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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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