Bergman Spaces Under Maps of Monomial Type

Abstract

For appropriate domains $\Omega _{1}, \Omega _{2}$, we consider mappings $\Phi _{\mathbf {A}}:\Omega _{1}\rightarrow \Omega _{2}$ of monomial type. We obtain an orthogonal decomposition of the Bergman space ${\mathcal {A}}^{2}(\Omega _{1})$ into finitely many closed subspaces indexed by characters of a finite abelian group associated to the mapping $\Phi _{\mathbf {A}}$. We then show that each subspace is isomorphic to a weighted Bergman space on $\Omega _{2}$. This leads to a formula for the Bergman kernel on $\Omega _{1}$ as a sum of weighted Bergman kernels on $\Omega _{2}$.