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
- Luigi Ambrosio and Bernd Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (2000), no. 3, 527–555. MR 1800768, DOI 10.1007/s002080000122
- Zoltán M. Balogh, Pekka Koskela, and Sari Rogovin, Absolute continuity of quasiconformal mappings on curves, Geom. Funct. Anal. 17 (2007), no. 3, 645–664. MR 2346270, DOI 10.1007/s00039-007-0607-x
- Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR 1835418, DOI 10.1090/gsm/033
- J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), no. 3, 428–517. MR 1708448, DOI 10.1007/s000390050094
- Jakub Duda, Absolutely continuous functions with values in a metric space, Real Anal. Exchange 32 (2007), no. 2, 569–581. MR 2369866
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- Bent Fuglede, Extremal length and functional completion, Acta Math. 98 (1957), 171–219. MR 97720, DOI 10.1007/BF02404474
- Piotr Hajłasz, Sobolev spaces on metric-measure spaces, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002) Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp. 173–218. MR 2039955, DOI 10.1090/conm/338/06074
- Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917, DOI 10.1007/978-1-4613-0131-8
- Juha Heinonen, Nonsmooth calculus, Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 2, 163–232. MR 2291675, DOI 10.1090/S0273-0979-07-01140-8
- Juha Heinonen and Pekka Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), no. 1, 1–61. MR 1654771, DOI 10.1007/BF02392747
- Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, and Jeremy T. Tyson, Sobolev classes of Banach space-valued functions and quasiconformal mappings, J. Anal. Math. 85 (2001), 87–139. MR 1869604, DOI 10.1007/BF02788076
- Stephen Keith, Measurable differentiable structures and the Poincaré inequality, Indiana Univ. Math. J. 53 (2004), no. 4, 1127–1150. MR 2095451, DOI 10.1512/iumj.2004.53.2417
- Bernd Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc. 121 (1994), no. 1, 113–123. MR 1189747, DOI 10.1090/S0002-9939-1994-1189747-7
- Pekka Koskela and Paul MacManus, Quasiconformal mappings and Sobolev spaces, Studia Math. 131 (1998), no. 1, 1–17. MR 1628655
- Shin-ichi Ohta, Cheeger type Sobolev spaces for metric space targets, Potential Anal. 20 (2004), no. 2, 149–175. MR 2032946, DOI 10.1023/A:1026359313080
- Nageswari Shanmugalingam, Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana 16 (2000), no. 2, 243–279. MR 1809341, DOI 10.4171/RMI/275
- Jeremy Tyson, Quasiconformality and quasisymmetry in metric measure spaces, Ann. Acad. Sci. Fenn. Math. 23 (1998), no. 2, 525–548. MR 1642158
- Jussi Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229, Springer-Verlag, Berlin-New York, 1971. MR 0454009
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