Bi-lipschitz decomposition of Lipschitz functions into a metric space

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 [7] 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 "Thirtythree 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.