Potential theory of Dirichlet forms degenerate at the boundary: the case of no killing potential

In this paper we consider the Dirichlet form on the half-space ${\mathbb{R}}_{+}^d$ defined by the jump kernel $J(x,y)=\mathrm{\mid }x-y{\mathrm{\mid }}^{-d-\alpha }\mathcal{B}(x,y)$, where $\mathcal{B}(x,y)$ can be degenerate at the boundary. Unlike our previous works [6,7] where we imposed critical killing, here we assume that the killing potential is identically zero. In case $\alpha \in (1,2)$ we first show that the corresponding Hunt process has finite lifetime and dies at the boundary. Then, as our main contribution, we prove the boundary Harnack principle and establish sharp two-sided Green function estimates. Our results cover the case of the censored $\alpha$-stable process, $\alpha \in (1,2)$, in the half-space studied in [2].