Intrinsic Ultracontractivity, Conditional Lifetimes and Conditional Gauge for Symmetric Stable Processes on Rough Domains

For a symmetric α-stable process X on Rn with 0 < α < 2, n ≥ 2 and a domain D ⊂ Rn, let LD be the infinitesimal generator of the subprocess ofX killed upon leaving D. For a Kato class function q, it is shown that LD + 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 Rn. It is also shown that the conditional lifetimes for symmetric stable process in a Hölder domain of order 0 are uniformly bounded.