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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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Quasiconformal mappings and Royden algebras in space
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by Lawrence G. Lewis PDF
Trans. Amer. Math. Soc. 158 (1971), 481-492 Request permission

Abstract:

On every open connected set $G$ in Euclidean $n$-space ${R^n}$ and for every index $p > 1$, we define the Royden $p$-algebra ${M_p}(G)$. We use results by F. W. Gehring and W. P. Ziemer to prove that two such sets $G$ and $G’$ are quasiconformally equivalent if and only if their Royden $n$-algebras are isomorphic as Banach algebras. Moreover, every such algebra isomorphism is given by composition with a quasiconformal homeomorphism between $G$ and $G’$. This generalizes a theorem by M. Nakai concerning Riemann surfaces. In case $p \ne n$, the only homeomorphisms which induce an isomorphism of the $p$-algebras are the locally bi-Lipschitz mappings, and for $1 < p < n$, every such isomorphism arises this way. Under certain restrictions on the domains, these results extend to the Sobolev space $H_p^1(G)$ and characterize those homeomorphisms which preserve the $H_p^1$ classes.
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 158 (1971), 481-492
  • MSC: Primary 30.47
  • DOI: https://doi.org/10.1090/S0002-9947-1971-0281912-5
  • MathSciNet review: 0281912