Advanced Search

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 $i\partial \bar{\partial }\phi$.