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Bergman spaces under maps of monomial type

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