On the Hardy number of a domain in terms of harmonic measure and hyperbolic distance

TL;DR

This paper investigates the relationship between the Hardy number of a conformal map, harmonic measure, and hyperbolic distance in the unit disk, providing conditions for their equality and examples illustrating these properties.

Contribution

It establishes connections between harmonic measure, hyperbolic distance, and Hardy spaces for conformal maps, addressing a problem posed by Poggi-Corradini and providing new examples.

Findings

Identifies conditions for the existence of limits L and μ.

Shows when L and μ equal the Hardy number h(ψ).

Provides an example where ψ belongs to H^μ, contrary to previous results.

Abstract

Let $\psi $ be a conformal map on $\mathbb{D}$ with $ \psi \left( 0 \right)=0$ and let ${F_\alpha }=\left\{ {z \in \mathbb{D}:\left| {\psi \left( z \right)} \right| = \alpha } \right\}$ for $\alpha >0$. Denote by ${H^p}\left( \mathbb{D} \right)$ the classical Hardy space with exponent $p>0$ and by ${\tt h}\left( \psi \right)$ the Hardy number of $\psi$. Consider the limits $$ L:= \lim_{\alpha\to+\infty}\left( \log \omega_{\mathbb D}(0,F_{\alpha})^{-1}/\log \alpha \right), \,\, \mu:= \lim_{\alpha\to+\infty}\left( d_{\mathbb D}(0,F_{\alpha})/\log\alpha \right),$$ where $\omega _\mathbb{D}\left( {0,{F_\alpha }} \right)$ denotes the harmonic measure at $0$ of $F_\alpha $ and $d_\mathbb{D} {\left( {0,{F_\alpha }} \right)}$ denotes the hyperbolic distance between $0$ and $F_\alpha$ in $\mathbb{D}$. We study a problem posed by P. Poggi-Corradini. What is the relation between $L$, $\mu$ and ${\tt h}\left( \psi \right)$? We also provide conditions for the existence of $L$ and $\mu$ and for the equalities $L=\mu={\tt h}\left( \psi \right)$. Poggi-Corradini proved that $\psi \notin {H^{\mu}}\left( \mathbb{D} \right)$ for a wide class of conformal maps $\psi$. We present an example of $\psi$ such that $\psi \in {H^\mu {\left( \mathbb{D} \right)} }$.