Properties of Lipschitz Smoothing Heat Semigroups

We prove several functional and geometric inequalities only assuming the linearity and a quantitative L∞-to-Lipschitz smoothing of the heat semigroup in metric-measure spaces. Our results comprise a Buser inequality, a lower bound on the size of the nodal set of a Laplacian eigenfunction, and different estimates involving the Wasserstein distance. The approach works in a large variety of settings, including Riemannian manifolds with a variable Kato-type lower bound on the Ricci curvature tensor, RCD(K,∞) spaces, and some sub-Riemannian structures, such as Carnot groups, the Grushin plane and the SU(2) group.