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.
Categories: math.CA