On weighted Bergman spaces of a domain with Levi-flat boundary II
On weighted Bergman spaces of a domain with Levi-flat boundary II
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具有列维平坦边界 II 的域的加权 Bergman 空间
DOI：
10.1007/s40627-022-00097-0
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
Masanori Adachi
中科院分区：
文献类型：
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作者：
Ryuichi Sato;Masanori Adachi
研究了双曲紧致黎曼曲面的极大Grauert管上的全纯函数。证明了它们的加权Bergman空间对于大于的任意权阶是无穷维的，尽管它们不允许任何非常数有界全纯函数.证明的关键成分是计算加权范数的解析延拓的特征函数的拉普拉斯超几何函数。这个结果补充了我们以前的工作（阿达奇，Trans Am Math Soc 374（10）：7499-7524，2021），其中证明了黎曼曲面上的测地线段的空间具有完全相同的性质。
Holomorphic functions on the maximal Grauert tube of a hyperbolic compact Riemann surface are studied. It is shown that their weighted Bergman spaces are infinite dimensional for arbitrary weight order greater thanin spite of the fact that they do not admit any non-constant bounded holomorphic functions. The key ingredient of the proof is a computation of weighted norms of analytic continuations of eigenfunctions of the Laplacian in terms of the hypergeometric function. This result complements our previous work (Adachi in Trans Am Math Soc 374(10):7499–7524, 2021) where it was shown that the space of geodesic segments on the Riemann surface has exactly the same property.