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The degree of regularity of a quasiconformal mapping


Author: Pekka Koskela
Journal: Proc. Amer. Math. Soc. 122 (1994), 769-772
MSC: Primary 30C65; Secondary 30C62
DOI: https://doi.org/10.1090/S0002-9939-1994-1204381-8
MathSciNet review: 1204381
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Abstract: T. Iwaniec has conjectured that the derivative of a locally $ \alpha $-Hölder continuous quasiconformal mapping of $ {\mathbb{R}^n}$ is locally integrable to any power $ p < \frac{n}{{1 - \alpha }}$. We disprove this conjecture by producing examples of quasiconformal mappings of the plane that are uniformly Hölder continuous with exponent $ \frac{1}{2} < \alpha < 1$ but whose derivatives are not locally integrable to the power $ \frac{1}{{1 - \alpha }}$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1204381-8
Keywords: Quasiconformal mapping, Sobolev embedding, global integrability of the derivative
Article copyright: © Copyright 1994 American Mathematical Society

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