Articles

THE PRECISE NORM OF A CLASS OF FORELLI-RUDIN TYPE OPERATORS ON THE SIEGEL UPPER HALF SPACE

  • Lifang ZHOU ,
  • Yun FAN ,
  • Jin LU
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  • 1. Department of Mathematics, Huzhou University, Huzhou 313000, China;
    2. School of Internet, Anhui University, Hefei 230039, China

Received date: 2020-02-25

  Revised date: 2021-04-27

  Online published: 2021-10-21

Supported by

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).

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.

Cite this article

Lifang ZHOU , Yun FAN , Jin LU . THE PRECISE NORM OF A CLASS OF FORELLI-RUDIN TYPE OPERATORS ON THE SIEGEL UPPER HALF SPACE[J]. Acta mathematica scientia, Series B, 2021 , 41(5) : 1537 -1546 . DOI: 10.1007/s10473-021-0508-3

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
[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
[3] Dostanić M. Two sided norm estimate of the Bergman projection on Lp spaces. Czechoslovak Math J, 2008, 58(133)(2):569-575
[4] Dostanić M. Norm of Berezin transform on Lp spaces. J Anal Math, 2008, 104:13-23
[5] Dostanić M. Integral operators induced by Bergman type kernels in the half plane. Asymptot Anal, 2010, 67(3/4):217-228
[6] Erdélyi A, Magnus W, Oberhettinger F, Tricomi F G. Higher Transcendental Functions, Vol I. New York:McGraw-Hill, 1953
[7] Forelli F, Rudin W. Projections on spaces of holomorphic functions in balls. Indiana Univ Math J, 1974, 24:593-602
[8] Hou X Y, Xu Y. Norm of a Bloch-type projection. Complex Anal Oper Theory, 2019, 13(5):2269-2276
[9] Kalaj D, Marković M. Norm of the Bergman projection. Math Scand, 2014, 115(1):143-160
[10] Kalaj D, Vujadinović D. Norm of the Bergman projection onto the Bloch space. J Operator Theory, 2015, 73(1):113-126
[11] Korányi A. The Poisson integral for generalized half-planes and bounded symmetric domains. Ann Math, 1965, 82:332-350
[12] Korányi A, Stein E M. Fatou's theorem for generalized half-planes. Ann Scuola Norm Sup Pisa,1968, 22:107-112
[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
[14] Korányi A, Wolf J A. Realization of hermitian symmetric spaces as generalized half-planes. Ann Math, 1965, 81:265-288
[15] Kures O, Zhu K H. A class of integral operators on the unit ball of $\mathbb{C}^n$. Integr Equ Oper Theory, 2006, 56(1):71-82
[16] Liu C W. Sharp Forelli-Rudin estimates and the norm of the Bergman projection. J Funct Anal, 2015, 268(2):255-277
[17] Liu C W. Norm estimates for the Bergman and the Cauchy-Szegö projections over the Siegel upper halfspace. Constr Approx, 2018, 48(3):385-413
[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
[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
[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
[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
[22] Liu C W, Zhou L F. On the p-norm of the Berezin transform. Illinois J Math, 2012, 56(2):497-505
[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
[24] Marković M. Semi-norms of the Bergman projection. Comput Methods Funct Theory, 2016, 16(1):65-78
[25] Melentijević P. Norm of the Bergman projection onto the Bloch space with $\mathcal {M}$-invariant gradient norm. Ann Acad Sci Fenn Math, 2019, 44(1):211-220
[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
[27] Perälä A. Bloch space and the norm of the Bergman projection. Ann Acad Sci Fenn Math, 2013, 38(2):849-853
[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
[30] Zhao R H. Generalization of Schur's test and its application to a class of integral operators on the unit ball of $\mathbb{C}^n$. Integr Equ Oper Theroy, 2015, 82(4):519-532
[31] Zhou L F. On the boundedness and the norm of a class of integral operators. Acta Math Sci, 2015, 35B(6):1475-1482
[32] Zhou L F. Norm estimates for weighted Bergman projection on the upper half plane. Complex Anal Oper Theory, 2018, 12(1):217-233
[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:http://dx.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
[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
[36] Zhu K H. Operator Theory in Function Spaces. 2nd ed. Providence, RI:Amer Math Soc, 2007
[37] Zhu K H. Analysis on Fock Spaces. Graduate Texts in Mathematics, Vol 263. New York:Springer-Verlag, 2012
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