Intrinsic ultracontractivity for general Lévy processes on bounded open sets

Abstract

We prove that a general (not necessarily symmetric) Lévy process killed on exiting a bounded open set (without regular condition on the boundary) is intrinsically ultracontractive, provided that $B(0,R_{0})\subseteq\operatorname{supp} (\nu)$ for some constant $R_{0}>0$, where $\operatorname{supp} (\nu)$ denotes the support of the associated Lévy measure $\nu$. For a symmetric Lévy process killed on exiting a bounded Hölder domain of order $0$, we also obtain the intrinsic ultracontractivity under much weaker assumption on the associated Lévy measure.