  We study smoothing properties of the Bergman projection and also of weighted
Bergman projections. In particular, we relate these properties to the
hyperconvexity index of a pseudoconvex domain in $\mathbb{C}^{n}$. The notion
of a hyperconvexity index was first introduced by B.Y. Chen, which provides a
flexible criterion for studying geometric properties of hyperconvex domains. We
also obtain a new estimate of weighted Bergman projections, which improves a
well-known estimate of Berndtsson and Charpentier. We give several applications
of this estimate, including the study of smoothing properties of weighted
Bergman projections.
