Carleson Measures on the Weighted Bergman Spaces with Békollé Weights

Abstract

In this paper, the authors characterize Carleson measures for the weighted Bergman spaces with Békollé weights on the unit ball. They apply the Carleson embedding theorem to study the properties of Toeplitz-type operators and composition operators acting on such spaces.

Keywords

Békollé weight / Bergman space / Carleson measure / Toeplitz operator / Composition operator

References

[1] Bekollé D. Inégalité à poids pour le projecteur de Bergman dans la boule unité de ℂ ⁿ. Studia Math., 1981, 71(3): 305-323 1982

[2] Bekollè D, Bonami A. Inégalités à poids pour le noyau de Bergman. C. R. Acad. Sci. Paris Sér. A-B, 1978, 286(18): A775-A778 (in French)

[3] Constantin O. Carleson embeddings and some classes of operators on weighted Bergman spaces. J. Math. Anal. Appl., 2010, 365: 668-682

[4] Cowen C, MacCluer B D. Composition Operators on Spaces of Analytic Functions, 1995, Boca Raton: CRC Press

[5] Lin P, Rochberg R. Trace ideal criteria for Toeplitz and Hankel operators on the weighted Bergman spaces with exponential type weights. Pacific J. Math., 1996, 173: 127-146

[6] Luecking D. Representation and duality in weighted spaces of analytic functions. Indiana Univ. Math. J., 1985, 34: 319-336

[7] Luecking D. Trace ideal criteria for Toeplitz operators. J. Funct. Anal., 1987, 73(2): 345-368

[8] Luecking D. Embedding theorems for spaces of analytic functions via Khinchine’s inequality. Michigan Math. J., 1993, 40(2): 333-358

[9] Pau J. A remark on Schatten class Toeplitz operators on Bergman spaces. Proc. Amer. Math. Soc., 2014, 142(8): 2763-2768

[10] Pott S, Reguera M C. Sharp Békollé estimates for the Bergman projection. J. Funct. Anal., 2013, 265(12): 3233-3244

[11] Rahm R, Tchoundja E, Wick B. Weighted estimates for the Berezin transform and Bergman projection on the unit ball in ℂ ⁿ. Math. Z., 2017, 286: 1465-1478

[12] Tong, C., Li, J. and Arroussi, H., The Berezin transform of Toeplitz operators on the weighted Bergman space, submitted.

[13] Zhu K. Spaces of Holomorphic Functions in the Unit Ball, 2005, New York: Springer-Verlag

[14] Zhu K. Schatten class Toeplitz operators on weighted Bergman spaces of the unit ball. New York J. Math., 2007, 13: 299-316