Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality

TL;DR

This paper presents a novel proof of specific cases of the sharp Hardy-Littlewood-Sobolev inequality using reflection positivity, extending previous characterizations of minimizing functions.

Contribution

It introduces a new proof technique based on reflection positivity instead of symmetric rearrangement, expanding understanding of the inequality's minimizers.

Findings

New proof of sharp HLS inequality cases

Extension of Li and Zhu's characterization of minimizers

Demonstration of reflection positivity method effectiveness

Abstract

We give a new proof of certain cases of the sharp HLS inequality. Instead of symmetric decreasing rearrangement it uses the reflection positivity of inversions in spheres. In doing this we extend a characterization of the minimizing functions due to Li and Zhu.