On estimates for weighted Bergman projections

TL;DR

This paper extends weighted Sobolev estimates for Bergman projections from specific measures to more general measures on smoothly bounded pseudoconvex domains, also establishing stronger directional estimates and generalizations for various domains.

Contribution

It generalizes existing weighted Sobolev estimates for Bergman projections to broader classes of measures and domains, including stronger directional estimates and extensions to $L^{p}$-Sobolev and Lipschitz estimates.

Findings

Weighted Sobolev estimates hold for more general measures.

Established stronger directional Sobolev estimates.

Extended results to $L^{p}$-Sobolev and Lipschitz estimates for certain domains.

Abstract

In this note we show that the weighted $L^{2}$-Sobolev estimates obtained by P. Charpentier, Y. Dupain & M. Mounkaila for the weighted Bergman projection of the Hilbert space $L^{2}\left(\Omega,d\mu_{0}\right)$ where $\Omega$ is a smoothly bounded pseudoconvex domain of finite type in $\mathbb{C}^{n}$ and $\mu_{0}=\left(-\rho_{0}\right)^{r}d\lambda$, $\lambda$ being the Lebesgue measure, $r\in\mathbb{Q}_{+}$ and $\rho_{0}$ a special defining function of $\Omega$, are still valid for the Bergman projection of $L^{2}\left(\Omega,d\mu\right)$ where $\mu=\left(-\rho\right)^{r}d\lambda$, $\rho$ being any defining function of $\Omega$. In fact a stronger directional Sobolev estimate is established. Moreover similar generalizations are obtained for weighted $L^{p}$-Sobolev and lipschitz estimates in the case of pseudoconvex domain of finite type in $\mathbb{C}^{2}$ and for some convex domains of finite type.