Hardy Number of Koenigs Domains: Sharp Estimate

TL;DR

This paper establishes a sharp lower bound of 1/2 for the Hardy number of regular Koenigs domains, indicating the integrability properties of holomorphic functions mapping the unit disc into these domains.

Contribution

It provides the first sharp estimate for the Hardy number of Koenigs domains, advancing understanding of their boundary behavior and function theory.

Findings

Hardy number of Koenigs domains is at least 1/2

Holomorphic functions into these domains are in all Hardy spaces H^p for p<1/2

The estimate is sharp, confirming the bound cannot be improved

Abstract

Let $\Omega$ be a regular Koenigs domain in the complex plane $\mathbb{C}$. We prove that the Hardy number of $\Omega$ is greater or equal to $1/2$. That is, every holomorphic function in the unit disc $f \colon \mathbb{D} \to \Omega$ belongs to the Hardy space $H^{p}(\mathbb{D})$ for all $p<1/2$.