Proper holomorphic mappings, Bells formula and the Lu Qi-Keng problem on tetrablock

TL;DR

This paper investigates proper holomorphic mappings between domains in complex space, deriving a transformation formula for Bergman kernels, and concludes that the tetrablock domain does not possess the Lu Qi-Keng property.

Contribution

It establishes a Bells transformation formula for Bergman kernels under proper holomorphic maps and applies it to analyze the Lu Qi-Keng problem on the tetrablock.

Findings

Derived Bells formula for Bergman kernels under proper mappings

Constructed orthogonal projections related to these mappings

Proved the tetrablock is not a Lu Qi-Keng domain

Abstract

We consider a proper holomorphic map form D to G domains in C^n and show that it induces a unitary isomorphism between the Bergman space A^2(G) and some subspace of A^2(D). Using this isomorphism we construct orthogonal projection onto that subspace and we derive Bells transformation formula for the Bergman kernel under proper holomorphic mappings. As a consequence of the formula we get that the tetrablock is not a Lu Qi-Keng domain.