On Conditions for Unrectifiability of a Metric Space

Piotr Hajłasz; Soheil Malekzadeh

Analysis and Geometry in Metric Spaces (2015)

  • Volume: 3, Issue: 1, page 1-14, electronic only
  • ISSN: 2299-3274

Abstract

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We find necessary and sufficient conditions for a Lipschitz map f : E ⊂ ℝk → X into a metric space to satisfy ℋk(f(E)) = 0. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a variant of the classical implicit function theorem. Applications include pure unrectifiability of the Heisenberg groups.

How to cite

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Piotr Hajłasz, and Soheil Malekzadeh. "On Conditions for Unrectifiability of a Metric Space." Analysis and Geometry in Metric Spaces 3.1 (2015): 1-14, electronic only. <http://eudml.org/doc/268901>.

@article{PiotrHajłasz2015,
abstract = {We find necessary and sufficient conditions for a Lipschitz map f : E ⊂ ℝk → X into a metric space to satisfy ℋk(f(E)) = 0. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a variant of the classical implicit function theorem. Applications include pure unrectifiability of the Heisenberg groups.},
author = {Piotr Hajłasz, Soheil Malekzadeh},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {geometric measure theory; unrectifiability; metric spaces; Sard theorem; Carnot-Carathéodory spaces; Carnot-Carathéodory spaces},
language = {eng},
number = {1},
pages = {1-14, electronic only},
title = {On Conditions for Unrectifiability of a Metric Space},
url = {http://eudml.org/doc/268901},
volume = {3},
year = {2015},
}

TY - JOUR
AU - Piotr Hajłasz
AU - Soheil Malekzadeh
TI - On Conditions for Unrectifiability of a Metric Space
JO - Analysis and Geometry in Metric Spaces
PY - 2015
VL - 3
IS - 1
SP - 1
EP - 14, electronic only
AB - We find necessary and sufficient conditions for a Lipschitz map f : E ⊂ ℝk → X into a metric space to satisfy ℋk(f(E)) = 0. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a variant of the classical implicit function theorem. Applications include pure unrectifiability of the Heisenberg groups.
LA - eng
KW - geometric measure theory; unrectifiability; metric spaces; Sard theorem; Carnot-Carathéodory spaces; Carnot-Carathéodory spaces
UR - http://eudml.org/doc/268901
ER -

References

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  1. [1] Ambrosio, L., Kirchheim, B.: Rectifiable sets in metric and Banach spaces. Math. Ann. 318 (2000), 527–555.  Zbl0966.28002
  2. [2] Ambrosio, L., Kirchheim, B.: Currents in metric spaces. Acta Math. 185 (2000),1–80.  Zbl0984.49025
  3. [3] Balogh, Z. M., Hajłasz, P., Wildrick, K.: Weak contact equations for mappings into Heisenberg groups. Indiana Univ. Math. J. (to appear).  Zbl1316.53034
  4. [4] David, G., Semmes, S.: Fractured fractals and broken dreams. Self-similar geometry through metric and measure. Oxford Lecture Series in Mathematics and its Applications, 7. The Clarendon Press, Oxford University Press, New York, 1997.  Zbl0887.54001
  5. [5] DiBenedetto, E.: Real analysis. Birkhäuser Advanced Texts: Basler Lehrbücher.  
  6. [Birkhäuser Advanced Texts: Basel Textbooks] Birkhäuser Boston, Inc., Boston, MA, 2002.  
  7. [6] Evans, L. C., Gariepy, R. F.: Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992.  Zbl0804.28001
  8. [7] Federer, H.: Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969  
  9. [8] Franchi, B., Gutiérrez, C. E., Wheeden, R. L.: Weighted Sobolev-Poincaré inequalities for Grushin type operators. Comm. Partial Differential Equations 19 (1994), 523–604.  Zbl0822.46032
  10. [9] Gilbarg, D., Trudinger, N. S.: Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.  Zbl1042.35002
  11. [10] Gromov, M.: Partial differential relations. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 9. Springer-Verlag, Berlin, 1986.  Zbl0651.53001
  12. [11] Hajłasz, P.: Change of variables formula under minimal assumptions. Colloq. Math. 64 (1993), 93–101.  Zbl0840.26009
  13. [12] Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101 pp.  Zbl0954.46022
  14. [13] Heinonen, J.: Lectures on analysis on metric spaces. Universitext. Springer-Verlag, New York, 2001.  Zbl0985.46008
  15. [14] Kirchheim, B.: Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. Proc. Amer. Math. Soc. 121 (1994), 113–123.  Zbl0806.28004
  16. [15] Le Donne, E.: Lipschitz and path isometric embeddings of metric spaces. Geom. Dedicata 166 (2013), 47–66.  Zbl1281.30041
  17. [16] Magnani, V.: Unrectifiability and rigidity in stratified groups. Arch. Math. (Basel) 83 (2004), 568–576.  Zbl1062.22019
  18. [17] Malý, J., Ziemer,W. P.: Fine regularity of solutions of elliptic partial differential equations.Mathematical Surveys and Monographs, 51. American Mathematical Society, Providence, RI, 1997.  Zbl0882.35001
  19. [18] Martio, O., Väisälä, J.: Elliptic equations and maps of bounded length distortion. Math. Ann. 282 (1988), 423–443.  Zbl0632.35021
  20. [19] Mattila, P.: Geometry of sets and measures in Euclidean spaces. Cambridge Studies in Advanced Mathematics, Vol. 44. Cambridge University Press, Cambridge, 1995.  Zbl0819.28004
  21. [20] Monti R.: Distances, boundaries and surface measures in Carnot-Carathéodory spaces, PhD thesis 2001. Available at http://www.math.unipd.it/ monti/PAPERS/TesiFinale.pdf  
  22. [21] Sternberg, S.: Lectures on differential geometry. Second edition. With an appendix by Sternberg and Victor W. Guillemin. Chelsea Publishing Co., New York, 1983.  
  23. [22] Varopoulos, N. Th., Saloff-Coste, L., Coulhon, T.: Analysis and geometry on groups. Cambridge Tracts inMathematics, 100. Cambridge University Press, Cambridge, 1992.  
  24. [23] Whitney, H.: On totally differentiable and smooth functions. Pacific J. Math. 1 (1951), 143–159. Zbl0043.05803

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