Hausdorff measure and capacity associated with Cauchy potentials

V. Ya. Èiderman

Matematičeskie zametki, Tome 63 (1998) no. 6, pp. 923-934

Résumé

In the paper the connection between the Hausdorff measure $\Lambda_h(E)$ of sets $E\subset\mathbb C$ and the analytic capacity $\gamma(E)$, and also between $\Lambda_h(E)$ and the capacity $\gamma^+(E)$ generated by Cauchy potentials with nonnegative measures is studied. It is shown that if the integral $\int_0t^{-3}h^2(t)dt$ is divergent and $h$ satisfies the regularity condition, then there exists a plane Cantor set $E$ for which $\Lambda_h(E)>0$, but $\gamma^+(E)=0$. The proof is based on the estimate of $\gamma^+(E_n)$, where $E_n$ is the set appearing at the $n$th step in the construction of a plane Cantor set.