Bi-Lipschitz parts of quasisymmetric mappings

Abstract

A natural quantity that measures how well a map $f:{\mathbb{R}}^d\to {\mathbb{R}}^D$ is approximated by an affine transformation is

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 $f:{\mathbb{R}}^d\to {\mathbb{R}}^D$ 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.