*(English)*Zbl 0903.60063

For $0<\alpha <2$ and $d\ge 3$, let $({X}_{t},{P}_{x})$ be the symmetric $\alpha $-stable process on ${\mathbb{R}}^{d}$. Denote by ${G}_{D}(x,y)$ the Green function for a bounded open set $D$ in ${\mathbb{R}}^{d}$. The main purpose of this paper is to show the following upper and lower estimates of ${G}_{D}(x,y)$ for a bounded domain $D$ with ${C}^{1,1}$ boundary: There exist constants ${C}_{1}$ and ${C}_{2}$ depending on $D$, $d$ and $\alpha $ such that

where $\delta \left(x\right)=\text{dist}(x,\partial D)$. This is first shown by using the explicit expression of the Green function in the case that $D$ is the ball $B(0,r)$ of center 0 and radius $r$ and then it is used to get the result for general bounded ${C}^{1,1}$ domains. Some approach is taken from *Z. Zhao* [J. Math. Anal. Appl. 116, 309-334 (1986; Zbl 0608.35012)] in which Brownian motion case is concerned. Hence the essential change of the proof is necessary according to the jumping property of the sample paths. As applications, 3G Theorem and the estimates of the mean exit times are given.