| Title:
|
Weighted generalization of the Ramadanov's theorem and further considerations (English) |
| Author:
|
Pasternak-Winiarski, Zbigniew |
| Author:
|
Wójcicki, Paweł |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
68 |
| Issue:
|
3 |
| Year:
|
2018 |
| Pages:
|
829-842 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the limit behavior of weighted Bergman kernels on a sequence of domains in a complex space $\mathbb C^N$, and show that under some conditions on domains and weights, weighed Bergman kernels converge uniformly on compact sets. Then we give a weighted generalization of the theorem given by M. Skwarczyński (1980), highlighting some special property of the domains, on which the weighted Bergman kernels converge uniformly. Moreover, we show that convergence of weighted Bergman kernels implies this property, which will give a characterization of the domains, for which the inverse of the Ramadanov's theorem holds. (English) |
| Keyword:
|
weighted Bergman kernel |
| Keyword:
|
admissible weight |
| Keyword:
|
sequence of domains |
| MSC:
|
32A25 |
| MSC:
|
32A36 |
| idZBL:
|
Zbl 06986975 |
| idMR:
|
MR3851894 |
| DOI:
|
10.21136/CMJ.2018.0010-17 |
| . |
| Date available:
|
2018-08-09T13:15:41Z |
| Last updated:
|
2020-10-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147371 |
| . |
| Reference:
|
[1] Aronszajn, N.: Theory of reproducing kernels.Trans. Am. Math. Soc. 68 (1950), 337-404. Zbl 0037.20701, MR 0051437, 10.2307/1990404 |
| Reference:
|
[2] Bergman, S.: The Kernel Function and Conformal Mapping.Mathematical Surveys 5, American Mathematical Society, Providence (1970). Zbl 0208.34302, MR 0507701, 10.1090/surv/005 |
| Reference:
|
[3] Boas, H. P.: Counterexample to the Lu Qi-Keng conjecture.Proc. Am. Math. Soc. 97 (1986), 374-375. Zbl 0596.32032, MR 0835902, 10.2307/2046535 |
| Reference:
|
[4] Engliš, M.: Weighted Bergman kernels and quantization.Commun. Math. Phys. 227 (2002), 211-241. Zbl 1010.32002, MR 1903645, 10.1007/s002200200634 |
| Reference:
|
[5] Engliš, M.: Toeplitz operators and weighted Bergman kernels.J. Funct. Anal. 255 (2008), 1419-1457. Zbl 1155.32001, MR 2565714, 10.1016/j.jfa.2008.06.026 |
| Reference:
|
[6] Forelli, F., Rudin, W.: Projections on spaces of holomorphic functions in balls.Indiana Univ. Math. J. 24 (1974), 593-602. Zbl 0297.47041, MR 0357866, 10.1512/iumj.1974.24.24044 |
| Reference:
|
[7] Jacobson, R. L.: Weighted Bergman Kernel Functions and the Lu Qi-keng Problem. Thesis (Ph.D.).A&M University, Texas (2012). MR 3068007 |
| Reference:
|
[8] Jarnicki, M., Pflug, P.: Invariant Distances and Metrics in Complex Analysis.De Gruyter Expositions in Mathematics 9, Walter de Gruyter, Berlin (2013). Zbl 1273.32002, MR 3114789, 10.1515/9783110253863 |
| Reference:
|
[9] Krantz, S. G.: Function Theory of Several Complex Variables.American Mathematical Society, Providence (2001). Zbl 1087.32001, MR 1846625, 10.1090/chel/340 |
| Reference:
|
[10] Krantz, S. G.: Geometric Analysis of the Bergman Kernel and Metric.Graduate Texts in Mathematics 268, Springer, New York (2013). Zbl 1281.32004, MR 3114665, 10.1007/978-1-4614-7924-6 |
| Reference:
|
[11] Ligocka, E.: On the Forelli-Rudin construction and weighted Bergman projections.Stud. Math. 94 (1989), 257-272. Zbl 0688.32020, MR 1019793, 10.4064/sm-94-3-257-272 |
| Reference:
|
[12] Odzijewicz, A.: On reproducing kernels and quantization of states.Commun. Math. Phys. 114 (1988), 577-597. Zbl 0645.53044, MR 0929131, 10.1007/BF01229456 |
| Reference:
|
[13] Pasternak-Winiarski, Z.: On the dependence of the reproducing kernel on the weight of integration.J. Funct. Anal. 94 (1990), 110-134. Zbl 0739.46010, MR 1077547, 10.1016/0022-1236(90)90030-O |
| Reference:
|
[14] Pasternak-Winiarski, Z.: On weights which admit the reproducing kernel of Bergman type.Int. J. Math. Math. Sci. 15 (1992), 1-14. Zbl 0749.32019, MR 1143923, 10.1155/S0161171292000012 |
| Reference:
|
[15] 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:
|
[16] Shabat, B. V.: Introduction to Complex Analysis. Part II: Functions of Several Variables.Translations of Mathematical Monographs 110, American Mathematical Society, Providence (1992). Zbl 0799.32001, MR 1192135, 10.1090/mmono/110 |
| Reference:
|
[17] Skwarczyński, M.: Biholomorphic invariants related to the Bergman function.Diss. Math. 173 (1980), 59 pages. Zbl 0443.32014, MR 0575756 |
| Reference:
|
[18] Skwarczyński, M., Iwiński, T.: The convergence of Bergman functions for a decreasing sequence of domains.Approximation Theory. Proc. Conf. Poznan 1972 Z. Ciesielski, J. Musielak D. Reidel, Dordrecht (1975), 117-120. Zbl 0328.30005, MR 0450534 |
| Reference:
|
[19] Skwarczyński, M., Mazur, T.: Wstepne twierdzenia teorii funkcji wielu zmiennych zespolonych.Wydawnictwo Krzysztof Biesaga, Warszawa (2001), Polish. |
| Reference:
|
[20] Wójcicki, P. M.: Weighted Bergman kernel function, admissible weights and the Ramadanov theorem.Mat. Stud. 42 (2014), 160-164. Zbl 1327.32005, MR 3381259 |
| . |
