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.
Cite
@article{arxiv.math/0702630,
title = {Bi-Lipschitz Decomposition of Lipschitz functions into a Metric space},
author = {Raanan Schul},
journal= {arXiv preprint arXiv:math/0702630},
year = {2008}
}
Comments
11 pages. no figures. Paragraph surveying history has been corrected after referee report!