Boundedness of the Bergman Projection and Some Properties of Bergman Type Spaces

  • Alexey Karapetyants
  • Humberto Rafeiro
  • Stefan Samko


We give a simple proof of the boundedness of Bergman projection in various Banach spaces of functions on the unit disc in the complex plain. The approach of the paper is based on the idea of Zaharyuta and Yudovich (Uspekhi Mat Nauk 19(2):139–142, 1964) where the boundedness of the Bergman projection in Lebesgue spaces was proved using Calderón–Zygmund operators. We exploit this approach and treat the cases of variable exponent Lebesgue space, Orlicz space and variable exponent generalized Morrey spaces. In the case of variable exponent Lebesgue space the boundedness result is known, so in that case we provide a simpler proof, whereas the other cases are new. The major idea of this paper is to show that the approach can be applied to a wide range of function spaces. We also study the rate of growth of functions near the boundary in spaces under consideration and their approximation by mollified dilations.


Bergman space Bergman projection Variable exponent Lebesgue space Orlicz space Variable exponent generalized Morrey space 

Mathematics Subject Classification

30H20 46E30 46E15 



A. Karapetyants was partially supported by Southern Federal University Project No. 07/2017-31 and partially supported by the Grant 18-51-05009-Apm_a of Russian Foundation of Basic Research. H. Rafeiro was partially supported by Pontificia Universidad Javeriana. S. Samko was partially supported by the RFBR Grant 15-01-02732 and partially supported by the Grant 18-01-00094-a of Russian Foundation of Basic Research.


  1. 1.
    Aleman, A., Pott, S., Reguera, M.C.: Sarason conjecture on the Bergman space. In: International Mathematics Research Notices (2016), rnw134Google Scholar
  2. 2.
    Bekolle, D., Bonami, A.: Inegalites a poids pour le noyau de Bergman. CR Acad. Sci. Paris Ser. AB 286(18), 775–778 (1978)MathSciNetMATHGoogle Scholar
  3. 3.
    Zaharyuta, V.P., Yudovich, V.I.: The general form of a linear functional in \(H_p^{\prime }\). 19(2), 139–142 (1964)Google Scholar
  4. 4.
    Milutin, D.: Boundedness of the Bergman projections on \(L^p\) spaces with radial weights. Publ. de l’Institut Math. 86(100), 5–20 (2009)CrossRefMATHGoogle Scholar
  5. 5.
    Guliyev, V.S., Hasanov, J.J., Samko, S.G.: Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces. Math. Scand. 107, 285–304 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Diening, L., Ruzichka, M.: Calderón–Zygmund operators on generalized Lebesgues spaces \(L^{p(\cdot )}\) and problems related to fluid dynemics. J. Reine Angew. Math. 563, 197–220 (2003)MathSciNetGoogle Scholar
  7. 7.
    Chacón, G.R., Rafeiro, H.: Variable exponent Bergman spaces. Nonlinear Anal.: Theory Methods Appl. 105, 41–49 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chacón, G.R., Rafeiro, H.: Toeplitz operators on variable exponent Bergman spaces. Mediterr. J. Math. 13(5), 3525–3536 (2016)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Almeida, A., Hasanov, J., Samko, S.: Maximal and potential operators in variable exponent Morrey spaces. Georgian Math. J. 15, 195–208 (2008)MathSciNetMATHGoogle Scholar
  10. 10.
    Kokilashvili, H., Meskhi, A., Rafeiro, H., Samko, S.: Integral Operators in Non-Standard Function Spaces. Vol. I: Variable Exponent Lebesgue and Amalgam Spaces. Birkhäuser Basel: Operator Theory: Advances and Applications; (2016)Google Scholar
  11. 11.
    Kokilashvili, V., Meskhi, A., Rafeiro, H., Samko, S.: Integral Operators in Non-Standard Function Spaces. Vol. II: Variable Exponent Hölder, Morrey–Campanato and Grand Spaces. Birkhäuser Basel: Operator Theory: Advances and Applications (2016)Google Scholar
  12. 12.
    Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988)MATHGoogle Scholar
  13. 13.
    Liu, P.D., Hou, Y.L., Wang, M.F.: Weak Orlicz space and its applications to the martingale theory. Sci. China Math. 53(4), 905–916 (2010)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gallardo, D.: Orlicz spaces for which the Hardy–Littlewood maximal operator is bounded. Publ. Mat. Barc. 32(2), 261–266 (1988)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Diening, L., Harjulehto, P., Hasto, P., Ruzicka, M.: Lebesgue and Sobolev Spaces with Variable Exponents, vol. 2017. Springer, Heidelberg, Lecture Notes in Mathematics (2011)Google Scholar
  16. 16.
    Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces. Applied and Numerical Harmonic Analysis, Foundations and Harmonic Analysis. Springer, Basel (2013)Google Scholar
  17. 17.
    Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman Spaces. Springer, New York (2000)CrossRefMATHGoogle Scholar
  18. 18.
    Duren, P., Schuster, A.: Bergman Spaces, Providence, RI, Mathematical Surveys and Monographs, 100 (2004)Google Scholar
  19. 19.
    Zhu, K.: Spaces of Holomorphic Functions in The Unit Ball. Springer, Graduate texts in Mathematics (2004)Google Scholar
  20. 20.
    Zhu, K.: Operator Theory in Function Spaces, AMS Mathematical Surveys and Monographs, vol. 138 (2007)Google Scholar
  21. 21.
    Triebel, H.: Local Function Spaces, Heat and Navier–Stokes Equations. EMS Tracts in Mathematics Vol. 20 (2013)Google Scholar
  22. 22.
    Pick, L., Kufner, A., John, O., Fucik, S.: Function Spaces, 1. De Gruyter Series in Nonlinear Analysis and Applications 14 (2013)Google Scholar
  23. 23.
    Rafeiro, H., Samko, N., Samko, S.: Morrey-Campanato Spaces: an Overview, Operator Theory: Advances and Applications, vol. 228, pp. 293–323. Springer, Basel (2013)MATHGoogle Scholar
  24. 24.
    Kufner, A., John, O., Fucik, S.: Function Spaces, Leyden, Noordhoff International Publishing. Monographs and Textbooks on Mechanics of Solids and Fluids. Mechanics: Analysis (1977)Google Scholar
  25. 25.
    Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princenton, NJ, Annals of Mathematics Studies, vol. 105. Princenton University Press, Princenton (1977)Google Scholar
  26. 26.
    Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 146. Marcel Dekker, NY (1991)Google Scholar
  27. 27.
    Kokilashvili, V., Krbec, M.M.: Weighted Inequalities in Lorentz and Orlicz Spaces. World Scientific, Singapore (1991)CrossRefMATHGoogle Scholar
  28. 28.
    Krasnoselskii, M.A., Rutitski, Ya.B.: Convex Functions and Orlicz Spaces. Translated from the First Russian Edition. Noordhoff, Groningen (1961)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsSouthern Federal UniversityRostov-na-DonuRussia
  2. 2.Department of MathematicsDon State Technical UniversityRostov-na-DonuRussia
  3. 3.Departamento de Matemáticas, Facultad de CienciasPontificia Universidad JaverianaBogotáColombia
  4. 4.Department of Mathematics, Faculty of SciencesUniversidade do AlgarveFaroPortugal

Personalised recommendations