The Fock-Bargmann-Hartogs domain $D_{ n, m}(\mu)$ is defined by $$ D_{ n, m}(\mu):=\{ (z, w)\in\mathbb{ C}^{ n}\times\mathbb{ C}^m:\Vert w \Vert^2<e^{ -\mu\Vert z \Vert^2}\},$$ 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 $\rho(z,w):=\|w\|^2-e^{ -\mu \|z\|^2}$ is a defining function for $D_{ n, m}(\mu)$ and $\alpha>-1$. 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 $p\in [1,\infty)\setminus\{ 2\}$. This result turns out to be completely different from the well-known positive $L^p$ regularity result on bounded strongly pseudoconvex domain.
Categories: math.CV