Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access December 30, 2016

Weak Capacity and Modulus Comparability in Ahlfors Regular Metric Spaces

  • Jeff Lindquist


Let (Z, d, μ) be a compact, connected, Ahlfors Q-regular metric space with Q > 1. Using a hyperbolic filling of Z,we define the notions of the p-capacity between certain subsets of Z and of theweak covering p-capacity of path families Γ in Z.We show comparability results and quasisymmetric invariance.As an application of our methodswe deduce a result due to Tyson on the geometric quasiconformality of quasisymmetric maps between compact, connected Ahlfors Q-regular metric spaces.


[1] M. Bourdon and B. Kleiner, Combinatorial modulus, the combinatorial Loewner property, and Coxeter groups, Groups Geom. Dyn. 7 (2013), 39–107. 10.4171/GGD/177Search in Google Scholar

[2] M. Bonk and E. Saksman, Sobolev spaces and hyperbolic fillings, J. Reine Angew. Math., to appear. Search in Google Scholar

[3] M. Bourdon and H. Pajot, Cohomologie lp et espaces de Besov, J. Reine Angew. Math. 558 (2003), 85–108. Search in Google Scholar

[4] S. Buyalo and V. Schroeder, Elements of asymptotic geometry, Europ. Math. Soc., Zürich, 2007. 10.4171/036Search in Google Scholar

[5] M. Carrasco Piaggio, On the conformal gauge of a compact metric space, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 3, 495–548. Search in Google Scholar

[6] J. Heinonen, Lectures on analysis on metric spaces, Springer, New York, 2001. 10.1007/978-1-4613-0131-8Search in Google Scholar

[7] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), 1–61. 10.1007/BF02392747Search in Google Scholar

[8] J. Heinonen, P. Koskela, N. Shanmugalingam, and J. Tyson, Sobolev spaces on metric measure spaces, Cambridge University Press, Cambridge, 2015. 10.1017/CBO9781316135914Search in Google Scholar

[9] S. Keith and X. Zhong, The Poincaré inequality is an open ended condition, Ann. of Math. (2) 167 (2008), 575–599. 10.4007/annals.2008.167.575Search in Google Scholar

[10] J. Mackay and J. Tyson, Conformal dimension theory and application, University Lecture Series, Vol. 54, American Mathematical Society, Providence, Rhode Island, 2010. Search in Google Scholar

[11] P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. of Math. (2) 129 (1989), 1–60. Search in Google Scholar

[12] P. Pansu, Dimension conforme et sphère à l’infini des variétés à courbure négative. Ann. Acad. Sci. Fenn., Ser. A I, 14, (1989), 177–212. 10.5186/aasfm.1989.1424Search in Google Scholar

[13] J. Tyson, Quasiconformality and quasisymmetry in metric measure spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 23 (1998), 525–548. Search in Google Scholar

[14] M. Williams, Geometric and analytic quasiconformality in metric measure spaces, Proc. Amer. Math. Soc. 140 (2012), 1251– 1266. 10.1090/S0002-9939-2011-11035-9Search in Google Scholar

Received: 2016-8-5
Accepted: 2016-12-20
Published Online: 2016-12-30

© 2016 Jeff Lindquist

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 27.3.2023 from
Scroll Up Arrow