Abstract
We study the relation between the geometric properties of a quasicircle \(\Gamma \) and the complex dilatation \(\mu \) of a quasiconformal mapping that maps the real line onto \(\Gamma \). Denoting by S the Beurling transform, we characterize Bishop–Jones quasicircles in terms of the boundedness of the operator \((I-\mu S)\) on a particular weighted \(L^2\) space, and chord-arc curves in terms of its invertibility. As an application we recover the \(L^2\) boundedness of the Cauchy integral on chord-arc curves.
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K. Astala was supported by Academy of Finland, grants SA-12719831 and SA-1218321, and M.J. González by Ministerio de Economía y Competitividad, grants MTM2014-51824-P and MTM2011-24606.
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Astala, K., González, M.J. Chord-arc curves and the Beurling transform. Invent. math. 205, 57–81 (2016). https://doi.org/10.1007/s00222-015-0630-8
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DOI: https://doi.org/10.1007/s00222-015-0630-8