Bergman spaces under maps of monomial type

Abstract

For appropriate domains ${\Omega }_1,{\Omega }_2$ we consider mappings ${\Phi }_{\mathbf{A}}:{\Omega }_1\to {\Omega }_2$ 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 ${\Omega }_2$

26 pages; Added references