Proper holomorphic mappings, Bell’s formula, and the Lu Qi-Keng problem on the tetrablock

Abstract

We consider proper holomorphic maps ${\pi : D\rightarrow G}$, where D and G are domains in ${\mathbb{C}^{n}}$. Let ${\alpha\in \mathcal{C}(G,\mathbb{R}_{ > 0})}$. We show that every π induces some subspace H of ${\mathbb{A}^{2}_{\alpha\circ\pi}(D)}$ such that ${\mathbb{A}^{2}_{\alpha}(G)}$ is isometrically isomorphic to H via some unitary operator Γ. Using this isomorphism we construct the orthogonal projection onto H, and we derive Bell’s transformation formula for the weighted Bergman kernel function under proper holomorphic mappings. As a consequence of the formula, we get that the tetrablock is not a Lu Qi-Keng domain.