Bi-Lipschitz parts of quasisymmetric mappings
Jonas Azzam
Universitat Autònoma de Barcelona, Bellaterra (Barcelona), Spain
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Abstract
A natural quantity that measures how well a map is approximated by an affine transformation is
where the infimum ranges over all non-zero affine transformations . This is natural insofar as it is invariant under rescaling in either its domain or image. We show that if is quasisymmetric and its image has a sufficient amount of rectifiable structure (although not necessarily -finite), then is a Carleson measure on . Moreover, this is an equivalence: if this is a Carleson measure, then, in every ball , there is a set occupying 90 of , say, upon which 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 into are bi-Lipschitz on a large subset quantitatively.
Cite this article
Jonas Azzam, Bi-Lipschitz parts of quasisymmetric mappings. Rev. Mat. Iberoam. 32 (2016), no. 2, pp. 589–648
DOI 10.4171/RMI/896