Positive Toeplitz Operators on the Bergman Spaces of the Siegel Upper Half-Space

Abstract

We characterize bounded and compact positive Toeplitz operators defined on the Bergman spaces over the Siegel upper half-space.

Keywords

Toeplitz operators / Bergman spaces / Siegel upper half-space / Carleson measure / Berezin transform

References

[1] Axler S. Bergman spaces and their operators, surveys of some recent results in operator theory, vol. 1. Pitman Res. Notes Math. Ser.. 1988, 171 1-50

[2] Choe B, Koo H, Yi H. Positive Toeplitz operators between the harmonic Bergman spaces. Potential Anal.. 2002, 17 4 307-335

[3] Choe B, Nam K. Toeplitz operators and Herz spaces on the half-space. Integral Equ. Oper. Theory. 2007, 59 4 501-521

[4] Coifman RR, Rochberg R. Representation theorems for holomorphic and harmonic functions in $L^p$. Astérisque. 1980, 77 11-66

[5] Djrbashian MM, Karapetyan AH. Integral representations for some classes of functions holomorphic in a Siegel domain. J. Math. Anal. Appl.. 1993, 179 91-109

[6] Gindikin SG. Analysis in homogeneous domains. Russ. Math. Surv.. 1964, 19 4 1-89

[7] Krantz SG. Function theory of several complex variables. Reprint of the 1992 edition. 2001 Providence, RI: AMS Chelsea Publishing

[8] Liu C. Sharp Forelli-Rudin estimates and the norm of the Bergman projection. J. Funct. Anal.. 2015, 268 255-277

[9] Liu C. Norm estimates for the Bergman and Cauchy–Szegö projections over the Siegel upper half-space. Constr. Approx.. 2018, 48 3 385-413

[10] Liu C, Liu Y, Hu P, Zhou L. Two classes of integral operators over the Siegel upper half-space. Complex Anal. Oper. Theory. 2019, 3 685-701

[11] Stein EM. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. 1993 Princeton, NJ: Princeton University Press

[12] Yamaji S. Positive Toeplitz operators on weighted Bergman spaces of a minimal bounded homogeneous domain. J. Math. Soc. Jpn.. 2013, 65 4 1101-1115

[13] Zhang X, Li S, Shang Q, Guo Y. An integral estimate and the equivalent norms on $F(p, q, s, k)$ spaces in the unit ball. Acta Math. Sci. Ser. B. 2018, 38 6 1861-1880

[14] Zhu K. Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains. J. Oper. Theory. 1988, 20 329-357

[15] Zhu K. Spaces of Holomorphic Functions in the Unit Ball. 2005 New York: Springer

[16] Zhu K. Operator Theory in Function Spaces. 2007 2 Providence, RI: American Mathematical Society

Funding

National Natural Science Foundation of China(11571333)