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Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces

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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.
Twórcy
  • Department of Mathematics, University of Illinois at Urbana-
    Champaign, 1409 W Green Street, Urbana, IL 61801, USA, tyson@math.uiuc.edu
Bibliografia
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  • [5] Balogh, Z. M., Monti, R., and Tyson, J. T. Frequency of Sobolev and quasiconformal dimension distortion. J. Math.Pures Appl. (9) 99, 2 (2013), 125–149.[WoS]
  • [6] Balogh, Z. M., Tyson, J. T., and Warhurst, B. Sub-Riemannian vs. Euclidean dimension comparison and fractalgeometry in Carnot groups. Adv. Math. 220 (2009), 560–619.[WoS]
  • [7] Balogh, Z. M., Tyson, J. T., and Wildrick, K. Frequency of Sobolev dimension distortion of horizontal subgroups ofHeisenberg groups. (preprint, arXiv:1303.7094 [math.MG]).
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  • [24] Mackay, J. M., Tyson, J. T., and Wildrick, K. Modulus and Poincaré inequalities on non-self-similar Sierpinskicarpets. Geom. Funct. Anal. 23, 3 (2013), 985-1034[WoS][Crossref]
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Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_agms-2013-0005
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