On weighted Bergman spaces of a domain with Levi-flat boundary
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 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 in spite of the fact that the domain does not admit any non-constant bounded holomorphic functions.
Keywords
Cite
@article{arxiv.1703.08165,
title = {On weighted Bergman spaces of a domain with Levi-flat boundary},
author = {Masanori Adachi},
journal= {arXiv preprint arXiv:1703.08165},
year = {2021}
}
Comments
The title was changed from "Weighted Bergman spaces of domains with Levi-flat boundary: geodesic segments on compact Riemann surfaces". 25 pages, final version, to appear in Transactions of the American Mathematical Society