Quadrature Domains for the Bergman Space in Several Complex Variables

Abstract

We make use of the Bergman kernel function to study quadrature domains whose quadrature identities hold for \(L^2\) holomorphic functions of several complex variables. We generalize some mapping properties of planar quadrature domains and point out some differences from the planar case. We then show that every smooth bounded convex domain in \({\mathbb {C}}^n\) is biholomorphic to a quadrature domain. Finally, the possibility of continuous deformations within the class of planar quadrature domains is examined.