It is known that the couple formed by the two dimensional Brownian motion and its L\'evy area leads to the heat kernel on the Heisenberg group, which is one of the simplest sub-Riemannian space. The associated diffusion operator is hypoelliptic but not elliptic, which makes difficult the derivation of functional inequalities for the heat kernel. However, Driver and Melcher and more recently H.-Q. Li have obtained useful gradient bounds for the heat kernel on the Heisenberg group. We provide in this paper simple proofs of these bounds, and explore their consequences in terms of functional inequalities, including Cheeger and Bobkov type isoperimetric inequalities for the heat kernel.Comment: Minor correction

Let M be a smooth connected manifold endowed with a smooth measure µ and a smooth locally subelliptic diffusion operator L satisfying L1 = 0, and which is symmetric with respect to µ. 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 [BG1], then the following properties hold:• The volume doubling property; • The Poincaré 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.

Let 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-Garofalo [BG1]. Our goal is to discuss functional inequalities for µ like the Poincaré inequality, the log-Sobolev inequality or the Gaussian logarithmic isoperimetric inequality.