On the Bergman projection and kernel in periodic planar domains

TL;DR

This paper investigates Bergman kernels and projections in unbounded periodic planar domains, introducing a Floquet transform approach and deriving new formulas and estimates for these operators.

Contribution

It develops a Floquet transform technique for Bergman spaces in periodic domains and connects kernels to bounded domain projections, extending classical formulas.

Findings

Derived a formula relating $P_\Pi$ to bounded domain projections

Established boundedness properties of Floquet transform in Bergman spaces

Obtained weighted $L^p$ estimates for Bergman projections in periodic domains

Abstract

We study Bergman kernels $K_\Pi$ and projections $P_\Pi$ in unbounded planar domains $\Pi$, which are periodic in one dimension. In the case $\Pi$ is simply connected we write the kernel $K_\Pi$ in terms of a Riemann mapping $\varphi$ related to the bounded periodic cell $\varpi$ of the domain $\Pi$. We also introduce and adapt to the Bergman space setting the Floquet transform technique, which is a standard tool for elliptic spectral problems in periodic domains. We investigate the boundedness properties of the Floquet transform operators in Bergman spaces and derive a general formula connecting $P_\Pi$ to a projection on a bounded domain. We show how this theory can be used to reproduce the above kernel formula for $K_\Pi$. Finally, we consider weighted $L^p$-estimates for $P_\Pi$ in periodic domains.