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 R+d defined by the jump kernel J(x, y) = | x- y| ^-d-αB(x, y) , where B(x, y) can be degenerate at the boundary. Unlike our previous works [16, 17] where we imposed critical killing, here we assume that the killing potential is identically zero. In case α∈ (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 α -stable process, α∈ (1 , 2) , in the half-space studied in [2].