Sharp Estimates on the Green Functions of Perturbations of Subordinate Brownian Motions in Bounded κFat Open Sets

In this paper we study perturbations of a large class of subordinate Brownian motions in bounded κ-fat open sets, which include bounded John domains. Suppose that X is such a subordinate Brownian motion and that J is the Lévy density of X. The main result of this paper implies, in particular, that if Y is a symmetric Lévy process with Lévy density J^Y satisfying {pipe}J^Y(x) - J(x){pipe} ≤ c max {{pipe}x{pipe}^{-d + ρ}, 1} for some c > 0,ρ ∈ (0, d), then for any bounded John domain D the Green function G_D^Y of Y in D is comparable to the Green function G_D of X in D. One of the main tools of this paper is the drift transform introduced in Chen and Song (J Funct Anal 201:262-281, 2003). To apply the drift transform, we first establish a generalized 3G theorem for X.