Intrinsic ultracontractivity for non-symmetric Lévy processes

Recently in Kim and Song, Intrinsic ultracontractivity of non-symmetric diffusion semigroups in bounded domains: 2006, Kim and Song, Intrinsic ultracontractivity of non-symmetric diffusion with measure-valued drifts and potentials: 2006, 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 Lévy processes. We prove that, for a large class of non-symmetric discontinuous Lévy processes X such that the Lebesgue measure is absolutely continuous with respect to the Lévy 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 Vondraek, Glas. Mat. Ser. 37: 211-233, 2002, the semigroup of X^D is intrinsic ultracontractive for any bounded set D. Using the intrinsic ultracontractivity, we show that the parabolic boundary Harnack principle is true for those processes. Moreover, we get that the supremum 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.