On smoothing properties of the Bergman projection

TL;DR

This paper investigates the smoothing properties of the Bergman projection and weighted variants on pseudoconvex domains, linking these properties to the hyperconvexity index and providing improved estimates with various applications.

Contribution

It introduces new connections between smoothing properties and the hyperconvexity index, and offers an improved estimate for weighted Bergman projections with multiple applications.

Findings

Established a relation between smoothing properties and hyperconvexity index.

Derived a new, sharper estimate for weighted Bergman projections.

Applied the estimate to study smoothing properties in various contexts.

Abstract

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.