$Q_K$ SPACES: A BRIEF AND SELECTIVE SURVEY

doi:10.1007/s10473-021-0615-1

摘要/Abstract

摘要： This article traces several prominent trends in the development of Möbius invariant function spaces $Q_K$ with emphasis on theoretic aspects.

关键词: Möbius invariant function spaces, $Q_p$ spaces, $Q_K$ spaces, $K$-Carleson measures

Abstract: This article traces several prominent trends in the development of Möbius invariant function spaces $Q_K$ with emphasis on theoretic aspects.

Key words: Möbius invariant function spaces, $Q_p$ spaces, $Q_K$ spaces, $K$-Carleson measures

中图分类号:

30H25

鲍官龙, 乌兰哈斯. $Q_K$ SPACES: A BRIEF AND SELECTIVE SURVEY[J]. 数学物理学报(英文版), 2021, 41(6): 2039-2054.

Guanlong BAO, Hasi WULAN. $Q_K$ SPACES: A BRIEF AND SELECTIVE SURVEY[J]. Acta mathematica scientia,Series B, 2021, 41(6): 2039-2054.

参考文献 86

[1]	Wulan H, Zhu K. Möbius Invariant QK Spaces. Berlin:Springer-Verlag, 2017
[2]	Aulaskari R, Xiao J, Zhao R. On subspaces and subsets of BMOA and UBC. Analysis, 1995, 15(2):101-121
[3]	Xiao J. Holomorphic Q Classes. Lecture Notes in Mathematics 1767. Berlin:Springer-Verlag, 2001
[4]	Xiao J. Geometric Qp Functions. Basel, Boston, Berlin:Birkhäuser Verlag, 2006
[5]	Essén M, Wulan H. On analytic and meromorphic functions and spaces of QK type. Uppsala University, Department of mathematics Report, 2000:1-26. Illinois J Math, 2002, 46(4):1233-1258
[6]	Wu P, Wulan H. Characterizations of QT spaces. J Math Anal Appl, 2001, 254(2):484-497
[7]	Axler S. The Bergman spaces, the Bloch space, and commutators of multiplication operators. Duke Math J, 1986, 53(2):315-332
[8]	Baernstein II A. Analytic functions of bounded mean oscillation//Aspects of Contemporary Complex Analysis. London:Academic Press, 1980:3-36
[9]	Aulaskari R, Lappan P. Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal//Complex Analysis and Its Applications. Pitman Research Notes in Mathematics, 305. Harlow:Longman Scientific & Technical, 1994:136-146
[10]	Bao G, Mashreghi J, Pouliasis S, Wulan H. Möbius invariant function spaces and Dirichlet spaces with superharmonic weights. J Aust Math Soc, 2019, 106(1):1-18
[11]	Wulan H, Zhu K. Derivative free characterizations of QK spaces. J Aust Math Soc, 2007, 82(2):283-295
[12]	Zhou J, Han J. Interpolating sequences in QK spaces. J of Math (PRC), 2016, 36(3):511-518
[13]	Aleman A, Simbotin A. Estimates in Möbius invariant spaces of analytic functions. Complex Var Theory Appl, 2004, 49:487-510
[14]	Essén M, Wulan H, Xiao J. Function-theoretic aspects of Möbius invariant QK spaces. J Funct Anal, 2006, 230(1):78-115
[15]	Wulan H, Ye F. Universal Teichmüller space and QK spaces. Ann Acad Sci Fenn Math, 2014, 39(2):691-709
[16]	Wulan H, Zhu K. QK spaces via higher order derivatives. Rocky Mountain J Math, 2008, 38(1):329-350
[17]	Wulan H, Zhou J. QK and Morrey type spaces. Ann Acad Sci Fenn Math, 2013, 38(1):193-207
[18]	Wu Z, Xie C. Decomposition theorems for Qp spaces. Ark Mat, 2002, 40(2):383-401
[19]	Wulan H, Zhou J. Decomposition theorems for QK spaces and applications. Forum Math, 2014, 26(2):467-495
[20]	Rochberg R. Decomposition theorems for Bergman space and their applications in operators and function theory. Operator and Function Theory (Lancaster), NATO ASI Series C, 1984:225-277
[21]	Rochberg R, Semmes S. A decomposition theorem for BMO and applications. J Funct Anal, 1986, 67(2):228-263
[22]	Rochberg R, Wu Z. A new characterization of Dirichlet type spaces and applications. Illinois J. Math, 1993, 37(1):101-122
