Two necessary and sufficient conditions for uniform domains

Abstract

Let $D$ be a proper subdomain of Euclidean $n$-space $R^n\ (n\geq2)$. The following two necessary and sufficient conditions for uniform domains are obtained in this paper: (1) $D$ is a uniform domain if and only if there exists a constant $m=m(D)$ such that $k_D(x_1,x_2)\leq mj_D(x_1,x_2)$ for any $x_1, x_2\in D$, where $k_D$ is the quasi-hyperbolic metric in $D$, $j_D(x_1,x_2)=\frac12\log\left(\frac{|x_1-x_2|}{d(x_1,\partial D)}+1\right)\left(\frac{|x_1-x_2|}{d(x_2,\partial D)}+1\right)$. (2) $D$ is a uniform domain if and only if there exists a constant $M=M(D)$ such that each pair of points $x_1,x_2\in D$ can be joined by a rectifiable arc $\gamma\subset D$ which satisfies $\frac1{\left(c_2^\alpha-c_1^\alpha\right)}\int_{\gamma_{j,[c_1,c_2]}}d(x,\partial D)^{\alpha-1}\mbox{d}s\leq\frac M\alpha|x_1-x_2|^\alpha$ for any $0<\alpha\leq1$ and $0\leq c_1< c_2\leq1/2$, $j=1,2$, where $\gamma_{j,[c_1,c_2]}$ denotes the subarc between $\gamma_j(c_1l(\gamma))$ and $\gamma_j(c_2l(\gamma))$, $\gamma_j$ is the arc $\gamma$ which starts from $x_j$ and use arc length $s$ as parameter, $l(\gamma)$ is the Euclidean length of $\gamma$.