A family of Beckner inequalities under various curvature-dimension conditions

Abstract

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