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The purpose of this paper is to find optimal estimates for the Green functionof a half-space of {\it the relativistic $\alpha$-stable process} withparameter $m$ on $\Rd$ space. This process has an infinitesimal generator ofthe form $mI-(m^{2/\alpha}I-\Delta)^{\alpha/2},$ where $0<\alpha<2$, $m>0$, andreduces to the isotropic $\alpha$-stable process for $m=0$. Its potentialtheory for open bounded sets has been well developed throughout the recentyears however almost nothing was known about the behaviour of the process onunbounded sets. The present paper is intended to fill this gap and we providetwo-sided sharp estimates for the Green function for a half-space. As abyproduct we obtain some improvements of the estimates known for bounded setsspecially for balls. The advantage of these estimates is a clarification of therelationship between the diameter of the ball and the parameter $m$ of theprocess.  The main result states that the Green function is comparable with the Greenfunction for the Brownian motion if the points are away from the boundary of ahalf-space and their distance is greater than one. On the other hand for theremaining points the Green function is somehow related the Green function forthe isotropic $\alpha$-stable process. For example, for $d\ge3$, it iscomparable with the Green function for the isotropic $\alpha$-stable process,provided that the points are close enough.

Two-sided optimal bounds for Green function of half-spaces for relativistic $\alpha$-stable process