On the asymptotic behavior of condenser capacity under Blaschke products and universal covering maps

Abstract:

We prove an estimate for the capacity of the condenser $(\mathbb {D},K_{r})$, $r\in (0,1)$, where $\mathbb {D}$ is the open unit disc and $\{K_{r}\}$ is a compact exhaustion of the inverse image of a compact set under a Blaschke product $B$, involving weighted logarithmic integral means of the Frostman shifts of $B$. Also, we describe the asymptotic behavior of the capacity of condensers $(\mathbb {D},E_{r})$, where $E_{r}$ is a connected component of the inverse image of a closed disc with radius $r$ under universal covering maps as $r\to 0$.