The Bergman number of a plane domain

TL;DR

This paper introduces the Bergman number of a domain in the complex plane, establishes its equality with the Hardy number for regular domains, and explores related properties of Hardy and Bergman spaces.

Contribution

It defines the Bergman number as an analogue to the Hardy number and proves their equality for regular domains, extending previous work to more general domains.

Findings

Bergman number equals Hardy number for regular domains

Characterization of Hardy and Bergman numbers via harmonic measure

New criteria for membership in weighted Bergman spaces

Abstract

Let $D$ be a domain in the complex plane $\mathbb{C}$. The Hardy number of $D$, which first introduced by Hansen, is the maximal number $h(D)$ in $[0,+\infty]$ 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 $[0,+\infty]$ such that $f$ belongs to the weighted Bergman space $A^p_{\alpha} (\mathbb{D})$ whenever $p>0$ and $\alpha>-1$ satisfy $0<\frac{p}{\alpha+2}<b(D)$ 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.