Hardy-Littlewood-Sobolev and Stein-Weiss inequalities and integral systems on the Heisenberg group

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• 作者：Xiaolong Han ^{xlhan@math.wayne.edu} ; Guozhen Lu ; ^{gzlu@math.wayne.edu} ; Jiuyi Zhu ^{jiuyi.zhu@wayne.edu}
• 关键词：42B37 ; 42B20 ; 45G15 ; 45M05 ; 45E10
• 刊名：Nonlinear Analysis
• 出版年：2012
• 期刊代码：43_0362546x
• 类别：mt
• 出版时间：July, 2012
• 卷：75
• 期：11
• 页码：4296-4314
• 文件大小：315 K

摘要

In this paper, we study two types of weighted Hardy-Littlewood-Sobolev (HLS) inequalities, also known as Stein-Weiss inequalities, on the Heisenberg group. More precisely, we prove the weighted HLS inequality in and the weighted HLS inequality in (where we have denoted as points on the Heisenberg group). Then we provide regularity estimates of positive solutions to integral systems which are Euler-Lagrange equations of the possible extremals to the Stein-Weiss inequalities. Asymptotic behavior is also established for integral systems associated to the weighted HLS inequalities around the origin. By these a priori estimates, we describe asymptotically the possible optimizers for sharp versions of these inequalities.