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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

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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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

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-11874-0
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.