Estimates for the products of the Green function and the Martin kernel

Abstract

Let $\Omega$ be a proper subdomain of $\mathbb{R}^{n}$, $n \ge 2$, and let $x_{0} \in \Omega$ be fixed. By $G_{\Omega}$ and $K_{\Omega}$ we denote the Green function and the Martin kernel for $\Omega$, respectively. Under a certain assumption on $\Omega$ near a boundary point $\xi$, we show that the product $G_{\Omega}(x, x_{0})K_{\Omega}(x, \xi)$ is comparable to $|x-\xi|^{2-n}$ for $x$ in a nontangential cone with vertex at $\xi$. We also give an estimate for the product $K_{\Omega}(x, \xi)K_{\Omega}(x, \eta)$ in a uniform domain, where $\eta$ is another boundary point.