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

Delin, Henrik

Annales de l'Institut Fourier, Tome 48 (1998) no. 4, pp. 967-997.

Nous obtenons des estimations $L^2$ à poids pour la solution canonique de l’équation $\overline{\partial }$ dans $L^2({ℂ}^n,e^{-\varphi }d\lambda )$, 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 $L^2({ℂ}^n,e^{-\varphi }d\lambda )$. Le poids est utilisé pour obtenir un facteur $e^{-ϵ\rho (z,\zeta )}$ dans l’estimation du noyau, où $\rho$ est la distance associée à la métrique kählérienne définie par $i\partial \overline{\partial }\varphi$.

Weighted $L^2$ estimates are obtained for the canonical solution to the $\overline{\partial }$ equation in $L^2({ℂ}^n,e^{-\varphi }d\lambda )$, 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 $L^2({ℂ}^n,e^{-\varphi }d\lambda )$. The weight is used to obtain a factor $e^{-ϵ\rho (z,\zeta )}$ in the estimate of the kernel, where $\rho$ is the distance function in the Kähler metric given by the metric form $i\partial \overline{\partial }\varphi$.

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