  We establish a weighted $L^p$ norm estimate for the Bergman projection for a
class of pseudoconvex domains. We obtain an upper bound for the weighted $L^p$
norm when the domain is, for example, a bounded smooth strictly pseudoconvex
domain, a pseudoconvex domain of finite type in $\mathbb C^2$, a convex domain
of finite type in $\mathbb C^n$, or a decoupled domain of finite type in
$\mathbb C^n$. The upper bound is related to the Bekoll\'e-Bonami constant and
is sharp. When the domain is smooth, bounded, and strictly pseudoconvex, we
also obtain a lower bound for the weighted norm.
