Skip to main content
SpringerLink
Log in

Hausdorff measure and capacity associated with Cauchy potentials

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

In the paper the connection between the Hausdorff measure Λ h (E) of setsE ⊂ ℂ and the analytic capacityγ(E), and also between Λ h (E) and the capacityγ +(E) generated by Cauchy potentials with nonnegative measures is studied. It is shown that if the integral ∫0t −3 h 2(t)dt is divergent andh satisfies the regularity condition, then there exists a plane Cantor setE for which Λ h (E)>0, butγ +(E)=0. The proof is based on the estimate ofγ +(E n ), whereE n is the set appearing at thenth step in the construction of a plane Cantor set.

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

  1. J. Garnett,Analytic Capacity and Measure, Vol. 297, Lecture Notes in Math., Springer-Verlag, Berlin-Heidelberg-New York (1972).

    Google Scholar 

  2. T. Murai,A Real Variable Method for the Cauchy Transform, and Analytic Capacity, Vol. 1307, Lecture Notes in Math., Springer-Verlag, Berlin (1988).

    Google Scholar 

  3. L. Carleson,Selected Problems on Exceptional Sets, Van Nostrand, (1967).

  4. V. Ya. Éiderman, “On the comparison of Hausdorff measure and capacity,”Algebra i Analiz [St. Petersburg Math. J.],3, No. 6, 173–188 (1991).

    Google Scholar 

  5. V. Ya. Éiderman, “Estimates of potentials andδ-subharmonic functions outside of exceptional sets,”Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.],61, No. 6, 181–218 (1997).

    MATH  MathSciNet  Google Scholar 

  6. A. G. Vitushkin, “Example of a set of positive length but of zero analytic capacity,”Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.],127, No. 2, 246–249 (1959).

    MATH  MathSciNet  Google Scholar 

  7. J. Garnett, “Positive length but zero analytic capacity,”Proc. Amer. Math. Soc.,24, 696–699 (1970).

    MATH  MathSciNet  Google Scholar 

  8. P. Mattila, “A class of sets with positive length and zero analytic capacity,”Ann. Acad. Sci. Fenn. Ser. A. I Math.,10, 387–395 (1985).

    MATH  MathSciNet  Google Scholar 

  9. Linear and Complex Analysis Problem Book 3. Part 2 (V. P. Havin and N. K. Nikolski, editors), Vol. 1574, Lecture Notes in Math., Springer-Verlag, Berlin (1994).

    Google Scholar 

  10. P. Mattila, “On the analytic capacity and curvature of some Cantor sets with non-σ-finite length,”Publ. Mat.,40, No. 1, 195–204 (1996).

    MATH  MathSciNet  Google Scholar 

  11. M. S. Melnikov, “Analytic capacity: discrete approach and curvature of measure,”Mat. Sb. [Russian Acad. Sci. Sb. Math.],186, No. 6, 57–76 (1995).

    MATH  MathSciNet  Google Scholar 

  12. M. Christ,Lectures on Singular Integral Operators, Expository Lectures from the CBMS Regional Conference Held at the University of Montana, Missoula (M.T.), August 28–September 1, 1989, CBMS Regional Conf. Ser. in Math, Vol. 77, Amer. Math. Soc., Providence (R.I.) (1990).

    Google Scholar 

  13. V. Ya. Éiderman, “Analytic capacity of Cantor sets,” in:Proceedings of the International Conference on Potential Theory (Kouty, Czech Republic, 1994) (Netuka et al., editor), Walter de Gruyter, Berlin-New York (1995).

    Google Scholar 

  14. T. Murai, “Construction ofH 1 functions concerning the estimate of analytic capacity,”Bull. London Math. Soc.,19, 154–160 (1987).

    MATH  MathSciNet  Google Scholar 

  15. M. S. Melnikov and J. Verdera, “A geometric proof of theL 2-boundedness of the Cauchy integral on Lipschitz graphs,”Internat. Math. Res. Notices, No.7, 325–331 (1995).

    MathSciNet  Google Scholar 

  16. P. Mattila, M. S. Melnikov, and J. Verdera, “The Cauchy integral, analytic capacity, and uniform rectifiability,”Ann. of Math. (2),144, No. 1, 127–136 (1996).

    MathSciNet  Google Scholar 

  17. X. Tolsa,Curvature of Measures, Cauchy Singular Integral and Analytic Capacity, Thesis, Departament de Matemàtigues, Universitat Autònoma de Barcelona, Barcelona (1998).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow State University of Civil Engineering, USSR

    V. Ya. Éiderman

Authors
  1. V. Ya. Éiderman
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 923–934, June, 1998.

I am grateful to M. S. Melnikov and X. Tolsa for information about the work [17] and for discussion of open problems.

I wish to thank the Department of Mathematics of Uppsala University (Sweden) for the excellent conditions under which this research was begun, and to Professor M. Essén for the hospitality and support of this undertaking. I am grateful also to Professor P. Mattila and Professor S. Ya. Khavinson for very useful discussions of these problems.

Research partially supported by the Royal Swedish Academy of Sciences, International Science Foundation (Grant MB 6000) and by Ministry of Education of Russia (Grant 95-0-1.7-52)

This research was partially supported by the Royal Swedish Academy of Sciences, by the International Science Foundation under grant No. MB 6000 and by the Ministry of Education of Russia under grant No. 95-0-1.7-52.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Éiderman, V.Y. Hausdorff measure and capacity associated with Cauchy potentials. Math Notes 63, 813–822 (1998). https://doi.org/10.1007/BF02312776

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02312776

Key words

Search

Navigation