Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians (Q1060403)

The authors investigate connections between integral kernels of positivity preserving semigroups and \(L^ p\)-contractivity properties. There are treated essentially four connected topics: (1) Extension properties for \(e^{-tA}\) from \(L^ 2\) to \(L^{\infty}\) where A is a Schrödinger operator generated by its ground state. (2) The same problem for the Dirichlet Laplacian for certain subsets of \({\mathbb{R}}^ n.\) (3) Sobolev estimates up to the boundary. (4) Pointwise bounds for the integral kernel of \(e^{-Nt}\) in terms of the ground state of H.