Israel Journal of Mathematics

, Volume 158, Issue 1, pp 129–157 | Cite as

Uniform domains and capacity

  • Stephen M. Buckley
  • David A. Herron


We characterize the class of uniform domains in terms of capacity. As a byproduct of this investigation we provide results describing when a Loewner domain will be a quasiextremal distance domain.


Quasiconformal Mapping Carnot Group Cross Ratio Uniform Domain Disjoint Ball 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [BB03]
    Z. M. Balogh and S. M. Buckley, Geometric characterizations of Gromov hyperbolicty, Inventiones Mathematicae 153 (2003), 261–301.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [BB05]
    Z. M. Balogh and S. M. Buckley, Sphericalization and flattening, Conformal Geometry and Dynamics. An Electronic Journal of the American Mathematical Society 9 (2005), 76–101.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [BHK01]
    M. Bonk, J. Heinonen and P. Koskela, Uniformizing Gromov hyperbolic spaces, Astérisque 270 (2001), 1–99.Google Scholar
  4. [BK05]
    M. Bonk and B. Kleiner, Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary, Geometry & Toplogy 9 (2005), 219–246.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [BY02]
    A. Brania and S. Yang, Domains with controlled modulus and quasiconformal mappings, Nonlinear Stud 9 (2002), 57–73.MathSciNetGoogle Scholar
  6. [Buc03]
    S. Buckley, Slice conditions and their applications, in Future Trends In Geometric Function Theory (Univ. Jyväskylä), Vol. 92, Rep. Univ. Jyväskylä Dept. Math. Stat., 2003, RNC Workshop held in Jyvaskyla, June 15–18, 2003, pp. 63–76.MathSciNetGoogle Scholar
  7. [Buc04]
    S. M. Buckley, Quasiconfomal images of Hölder domains, Annales Academiæ Scientiarium Fennicæ. Mathematica 29 (2004), 21–42.zbMATHMathSciNetGoogle Scholar
  8. [BH06]
    S. Buckley and D. A. Herron, Uniform domains and weak slice conditions in metric spaces, in preparation (2006).Google Scholar
  9. [CGN00]
    L. Capogna, N. Garofalo and D-M Nhieu, Examples of uniform and NTA domains in Carnot groups, Proceedings on Analysis and Geometry (Novosibirsk), Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., 2000, (Novosibirsk Akad., 1999), pp. 103–121.Google Scholar
  10. [CT95]
    L. Capogna and P. Tang, Uniform domains and quasiconformal mappings on the Heisenberg group, Manuscripta Mathematica 86 (1995), 267–281.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [Geh87]
    F. W. Gehring, Uniform domains and the ubiquitous quasidisk, Jahresbericht der Deutscher Mathematiker Vereinigung 89 (1987), 88–103.zbMATHMathSciNetGoogle Scholar
  12. [GM85]
    F. W. Gehring and O. Martio, Quasiextremal distance domains and extension of quasiconformal mappings, Journal d’Analyse Mathématique 45 (1985), 181–206.zbMATHMathSciNetGoogle Scholar
  13. [GO79]
    F. W. Gehring and B. G. Osgood, Uniform domains and the quasi-hyperbolic metric, Journal d’Analyse Mathématique 36 (1979), 50–74.zbMATHMathSciNetGoogle Scholar
  14. [GP76]
    F. W. Gehring and B. P. Palka, Quasiconformally homogeneous domains, Journal d’Analyse Mathématique 30 (1976), 172–199.zbMATHMathSciNetCrossRefGoogle Scholar
  15. [GLV79]
    V. M. Gol’dshtein, T. G. Latfullin and S. K. Vodop’yanov, Criteria for extension of functions of the class ℓ1/2 from unbounded plane domains, Siberian Mathematical Journal 20 (1979), 298–301.CrossRefMathSciNetGoogle Scholar
  16. [Gre01]
    A. V. Greshnov, On uniform and NTA-domains on Carnot groups, Siberian Mathematical Journal 42 (2001), 851–864.CrossRefMathSciNetGoogle Scholar
  17. [Hei01]
    J. Heinonen, Lectures on Analysis on Metric Spaces, Springer-Verlag, New York, 2001.zbMATHGoogle Scholar
  18. [HK98]
    J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Mathematica 181 (1998), 1–61.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [HK90]
    D. A. Herron and P. Koskela, Quasiextremal distance domains and conformal mappings onto circle domains, Complex Variables 15 (1990), 167–179.zbMATHMathSciNetGoogle Scholar
  20. [HK91]
    D. A. Herron and P. Koskela, Uniform, Sobolev extension and quasiconformal circle domains, Journal d’Analyse Mathématique 57 (1991), 172–202.zbMATHMathSciNetGoogle Scholar
  21. [HK96]
    D. A. Herron and P. Koskela, Conformal capacity and the quasihyperbolic metric, Indiana University Mathematics Journal 45 (1996), 333–359.zbMATHCrossRefMathSciNetGoogle Scholar
  22. [Joh61]
    F. John, Rotation and strain, Communications on Pure and Applied Mathematics 14 (1961), 391–413.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [Jon81]
    P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Mathematica 147 (1981), 71–88.zbMATHCrossRefMathSciNetGoogle Scholar
  24. [Kos99]
    P. Koskela, Removable sets for Sobolev spaces, Arkiv för Matematik 37 (1999), 291–304.zbMATHCrossRefMathSciNetGoogle Scholar
  25. [MS79]
    O. Martio and J. Sarvas, Injectivity theorems in plane and space, Suomalaisen Tiedeakatemian Toimituksia. Sarja A. Annales Academiae Scientiarum Fennicae. Series A I. Mathematica 4 (1978/79), 383–401.MathSciNetGoogle Scholar
  26. [Väi71]
    J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Lecture Notes in Mathematics 229, Springer-Verlag, Berlin, 1971.Google Scholar
  27. [Väi88]
    J. Väisälä, Uniform domains, Tôhoku Mathematical Journal 40 (1988), 101–118.zbMATHGoogle Scholar
  28. [Väi91]
    J. Väisälä, Free quasiconformality in Banach spaces II, Annales Academiae Scientiarum Fennicae. Series A I. Mathematica 16 (1991), 255–310.zbMATHMathSciNetGoogle Scholar
  29. [Vuo88]
    M. Vuorinen, Conformal Geometry and Quasiregular Mappings, Lecture Notes in Mathematics 1319, Springer-Verlag, Berlin, 1988.Google Scholar

Copyright information

© The Hebrew University of Jerusalem 2007

Authors and Affiliations

  • Stephen M. Buckley
    • 1
  • David A. Herron
    • 2
  1. 1.Department of MathematicsNational University of IrelandMaynooth, Co. KildareIreland
  2. 2.Department of MathematicsUniversity of CincinnatiCincinnatiUSA

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