Resumen de Bi-Lipschitz parts of quasisymmetric mappings

Jonas Azzam

• A natural quantity that measures how well a map f:Rd→RD is approximated by an affine transformation is ωf(x,r)=infA(∫B(x,r)(|f−A||A′|r)2)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:Rd→RD is quasisymmetric and its image has a sufficient amount of rectifiable structure (although not necessarily Hd-finite), then ωf(x,r)2dxdr/r is a Carleson measure on Rd×(0,∞). Moreover, this is an equivalence: if this is a Carleson measure, then, in every ball B(x,r)⊆Rd, 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 Rd into Rd are bi-Lipschitz on a large subset quantitatively.