Affine Hardy--Littlewood--Sobolev inequalities

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 ${\mathbb{R}}^n$.