Parole chiave

• Bloch space
• Hardy space
• Hardy number
• Bergman space
• Bergman number

Abstract This talk surveys the problem of characterizing the $X$-domains: planar domains $\Omega \subset \mathbb{C}$ such that every holomorphic map $f: \mathbb{D} \to \Omega$ 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 $\mathrm{BMOA}$ (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 $\mathrm{Hol}(\mathbb{D},\Omega) \subset H^p$. 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.s