Estimates for some Weighted Bergman Projections

Philippe Charpentier (1) , Yves Dupain (1) , Modi Mounkaila, (2)

1 IMB - Institut de Mathématiques de Bordeaux
2 Chercheur indépendant

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 $\rho$ 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.

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

Domaines: Variables complexes [math.CV]

Philippe Charpentier, Yves Dupain, Modi Mounkaila,. Estimates for some Weighted Bergman Projections. 2012. ⟨hal-00761375v2⟩