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-(m^2/α-Δ)^α/2) in half-space-like C^1,1 open sets. The estimates are uniform in m∈(0,M] for each fixed M ∈ (0,∞). When m ↓ 0, our estimates reduce to the sharp Green function estimates for -(-Δ) ^α/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 X^m, which is uniform for all m ∈ (0,∞), holds for a large class of non-smooth open sets.