Mathematical Notes

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

Hausdorff measure and capacity associated with Cauchy potentials

  • V. Ya. Éiderman


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 


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Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

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

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