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/α−Δ)α/2m−(m2/α−Δ)α/2) in half-space-like C1,1C1,1 open sets. The estimates are uniform in m∈(0,M]m∈(0,M] for each fixed M∈(0,∞)M∈(0,∞). When m↓0m↓0, our estimates reduce to the sharp Green function estimates for −(−Δ)α/2−(−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 XmXm, which is uniform for all m∈(0,∞)m∈(0,∞), holds for a large class of non-smooth open sets.