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<p>We consider <i>p</i>-Bergman kernels, i.e., a generalization of the classical Bergman kernel for Banach spaces of integrable in <i>p</i>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 <i>p</i>-Bergman kernel. Then we give an example of a weight for which the corresponding Banach space with weighted norm does not admit the <i>p</i>-Bergman kernel. Next, using biholomorphisms we show that such weights exist for a large class of domains. Later we give a formula for the <i>p</i>-Bergman kernel for a specific case of weight being <i>p</i>th power of modulus of a holomorphic function in dependence on <i>p</i>-Bergman kernel with weight 1. Then we show estimates for <i>p</i>-Bergman kernels. In the end we prove that the <i>p</i>-Bergman kernel depends continuously on a sequence of domains and a weight of integration in precisely defined sense.</p>

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p-Bergman kernels. Admissible weights, formulas, estimates, Ramadanov theorem and dependence on a weight of integration

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月3日

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  • Tomasz Łukasz Żynda

摘要

We consider p-Bergman kernels, i.e., a generalization of the classical Bergman kernel for Banach spaces of integrable in pth 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 pth 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.