[23]	Stegenga D. Multipliers of the Dirichlet space. Illinois J Math, 1980, 24:113-139
[24]	Pau J, Peláez J. Multipliers of Möbius invariant Qs spaces. Math Z, 2009, 261(3):545-555
[25]	Li D, Wulan H. Corona and Wolff theorems for the multiplier algebras of QK spaces (in Chinese). Sci Sin Math, 2021, 51:301-314
[26]	Dyakonov K. Self-improving behaviour of inner functions as multipliers. J Funct Anal, 2006, 240(2):429- 444
[27]	Peláez J. Inner functions as improving multipliers. J Funct Anal, 2008, 255(6):1403-1418
[28]	Bao G, Lou Z, Qian R, Wulan H. Improving multipliers and zero sets in QK spaces. Collect Math, 2015, 66(3):453-468
[29]	Wulan H, Ye F. Some results in Möbius invariant QK spaces. Complex Var Elliptic Equ, 2015, 60(11):1602-1611
[30]	Li S, Wulan H. Derivative free characterizations of QK spaces II. Sarajevo J Math, 2006, 14(1):63-71
[31]	Bao G, Lou Z, Qian R, Wulan H. On absolute values of QK functions. Bull Korean Math Soc, 2016, 53(2):561-568
[32]	Ahlfors L. Lectures on Quasi-conformal Mappings. Princeton:Van Nostrand, 1966
[33]	Ahlfors L, Bers L. Riemann's mapping theorem for variable metrics. Ann Math, 1960, 72:385-404
[34]	Gehring F. Univalent functions and the Schwarzian derivative. Comment Math Helv, 1977, 52:561-572
[35]	Astala K, Gehring F. Injectivity, the BMO norm, and the universal Teichmüller space. J Anal Math, 1986, 46:16-57
[36]	Lehto O. Univalent Functions and Teichmüller Spaces. Springer-Verlag, 1987
[37]	Astala K, Zinsmeister M. Teichmüller spaces and BMOA. Math Ann, 1991, 289(4):613-625
[38]	Shen Y, Wei H. Universal Teichmüller space and BMO. Adv Math, 1999, 234:129-148
[39]	Wei H, Shen Y. On the tangent space to the BMO-Teichmüller space. J Math Anal Appl, 2014, 419(2):715-726
[40]	Jin J, Tang S. On QK-Teichmüller spaces. J Math Anal Appl, 2018, 467(1):622-637
[41]	Pommerenke Ch. On the mean growth of the solutions of complex linear differential equations in the disk. Complex Variables Theory Appl, 1982, 1(1):23-38
[42]	Li H, Li S. Nonlinear differential equation and analytic function spaces. Complex Var Elliptic Equ, 2018, 63(1):136-149
[43]	Li H, Wulan H. Linear differential equations with solutions in the QK spaces. J Math Anal Appl, 2011, 375(2):478-489
[44]	Shapiro J. The essential norm of a composition operator. Ann Math, 1987, 125:375-404
[45]	Shapiro J. Composition Operators and Classical Function Theory. New York:Springer-Verlag, 1993
[46]	De Branges L. A proof of the Bieberbach conjecture. Acta Math, 1985, 154:137-152
[47]	De Branges L. Underlying concepts in the proof of the Bieberbach conjecture. Proceedings of the International Congress of Mathematicians (Berkeley), 1986:25-42
[48]	Smith W, Zhao R. Composition operators mapping into the Qp spaces. Analysis, 1997, 17:239-263
[49]	Wu Y, Wulan H. Products of differentiation and composition operators on the Bloch space. Collect Math, 2012, 63(1):93-107
[50]	Bourdon P, Cima J, Matheson A. Compact composition operators on BMOA. Trans Amer Math Soc, 1999, 351(6):2183-2196
[51]	Smith W. Compactness of composition operators on BMOA. Proc Amer Math Soc, 1999, 127(9):2715-2725
[52]	Wulan H. Compactness of composition operators on BMOA and VMOA. Sci China Ser A, 2007, 50(7):997-1004
[53]	Wulan H, Zheng D, Zhu K. Compact composition operators on the Bloch and BMOA spaces. Proc Amer Math Soc, 2009, 137(11):3861-3868
[54]	Zhou J. Predual of QK spaces. J Funct Spaces Appl, 2013, 2013:Article ID 252735, 6 pages
