Hyperbolic distance and membership of conformal maps in the Hardy space

TL;DR

This paper characterizes when a conformal map of the unit disk belongs to a Hardy space using hyperbolic distances to level sets, providing a new integral criterion that addresses a question by Poggi-Corradini.

Contribution

It establishes a novel integral condition involving hyperbolic distances that characterizes Hardy space membership of conformal maps, answering an open question.

Findings

Conformal maps in Hardy spaces are characterized by an integral involving hyperbolic distances.

The criterion provides a new geometric perspective on Hardy space membership.

The result connects conformal geometry with function space theory.

Abstract

Let $\psi$ be a conformal map of the unit disk $\mathbb{D}$ onto an unbounded domain and, for $\alpha >0$, let ${F_\alpha }=\left\{ {z \in \mathbb{D}:\left| {\psi \left( z \right)} \right| = \alpha } \right\}$. If ${H^p}\left( \mathbb{D} \right)$ denotes the classical Hardy space and $d_\mathbb{D} {\left( {0,{F_\alpha }} \right)}$ denotes the hyperbolic distance between $0$ and $F_\alpha$ in $\mathbb{D}$, we prove that $\psi$ belongs to ${H^p}\left( \mathbb{D} \right)$ if and only if \[\int_0^{ + \infty } {{\alpha ^{p - 1}}{e^{ - {d_{\mathbb{D}}}\left( {0,{F_\alpha }} \right)}}d\alpha } < + \infty .\] This result answers a question posed by P. Poggi-Corradini.