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.