Hardy-Littlewood-Sobolev Inequalities via Fast Diffusion Flows

TL;DR

This paper provides a straightforward proof of certain cases of the sharp Hardy-Littlewood-Sobolev inequalities using a monotone flow derived from the fast diffusion equation, simplifying previous approaches.

Contribution

It introduces a simple proof method for specific cases of the Hardy-Littlewood-Sobolev inequalities utilizing fast diffusion flows, offering a new perspective.

Findings

Proof of the $oxed{ ext{d-2}}$ case of the sharp Hardy-Littlewood-Sobolev inequality for $d extgreater 2$

Proof of the sharp Logarithmic Hardy-Littlewood-Sobolev inequality for $d=2$

Demonstrates the effectiveness of fast diffusion flows in proving functional inequalities.

Abstract

We give a simple proof of the $\lambda = d-2$ cases of the sharp Hardy-Littlewood-Sobolev inequality for $d\geq 3$, and the sharp Logarithmic Hardy-Littlewood-Sobolev inequality for $d=2$ via a monotone flow governed by the fast diffusion equation.