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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On quasiconformal self-mappings of the unit disk satisfying Poisson’s equation
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by David Kalaj and Miroslav Pavlović PDF
Trans. Amer. Math. Soc. 363 (2011), 4043-4061 Request permission

Abstract:

Let $\mathcal {QC}(K,g)$ be a family of $K$-quasiconformal mappings of the open unit disk onto itself satisfying the PDE $\Delta w =g$, $g\in C(\overline {\mathbb {U}})$, $w(0)=0$. It is proved that $\mathcal {QC}(K,g)$ is a uniformly Lipschitz family. Moreover, if $|g|_\infty$ is small enough, then the family is uniformly bi-Lipschitz. The estimations are asymptotically sharp as $K \to 1$ and $|g|_\infty \to 0$, so $w\in \mathcal {QC}(K,g)$ behaves almost like a rotation for sufficiently small $K$ and $|g|_\infty$.
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Additional Information
  • David Kalaj
  • Affiliation: Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put b.b. 81000 Podgorica, Montenegro
  • MR Author ID: 689421
  • Email: davidkalaj@gmail.com
  • Miroslav Pavlović
  • Affiliation: Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Belgrade, Serbia
  • Email: pavlovic@matf.bg.ac.rs
  • Received by editor(s): May 7, 2008
  • Received by editor(s) in revised form: April 12, 2009
  • Published electronically: March 23, 2011
  • © Copyright 2011 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 363 (2011), 4043-4061
  • MSC (2010): Primary 30C62
  • DOI: https://doi.org/10.1090/S0002-9947-2011-05081-6
  • MathSciNet review: 2792979