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New formulas of the Bergman kernels for complex ellipsoids in $ \mathbb{C}^2$


Author: Jong-Do Park
Journal: Proc. Amer. Math. Soc. 136 (2008), 4211-4221
MSC (2000): Primary 32A25; Secondary 33D70
DOI: https://doi.org/10.1090/S0002-9939-08-09576-2
Published electronically: July 15, 2008
MathSciNet review: 2431034
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Abstract: We compute the explicit formula of the Bergman kernel for a nonhomogeneous domain $ \{(z_1,z_2)\in\mathbb{C}^2:\vert z_1\vert^{4/q_1}+\vert z_2\vert^{4/q_2}<1\}$ for any positive integers $ q_1$ and $ q_2$. We also prove that among the domains $ D_p:=\{(z_1,z_2)\in\mathbb{C}^2:\vert z_1\vert^{2p_1}+\vert z_2\vert^{2p_2}<1\}$ in $ \mathbb{C}^2$ with $ p=(p_1,p_2)\in\mathbb{N}^2$, the Bergman kernel is represented in terms of closed forms if and only if $ p=(p_1,1),(1,p_2)$, or $ p=(2,2)$.


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  • 1. P. Appell and J. Kampé de Fériet, Fonctions hypergé ométriques et hypersphériques, Gauthier-Villars, Paris, (1926).
  • 2. S. Bell, Proper holomorphic mappings and the Bergman projection, Duke Math. J. 48 (1981), 167-175. MR 610182 (82d:32011)
  • 3. S. Bell, The Bergman kernel function and proper holomorphic mappings, Trans. Amer. Math. Soc. 270 (1982), no. 2, 685-691. MR 645338 (83i:32033)
  • 4. S. Bergman, Zur Theorie von pseudokonformen Abbildungen, Mat. Sb. (N.S.) 1(43) (1936), no. 1, 79-96.
  • 5. H. P. Boas, S. Fu and E. J. Straube, The Bergman kernel function: Explicit formulas and zeroes, Proc. Amer. Math. Soc. 127 (1999), no. 3, 805-811. MR 1469401 (99f:32037)
  • 6. J. P. D'Angelo, A note on the Bergman kernel, Duke Math. J. 45 (1978), 259-265. MR 0473231 (57:12906)
  • 7. J. P. D'Angelo, An explicit computation of the Bergman kernel function, J. Geom. Anal. 4 (1994), 23-34. MR 1274136 (95a:32039)
  • 8. G. Francsics and N. Hanges, The Bergman kernel of complex ovals and multivariable hypergeometric functions, J. Funct. Anal. 142 (1996), 494-510. MR 1423042 (97m:32039)
  • 9. K. Fujita, Bergman transformation for analytic functionals on some balls. Microlocal Analysis and Complex Fourier Analysis, 81-98, World Scientific Publisher, River Edge, NJ, 2002. MR 2068530 (2005e:32002)
  • 10. K. Fujita, Bergman kernel for the two-dimensional balls, Complex Var. Theory Appl. 49 (2004), no. 3, 215-225. MR 2046397 (2005b:32004)
  • 11. S. Gong and X. Zheng, The Bergman kernel function of some Reinhardt domains, Trans. Amer. Math. Soc. 348 (1996), no. 5, 1771-1803. MR 1329534 (96h:32032)
  • 12. I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series, and products, translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. Seventh edition. Elsevier/Academic Press, Amsterdam, 2007. MR 2360010
  • 13. L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domain (in Russian), Translations of Math. Monographs, Vol. 6, Amer. Math. Soc., Providence, RI, 1979. MR 598469 (82c:32032)
  • 14. S. G. Krantz and J. Yu, On the Bergman invariant and curvatures of the Bergman metric, Illinois J. Math. 40 (1996), no. 2, 226-244. MR 1398092 (97g:32026)
  • 15. S. B. Opps, N. Saad and H. M. Srivastava, Some reduction and transformation formulas for the Appell hypergeometric function $ F_2$, J. Math. Anal. Appl. 302 (2005), 180-195. MR 2107356 (2005g:33028)
  • 16. H. Valencourt, Projecteurs sur les espaces de fonctions holomorphes: Propriétés et applications, docteur de l'Université de Poitiers, 2002.
  • 17. W. Yin, Two problems on Cartan domains, J. China Univ. Sci. Tech. 16 (1986), no. 2, 130-146. MR 900957 (88k:32061)
  • 18. E. H. Youssfi, Proper holomorphic liftings and new formulas for the Bergman and Szegő kernels, Studia Math. 152(2) (2002), 161-186. MR 1916548 (2003e:32004)
  • 19. B. S. Zinov$ ^{\prime}$ev, Reproducing kernels for multicircular domains of holomorphy (Russian), Sibirsk. Mat. Ž. 15 (1974), 35-48, 236. MR 0333230 (48:11555)

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Additional Information

Jong-Do Park
Affiliation: Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea
Address at time of publication: Department of Mathematics, Pohang University of Science and Technology, San 31, Hyoja-dong, Namgu, Pohang, Kyungbuk, 790-784, Korea
Email: jongdopark@gmail.com, jdpark@postech.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-08-09576-2
Keywords: Bergman kernel, homogeneous domains, hypergeometric function, complex ellipsoids
Received by editor(s): February 28, 2007
Published electronically: July 15, 2008
Additional Notes: The author was supported by Korea Research Foundation Grant 2005-070-C00007 and partially supported by BK21 CoDiMaRO
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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