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

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For a symmetric α-stable process X on ⁿ with 0<α<2, n≥ 2 and a domain D ⊂ ⁿ, let Lᵈ be the infinitesimal generator of the subprocess of X 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 ⁿ. It is also shown that the conditional lifetimes for symmetric stable process in a Hölder domain of order 0 are uniformly bounded.