Lipschitz smoothing heat semigroup and functional-geometric inequalities

In this seminar, we present several functional and geometric inequalities, including a Buser-type inequality, a lower bound on the size of the nodal set of a Laplacian eigenfunction, and various estimates involving the 1-Wasserstein distance. Emphasis will be placed on the proof strategy rather than the results, highlighting how all the inequalities are derived using heat semigroup techniques through a key property: a Lipschitz smoothing estimate. This estimate will be presented in detail, and we will show its validity across a broad class of spaces, including Riemannian manifolds with Ricci curvature lower bounds and various sub-Riemannian structures.
The seminar is based on joint work with G. Stefani.