Properties of Lipschitz smoothing heat semigroups

We prove several functional and geometric inequalities only assuming the linearity and a quantitative ${\mathrm{L}}^{\mathrm{\infty }}$-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 large variety settings, including Riemannian manifolds with a variable Kato-type lower bound on the Ricci curvature tensor, $\mathsf{R}\mathsf{C}\mathsf{D}(K,\mathrm{\infty })$ spaces, and some sub-Riemannian structures, such as Carnot groups, the Grushin plane and the $\mathbb{S}\mathbb{U}(2)$ group.