The Journal of Geometric Analysis

, Volume 29, Issue 1, pp 217–223 | Cite as

Equivalent Bergman Spaces with Inequivalent Weights

  • Blake J. BoudreauxEmail author


We give a proof that every space of weighted square-integrable holomorphic functions admits an equivalent weight whose Bergman kernel has zeroes. Here the weights are equivalent in the sense that they determine the same space of holomorphic functions. Additionally, a family of radial weights in \(L^1(\mathbb {C})\) whose associated Bergman kernels that have infinitely many zeroes is exhibited.


Admissible weight Bergman space Bergman kernel Radial weight 

Mathematics Subject Classification

Primary 32A25 Secondary 32A36 


  1. 1.
    Bommier-Hato, H., Engliš, M., Youssfi, E.: Bergman-type projections in generalized Fock spaces. J. Math. Anal. Appl. 389(2), 1086–1104 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Conway, J.B.: Functions of One Complex Variable. Graduate Texts in Mathematics, vol. 11, 2nd edn. Springer, New York (1978). CrossRefGoogle Scholar
  3. 3.
    Duran, Antonio J.: The Stieltjes moments problem for rapidly decreasing functions. Proc. Am. Math. Soc. 107(3), 731–741 (1989). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Leffler Functions Related Topics and Applications. Springer Monographs in Mathematics. Springer, Heidelberg (2014). CrossRefzbMATHGoogle Scholar
  5. 5.
    Krantz, S.G.: Function Theory of Several Complex Variables. AMS Chelsea Publishing, Providence, RI. Reprint of the 1992 edition. (2001).
  6. 6.
    Lu, Q-k: On Kaehler manifolds with constant curvature. Chin. Math. Acta 8, 283–298 (1966)MathSciNetGoogle Scholar
  7. 7.
    Pasternak-Winiarski, Z., Wójcicki, P.M.: Weighted generalization of the Ramadanov theorem and further considerations (2016). ArXiv: 1612.05619
  8. 8.
    Pasternak-Winiarski, Z.: On the dependence of the reproducing kernel on the weight of integration. J. Funct. Anal. 94(1), 110–134 (1990). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Pasternak-Winiarski, Z.: On weights which admit the reproducing kernel of Bergman type. Int. J. Math. Math. Sci. 15(1), 1–14 (1992). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Perälä, A.: Vanishing Bergman kernels on the disk. J. Geom. Anal. (2017).
  11. 11.
    Michael Range, R.: Holomorphic Functions and Integral Representations in Several Complex Variables. Graduate Texts in Mathematics, vol. 108. Springer, New York (1986). CrossRefzbMATHGoogle Scholar
  12. 12.
    Shohat, J.A., Tamarkin, J.D.: The Problem of Moments. American Mathematical Society Mathematical Surveys, vol. I. American Mathematical Society, New York (1943)Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA

Personalised recommendations