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Bergman projection induced by kernel with integral representation

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Abstract

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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References

  1. A. Aleman, S. Pott and M. C. Reguera, Sarason conjecture on the Bergman space, Int. Math. Res. Not. IMRN 2017 (2017), 4320–4349.

    MathSciNet  MATH  Google Scholar 

  2. H. Arrousi, Function and Operator Theory on Large Bergman Spaces, PhD thesis, University of Barcelona, Barcelona, 2016.

    Google Scholar 

  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.

    Article  MATH  Google Scholar 

  4. D. Bekollé, Projections sur des espaces de fonctions holomorphes dans des domaines plans, Canad. J. Math. 38 (1986), 127–157.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  7. Y. Deang, L. Huang, T. Zhao and D. Zeng, Bergman projection and Bergman spaces, J. Operator Theory 46 (2001), 3–24.

    MathSciNet  Google Scholar 

  8. J. B. Garnett and P. Jones, BMO from dyadic BMO, Pacific J. Math. 99 (1982), 351–371.

    Article  MathSciNet  MATH  Google Scholar 

  9. T. Mei, BMO is the intersection of two translates of dyadic BMO, C. R. Math. Acad. Sci. Paris 336 (2003), 1003–1006.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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. J. A. Peláez and J. Rättyä, Weighted Bergman spaces induced by rapidly increasing weights, Mem. Amer. Math. Soc. 227 (2014).

  13. J. A. Peláez and J. Rättyä, Two weight inequality for Bergman projection, J. Math. Pures. Appl. 105 (2016), 102–130.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. A. Peláez and J. Rättyä, Bergman projection induced by radial weight, arXiv:1902.09837.

  15. S. Pott and M. C. Reguera, Sharp Bekollé esrtimate for the Bergman projection, J. Funct. Anal. 265 (2013), 3233–3244.

    Article  MathSciNet  MATH  Google Scholar 

  16. E. T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), 1–11.

    Article  MathSciNet  MATH  Google Scholar 

  17. E. T. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 308 (1998), 533–545.

    Article  MathSciNet  MATH  Google Scholar 

  18. R. L. Schilling, R. Song and Z. Vondraćek, Bernstein Functions: Theory and Applications, De Gruyter, Berlin, 2012.

    Book  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  20. S. Shimorin, An integral formula for weighted Bergman Reproducing kernels, Comp. Var. 47 (2002), 1015–1028.

    MathSciNet  MATH  Google Scholar 

  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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  23. Y. E. Zeytuncu, L p-regularity of weighted Bergman projections, Trans. Amer. Math. Soc. 365 (2013), 2959–2976.

    Article  MathSciNet  MATH  Google Scholar 

  24. K. Zhu, Operator Theory in Function Spaces, American Mathematical Society, Providence, RI, 2007.

    Book  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Departamento de Análisis Matemático, Universidad de Málaga, Campus de Teatinos, 29071, Málaga, Spain

    José Ángel Peláez

  2. University of Eastern Finland, P.O.Box 111, 80101, Joensuu, Finland

    Jouni Rättyä

  3. Department of Mathematics, Washington University — St. Louis, One Brookings Drive, St. Louis, MO, 63130, USA

    Brett D. Wick

Authors
  1. José Ángel Peláez
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  2. Jouni Rättyä
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  3. Brett D. Wick
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Corresponding author

Correspondence to Brett D. Wick.

Additional information

This research was supported in part by the Ramón y Cajal program of MICINN (Spain); by Ministerio de Economía y Competitivivad, Spain, project MTM2014-52865-P; by La Junta de Andalucía, (FQM210) and (P09-FQM-4468); by Academy of Finland project no. 268009, and by Faculty of Science and Forestry of University of Eastern Finland project no. 930349; by National Science Foundation Grants DMS # 1560955 and # 1603246.

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Peláez, J.Á., Rättyä, J. & Wick, B.D. Bergman projection induced by kernel with integral representation. JAMA 138, 325–360 (2019). https://doi.org/10.1007/s11854-019-0035-5

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  • DOI: https://doi.org/10.1007/s11854-019-0035-5

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