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Quasiconformal maps in metric spaces with controlled geometry

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References

  1. Assouad, P., Plongements lipschitziens dansR n.Bull. Soc. Math. France, 111 (1983), 429–448.

    MATH  MathSciNet  Google Scholar 

  2. Beurling, A. &Ahlfors, L. V., The boundary correspondence under quasiconformal mappings,Acta Math., 96 (1956), 125–142.

    Article  MATH  MathSciNet  Google Scholar 

  3. Bojarski, B., Homeomorphic solutions of Beltrami systems.Dokl. Akad. Nauk SSSR, 102 (1955), 661–664 (Russian).

    MathSciNet  Google Scholar 

  4. Bourdon, M. & Pajot, H., Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings. To appear inProc. Amer. Math. Soc.

  5. Buser, P., A note on the isoperimetric constant.Ann. Sci. École Norm. Sup. (4), 15 (1982), 213–230.

    MATH  MathSciNet  Google Scholar 

  6. Cannon, J. W., The combinatorial Riemann mapping theorem.Acta Math., 173 (1994), 155–234.

    Article  MATH  MathSciNet  Google Scholar 

  7. Cheeger, J. &Colding, T. H., Almost rigidity of warped products and the structure of spaces with Ricci curvature bounded below.C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 353–357.

    MATH  MathSciNet  Google Scholar 

  8. Coifman, R. &Fefferman, C., Weighted norm inequalities for maximal functions and singular integrals.Studia Math., 51 (1974), 241–250.

    MATH  MathSciNet  Google Scholar 

  9. Cannon, J. W., Floyd, W. J. &Parry, W. R., Squaring rectangles: The finite Riemann mapping theorem.Contemp. Math., 169 (1994), 133–212.

    MATH  MathSciNet  Google Scholar 

  10. Chavel, I.,Riemannian Geometry—A Modern Introduction. Cambridge Tracts in Math., 108. Cambridge Univ. Press, Cambridge, 1993.

    MATH  Google Scholar 

  11. Cannon, J. W. &Swenson, E. L., Recognizing constant curvature discrete groups in dimension 3.Trans. Amer. Math. Soc., 350 (1998), 809–849.

    Article  MATH  MathSciNet  Google Scholar 

  12. Coulhon, T. &Saloff-Coste, L., Variétés riemanniennes isométriques à l'infini.Rev. Mat. Iberoamericana, 11 (1995), 687–726.

    MATH  MathSciNet  Google Scholar 

  13. Coifman, R. &Weiss, G.,Analyse harmonique non-commutative sur certains espaces homogènes. Lecture Notes in Math., 242. Springer-Verlag, Berlin-New York, 1971.

    MATH  Google Scholar 

  14. David, G. &Semmes, S., StrongA ∞-weights, Sobolev inequalities and quasiconformal mappings, inAnalysis and Partial Differential Equations, pp. 101–111. Lecture Notes in Pure and Appl. Math., 122. Dekker, New York, 1990.

    Google Scholar 

  15. —,Singular Integrals and Rectifiable Sets in R n:Au-delà des graphes lipschitziens. Astérisque, 193. Soc. Math. France, Paris, 1991.

    Google Scholar 

  16. —,Analysis of and on Uniformly Rectifiable Sets. Math. Surveys Monogr., 38. Amer. Math. Soc., Providence, RI, 1993.

    MATH  Google Scholar 

  17. Federer, H.,Geometric Measure Theory. Grundlehren Math. Wiss., 153. Springer-Verlag, New York, 1969.

    MATH  Google Scholar 

  18. Fuglede, B., Extremal length and functional completion.Acta Math., 98 (1957), 171–219.

    Article  MATH  MathSciNet  Google Scholar 

  19. Gehring, F. W., The definitions and exceptional sets for quasiconformal mappings.Ann. Acad. Sci. Fenn. Math., 281 (1960), 1–28.

    MathSciNet  Google Scholar 

  20. —, Symmetrization of rings in space.Trans. Amer. Math. Soc., 101 (1961), 499–519.

    Article  MATH  MathSciNet  Google Scholar 

  21. —, TheL P-integrability of the partial derivatives of a quasiconformal mapping.Acta Math., 130 (1973), 265–277.

    Article  MATH  MathSciNet  Google Scholar 

  22. Ghys, E. &Harpe, P. de la,Sur les groupes hyperboliques d'après Mikhael Gromov. Progr. Math., 83. Birkhäuser, Boston, MA, 1990.

    MATH  Google Scholar 

  23. Gromov, M.,Structures métriques pour les variétés riemanniennes. Edited by J. Lafontaine and P. Pansu. Textes Mathématiques, 1. CEDIC, Paris, 1981.

    Google Scholar 

  24. Gehring, F. W. &Martio, O., Quasiextremal distance domains and extension of quasiconformal mappings.J. Analyse Math., 45 (1985), 181–206.

    MATH  MathSciNet  Google Scholar 

  25. Gromov, M. &Pansu, P., Rigidity of lattices: An introduction, inGeometric Topology: Recent Developments. Lecture Notes in Math., 1504. Springer-Verlag, Berlin-New York, 1991.

