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

Abstract

top
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.

How to cite

top

Zoltán 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 -

References

top
  1. [1] Arcozzi, N., and Baldi, A. From Grushin to Heisenberg via an isoperimetric problem. J. Math. Anal. Appl. 340, 1 (2008), 165–174. [WoS] Zbl1134.53016
  2. [2] Aronszajn, N. Differentiability of Lipschitzian mappings between Banach spaces. Studia Math. 57, 2 (1976), 147–190. [WoS] Zbl0342.46034
  3. [3] Astala, K. Area distortion of quasiconformal mappings. Acta Math. 173, 1 (1994), 37–60. Zbl0815.30015
  4. [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. [WoS] Zbl1260.28007
  5. [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] Zbl1266.28003
  6. [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. [WoS] Zbl1155.22011
  7. [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]). 
  8. [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. Zbl0862.53031
  9. [9] Bishop, C., and Hakobyan, H. Frequency of dimension distortion under quasisymmetric mappings. (preprint, 2012). 
  10. [10] Cheeger, J. Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9, 3 (1999), 428–517. [Crossref] Zbl0942.58018
  11. [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. Zbl0302.43001
  12. [12] Csörnyei, M. Aronszajn null and Gaussian null sets coincide. Israel J. Math. 111 (1999), 191–201. Zbl0952.46030
  13. [13] David, G., and Semmes, S. Regular mappings between dimensions. Publ. Mat. 44, 2 (2000), 369–417. [Crossref] Zbl1041.42010
  14. [14] Gehring, F. W. The Lp-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130 (1973), 265–277. Zbl0258.30021
  15. [15] Gehring, F. W., and Väisälä, J. Hausdorff dimension and quasiconformal mappings. J. London Math. Soc. (2) 6 (1973), 504–512. Zbl0258.30020
  16. [16] Hajłasz, P., and Koskela, P. Sobolev met Poincaré. Mem. Amer. Math. Soc. 145, 688 (2000), x+101. Zbl0954.46022
  17. [17] Heinonen, J. Lectures on analysis on metric spaces. Universitext. Springer-Verlag, New York, 2001. Zbl0985.46008
  18. [18] Heinonen, J., and Koskela, P. Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181, 1 (1998), 1–61. Zbl0915.30018
  19. [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. Zbl1013.46023
  20. [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. Zbl1245.28006
  21. [21] Hunt, B. R., and Kaloshin, V. Y. How projections affect the dimension spectrum of fractal measures. Nonlinearity 10, 5 (1997), 1031–1046. [Crossref] Zbl0903.28008
  22. [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. [Crossref] Zbl0763.28009
  23. [23] Kaufman, R. P. Sobolev spaces, dimension, and random series. Proc. Amer. Math. Soc. 128, 2 (2000), 427–431. Zbl0938.28001
  24. [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 [WoS][Crossref] Zbl1271.30032
  25. [25] Mattila, P. Geometry of sets and measures in Euclidean spaces, vol. 44 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995. Zbl0819.28004
  26. [26] Ott, W., and Yorke, J. A. Prevalence. Bull. Amer. Math. Soc. (N.S.) 42, 3 (2005), 263–290 (electronic). [Crossref] 
  27. [27] Rothschild, L. P., and Stein, E. M. Hypoelliptic differential operators and nilpotent groups. Acta Math. 137, 3-4 (1976), 247–320. Zbl0346.35030
  28. [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. Zbl0884.28006
  29. [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. [Crossref] Zbl1273.30054
  30. [30] Shanmugalingam, N. Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana 16, 2 (2000), 243–279. [Crossref] Zbl0974.46038
  31. [31] Ziemer, W. P. Weakly differentiable functions, vol. 120 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1989. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.