arXiv 1502.021812015-02-07math.CAmath.CV

Chord-arc curves and the Beurling transform

K. Astala, M. J. González

备注：: 27 pages
DOI：: 10.1007/s00222-015-0630-8

英文摘要：

We study the relation between the geometric properties of a quasicircle~$Γ$ and the complex dilatation~$μ$ of a quasiconformal mapping that maps the real line onto~$Γ$. Denoting by~$S$ the Beurling transform, we characterize Bishop-Jones quasicircles in terms of the boundedness of the operator~$(I-μS)$ 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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