Skip to main content
SpringerLink
Log in

Fractal inner chordarc disks

  • Published:
Journal d’Analyse Mathématique Aims and scope

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

We introduce a class of fractal inner chordarc domains and characterize them as inner Ahlfors regular John disks. We analyze ‘global’ parametrizations for such regions, describe the associated Riemann maps and exhibit several illustrative examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • [Dav84] G. David,Opérateurs intégraux singuliers sur certaines courbes du plan complexe, Ann. Sci. École Norm. Sup.17 (1984), 157–189.

    MATH  Google Scholar 

  • [GH98] M. Ghamsari and D. A. Herron,Higher dimensional Ahlfors regular sets and chordarc curves in R n Rocky Mountain J. Math.28 (1998), 191–222.

    Article  MATH  MathSciNet  Google Scholar 

  • [Gha90] M. Ghamsari,Extension Domains, PhD thesis, University of Michigan, 1990.

  • [GHM89] F. W. Gehring, K. Hag and O. Martio,Quasihyperbolic geodesics in John domains, Math. Scand.65 (1989), 75–92.

    MATH  MathSciNet  Google Scholar 

  • [GNV94] M. Ghamsari, R. Näkki and J. Väisälä,John disks and extension of maps, Monatsh. Math.117 (1994), 63–94.

    Article  MATH  MathSciNet  Google Scholar 

  • [HH88] B. Hanson and D. A. Herron,The interior distance ratio in quadrilaterals and inequalities for rings, Complex Variables9 (1988), 321–325.

    MATH  MathSciNet  Google Scholar 

  • [JK82] D. S. Jerison and C. E. Kenig,Hardy spaces, A ∞,and singular integrals on chord-arc domains, Math. Scand.50 (1982), 221–247.

    MATH  MathSciNet  Google Scholar 

  • [Joh61] F. John,Rotation and strain, Comm. Pure Appl. Math.14 (1961), 391–413.

    Article  MATH  MathSciNet  Google Scholar 

  • [Lat82] T. G. Latfullin,Geometric characterization of the quasi-isometric image of a half plane (Russian), inTheory of Mappings, its Generalizations and Applications, Naukova Dumka, Kiev, 1982, pp. 116–126.

    Google Scholar 

  • [MS79] O. Martio and J. Sarvas,Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Ser. A I Math.4 (1978/79), 383–401.

    MathSciNet  Google Scholar 

  • [New51] M. H. A. Newman,Elements of the Topology of Plane Sets of Points, Cambridge University Press, London, 1951.

    MATH  Google Scholar 

  • [NV91] R. Näkki and J. Väisälä,John disks, Exposition. Math.9 (1991), 3–43.

    MATH  MathSciNet  Google Scholar 

  • [Pom82] Ch. Pommerenke,One-sided smoothness conditions and conformal mapping, J. London Math. Soc.26 (1982), 77–88.

    Article  MATH  MathSciNet  Google Scholar 

  • [Pom92] Ch. Pommerenke,Boundary Behavior of Conformal Maps, Grundlehren der mathematischen Wissenschaften299, Springer-Verlag, Berlin, 1992.

    Google Scholar 

  • [Sul00] T. Sullivan,Alpha-Dimensional Inner ChordArc Domains, PhD thesis, University of Cincinnati, 2000.

  • [Tuk81] P. Tukia,Extension of quasisymmetric and lipschitz embeddings of the real line into the plane, Ann. Acad. Sci. Fenn. Ser. A I Math.,6 (1981), 89–94.

    MathSciNet  MATH  Google Scholar 

  • [Väi71] J. Väisälä,Lectures on n-Dimensional Quasiconformal Mappings, Lecture Notes in Math.229, Springer-Verlag, Berlin, 1971.

    MATH  Google Scholar 

  • [Väi87] J. Väisälä,Homeomorphisms of bounded length distortion, Ann. Acad. Sci. Fenn. Ser. A I Math.12 (1987), 303–312.

    MATH  MathSciNet  Google Scholar 

  • [Väi89] J. Väisälä,Quasiconformal maps of cylindrical domains, Acta Math.162 (1989), 201–225.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Cincinnati, 45221, Cincinnati, OH, USA

    David A. Herron & Theresa S. Sullivan

Authors
  1. David A. Herron
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Theresa S. Sullivan
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Both authors were supported by the Charles Phelps Taft Memorial Fund.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Herron, D.A., Sullivan, T.S. Fractal inner chordarc disks. J. Anal. Math. 84, 173–205 (2001). https://doi.org/10.1007/BF02788109

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02788109

Keywords

Search

Navigation