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BY 4.0 license Open Access Published by De Gruyter Open Access August 24, 2020

Duality of Moduli and Quasiconformal Mappings in Metric Spaces

  • Rebekah Jones EMAIL logo and Panu Lahti


We prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincaré inequality. Then we apply this to show that quasiconformal mappings can be characterized by the fact that they quasi-preserve the modulus of certain families of surfaces.


[1] H. Aikawa and M. Ohtsuka, Extremal length of vector measures, Ann. Acad. Sci. Fenn. Math. 24 1999, no. 1, 61–88.Search in Google Scholar

[2] L. Ambrosio, Fine properties of sets of finite perimeter in doubling metric measure spaces, Calculus of variations, nonsmooth analysis and related topics. Set-Valued Anal. 10 2002, no. 2–3, 111–128.Search in Google Scholar

[3] C. Bishop, H. Hakobyan, and M. Williams, Quasisymmetric dimension distortion of Ahlfors regular subsets of a metric space, Geom. Funct. Anal. 26 2016, no. 2, 379–421.10.1007/s00039-016-0368-5Search in Google Scholar

[4] A. Björn and J. Björn, Nonlinear potential theory on metric spaces, EMS Tracts in Mathematics, 17. European Mathematical Society (EMS), Zürich, 2011, xii+403 pp.10.4171/099Search in Google Scholar

[5] A. Björn and J. Björn, Obstacle and Dirichlet problems on arbitrary nonopen sets in metric spaces, and fine topology, Rev. Mat. Iberoam. 31 2015, no. 1, 161–214.10.4171/RMI/830Search in Google Scholar

[6] A. Björn, J. Björn, and V. Latvala, The Cartan, Choquet and Kellogg properties for the fine topology on metric spaces, J. Anal. Math. 135 2018, no. 1, 59–83.10.1007/s11854-018-0029-8Search in Google Scholar

[7] J. Björn, Fine continuity on metric spaces, Manuscripta Math. 125 2008, no. 3, 369–381.10.1007/s00229-007-0154-7Search in Google Scholar

[8] B. Fuglede, Extremal length and functional completion, Acta Math. 98 1957, 171–219.10.1007/BF02404474Search in Google Scholar

[9] F. Gehring and J. C. Kelly, Quasi-conformal mappings and Lebesgue density, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), Ann. of Math. Studies 79, Princeton Univ. Press, Princeton, N.J., 1974, pp. 171–179.10.1515/9781400881642-015Search in Google Scholar

[10] P. Hajłasz, Sobolev spaces on metric-measure spaces, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), 173–218, Contemp. Math., 338, Amer. Math. Soc., Providence, RI, 2003.10.1090/conm/338/06074Search in Google Scholar

[11] J. Heinonen and P. Koskela, Definitions of quasiconformality, Invent. Math. 120 1995, no. 1, 61–79.10.1007/BF01241122Search in Google Scholar

[12] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 1998, no. 1, 1–61.10.1007/BF02392747Search in Google Scholar

[13] J. Heinonen, P. Koskela, N. Shanmugalingam, and J. Tyson, Sobolev classes of Banach space-valued functions and quasi-conformal mappings, J. Anal. Math. 85 2001, 87–139.10.1007/BF02788076Search in Google Scholar

[14] J. Heinonen, P. Koskela, N. Shanmugalingam, and J. Tyson, Sobolev spaces on metric measure spaces: An approach based on upper gradients, New Mathematical Monographs, 27. Cambridge University Press, Cambridge, 2015, xii+434 pp.10.1017/CBO9781316135914Search in Google Scholar

[15] R. Jones, P. Lahti, and N. Shanmugalingam, Modulus of families of sets of finite perimeter and quasiconformal maps between metric spaces of globally Q-bounded geometry, to appear in Indiana University Mathematics Journal. in Google Scholar

[16] E. Järvenpää, M. Järvenpää, K. Rogovin, S. Rogovin, and N. Shanmugalingam, Measurability of equivalence classes and MECp-property in metric spaces, Rev. Mat. Iberoam. 23 2007, no. 3, 811–830.10.4171/RMI/514Search in Google Scholar

[17] S. Kallunki and N. Shanmugalingam, Modulus and continuous capacity, Ann. Acad. Sci. Fenn. Math. 26 2001, no. 2, 455–464.Search in Google Scholar

[18] J. C. Kelly, Quasiconformal mappings and sets of finite perimeter, Trans. Amer. Math. Soc. 180 1973, 367–387.10.1090/S0002-9947-1973-0357783-7Search in Google Scholar

[19] R. Korte, A Caccioppoli estimate and fine continuity for superminimizers on metric spaces, Ann. Acad. Sci. Fenn. Math. 33 2008, no. 2, 597–604.Search in Google Scholar

[20] R. Korte and P. Lahti, Relative isoperimetric inequalities and sufficient conditions for finite perimeter on metric spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 2014, no. 1, 129–154.10.1016/j.anihpc.2013.01.005Search in Google Scholar

[21] R. Korte, N. Marola, and N. Shanmugalingam, Quasiconformality, homeomorphisms between metric measure spaces preserving quasiminimizers, and uniform density property, Ark. Mat. 50 2012, 111–134.10.1007/s11512-010-0137-xSearch in Google Scholar

[22] A. Lohvansuu and K. Rajala, Duality of moduli in regular metric spaces, preprint 2018.Search in Google Scholar

[23] M. Miranda, Jr., Functions of bounded variation on “good” metric spaces, J. Math. Pures Appl. (9) 82 2003, no. 8, 975–1004.10.1016/S0021-7824(03)00036-9Search in Google Scholar

[24] N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana 16(2) 2000, 243–279.10.4171/RMI/275Search in Google Scholar

[25] M. Williams, Geometric and analytic quasiconformality in metric measure spaces, Proc. Amer. Math. Soc. 140 2012, no. 4, 1251–1266.10.1090/S0002-9939-2011-11035-9Search in Google Scholar

[26] W. Ziemer, Extremal length and conformal capacity, Trans. Amer. Math. Soc. 126 1967 460–473.10.1090/S0002-9947-1967-0210891-0Search in Google Scholar

[27] W. Ziemer, Extremal length and p-capacity, Michigan Math. J. 16 1969 43–51.10.1307/mmj/1029000164Search in Google Scholar

Received: 2019-03-06
Accepted: 2020-06-05
Published Online: 2020-08-24

© 2020 Rebekah Jones et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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