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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.

28A75 Length, area, volume, other geometric measure theory
42C99 Nontrigonometric harmonic analysis
51F99 Metric geometry
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[6] Heinonen, J. and Semmes, S.: Thirty-three yes or no questions about mappings, measures, and metrics. Conform. Geom. Dyn. 1 (1997), 1-12 (electronic). · Zbl 0885.00006
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[9] Schul, R.: Ahlfors-regular curves in metric spaces. Ann. Acad. Sci. Fenn. Math. 32 (2007), no. 2, 437-460. · Zbl 1122.28006
[10] Schul, R.: Analyst’s traveling salesman theorems. A survey. In: In the tradition of Ahlfors and Bers. IV , 209-220. Contemp. Math. 432 . Amer. Math. Soc., Providence, RI, 2007. · Zbl 1187.49039
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