Sharp Beckner-type inequalities for Cauchy and spherical distributions

TL;DR

This paper derives new optimal Beckner-type inequalities for Cauchy and spherical distributions using harmonic extensions, probabilistic methods, and curvature-dimension inequalities, revealing connections to Sobolev inequalities.

Contribution

It introduces a novel family of optimal Beckner inequalities for Cauchy and spherical distributions, linking them to Sobolev inequalities and employing harmonic and probabilistic techniques.

Findings

New optimal inequalities for Cauchy distributions

Equivalent non-tight inequalities on the sphere

Connection to Sobolev inequalities

Abstract

Using some harmonic extensions on the upper-half plane, and probabilistic representations, and curvature-dimension inequalities with some negative dimensions, we obtain some new opimal functional inequalities of the Beckner type for the Cauchy type distributions on the Euclidean space. These optimal inequalities appear to be equivalent to some non tight optimal Beckner inequalities on the sphere, and the family appear to be a new form of the Sobolev inequality.