Heat kernel estimates with applications to analysis on manifolds

Abstract:

Upper and lower Gaussian estimates for the heat kernel are the key to a number of analytic results on doubling metric measure spaces endowed with a Dirichlet form, and they also have a probabilistic interpretation. We will describe recent characterisations of the upper estimates in terms of new global Sobolev type inequalities, which simplify and extend known results in this direction. We will also present a new approach to the lower estimates, which opens the way to a better understanding of Lp boundedness of the Riesz transform on Riemannian manifolds and more general spaces. This relies on joint works with Frédéric Bernicot, Dorothée Frey, and Adam Sikora.