Toeplitz operators on Bergman spaces of polygonal domains

TL;DR

This paper investigates the boundedness of Toeplitz operators with locally integrable symbols on Bergman spaces over polygonal domains, providing new criteria and examples even when the Bergman projection is unbounded.

Contribution

It introduces sufficient conditions for Toeplitz operator boundedness on Bergman spaces of polygonal domains using Whitney decomposition and average-based criteria.

Findings

Established criteria for boundedness of Toeplitz operators on polygonal domains.

Provided examples of bounded Toeplitz operators despite unbounded Bergman projections.

Utilized Whitney decomposition to analyze operator boundedness.

Abstract

We study the boundedness of Toeplitz operators with locally integrable symbols on Bergman spaces $A^p(\Omega),$ $1<p<\infty,$ where $\Omega\subset \mathbb{C}$ is a bounded simply connected domain with polygonal boundary. We give sufficient conditions for the boundedness of generalized Toeplitz operators in terms of "averages" of symbol over certain Cartesian squares. We use the Whitney decomposition of $\Omega$ in the proof. We also give examples of bounded Toeplitz operators on $A^p(\Omega)$ in the case where polygon $\Omega$ has such a large corner that the Bergman projection is unbounded.