A quantitative metric differentiation theorem

Azzam, Jonas; Schul, Raanan

doi:10.1090/S0002-9939-2014-11874-0

A quantitative metric differentiation theorem
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by Jonas Azzam and Raanan Schul

Proc. Amer. Math. Soc. 142 (2014), 1351-1357

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

Published electronically: January 29, 2014

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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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