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

Abstract

Let $\mathbb{M}$ 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 \cite{BG}, then the following properties hold: 1 The volume doubling property; 2 The Poincaré inequality; 3 The parabolic Harnack inequality. The key ingredient is the study of dimensional 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 non negative, all Carnot groups with step two, and to wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is non negative.

References in corpus (1)

• Generalized Ricci Curvature Bounds for Three Dimensional Contact Subriemannian manifolds

Cited by in corpus (10)

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• Poincare inequality and the uniqueness of solutions for the heat equation associated with subelliptic diffusion operators
• Curvature dimension inequalities and subelliptic heat kernel gradient bounds on contact manifolds
• A note on the boundedness of Riesz transform for some subelliptic operators
• Derivative Formula and Gradient Estimates for Gruschin Type Semigroups
• Heat kernel upper bounds under the generalized curvature(-dimension) inequality
• Diffusion semigroup on manifolds with time-dependent metrics
• Derivative Formulae and Poincaré Inequality for Kohn-Laplacian Type Semigroups
• Intrinsic complements of equiregular sub-Riemannian manifolds