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

作者：Baudoin Fabrice; Bonnefont Michel; Garofalo Nicola^*

来源：Mathematische Annalen, 2014, 358(3-4): 833-860.

DOI：10.1007/s00208-013-0961-y

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.

• 出版日期2014-4