On (r,p)-capacities for Markov processes
Let X be a separable metric space and \(m\geq 0\) be a Radon measure with support X. Let \((P_ t)\) be a Markovian contractive semigroup on \(L_ p(X;m)\) \((1<p<\infty)\) and define, \[ V_ r:=\ell (r/2)^{- 1}\int^{\infty}_{0}s^{r/2-\ell}\cdot e^{-s}\cdot P_ sds,\quad F_{r,p}:=V_ r(L_ p). \] The (r,p)-capacity \(C_{r,p}\) is a set function defined by: \[ C_{r,p}(A)=\inf \{\| u\|^ p_{r,p};\quad u\in F_{r,p},\quad u\geq 1\quad m-a.e.\quad on\quad U\}, \] where \(\| u\|_{r,p}=\| f\|_{L_ p}\) for \(u=V_ rf\) and U any open set \(\supset A\). When \((P_ t)\) is a symmetric semigroup on \(L_ 2\) then the well-known \(C_{1,2}\) capacity on Dirichlet spaces introduced by Fukushima is recovered. The aim of the paper under review is to study fundamental properties of (r,p)-capacities associated with a non-necessarily symmetric Markovian semigroup. As application the author refines an ergodic theorem of \textit{G. C. Rota} [Bull. Am. Math. Soc. 68, 95-102 (1962; Zbl 0116.104)] in the sense that m-a.e. statements can be replaced by null \(C_{r,p}\)- capacity statements. Furthermore, the existence theorem for Hunt semigroups associated with a Dirichlet space can also be extended to the case where \(r\neq 1\) and \(p\neq 2\).
scientific article; zbMATH DE number 4030663