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