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 $\mathtt{h}\left(\psi \right)$ the Hardy number of $\psi$. Consider the limits $$L:=\underset{\alpha \to +\infty }{\lim }\left(\log {\omega }_{\mathbb{D}}(0,F_{\alpha }{)}^{-1}/\log \alpha \right),\mu :=\underset{\alpha \to +\infty }{\lim }\left(d_{\mathbb{D}}(0,F_{\alpha })/\log \alpha \right),$$ 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 $\mathtt{h}\left(\psi \right)$? We also provide conditions for the existence of $L$ and $\mu$ and for the equalities $L=\mu =\mathtt{h}\left(\psi \right)$. 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 $\psi \in H^{\mu }\left(\mathbb{D}\right)$.

19 pages, 15 figures