$L^p$ regularity of weighted Bergman projection on Fock-Bargmann-Hartogs domain

The Fock-Bargmann-Hartogs domain $D_{n,m}(\mu )$ is defined by $$D_{n,m}(\mu ):=\{(z,w)\in {\mathbb{C}}^n\times {\mathbb{C}}^m:\mathrm{\parallel }w{\mathrm{\parallel }}^2<e^{-\mu \mathrm{\parallel }z{\mathrm{\parallel }}^2}\},$$ where $\mu >0.$ The Fock-Bargmann-Hartogs domain $D_{n,m}(\mu )$ is an unbounded strongly pseudoconvex domain with smooth real-analytic boundary. In this paper, we first compute the weighted Bergman kernel of $D_{n,m}(\mu )$ with respect to the weight $(-\rho {)}^{\alpha }$, where $\rho (z,w):=\mathrm{\parallel }w{\mathrm{\parallel }}^2-e^{-\mu \mathrm{\parallel }z{\mathrm{\parallel }}^2}$ is a defining function for $D_{n, m}(\mu)$and$\alpha>-1$\mathrm{.}Then,for$p\in [1,\infty),$ we show that the corresponding weighted Bergman projection $P_{D_{n,m}(\mu ),(-\rho {)}^{\alpha }}$ is unbounded on $L^p(D_{n,m}(\mu ),(-\rho {)}^{\alpha })$, except for the trivial case $p=2$. In particular, this paper gives an example of an unbounded strongly pseudoconvex domain whose ordinary Bergman projection is $L^p$ irregular when $p\in [1,\mathrm{\infty })\setminus \{2\}$. This result turns out to be completely different from the well-known positive $L^p$ regularity result on bounded strongly pseudoconvex domain.