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

In this paper we obtain several characterizations of the pointwise multipliers of the space $BMOA$ in the unit ball $B$ of ${ℂ}^n$. Moreover, if $g_1,...,g_m$ are holomorphic functions on $B$, we prove that $M_g\left(f\right)\left(z\right)=\sum _{j=1}^m{g}_j\left(z\right)f_j\left(z\right)$ maps $BMOA\times ...\times BMOA$ onto $BMOA$ if and only if the functions $g_j$ are multipliers of the space $BMOA$ and satisfy $\sum _{j=1}^m\left|g_j\left(z\right)\right|\ge \delta \>0.$

A family of holomorphic function spaces can be defined with reproducing kernels $B_{\alpha }\left(z,w\right)$, obtained as real powers of the Cauchy-Szegö kernel. In this paper we study properties of the associated Poisson-like kernels: $P_{\alpha }\left(z,w\right)={\left|B_{\alpha }\left(z,w\right)\right|}^2/B_{\alpha }\left(z,z\right)$. In particular, we show boundedness of associated maximal operators, and obtain formulas for the limit of Poisson integrals in the topological boundary of the cone.