×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI