AUTHORS: Shanli Ye
Download as PDF
In recent years the composition operator Cφ has been received much attention and appear in various settings in the literature. It is interesting to provide a function theoretic characterization when φ induces a bounded or compact composition operator on various function spaces. In this paper we consider the products of Volterratype operators and composition operators. We characterize the boundedness and compactness of the products of Volterra-type operators and composition operators TgCφ and IgCφ from the analytic Morrey spaces L 2,λ to the Zygmund space Z, and the little analytic Morrey spaces L 2,λ 0 to the little Zygmund space Z0 over the unit disk, respectively
KEYWORDS: -Analytic Morrey space, Zygmund space, Volterra-type operators, Composition operators, Boundedness, Compactness
REFERENCES:
[
1] A. Aleman and A. G. Siskakis, An integral operator on Hp
, Complex Variables. 28, 1995, pp. 149-158.
[2] A. Aleman and A. G. Siskakis, Intergration operators on Bergman spaces, Indiana University
Math. J. 46, 1997, pp. 337-356.
[3] B.R.Choe, H. Koo and W. Smith, Composition
operators on small spaces, Integr. equ. oper. Theory, 56, 2006, pp. 357-380.
[4] C.C. Cowen and B.D. Maccluer, Composition
Operator on Spaces of Analytic Functions, CRC
Press, Boca Raton, 1995.
[5] P. Duren, Theory of Hp Spaces, Academic Press,
New York, 1970.
[6] D. Girela, Analytic functions of bounded mean
oscillation, Complex Function Spaces, Mekrijarvi, 1999, pp. 61-170, Univ. Joensuu Dept.
Rep. Ser., 4, Univ. Joensuu, Joensuu, 2001.
[7] S. Li and S. Stevic´, Gerneralized composition
operators on the Zygmund spaces and Bloch
type spaces, J. Math Anal. Appl. 338, 2008, pp. 1282-1295.
[8] S. Li and S. Stevic´, Products of Volterra type operator and composition operator from H∞ and
Bloch spaces to Zygmund spaces, J. Math Anal.
Appl. 345, 2008, pp. 40-52.
[9] P. Li, J. Liu and Z. Lou, Integral operators on analytic Morrey spaces, Sci China Math, 57, 2014,
pp. 1961-1974.
[10] K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans, Amer.
Math. soc. 347, 1995, pp. 2679-2687.
[11] K. Madigan, Composition operators on analytic Lipschitz spaces, Proc. Amer. Math. Soc. 119,
1993, pp. 465-473.
[12] Ch. Pommerenke, Schlichte funktionen und analytische funktionen vonbeschra¨nkter mittlerer
oszillation, Ciomment. Math. Helv. 52, 1997, pp. 591-602.
[13] S. Ohno and R. Zhao, Weighted composition operators on the Bloch space, Bull. Austral. Math.
Soc. 63, 2001, pp. 177-185.
[14] S. Ohno, K. Stroethoff and R. Zhao, Weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math. 33, 2003, pp. 191-215.
[15] J. H. Shapiro, Composition operators and classical function theory, Springer Verlag New York,
1993.
[16] A. G. Siskakis and R. Zhao, A Volterra type operator on spaces of analytic functions, Contemp.
Math. 232, 1999, pp. 299-311.
[17] W. Smith, Composition operators between
Bergman and Hardy spaces, Trans. Amer. Math.
Soc. 348, 1996, pp. 2331-2348.
[18] S. Stevic, On a new operator from the logarith- ´
mic Bloch space to the Bloch-type space on the
unit ball, Appl. Math. Comput. 206, 2008, pp. 313-320.
[19] S. Stevic, Boundedness and compactness of an ´
integral operator on mixed norm spaces on the
polydisc, Sibirsk. Mat. Zh. 48, 2007, pp. 694-
706.
[20] S. Stevic, Composition operators between ´ H1
and the α-Bloch spaces on the polydisc, Z. Anal.
Anwendungen, 25, 2006, pp. 457-466.
[21] H. Wulan and J. Zhou, QK and Morrey type spaces. Ann Acad Sci Fenn Math. 38, 2013, pp. 193-207.
[22] J. Xiao, Riemann-Sttieltjes operators on weighted Bloch and Bergman spaces of the unit ball, J.
London Math. Soc. 70, 2004, pp. 1045-1061.
[23] J. Xiao, Geometric Qp Functions, Frontiers in
Mathematics. Birkha¨auser Verlag, Basel, 2006.
[24] J. Xiao and W. Xu, Composition operators between analytic Campanato spaces, J. Geom.
Anal. 24, 2014, pp. 649-666.
[25] S. Ye, A weighted composition operators on
the logarithmic Bloch space, Bull. Korean Math.
Soc. 47, 2010, pp. 527-540.
[26] S. Ye and Q. Hu, Weighted composition operators on the Zygmund space, Abstr. Appl. Anal.
Article ID 462482, 2012, 18 pages.
[27] S. Ye and Z. Zhuo, Weighted composition operators from Hardy to Zygmund type spaces, Abstr.
Appl. Anal. Article ID 365286, (2013), 10 pages.
[28] S. Ye, Products of Volterra-type operators and
composition operators on logarithmic Bloch space, WESEAS Trans. Math. 12, 2013, pp. 180-
188.
[29] Z. Zhuo and S. Ye, Volterra-types from analytic
Morrey spaces to Bloch space, J. Integral Equ.
Appl. 27, 2015, pp. 289-308.
[30] A. Zygmund, Trigonometric Series, Cambridge,
1959.