    Google Scholar 

  26. Gromov, M., Carnot-Carathéodory spaces seen from within, inSub-Riemannian Geometry, pp. 79–323. Progr. Math., 144. Birkhäuser, Basel, 1996.

    Google Scholar 

  27. Gilbarg, D. &Trudinger, N. S.,Elliptic Partial Differential Equations of Second Order, 2nd edition. Grundlehren Math. Wiss., 224. Springer-Verlag, Berlin-New York, 1983.

    Google Scholar 

  28. Heinonen, J., Quasiconformal mappings onto John domains.Rev. Mat. Iberoamericana, 5 (1989), 97–123.

    MATH  MathSciNet  Google Scholar 

  29. —, A capacity estimate on Carnot groups.Bull. Sci. Math., 119 (1995), 475–484.

    MATH  MathSciNet  Google Scholar 

  30. Haj⦊asz, P., Sobolev spaces on an arbitrary metric space.Potential Anal., 5 (1996), 403–415.

    MathSciNet  Google Scholar 

  31. Haj⦊asz, P. &Koskela, P., Sobolev meets Poincaré.C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 1211–1215.

    Google Scholar 

  32. Hesse, J., Ap-extremal length andp-capacity equality.Ark. Mat., 13 (1975), 131–144.

    Article  MATH  MathSciNet  Google Scholar 

  33. Heinonen, J. &Koskela, P., Definitions of quasiconformality.Invent. Math., 120 (1995), 61–79.

    Article  MATH  MathSciNet  Google Scholar 

  34. —, From local to global in quasiconformal structures.Proc. Nat. Acad. Sci. U.S.A., 93 (1996), 554–556.

    Article  MATH  MathSciNet  Google Scholar 

  35. Heinonen, J., Kilpeläinen, T. &Martio, O.,Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Univ. Press, New York, 1993.

    MATH  Google Scholar 

  36. Hocking, J. G. &Young, G. S.,Topology. Academic Press, Reading, MA, 1959.

    MATH  Google Scholar 

  37. Jerison, D., The Poincaré inequality for vector fields satisfying Hörmander's condition.Duke Math. J., 53 (1986), 503–523.

    Article  MATH  MathSciNet  Google Scholar 

  38. Koskela, P., Removable sets for Sobolev spaces. To appear inArk. Mat.

  39. Koskela, P. &MacManus, P., Quasiconformal mappings and Sobolev spaces.Studia Math., 131 (1998), 1–17.

    MATH  MathSciNet  Google Scholar 

  40. Korányi, A. &Reimann, H. M., Quasiconformal mappings on the Heisenberg group.Invent. Math., 80 (1985), 309–338.

    Article  MATH  MathSciNet  Google Scholar 

  41. —, Foundations for the theory of quasiconformal mappings on the Heisenberg group.Adv. in Math., 111 (1995), 1–87.

    Article  MATH  Google Scholar 

  42. Korevaar, J. N. &Schoen, R. M., Sobolev spaces and harmonic maps for metric space targets.Comm. Anal. Geom., 1 (1993), 561–659.

    MATH  MathSciNet  Google Scholar 

  43. Loewner, C., On the conformal capacity in space.J. Math. Mech., 8 (1959), 411–414.

    MATH  MathSciNet  Google Scholar 

  44. Mattila, P.,Geometry of Sets and Measures in Euclidean Spaces. Cambridge Stud. Adv. Math., 44. Cambridge Univ. Press, Cambridge, 1995.

    MATH  Google Scholar 

  45. Margulis, G. A. &Mostow, G. D., The differential of a quasiconformal mapping of a Carnot-Carathéodory space.,Geom. Funct. Anal., 5 (1995), 402–433.

    Article  MATH  MathSciNet  Google Scholar 

  46. Mostow, G. D.,Strong Rigidity of Locally Symmetric Spaces. Princeton Univ. Press, Princeton, NJ, 1973.

    MATH  Google Scholar 

  47. —, A remark on quasiconformal mappings on Carnot groups.Michigan Math. J., 41 (1994), 31–37.

    Article  MathSciNet  Google Scholar 

  48. Maheux, P. &Saloff-Coste, L., Analyse sur les boules d'un opérateur souselliptique.Math. Ann., 303 (1995), 713–740.

    Article  MATH  MathSciNet  Google Scholar 

  49. Näkki, R., Extension of Loewner's capacity theorem.Trans. Amer. Math. Soc., 180 (1973), 229–236.

    Article  MATH  MathSciNet  Google Scholar 

  50. Pansu, P., Quasiisométries des variétés à courbure négative. Thesis, Université Paris VII, 1987.

  51. —, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un.Ann. of Math., 129 (1989), 1–60.

    Article  MATH  MathSciNet  Google Scholar 

  52. —, Dimension conforme et sphère à l'infini des variétés à courbure négative.Ann. Acad. Sci. Fenn. Math., 14 (1989), 177–212.

