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 $\mu _{0}=\left (-\rho _{0}\right )^{r}d\lambda$, with $\lambda$ 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 $\mu =\left (-\rho \right )^{r}d\lambda$, with $\rho$ any defining function of $\Omega$ and $r\in \mathbb {R}_{+}$. In fact a stronger directional Sobolev estimate is established. Moreover similar generalizations (for $r\in \mathbb {Q}_{+}$) are obtained for weighted $L^{p}$-Sobolev and Lipschitz estimates in the case of the pseudoconvex domain of finite type in $\mathbb {C}^{2}$ (or, more generally, when the rank of the Levi form is $\geq n-2$).