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Tytuł artykułu

On Conditions for Unrectifiability of a Metric Space

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Rocznik
Tom
3
Numer
1
Opis fizyczny
Daty
otrzymano
2014-03-07
zaakceptowano
2014-11-15
online
2014-12-17
Twórcy
  • Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA, hajlasz@pitt.edu
  • Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA, som13@pitt.edu
Bibliografia
  • [1] Ambrosio, L., Kirchheim, B.: Rectifiable sets in metric and Banach spaces. Math. Ann. 318 (2000), 527–555.
  • [2] Ambrosio, L., Kirchheim, B.: Currents in metric spaces. Acta Math. 185 (2000),1–80.
  • [3] Balogh, Z. M., Hajłasz, P., Wildrick, K.: Weak contact equations for mappings into Heisenberg groups. Indiana Univ. Math. J. (to appear).
  • [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.
  • [5] DiBenedetto, E.: Real analysis. Birkhäuser Advanced Texts: Basler Lehrbücher.
  • [Birkhäuser Advanced Texts: Basel Textbooks] Birkhäuser Boston, Inc., Boston, MA, 2002.
  • [6] Evans, L. C., Gariepy, R. F.: Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992.
  • [7] Federer, H.: Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969
  • [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.
  • [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.
  • [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.
  • [11] Hajłasz, P.: Change of variables formula under minimal assumptions. Colloq. Math. 64 (1993), 93–101.
  • [12] Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101 pp.
  • [13] Heinonen, J.: Lectures on analysis on metric spaces. Universitext. Springer-Verlag, New York, 2001.
  • [14] Kirchheim, B.: Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. Proc. Amer. Math. Soc. 121 (1994), 113–123.
  • [15] Le Donne, E.: Lipschitz and path isometric embeddings of metric spaces. Geom. Dedicata 166 (2013), 47–66.
  • [16] Magnani, V.: Unrectifiability and rigidity in stratified groups. Arch. Math. (Basel) 83 (2004), 568–576.
  • [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.
  • [18] Martio, O., Väisälä, J.: Elliptic equations and maps of bounded length distortion. Math. Ann. 282 (1988), 423–443.
  • [19] Mattila, P.: Geometry of sets and measures in Euclidean spaces. Cambridge Studies in Advanced Mathematics, Vol. 44. Cambridge University Press, Cambridge, 1995.
  • [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
  • [21] Sternberg, S.: Lectures on differential geometry. Second edition. With an appendix by Sternberg and Victor W. Guillemin. Chelsea Publishing Co., New York, 1983.
  • [22] Varopoulos, N. Th., Saloff-Coste, L., Coulhon, T.: Analysis and geometry on groups. Cambridge Tracts inMathematics, 100. Cambridge University Press, Cambridge, 1992.
  • [23] Whitney, H.: On totally differentiable and smooth functions. Pacific J. Math. 1 (1951), 143–159.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_agms-2015-0001
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