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

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.