Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians

Abstract

Abstract connections between integral kernels of positivity preserving semigroups and suitable L^p contractivity properties are established. Then these questions are studied for the semigroups generated by $\text{−Δ + V}\text{and}\text{H}{}_{\mathrm{\Omega }}$, the Dirichlet Laplacian for an open, connected region Ω. As an application under a suitable hypothesis, Sobolev estimates are proved valid up to ∂Ω, of the form $\text{¦η(x)¦⩽ cϑ}{}_0\text{(x) ∥H}{}_{\mathrm{\Omega }}{}^{\mathrm{k}}\text{η∥}{}_2$, where ϑ₀ is the unique positive L² eigenfunction of $\text{H}{}_{\mathrm{\Omega }}$.