Bi-Lipschitz parts of quasisymmetric mappings
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Jonas Azzam [1]
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[1] Universitat Autònoma de Barcelona
Universitat Autònoma de Barcelona
Barcelona, España
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- Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 32, Nº 2, 2016, págs. 589-648
- Idioma: inglés
- DOI: 10.4171/RMI/896
- Texto completo no disponible (Saber más ...)
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Resumen
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.
