摘要
1998年伍鹏程在文章《On increasing functions, Bloch functions and normal functions》中研究了Bloch函数和normal函数的判别准则时引入了一个增函数,2001年伍鹏程和乌兰哈斯在此文的基础上于文《Characterizations of QK spaces》中提出了QK空间的概念。至今QK空间及QK型空间是复函数几何理论研究的热点,有很多问题尚未解决。最近,乌兰哈斯和朱克和等人做了很多关于QK空间与其他空间之间复合算子性质的研究,如乌兰哈斯和伍鹏程给出了从Bloch空间到QK空间上复合算子有界性及紧性刻画,程训辉在文《加权Bloch空间到QK空间的复合算子》中讨论了加权Bloch空间(小加权Bloch空间)和QK间的复合算子Cφ的有界性。2009年周江河在文《关于解析QT,s空间》中提出了QT,s空间的概念和研究了这个空间上的包含关系。近年来,双曲函数空间的研究也成为研究热点,2003年李晓南在其硕士论文中提出了双曲Qp空间并就其初等性质进行了初探。
本论文主要由四部分组成：
第一章与第二章介绍了函数空间及其算子理论的研究背景和一些主要研究成果。
第三章将双曲导数与QT,s空间相结合引入了双曲QT,s空间QT,s*,系统研究了该空间的一些基本性质,其中包括利用Carleson测度来对该空间进行刻画。
第四章主要研究了单位圆上对数Bloch空间Blog(B0,log)和空间QT,S的复合算子Cφ的有界性和紧性的充分必要条件。
第五章主要讨论了从Bloch型空间Bα到加权Bloch型空间Blogβ的Volterra算子的有界性与紧性的充要条件。
P.Wu studied the criterion of normal and Bloch functions with a nondecreasing function in the paper On increasing functions,Bloch functions and normal functions in 1998, on the basic of this paper, P.Wu and H.Wulan formally introduce the definition of QK spaces in their paper Characterizations of QK spaces. By now QK spaces also attracts a lot of attentions of many people. At the same time, there is a lot of unsolved problems in this field. Recently, P.Wu and K.Zhu made a lot of research in the properties of Composition Operators between the QK spaces and other spaces. Such as, P.Wu and H.Wulan gave the characterizations of bounded and compact composition operators from Bloch spaces to QK spaces, X.Cheng in the paper Composition operator between the weighted Bloch space and QK space studied the boundedness of the composition operators between weighted bloch space and QK space. J.Zhou in the paper About the analytic QT,s spaces in 2009 formally introduce the definition of QT,s spaces and the inclusion relations in this space. In recent years, Space of hyperbolic functions has become a research focus, X.Li introduce the definition of hyberbolic Qp spaces in his master's thesis in 2003, and investigate its primary properties.
This thesis mainly consite of four parts:
Chapter 1 and chapter 2 introduce the background of the Function spaces and operator theory and some main research results.
Chapter 3 we have the hyperbolic QT,s space QT,s* by combining the hyperbolic derivative and the classic QT,s space, investigate some basic properties of hyberbolic QT,s space, include use Carleson measure to charactering hyberbolic QT,s space.
Chapter 4 mainly discuss the necessary and sufficient condition when the composition operators Cφfrom Blog (B0,log) space to QT,s space are bounded and compact.
Chapter 5 mainly discuss the necessary and sufficient conditions when the Volterra Operator from Bloch type analytic function spaces Bαto weighted bloch type analytic function spaces Blogβ are bounded and compact.
引文
[1]M.Kotilainen.On composition oerators in QK type spaces[J]. Funct.spaces. Appl. 2007,23(5):103-112.
[2]周江河.解析的QT,s空间[J].五邑大学学报(自然科学版),2009年02期,128-131.
[3]E.Nordgren. Composition operators. Can[J]. Math.1968,20:442-449.
[4]C.C.Cowen and B.Maccluer. Composition operators on spaces of Analytic Functions[M]. CRC Press, Boca Raton, FL,1995.
[5]J.H.Shapiro. Composition operators and Clsssical Function Theory[M]. Springer-Verlag. New York.1993.
[6]Zhu K. Operator theory in functions spaces [M]. New York:Springer,1990.
[7]B.D.MacCluer snd J.Shapiro.Angular derivatives and compact comporition operators on the Hardy and Bergman spaces[J]. Canad.J.Math.1986,38:878-906.
[8]K.Madigan and A.Matheson. Compact composition operators on the Bloch space[J]. Trans.Amer.Math.Soc.1995,347(7):2679-2687.
[9]A.Aleman, J.A.Cima. An integral operatoron Hp and Hardy's inequality[J]. Anal.Math.2001,85:157-176.
[10]A.Aleman, A.G.Siskakis, An integral operator on Hp[J]. Complex Variables, 1995,28:149-158.
[11]A.Aleman,A.G.Siskakis,Integration operators on Bergman spaces [J]. Indiana Univ. Math.1997,46:337-356.
[12]A.G.Siskakis and R.Zhao. A Volterra type operators on spaces of analytic functions[J]. Contemporary Math.1999,232:299-311.
[13]瞿丹.Bloch型空间上的Volterra算子[J].韩山师范学院学报：2008,年06期.
