Green function estimates for relativistic stable processes in half-space-like open sets

In this paper, we establish sharp two-sided estimates for the Green functions of relativistic stable processes (i.e. Green functions for non-local operators m-(m2/[alpha]-[Delta])[alpha]/2) in half-space-like C1,1 open sets. The estimates are uniform in m[set membership, variant](0,M] for each fixed M[set membership, variant](0,[infinity]). When m[downwards arrow]0, our estimates reduce to the sharp Green function estimates for -(-[Delta])[alpha]/2 in such kind of open sets that were obtained recently in Chen and Tokle [12]. As a tool for proving our Green function estimates, we show that a boundary Harnack principle for Xm, which is uniform for all m[set membership, variant](0,[infinity]), holds for a large class of non-smooth open sets.