Analytic continuation of weighted Bergman kernels (Q616304)

The main result states that, on a bounded, smooth, strictly pseudoconvex domain, the reproducing kernel function for the space of holomorphic functions that are square integrable with respect to the weight \(|\rho|^{\alpha}\), where \(\rho\)~is a defining function for the domain and \(\alpha>-1\), admits an extension to a meromorphic function of~\(\alpha\) in the whole complex plane. The author's theory handles even somewhat more general weights. When \(\alpha\)~is outside the set of poles, the author determines the singularity of the corresponding kernel function at the boundary diagonal, thereby obtaining a new interpretation of the ``local Sobolev-Bergman kernels'' of \textit{K.~Hirachi} and \textit{G.~Komatsu} [Trends in Mathematics. 63--96 (1999; Zbl 0951.32002)].