Mathematics

Starting from a quantitative L ∞ to Lipschitz regularization of the heat semigroup, we show how to derive in a simple way various functional and geometrical inequalities like: Buser's inequality; indeterminacy estimates for the Wasserstein distance; lower bounds on the size of nodal sets and on the Wasserstein distance between the positive and negative parts of an eigenfunction. All the results apply to a large class of spaces, including but not limited to compact Riemannian manifolds and, more generally, RCD(K, ∞) spaces with nite measure. Based on joint works with Sara Farinelli, Andrea Mondino, Giorgio Stefani.