A geometric characterization of $N^{+}$ domains

Abstract:

A connected open set $\mathcal {O} \subseteq {\mathbf {C}}$ is called an ${N^ + }$ domain if every holomorphic function defined in the unit disc and taking values in $\mathcal {O}$ is necessarily in the Smirnov class ${N^ + }$. We show that $\mathcal {O}$ is an ${N^ + }$ domain if and only if $\infty$ is a regular point for the solution of the Dirichlet problem for $\mathcal {O}$. We get a similar characterization when ${N^ + }$ is replaced by the class of outer functions.