Estimates the Bergman kernel for classical domains ´E. Cartan’s
https://doi.org/10.22405/2226-8383-2021-22-3-20-31
Abstract
The aim of this work is to find optimal estimates for the Bergman kernels for the classical domains ℜ𝐼 (𝑚, 𝑘) ,ℜ𝐼𝐼 (𝑚) ,ℜ𝐼𝐼𝐼 (𝑚) and ℜ𝐼𝑉 (𝑛) through the Bergman kernels of balls in the spaces C𝑚𝑘,C𝑚(𝑚+1) 2 ,C𝑚(𝑚−1) 2 and C𝑛, respectively. For this, we use the statements of the Summer-Mehring theorem on the extension of the Bergman kernel and some properties of the Bergman kernel.
About the Author
Jonibek Shokirovich Abdullayev
National University of Uzbekistan named after M. Ulugbek
Russian Federation
Russian Federation
PhD-Student
References
1. Cartan ´E. Sur les domaines bornes homogenes de l ’espace de 𝑛 variables complexes // Abh. Math. Sem. Univ. Hamburg, 11 (1935), pp. 116-162.
2. C.L.Siegel, Automorphic functions of several complex variables // M.: Publishing house of foreign literature, 1954. p. 168. (In Russian).
3. Hua Luogeng. Harmonic analysis of functions of several complex variables in classical domains // Inostr. Lit., M., 1959 (In Russian)
4. Pjateckiˇi-ˇSapiro I.I. Geometry of classical domains and the theory of automorphic functions // Moscow: State publishing house of physical and mathematical literature, 1961. p. 191. (In
5. Russian).
6. Henkin G.M. The method of integral representations in complex analysis, Complex analysis-several variables – 1 // Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 7, VINITI,
7. Moscow, 1985, 23–124
8. Xiao Ming. Regularity of mappings into classical domains // Mathematische Annalen, 2020, 378(3-4), pp. 1271-1309
9. Xiao M. Bergman-Harmonic Functions on Classical Domains // International Mathematics Research Notices, Vol. 00, No. 0, pp. 1–36 (2019).
10. Fuks B. A. Special Chapters in the Theory of Analytic Functions of Several Complex Variables // Физматгиз, 1963.
11. Aizenberg L.A. Carleman Formulas in complex analysis // Novosibirsk, Science. 1990. 248 p. (In Russian)
12. Aizenberg, L.A., Yuzhakov, A.P. Integral Representations and Residues in Multidimensional Complex Analysis. Novosibirsk: Nauka, 366 pp. (1979):
13. Khudayberganov G., Hidirov B.B., Rakhmonov U.S., Automorphisms of matrix balls Acta NUUz, 2010, no. 3, 205-210 (In Russian)
14. Rudin W. Function Theory in the Unit Ball of C𝑛 // New York, Berlin, Heidelberg: Springer-Verlag, (1980) 436 p.
15. Krantz S. G. Harmonic and complex analysis in several variables, Springer Monographs in Mathematics, Gewerbestrasse // 11, 6330 Cham, Switzerland (2017), 429 p.
16. Sergeev A. G . On matrix and Reinhardt domains // Preprint, Inst. Mittag-Leffler, Stockholm, 7 pp. (1988).
17. Khudayberganov G., Kytmanov A.M., Shaimkulov B.A., Analysis in matrix domains // Monograph. Krasnoyarsk: Siberian Federal University, 2017. p. 296 (In Russian)
18. Khudayberganov G., Rakhmonov U. S. Carleman Formula for Matrix Ball of the Third Type // Algebra, Complex Analysis, and Pluripotential Theory. USUZCAMP 2017. Springer Proceedingsin Mathematics & Statistics, vol. 264 (2017), pp. 101-108, Springer, Cham.
19. Khudayberganov G., Rakhmonov U. The Bergman and Cauchy-Szeg¨o kernels for matrix ball of the second type // J. Sib. Fed. Univ. Math. Phys., 7:3 (2014), 305-310.
20. Myslivets S. G. Construction of Szeg¨o and Poisson kernels in convex domains // J. Sib. Fed. Univ. Math. Phys., 2018, Volume 11, Issue 6, 792-795.
21. Khudayberganov G., Abdullayev J. Sh. Relationship between the Bergman and Cauchy-Szeg¨o kernels in the domains 𝜏+ (𝑛 − 1) and ℜ𝑛𝐼 𝑉 // J. Sib. Fed. Univ. Math. Phys., 13:5 (2020),
22. -567.
23. Myslivets S.G. On the Szeg˝o and Poisson kernels in the convex domains in C𝑛 // Russian Mathematics (Izv. Vyssh. Uchebn. Zaved. Mat) 2019, no. 1, pp. 42-48.
24. Rakhmonov U. S., Abdullayev J. Sh. On volumes of matrix ball of third type and generalized Lie balls // Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Kompyuternye Nauki, 2019, vol. 29, issue 4, pp. 548-557.
25. Khudayberganov G., Khalknazarov A.M., Abdullayev J.Sh., Laplace and Hua Luogeng operators // Russian Mathematics (Izv. Vyssh. Uchebn. Zaved. Mat) 2020, Vol 64 , no. 3,
26. pp. 66-71. Allerton Press, Inc., 2020
27. Shabat B.V. , Introduction to Complex Analysis Part II Functions of Several Variables, Nauka, Fiz. Mat. Lit., M., pp. 464. 1985 (in Russian)
28. Bremermann H.-J. Die charakterisierung von regularit¨atsgebieten durch pseudokonvexe runktionen // Schriftrenreihe Math. Inst. Munster, № 5 (1951).
29. Hua Luogeng. On the theory of automorphic functions of a matrix variable I-geometrical basis // American Journal of Mathematics, Vol. 66, No. 3 , pp. 470-488 (1944).
Review
For citations:
Abdullayev J.Sh. Estimates the Bergman kernel for classical domains ´E. Cartan’s. Chebyshevskii Sbornik. 2021;22(3):20-31. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-3-20-31
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