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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Quasiconformal mappings and Ahlfors-David curves


Author: Paul MacManus
Journal: Trans. Amer. Math. Soc. 343 (1994), 853-881
MSC: Primary 30C65
MathSciNet review: 1202420
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Abstract: We show that if $ \rho $ is a quasiconformal mapping of the plane whose dilatation satisfies a certain quadratic Carleson measure condition relative to an Ahlfors-David curve $ \Gamma $ then $ \rho $ is differentiable almost everywhere on $ \Gamma $ and $ \log \vert\rho \prime \vert \in {\text{BMO}}$. When $ \Gamma $ is chord-arc we show that its image is a Bishop-Jones curve. If the Carleson norm is small then we show that $ \rho $ is absolutely continuous on $ \Gamma $, the image of $ \Gamma $ is an Ahlfors-David curve, and $ \rho \prime = {e^a}$, where $ a \in {\text{BMO}}$ with a small norm.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1202420-6
PII: S 0002-9947(1994)1202420-6
Keywords: Quasiconformal mapping, Ahlfors-David curve, Carleson measures, BMO
Article copyright: © Copyright 1994 American Mathematical Society