Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion

TL;DR

This paper explores the duality between Sobolev and Hardy-Littlewood-Sobolev inequalities using fast diffusion flows, leading to improved inequalities in higher dimensions and connections with Onofri's inequality in two dimensions.

Contribution

It introduces a novel approach linking these inequalities through diffusion equations, enhancing understanding and bounds in specific dimensions.

Findings

Improved Sobolev inequalities for dimensions d≥5.

Established duality relations via fast diffusion flows.

Connected Onofri's inequality with logarithmic Hardy-Littlewood-Sobolev inequality.

Abstract

In the euclidean space, Sobolev and Hardy-Littlewood-Sobolev inequalities can be related by duality. In this paper, we investigate how to relate these inequalities using the flow of a fast diffusion equation in dimension $d\ge3$. The main consequence is an improvement of Sobolev's inequality when $d\ge5$, which involves the various terms of the dual Hardy-Littlewood-Sobolev inequality. In dimension $d=2$, Onofri's inequality plays the role of Sobolev's inequality and can also be related to its dual inequality, the logarithmic Hardy-Littlewood-Sobolev inequality, by a super-fast diffusion equation.