Carleson measures on planar sets

Abstract: In this paper, we investigate what are Carleson measures on open subsets in the complex plane. A circular domain is a connected open subset whose boundary consists of finitely many disjoint circles. We call a domain $G$ multi-nicely connected if there exists a circular domain $W$ and a conformal map $\psi$ from $W$ onto $G$ such that $\psi$ is almost univalent with respect the arclength on $\partial W$. We characterize all Carleson measures for those open subsets so that each of their components is multi-nicely connected and harmonic measures of the components are mutually singular. Our results suggest the extend of Carleson measures probably is up to this class of open subsets.