Intrinsic ultracontractivity for non-symmetric Levy processes

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-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 Levy measure is compactly supported.