    MATH  MathSciNet  Google Scholar 

  53. Paulin, P., Un groupe hyperbolique est déterminé par son bord.J. London Math. Soc., 54 (1996), 50–74.

    MATH  MathSciNet  Google Scholar 

  54. Reimann, H. M., An estimate for pseudoconformal capacities on the sphere.Ann. Acad. Sci. Fenn. Math., 14 (1989), 315–324.

    MATH  MathSciNet  Google Scholar 

  55. Rudin, W.,Real and Complex Analysis, 2nd edition. McGraw-Hill, New York-Düsseldorf-Johannesburg, 1974.

    MATH  Google Scholar 

  56. Semmes, S., Bilipschitz mappings and strongA ∞-weights.Ann. Acad. Sci. Fenn. Math., 18 (1993), 211–248.

    MathSciNet  Google Scholar 

  57. —, On the nonexistence of bilipschitz parameterizations and geometric problems aboutA ∞-weights.Rev. Mat. Iberoamericana, 12 (1996), 337–410.

    MATH  MathSciNet  Google Scholar 

  58. —, Good metric spaces without good parameterizations.Rev. Math. Iberoamericana, 12 (1996), 187–275.

    MATH  MathSciNet  Google Scholar 

  59. —, Finding curves on general spaces through quantitative topology with applications to Sobolev and Poincarè inequalities.Selecta Math. (N.S.), 2 (1996), 155–295.

    Article  MATH  MathSciNet  Google Scholar 

  60. —, Some remarks about metric spaces, spherical mappings, functions and their derivatives.Publ. Math., 40 (1996), 411–430.

    MATH  MathSciNet  Google Scholar 

  61. Saloff-Coste, L., Uniformly elliptic operators on Riemannian manifolds.J. Differential Geom., 36 (1992), 417–450.

    MATH  MathSciNet  Google Scholar 

  62. Stein, E. M.,Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton, NJ, 1970.

    MATH  Google Scholar 

  63. —,Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton Univ. Press, Princeton, NJ, 1993.

    MATH  Google Scholar 

  64. Strömberg, J. D. &Torchinsky, A.,Weighted Hardy Spaces, Lecture Notes in Math., 1381. Springer-Verlag, Berlin-New York, 1989.

    MATH  Google Scholar 

  65. Sullivan, D. P., Hyperbolic geometry and homeomorphisms, inGeometric Topology (Athens, GA, 1977), pp. 543–555. Academic Press, New York-London, 1979.

    Google Scholar 

  66. Tukia, P. &Väisälä, J., Quasisymmetric embeddings of metric spaces.Ann. Acad. Sci. Fenn. Math., 5 (1980), 97–114.

    MATH  Google Scholar 

  67. Tyson, J., Quasiconformality and quasisymmetry in metric measure spaces. To appear inAnn. Acad. Sci. Fenn. Math.

  68. Väisälä, J.,Lectures on n-Dimensional Quasiconformal Mappings. Lecture Notes in Math., 229. Springer-Verlag, Berlin-New York, 1971.

    MATH  Google Scholar 

  69. —, Quasisymmetric embeddings in Euclidean spaces.Trans. Amer. Math. Soc., 264 (1981), 191–204.

    Article  MATH  MathSciNet  Google Scholar 

  70. —, Quasimöbius maps.J. Analyse Math., 44 (1984/85), 218–234.

    Article  MathSciNet  Google Scholar 

  71. —, Quasiconformal maps of cylindrical domains.Acta Math., 162 (1989), 201–225.

    Article  MATH  MathSciNet  Google Scholar 

  72. —, Free quasiconformality in Banach spaces, I.Ann. Acad. Sci. Fenn. Math., 15 (1990), 355–379.

    MATH  Google Scholar 

  73. —, Free quasiconformality in Banach spaces, II.Ann. Acad. Sci. Fenn. Math., 16 (1991), 255–310.

    MATH  Google Scholar 

  74. Vodop'yanov, S. K. &Greshnov, A. V., Analytic properties of quasiconformal mappings on Carnot groups.Siberian Math. J., 36 (1995), 1142–1151.

    Article  MATH  MathSciNet  Google Scholar 

  75. Varopoulos, N. Th., Saloff-Coste, L. &Coulhon, T.,Analysis and Geometry on Groups. Cambridge Tracts in Math., 100. Cambridge Univ. Press, Cambridge, 1992.

    Google Scholar 

  76. Vuorinen, M.,Conformal Geometry and Quasiregular Mappings. Lecture Notes in Math., 1319. Springer-Verlag, New York, 1988.

    MATH  Google Scholar 

  77. Ziemer, W. P., Extremal length andp-capacity.Michigan Math. J., 16 (1969), 43–51.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. University of Michigan, Ann Arbor, MI, U.S.A.

    Juha Heinonen

  2. University of Jyväskylä, Jyväskylä, Finland

    Pekka Koskela

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  1. Juha Heinonen
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  2. Pekka Koskela
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Both authors were supported in part by the NSF and the Academy of Finland. The first author is a Sloan Fellow.

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Heinonen, J., Koskela, P. Quasiconformal maps in metric spaces with controlled geometry. Acta Math 181, 1–61 (1998). https://doi.org/10.1007/BF02392747

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