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Reversed Stein–Weiss Inequalities with Poisson-Type Kernel and Qualitative Analysis of Extremal Functions
Advanced Nonlinear Studies ( IF 1.8 ) Pub Date : 2021-02-01 , DOI: 10.1515/ans-2020-2112
Chunxia Tao 1
Affiliation  

Through conformal map, isoperimetric inequalities are equivalent to the Hardy–Littlewood–Sobolev (HLS) inequalities involved with the Poisson-type kernel on the upper half space. From the analytical point of view, we want to know whether there exists a reverse analogue for the Poisson-type kernel. In this work, we give an affirmative answer to this question. We first establish the reverse Stein–Weiss inequality with the Poisson-type kernel, finding that the range of index 𝑝,q′q^{\prime} appearing in the reverse inequality lies in the interval (0,1)(0,1), which is perfectly consistent with the feature of the index for the classical reverse HLS and Stein–Weiss inequalities. Then we give the existence and asymptotic behaviors of the extremal functions of this inequality. Furthermore, for the reverse HLS inequalities involving the Poisson-type kernel, we establish the regularity for the positive solutions to the corresponding Euler–Lagrange system and give the sufficient and necessary conditions of the existence of their solutions. Finally, in the conformal invariant index, we classify the extremal functions of the latter reverse inequality and compute the sharp constant. Our methods are based on the reversed version of the Hardy inequality in high dimension, Riesz rearrangement inequality and moving spheres.

中文翻译:

具有泊松型核的反向Stein-Weiss不等式和极值函数的定性分析

通过共形图,等距不等式等于上半空间上的Poisson型核所涉及的Hardy–Littlewood–Sobolev(HLS)不等式。从分析的角度来看,我们想知道泊松型核是否存在反向类似物。在这项工作中,我们对这个问题给出肯定的答案。我们首先用泊松型核建立反向Stein-Weiss不等式,发现出现在反向不等式中的索引𝑝,q'q ^ {\ prime}的范围位于区间(0,1)(0,1 ),这与经典反向HLS和Stein-Weiss不等式的索引特征完全一致。然后我们给出了该不等式极值函数的存在性和渐近性。此外,对于涉及Poisson型核的反向HLS不等式,我们建立了相应的Euler-Lagrange系统正解的正则性,并给出了其解存在的充分必要条件。最后,在共形不变索引中,我们对后者的反向不等式的极值函数进行分类,并计算出尖锐常数。我们的方法基于高维Hardy不等式,Riesz重排不等式和运动球的反向版本。
更新日期:2021-03-16
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