[14]EssenM, Wulan H S, Xiao J. Several function theoretic characterizations of Mobius invariant QK spaces. Journal of Functional Analysis[J].2006,230: 78-115.
[15]谭海鸥.一类加权α-Bloch空间上复合算子的有界性和紧性[J].五邑大学学报(自然科学版)：2004年04期.
[16]X.Li, F.P.Gonzalez and Rattya, Composition operators in Hyperbolic Q-classes[J]. Annales Acade Sci Fennecs Mathematical,2006,31:391-404.
[17]S.Yamashita, Holomorphic functions of Hyperbolically bounded mean oscillation[J], Bollettino U.M.I.1986,5 (6)-B:986-1000.
[18]Wu P C, Wulan H S. Characterizations of QT spaces [J]. Math. Anal Appl 2001, 254:484-497.
[19]唐树安.双曲的QK空间[D].贵州师范大学：2008.
[20]J.B.Garnett, Bounded analytic functions,Academic Press, New York,1981.
[21]张芳,刘永民,王峰.从Ba到QK的复合算子[J].数学研究,2007年04期.
[22]Wulan H., Some problems on normal and little normal functions[J]. Complex Variables Theory and Appl.,1998,36:301-309.
[23]Wulan H S, Zhu K H. Derivative free characterization of QK spaces Preprint.
[24]程训辉.加权Bloch空间到QK空间的复合算子[J].徐州工程学院学报(自然科学版),2008年04期。
[25]叶善力.不同权的Bloch型空间之间的加权复合算子[J].数学学报.2007,50：927-942.
[26]Yanyan Yu, Yongmin Liu. On a Li-Stevic integral-type operators between different weighted Bloch-type spaces[J]. Inequalities and Applications,2008, Article ID 780845,14pages.
[27]李颂孝,有界对称域上加权Dirichlet空间上的加权复合算子[J].湖州师范学院学报,2003年06期,22-25
[28]张南岳.陈怀惠,复变函数论选讲[M].北京大学出版社,1995.8.
[29]EssenM, W ulan H S. On analytic and meromorphic functions and spaces of QK type Illinois [J]. Math.2002,46:1233-1258.
[30]W ulan H S. Compactness of the composition operators from the Bloch space B to QK space[J] Acta Math.Sin.(Engl Ser),2005,21:1415-1424.
[31]Wu P C, W ulan H S.Composition oerators from the Bloch space in to the spaces QK [J]. Internal J.Math.Scie 2003,31:1973-1979.
[32]高进寿,贾厚玉.Bergman空间上复合算子的紧性刻划[J]. Bergman空间上复合算子的紧性刻画.高校应用数学学报A辑(中文版),2003年04期,446-454.
[33]李觉先,吕方,杨瑞梅.加权复合算子的紧性[J].吉林大学自然科学学报,1998年01期,14-18.
[34]李颂孝,加权Bergman空间上复合算子的范数[J].嘉兴学院学报,2003年06期,9-10.
[35]李颂孝,有界对称域上加权Dirichlet空间上的加权复合算子[J].湖州师范学院学报,2003年06期,22-25.
[36]李颂孝,胡俊云,指数型加权Bergman空间上Schatten类复合算子[J].南昌大学学报(理科版)第29卷第4期,2005年8月.
[37]李颂孝,不同加权Bergman空间之间的加权复合算子[J]嘉应学院学报,2004年06期,5-8.
[38]李颂孝,从Hardy空间到Bers-型空间的加权复合算子[J].湖州师范学院学报,2004年01期,8-11.
[39]X.J.Zhang. Weighted composition operators on the little Bloch Type Spaces [J]. Journal of Math.2005,25 (4):361-367.
[40]R.H. Zhao,On a general family of function spaces[J].Ann. Acad. Sci Fenn. Math. Diss.1996,105.
[41]R.H. Zhao. On Logarithmic Carleson measures[J]. Acta. Sci. Math.(Szeged), 2003,69:605-618.
[42]C.Cowen.B.Maculure. Composition operators on spaces of analytic functions [M]. Studies in Advanced Mathematics. CRC Press. Boca. Raton,1995:222-247.
[43]Z.J.Hu and T.S.Liu, Weighted integrals of holomorphic functions [J]. Appl.Math.J. Chinese Univ.Ser.B.2004,19(4):474-480.
[44]Jouni Rattya, On some Complex Function Spaces and Classes[J]. Ann. Acad. Sci Fenn. Math. Diss. No.124,2001.
[45]Y.M.Liu, Compact Composition Operators Mapping into thE0(p,q) [J].Journal of Mathematical Research and Exposition. Vol.22, No.4, Nov.2002,538-554.
[46]Y.M.Liu, Schatten class of composition operator on A2(Ω,dVa) [J]. Chinese Ann.Math.,2001,22A:153-56.
[47]L.Luo,J.H.Shi, Composition operators between the weighted Bergman spaces on bounded symmetric domains of Cn[J]. Chinese Ann.Math.2000,21A:45-52.
[48]B.Maccluer and R.zhao, Vanishing Logarithmic Carleson Measures[J]. Illinois Journal of Mathematica.2002,46(2):507-518.