The Bergman kernel on uniformly extendable pseudoconvex domains
The authors consider the following conjecture of \textit{J. J. Kohn} [J. Differ. Geom. 6, 523-542 (1972; Zbl 0256.35060)]: Let \(\Omega\) be a bounded weakly pseudoconvex domain in \({\mathbb{C}}^ n\) with real-analytic smooth boundaries, and let \(d_{\Omega}(z)\) denote the distance from a point \(z\in \Omega\) to \(\partial \Omega\). Suppose the Levi-form of \(\partial \Omega\) at the point \(z_ 0\in \partial \Omega\) has exactly k positive eigenvalues. Then the Bergman kernel \(K_{\Omega}(z)=K_{\Omega}(z,z)\) grows near \(z_ 0\) at least like \(K_{\Omega}(z)\geq C d_{\Omega}^{-(k+2+\epsilon)}(z),\) where \(\epsilon =0\) if \(k=n-1\) and \(\epsilon >0\) if \(k<n-1.\) This conjecture has been verified by \textit{D. Catlin} [Several complex variables, Proc. 1981 Hangzhou Conf., 7-12 (1984)] for \(n=2\). For \(n>2\), the second author of this paper showed in Manuscr. Math. 45, 69-76 (1983; Zbl 0559.32006) that the principal part of the singularity of \(K_{\Omega}\) does not have, in general, a rational order and it may contain logarithmic factors. Also, the last author of this paper [Publ. Res. Inst. Math. Sci. 20, 897-902 (1984; Zbl 0569.32013)] has proved the above conjecture for all pseudoconvex \(C^{\infty}\)-smooth domains in \({\mathbb{C}}^ n\), but with \(\epsilon\) replaced by -\(\epsilon\). In this paper, the authors verify this conjecture for a class of bounded pseudoconvex domains in \({\mathbb{C}}^ n\), the so-called ''uniformly extendable pseudo-convex domains'', which contains, in particular, all \(\Omega\) with real-analytic smooth boundaries.
scientific article; zbMATH DE number 3931398
scientific article