Hardy-Littlewood-Sobolev（简记H-L-S）不等式在调和分析、偏微分方程、几何分析和概率论中有着广泛的应用。申请人及其合作者已建立了欧式空间上逆Stein-Weiss（简记S-W）不等式、极值函数存在性以及带权的H-L-S方程正解的正则性、径向对称性和单调性。本项目拟进一步研究：欧式空间上逆H-L-S不等式和逆S-W不等式对应欧拉拉格朗日方程正解的径向对称性和单调性以及半空间上更一般非线性项共形不变负指标积分方程正解的Liouville型定理；欧式空间上带权的Possion核H-L-S不等式、极值函数存在性、最佳常数以及相应欧拉拉格朗日方程正解的正则性、径向对称性及其渐近行为；Heisenberg群上逆H-L-S不等式和逆S-W不等式、极值函数存在性、最佳常数以及相应欧拉拉格朗日方程正解的渐近行为。本项目的研究是现代分析领域热点问题之一，具有重要的理论意义和研究价值。

Hardy-Littlewood-Sobolev inequalities are widely used in harmonic analysis, partial differential equations, geometric analysis and probability theory. The applicant and his collaborators have established reversed Stein-Weiss inequality, existence of extremal functions, and regularity, radial symmetry and monotonicity of positive solutions for the weighted Hardy-Littlewood-Sobolev equations on Euclidean space. This project aims, on Euclidean space, to further study, radial symmetry and monotonicity of positive solutions of corresponding Euler-Lagrange equations for reversed Hardy-Littlewood-Sobolev inequality and reversed Stein-Weiss inequality as well as Liouville type theorems of positive solutions for general integral equations with negative exponents satisfying conformally invariant conditions on half space; to establish Possion type weighted Hardy-Littlewood-Sobolev inequality, existence of its extremal functions, sharp constants as well as regularity, radial symmetry and asymptotic behaviors of positive solutions to its related Euler-Lagrange equations; on Heisenberg group, to study reversed Hardy-Littlewood-Sobolev inequality and Stein-Weiss inequality, existence of their extremal functions, sharp constants as well as asymptotic behaviors of positive solutions to their related Euler-Lagrange equations. The research of this project, which has important theoretical significance and research value, is one of the focus in the field of mordern analysis.