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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−α}; ; ; B(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 studiedin [2].

Jump processes, jump kernel, jump kernel degenerate at the boundary, Carleson estimate, boundary Harnack principle, Green function

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