A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincare inequality

Let M be a smooth connected manifold endowed with a smooth measure mu and a smooth locally subelliptic diffusion operator L satisfying L1 = 0, and which is symmetric with respect to mu. We show that if L satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced by the first and third named authors in http://arxiv.org/abs/1101.3590, then the following properties hold: . The volume doubling property; . The Poincare inequality; . The parabolic Harnack inequality. The key ingredient is the study of dimension dependent reverse log-Sobolev inequalities for the heat semigroup and corresponding non-linear reverse Harnack type inequalities. Our results apply in particular to all Sasakian manifolds whose horizontal Webster-Tanaka-Ricci curvature is nonnegative, all Carnot groups of step two, and to wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is nonnegative.