Intrinsic Ultracontractivity, Conditional Lifetimes and Conditional Gauge for Symmetric Stable Processes on Rough Domains
Zhen-Qing Chen and Renming Song
For a symmetric $\alpha$-stable process $X$ on ${\mathbb R}^n$ with $0\lt \alpha \lt 2$, $n\geq 2$ and a domain $D \subset {\mathbb R}^n$, let $L^D$ be the infinitesimal generator of the subprocess of $X$ killed upon leaving $D$. For a Kato class function $q$, it is shown that $L^D+q$ is intrinsic ultracontractive on a Hölder domain $D$ of order 0. This is then used to establish the conditional gauge theorem for $X$ on bounded Lipschitz domains in ${\mathbb R}^n$. It is also shown that the conditional lifetimes for symmetric stable process in a Hölder domain of order 0 are uniformly bounded.