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Analysis and Geometry in Metric Spaces

Ed. by Ritoré, Manuel

CiteScore 2017: 0.65

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Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces

Zoltán M. Balogh / Jeremy T. Tyson
  • Department of Mathematics, University of Illinois at Urbana- Champaign, 1409 W Green Street, Urbana, IL 61801, USA
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/ Kevin Wildrick
Published Online: 2013-07-16 | DOI: https://doi.org/10.2478/agms-2013-0005


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.

Keywords: Sobolev mapping; Ahlfors regularity; Poincaré inequality; foliation; David–Semmes regular mapping

MSC: 46E35; 28A78; 46E40; 53C17; 30L99

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About the article

Received: 2013-04-16

Accepted: 2013-06-21

Published Online: 2013-07-16

Citation Information: Analysis and Geometry in Metric Spaces, Volume 1, Pages 232–254, ISSN (Online) 2299-3274, DOI: https://doi.org/10.2478/agms-2013-0005.

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