On estimates for weighted Bergman projections

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 $L^{2}left(Omega,dmu_{0}right)$ where $Omega$ is a smoothly bounded pseudoconvex domain of finite type in $mathbb{C}^{n}$ and $mu_{0}=left(-rho_{0}right)^{r}dlambda$, $lambda$ being the Lebesgue measure, $rinmathbb{Q}_{+}$ and $rho_{0}$ a special defining function of $Omega$, are still valid for the Bergman projection of $L^{2}left(Omega,dmuright)$ where $mu=left(-rhoright)^{r}dlambda$, $rho$ 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.