Recently in [17, 18], we extended the concept of intrinsic ultracontractivity to non-symmetric semigroups and proved that for a large class of non-symmetric diffusions Z with measure-valued drift and potential, the semigroup of Z~D (the process obtained by killing Z upon exiting D) in a bounded domain is intrinsic ultracontractive under very mild assumptions. In this paper, we study the intrinsic ultracontractivity for non-symmetric discontinuous Levy processes. We prove that, for a large class of non-symmetric discontinuous Levy processes X such that the Lebesgue measure is absolutely continuous with respect to the Levy measure of X, the semigroup of X~D in any bounded open set D is intrinsic ultracontractive. In particular, for the non-symmetric stable process X discussed in [25], the semigroup of X° is intrinsic ultra-contractive for any bounded set D. Using the intrinsic ultracontractivity, we show that the par_abolic boundary Harnack principle is true for those processes. Moreover, we get that the su_premum of the expected conditional lifetimes in a bounded open set is finite. We also have results of the same nature when the Lévy measure is compactly supported.
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