Spherical reflection positivity and the Hardy-Littlewood-Sobolev inequality

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 $(1+|x|^2)^{-p}$.