Weighted Bergman spaces of domains with Levi-flat boundary: geodesic segments on compact Riemann surfaces

Masanori Adachi

Published 2017-03-23Version 1

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 the case of the space of all the geodesic segments on a hyperbolic compact Riemann surface, which is a typical example of such a domain in the sense that its realization as a holomorphic disk bundle has the best possible Diederich-Fornaess index $1/2$. Our main finding is an integral formula that produces holomorphic functions on the domain from holomorphic differentials on the base Riemann surface via optimal $L^2$-jet extension, and, in particular, it is shown that the weighted Bergman spaces of the domain is infinite dimensional for all the order greater than $-1$ beyond $-1/2$, the limiting order until which known $L^2$-estimates for the $\overline{\partial}$-equation work. Some applications are also given thanks to the generalized hypergeometric function $_3F_2$ expressing the norm of the optimal $L^2$-jet extension: a proof for the Liouvilleness which does not appeal to the ergodicity of the Levi foliation, and a Forelli-Rudin construction for the disk bundle over the Riemann surface.