Sobolev classes of Banach space-valued functions and quasiconformal mappings
type: Item
instance of: scholarly article
title: Sobolev classes of Banach space-valued functions and quasiconformal mappings
zbMATH Open document ID: 1013.46023
DOI: 10.1007/BF02788076
author: Juha Heinonen
author: Pekka Koskela
author: Nageswari Shanmugalingam
author: Jeremy T. Tyson
published in: Journal d'Analyse Mathématique
publication date: 2002-07-10T00:00:00Z
review text: A triple \((X,d,\mu)\) is called a metric measure space if \((X,d)\) is a metric space and \(u\) is a complete, regular Borel measure on \(X\) for which \(\mu(B(a,r)) > 0\) for each \(a\in X\), \(r>0\). If \(\mu\) has finite total mass, then for each Banach space \(V\) and \(1\leq p <\infty\) the Sobolev space \(N^{1,p}(X:V)\) is introduced. The elements of \(N^{1,p}(X:V)\) are classes of functions on \(X\) with values in \(V\) which are uniquely defined on \(X\) up to a set of \(p\)-capacity zero. They get characterized by various properties. Further, Poincaré inequalities for certain classes of Banach space valued functions on \(X\) are considered and it is shown that they do not depend on the Banach space if \(\mu\) is doubling, i.e., there exists \(C\geq 1\) such that \(\mu(B(a,2r))\leq C\mu(B(a,r))\) for all \(a\in X\), \(r > 0\). Among other things, \(N^{1,p}(X : V)\) is compared with the Sobolev spaces considered by Korevaar-Schoen and the role of quasisymmetric embeddings \(F : X \to V\) and the quasiconformal mappings \((X,d)\to (X',d)\) in connection with Sobolev spaces is investigated.
reviewed by: Reinhold Meise
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 1765855
zbMATH Keywords: metric measure space
zbMATH Keywords: Poincaré inequality
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: quasiconformal mappings
Wikidata QID: Q112013280
MaRDI profile type: Publication
cites work: On quasiconformal mappings
cites work: Q3211883
cites work: Quasiconformality, quasisymmetry, and removability in Loewner spaces. Appendix by Jussi Väisälä
cites work: The approximation problem for Sobolev maps between two manifolds
cites work: Uniformizing Gromov hyperbolic spaces
cites work: Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings
cites work: Differentiability of Lipschitz functions on metric measure spaces
cites work: Q5565773
cites work: Extremal length and functional completion
cites work: Q3262941
cites work: Rings and Quasiconformal Mappings in Space
cites work: The L\(^p\)-integrability of the partial derivatives of a quasiconformal mapping
cites work: Lower dimensional absolute continuity properties of quasiconformal mappings
cites work: Sobolev spaces on an arbitrary metric space
cites work: Sobolev met Poincaré
cites work: An 𝑛-dimensional space that admits a Poincaré inequality but has no manifold points
cites work: Quasiconformal maps between compact polyhedra are quasisymmetric
cites work: Definitions of quasiconformality
cites work: From local to global in quasiconformal structures.
cites work: Quasiconformal maps in metric spaces with controlled geometry
cites work: Q4526387
cites work: Generalized Dirichlet forms and harmonic maps
cites work: Q4215798
cites work: Quasiconformal mappings on the Heisenberg group
cites work: Foundations for the theory of quasiconformal mappings on the Heisenberg group
cites work: Sobolev spaces and harmonic maps for metric space targets
cites work: Removable sets for Sobolev spaces
cites work: Q4238869
cites work: Removable sets for the poincare inequality on metric spaces
cites work: Ahlfors \(Q\)-regular spaces with arbitrary \(Q>1\) admitting weak Poincaré inequality
cites work: Extensions of Lipschitz maps into Hadamard spaces
cites work: Lusin's condition (N) and mappings of the class W1, n.
cites work: The differential of a quasi-conformal mapping of a Carnot-Carathéodory space
cites work: Q4328334
cites work: Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms
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: An integral type characterization of constant functions on metric-measure spaces
cites work: Q3772469
cites work: Sobolev-type classes of functions with values in a metric space
cites work: Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities
cites work: Harmonic functions on metric spaces
cites work: Newtonian spaces: An extension of Sobolev spaces to metric measure spaces
cites work: Q4328597
cites work: Quasisymmetric embeddings of metric spaces
cites work: Q4354145
cites work: Analytic properties of locally quasisymmetric mappings from Euclidean domains
cites work: Quasi-Symmetric Embeddings in Euclidean Spaces
cites work: Analytic properties of quasiconformal mappings on Carnot groups
full work available at URL: https://doi.org/10.1007/bf02788076
OpenAlex ID: W1984151431
description: scientific article; zbMATH DE number 1765855