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

Abstract

Let ψ 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 ψ. Consider the limits

\[L:=\underset{\alpha \to +\infty }{\lim }\left(\log {\omega _{\mathbb{D}}}{\left(0,{F_{\alpha }}\right)^{-1}}\Big/ \log \alpha \right),\hspace{1em}\mu :=\underset{\alpha \to +\infty }{\lim }\left({d_{\mathbb{D}}}\left(0,{F_{\alpha }}\right)\big/ \log \alpha \right),\]

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, μ and $\mathtt{h}\left(\psi \right)$? Motivated by the result of Kim and Sugawa that $\mathtt{h}\left(\psi \right)={\liminf _{\alpha \to +\infty }}(\log {\omega _{\mathbb{D}}}{\left(0,{F_{\alpha }}\right)^{-1}}{}\log \alpha )$, we show that $\mathtt{h}\left(\psi \right)={\liminf _{\alpha \to +\infty }}\left({d_{\mathbb{D}}}\left(0,{F_{\alpha }}\right)\big/\log \alpha \right)$. We also provide conditions for the existence of L and μ 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 ψ. We present an example of ψ such that $\psi \in {H^{\mu}}\left(\mathbb{D}\right)$.