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

Abstract

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.

Two classical properties of the Bessel quotient 𝐼_{𝜈+1}/𝐼_{𝜈} and their implications in pde’s

“…A remarkable aspect of (3.1) is that it is equivalent to the lower bound Ricci ≥ ρ on the Ricci tensor of M. In the paper [15] it was shown that many global properties of the heat semigroup, in a setting which includes the Riemannian one, can be derived exclusively from a generalization of the curvature-dimension inequality (3.1). In this connection one should also see [14], [11] and [12].…”

Hamilton Type Gradient Estimate for the Sub-Elliptic Operators

“…Recall that the doubling volume property and two-sided bounds for the heat kernel p(x, y, t) has been established by Baudoin et al, see Theorem 3.8 (or Theorem 3.1) and Theorem 4.1 respectively in [8]: for every ε > 0, there exist constants C 1 = C 1 (ρ 2 , k, d) > 0, C 2 (ε) = C 2 (ε, ρ 2 , k, d) > 0 such that for every x, y ∈ M and t > 0,…”

“…Under the generalized curvature-dimension inequality CD(ρ 1 , ρ 2 , k, d) introduced in [9], we first establish the Hamilton type gradient estimate for positive and upper bounded solutions to heat equations, see Theorem 2.1. Combining with the doubling volume property and the lower (upper) bounds for the heat kernel established by Baudoin et al [8], we can get the gradient estimate for the associated heat kernel, see Theorem 3.1.…”

On the Cheng-Yau gradient estimate for Carnot groups and sub-Riemannian manifolds

“…Preliminaries. In this section we study the setting similar to [5]. We state relevant details here for completeness.…”