Revista Matemática Iberoamericana

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Volume 25, Issue 2, 2009, pp. 521–531
DOI: 10.4171/RMI/574

Published online: 2009-08-31

Bi-Lipschitz decomposition of Lipschitz functions into a metric space

Raanan Schul[1]

(1) Stony Brook University, USA

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.

Keywords: Lipschitz, Bi-Lipschitz, metric space, uniform rectifiability, Sard’s theorem

Schul Raanan: Bi-Lipschitz decomposition of Lipschitz functions into a metric space. Rev. Mat. Iberoam. 25 (2009), 521-531. doi: 10.4171/RMI/574