Mappings with bounded distortion and with finite distortion on Carnot groups
type: Item
instance of: scholarly article
title: Mappings with bounded distortion and with finite distortion on Carnot groups
zbMATH Open document ID: 0973.30021
DOI: 10.1007/BF02675667
author: S. K. Vodop'yanov
published in: Siberian Mathematical Journal
publication date: 2000-05-04T00:00:00Z
review text: The goal of the article is to develop a theory of mappings with bounded distortion (quasiregular mappings) on a Carnot group equipped with a Carnot-Carathéodory metric, starting from the weakest analytical assumption. A Carnot group is said to be a connected simply connected nilpotent Lie group \(\mathbb G\) whose Lie algebra is decomposed into the direct sum \(V_1\oplus\dots\oplus V_m\) of vector spaces such that \(\dim V_1\geq 2\), \([V_1,V_k]=V_{k+1}\) for \(1\leq k\leq m-1\) and \([V_1,V_m]=\{0\}\); the Carnot-Carathéodory distance \(d(x,y)\) between \(x,y\in\mathbb G\) is defined as the infimum of the lengths of the curves tangent to \(V_1\) a.e. [see \textit{P.~Pansu}, Ann. Math., II. Ser. 129, No. 1, 1-60 (1989; Zbl 0678.53042)]. Definition 1. A continuous mapping \(\varphi: \Omega \to \mathbb G\) is a mapping with bounded distortion on a domain \(\Omega\) of a Carnot group if \(\varphi\) belongs to \(HW_{\nu,\text{loc}}^1(\Omega)\), \(\nu\) is the Hausdorff dimension of \(\mathbb G\), and, for a.e. \(x\in\Omega\), the formal horizontal differential \(|D_H \varphi(x)|\) determined by the matrix \(X_k\varphi\) (\(X_k\), \(1\leq\dim V_1\), is the left-invariant basis of \(V_1\)) satisfies the inequality \[ |D_H \varphi(x)|^{\nu}\leq KJ(x,\varphi). \] Foundation of the theory is based on a broader class of mappings with finite distortion. Definition 2. A mapping \(\varphi: \Omega \to \mathbb G\) is a mapping with finite distortion on a domain \(\Omega\) of a Carnot group if \(\varphi\) belongs to \(HW_{\nu,\text{loc}}^1(\Omega)\), its Jacobian \(J(x,\varphi)\) is nonnegative, and \(D_H \varphi(x)\neq 0\) implies \(J(x,\varphi)\geq 0\) for a.e. \(x\in\Omega\). It is proven that mappings with finite distortion are continuous, quasimonotone, \(P\)-differentiable, and satisfy the \(N\)-Luzin property. This makes it possible to establish a connection between solutions to quasilinear subelliptic equations and mappings with bounded distortion in the case when the \(\nu…
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 43A80
zbMATH DE Number: 1439065
zbMATH Keywords: Carnot group
zbMATH Keywords: mapping with bounded distortion
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1439065