Affine Hardy--Littlewood--Sobolev inequalities

TL;DR

This paper establishes new sharp affine Hardy--Littlewood--Sobolev inequalities that improve upon classical results, along with reverse inequalities for log-concave functions, advancing the understanding of functional inequalities in analysis.

Contribution

The paper introduces stronger affine Hardy--Littlewood--Sobolev inequalities and their reverse forms, extending classical inequalities with affine invariance and sharp constants.

Findings

Established sharp affine Hardy--Littlewood--Sobolev inequalities.

Derived sharp reverse inequalities for log-concave functions.

Connected affine inequalities to classical Hardy--Littlewood--Sobolev results.

Abstract

Sharp affine Hardy--Littlewood--Sobolev inequalities for functions on $\mathbb R^n$ are established, which are significantly stronger than (and directly imply) the sharp Hardy--Littlewood--Sobolev inequalities by Lieb and by Beckner, Dou, and Zhu. In addition, sharp reverse inequalities for the new inequalities and the affine fractional $L^2$ Sobolev inequalities are obtained for log-concave functions on $\mathbb R^n$.