Bi-Lipschitz decomposition of Lipschitz functions into a metric space

Abstract

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 $f([0,1{]}^k\setminus \cup F_i)$ 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 coarse Lipschitz functions.