The Quasi Curvature-Dimension Condition with Applications to Sub-Riemannian Manifolds

We obtain the best known quantitative estimates for the L-p-Poincare and log-Sobolev inequalities on domains in various sub-Riemannian manifolds, including ideal Carnot groups and in particular ideal generalized H-type Carnot groups and the Heisenberg groups, corank 1 Carnot groups, the Grushin plane, and various H-type foliations, Sasakian and 3-Sasakian manifolds. Moreover, this constitutes the first time that a quantitative estimate independent of the dimension is established on these spaces. For instance, the Li-Yau / Zhong-Yang spectral-gap estimate holds on all Heisenberg groups of arbitrary dimension up to a factor of 4. We achieve this by introducing a quasi-convex relaxation of the Lott-Sturm-Villani CD(K, N) condition we call the "quasi curvature-dimension condition" QCD(Q, K, N). Our motivation stems from a recent interpolation inequality along Wasserstein geodesics in the ideal sub-Riemannian setting due to Barilari and Rizzi. We show that on an ideal sub-Riemannian manifold of dimension n, the measure contraction property MCP(K, N) implies QCD(Q, K, N) with Q = 2(N - n) >= 1, thereby verifying the latter property on the aforementioned ideal spaces; a result of Balogh-Kristaly-Sipos is used instead to handle nonideal corank 1 Carnot groups. By extending the localization paradigm to completely general interpolation inequalities, we reduce the study of various analytic and geometric inequalities on QCD spaces to the one-dimensional case. Consequently, we deduce that while (strictly) sub-Riemannian manifolds do not satisfy any type of CD condition, many of them satisfy numerous functional inequalities with exactly the same quantitative dependence (up to a factor of Q) as their CD counterparts. (c) 2020 Wiley Periodicals LLC