从混合模空间到 Bloch -型空间微分算子与乘子的积

数学物理学报 ›› 2012, Vol. 32 ›› Issue (1): 68-79.

从混合模空间到 Bloch -型空间微分算子与乘子的积

于燕燕, |刘永民^*

徐州工程学院数学与物理科学学院江苏徐州 221111|徐州师范大学数学科学学院江苏徐州 221116

收稿日期:2010-03-13 修回日期:2011-04-28 出版日期:2012-02-25 发布日期:2012-02-25
通讯作者: 刘永民, minliu@xznu.edu.cn E-mail:minliu@xznu.edu.cn
基金资助:
国家自然科学基金(11171285, 10471039)和江苏省普通高校自然科学研究计划项目(06KJD110175, 07KJB110115)资助

The Product of Differentiation and Multiplication Operator from the Mixed-norm to the Bloch-type Space

YU Yan-Yan, |Liu-Yong-Min^*

School of Mathematics and Physics Science, Xuzhou Institute of Technology, Jiangsu Xuzhou 221111)|Department of Mathematics, Xuzhou Normal University, Jiangsu Xuzhou 221116

Received:2010-03-13 Revised:2011-04-28 Online:2012-02-25 Published:2012-02-25
Contact: Liu-Yong-Min, minliu@xznu.edu.cn E-mail:minliu@xznu.edu.cn
Supported by:
国家自然科学基金(11171285, 10471039)和江苏省普通高校自然科学研究计划项目(06KJD110175, 07KJB110115)资助

摘要/Abstract

摘要：

设H(D) 表示单位圆盘D上的解析函数空间, u ∈ H(D). 该文研究了从混合模空间到Bloch -型空间微分算子与乘子的积DM_u 的有界性与紧性.

关键词: 微分算子, 乘子, 混合模空间, Bloch-型空间

Abstract:

Let H(D) be the space of analytic functions on the unit disk D and u ∈ H(D). The boundedness and compactness of the product DM_u of differentiation operator and multiplica-tion operator from the mixed norm to the Bloch-type space are investigated in this paper.

Key words: Differentiation operator, Multiplication operator, Mixed norm space, Bloch-type space

中图分类号:

47B38

于燕燕, 刘永民. 从混合模空间到 Bloch -型空间微分算子与乘子的积[J]. 数学物理学报, 2012, 32(1): 68-79.

YU Yan-Yan, Liu-Yong-Min. The Product of Differentiation and Multiplication Operator from the Mixed-norm to the Bloch-type Space[J]. Acta mathematica scientia,Series A, 2012, 32(1): 68-79.

参考文献

[1] Choe B, Koo H, Smith W. Composition operators on small spaces. Integral Equations Operator Theory, 2006, 56(3): 357--380

[2] Hedenmalm H, Korenblum B, Zhu K. Theory of Bergman Spaces. Graduate Text in Mathematics. New York: Springer, 2000

[3] Hibschweiler R A, Portnoy N. Composition followed by differentiation between Bergman and Hardy spaces.
Rocky Mountain J Math, 2005, 35(3): 843--855

[4] Krantz S, Stevi\' c S. On the iterated logarithmic Bloch space on the unit ball. Nonlinear Anal TMA, 2009, 71(5/6): 1772--1795

[5] Li S, Stevi\' c S. Composition followed by differentiation between Bloch type spaces (English summary). J Comput Anal Appl, 2007, 9(2): 195--205

[6] Li S, Stevi\' c S. Composition followed by differentiation between weighted Bergman spaces and Bloch type
spaces (English summary). J Appl Funct Anal, 2008, 3(3): 333--340

[7] Li S, Stevi\' c S. Products of composition and integral type operators from $H^\infty$ to the Bloch space. Complex Var Elliptic Equ, 2008, 53(5): 463--474

[8] Li S, Stevi\' c S. Products of Volterra type operator and composition operator from $H^\infty$ and Bloch spaces to Zygmund spaces. J Math Anal Appl, 2008, 345(1): 40--52

[9] Li S, Stevi\' c S. Differentiation of a composition as an operator from spaces with mixed norm to Bloch α-spaces. Translated from the English (Russian summary). Mat Sb, 2008, 199(12): 117--128;
Translation in Sb Math, 2008, 199(11/12): 1847--1857

[10] Li S, Stevi\' c S. Products of integral-type operators and composition operators between Bloch-type spaces.
J Math Anal Appl, 2009, 349(2): 596--610

[11] Li S, Stevi\' c S. Composition followed by differentiation between $H^\infty$ and $\alpha$-Bloch spaces (English summary). Houston J Math, 2009, 35(1): 327--340

[12] Li S, Stevi\' c S. Generalized composition operators on Zygmund spaces and Bloch type spaces. J Math Anal Appl, 2008, 338(2): 1282--1295

[13] 刘永民, 刘浩. 从混合模空间到 Zygmund 空间的 Volterra 型复合算子. 数学学报, 2011, 54(3): 381--396

[14] Liu Y, Yu Y. On a Li-Stevi\' c integral-type operators from the Bloch-type spaces into the logarithmic Bloch spaces. Integral Transforms Spec Funct, 2010, 21(2): 93--103

