Revista Matemática Iberoamericana

Bi-Lipschitz decomposition of Lipschitz functions into a metric space

Raanan Schul

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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 be decomposed $f$ into a finite number of BiLipschitz functions $f|_{F_i}$ so that the k-Hausdorff content of $f([0,1]^k\smallsetminus \cup F_i)$ 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 $\mathbb{R}^d$ 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.

Article information

Source
Rev. Mat. Iberoamericana, Volume 25, Number 2 (2009), 521-531.

Dates
First available in Project Euclid: 13 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1255440066

Mathematical Reviews number (MathSciNet)
MR2554164

Zentralblatt MATH identifier
1228.28004

Subjects
Primary: 28A75: Length, area, volume, other geometric measure theory [See also 26B15, 49Q15]
Secondary: 42C99: None of the above, but in this section 51F99: None of the above, but in this section

Keywords
Lipschitz bi-Lipschitz metric space uniform rectifiability Sard's theorem

Citation

Schul, Raanan. Bi-Lipschitz decomposition of Lipschitz functions into a metric space. Rev. Mat. Iberoamericana 25 (2009), no. 2, 521--531. https://projecteuclid.org/euclid.rmi/1255440066


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References

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