Hausdorff measure and capacity associated with Cauchy potentials
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 wordsHausdorff mesure Cauchy potentials capacity Cantor set
Unable to display preview. Download preview PDF.
- 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).Google Scholar
- 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
- 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
- 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
- 17.X. Tolsa,Curvature of Measures, Cauchy Singular Integral and Analytic Capacity, Thesis, Departament de Matemàtigues, Universitat Autònoma de Barcelona, Barcelona (1998).Google Scholar