Bessel potentials, hitting distributions and Green functions

Abstract

The purpose of the paper is to find explicit formulas for basic objects pertaining to the potential theory of the operator ( I − Δ ) α / 2 (I-\Delta )^{\alpha /2} , which is based on Bessel potentials J α = ( I − Δ ) − α / 2 J_{\alpha }=(I-\Delta )^{-\alpha /2} , 0 > α > 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 − Δ ) α / 2 (I-\Delta )^{\alpha /2} . As an application we provide sharp estimates for the Green function of the half-space for the relativistic process.