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Bi-Lipschitz decomposition of Lipschitz functions into a metric space. (English) Zbl 1228.28004
Summary: 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 bi-Lipschitz functions \(f|_{F_i}\) so that the \(k\)-Hausdorff content of \(f([0,1]^k\setminus \bigcup F_i)\) is small. We thus generalize a theorem of P. W. Jones [Rev. Mat. Iberoam. 4, No. 1, 115–121 (1988; Zbl 0782.26007)] 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. Self-similar geometry through metric and measure. Oxford: Clarendon Press (1997; Zbl 0887.54001)] by G. David and S. Semmes, or equivalently, question 9 from [Conform. Geom. Dyn. 1, No. 1, 1–12 (1997; Zbl 0885.00006)] by J. Heinonen and S. Semmes. Our statements extend to the case of coarse Lipschitz functions.

MSC:
28A75 Length, area, volume, other geometric measure theory
42C99 Nontrigonometric harmonic analysis
51F99 Metric geometry
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References:
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