Journal of Integral Equations and Applications

Volterra-type operators from analytic Morrey spaces to Bloch space

Zhengyuan Zhuo and Shanli Ye

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Abstract

In this note, we study the boundedness and compactness of integral operators $I_g$ and $T_g $ from analytic Morrey spaces to Bloch space. Furthermore, the norm and essential norm of those operators are given.

Article information

Source
J. Integral Equations Applications, Volume 27, Number 2 (2015), 289-309.

Dates
First available in Project Euclid: 9 September 2015

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1441790290

Digital Object Identifier
doi:10.1216/JIE-2015-27-2-289

Mathematical Reviews number (MathSciNet)
MR3395972

Zentralblatt MATH identifier
1329.47035

Subjects
Primary: 47B38: Operators on function spaces (general) 30H30: Bloch spaces 30H99: None of the above, but in this section

Keywords
Analytic Morrey space Bloch space Volterra type operator essential norm

Citation

Zhuo, Zhengyuan; Ye, Shanli. Volterra-type operators from analytic Morrey spaces to Bloch space. J. Integral Equations Applications 27 (2015), no. 2, 289--309. doi:10.1216/JIE-2015-27-2-289. https://projecteuclid.org/euclid.jiea/1441790290


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References

  • D. Adams and J. Xiao, Nonlinear potential analysis on Morrey spaces and their capacities, Indiana Univ. Math. J. 53 (2004), 1629–1663.
  • ––––, Morrey spaces in harmonic analysis, Ark. Mat. 50 (2012), 201–230.
  • A. Aleman and J.A. Cima, An integral operator on $H^p$ and Hardy's inequality, J. Anal. Math. 85 (2001), 157–176.
  • A. Aleman and A.G. Siskakis, Integration operators on Bergman spaces, Indiana Univ. Math. J. 46 (1997), 337–356.
  • J.M. Anderson, Bloch functions: The basic theory, operators and function theory, Math. Phys. Sci. 153 (1985), 1–17.
  • J.M. Anderson, J. Clunie and Ch. Pommerenke, On Bloch functions and normal functions, J. reine angew. Math. 270 (1974), 12–37.
  • R. Aulaskari and P. Lappan, Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal, in Complex analysis and its applications, Pitman Res. Not. Math. 305 (1994), 136–146.
  • R. Aulaskari, D.A. Stegenga and J. Xiao, Some subclasses of BMOA and their characterizations in terms of Carleson measures, Rocky. Mountain J. Math. 26 (1996), 485–506.
  • K.D. Bierstedt, J. Bonet and A. Galbis, Weighted spaces of holomorphic functions on bounded domains, Michigan Math. J. 40 (1993), 271–297.
  • K.D. Bierstedt and W.H. Summers, Biduals of weighted Banach spaces of analytic functions, J. Austr. Math. Soc. 54 (1993), 70–79.
  • C. Cascante, J. Fàbrega and J.M. Ortega, The Corona theorem in weighted Hardy and Morrey spaces, Ann. Sc. Norm. Super. Pisa, DOI 10.2422/2036-2145.201202 006.
  • O. Constantin, A Volterra-type integration operator on Fock spaces, Proc. Amer. Math. Soc. 140 (2012), 4247–4257.
  • C. Cowen and D. MacCluer, Composition operators on spaces of analytic functions, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1995.
  • P. Duren, Theory of $H^p$ paces, Academic Press, New York, (1970).
  • D. Girela, Analytic functions of bounded mean oscillation, Complex function spaces, Univ. Joensuu Dept. Rep. Ser. 4, University of Joensuu, Joensuu, 2001.
  • J. Laitila, S. Miihkinen and P. Nieminen, Essential norms and weak compactness of integration operators, Arch. Math. 97 (2011), 39–48.
  • P. Li, J. Liu and Z. Lou, Integral operators on analytic Morrey spaces, Sci. China Math. 57 (2014), 1961–1974.
  • J. Liu, Z. Lou and C. Xiong, Essential norms of integral operators on spaces of analytic functions, Nonlin. Anal. 75 (2012), 5145–5156.
  • S. Li and S. Stević, Volterra-type operators on Zygmund spaces, J. Inequality Appl. Article ID 32124, (2007), 10 pages.
  • K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347 (1995), 2679–2687.
  • C.B. Morrey, Jr., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), 126–166.
  • J. Peetre, On the theory of $\mathcal{L}_{p,\lambda} $ spaces, J. Funct. Anal. 4 (1964), 71–87.
  • Ch. Pommerenke, Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation, Comment. Math. Helv. 52 (1977), 591–602.
  • J. Shapiro, The essential norm of a composition operator, Ann. Math. 125 (1987), 375–404.
  • A.L. Shields and D.L. Williams, Bounded projections, duality, and multipliers in spaces of harmonic functions, J. reine angew. Math. 299 (1978), 256–279.
  • ––––, Bounded projections and the growth of harmonic conjugates in the disk, Michigan Math. J. 29 (1982), 3–25.
  • A. Siskakis and R. Zhao, A Volterra type operator on spaces of analytic functions, Contemp. Math. 232 (1999), 299–311.
  • J. Wang and J. Xiao, Holomorphic Campanato spaces on the unit ball, arXiv:1405.6192v1 [math.CV].
  • ––––, Two predualities and three operators over analytic Campanato spaces, arXiv:1402.4377v1 [math.CV].
  • Z. Wu, A new characterization for Carleson measure and some applications, Int. Equat. Oper. Theor. 71 (2011), 161–180.
  • Z. Wu and C. Xie, $Q$ spaces and Morrey spaces, J. Funct. Anal. 297 (2003), 282–297.
  • H. Wulan and J. Zhou, $Q_K$ and Morrey type spaces, Ann Acad. Sci. Fenn. Math. 38 (2013), 193–207.
  • J. Xiao, The $Q_p$ Carleson measure problem, Adv. Math. 217 (2008), 2075–2088.
  • ––––, Holomorphic $Q$ classes, Lect. Notes Math. 1767 (2001), Springer-Verlag, Berlin.
  • ––––, Geometric $Q_p$ functions, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006.
  • J. Xiao and W. Xu, Composition operators between analytic Campanato spaces, J. Geom. Anal, doi 10.1007/s12220-012-9349-6.
  • J. Xiao and C. Yuan, Analytic Campanato spaces and their compositions, arXiv: 1303.5032v2 [math.CV].
  • S. Ye, Norm and essential norm of composition followed by differentiation from logarithmic Bloch spaces to $H^\infty_\mu$, Abst. Appl. Anal. Art. ID 725145, (2014), 6 pages.
  • ––––, Products of Volterra-type operators and composition operators on logarithmic Bloch space, WSEAS Trans. Math. 12 (2013), 180–188.
  • S. Ye and J. Gao, Extended Cesáro operators between different weighted Bloch-type spaces, Acta Math. Sci. 28 (2008), 349–358 (in Chinese).
  • K. Zhu, Operator theorey in function spaces, Second edition, Math. Surv. Mono. 138 (2007).
  • C.T. Zorko, Morrey space, Proc. Amer. Math. Soc. 98 (1986), 586–592.