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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Quasiconformal mappings and chord-arc curves

Author: Stephen W. Semmes
Journal: Trans. Amer. Math. Soc. 306 (1988), 233-263
MSC: Primary 30C60
MathSciNet review: 927689
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Abstract: Given a quasiconformal mapping $ \rho $ on the plane, what conditions on its dilatation $ \mu $ guarantee that $ \rho ({\mathbf{R}})$ is rectifiable and $ \rho {\vert _{\mathbf{R}}}$ is locally absolutely continuous? We show in this paper that if $ \mu $ satisfies certain quadratic Carleson measure conditions, with small norm, then $ \rho ({\mathbf{R}})$ is a chord-arc curve with small constant, and $ \rho (x) = \rho (0) + \int_0^x {{e^{a(t)}}dt} $ for $ x \in {\mathbf{R}}$, with $ a \in \operatorname{BMO} $ having small norm. Conversely, given any such map from $ {\mathbf{R}} \to {\mathbf{C}}$, we show that it has an extension to $ {\mathbf{C}}$ with the right kind of dilatation. Similar results hold with $ {\mathbf{R}}$ replaced by a chord-arc curve. Examples are given that show that it would be hard to improve these results. Applications are given to conformal welding and the theorem of Coifman and Meyer on the real analyticity of the Riemann mapping on the manifold of chord-arc curves.

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Keywords: Quasiconformal mapping, chord-arc curves, BMO, Carleson measures
Article copyright: © Copyright 1988 American Mathematical Society