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June, 2009 Bi-Lipschitz decomposition of Lipschitz functions into a metric space
Raanan Schul
Rev. Mat. Iberoamericana 25(2): 521-531 (June, 2009).


We prove a quantitative version of the following statement. Given a Lipschitz function $f$ from the k-dimensional unit cube into a general metric space, one can be decomposed $f$ into a finite number of BiLipschitz functions $f|_{F_i}$ so that the k-Hausdorff content of $f([0,1]^k\smallsetminus \cup F_i)$ is small. We thus generalize a theorem of P. Jones [Lipschitz and bi-Lipschitz functions. Rev. Mat. Iberoamericana 4 (1988), no. 1, 115-121] from the setting of $\mathbb{R}^d$ to the setting of a general metric space. This positively answers problem 11.13 in "Fractured Fractals and Broken Dreams" by G. David and S. Semmes, or equivalently, question 9 from "Thirty-three yes or no questions about mappings, measures, and metrics" by J. Heinonen and S. Semmes. Our statements extend to the case of {\it coarse} Lipschitz functions.


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Raanan Schul . "Bi-Lipschitz decomposition of Lipschitz functions into a metric space." Rev. Mat. Iberoamericana 25 (2) 521 - 531, June, 2009.


Published: June, 2009
First available in Project Euclid: 13 October 2009

zbMATH: 1228.28004
MathSciNet: MR2554164

Primary: 28A75
Secondary: 42C99 , 51F99

Keywords: bi-Lipschitz , Lipschitz , metric space , Sard's theorem , uniform rectifiability

Rights: Copyright © 2009 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.25 • No. 2 • June, 2009
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