Pointwise estimates for the weighted Bergman projection kernel in ${\mathbb {C}}^n$, using a weighted $L^2$ estimate for the $\bar{\partial }$ equation

Delin, Henrik

Annales de l'Institut Fourier, Tome 48 (1998), p. 967-997 / Harvested from Numdam

Résumé

Nous obtenons des estimations $L^2$ à poids pour la solution canonique de l’équation $\bar{\partial }$ dans $L^2({\Bbb C}^n,e^{-\varphi }d\lambda )$, où $\Omega $ est un domaine pseudoconvexe et $\varphi $ une fonction strictement plurisousharmonique. Ces estimations sont ensuite utilisées pour démontrer des estimations ponctuelles pour le noyau du projecteur de Bergman dans $L^2({\Bbb C}^n,e^{-\varphi }d\lambda )$. Le poids est utilisé pour obtenir un facteur $e^{-\epsilon \rho (z,\zeta )}$ dans l’estimation du noyau, où $\rho $ est la distance associée à la métrique kählérienne définie par $i\partial \bar{\partial }\varphi $.

Abstract

Weighted $L^2$ estimates are obtained for the canonical solution to the $\bar{\partial }$ equation in $L^2({\Bbb C}^n,e^{-\varphi }d\lambda )$, where $\Omega $ is a pseudoconvex domain, and $\varphi $ is a strictly plurisubharmonic function. These estimates are then used to prove pointwise estimates for the Bergman projection kernel in $L^2({\Bbb C}^n,e^{-\varphi }d\lambda )$. The weight is used to obtain a factor $e^{-\epsilon \rho (z,\zeta ) }$ in the estimate of the kernel, where $\rho $ is the distance function in the Kähler metric given by the metric form $i\partial \bar{\partial }\varphi $.

Détail

Publié le : 1998-01-01

MR 99j:32027 | Zbl 0918.32007

DOI : https://doi.org/10.5802/aif.1645

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