Duality and approximation of Bergman spaces

TL;DR

This paper investigates the duality and approximation properties of Bergman spaces, showing their failure in general but establishing positive results under certain conditions, with applications to Hartogs triangles and Reinhardt domains.

Contribution

It demonstrates the failure of duality and approximation in Bergman spaces and constructs operators to recover these properties on specific domains.

Findings

Duality and approximation fail on general Bergman spaces.

Constructed operators restore duality and approximation on Hartogs triangles.

Laurent series of Bergman functions converge on bounded Reinhardt domains.

Abstract

Expected duality and approximation properties are shown to fail on Bergman spaces of domains in $\mathbb{C}^n$, via examples. When the domain admits an operator satisfying certain mapping properties, positive duality and approximation results are proved. Such operators are constructed on generalized Hartogs triangles. On a general bounded Reinhardt domain, norm convergence of Laurent series of Bergman functions is shown. This extends a classical result on Hardy spaces of the unit disc.