Bergman projection induced by kernel with integral representation

  • José Ángel Peláez
  • Jouni Rättyä
  • Brett D. WickEmail author


Bounded Bergman projections \(P_\omega:L_\omega^p(v)\rightarrow{L_\omega^p(v)}\), induced by reproducing kernels admitting the representation
$$\frac{1}{(1-\overline{z}\zeta)^\gamma}\int_{0}^{1} \frac{dv(r)}{1-r\overline{z}\zeta},\;\;0\leq{r}<1,$$
and the corresponding (1,1)-inequality are characterized in terms of Bekollé-Bonami-type conditions. The two-weight inequality for the maximal Bergman projection \(P_\omega^+:L_\omega^p(u)\rightarrow{L_\omega^p(v)}\) in terms of Sawyer-testing conditions is also discussed.


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  1. [1]
    A. Aleman, S. Pott and M. C. Reguera, Sarason conjecture on the Bergman space, Int. Math. Res. Not. IMRN 2017 (2017), 4320–4349.MathSciNetzbMATHGoogle Scholar
  2. [2]
    H. Arrousi, Function and Operator Theory on Large Bergman Spaces, PhD thesis, University of Barcelona, Barcelona, 2016.Google Scholar
  3. [3]
    D. Bekollé, Inégalités á poids pour le projecteur de Bergman dans la boule unité de C n, Studia Math. 71 (1981/82), 305–323.CrossRefzbMATHGoogle Scholar
  4. [4]
    D. Bekollé, Projections sur des espaces de fonctions holomorphes dans des domaines plans, Canad. J. Math. 38 (1986), 127–157.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    D. Bekollé and A. Bonami, Inégalités á poids pour le noyau de Bergman, C. R. Acad. Sci. Paris Sèr. A-B 286 (1978), 775–778.MathSciNetzbMATHGoogle Scholar
  6. [6]
    O. Constantin and J. A. Peláez, Boundedness of the Bergman projection on Lp-spaces with exponential weights, Bull. Sci. Math. 139 (2015), 245–268.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Y. Deang, L. Huang, T. Zhao and D. Zeng, Bergman projection and Bergman spaces, J. Operator Theory 46 (2001), 3–24.MathSciNetGoogle Scholar
  8. [8]
    J. B. Garnett and P. Jones, BMO from dyadic BMO, Pacific J. Math. 99 (1982), 351–371.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    T. Mei, BMO is the intersection of two translates of dyadic BMO, C. R. Math. Acad. Sci. Paris 336 (2003), 1003–1006.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    F. Nazarov, S. Treil and A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc. 12 (1999), 909–928.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    J. A. Peláez, Small weighted Bergman spaces, in Proceedings of the Summer School in Complex and Harmonic Analysis, and Related Topics, University of Eastern Finland, Faculty of Science and Forestry, Joensuu, 2016, pp. 29–98.Google Scholar
  12. [12]
    J. A. Peláez and J. Rättyä, Weighted Bergman spaces induced by rapidly increasing weights, Mem. Amer. Math. Soc. 227 (2014).Google Scholar
  13. [13]
    J. A. Peláez and J. Rättyä, Two weight inequality for Bergman projection, J. Math. Pures. Appl. 105 (2016), 102–130.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    J. A. Peláez and J. Rättyä, Bergman projection induced by radial weight, arXiv:1902.09837.Google Scholar
  15. [15]
    S. Pott and M. C. Reguera, Sharp Bekollé esrtimate for the Bergman projection, J. Funct. Anal. 265 (2013), 3233–3244.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    E. T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), 1–11.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    E. T. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 308 (1998), 533–545.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    R. L. Schilling, R. Song and Z. Vondraćek, Bernstein Functions: Theory and Applications, De Gruyter, Berlin, 2012.CrossRefzbMATHGoogle Scholar
  19. [19]
    A. Shields and D. Williams, Bounded projections and the growth of harmonic conjugates in the unit disc, Michigan Math. J. 29 (1982), 3–25.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    S. Shimorin, An integral formula for weighted Bergman Reproducing kernels, Comp. Var. 47 (2002), 1015–1028.MathSciNetzbMATHGoogle Scholar
  21. [21]
    S. Treil, A remark on two weight estimates for positive dyadic operators, in Operator-Related Function Theory and Time-Frequency Analysis, Springer, Cham, 2015, pp. 85–195.Google Scholar
  22. [22]
    D. V. Widder, Necessary and sufficient conditions for the representation of a function as a Laplace integral, Trans. Amer. Math. Soc. 33 (1931), 851–892.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Y. E. Zeytuncu, L p-regularity of weighted Bergman projections, Trans. Amer. Math. Soc. 365 (2013), 2959–2976.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    K. Zhu, Operator Theory in Function Spaces, American Mathematical Society, Providence, RI, 2007.CrossRefzbMATHGoogle Scholar

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© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  • José Ángel Peláez
    • 1
  • Jouni Rättyä
    • 2
  • Brett D. Wick
    • 3
    Email author
  1. 1.Departamento de Análisis MatemáticoUniversidad de MálagaMálagaSpain
  2. 2.University of Eastern FinlandJoensuuFinland
  3. 3.Department of MathematicsWashington University — St. LouisSt. LouisUSA

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