Let $D_{n,m}=\left\{\right(z,w)\in C^n\times C^m:|w{|}^2<{\mathrm{e}}^{-\mu |z{|}^2}\}$ be the Fock–Bargmann–Hartogs domain, where $\mu >0$. 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 $n\ge n_0$, which verifies a conjecture posed by Yamamori. Meanwhile, we show that the Bergman projection is bounded on $L^p\left(D_{n,m}\right)$ if and only if p = 2. We also obtain the optimal rate of growth for holomorphic functions in $L^p\left(D_{n,m}\right)$.

中文翻译：

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Fock-Bargmann-Hartogs 域上的 Bergman 核和投影

让$D_{n,米}=\left\{\right(z,w)\in C^n\times C^{米}：|w{|}^2<{\mathrm{e}}^{-\mu |z{|}^2}\}$是 Fock–Bargmann–Hartogs 域，其中$\mu >0$. 对于任何固定的正整数m，我们证明存在唯一的正整数$n_0$这样$D_{n,米}$不是陆启坑域当且仅当$n\ge n_0$，这验证了 Yamamori 提出的猜想。同时，我们证明了伯格曼投影是有界的${\mathrm{大号}}^p\left(D_{n,米}\right)$当且仅当p  = 2。我们还获得了全纯函数的最佳增长率${\mathrm{大号}}^p\left(D_{n,米}\right)$.