Bi-Lipschitz decomposition of Lipschitz functions into a metric space
Rev. Mat. Iberoamericana, Tome 25 (2009) no. 1, p. 521-531 / Harvested from Project Euclid
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.
Publié le : 2009-06-15
Classification:  Lipschitz,  bi-Lipschitz,  metric space,  uniform rectifiability,  Sard's theorem,  28A75,  42C99,  51F99
@article{1255440066,
     author = {Schul
,  
Raanan},
     title = {Bi-Lipschitz decomposition of Lipschitz functions into a metric space},
     journal = {Rev. Mat. Iberoamericana},
     volume = {25},
     number = {1},
     year = {2009},
     pages = { 521-531},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1255440066}
}
Schul
,  
Raanan. Bi-Lipschitz decomposition of Lipschitz functions into a metric space. Rev. Mat. Iberoamericana, Tome 25 (2009) no. 1, pp.  521-531. http://gdmltest.u-ga.fr/item/1255440066/