We study a generalization of the Bakry-\'Emery pointwise gradient estimate for heat semigroup and its equivalence with some entropic inequalities along flow Wasserstein geodesics metric-measure spaces suitable group structure. Our main result applies to Carnot groups any step $\mathbb{SU}(2)$ group.

Book Response: Education, Arts and Sustainability: Emerging Practice for a Changing World, by Mary Ann Hunter, Arnold Aprill, Allen Hill & Sherridan Emery. Springer Singapore, Springer, 2018, 120 pp., ISBN: 9811077096 Keywords: Australia; Arts; Critical Thinking; Education; Partnership; Sustainability. Réaction à un ouvrage : par et Mots-clés Australie, arts, raisonnement critique, éducatio...

We prove a Yau’s type gradient estimate for positive f-harmonic functions with the Dirichlet boundary condition on smooth metric measure spaces compact when infinite dimensional Bakry–Emery Ricci tensor and weighted mean curvature are bounded below. As an application, we give Liouville result condition. Our results do not depend any assumption potential function f.