Mathematical Notes

, Volume 63, Issue 6, pp 813–822

Hausdorff measure and capacity associated with Cauchy potentials

  • V. Ya. Éiderman

DOI: 10.1007/BF02312776

Cite this article as:
Éiderman, V.Y. Math Notes (1998) 63: 813. doi:10.1007/BF02312776


In the paper the connection between the Hausdorff measure Λh(E) of setsE ⊂ ℂ and the analytic capacityγ(E), and also between Λh(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 Λh(E)>0, butγ+(E)=0. The proof is based on the estimate ofγ+(En), whereEn is the set appearing at thenth step in the construction of a plane Cantor set.

Key words

Hausdorff mesureCauchy potentialscapacityCantor set

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

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