Functional Inequalities on sub-Riemannian manifolds via QCD

We are interested in obtaining Poincar\’e 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. We show that
while ideal (strictly) sub-Riemannian manifolds do not satisfy any type of CD condition, they do satisfy a
quasi-convex relaxation thereof, which we name QCD(Q,K,N). As a consequence, these spaces satisfy numerous
functional inequalities with exactly the same quantitative dependence (up to a factor of Q) 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-Poincar\’e and log-Sobolev inequalities on domains in
the ideal sub-Riemannian setting, 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.
No prior knowledge will be assumed, and we will (hopefully) explain all of the above notions during the talk.