# Bi-Lipschitz decomposition of Lipschitz functions into a metric space

### Raanan Schul

Stony Brook University, USA

## 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 $f([0,1]_{k}∖∪F_{i})$ 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 $R_{d}$ 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.

## Cite this article

Raanan Schul, Bi-Lipschitz decomposition of Lipschitz functions into a metric space. Rev. Mat. Iberoam. 25 (2009), no. 2, pp. 521–531

DOI 10.4171/RMI/574