Dirichlet forms associated with hypercontractive semigroups

Abstract:

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