Mathematical Notes

, Volume 63, Issue 6, pp 813–822 | Cite as

Hausdorff measure and capacity associated with Cauchy potentials

  • V. Ya. Éiderman
Article

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−3h2(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.

Key words

Hausdorff mesure Cauchy potentials capacity Cantor set 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Garnett,Analytic Capacity and Measure, Vol. 297, Lecture Notes in Math., Springer-Verlag, Berlin-Heidelberg-New York (1972).Google Scholar
  2. 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. 3.
    L. Carleson,Selected Problems on Exceptional Sets, Van Nostrand, (1967).Google Scholar
  4. 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. 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).MATHMathSciNetGoogle Scholar
  6. 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).MATHMathSciNetGoogle Scholar
  7. 7.
    J. Garnett, “Positive length but zero analytic capacity,”Proc. Amer. Math. Soc.,24, 696–699 (1970).MATHMathSciNetGoogle Scholar
  8. 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).MATHMathSciNetGoogle Scholar
  9. 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. 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).MATHMathSciNetGoogle Scholar
  11. 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).MATHMathSciNetGoogle Scholar
  12. 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. 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. 14.
    T. Murai, “Construction ofH 1 functions concerning the estimate of analytic capacity,”Bull. London Math. Soc.,19, 154–160 (1987).MATHMathSciNetGoogle Scholar
  15. 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).MathSciNetGoogle Scholar
  16. 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).MathSciNetGoogle Scholar
  17. 17.
    X. Tolsa,Curvature of Measures, Cauchy Singular Integral and Analytic Capacity, Thesis, Departament de Matemàtigues, Universitat Autònoma de Barcelona, Barcelona (1998).Google Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • V. Ya. Éiderman
    • 1
  1. 1.Moscow State University of Civil EngineeringUSSR

Personalised recommendations