Asymptotics of the fast diffusion equation via entropy estimates

We consider non-negative solutions of the fast diffusion equation in the Euclidean space R^d, and study the asymptotic behavior of a natural class of solutions foir large times, where "large" means that t is approaching infinity in case the parameter m appearing in the equation is close to one, or that t approaches the finite extinction time in case the diffusion is "very fast". For a class of initial data, we prove that the solution converges with a polynomial rate to a self-similar solution. Such results are new in the "very fast" case, whereas when m is close to one we improve on known results. A fundamental role in this studied is played by suitable new functional inequalities which are related to the spectral properties of the linearized generator of the evolution considered, after suitable rescaling.