On estimates for weighted Bergman projections

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.