We introduce and investigate systematically Bessel potential spaces associated with a real-valued continuous negative definite function. These spaces, which appear in a natural way as domains of definition for some Lp-sub-Markovian semigroups, 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.
Our investigation, which is a joint work with Niels Jacob (Swansea) and Rene L. Schilling (Nottingham) is motivated by the problem of constructing a Markov process which can start everywhere in n.