Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Analysis and Geometry in Metric Spaces

Ed. by Ritoré, Manuel


CiteScore 2017: 0.65

SCImago Journal Rank (SJR) 2017: 1.063
Source Normalized Impact per Paper (SNIP) 2017: 0.833

Mathematical Citation Quotient (MCQ) 2017: 0.86

Open Access
Online
ISSN
2299-3274
See all formats and pricing
More options …

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
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Kevin Wildrick
Published Online: 2013-07-16 | DOI: https://doi.org/10.2478/agms-2013-0005

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.

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

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

  • [1] Arcozzi, N., and Baldi, A. From Grushin to Heisenberg via an isoperimetric problem. J. Math. Anal. Appl. 340, 1 (2008), 165–174. Google Scholar

  • [2] Aronszajn, N. Differentiability of Lipschitzian mappings between Banach spaces. Studia Math. 57, 2 (1976), 147–190. Google Scholar

  • [3] Astala, K. Area distortion of quasiconformal mappings. Acta Math. 173, 1 (1994), 37–60. Google Scholar

  • [4] Balogh, Z. M., Fässler, K., Mattila, P., and Tyson, J. T. Projection and slicing theorems in Heisenberg groups. Adv. Math. 231, 2 (2012), 569–604. Web of ScienceGoogle Scholar

  • [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. Google Scholar

  • [6] Balogh, Z. M., Tyson, J. T., and Warhurst, B. Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry in Carnot groups. Adv. Math. 220 (2009), 560–619. Web of ScienceGoogle Scholar

  • [7] Balogh, Z. M., Tyson, J. T., and Wildrick, K. Frequency of Sobolev dimension distortion of horizontal subgroups of Heisenberg groups. (preprint, arXiv:1303.7094 [math.MG]). Google Scholar

  • [8] Bellaïche, A. The tangent space in sub-Riemannian geometry. In Sub-Riemannian geometry, vol. 144 of Progr. Math. Birkhäuser, Basel, 1996, pp. 1–78. Google Scholar

  • [9] Bishop, C., and Hakobyan, H. Frequency of dimension distortion under quasisymmetric mappings. (preprint, 2012). Google Scholar

  • [10] Cheeger, J. Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9, 3 (1999), 428–517. Google Scholar

  • [11] Christensen, J. P. R. Measure theoretic zero sets in infinite dimensional spaces and applications to differentiability of Lipschitz mappings. Publ. Dép. Math. (Lyon) 10, 2 (1973), 29–39. Actes du Deuxième Colloque d’Analyse Fonctionnelle de Bordeaux (Univ. Bordeaux, 1973), I, pp. 29–39. Google Scholar

  • [12] Csörnyei, M. Aronszajn null and Gaussian null sets coincide. Israel J. Math. 111 (1999), 191–201. Google Scholar

  • [13] David, G., and Semmes, S. Regular mappings between dimensions. Publ. Mat. 44, 2 (2000), 369–417. Google Scholar

  • [14] Gehring, F. W. The Lp-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130 (1973), 265–277. Google Scholar

  • [15] Gehring, F. W., and Väisälä, J. Hausdorff dimension and quasiconformal mappings. J. London Math. Soc. (2) 6 (1973), 504–512. CrossrefGoogle Scholar

  • [16] Hajłasz, P., and Koskela, P. Sobolev met Poincaré. Mem. Amer. Math. Soc. 145, 688 (2000), x+101. Google Scholar

  • [17] Heinonen, J. Lectures on analysis on metric spaces. Universitext. Springer-Verlag, New York, 2001. Google Scholar

  • [18] Heinonen, J., and Koskela, P. Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181, 1 (1998), 1–61. Google Scholar

  • [19] Heinonen, J., Koskela, P., Shanmugalingam, N., and Tyson, J. T. Sobolev classes of Banach space-valued functions and quasiconformal mappings. J. Anal. Math. 85 (2001), 87–139. Google Scholar

  • [20] Hencl, S., and Honzík, P. Dimension of images of subspaces under Sobolev mappings. Ann. Inst. H. Poincaré Anal. Non Linéaire 29, 3 (2012), 401–411. Google Scholar

  • [21] Hunt, B. R., and Kaloshin, V. Y. How projections affect the dimension spectrum of fractal measures. Nonlinearity 10, 5 (1997), 1031–1046. CrossrefGoogle Scholar

  • [22] Hunt, B. R., Sauer, T. D., and Yorke, J. A. Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces. Bull. Amer. Math. Soc. (N.S.) 27, 2 (1992), 217–238. CrossrefGoogle Scholar

  • [23] Kaufman, R. P. Sobolev spaces, dimension, and random series. Proc. Amer. Math. Soc. 128, 2 (2000), 427–431. Google Scholar

  • [24] Mackay, J. M., Tyson, J. T., and Wildrick, K. Modulus and Poincaré inequalities on non-self-similar Sierpinski carpets. Geom. Funct. Anal. 23, 3 (2013), 985-1034 CrossrefGoogle Scholar

  • [25] Mattila, P. Geometry of sets and measures in Euclidean spaces, vol. 44 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995. Google Scholar

  • [26] Ott, W., and Yorke, J. A. Prevalence. Bull. Amer. Math. Soc. (N.S.) 42, 3 (2005), 263–290 (electronic). CrossrefGoogle Scholar

  • [27] Rothschild, L. P., and Stein, E. M. Hypoelliptic differential operators and nilpotent groups. Acta Math. 137, 3-4 (1976), 247–320. Google Scholar

  • [28] Sauer, T. D., and Yorke, J. A. Are the dimensions of a set and its image equal under typical smooth functions? Ergodic Theory Dynam. Systems 17, 4 (1997), 941–956. Google Scholar

  • [29] Seo, J. A characterization of bi-Lipschitz embeddable metric spaces in terms of local bi-Lipschitz embeddability. Math. Res. Lett. 18, 6 (2011), 1179–1202. CrossrefGoogle Scholar

  • [30] Shanmugalingam, N. Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana 16, 2 (2000), 243–279. Google Scholar

  • [31] Ziemer, W. P. Weakly differentiable functions, vol. 120 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1989.Google Scholar

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.

Export Citation

©2013 Versita Sp. z o.o.. This content is open access.

Comments (0)

Please log in or register to comment.
Log in