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Two necessary and sufficient conditions for uniform domains. (English) Zbl 1173.30315
Summary: Let $$D$$ be a proper subdomain of the Euclidean $$n$$-space $$\mathbb R^n$$ $$(n\geq 2)$$. The following 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 $$\frac1{\left(c_2^\alpha-c_1^\alpha\right)}\int_{\gamma_{j,[c_1,c_2]}}d(x,\partial D)^{\alpha-1}\text{d}s\leq\frac M\alpha|x_1-x_2|^\alpha$$ 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 uses arc length $$s$$ as parameter, $$l(\gamma)$$ is the Euclidean length of $$\gamma$$.
##### MSC:
 30C99 Geometric function theory
##### Keywords:
uniform domain; quasi-hyperboloc metric; rectifiable arc
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