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On estimates for weighted Bergman projections


Authors: P. Charpentier, Y. Dupain and M. Mounkaila
Journal: Proc. Amer. Math. Soc. 143 (2015), 5337-5352
MSC (2010): Primary 32T25, 32T27
DOI: https://doi.org/10.1090/proc/12660
Published electronically: June 18, 2015
MathSciNet review: 3411150
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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$).


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P. Charpentier
Affiliation: Institut de Mathématiques de Bordeaux, Université Bordeaux I, 351, Cours de la Libération, 33405, Talence, France
Email: philippe.charpentier@math.u-bordeaux1.fr

Y. Dupain
Affiliation: Institut de Mathématiques de Bordeaux, Université Bordeaux I, 351, Cours de la Libération, 33405, Talence, France

M. Mounkaila
Affiliation: Faculté des Sciences, Université Abdou Moumouni, B.P. 10662, Niamey, Niger
Email: modi.mounkaila@yahoo.fr

DOI: https://doi.org/10.1090/proc/12660
Keywords: Pseudo-convex, finite type, Levi form locally diagonalizable, weighted Bergman projection, $\overline{\partial}$-Neumann problem
Received by editor(s): February 14, 2014
Received by editor(s) in revised form: August 21, 2014, and October 6, 2014
Published electronically: June 18, 2015
Communicated by: Franc Forstneric
Article copyright: © Copyright 2015 American Mathematical Society

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