Geodesic
An analogue of Bremermann's theorem on finding the Bergman kernel for the Cartesian product of the classical domains ${{\Re }_{I}}\left( m,k \right)$ and ${{\Re }_{II}}\left( n \right)$
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2020), pp. 88-96 
Jonibek Sh. Abdullayev. An analogue of Bremermann's theorem on finding the Bergman kernel for the Cartesian product of the classical domains ${{\Re }_{I}}\left( m,k \right)$ and ${{\Re }_{II}}\left( n \right)$. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2020), pp. 88-96. http://geodesic.mathdoc.fr/item/BASM_2020_3_a5/
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     title = {An analogue of {Bremermann's} theorem on finding the {Bergman} kernel for the {Cartesian} product of the classical domains ${{\Re }_{I}}\left( m,k \right)$ and ${{\Re }_{II}}\left( n \right)$},
     journal = {Buletinul Academiei de \c{S}tiin\c{t}e a Republicii Moldova. Matematica},
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, an analogue of Bremermann's theorem on finding the Bergman kernel is obtained for the Cartesian product of classical domains. For this purpose, the groups of automorphisms of the considered domains are used, i.e., the Bergman kernels are constructed for the Cartesian product of classical domains, without applying complete orthonormal systems.

[1] Shabat B. V., Introduction to Complex Analysis, v. II, Functions of Several Variables, Nauka, Physical and mathematical literature, M., 1985, 464 pp. (in Russian) | MR

[2] Transl. Math. Monogr., 58, Providence, 1983, 283 pp. | DOI | MR | Zbl | Zbl

[3] Khudayberganov G., Kytmanov A. M., Shaimkulov B. A., Analysis in matrix domains, Monograph., Siberian Federal University, Krasnoyarsk, 2017, 297 pp. (in Russian)

[4] Hua Luogeng, Harmonic analysis of functions of several complex variables in classical domains, Inostr. Lit., M., 1959 (in Russian) | MR | Zbl

[5] Khenkin G. M., “The method of integral representations in complex analysis”, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 7, VINITI, M., 1985, 23–124 (in Russian)

[6] Math. Notes, 45:3 (1989), 243–248 | DOI | MR | Zbl

[7] Khudayberganov G., Abdullayev J. Sh., “Relationship between the Kernels Bergman and Cauchy-Szegő in the domains $\tau ^{+} \left(n-1\right)$ and $\Re _{IV}^{n} $”, J. Sib. Fed. Univ. Math. Phys., 13:5 (2020), 559–567 | DOI | MR | Zbl

[8] Khudayberganov G., Rakhmonov U. S., “The Bergman and Cauchy–Szegő kernels for matrix ball of the second type”, J. Sib. Fed. Univ. Math. Phys., 7:3 (2014), 305–310 | MR | Zbl

[9] Khudayberganov G., Rakhmonov U. S., “Carleman Formula for Matrix Ball of the Third Type”, Algebra, Complex Analysis, and Pluripotential Theory, USUZCAMP 2017, Springer Proceedings in Mathematics Statistics, 264, Springer, Cham, 2017, 101–108 | DOI | MR

[10] Russian Math. Surveys, 19:4 (1964), 1–89 | DOI | MR | Zbl

[11] Khudayberganov G., Rakhmonov U. S., Matyakubov Z. Q., “Integral formulas for some matrix domains”, Topics in Several Complex Variables, AMS, 2016, 89–95 | DOI | MR | Zbl

[12] Myslivets S. G., “Construction of Szegő and Poisson kernels in convex domains”, J. Sib. Fed. Univ. Math. Phys., 11:6 (2018), 792–795 | DOI | MR | Zbl

[13] Rakhmonov U. S., Abdullayev J. Sh., “On volumes of matrix ball of third type and generalized Lie balls”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 29:4 (2019), 548–557 | DOI | MR | Zbl

[14] Myslivets S. G., “On the Szegő and Poisson kernels in the convex domains in ${{\mathbb{C}}^{n}}$”, Izv. Vyssh. Uchebn. Zaved. Mat., 2019, no. 1, 42–48 | DOI | MR | Zbl

[15] Khudayberganov G., Khalknazarov A. M., Abdullayev J. Sh., “Laplace and Hua Luogeng operators”, Izv. Vyssh. Uchebn. Zaved. Mat., 2020, no. 3, 74–79 | DOI | MR | Zbl

[16] Cartan E., “Sur les domaines bornes homogenes de l'espace de variables complexes”, Abh. Math. Sern. Univ. Hamburg, 11 (1935), 116–162 | DOI | MR

[17] Krantz S., Harmonic and Complex Analysis in Several Variables, Springer Monographs in Mathematics, Springer, 2017, 424 pp. | DOI | MR | Zbl

[18] Bremermann H. J., “Die Holomorphiehüllen der Tuben- und Halbtubengebiete”, Math. Ann., 127 (1954), 406–423 | DOI | MR | Zbl

[19] Translations of Mathematical Monographs, 14, AMS, 1965, 364 pp. | DOI | MR | Zbl