Pointwise estimates for the weighted Bergman projection kernel in \({\mathbb C}^n\), using a weighted \(L^2\) estimate for the \(\bar\partial\) equation (Q1271449)

Let \(S_\varphi\) be the Bergman kernel on a domain \(\Omega\) with metric \(e^{-\varphi}d\lambda\) (\(\lambda\) Lebesgue measure; \(\varphi\) a strictly plurisubharmonic \(C^2\) function). The main result of this paper is an estimate of \(S_\varphi\) of the form \[ | S_\varphi(z,\zeta)| ^2\leq C(\varepsilon,\zeta)\det(\mu_z)e^{\varphi(z)-\varepsilon \rho_\omega(z,\zeta)}. \] \(C(\varepsilon,\zeta)\) is explicitly given in the paper, \(\mu_z\) is a constant Hermitian \((1,1)\)-form majorating the Kähler form \(i\partial\overline\partial\varphi\) in the unit ball for the \(\mu_z\) metric centred in \(z\) and \(\varepsilon\) is a parameter, \(0<\varepsilon<\sqrt 2\). To obtain this estimate the author uses an idea of Kerzman in which a solution of a \(\overline\partial\)-Neumann problem is used to find the kernel. Therefore he first proves a continuity estimate on the \(\overline\partial\)-Neumann solution operator which is of interest in itself.