Basic potential theory of certain nonsymmetric strictly alpha-stable processes

We study potential theoretic properties of strictly α-stable processes whose Levy measure is comparable to that of a symmetric α-stable process. We show the existence, continuity and strict positivity of transition densities and Green function of the process killed upon exiting a bounded domain. We further show that the exit distributions of the process from a domain satisfying the uniform volume condition have a density. The density is used to establish a representation of regular harmonic functions of the process. Finally, we indicate that the Harnack inequality is true for nonnegative harmonic functions