisson Kernel and Their Applications in PDEs

作     者：Lu CHEN Guozhen LU Chunxia TAO 

作者机构：School of Mathematics and Statistics Beijing Institute of Technology Department of Mathematics University of Connecticut School of Mathematical Sciences Beijing Normal University 

出 版 物：《Acta Mathematica Sinica,English Series》 (数学学报（英文版）)

年 卷 期：2019年第35卷第6期

页      面：853-875页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：partly supported by a US NSF grant a Simons Collaboration grant from the Simons Foundation 

主　　题：Existence of extremal functions Hardy–Littlewood–Sobolev inequality Moving plane method Poisson kernel 

摘      要：The purpose of this paper is five-fold. First, we employ the harmonic analysis techniques to establish the following Hardy–Littlewood–Sobolev inequality with the fractional Poisson kernel on the upper half space ■ where f ∈ L^p(?R_+~n), g ∈ Lq(R_+~n) and p, q ∈(1, +∞), 2 ≤α n satisfying (n-1)/np+1/q +(2-α)/n= ***, we utilize the technique combining the rearrangement inequality and Lorentz interpolation to show the attainability of best constant C_(n,α,p,q ). Third, we apply the regularity lifting method to obtain the smoothness of extremal functions of the above inequality under weaker assumptions. Furthermore,in light of the Pohozaev identity, we establish the sufficient and necessary condition for the existence of positive solutions to the integral system of the Euler–Lagrange equations associated with the extremals of the fractional Poisson kernel. Finally, by using the method of moving plane in integral forms, we prove that extremals of the Hardy–Littlewood–Sobolev inequality with the fractional Poisson kernel must be radially symmetric and decreasing about some point ξ_0 ∈ ?R_+~n. Our results proved in this paper play a crucial role in establishing the Stein–Weiss inequalities with the Poisson kernel in our subsequent paper.