JournalsrmiVol. 25, No. 2pp. 521–531

Bi-Lipschitz decomposition of Lipschitz functions into a metric space

  • Raanan Schul

    Stony Brook University, USA
Bi-Lipschitz decomposition of Lipschitz functions into a metric space cover
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Abstract

We prove a quantitative version of the following statement. Given a Lipschitz function ff from the k-dimensional unit cube into a general metric space, one can be decomposed ff into a finite number of BiLipschitz functions fFif|_{F_i} so that the k-Hausdorff content of f([0,1]kFi)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 Rd\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 {\it coarse} Lipschitz functions.

Cite this article

Raanan Schul, Bi-Lipschitz decomposition of Lipschitz functions into a metric space. Rev. Mat. Iberoam. 25 (2009), no. 2, pp. 521–531

DOI 10.4171/RMI/574