Dirichlet forms associated with hypercontractive semigroups

Abstract

We exhibit a class of probability measures on R n {\textbf {R}^n} such that the associated Dirichlet form is represented by a selfadjoint operator A and such that e − t A {e^{ - tA}} is a hypercontractive semigroup of operators. The measures are of the form d μ = Ω 2 d x d\mu \, = \,{\Omega ^2}\,dx where Ω \Omega has classical first derivatives and L p {L^p} second derivatives, p determined by n.