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The Precise Norm of a Class of Forelli-Rudin Type Operators on the Siegel Upper Half Space

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Abstract

The precise Lp norm of a class of Forelli-Rudin type operators on the Siegel upper half space is given in this paper. The main result not only implies the upper Lp norm estimate of the Bergman projection, but also implies the precise Lp norm of the Berezin transform.

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References

  1. Cheng G Z, Fang X, Wang Z P, Yu J Y. The hyper-singular cousin of the Bergman projection. Trans Amer Math Soc, 2017, 369(12): 8643–8662

    Article  MathSciNet  Google Scholar 

  2. Choe B R, Koo H, Nam K. Optimal norm estimate of operators related to the harmonic Bergman projection on the unit ball. Tohoku Math J, 2010, 62(3): 357–374

    Article  MathSciNet  Google Scholar 

  3. Dostanić M. Two sided norm estimate of the Bergman projection on Lp spaces. Czechoslovak Math J, 2008, 58(133)(2): 569–575

    Article  MathSciNet  Google Scholar 

  4. Dostanić M. Norm of Berezin transform on Lp spaces. J Anal Math, 2008, 104: 13–23

    Article  MathSciNet  Google Scholar 

  5. Dostanić M. Integral operators induced by Bergman type kernels in the half plane. Asymptot Anal, 2010, 67(3/4): 217–228

    Article  MathSciNet  Google Scholar 

  6. Erdélyi A, Magnus W, Oberhettinger F, Tricomi F G. Higher Transcendental Functions, Vol I. New York: McGraw-Hill, 1953

    MATH  Google Scholar 

  7. Forelli F, Rudin W. Projections on spaces of holomorphic functions in balls. Indiana Univ Math J, 1974, 24: 593–602

    Article  MathSciNet  Google Scholar 

  8. Hou X Y, Xu Y. Norm of a Bloch-type projection. Complex Anal Oper Theory, 2019, 13(5): 2269–2276

    Article  MathSciNet  Google Scholar 

  9. Kalaj D, Marković M. Norm of the Bergman projection. Math Scand, 2014, 115(1): 143–160

    Article  MathSciNet  Google Scholar 

  10. Kalaj D, Vujadinović D. Norm of the Bergman projection onto the Bloch space. J Operator Theory, 2015, 73(1): 113–126

    Article  MathSciNet  Google Scholar 

  11. Korányi A. The Poisson integral for generalized half-planes and bounded symmetric domains. Ann Math, 1965, 82: 332–350

    Article  MathSciNet  Google Scholar 

  12. Korányi A, Stein E M. Fatou’s theorem for generalized half-planes. Ann Scuola Norm Sup Pisa, 1968, 22: 107–112

    MathSciNet  MATH  Google Scholar 

  13. Korányi A, Vági S. Singular integrals on homogeneous spaces and some problems of classical analysis. Ann Scuola Norm Sup Pisa, 1971, 25: 575–648

    MathSciNet  MATH  Google Scholar 

  14. Korányi A, Wolf J A. Realization of hermitian symmetric spaces as generalized half-planes. Ann Math, 1965, 81: 265–288

    Article  MathSciNet  Google Scholar 

  15. Kures O, Zhu K H. A class of integral operators on the unit ball of ℂn. Integr Equ Oper Theory, 2006, 56(1): 71–82

    Article  MathSciNet  Google Scholar 

  16. Liu C W. Sharp Forelli-Rudin estimates and the norm of the Bergman projection. J Funct Anal, 2015, 268(2): 255–277

    Article  MathSciNet  Google Scholar 

  17. Liu C W. Norm estimates for the Bergman and the Cauchy-Szegö projections over the Siegel upper half-space. Constr Approx, 2018, 48(3): 385–413

    Article  MathSciNet  Google Scholar 

  18. Liu C W, Liu Y, Hu P Y, Zhou L F. Two classes of integral operators over the Siegel upper half-space. Complex Anal Oper Theory, 2019, 13(3): 685–701

    Article  MathSciNet  Google Scholar 

  19. Liu C W, Perälä A, Zhou L F. Two-sided norm estimates for Bergman-type projections with an asymptotically sharp lower bound. Rev Mat Iberoam, 2018, 34(4): 1427–1441

    Article  MathSciNet  Google Scholar 

  20. Liu C W, Si J J, Hu P Y. Lp-Lq boundedness of Bergman-type operators over the Siegel upper-space. J Math Anal Appl, 2018, 464(2): 1203–1212

