Quadrature domains for the Bergman space in several complex variables (Q1653349)

The author uses the Bergman kernel function to study quadrature domains whose quadrature identities hold for \(L^{2}\) holomorphic functions in several complex variables. Based on his doctoral dissertation at Purdue University under the guidance of Steve Bell, who is a leading expert in the field of quadrature domains, this article unifies earlier work and generalizes it to several complex variables. The article is organised in six sections. After a satisfying introduction, the author proceeds with product domains, mapping properties, multidimensional counterexamples, quadrature domains, density and deformation of quadrature domains. Mainly, the author generalizes some mapping properties of planar quadrature domains and points out some differences to the planar case. He also shows that every bounded convex domain in \(\mathbb{C}^{n}\) is biholomorphic to a quadrature domain. Finally, he examines the possibility of continuous deformations within the class of planar quadrature domains.