Bi-Lipschitz parts of quasisymmetric mappings

TL;DR

This paper investigates how quasisymmetric maps can be approximated by affine transformations and establishes a link between a measure of this approximation and the existence of bi-Lipschitz pieces within the map.

Contribution

It introduces a new quantity to measure affine approximation of quasisymmetric maps and proves its relation to rectifiable structures and bi-Lipschitz pieces.

Findings

The quantity  measures affine approximation and is invariant under rescaling.

A Carleson measure condition on  characterizes the presence of bi-Lipschitz pieces.

Quasisymmetric maps are bi-Lipschitz on large subsets, with quantitative bounds.

Abstract

A natural quantity that measures how well a map $f:\mathbb{R}^{d}\rightarrow \mathbb{R}^{D}$ is approximated by an affine transformation is \[\omega_{f}(x,r)=\inf_{A}\left(\frac{1}{|B(x,r)|}\int_{B(x,r)}\left(\frac{|f-A|}{|A'|r}\right)^{2}\right)^{\frac{1}{2}},\] where the infimum ranges over all non constant affine transformations. This is natural insofar as it is invariant under rescaling $f$ in either its domain or image. We show that if $f:\mathbb{R}^{d}\rightarrow \mathbb{R}^{D}$ is quasisymmetric and its image has a sufficient amount of rectifiable structure (although not necessarily $\mathcal{H}^{d}$-finite), then $\omega_{f}(x,r)^{2}\frac{dxdr}{r}$ is a Carleson measure on $\mathbb{R}^{d}\times(0,\infty)$. Moreover, this is an equivalence: the existence of such a Carleson measure implies that, 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.