Sharp estimates on the Green functions of perturbations of subordinate Brownian motions in bounded κ-fat 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 satisfying |J (x) − J(x)| ≤ cmax{|x|−d+ρ, 1} for some c > 0, ρ ∈ (0, d), then for any bounded John domain D the Green function GD of Y in D is comparable to the Green function GD of X in D. One of the main tools of this paper is the drift transform introduced in [11]. To apply the drift transform, we first establish a generalized 3G theorem for X. AMS 2010 Mathematics Subject Classification: Primary 60J45, Secondary 60J25, 60J51.