Space filling with metric measure spaces
type: Item
instance of: scholarly article
title: Space filling with metric measure spaces
zbMATH Open document ID: 1242.46047
author: Kevin Wildrick
author: Thomas Zürcher
published in: Mathematische Zeitschrift
publication date: 2012-02-14T00:00:00Z
review text: \textit{P. Hajłasz} and \textit{J. T. Tyson} [Mich. Math. J. 56, No. 3, 687--702 (2008; Zbl 1165.28004)] proved the following version of the Hahn-Mazurkiewicz theorem: If \(Y\) is a length-compact metric space and \(n\geq2\), then there is a continuous surjection \(f:[0,1]^n\to Y\) in the Sobolev class \(W^{1,n}([0,1]^n,Y)\). In [Ill. J. Math. 53, No. 2, 365--378 (2009; Zbl 1223.46038)], \textit{K. Wildrick} and \textit{T. Zürcher} proved a version of the above result involving Sobolev-Lorentz spaces \(W^{1,n,q}([0,1]^n,Y)\), \(1<q\leq n\). In addition, they proved that the existence of a continuous surjection \(f:[0,1]^n\to Y\) in the class \(W^{1,n,1}([0,1]^n,Y)\) implies that the Hausdorff dimension of \(Y\) is at most \(n\). In the present paper, they tackle similar questions in the situation when a general metric space (the Heisenberg group, in particular) is considered instead of \([0,1]^n\).
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 26B35
zbMATH DE Number: 6006394
zbMATH Keywords: filling mappings
zbMATH Keywords: metric measure space
zbMATH Keywords: Poincaré inequality
zbMATH Keywords: Lorentz space
zbMATH Keywords: length-compact metric space
zbMATH Keywords: upper gradient
zbMATH Keywords: Heisenberg group
zbMATH Keywords: Cantor diamond space
reviewed by: Jiří Rákosník
MaRDI profile type: Publication
full work available at URL: s00209-010-0787-1
OpenAlex ID: W2069309736
Wikidata QID: Q110125249
cites work: The Stepanov differentiability theorem in metric measure spaces
cites work: Q4039801
cites work: An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem
cites work: Differentiability of Lipschitz functions on metric measure spaces
cites work: Scaling invariant Sobolev-Lorentz capacity on $\mathbb{R}^{n}$
cites work: Q4453425
cites work: Sobolev met Poincaré
cites work: Sobolev Peano cubes
cites work: Lectures on analysis on metric spaces
cites work: Quasiconformal maps in metric spaces with controlled geometry
cites work: Sobolev classes of Banach space-valued functions and quasiconformal mappings
cites work: A singular map of a cube onto a square
cites work: On functions with derivatives in a Lorentz space
cites work: A differentiable structure for metric measure spaces
cites work: The Poincaré inequality is an open ended condition
cites work: Geometric implications of the Poincaré inequality
cites work: Q4238869
cites work: Q4354419
cites work: Q3227685
cites work: An embedding theorem for Sobolev type functions with gradients in a Lorentz space
cites work: Q3068291
cites work: Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities
cites work: Newtonian spaces: An extension of Sobolev spaces to metric measure spaces
cites work: Editor's note: the differentiability of functions in \({\mathbb{R}}^ n\)
cites work: Q4328597
cites work: Q3157281
cites work: Peano cubes with derivatives in a Lorentz space
DOI: 10.1007/S00209-010-0787-1
description: scientific article; zbMATH DE number 6006394