Abstract
In this paper, we give a survey of some recent progress in terms of verifying Carleson measures; this includes the difference between two definitions of a Carleson measure, the Bergman tree condition, the T1 condition for Besov-Sobolev spaces on a complex ball, vector-valued Carleson measures, Carleson measures in strongly pseudoconvex domains and reverse Carleson measures.
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Abate M, Mongodi S, Raissy J. Toeplitz operators and skew Carleson measures for weighted Bergman spaces on strongly pseudoconvex domains. J Operator Theory, 2020, 84(2): 339–364
Abate M, Raissy J, Saracco A. Toeplitz operators and Carleson measures in strongly pseudoconvex domains. J Funct Anal, 2012, 263(11): 3449–3491
Abate M, Saracco A. Carleson measures and uniformly discrete sequences in strongly pseudocomvex domains. J London Math Soc, 2011, 83(2): 587–605
Arcozzi N, Rochberg R, Sawyer E. Carleson measures for analytic Besov spaces. Rev Mat Iberoamericana, 2002, 2: 443–510
Arcozzi N, Rochberg R, Sawyer E. Carleson measures and interpolating sequences. Mem Amer Math Soc, 2006, 182(859)
Arcozzi N, Rochberg R, Sawyer E. Carleson measures for the Drury-Arveson Hardy space and other Besov-Sobolev spaces on complex balls. Adv Math, 2008, 2: 1107–1180
Carleson L. An interpolation problem for bounded analytic functions. Amer J Math, 1958, 2: 921–930
Carleson L. Interpolations by bounded analytic functions and the corona problem. Ann of Math, 1962, 2: 547–559
Cima J A, Wogen W R. A Carleson measure theorem for the Bergman space on the ball. J Operator Theory, 1982, 7 (1): 157–165
Duren P L, Weir R. The pseudohyperbolic metric and Bergman spaces in the ball. Trans Amer Math Soc, 2007, 2: 63–76
Hartmann A, Massaneda X, Nicolau A, Ortega-Cerda J. Reverse Carleson measures in Hardy spaces. Collectanea Math, 2014, 2: 357–365
Hastings W W. A Carleson measure theorem for Bergman spaces. Proc Amer Math Soc, 1975, 2: 237–241
Lefèvre P, Li D, Queffelec H, Rodriguez-Piazza L. Nevanlinna counting function and Carleson function of analytic maps. Math Ann, 2011, 2: 305–326
Lefèvre P, Li D, Queffelec H, Rodriguez-Piazza L. Some revisited results about composition operators on Hardy spaces. Rev Mat Iberoamericana, 2012, 2: 57–76
Luecking D. A technique for characterizing Carleson measure on Bergman spaces. Proc Amer Math Soc, 1983, 2: 656–660
Nazarov F, Treil S, Volberg A. The Tb-theorem on non-homogeneous spaces. Acta Math, 2003, 2: 151–239
Ouyang C, Xu Q. BMO functions and Carleson measures with values in uniformly convex spaces. Canad J Math, 2010, 2: 827–844
Ouyang C, Yang W, Zhao R. Characterizations of Bergman spaces and Bloch space in the unit ball of Cn. Trans Amer Math Soc, 1995, 2: 4301–4313
Pau J, Zhao R. Carleson measures and Toeplitz operators for weighted Bergman spaces on the unit ball. Michigan Math J, 2015, 2: 759–796
Peng R, Ouyang C. Carleson measures for Besov-Sobolev spaces with applications in the unit ball of Cn. Acta Math Sci, 2013, 33B: 1219–1230
Volberg A, Wick B D. Bergman-type singular integral operators and the characterization of Carleson measures for Besov-Sobolev spaces on the complex ball. Amer J Math, 2012, 2: 949–992
Zhu K. Spaces of Holomorphic Functions in the Unit Ball. Graduate Texts in Mathematics, Vol 226. New York: Springer-Verlag, 2005
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Dedicated to the memory of Professor Jiarong YU
Supported by the National Natural Science Foundation of China (11771441, 11601400).
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Deng, F., Ouyang, C. & Peng, R. Some Questions Regarding Verification of Carleson Measures. Acta Math Sci 41, 2136–2148 (2021). https://doi.org/10.1007/s10473-021-0620-4
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DOI: https://doi.org/10.1007/s10473-021-0620-4