Carleson measures on planar sets (Q2655592)

The author extends the definition of Carleson measures to general domains in \(\mathbb C\). A positive finite Borel measure \(\mu\) on a connected, open subset \(G\) is called a Carleson measure, if there exists a constant \(C>0\) such that for all \(f\) analytic in \(G\) and continuous on \(\overline{G}\), \[ \int_G |f|^q\,d\mu\leq C\int_{\partial G} |f|^q\,d\omega\qquad \text{for all } q\in [1,\infty), \] where \(\omega\) is a harmonic measure for \(G\). A domain \(G\) is said to be \textit{multi-nicely connected}, if there exists a circular domain W and a conformal mapping \(\psi\) from \(W\) onto \(G\) such that \(\psi\) is almost univalent with respect to arclength on \(\partial W\). One of the main results of the paper is Theorem 4: Let \(G\) be a multi-nicely connected domain conformally equivalent to a circular domain W, and let \(\alpha\) denote a conformal map of \(W\) onto \(G\). Then a positive measure \(\mu\) on \(G\) is a Carleson measure if and only if \(\mu\circ\alpha\) is a Carleson measure on \(W\). The author also characterizes all Carleson measures for those open subsets for which each component is multi-nicely connected and the harmonic measures of the components are mutually singular.