Two-sided Green function estimates for killed subordinate Brownian motions

Abstract

A subordinate Brownian motion is a Lévy process which can obtained by replacing the time of Brownian motion by an independent increasing Lévy process. The infinitesimal generator of a subordinate Brownian motion is −φ(−∆), where φ is the Laplace exponent of the subordinator. In this paper, we consider a large class of subordinate Brownian motions without diffusion term. This class of processes include symmetric stable processes, relativistic stable processes, sums of independent symmetric stable processes, sums of independent relativistic stable processes, and much more. We give sharp two-sided estimates on the Green functions of these subordinate Brownian motions in any bounded κ-fat open set D. When D is a bounded C 1,1 open set, we establish an explicit form of the estimates in terms of the distance to the boundary. As a consequence of such sharp Green function estimates, we obtain the boundary Harnack principle in C 1,1 open set with explicit decay rate