    Article  MathSciNet  Google Scholar 

  21. Liu C W, Zhou L F. Norm of an integral operator related to the harmonic Bergman projection. Integr Equ Oper Theroy. 2011, 69(4): 557–566

    Article  MathSciNet  Google Scholar 

  22. Liu C W, Zhou L F. On the p-norm of the Berezin transform. Illinois J Math, 2012, 56(2): 497–505

    Article  MathSciNet  Google Scholar 

  23. Liu C W, Zhou L F. On the p-norm of an integral operator in the half plane. Canad Math Bull, 2013, 56(3): 593–601

    Article  MathSciNet  Google Scholar 

  24. Marković M. Semi-norms of the Bergman projection. Comput Methods Funct Theory, 2016, 16(1): 65–78

    Article  MathSciNet  Google Scholar 

  25. Melentijević P. Norm of the Bergman projection onto the Bloch space with M-invariant gradient norm. Ann Acad Sci Fenn Math, 2019, 44(1): 211–220

    Article  MathSciNet  Google Scholar 

  26. Perälä A. On the optimal constant for the Bergman projection onto the Bloch space. Ann Acad Sci Fenn Math, 2012, 37(1): 245–249

    Article  MathSciNet  Google Scholar 

  27. Perälä A. Bloch space and the norm of the Bergman projection. Ann Acad Sci Fenn Math, 2013, 38(2): 849–853

    Article  MathSciNet  Google Scholar 

  28. Sehba B. On the boundedness of the fractional Bergman operators. Abstr Appl Anal, 2017, https://doi.org/10.1155/2017/8363478

  29. Stein E M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton, New Jersey: Princeton University Press, 1993

    MATH  Google Scholar 

  30. Zhao R H. Generalization of Schur’s test and its application to a class of integral operators on the unit ball of ℂn. Integr Equ Oper Theroy, 2015, 82(4): 519–532

    Article  MathSciNet  Google Scholar 

  31. Zhou L F. On the boundedness and the norm of a class of integral operators. Acta Math Sci, 2015, 35B(6): 1475–1482

    Article  MathSciNet  Google Scholar 

  32. Zhou L F. Norm estimates for weighted Bergman projection on the upper half plane. Complex Anal Oper Theory, 2018, 12(1): 217–233

    Article  MathSciNet  Google Scholar 

  33. Zhou L F, Lu J. The improvement on the boundedness and norm of a class of integral operators on Lp space. J Funct Spaces, 2015: https://doi.org/10.1155/2015/362681

  34. Zhou L F, Lu J. The Forelli-Rudin type theorem and a class of integral operator related to hypergeometric function (in Chinese). Sci Sin Math, 2019, 49: 765–780

    Article  Google Scholar 

  35. Zhu K H. A sharp norm estimate of the Bergman projection on Lp spaces//Bergman Spaces and Related Topics in Complex Analysis. Contemp Math 404. Providence, RI: Amer Math Soc, 2006: 195–205

    Google Scholar 

  36. Zhu K H. Operator Theory in Function Spaces. 2nd ed. Providence, RI: Amer Math Soc, 2007

    Book  Google Scholar 

  37. Zhu K H. Analysis on Fock Spaces. Graduate Texts in Mathematics, Vol 263. New York: Springer-Verlag, 2012

    Book  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Huzhou University, Huzhou, 313000, China

    Lifang Zhou  (周立芳) & Yun Fan  (樊云)

  2. School of Internet, Anhui University, Hefei, 230039, China

    Jin Lu  (卢金)

Authors
  1. Lifang Zhou  (周立芳)
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  2. Yun Fan  (樊云)
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  3. Jin Lu  (卢金)
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Corresponding author

Correspondence to Yun Fan  (樊云).

Additional information

The first author was supported by the National Natural Science Foundation of China (11801172, 11771139, 12071130). The second author was supported by the Natural Science Foundation of Zhejiang Province (LQ21A010002). The third author was supported by the Natural Science Foundation of Zhejiang Province (LY20A010007).

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Zhou, L., Fan, Y. & Lu, J. The Precise Norm of a Class of Forelli-Rudin Type Operators on the Siegel Upper Half Space. Acta Math Sci 41, 1537–1546 (2021). https://doi.org/10.1007/s10473-021-0508-3

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  • DOI: https://doi.org/10.1007/s10473-021-0508-3

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