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Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. (English) Zbl 0568.47034

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.
Reviewer: H.Siedentop

MSC:

47D03 Groups and semigroups of linear operators
47F05 General theory of partial differential operators
35J10 Schrödinger operator, Schrödinger equation
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