Frequency of Sobolev and quasiconformal dimension distortion
type: Item
instance of: scholarly article
title: Frequency of Sobolev and quasiconformal dimension distortion
zbMATH Open document ID: 1266.28003
author: Zoltán M. Balogh
author: Roberto Monti
author: Jeremy T. Tyson
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2013-02-19T00:00:00Z
review text: The authors study Hausdorff and Minkowski dimension distortion for images of generic affine subspaces of Euclidean space under Sobolev and quasiconformal maps. \textit{P.~Hajłasz} and \textit{J.~Tyson} constructed [Mich. Math. J. 56, No. 3, 687--702 (2008; Zbl 1165.28004)] Sobolev mappings in the Reshetnyak-Sobolev space \({\mathbb R}^{1,n}\) from the \(n\)-dimensional cube onto any length-compact space, a class which includes certain infinite-dimensional spaces. On the other hand, it follows from the Sobolev embedding theorem that, in the case a Sobolev exponent \(p\) larger than \(n\), the dimension of the image is at most \(n\). (When \(p>n\), we always talk of the continuous representative.) However, subsets of the cube that have dimension less than \(n\) may still suffer an increase in dimension. \textit{R.~Kaufman} obtained [Proc. Am. Math. Soc. 128, No. 2, 427--431 (2000; Zbl 0938.28001)] better bounds than one would deduce by simply applying the Hölder continuity arising from the embedding theorem. The following version of one of Kaufman's results is the starting point for the paper under review. Denoting by \(G(n,m)\) the Grassmannian, by \(V^\perp\) the orthogonal complement of \(V\in G(n,m)\), and for \(a\in V^\perp\) setting \(V_a=V+a\): Let \(f\in W^{1,p}(\Omega,Y)\) for \(p>n\) and let \(V\) in the Grassmannian \(G(n,m)\). Then \(f(V_a\cap \Omega)\) has zero \({\mathcal{H}}^{pm/(p-n+m)}\) measure for each \(a\in V^\perp\), in particular, \[ \dim f(V_a\cap \Omega)\leq {pm\over p-n+m}. \] The authors generalize this result as follows by allowing an exceptional set in \(V^\perp\). For integers \(m\) and \(n\) satisfying \(1\leq m\leq n-1\), for \(p\geq 1\) and \(m\leq \alpha\leq {pm\over p-n+m}\), they set \[ \beta(p,\alpha):=(n-m)-(1-{m\over\alpha})p. \] Suppose \(\Omega\) in \(\mathbb R^n\) is a domain, \(f\in W^{1,p}(\Omega,Y)\), \(p>n\), \(V\in G(n,m)\), and \[ m<\alpha\leq {pm\over p-n+m}. \] Then \(f(V_\alpha\cap \Omega)\) has zero \({\mathcal H}^\a…
reviewed by: Thomas Zürcher
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 31B15
Mathematics Subject Classification ID: 30L99
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 30C65
zbMATH DE Number: 6137364
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: Sobolev mapping
zbMATH Keywords: potential theory
zbMATH Keywords: quasiconformal mapping
zbMATH Keywords: space-filling mapping
MaRDI profile type: Publication
full work available at URL: j.matpur.2012.06.005
OpenAlex ID: W2100889199
DOI: 10.1016/J.MATPUR.2012.06.005
description: scientific article; zbMATH DE number 6137364