Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,infinity) metric measure spaces

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 Lp-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-Émery 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 Carré 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.