Dimension distortion by sobolev mappings in foliated metric spaces

Zoltán M. Balogh, Jeremy T. Tyson, Kevin Wildrick

Research output: Contribution to journalArticlepeer-review

Abstract

We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincaré inequality. We show that the set of mappings that distort the dimensions of sets by the maximum possible amount is a prevalent subset of the relevant function space. For foliations of a metric space X defined by a David-Semmes regular mapping π : X → W, we quantitatively estimate, in terms of Hausdorff dimension in W, the size of the set of leaves of the foliation that are mapped onto sets of higher dimension. We discuss key examples of such foliations, including foliations of the Heisenberg group by left and right cosets of horizontal subgroups.

Original languageEnglish (US)
Pages (from-to)232-254
Number of pages23
JournalAnalysis and Geometry in Metric Spaces
Volume1
Issue number1
DOIs
StatePublished - 2013

Keywords

  • Ahlfors regularity
  • David-semmes regular mapping
  • Foliation
  • Poincaré inequality
  • Sobolev mapping

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology
  • Applied Mathematics

Fingerprint

Dive into the research topics of 'Dimension distortion by sobolev mappings in foliated metric spaces'. Together they form a unique fingerprint.

Cite this