Holomorphic function spaces and the geometry of image domains

This talk surveys the problem of characterizing the \(X-\)domains: planar domains \(\Omega\subset\mathbb C\) such that every holomorphic map \(f:\mathbb D\to\mathbb C\) belongs to a given holomorphic function space \(X\) on the unit disk \(\mathbb D\). Classical contributions include characterizations for \(X\) being the Bloch space (Seidel, Walsh, 1942), the space of analytic functions of bounded mean oscillation BMOA (Hayman, Pommerenke, 1978), and the Smirnov class (Ahern, Cohn, 1983).

The case of Hardy spaces \(H^p\) has remained particularly active. Hansen (1970) introduced the Hardy number of a domain \(\Omega\subset\mathbb C\), defined as the supremum of all \(p>0\) such that \(\operatorname{Hol}(\mathbb D,\Omega)\). Subsequent work, notably by Essén (1981) and Kim and Sugawa (2011), established connections between the Hardy number of a domain and harmonic measure. More relations between the Hardy number of a domain and other potential-theoretic quantities have been encountered recently. The Hardy number of domains with special geometric properties has also been explored, including star-like domains (Hansen, 1971), comb domains (Karafyllia, 2021), and Koenigs domains (Contreras, Cruz-Zamorano, Kourou, Rodr'iguez-Piazza, 2024).

Karafyllia (2023), building on a previous work with Karamanlis, extended this problem to (weighted) Bergman spaces, introducing the Bergman number of a domain. We will also discuss a close relationship between the Hardy number and the Bergman number of a planar domain, covering recent ideas on this topic obtained in collaboration with Betsakos.