Poincar{\'e} inequalities and integrated curvature-dimension criterion for generalised Cauchy and convex measures

TL;DR

This paper establishes new sharp weighted Poincaré inequalities on Riemannian manifolds for a broad class of measures, including generalized Cauchy measures, providing optimal spectral gaps and extremal functions.

Contribution

It introduces a unified approach to weighted Poincaré inequalities for generalized Cauchy measures with optimal constants and extremal functions.

Findings

Optimal spectral gap for generalized Cauchy measures

Unified proof of weighted Poincaré inequalities

Identification of extremal functions

Abstract

We obtain new sharp weighted Poincar{\'e} inequalities on Riemannian manifolds for a general class of measures. When specialised to generalised Cauchy measures, this gives a unified and simple proof of the weighted Poincar{\'e} inequality for the whole range of parameters, with the optimal spectral gap, the error term and the extremal functions.