Hausdorff measure and capacity associated with Cauchy potentials
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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 wordsHausdorff mesure Cauchy potentials capacity Cantor set
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