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

Potential theory of Dirichlet forms with jump kernels blowing up at the boundary