Functional Inequalities on sub-Riemannian manifolds via QCD

Abstract: We are interested in obtaining Poincarè and log-Sobolev inequalities on domains in sub-Riemannian manifolds (equipped with their natural sub-Riemannian metric and volume measure).

It is well-known that strictly sub-Riemannian manifolds do not satisfy any type of Curvature-Dimension condition CD(K,N), introduced by Lott-Sturm-Villani some 15 years ago, so we must follow a different path. Motivated by recent work of Barilari-Rizzi and Balogh-Kristàly-Sipos, we show that in the ideal setting or for general corank 1 Carnot groups, these spaces nevertheless do satisfy a quasi-convex relaxation of the CD condition, which we name QCD(Q,K,N). As a consequence, these spaces satisfy numerous functional inequalities with exactly the same quantitative dependence (up to the slack parameter Q>1) as their CD counterparts. We achieve this by extending the localization paradigm to completely general interpolation inequalities, and a one-dimensional comparison of QCD densities with their "CD upper envelope". We thus obtain the best known quantitative estimates for (say) the $L^p$-Poincare and log-Sobolev inequalities on domains in ideal sub-Riemannian manifolds and in general corank 1 Carnot groups, which in particular are independent of the topological dimension. For instance, the classical Li-Yau / Zhong-Yang spectral-gap estimate holds on all Heisenberg groups of arbitrary dimension up to a factor of 4.

analysis of PDEsdifferential geometrymetric geometryoptimization and controlspectral theory

Audience: researchers in the topic

( slides )

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Sub-Riemannian Seminars

Series comments: The "Sub-Riemannian seminars" are the union of the "Séminaire de géométrie et analyse sous-riemannienne" (held in Paris since 2011) and the "International Sub-Riemannian Seminars", which were born in spring 2020 as a reaction to the COVID-19 pandemic.

The new format will gather every 3 weeks on average, alternating between these types of sessions:

- physical session in Paris (Laboratoire Jacques-Louis Lions), also transmitted online on Zoom.

- fully online session on Zoom.

- special session hosted physically somewhere else, and transmitted online.

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