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 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 (1988) 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 {\it coarse} Lipschitz functions.