Monday Analysis Seminar: Intrinsic ultracontractivity and the boundary Harnack principle — A unified approach with capacitary width

2013-4-15 14:45 - 2013-4-15 16:15
Faculty of Science Buliding #3 Room 210
Hiroaki Aikawa (Hokkaido University)
We study intrinsic ultracontractivity (IU) for the semigroup associated with Dirichlet heat kernel. We give a sharp sufficient condition for IU, valid for arbitrary domains, in terms of capacitary width of sublevel sets of the Green function and the ground state. This condition, together with the Harnack inequality, yields more specific sufficient conditions for IU of nonsmooth domains. Our approach employs a parabolic box argument, a counterpart of the box argument for the boundary Harnack principle. We can treat IU and the boundary Harnack principle in parallel.
Our method is purely analytic and elementary; it enables us to dispense with logarithmic Sobolev inequalities.