论文标题
飞机域的伯格曼人数
The Bergman number of a plane domain
论文作者
论文摘要
令$ d $为复杂平面$ \ mathbb {c} $中的一个域。汉森首先引入的$ d $的hardy数字是$ [0，+\ fy infty] $的最大数量$ h（d）$，因此每当$ 0 <p <h（d）$ f $ in nime in nime in dieveach in Anth y nime in dis nit nime in dions of holomorphic in d in nime， $ D $。作为对$ \ Mathbb {c} $ in tomain $ d $ hardy数量的类似概念，我们介绍了$ d $的伯格曼号码，我们用$ b（d）$表示它。我们的主要结果是，如果$ d $是常规的，则$ h（d）= b（d）$。这概括了作者和卡拉曼利斯的早期工作，简单地连接域。伯格曼数字$ b（d）$是$ [0，+\ infty] $中的最大数字，因此每当$ f $属于加权的伯格曼空间$ a^p_α（\ mathbb {d}）$，每当$ p> 0 $ p> 0 $ p> $ p> $ p> 0 $ and $α> -1 $ -1 $ $ 0 $ 0 <\ frac $ $ 0 <\\ frac $ $ $ $ 2} $ \ mathbb {d} $带有$ d $中的值。我们还建立了有关耐力空间和加权伯格曼空间的几个结果，并给出了耐力数量的新特征，从而给出了相对于谐波度量的常规域的伯格曼数量。
Let $D$ be a domain in the complex plane $\mathbb{C}$. The Hardy number of $D$, which first introduced by Hansen, is the maximal number $h(D)$ in $[0,+\infty]$ such that $f$ belongs to the classical Hardy space $H^p (\mathbb{D})$ whenever $0<p<h(D)$ and $f$ is holomorphic on the unit disk $\mathbb{D}$ with values in $D$. As an analogue notion to the Hardy number of a domain $D$ in $\mathbb{C}$, we introduce the Bergman number of $D$ and we denote it by $b(D)$. Our main result is that, if $D$ is regular, then $h(D)=b(D)$. This generalizes earlier work by the author and Karamanlis for simply connected domains. The Bergman number $b(D)$ is the maximal number in $[0,+\infty]$ such that $f$ belongs to the weighted Bergman space $A^p_α (\mathbb{D})$ whenever $p>0$ and $α>-1$ satisfy $0<\frac{p}{α+2}<b(D)$ and $f$ is holomorphic on $\mathbb{D}$ with values in $D$. We also establish several results about Hardy spaces and weighted Bergman spaces and we give a new characterization of the Hardy number and thus of the Bergman number of a regular domain with respect to the harmonic measure.
