On the Hardy number and the Bergman number of a planar domain

TL;DR

This work analyzes Hardy and weighted Bergman spaces for holomorphic maps with a fixed target domain, introducing the Hardy number $h(D)$ and Bergman numbers and linking $h(D)$ to the Green function via $\psi_D(r)=\int_0^{2\pi} g_D(re^{i\theta},0)\,d\theta$. It establishes a new formula $h(D)=\liminf_{r\to\infty}(-\log\psi_D(r)/\log r)$ and identifies a natural class $\mathcal{D}$ of domains for which $h(D)=b(D)=h(\Omega)=b(\Omega)=b_{\alpha}(D)/(\alpha+2)$ for any $\alpha>-1$, with $\Omega$ the associated simply connected domain. The authors also demonstrate that $h(D)$ and $b(D)$ need not coincide in general by constructing domains (including regular ones) where $h(D)<\infty$ but $b(D)=\infty$, and provide both supporting lemmas and explicit geometric constructions. Through these results, the paper clarifies how domain geometry at infinity governs Hardy and Bergman integrability, and it extends known equalities from simply connected domains to a broader class while highlighting notable exceptions.

Abstract

This article deals with functions with a prefixed range and their inclusions in Hardy and weighted Bergman spaces. This idea was originally introduced by Hansen for Hardy spaces, and it was recently taken into weighted Bergman spaces by Karafyllia and Karamanlis. We provide a new characterization for the Hardy number of a domain in terms of its Green function. Based on this, we present a class of domains for which the Hardy number and the Bergman number coincide. However, in general, we show that the Hardy number and the Bergman number of a domain are not equal; even for domains which are regular for the Dirichlet problem.

Figures (1)

• Figure 1: Domain $D$ for some choice of the coefficients $\{\alpha_n\}$.

Theorems & Definitions (57)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 2.1
• Theorem 2.2
• Lemma 2.3
• proof
• Theorem 2.4
• Remark 2.5