Kato class functions of Markov processes under ultracontractivity

Abstract

We show that $f \in L^p (X ; m)$ implies $|f| dm \in S_{K}^{1}$ for $p \gt D$ with $D \gt 0$, where $S_{K}^{1}$ is a subfamily of Kato class measures relative to a semigroup kernel $p_t (x, y)$ of a Markov process associated with a (non-symmetric) Dirichlet form on $L^2 (X ; m)$. We only assume that $p_t (x, y)$ satisfies the Nash type estimate of small time depending on $D$. No concrete expression of $p_t (x,y)$ is needed for the result.