SELF-IMPROVEMENT OF THE BAKRY-EMERY CONDITION AND WASSERSTEIN CONTRACTION OF THE HEAT FLOW IN RCD(K, ∞) METRIC MEASURE SPACES

摘要

We prove that the linear "heat" flow in a RCD(K, ∞) metric measure space (X, d,m) satisfies a contraction property with respect to every L~p-Kantorovich-Rubinstein-Wasserstein distance, p ∈ [1, ∞]. In particular, we obtain a precise estimate for the optimal W_∞-coupling between two fundamental solutions in terms of the distance of the initial points. The result is a consequence of the equivalence between the RCD(K, ∞) lower Ricci bound and the corresponding Bakry-Emery condition for the canonical Cheeger-Dirichlet form in (X, d,m). The crucial tool is the extension to the non-smooth metric measure setting of the Bakry's argument, that allows to improve the commutation estimates between the Markov semigroup and the Carre du Champ Γ associated to the Dirichlet form. This extension is based on a new a priori estimate and a capacitary argument for regular and tight Dirichlet forms that are of independent interest.