The Fock–Bargmann–Hartogs domain
$D_{n,m}(\,\unicode[STIX]{x1D707}):=\{(z,w)\in \mathbb{C}^{n}\times \mathbb{C}^{m}:\Vert w\Vert ^{2}<e^{-\unicode[STIX]{x1D707}\Vert z\Vert ^{2}}\}$
, where
$\unicode[STIX]{x1D707}>0$
, is an unbounded strongly pseudoconvex domain with smooth real-analytic boundary. We compute the weighted Bergman kernel of
$D_{n,m}(\,\unicode[STIX]{x1D707})$
with respect to the weight
$(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}}$
, where
$\unicode[STIX]{x1D70C}(z,w):=\Vert w\Vert ^{2}-e^{-\unicode[STIX]{x1D707}\Vert z\Vert ^{2}}$
and
$\unicode[STIX]{x1D6FC}>-1$
. Then, for
$p\in [1,\infty ),$
we show that the corresponding weighted Bergman projection
$P_{D_{n,m}(\,\unicode[STIX]{x1D707}),(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}}}$
is unbounded on
$L^{p}(D_{n,m}(\,\unicode[STIX]{x1D707}),(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}})$
, except for the trivial case
$p=2$
. This gives an example of an unbounded strongly pseudoconvex domain whose ordinary Bergman projection is
$L^{p}$
irregular when
$p\in [1,\infty )\setminus \{2\}$
, in contrast to the well-known positive
$L^{p}$
regularity result on a bounded strongly pseudoconvex domain.