The Michigan Mathematical Journal

The ∞-Poincaré inequality on metric measure spaces

Estibalitz Durand-Cartagena, Jesús Jaramillo, and Nageswari Shanmugalingam

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Michigan Math. J., Volume 61, Issue 1 (2012), 63-85.

First available in Project Euclid: 8 March 2012

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Zentralblatt MATH identifier

Primary: 31E05: Potential theory on metric spaces 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems
Secondary: 30L10: Quasiconformal mappings in metric spaces


Durand-Cartagena, Estibalitz; Jaramillo, Jesús; Shanmugalingam, Nageswari. The ∞-Poincaré inequality on metric measure spaces. Michigan Math. J. 61 (2012), no. 1, 63--85. doi:10.1307/mmj/1331222847.

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  • S. G. Bobkov and C. Houdré, Some connections between isoperimetric and Sobolev-type inequalities, Mem. Amer. Math. Soc. 129 (1997).
  • M. Bourdon and H. Pajot, Quasi-conformal geometry and hyperbolic geometry, Rigidity in dynamics and geometry (Cambridge, 2000), pp. 1–17, Springer-Verlag, Berlin, 2002.
  • E. Durand-Cartagena and J. A. Jaramillo, Pointwise Lipschitz functions on metric spaces, J. Math. Anal. Appl. 363 (2010), 525–548.
  • G. B. Folland, Real analysis. Modern techniques and their applications, Pure Appl. Math. (N.Y.), Wiley, New York, 1999.
  • P. Hajłasz, Sobolev spaces on metric-measure spaces, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), Contemp. Math., 338, pp. 173–218, Amer. Math. Soc, Providence, RI, 2003.
  • P. Hajlasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000).
  • J. Heinonen, Lectures on analysis on metric spaces, Springer-Verlag, New York, 2001.
  • J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), 1–61.
  • –––, A note on Lipschitz functions, upper gradients, and the Poincaré inequality, New Zealand J. Math. 28 (1999), 37–42.
  • E. Järvenpää, M. Järvenpää, N. Shanmugalingam, K. Rogovin, and S. Rogovin, Measurability of equivalence classes and MEC$_p$-property in metric spaces, Rev. Mat. Iberoamericana 23 (2007), 811–830.
  • S. Keith and X. Zhong, The Poincaré inequality is an open ended condition, Ann. of Math. (2) 167 (2008), 575–599.
  • J. Kinnunen and R. Korte, Characterizations of Sobolev inequalities on metric spaces, J. Math. Anal. Appl. 344 (2008), 1093–1104.
  • R. Korte, Geometric implications of the Poincaré inequality, Licentiate thesis, Helsinki University of Technology, 2006.
  • T. J. Laakso, Ahlfors $Q$-regular spaces with arbitrary $Q>1$ admitting weak Poincaré inequality, Geom. Funct. Anal. 10 (2000), 111–123.
  • P. Mattila, Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability, Cambridge Stud. Adv. Math., 44, Cambridge Univ. Press, 1995.
  • M. Miranda, Functions of bounded variation on “good” metric spaces, J. Math. Pures Appl. (9) 82 (2003), 975–1004.
  • S. Semmes, Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities, Selecta Math. (N.S.) 2 (1996), 155–295.
  • N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Ph.D. thesis, University of Michigan, 1999, $\langle$\raise1pt$\MTSY{6.50pt{6.00pt}{5.00pt} \sim$}nages/papers.html$\rangle.$
  • –––, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana 16 (2000), 243–279.