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

For a symmetric alpha-stable process X on R-n with 0 < alpha < 2, n less th an or equal to 2 and a domain D subset of R-n,let L-D be the infinitesimal generator of the subprocess of X killed upon leaving D. For a Kato class fu nction q, it is shown that L-D + q is intrinsic ultracontractive on a Holde r domain D of order 0. Then this is used to establish the conditional gauge theorem for X on bounded Lipschitz domains in R-n. it is also shown that t he conditional lifetimes for symmetric stable process in a Holder domain of order 0 are uniformly bounded.