## Geometric and analytic quasiconformality in metric measure spaces

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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 $K_O(f)= \operatorname {ess sup} H_O(x,f)$.## References

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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: https://doi.org/10.1090/S0002-9939-2011-11035-9
- MathSciNet review: 2869110