Abstract:The aim of this study is to understand to what extent a 1-convex domain with Levi-flat boundary is capable of holomorphic functions with slow growth. This paper discusses a typical example of such domain, the space of all the geodesic segments on a hyperbolic compact Riemann surface. Our main finding is an integral formula that produces holomorphic functions on the domain from holomorphic differentials on the Riemann surface. This construction can be seen as a non-trivial example of $L^2$ jet extension of holomorphic functions with optimal constant. As its corollary, it is shown that the weighted Bergman spaces of the domain is infinite dimensional for any weight order greater than $-1$ in spite of the fact that the domain does not admit any non-constant bounded holomorphic functions.

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中文翻译：

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具有 Levi-flat 边界的域的加权 Bergman 空间

摘要：本研究的目的是了解具有 Levi-flat 边界的 1-凸域在多大程度上能够具有缓慢增长的全纯函数。本文讨论了此类域的一个典型示例，即双曲紧致黎曼曲面上所有测地线段的空间。我们的主要发现是一个积分公式，它从黎曼曲面上的全纯微分在域上产生全纯函数。这种构造可以看作是具有最优常数的全纯函数的 $L^2$ 射流扩展的非平凡示例。作为其推论，尽管域不允许任何非常量有界全纯函数，但对于任何大于 $-1$ 的权重阶，域的加权伯格曼空间都是无限维的。

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