[55]	Zhan M, Cao G. Duality of QK-type spaces. Bull Korean Math Soc, 2014, 51(5):1411-1423
[56]	Perfekt K. Duality and distance formulas in spaces defined by means of oscillation. Ark Mat, 2013, 51(2):345-361
[57]	Korenblum B. BMO Estimates and radial growth of Bloch functions. Bull of Amer Math Soc (New Series), 1985, 12(1):99-102
[58]	Anderson J, Clunie J, Pommerenke Ch. On Bloch functions and normal functions. J Reine Angew Math, 1974, 270:12-37
[59]	Anderson J. Bloch functions:The basic theory//Power S C, ed. Operators and Function Theory, 152. Bordrecht:Reidel, 1984:1-17
[60]	Ghatage P, Zheng D. Analytic functions of bounded mean oscillation and the Bloch space. Int Equ Oper Theory, 1993, 17(4):501-515
[61]	Axler S, Zhu K. Boundary behavior of derivatives of analytic function. Michigan Math J, 1992, 39(1):129-143
[62]	Lou Z, Chen W. Distances from Bloch functions to QK type spaces. Integr Equ Oper Theory, 2010, 67(2):171-181
[63]	Zhao R. Distances from Bloch functions to some Möbius invariant spaces. Ann Acad Sci Fenn Math, 2008, 33(1):303-313
[64]	Galán N, Nicolau A. The closure of the Hardy space in the Bloch space norm. St Petersburg Math J, 2011, 22(1):55-59
[65]	Galanopoulos P, Galán N, Pau J. Closure of Hardy spaces in the Bloch space. J Math Anal Appl, 2015, 429(2):1214-1221
[66]	Tjani M. Distance of a Bloch function to the little Bloch space. Bull Aust Math Soc, 2006, 74(1):101-119
[67]	Danikas N, Siskakis A G. The Cesáro operator on bounded analytic functions. Analysis, 1993, 13(3):295-299
[68]	Essén M, Xiao J. Some results on Qp spaces, 0
[69]	Bao G, Wulan H, Ye F. The range of the Cesáro operator acting on H∞. Canad Math Bull, 2020, 63(3):633-642
[70]	Pau J, Peláez J. On the zeros of functions in Dirichlet-type spaces. Trans Amer Math Soc, 2011, 363(4):1981-2002
[71]	Fefferman C. Characterizations of bounded mean oscillations. Bull Amer Math Soc, 1971, 77:587-588
[72]	Fefferman C, Stein E. Hp spaces of several variables. Acta Math, 1972, 129:137-193
[73]	Nicolau A, Xiao J. Bounded functions in Möbius invariant Dirichlet spaces. J Funct Anal, 1997, 150(2):383-425
[74]	Pau J. Bounded Möbius invariant QK spaces. J Math Anal Appl, 2008, 338(2):1029-1042
[75]	Garnett J. Bounded Analytic Functions. New York:Academic Press, 1982
[76]	Jones P. L∞ estimates for the $\overline \partial $-problem in a half plane. Acta Math, 1983, 150:137-152
[77]	Carleson L. An interpolation problem for bounded analytic functions. Amer J Math, 1958, 80:921-930
[78]	Carleson L. Interpolations by bounded analytic functions and the corona problem. Ann Math, 1962, 76:547-559
[79]	Pau J. Multipliers of Qs spaces and the corona theorem. Bull Lond Math Soc, 2008, 40(2):327-336
[80]	Garnett J, Jones P. BMO from dyadic BMO. Pacific J Math, 1982, 99:351-371
[81]	Janson S. On the space Qp and its dyadic counterpart. Complex Analysis and Differential Equations (Uppsala), 1997:194-205
[82]	Essén M, Janson S, Peng L, Xiao J. Q-spaces of several real variables. Indiana Univ Math J, 2000, 49(2):575-615
[83]	Bao G, Wulan H. QK spaces of several real variables. Abstr Appl Anal, 2014:Art. ID 931937, 14 pp
[84]	Bao G, Wulan H. John-Nirenberg type inequality and wavelet characterization for QK(Rn) spaces (in Chinese). Sci Sin Math, 2015, 45(11):1833-1846
[85]	Xu W. QK spaces on the unit ball of Cn. J Nanjing Norm Univ Nat Sc Ed, 2005, 28:21-28
[86]	Chen H, Xu W. Lacunary series and QK spaces on the unit ball. Ann Acad Sci Fenn Math, 2010, 35:47-57