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Keywords:
reproducing kernel Banach space; $p$-Bergman kernel; admissible weight; Ramadanov theorem
reproducing kernel Banach space; $p$-Bergman kernel; admissible weight; Ramadanov theorem
Summary:
We consider $p$-Bergman kernels, i.e., a generalization of the classical Bergman kernel for Banach spaces of integrable in $p$th power and holomorphic functions. This is done by the minimal norm property of a classical reproducing kernel. We show a sufficient condition which the weight of integration must satisfy in order, for the corresponding Banach space with weighted norm, to have $p$-Bergman kernel. Then we give an example of a weight for which the corresponding Banach space with weighted norm does not admit the $p$-Bergman kernel. Next, using biholomorphisms we show that such weights exist for a large class of domains. Later we give a formula for the $p$-Bergman kernel for a specific case of weight being $p$th power of modulus of a holomorphic function in dependence on $p$-Bergman kernel with weight $1$. Then we show estimates for $p$-Bergman kernels. In the end we prove that the $p$-Bergman kernel depends continuously on a sequence of domains and a weight of integration in precisely defined sense.
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