AI 中文总结
研究刻画一般区域伯格曼数的方法，通过在增长空间定义增长数，证明伯格曼数与增长数的等式，给出特定区域示例解决前人问题，还将类似思路用于布洛赫型空间。
AI 中文摘要
在本文中，我们完全刻画了一般区域的伯格曼数。即证明了双曲无界区域的伯格曼数可根据其双曲度量在无穷远处的渐近行为来计算。为得到此结果，我们先在增长空间的设定下定义区域的增长数，接着证明了一个将伯格曼数与区域增长数相关联的等式。我们还给出了具有规定伯格曼数且哈代数为零的区域示例，解决了贝萨科斯和克鲁兹 - 萨莫拉诺之前提出的一个问题。最后，对布洛赫型空间的情况也采用了类似思路。
英文摘要
In this article we completely characterize the Bergman number of general domains. Namely, we prove that the Bergman number of a hyperbolic unbounded domain can be calculated in terms of the asymptotic behavior of its hyperbolic metric near infinity. To obtain the proof of this result we first work in the setting of growth spaces, defining the growth number of a domain. Later, we prove an equality relating the Bergman number and the growth number of domains. We also provide examples of domains with prescribed Bergman number which have zero Hardy number, solving a question posed previously by Betsakos and Cruz-Zamorano. At the end, a similar idea is treated for the case of Bloch-type spaces.
CommentsTo appear in Journal of Mathematical Analysis and Applications
