On \(p\)-Bergman kernel for bounded domains in \(\mathbb{C}^n\)
The \(p\)-Bergman kernel in bounded open subsets of \(\mathbb C ^n\) is studied. Its definition is obtained from the one of the usual Bergman kernel (for \(p=2\)) by replacing the exponent \(2\) with \(p\). The authors prove some of the properties applying the \(L^p\) generalization of the Ohsawa-Takegoshi extension theorem. In particular they show that a bounded domain in \(\mathbb C ^n\) is pseudoconvex iff its \(p\)-Bergman kernel is an exhaustion function for all \(p\in (0,2)\). They also disprove a conjecture of H. Tsuji concerning the asymptotic behaviour of the Bergman kernels related to sections of \(m\) multiples of the canonical bundle of the punctured disc.
scientific article; zbMATH DE number 6674138