Estimates for some Weighted Bergman Projections

TL;DR

This paper studies the regularity of weighted Bergman projections on certain complex domains, providing new Sobolev and Lipschitz estimates for weights based on domain defining functions, extending previous results.

Contribution

It introduces new regularity estimates for weighted Bergman projections on specific complex domains, generalizing prior work with novel weighted Sobolev and Lipschitz bounds.

Findings

Weighted Sobolev-$L^{p}$ estimates for domains in $ ext{C}^2$

Lipschitz estimates for convex and decoupled domains

General weighted Sobolev-$L^{2}$ estimate

Abstract

In this paper we investigate the regularity properties of weighted Bergman projections for smoothly bounded pseudo-convex domains of finite type in $\mathbb{C}^{n}$. The main result is obtained for weights equal to a non negative rational power of the absolute value of a special defining function $\rho$ of the domain: we prove (weighted) Sobolev-$L^{p}$ and Lipchitz estimates for domains in $\mathbb{C}^{2}$ (or, more generally, for domains having a Levi form of rank $\geq n-2$ and for "decoupled" domains) and for convex domains. In particular, for these defining functions, we generalize results obtained by A. Bonami & S. Grellier and D. C. Chang & B. Q. Li. We also obtain a general (weighted) Sobolev-$L^{2}$ estimate.