A Koebe distortion theorem for quasiconformal mappings in the Heisenberg group

被引：0
作者：
Tomasz Adamowicz
Katrin Fässler
Ben Warhurst
机构：
[1] Polish Academy of Sciences,The Institute of Mathematics
[2] University of Fribourg,Department of Mathematics
[3] University of Warsaw,Institute of Mathematics
[4] University of Jyväskylä,Department of Mathematics and Statistics
来源：
Annali di Matematica Pura ed Applicata (1923 -) | 2020年 / 199卷
关键词：
Quasiconformal mappings; Heisenberg group; Modulus of curves; Koebe theorem; BMO; Primary 30L10; Secondary 30C65; 30F45;
D O I：
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学科分类号：
摘要：
We prove a Koebe distortion theorem for the average derivative of a quasiconformal mapping between domains in the sub-Riemannian Heisenberg group H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {H}}^{1}$$\end{document}. Several auxiliary properties of quasiconformal mappings between subdomains of H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {H}}^{1}$$\end{document} are proven, including BMO estimates for the logarithm of the Jacobian. Applications of the Koebe theorem include diameter bounds for images of curves, comparison of integrals of the average derivative and the operator norm of the horizontal differential, as well as the study of quasiconformal densities and metrics in domains in H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {H}}^{1}$$\end{document}. The theorems are discussed for the sub-Riemannian and the Korányi distances. This extends results due to Astala–Gehring, Astala–Koskela, Koskela and Bonk–Koskela–Rohde.
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页码：147 / 186
页数：39
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