Bergman subspaces and subkernels: Degenerate $L^p$ mapping and zeroes

Regularity and irregularity of the Bergman projection on $L^p$ spaces is established on a natural family of bounded, pseudoconvex domains. The family is parameterized by a real variable $\gamma$. A surprising consequence of the analysis is that, whenever $\gamma$ is irrational, the Bergman projection is bounded only for $p=2$.