We study Toeplitz operators from Hardy spaces to weighted Bergman spaces in the unit ball of $C^{n}$. Toeplitz operators are closely related to many classical mappings, such as composition operators, the Volterra type integration operators and Carleson embeddings. We characterize the boundedness and compactness of Toeplitz operators from Hardy spaces $H^{p}$ to weighted Bergman spaces $A_{\alpha}^{q}$ for the different values of $p$ and $q$ in the unit ball.
Ru PENG
,
Yaqing FAN
. TOEPLITZ OPERATORS FROM HARDY SPACES TO WEIGHTED BERGMAN SPACES IN THE UNIT BALL OF Cn[J]. Acta mathematica scientia, Series B, 2022
, 42(1)
: 349
-363
.
DOI: 10.1007/s10473-022-0119-7
[1] Carleson L. An interpolation problem for bounded analytic functions. Amer J Math, 1958, 80:921-930
[2] Carleson L. Interpolations by bounded analytic functions and the corona problem. Ann of Math, 1962, 76:547-559
[3] Choe B R, Koo H, Yi H. Positive Toeplitz operators between harmonic Bergman space. Potential Anal, 2002, 17:307-335
[4] Cowen C, MacCluer B. Composition Operators on Spaces of Analytic Functions. Boca Raton:CRC Press, 1995
[5] Duren P. Theory of Hp Spaces. New York-London:Academic Press, 1970. Reprint:New York:Dover, Mineola, 2000
[6] Feldman N S. Pointwise multipliers from the Hardy space to the Bergman space. Illinois J Math, 1999, 43:211-221
[7] Hastings W. A Carleson measure theorem for Bergman spaces. Proc Amer Math Soc, 1975, 52:237-241
[8] Hu P Y, Shi J H. Multipliers on Dirichlet type spaces. Acta Math Sinica (Engl Ser), 2001, 17(2):263-272
[9] Hu Z J. Extended Cesáro operators on the Bloch space in the unit ball of Cn. Acta Mathematica Scientia, 2003, 23B:561-566
[10] Li S, Stevic S. Riemann-Stieltjes operators between different weighted Bergman spaces. Bull Belg Math Soc Simon Stevin, 2008, 15:677-686
[11] Luecking D H. Trace ideal criteria for Toeplitz operators. J Funct Anal, 1987, 73:345-368
[12] Luecking D H. Embedding derivatives of Hardy spaces into Lebesgue spaces. Proc Lond Math Soc, 1991, 63:595-619
[13] Luecking D H. Embedding theorems for spaces of analytic functions via Khinchine's inequality. Michigan Math J, 1993, 40:333-358
[14] Miihkinen S, Pau J, Perälä A, et al. Volterra type integration operators from Bergman spaces to Hardy spaces. J Funct Anal, 2020, 279(4):108564
[15] Ortega J M, Fàbrega J. Pointwise multipliers and decomposition theorems in $F_{s}^{\infty,q}$. Math Ann, 2004, 329:247-277
[16] Ouyang C H, Yang W S, Zhao R H. Characterizations of Bergman spaces and Bloch space in the unit ball of Cn. Trans Amer Math Soc, 1995, 347:4301-4313
[17] Pau J, Zhao R H. Weak factorization and Hankel forms for weighted Bergman spaces on the unit ball. Mathematische Annalen, 2015, 363:363-383
[18] Pau J, Zhao R H. Carleson measures and Toeplitz operators for weighted Bergman spaces on the unit ball. Michigan Math J, 2015, 64:759-796
[19] Pau J. Characterization of Schatten class Hankel operations on weighted Bergman spaces. Duke Math J, 2016, 165:2771-2791
[20] Pau J. Integration operators between Hardy spaces on the unit ball of Cn. J Funct Anal, 2016, 270:134-176
[21] Pau J, Perälä A. A Toeplitz type operator on Hardy spaces in the unit ball. Tran Amer Math Soc, 2020, 373(5):3031-3062
[22] Peng R, Ouyang C H. Riemann-Stieltjes operators and multipliers on Qp spaces in the unit ball of Cn. J Math Anal Appl, 2011, 377:180-193
[23] Peng R, Xing X L, Jiang L Y. Pointwise multiplication operators from Hardy spaces to weighthed Bergman spaces in the unit ball of Cn. Acta Mathematica Scientia, 2019, 39B(4):1003-1016
[24] Pommerenke C H. Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation. Comment Math Helv, 1997, 52:591-602
[25] Rudin W. Function Theory in the Unit Ball of Cn. New York:Springer-Verlag, 1980
[26] Wu Z J. Carleson measures and multipliers for Dirichlet spaces. J Funct Anal, 1999, 169:148-163
[27] Xiao J. The Qp Carleson measure problem. Adv Math, 2008, 217:2075-2088
[28] Zhang X J. The pointwise multipliers of Bloch type space βp and Dirichlet type space Dq on the unit ball of Cn. J Math Anal Appl, 2003, 285:376-386
[29] Zhao R H. On a general family of function spaces. Ann Acad Sci Fenn Math Dissertations, 1996, 105:56 pp
[30] Zhao R H, Zhu K H. Theory of Bergman Spaces in the unit ball of Cn. Mem Soc Math Fr, 2008, 115
[31] Zhu K H. Spaces of Holomorphic Functions in the Unit Ball. Graduate Texts in Mathematics, Vol 226. New York:Springer-Verlag, 2005
