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Geometric and analytic quasiconformality in metric measure spaces


Author: Marshall Williams
Journal: Proc. Amer. Math. Soc. 140 (2012), 1251-1266
MSC (2010): Primary 30L10
DOI: https://doi.org/10.1090/S0002-9939-2011-11035-9
Published electronically: July 19, 2011
MathSciNet review: 2869110
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Abstract: 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

DOI: https://doi.org/10.1090/S0002-9939-2011-11035-9
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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.