On the dependence of the Bergman function on deformations of the Hartogs domain

Zbigniew Pasternak-Winiarski

Annales Polonici Mathematici (1991)

  • Volume: 55, Issue: 1, page 287-300
  • ISSN: 0066-2216

Abstract

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We apply the Rudin idea to represent the Bergman kernel of the Hartogs domain as the sum of a series of weighted Bergman functions in the study of the dependence of this kernel on deformations of the domain. We prove that the Bergman function depends smoothly on the function defining the Hartogs domain.

How to cite

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Zbigniew Pasternak-Winiarski. "On the dependence of the Bergman function on deformations of the Hartogs domain." Annales Polonici Mathematici 55.1 (1991): 287-300. <http://eudml.org/doc/262271>.

@article{ZbigniewPasternak1991,
abstract = {We apply the Rudin idea to represent the Bergman kernel of the Hartogs domain as the sum of a series of weighted Bergman functions in the study of the dependence of this kernel on deformations of the domain. We prove that the Bergman function depends smoothly on the function defining the Hartogs domain.},
author = {Zbigniew Pasternak-Winiarski},
journal = {Annales Polonici Mathematici},
keywords = {Hartogs domain; Bergman kernel function},
language = {eng},
number = {1},
pages = {287-300},
title = {On the dependence of the Bergman function on deformations of the Hartogs domain},
url = {http://eudml.org/doc/262271},
volume = {55},
year = {1991},
}

TY - JOUR
AU - Zbigniew Pasternak-Winiarski
TI - On the dependence of the Bergman function on deformations of the Hartogs domain
JO - Annales Polonici Mathematici
PY - 1991
VL - 55
IS - 1
SP - 287
EP - 300
AB - We apply the Rudin idea to represent the Bergman kernel of the Hartogs domain as the sum of a series of weighted Bergman functions in the study of the dependence of this kernel on deformations of the domain. We prove that the Bergman function depends smoothly on the function defining the Hartogs domain.
LA - eng
KW - Hartogs domain; Bergman kernel function
UR - http://eudml.org/doc/262271
ER -

References

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  1. [1] J. Burbea and P. Masani, Banach and Hilbert Spaces of Vector-Valued Functions, Res. Notes Math. 90, Pitman, Boston 1984. Zbl0555.46012
  2. [2] F. Forelli and W. Rudin, Projections on spaces of holomorphic functions in balls, Indiana Univ. Math. J. 24 (1974), 593-602. Zbl0297.47041
  3. [3] R. E. Greene and S. G. Krantz, Stability of the Bergman kernel and curvature properties of bounded domains, in: Recent Developments in Several Complex Variables, J. Fornaess (ed.), Ann. of Math. Stud. 100, Princeton Univ. Press, Princeton, N.J., 1981. Zbl0483.32014
  4. [4] R. E. Greene and S. G. Krantz, Deformation of complex structures, estimates for the ∂̅ equation, and stability of the Bergman kernel, Adv. in Math. 43 (1982), 1-86. Zbl0504.32016
  5. [5] S. G. Krantz, Function Theory of Several Complex Variables, Interscience-Wiley, New York 1982. Zbl0471.32008
  6. [6] E. Ligocka, The regularity of the weighted Bergman projection, in: Seminar of Deformation Theory 1982/84, Lecture Notes in Math. 1165, Springer, 1985, 197-203. 
  7. [7] E. Ligocka, On the Forelli-Rudin construction and weighted Bergman projections, Studia Math. 94 (1989), 257-272. Zbl0688.32020
  8. [8] K. Maurin, Analysis, Part 1, Elements, PWN-Reidel, Warszawa-Dordrecht 1976. 
  9. [9] K. Maurin, Analysis, Part 2, Integration, Distributions, Holomorphic Functions, Tensor and Harmonic Analysis, PWN-Reidel, Warszawa-Dordrecht 1980. 
  10. [10] T. Mazur, On the complex manifolds of Bergman type, to appear. Zbl1064.32500
  11. [11] Z. Pasternak-Winiarski, On the dependence of the reproducing kernel on the weight of integration, J. Funct. Anal. 94 (1990), 110-134. Zbl0739.46010
  12. [12] Z. Pasternak-Winiarski, On weights which admit the reproducing kernel of Bergman type, Internat. J. Math. and Math. Sci., to appear. Zbl0749.32019
  13. [13] W. Rudin, Function Theory in the Unit Ball in ℂ n , Springer, Berlin 1980. Zbl0495.32001
  14. [14] B. V. Shabat, Introduction to Complex Analysis, 3rd ed., Nauka, Moscow 1985 (in Russian). Zbl0574.30001

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