Boundary behavior of the Bergman kernel function on pseudoconvex domains (Q1061281)

Let D be a bounded pseudoconvex domain in \({\mathbb{C}}^ n\) and let \(K_ D(z,\bar w)\) be the Bergman kernel of D. Using a localization principle and a \(L^ 2\) extension technique, the following result is obtained. Theorem: If \(\partial D\) is \(C^{\infty}\)-smooth and the maximal corank of the Levi-form of \(\partial D\) is k, then for any \(\epsilon >0\) there exists a constant \(C>0\) such that \((*) K_ D(z,\bar z)\geq Cd_ D(z)^{-n-1+k+\epsilon},\) where \(d_ D(z)=\inf_{x\in \partial D}\| z-x\|.\) Recently the \(L^ 2\) extension technique was improved by the author and \textit{Takegoshi} [''An extension theorem for \(L^ 2\) holomorphic functions'' (to appear)] and it turned out that one needs no \(\epsilon\) in (*).