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A quantitative metric differentiation theorem

Authors: Jonas Azzam and Raanan Schul
Journal: Proc. Amer. Math. Soc. 142 (2014), 1351-1357
MSC (2010): Primary 26A16, 54E40
Published electronically: January 29, 2014
MathSciNet review: 3162255
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Abstract: The purpose of this note is to point out a simple consequence of some earlier work of the authors, ``Hard Sard: Quantitative implicit function and extension theorems for Lipschitz maps''. For $ f$, a Lipschitz function from a Euclidean space into a metric space, we give quantitative estimates for how often the pullback of the metric under $ f$ is approximately a seminorm. This is a quantitative version of Kirchheim's metric differentiation result from 1994. Our result is in the form of a Carleson packing condition.

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  • [AS12] J. Azzam and R. Schul, Hard Sard: Quantitative implicit function and extension theorems for Lipschitz maps, Geom. Funct. Anal. 22 (2012), no. 5, 1062-1123. MR 2989430
  • [BJL99] S. Bates, W. B. Johnson, J. Lindenstrauss, D. Preiss, and G. Schechtman, Affine approximation of Lipschitz functions and nonlinear quotients, Geom. Funct. Anal. 9 (1999), no. 6, 1092-1127. MR 1736929 (2000m:46021),
  • [Bou87] N. Bourbaki, Topological vector spaces. Chapters 1-5, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1987. Translated from the French by H. G. Eggleston and S. Madan. MR 910295 (88g:46002)
  • [Dor85] José R. Dorronsoro, A characterization of potential spaces, Proc. Amer. Math. Soc. 95 (1985), no. 1, 21-31. MR 796440 (86k:46046),
  • [DS91] G. David and S. Semmes, Singular integrals and rectifiable sets in $ {\bf R}^n$: Beyond Lipschitz graphs, Astérisque 193 (1991), 152 (English, with French summary). MR 1113517 (92j:42016)
  • [DS93] G. David and S. Semmes, Analysis of and on uniformly rectifiable sets, Mathematical Surveys and Monographs, vol. 38, American Mathematical Society, Providence, RI, 1993. MR 1251061 (94i:28003)
  • [DS00] G. David and S. Semmes, Regular mappings between dimensions, Publ. Mat. 44 (2000), no. 2, 369-417. MR 1800814 (2002e:53054),
  • [EFW07] Alex Eskin, David Fisher, and Kevin Whyte, Quasi-isometries and rigidity of solvable groups, Pure Appl. Math. Q. 3 (2007), no. 4, 927-947. MR 2402598 (2009b:20074)
  • [Gar07] John B. Garnett, Bounded analytic functions, 1st ed., Graduate Texts in Mathematics, vol. 236, Springer, New York, 2007. MR 2261424 (2007e:30049)
  • [Jon90] Peter W. Jones, Rectifiable sets and the traveling salesman problem, Invent. Math. 102 (1990), no. 1, 1-15. MR 1069238 (91i:26016),
  • [Kir94] Bernd Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc. 121 (1994), no. 1, 113-123. MR 1189747 (94g:28013),
  • [Kol34] A. Kolmogoroff, Zur normierbarkeit eines allgemeinen topologischen linearen Raumes, Studia Math. 5 (1934), 29-33.
  • [LN] S. Li and A. Naor, Discretization and affine approximation in high dimensions, Israel J. Math. 197 (2013), no. 1, 107-129. MR 3096609
  • [Mat02] Jiří Matoušek, Lectures on discrete geometry, Graduate Texts in Mathematics, vol. 212, Springer-Verlag, New York, 2002. MR 1899299 (2003f:52011)
  • [Oki92] Kate Okikiolu, Characterization of subsets of rectifiable curves in $ {\bf R}^n$, J. London Math. Soc. (2) 46 (1992), no. 2, 336-348. MR 1182488 (93m:28008),
  • [Sch09] Raanan Schul, Bi-Lipschitz decomposition of Lipschitz functions into a metric space, Rev. Mat. Iberoam. 25 (2009), no. 2, 521-531. MR 2554164 (2011e:28007),
  • [Ste93] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, with the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. MR 1232192 (95c:42002)

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Additional Information

Jonas Azzam
Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195-4350

Raanan Schul
Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794-3651

Keywords: Lipschitz map, metric differential, differentiability of Lipschitz maps, quantitative differentiation, Rademacher
Received by editor(s): January 10, 2012
Received by editor(s) in revised form: May 22, 2012
Published electronically: January 29, 2014
Additional Notes: The first author was supported by RTG grant DMS-0838212
The second author was supported by a fellowship from the Alfred P. Sloan Foundation and by NSF grant DMS 1100008
Communicated by: Tatiana Toro
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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