Article
| Title: | $p$-Bergman kernels. Admissible weights, formulas, estimates, Ramadanov theorem and dependence on a weight of integration (English) |
| Author: | Żynda, Tomasz Łukasz |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 76 |
| Issue: | 1 |
| Year: | 2026 |
| Pages: | 105-122 |
| Summary lang: | English |
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| Category: | math |
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| 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. (English) |
| Keyword: | reproducing kernel Banach space |
| Keyword: | $p$-Bergman kernel |
| Keyword: | admissible weight |
| Keyword: | Ramadanov theorem |
| MSC: | 46E22 |
| DOI: | 10.21136/CMJ.2026.0185-25 |
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| Date available: | 2026-03-13T09:29:45Z |
| Last updated: | 2026-03-16 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153563 |
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| Reference: | [6] Pasternak-Winiarski, Z., Wójcicki,, P.: Weighted generalization of the Ramadanov's theorem and further considerations.Czech. Math. J. 68 (2018), 829-842. Zbl 1513.32006, MR 3851894, 10.21136/CMJ.2018.0010-17 |
| Reference: | [7] Ramadanov, I.: Sur une propriété de la fonction de Bergman.C. R. Acad. Bulg. Sci. 20 (1967), 759-762 French. Zbl 0206.09002, MR 0226042 |
| Reference: | [8] Rudin, W.: Real and Complex Analysis.McGraw-Hill Series in Higher Mathematics. McGraw-Hill, New York (1974). Zbl 0278.26001, MR 0344043 |
| Reference: | [9] Żynda, T. Ł.: Reproducing kernel Banach space defined by the minimal norm property and applications to partial differential equation theory.Georgian Math. J. 31 (2024), 527-537. Zbl 1553.46026, MR 4751082, 10.1515/gmj-2023-2095 |
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