Bessel potentials, hitting distributions and Green functions

The purpose of the paper is to find explicit formulas for basic objects pertaining to the potential theory of the operator $(I-\Delta)^{\alpha/2}$, which is based on Bessel potentials $J_{\alpha}=(I-\Delta)^{-\alpha/2}$, $0<\alpha<2$. We compute the harmonic measure of the half-space and obtain a concise form for the corresponding Green function of the operator $(I-\Delta)^{\alpha/2}$. As an application we provide sharp estimates for the Green function of the half-space for the relativistic process.