We prove that, on any sub-Riemannian manifold endowed with a positive smooth measure,
the Bakry–Émery inequality for the corresponding sub-Laplacian, 1 2 Δ (‖∇ u‖ 2)≥ g (∇ …

Abstract The Lott–Sturm–Villani curvature-dimension condition CD (K, N) provides a
synthetic notion for a metric measure space to have curvature bounded from below by K and …

We establish geometric inequalities in the sub-Riemannian setting of the Heisenberg
group\mathbb H^ n H n. Our results include a natural sub-Riemannian version of the …

We prove a sharp isoperimetric inequality for the class of metric measure spaces verifying
the synthetic Ricci curvature lower bounds Measure Contraction property ($\mathsf {MCP}(0 …

We prove that H-type Carnot groups of rank k and dimension n satisfy the MCP (K, N) if and
only if K≤ 0 and N≥ k+ 3 (n− k). The latter integer coincides with the geodesic dimension of …

We develop a variational theory of geodesics for the canonical variation of the metric of a
totally geodesic foliation. As a consequence, we obtain comparison theorems for the …

We obtain the best known quantitative estimates for the L p‐Poincaré and log‐Sobolev
inequalities on domains in various sub‐Riemannian manifolds, including ideal Carnot …

We construct a canonical differential structure on the configuration space $\Upsilon $ over a
singular base space $ X $ and with a general invariant measure $\mu $ on $\Upsilon $. We …

We prove that no Brunn–Minkowski inequality from the Riemannian theories of curvature-
dimension and optimal transportation can be satisfied by a strictly sub-Riemannian structure …

The curvature dimension condition CD (K, N), pioneered by Sturm and Lott–Villani in Sturm
(2006a); Sturm (2006b); Lott and Villani (2009), is a synthetic notion of having curvature …