Harnack meets Hardy: intrinsic ultracontractivity revisited

In this talk, we reconsider the classical notion of intrinsic ultracontractivity for semigroups generated by elliptic operators with Dirichlet boundary conditions. The traditional approaches make use of logarithmic Sobolev inequalities and quadratic forms, or probabilistic methods to estimate heat kernels, together with specific assumptions on the geometry of the underlying domain. As a consequence, intrinsic ultracontractivity has been observed in many sophisticated examples.

By contrast, in our work, we identify an underlying structure of the phenomenon, and show in a transparent way how intrinsic ultracontractivity arises simply from basic functional inequalities: a Harnack-type inequality, a Hardy inequality, and the Sobolev inequality.

This is joint work in progress with Jochen Glück (University of Wuppertal).