Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality

TL;DR

This paper investigates functional inequalities such as Poincaré and log-Sobolev inequalities for subelliptic operators on manifolds that satisfy a generalized curvature dimension condition, extending classical results to more general geometric settings.

Contribution

It establishes log-Sobolev inequalities for subelliptic operators under a generalized curvature dimension inequality, broadening the scope of functional inequalities in geometric analysis.

Findings

Proves log-Sobolev inequalities for subelliptic operators

Extends curvature-dimension conditions to subelliptic settings

Provides tools for analyzing functional inequalities on manifolds

Abstract

Let $\M$ be a smooth connected manifold endowed with a smooth measure $\mu$ and a smooth locally subelliptic diffusion operator $L$ which is symmetric with respect to $\mu$. We assume that $L$ satisfies a generalized curvature dimension inequality as introduced by Baudoin-Garofalo \cite{BG1}. Our goal is to discuss functional inequalities for $\mu$ like the Poincar\'e inequality, the log-Sobolev inequality or the Gaussian logarithmic isoperimetric inequality.