Quasiregular maps on Carnot groups
type: Item
instance of: scholarly article
title: Quasiregular maps on Carnot groups
zbMATH Open document ID: 0905.30018
DOI: 10.1007/BF02921707
author: Juha Heinonen
author: Ilkka Holopainen
published in: The Journal of Geometric Analysis
publication date: 1998-08-20T00:00:00Z
review text: The authors develop a theory of quasiregular maps in a sub-Riemannian geometry of two-step Carnot groups. An analytic definition for quasiregularity is suggested and it is shown that conconstant quasiregular maps are open and discrete maps on Carnot groups which are two-step nilpotent and of Heisenberg type. Some results which are known to be valid in \(\mathbb{R}^n\) are extended to this setting of Carnot groups. In particular, it is established that the branch set of a nonconstant quasiregular map has Haar measure zero and quasiregular maps are almost everywhere differentiable in the sense of Pansu. The authors approach is that of nonlinear potential theory: the Harnack inequality and Hölder continuity of solutions are obtained.
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 31C45
Mathematics Subject Classification ID: 58J05
zbMATH DE Number: 1192506
zbMATH Keywords: Heisenberg group
zbMATH Keywords: nonlinear potential theory
zbMATH Keywords: quasiregular maps
zbMATH Keywords: Carnot groups
MaRDI profile type: Publication
cites work: Analytical foundations of the theory of quasiconformal mappings in R^n
cites work: Harnack's inequality for elliptic differential equations on minimal surfaces
cites work: An embedding theorem and the harnack inequality for nonlinear subelliptic equations
cites work: Q4692884
cites work: \(H\)-type groups and Iwasawa decompositions
cites work: Uniform limits of certain A-harmonic functions with applications to quasiregular mappings
cites work: Subelliptic estimates and function spaces on nilpotent Lie groups
cites work: Hardy Spaces on Homogeneous Groups. (MN-28), Volume 28
cites work: Q4106439
cites work: Multilinear Algebra
cites work: Some regularity theorems for Carnot-Caratheodory metrics
cites work: The Dirichlet problem for sublaplacians on nilpotent Lie groups - geometric criteria for regularity
cites work: A capacity estimate on Carnot groups
cites work: Definitions of quasiconformality
cites work: Q4693180
cites work: Q3996137
cites work: Positive Solutions of Quasilinear Elliptic Equations on Riemannian Manifolds
cites work: A Picard type theorem for quasiregular mappings of \(\mathbb{R}^ n\) into \(n\)- manifolds with many ends
cites work: Quasiregular mappings of the Heisenberg group
cites work: Wiener's criterion in potential theory with applications to nilpotent Lie groups
cites work: Quasiregular mappings in even dimensions
cites work: The Poincaré inequality for vector fields satisfying Hörmander's condition
cites work: Fundamental Solutions for a Class of Hypoelliptic PDE Generated by Composition of Quadratic Forms
cites work: Q3772867
cites work: Geometric properties of Heisenberg-type groups
cites work: Quasiconformal mappings on the Heisenberg group
cites work: Foundations for the theory of quasiconformal mappings on the Heisenberg group
cites work: Picard's Theorem and Rickman's Theorem by Way of Harnack's Inequality
cites work: Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications
cites work: The sharp Poincaré inequality for free vector fields: An endpoint result
cites work: Lusin's condition (N) and mappings with nonnegative Jacobians
cites work: Strong Rigidity of Locally Symmetric Spaces. (AM-78)
cites work: A remark on quasiconformal mappings on Carnot groups
cites work: Carnot-Carathéodory metrics and quasiisometries of symmetric spaces of rank 1
cites work: Dimension conforme et sphère à l'infini des variétés à courbure négative
cites work: An estimate for pseudoconformal capacities on the sphere
cites work: Q5546093
cites work: Q3992482
cites work: The analogue of Picard's theorem for quasiregular mappings in dimension three
cites work: Q3141918
cites work: Sub-Riemannian geometry
cites work: Quasiconformal homeomorphisms on CR 3-manifolds with symmetries
cites work: The Extension of Interiority, With Some Applications
cites work: Quasiconformal space mappings. A collection of surveys 1960-1990
description: scientific article; zbMATH DE number 1192506