# Estimates for some Weighted Bergman Projections

1 IMB - Institut de Mathématiques de Bordeaux 2 Chercheur Indépendant

Abstract : 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 $ho$ 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.

Keywords : pseudo-convex finite type Levi form locally diagonalizable convex extremal basis geometric separation weighted Bergman projection $\overline{\partial} {\varphi}$-Neumann problem}

Author: Philippe Charpentier - Yves Dupain - Modi Mounkaila, -

Source: https://hal.archives-ouvertes.fr/