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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 ,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.

Reversed Stein–Weiss Inequalities with Poisson-Type Kernel and Qualitative Analysis of Extremal Functions