Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces

• Autorzy: Zoltán M. Balogh , Jeremy T. Tyson , Kevin Wildrick
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

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

Kategorie tematyczne

• 46E40: Spaces of vector- and operator-valued functions
• 46E35: Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems
• 28A78: Hausdorff and packing measures
• 53C17: Sub-Riemannian geometry
• 30L99: None of the above, but in this section

• Wydawca: De Gruyter Open
• Czasopismo: Analysis and Geometry in Metric Spaces
• Rocznik: 2013
• Tom: 1
• Strony: 232-254

Daty

otrzymano

2013-04-16

zaakceptowano

2013-06-21

online

2013-07-16

Twórcy

autor

Zoltán M. Balogh

• Mathematisches Institut, Universität Bern, Sidlerstrasse 5,
  3012 Bern, Switzerland

autor

Jeremy T. Tyson

• Department of Mathematics, University of Illinois at Urbana-
  Champaign, 1409 W Green Street, Urbana, IL 61801, USA

autor

Kevin Wildrick

• Mathematisches Institut, Universität Bern, Sidlerstrasse 5,
  3012 Bern, Switzerland

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Identyfikatory

DOI

10.2478/agms-2013-0005