Weighted Bergman kernel function, admissible weights and the Ramadanov theorem

P. Wojcicki
Faculty of Mathematics and Information Science, Warsaw University of Technology
The Bergman kernel is an important tool in geometric function theory, both in one and several complex variables. It turned out that not only \regular" Bergman kernel, but also weighted one can be useful, particularly from quantum theory point of view ([3], [7]). But in general, it is difficult to say anything about the kernel of a given domain. One of the classic results is Ramadanov's theorem. Theorem. Let Let $\Omega_1\Subset\Omega_2\Subset\Omega_3\ldots$ be an increasing sequence of domains and set $\Omega\equiv\bigcup_j\Omega_j$. Then, {for all $j$,} $K_{\Omega_j}\rightarrow K_\Omega$ uniformly on compact subsets of $\Omega\times\Omega.$ Some versions of a weighted analog of this theorem are known ([4, Pr. 3.17; Th. 3.18] for instance), but considered weights are of a special form, as a modulus of holomorphic functions or $C^2$ functions, or as a product of one of those by a given weight $\psi$. In the present paper we provide a weighted analog of this theorem, for very general weights.
weighted Bergman kernel; Bergman space; Ramadanov theorem
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