Weighted $L^2$ estimates are obtained for the canonical solution to the $\bar{\partial }$ equation in $L^2({ℂ}^n,e^{-\phi }d\lambda )$, where $\Omega$ is a pseudoconvex domain, and $\phi$ is a strictly plurisubharmonic function. These estimates are then used to prove pointwise estimates for the Bergman projection kernel in $L^2({ℂ}^n,e^{-\phi }d\lambda )$. The weight is used to obtain a factor $e^{-ϵ\rho (z,\zeta )}$ in the estimate of the kernel, where $\rho$ is the distance function in the Kähler metric given by the metric form $i\partial \bar{\partial }\phi$.

We consider a large class of convex circular domains in $M_{m₁,n₁}\left(ℂ\right)\times ...\times M_{m_d,n_d}\left(ℂ\right)$ which contains the oval domains and minimal balls. We compute their Bergman and Szegő kernels. Our approach relies on the analysis of some proper holomorphic liftings of our domains to some suitable manifolds.