Bi-Lipschitz decomposition of Lipschitz functions into a metric space

• DOI: 10.4171/rmi/574
• Publication Date: 2011-12-06T22:45:36Z
• Authors: Raanan Schul
• Keywords: Sard's theorem; metric space; Metric Geometry (math.MG); uniform rectifiability; 01 natural sciences; 51F99; Mathematics - Metric Geometry; Mathematics - Classical Analysis and ODEs; Classical Analysis and ODEs (math.CA); FOS: Mathematics; 28A75; 28a75; Lipschitz; 0101 mathematics; bi-Lipschitz; 42C99

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.