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.

The paper is devoted to the study of polynomially convex hulls of compact subsets of ℂ², fibered over the boundary of the unit disc, such that all fibers are simple arcs in the plane and their endpoints form boundaries of two closed, not intersecting analytic discs. The principal question concerned is under what additional condition such a hull is a bordered topological hypersurface and, in particular, is foliated by a unique holomorphic motion. One of the main results asserts that this happens...