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

中文翻译：

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关于域的Hardy数，用调和测度和双曲距离表示

令ψ为$ \ psi \ left（0 \ right）= 0 $在$ \ mathbb {D} $上的保形图，并让$ {F _ {\ alpha}} = \ left \ {z \ in \ mathbb {D }：\ left | \ psi \ left（z \ right）\ right | = \ alpha \ right \} $表示$ \ alpha> 0 $。用$ {H ^ {p}} \ left（\ mathbb {D} \ right）$表示指数为$ p> 0 $的经典Hardy空间，用$ \ mathtt {h} \ left（\ psi \ right）$表示ψ的Hardy数。考虑极限\ [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），\]其中$ {\ omega _ {\ mathbb {D} }} \ left（0，{F _ {\ alpha}} \ right）$表示$ {F _ {\ alpha}} $和$ {d _ {\ mathbb {D}}} \ left（0 ，{F _ {\ alpha}} \ right）$表示$ \ mathbb {D} $中0和$ {F _ {\ alpha}} $之间的双曲线距离。我们研究了P. Poggi-Corradini提出的问题。之间有什么关系L，μ和$ \ mathtt {h} \ left（\ psi \ right）$？根据Kim和Sugawa的结果，$ \ mathtt {h} \ left（\ psi \ right）= {\ liminf _ {\ alpha \ to + \ infty}}（\ log {\ omega _ {\ mathbb {D }}} {\ left（0，{F _ {\ alpha}} \ right）^ {-1}} {} \ log \ alpha）$，我们显示$ \ mathtt {h} \ left（\ psi \ right ）= {\ liminf _ {\ alpha \ to + \ infty}} \ left（{d _ {\ mathbb {D}}} \ left（0，{F _ {\ alpha}} \ right）\ big / \ log \ alpha \ right）$。我们还提供了存在L和μ以及等式$ L = \ mu = \ mathtt {h} \ left（\ psi \ right）$的条件。Poggi-Corradini证明了针对各种共形映射ψ的$ \ psi \ notin {H ^ {\ mu}} \ left（\ mathbb {D} \ right）$。我们给出一个ψ的例子，其中$ \ psi \ in {H ^ {\ mu}} \ left（\ mathbb {D} \ right）$。