Generalized Bakry–Émery Curvature Condition and Equivalent Entropic Inequalities in Groups

Abstract

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