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 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 Rⁿ. It is also shown that the conditional lifetimes for symmetric stable process in a Hölder domain of order 0 are uniformly bounded.