Spherical reflection positivity and the Hardy-Littlewood-Sobolev inequality

TL;DR

This paper introduces spherical reflection positivity and uses it to provide a new proof for the sharp constants in the Hardy-Littlewood-Sobolev inequality, extending previous characterizations of minimizing functions.

Contribution

It presents a novel concept of spherical reflection positivity and applies it to derive new proofs for key inequalities, enhancing understanding of their sharp constants.

Findings

Established spherical reflection positivity as a useful tool.

Provided new proofs for sharp constants in HLS inequalities.

Extended previous work on minimizing functions.

Abstract

We introduce the concept of spherical (as distinguished from planar) reflection positivity and use it to obtain a new proof of the sharp constants in certain cases of the HLS and the logarithmic HLS inequality. Our proofs relies on an extension of a work by Li and Zhu which characterizes the minimizing functions of the type $(1+|x|^2)^{-p}$.