Affine Hardy--Littlewood--Sobolev inequalities

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