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

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 $\psi \in {H^\mu {\left( \mathbb{D} \right)} }$.