  In this paper we study the potential theory of Dirichlet forms on the
half-space $\mathbb{R}^d_+$ defined by the jump kernel
$J(x,y)=|x-y|^{-d-\alpha}\mathcal{B}(x,y)$ and the killing potential $\kappa
x_d^{-\alpha}$, where $\alpha\in (0, 2)$ and $\mathcal{B}(x,y)$ can blow up to
infinity at the boundary. The jump kernel and the killing potential depend on
several parameters. For all admissible values of the parameters involved and
all $d \ge 1$, we prove that the boundary Harnack principle holds, and
establish sharp two-sided estimates on the Green functions of these processes.
