The Bergman kernel and projection on the Fock–Bargmann–Hartogs domain

Let Dn,m={(z,w)∈Cn×Cm:|w|20. For any fixed positive integer m, we prove that there exists a unique positive integer n0 such that Dn,m is not a Lu Qi-Keng domain if and only if n≥n0, which verifies a conjecture posed by Yamamori. Meanwhile, we show that the Bergman projection is bounded on Lp(Dn,m) if and only if p = 2. We also obtain the optimal rate of growth for holomorphic functions in Lp(Dn,m).