Revista Matemática Iberoamericana

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Volume 32, Issue 2, 2016, pp. 589–648
DOI: 10.4171/RMI/896

Published online: 2016-06-08

Bi-Lipschitz parts of quasisymmetric mappings

Jonas Azzam[1]

(1) Universitat Autònoma de Barcelona, Bellaterra (Barcelona), Spain

A natural quantity that measures how well a map $f\colon \mathbb{R}^{d}\rightarrow \mathbb{R}^{D}$ is approximated by an affine transformation is \[\omega_{f}(x,r)=\inf_{A}\Big(\, \int_{B(x,r)}\Big(\frac{|f-A|}{|A'|r}\Big)^{2}\,\Big)^{{1}/{2}},\] where the infimum ranges over all non-zero affine transformations $A$. This is natural insofar as it is invariant under rescaling $f$ in either its domain or image. We show that if $f\colon \mathbb{R}^{d}\rightarrow \mathbb{R}^{D}$ is quasisymmetric and its image has a sufficient amount of rectifiable structure (although not necessarily $\mathcal{H}^{d}$-finite), then $\omega_{f}(x,r)^{2}{dxdr}/{r}$ is a Carleson measure on $\mathbb{R}^{d}\times(0,\infty)$. Moreover, this is an equivalence: if this is a Carleson measure, then, in every ball $B(x,r)\subseteq \mathbb{R}^{d}$, there is a set $E$ occupying 90$\%$ of $B(x,r)$, say, upon which $f$ is bi-Lipschitz (and hence guaranteeing rectifiable pieces in the image).

En route, we make a minor adjustment to a theorem of Semmes to show that quasisymmetric maps of subsets of $\mathbb R^{d}$ into $\mathbb R^{d}$ are bi-Lipschitz on a large subset quantitatively.

Keywords: Quantitative differentiation, coarse differentiation, uniform approximation by affine property, quasisymmetric maps, quasiconformal maps, Carleson measures, affine approximation, uniform rectifiability, rectifiable sets, big pieces of bi-Lipschitz images, Dorronsoro’s theorem

Azzam Jonas: Bi-Lipschitz parts of quasisymmetric mappings. Rev. Mat. Iberoamericana 32 (2016), 589-648. doi: 10.4171/RMI/896