On Uniformity Exponents of $$\varphi $$ -Uniform Domains

IF 0.6 4区 数学 Q3 MATHEMATICS
Yahui Sheng, Fan Wen, Kai Zhan
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引用次数: 0

Abstract

Let \(G\subsetneq {\mathbb {R}}^n\) be a domain, where \(n\ge 2\). Let \(k_G\) and \(j_G\) be the quasihyperbolic metric and the distance ratio metric on G, respectively. In the present paper, we prove that the identity map of \((G,k_G)\) onto \((G,j_G)\) is quasisymmetric if and only if it is bilipschitz. To classify domains of \({\mathbb {R}}^n\) into various types according to the behaviors of their quasihyperbolic metrics, we define a uniformity exponent for every proper subdomain of \({\mathbb {R}}^n\) and prove that this exponent may assume any value in \(\{0\}\cup [1,\infty ]\). Moreover, we study the properties of domains of uniformity exponent 1 and show by an example that such a domain may be neither quasiconvex nor accessible.

Abstract Image

论 $$\varphi $$ -Uniform 域的均匀性指数
让(G/subsetneq {\mathbb {R}}^n\) 是一个域,其中(n/ge 2\).让 \(k_G\) 和 \(j_G\) 分别是 G 上的准双曲度量和距离比度量。在本文中,我们将证明当且仅当 \((G,k_G)\ 到 \((G,j_G)\) 的标识映射是双双曲的时候,它是准对称的。为了根据准双曲度量的行为将 \({\mathbb {R}}^n\) 的域划分为各种类型,我们为 \({\mathbb {R}}^n\) 的每个适当子域定义了一个均匀性指数,并证明这个指数可以在 \({0\}\cup [1,\infty ]\) 中取任意值。此外,我们还研究了均匀性指数为 1 的域的性质,并通过一个例子证明了这样的域可能既不是准凸的,也不是可及的。
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来源期刊
Computational Methods and Function Theory
Computational Methods and Function Theory MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
3.20
自引率
0.00%
发文量
44
审稿时长
>12 weeks
期刊介绍: CMFT is an international mathematics journal which publishes carefully selected original research papers in complex analysis (in a broad sense), and on applications or computational methods related to complex analysis. Survey articles of high standard and current interest can be considered for publication as well.
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