# Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces

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

Analysis and Geometry in Metric Spaces (2013)

- Volume: 1, page 232-254
- ISSN: 2299-3274

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topZoltán M. Balogh, Jeremy T. Tyson, and Kevin Wildrick. "Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces." Analysis and Geometry in Metric Spaces 1 (2013): 232-254. <http://eudml.org/doc/266674>.

@article{ZoltánM2013,

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.},

author = {Zoltán M. Balogh, Jeremy T. Tyson, Kevin Wildrick},

journal = {Analysis and Geometry in Metric Spaces},

keywords = {Sobolev mapping; Ahlfors regularity; Poincaré inequality; foliation; David–Semmes regular mapping; David-Semmes regular mapping},

language = {eng},

pages = {232-254},

title = {Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces},

url = {http://eudml.org/doc/266674},

volume = {1},

year = {2013},

}

TY - JOUR

AU - Zoltán M. Balogh

AU - Jeremy T. Tyson

AU - Kevin Wildrick

TI - Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces

JO - Analysis and Geometry in Metric Spaces

PY - 2013

VL - 1

SP - 232

EP - 254

AB - 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.

LA - eng

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

UR - http://eudml.org/doc/266674

ER -

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