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Estimates for some Weighted Bergman Projections

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 $ρ$ 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.