Intrinsic ultracontractivity, conditional lifetimes and conditional gauge for symmetric stable processes on rough domains

Abstract

For a symmetric $\alpha$-stable process $X$ on $\mathbf{R}^{n}$ with $0 \lt \alpha \lt 2$, $n \geq 2$ and a domain $D \subset \mathbf{R}^{n}$, 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. Then this is used to establish the conditional gauge theorem for $X$ on bounded Lipschitz domains in $\mathbf{R}^{n}$. It is also shown that the conditional lifetimes for symmetric stable process in a Hölder domain of order 0 are uniformly bounded.