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

Let $\mathbb{M}$ be a smooth connected manifold endowed with a smooth measure μ and a smooth locally subelliptic diffusion operator L which is symmetric with respect to μ. We assume that L satisfies a generalized curvature dimension inequality as introduced by Baudoin and Garofalo (2009) 9]. Our goal is to discuss functional inequalities for μ like the Poincaré inequality, the log-Sobolev inequality or the Gaussian logarithmic isoperimetric inequality.