Function spaces related to continuous negative definite functions: ψ-Bessel potential spaces

Abstract

We introduce and systematically investigate Bessel potential spaces associated with a real-valued continuous negative definite function. These spaces can be regarded as (higher order) Lp-variants of translation invariant Dirichlet spaces and in general they are not covered by known scales of function spaces. We give equivalent norm characterizations, determine the dual spaces and prove embedding theorems. Furthermore, complex interpolation spaces are calculated. Capacities are introduced and the existence of quasi-continuous modifications is shown.