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

Let D n , m = { (z , w) ∈ C n × C m : | w | 2 < e − μ | z | 2 } be the Fock–Bargmann–Hartogs domain, where μ > 0. For any fixed positive integer m, we prove that there exists a unique positive integer n 0 such that D n , m is not a Lu Qi-Keng domain if and only if n ≥ n 0 , which verifies a conjecture posed by Yamamori. Meanwhile, we show that the Bergman projection is bounded on L p (D n , m) if and only if p = 2. We also obtain the optimal rate of growth for holomorphic functions in L p (D n , m). [ABSTRACT FROM AUTHOR]