Towards an L p Potential Theory for Sub–Markovian Semigroups: Kernels and Capacities

Abstract

We study in fairly general measure spaces (X, μ) the (non–linear) potential theory of Lp sub–Markovian semigroups which are given by kernels having a density with respect to the underlying measure. In terms of mapping properties of the operators we provide sufficient conditions for the existence (and regularity) of such densities. We give various (dual) representations for several associated capacities and, in the corresponding abstract Bessel potential spaces, we study the role of the truncation property. Examples are discussed in the case of ℝn, where abstract Bessel potential spaces can be identified with concrete function spaces.