Weighted Bergman kernel function, admissible weights and the Ramadanov theorem 

Author 
p.wojcicki@mini.pw.edu.pl
Faculty of Mathematics and Information Science,
Warsaw University of Technology

Abstract 
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.

Keywords 
weighted Bergman kernel; Bergman space; Ramadanov theorem

Reference 
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Pages 
160164

Volume 
42

Issue 
2

Year 
2014

Journal 
Matematychni Studii

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