# On weighted Bergman spaces of a domain with Levi-flat boundary

## TL;DR

This paper explores the properties of weighted Bergman spaces on a specific 1-convex domain with Levi-flat boundary, revealing infinite dimensionality despite the absence of non-constant bounded holomorphic functions, through an integral formula linking holomorphic differentials.

## Contribution

It introduces an integral formula for constructing holomorphic functions on the domain from differentials, demonstrating an $L^2$ jet extension with optimal constant and showing the infinite dimensionality of weighted Bergman spaces.

## Key findings

- Weighted Bergman spaces are infinite dimensional for weights > -1.
- Constructs holomorphic functions via an integral formula from differentials.
- Shows the domain admits no non-constant bounded holomorphic functions.

## 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.