Mathematical Notes

, Volume 63, Issue 6, pp 813–822

# 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

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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