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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Geometric and analytic quasiconformality in metric measure spaces
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by Marshall Williams PDF
Proc. Amer. Math. Soc. 140 (2012), 1251-1266 Request permission


We prove the equivalence between geometric and analytic definitions of quasiconformality for a homeomorphism $f\colon X\rightarrow Y$ between arbitrary locally finite separable metric measure spaces, assuming no metric hypotheses on either space. When $X$ and $Y$ have locally $Q$-bounded geometry and $Y$ is contained in an Alexandrov space of curvature bounded above, the sharpness of our results implies that, as in the classical case, the modular and pointwise outer dilatations of $f$ are related by $K_O(f)= \operatorname {ess sup} H_O(x,f)$.
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Additional Information
  • Marshall Williams
  • Affiliation: Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, Illinois 60607-7845
  • Received by editor(s): August 13, 2010
  • Received by editor(s) in revised form: December 21, 2010
  • Published electronically: July 19, 2011
  • Additional Notes: Partially supported under NSF awards 0602191, 0353549 and 0349290.
  • Communicated by: Mario Bonk
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 140 (2012), 1251-1266
  • MSC (2010): Primary 30L10
  • DOI:
  • MathSciNet review: 2869110