[15] Liu X, Yu Y. The product of differentiation operator and multiplication operator from $H^\infty$ to Zygmund spaces. J Xuzhou Norm Univ Nat Sci Ed, 2011, 29(1): 37--39

[16] Madigan K, Matheson A. Compact composition operators on the Bloch space. Trans Amer Math Soc, 1995, 347(7): 2679--2687

[17] Ohno S. Products of composition and differentiation between Hardy spaces (English summary). Bull Austral Math Soc, 2006, 73(2): 235--243

[18] Shields A, Williams D. Bounded projections, duality, and multipliers in spaces of analytic functions. Trans Amer Math Soc, 1971, 162: 287--302

[19] Stevi\' c S. Composition operators between $H^\infty$ and the $\alpha$-Bloch spaces on the polydisc. Z Anal Anwendungen, 2006, 25(4): 457--466

[20] Stevi\' c S. Generalized composition operators between mixed-norm and some weighted spaces. Numer Funct Anal Optim, 2008, 29(7/8): 959--978

[21] Stevi\' c S. On a new integral-type operator from the weighted Bergman space to the Bloch-type space on the unit ball (Article ID 154263). Discrete Dyn Nat Soc, 2008, 2008: 14

[22] Stevi\' c S. On a new operator from $H^\infty$ to the Bloch-type space on the unit ball. Util Math, 2008, 77: 257--263

[23] Stevi\' c S. Integral-type operators from the mixed-norm space to the Bloch-type space on the unit ball. Siberian J Math, 2009, 50(6): 1098--1105

[24] Stevi\' c S. Products of composition and differentiation operators on the weighted Bergman space. Bull Belg Math Soc Simon Stevin, 2009, 16: 623--635

[25] Stevi\' c S. Norm and essential norm of composition followed by differentiation from $\alpha$-Bloch spaces to $H^\infty_\mu$. Appl Math Comput, 2009, 207(1): 225--229

[26] Stevi\' c S. Weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces. Appl Math Comput, 2009, 211(1): 222--233

[27] Stevi\' c S. Weighted composition operators from mixed norm spaces into weighted Bloch spaces. J Comput Anal Appl, 2009, 11(1): 70--80

[28] Stevi\' c S. On an integral operator between Bloch-type spaces on the unit ball. Bull Sci Math, 2010, 134: 329--339

[29] Stevi\' c S. On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball. J
Math Anal Appl, 2009, 354(2): 426--434

[30] Stevi\' c S. On a new operator from $H^\infty$ to the Bloch-type space on the unit ball. Util Math, 2008, 77: 257--263

[31] Stevi\' c S. On a new operator from the logarithmic Bloch space to the Bloch-type space on the unit ball. Appl Math Comput, 2008, 206(1): 313--320

[32] Stevi\' c S. Products of integral-type operators and composition operators from the mixed norm space to Bloch-type spaces. Siberian J Math, 2009, 50(4): 726--736

[33] Stevi\' c S. On an integral-type operator from logarithmic Bloch-type and mixed-norm spaces to Bloch-type spaces. Nonlinear Anal TMA, 2009, 71: 6323--6342

[34] Stevi\' c S, Ueki S I. Integral-type operators acting between weighted-type spaces on the unit ball. Appl Math Comput, 2009, 215: 2464--2471

[35] Wang X, Liu T. Composition type operators from Hardy spaces to $\mu$-Bloch spaces on the unit ball. Acta Math Sci, 2009, 29B(5): 1430--1438

[36] Yang W. Products of composition and differentiation operators from Q_K(p, q) spaces to Bloch-type spaces (Article ID 741920). Abstr Appl Anal, 2009, 2009: 14

[37] Yang W. On an integral-type operator between Bloch-type spaces. Appl Math Comput, 2009, 215(3): 954--960

[38] 叶善力, 高进寿. 不同权 Bloch -型空间之间的加权 Ces\' aro 算子. 数学物理学报, 2008, 28A(2): 349--358

[39] 于燕燕. 加权 Bergman 空间上复合算子的积. 徐州师范大学学报(自然科学版), 2007, 25(1): 22--25

[40] Yu Y. Volterra-type composition operators from logarithmic Bloch spaces into Bloch-type spaces. Journal of Xuzhou Normal University(Nat Sci Ed), 2009, 27(3): 14--18

[41] 于燕燕, 刘永民. 从双曲 α-Bloch 空间到双曲Q_K 型空间上的复合算子. 数学物理学报, 2009, 29A(5): 1311--1320

[42] Zhu X. Generalized weighted composition operators from Bloch-type spaces to weighted Bergman spaces. Indian J Math, 2007, 49(2): 139--149

[43] Zhu X. Products of differentiation, composition and multiplication from Bergman type spaces to Bers type spaces (English summary). Integral Transforms Spec Funct, 2007, 18(3/4): 223--231