The Bergman number of a plane domain

Abstract

Let D be a domain in the complex plane $\mathbb{C}$. The Hardy number of D, which was first introduced by Hansen, is the maximal number $h(D)$ in $[0,+\mathrm{\infty }]$ such that f belongs to the classical Hardy space $H^p(\mathbb{D})$ whenever $0<p<h(D)$ and f is holomorphic on the unit disk $\mathbb{D}$ with values in D. As an analogue notion to the Hardy number of a domain D in $\mathbb{C}$, we introduce the Bergman number of D, and we denote it by $b(D)$. Our main result is that if D is regular, then $h(D)=b(D)$. This generalizes earlier work by the author and Karamanlis for simply connected domains. The Bergman number $b(D)$ is the maximal number in $[0,+\mathrm{\infty }]$ such that f belongs to the weighted Bergman space $A_{\mathit{\alpha }}^p(\mathbb{D})$ whenever $p>0$ and $\mathit{\alpha }>-1$ satisfy $0<\frac{p}{\mathit{\alpha }+2}<b(D)$ and f is holomorphic on $\mathbb{D}$ with values in D. We also establish several results about Hardy spaces and weighted Bergman spaces, and we give a new characterization of the Hardy number and thus of the Bergman number of a regular domain with respect to the harmonic measure.