Chord-arc curves and the Beurling transform

DOI10.1007/S00222-015-0630-8zbMATH Open1355.30017arXiv1502.02181OpenAlexW3104567522MaRDI QIDQ2630281

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Publication date: 26 July 2016

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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~$(I-muS)$ 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.

Full work available at URL: https://arxiv.org/abs/1502.02181

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