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EXPLICIT FORMULAS FOR THE BERGMAN KERNEL ON CERTAIN FORELLI-RUDIN CONSTRUCTION

  • Zhang, Liyou ;
  • Wang, An ;
  • Li, Qingbin
  • Received : 2010.08.20
  • Published : 2012.01.01

Abstract

In this note, we present certain circular domain, named Forelli-Rudin construction or Hua construction, which is built on Cartan domains. We compute the explicit Bergman kernel for it and get the corresponding weighted Bergman kernel on its base.

Keywords

Bergman kernel;weighted Bergman kernel;Forelli-Rudin construction;Hua construction

References

  1. S. Bergman, Uber die Entwickling der harmonischen Funktionen der Ebene und Raumes nach Orthogonal funktionen, Math. Ann. 96 (1922), 237.
  2. S. Bergman, Uber die kernfunktion eines Bereiches und ihre Verhalten am Rande, J. Reine Angew. Math. 169 (1933), 1
  3. S. Bergman, Uber die kernfunktion eines Bereiches und ihre Verhalten am Rande, J. Reine Angew. Math. 172 (1935), 89.
  4. S. Bergman, Zur theorie Von pseudokonformen abbildungen, Mat. sb (N.S) 1 (1963), no. 1, 79-96.
  5. H. P. Boas, Lu Qi-Keng's problem, J. Korean Math. Soc. 37 (2000), no. 2, 253-267.
  6. H. P. Boas, S. Q. Fu, and E. J. Straube, The Bergman kernel function: Explicit formulas and zeroes, Proc. Amer. Math. Soc. 127 (1999), no. 3, 805-811. https://doi.org/10.1090/S0002-9939-99-04570-0
  7. J. P. D'Angelo, A note on the Bergman kernel, Duke Math. J. 45 (1978), no. 2, 259-265. https://doi.org/10.1215/S0012-7094-78-04515-5
  8. J. P. D'Angelo, An explicit computation of the Bergman kernel function, J. Geom. Anal. 4 (1994), no. 1, 23-34. https://doi.org/10.1007/BF02921591
  9. A. Edigarian and W. Zwonek, Geometry of the symmetrized polydisc, Arch. Math. (Basel) 84 (2005), no. 4, 364-374. https://doi.org/10.1007/s00013-004-1183-z
  10. M. Englis, A Forelli-Rudin construction and asymptotics of weighted Bergman kernels, J. Funct. Anal. 177 (2000), no. 2, 257-281. https://doi.org/10.1006/jfan.2000.3642
  11. M. Englis and G. K. Zhang, On a generalized Forelli-Rudin construction, Complex Var. Elliptic Equ. 51 (2006), no. 3, 277-294. https://doi.org/10.1080/17476930500515017
  12. F. Forelli and W. Rudin, Projections on spaces of holomorphic functions in balls, Indiana Univ. Math. J. 24 (1974), 593-602. https://doi.org/10.1512/iumj.1974.24.24044
  13. G. Francsics and N. Hanges, The Bergman kernel and hypergeometric functions, J. Funct. Anal. 142 (1996), no. 2, 494-510. https://doi.org/10.1006/jfan.1996.0157
  14. L. G. Hua, Harmonic Analysis of Function of several Complex Variables in Classical Domains, Beijing, Science Press, 1959.
  15. E. Ligocka, On the Forelli-Rudin construction and weighted Bergman projections, Studia Math. 94 (1989), no. 3, 257-272. https://doi.org/10.4064/sm-94-3-257-272
  16. Q. K. Lu, The Classical Manifolds and Classical Domains, Shanghai: Shanghai Scientific and Technical Publisher, 1963.
  17. K. Oeljeklaus, P. P ug, and E.H. Youssfi, The Bergman kernel of the minimal ball and applications, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 3, 915-928. https://doi.org/10.5802/aif.1585
  18. J. D. Park, New formulas of the Bergman kernel for complex ellipsoids in $C^2$, Proc. Amer. Math. Soc. 136 (2008), no. 12, 4211-4221. https://doi.org/10.1090/S0002-9939-08-09576-2
  19. G. Roos, Weighted Bergman kernels and virtual Bergman kernels, Sci. China Ser. A 48 (2005), suppl., 387-399.
  20. A. Wang, W. P. Yin, L. Y. Zhang, and G. Roos, The Kahler-Einstein metric for some Hartogs domains over bounded symmetric domains, Sci. China Ser. A 49 (2006), no. 9, 1175-1210. https://doi.org/10.1007/s11425-006-0230-6
  21. W. P. Yin, The Bergman Kernels on Cartan-Hartogs domains, Chinese Sci. Bull. 44 (1999), no. 21, 1947-1951. https://doi.org/10.1007/BF02887114
  22. W. P. Yin, The Bergman kernels on super-Cartan domains of the first type, Sci. China Ser. A 43 (2000), no. 1, 13-21. https://doi.org/10.1007/BF02903843
  23. L. Y. Zhang and W. P. Yin, Lu Qi-Kengs problem on some complex ellipsoids, J. Math. Anal. Appl. 357 (2009), no. 2, 364-370. https://doi.org/10.1016/j.jmaa.2009.04.018

Acknowledgement

Supported by : NSFC, BNSF