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\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
coarse
Lipschitz functions.