A family of Beckner inequalities under various curvature-dimension conditions

TL;DR

This paper proves a family of functional inequalities that interpolate between Poincaré and logarithmic Sobolev inequalities, using entropy flows and curvature-dimension conditions, applicable to Riemannian manifolds.

Contribution

It introduces a unified proof for Beckner inequalities under various curvature-dimension conditions, extending previous results to more general geometric settings.

Findings

Validated inequalities on Riemannian manifolds with different curvature conditions

Unified approach using entropy flows and CD conditions

Generalization of previous inequalities to broader geometric contexts

Abstract

In this paper, we offer a proof for a family of functional inequalities interpolating between the Poincar{\'e} and the logarithmic Sobolev (standard and weighted) inequalities. The proofs rely both on entropy flows and on a CD($\rho$, n) condition, either with $\rho$ = 0 and n > 0, or with $\rho$ > 0 and n $\in$ R. As such, results are valid in the case of a Riemannian manifold, which constitutes a generalization to what was proved in [BGS18, Ngu18].