Some recent progress on functional inequalities and applications

Abstract

In this survey paper we introduce some recent progresses on functional inequalities and applications. We start from a Poincaré type inequality on Hilbert spaces which is corresponding to upper bound estimates of the essential spectrum for the infinitesimal generator, then clarify the inequality and further applications in the framework of Dirichlet form. Applications to the exponential decay of semigroups in the tail norm and transportation-cost inequalities with super cost functions will be presented. Finally, functional inequalities on Riemannian manifolds with unbounded below curvature or non-convex boundary will be investigated.