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Tytuł artykułu

BiLipschitz Decomposition of Lipschitz Maps between Carnot Groups

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let f : G → H be a Lipschitz map between two Carnot groups. We show that if B is a ball of G, then there exists a subset Z ⊂ B, whose image in H under f has small Hausdorff content, such that B\Z can be decomposed into a controlled number of pieces, the restriction of f on each of which is quantitatively biLipschitz. This extends a result of [14], which proved the same result, but with the restriction that G has an appropriate discretization. We provide an example of a Carnot group not admitting such a discretization.
Wydawca
Rocznik
Tom
3
Numer
1
Opis fizyczny
Daty
otrzymano
2015-01-20
zaakceptowano
2015-07-08
online
2015-09-01
Twórcy
autor
  • Department of Mathematics, The University of Chicago, Chicago, IL 60637
Bibliografia
  • [1] E. Breuillard. Geometry of locally compact groups of polynomial growth and shape of large balls, 2007. arXiv:0704.0095.
  • [2] M. Christ. A T(b) theorem with remarks on analytic capacity and the Cauchy integral. Colloq.Math., 60/61(2):601–628, 1990.
  • [3] G. David. Morceaux de graphes Lipschitziens et intégrales singulières sur un surface. Rev. Mat. Iberoam., 4(1):73–114, 1988. [Crossref]
  • [4] G. David. Wavelets and singular integrals on curves and surfaces, volume 1465 of Lecture Notes in Mathematics. Springer- Verlag, 1991.
  • [5] G. David and S. Semmes. Quantitative rectifiability and Lipschitz mappings. Trans. Amer. Math. Soc., 337(2):855–889, 1993.
  • [6] G.C. David. Bi-Lipschitz pieces between manifolds. Rev. Mat. Iberoam. To appear.
  • [7] Y. Guivarc’h. Croissance polynômiale et périodes des fonctions harmoniques. Bull. Sc. Math. France, 101:353–379, 1973.
  • [8] J. Heinonen and S. Semmes. Thirty-three yes or no questions about mappings, measures, and metrics. Conform. Geom. Dyn., 1:1–12, 1997. [Crossref]
  • [9] P. Jones. Lipschitz and bi-Lipschitz functions. Rev. Mat. Iberoam., 4(1):115–121, 1988. [Crossref]
  • [10] E. Le Donne. A metric characterization of Carnot groups. Proc. Amer. Math. Soc., 132:845–849, 2015.
  • [11] E. Le Donne, S. Li, and T. Rajala. Ahlfors-regular distances on the Heisenberg group without biLipschitz pieces, 2015. Preprint.
  • [12] S. Li. Coarse differentiation and quantitative nonembeddability for Carnot groups. J. Funct. Anal., 266:4616–4704, 2014. [WoS]
  • [13] V. Magnani. Differentiability and area formula on stratified Lie groups. Houston J. Math., 27(2):297–323, 2001.
  • [14] W. Meyerson. Lipschitz and bilipschitz maps on Carnot groups. Pac. J. Math, 263(1):143–170, 2013. [WoS]
  • [15] R. Montgomery. A tour of sub-Riemannian geometries, their geodesics and applications, volume 91 of Mathematical Surveys and Monographs. American Mathematical Society, 2002.
  • [16] P. Pansu. Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. Ann.Math. (2), 129(1):1– 60, 1989.
  • [17] R. Schul. Ahlfors-regular curves in metric spaces. Ann. Acad. Sci. Fenn. Math., 32:437–460, 2007.
  • [18] R. Schul. Bi-Lipschitz decomposition of Lipschitz functions into a metric space. Rev. Mat. Iberoam., 25(2):521–531, 2009. [Crossref]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_agms-2015-0014
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