Function Spaces and Symmetric Markov Processes

Abstract

We exhibit some mutual interactions between potential theory for concrete function spaces on $\mathbb{R}^n$ and the Dirichlet space theory associated with symmetric Markov processes. Our first concern is the role of the Dirichlet form version of the capacitary strong type inequality in the study of the ultracontractivity of the transition semigroup of time changed symmetric Markov processes. In particular, we study time changes of symmetric stable processes in relation to $d$-bounds of measures. We next show how the theory on capacity and the spectral synthesis for the Dirichlet space can be well inherited to a general function space with contractive $p$-norm. A link connecting those two topics is a contractive Besov space over a $d$-set of $\mathbb{R}^n$.