Research article

Volterra integral operator and essential norm on Dirichlet type spaces

  • Received: 22 April 2021 Accepted: 06 July 2021 Published: 07 July 2021
  • MSC : 30D45, 30D50

  • In this paper, we study the boundedness and essential norm of Volterra integral operator $ V_g $ and integral operator $ S_g $ on Dirichlet type spaces $ {\mathcal{D}_{K, \alpha}} $.

    Citation: Liu Yang, Ruishen Qian. Volterra integral operator and essential norm on Dirichlet type spaces[J]. AIMS Mathematics, 2021, 6(9): 10092-10104. doi: 10.3934/math.2021586

    Related Papers:

  • In this paper, we study the boundedness and essential norm of Volterra integral operator $ V_g $ and integral operator $ S_g $ on Dirichlet type spaces $ {\mathcal{D}_{K, \alpha}} $.



    加载中


    [1] A. Aleman, A. Siskakis, An integral operator on $H^p$, Complex Var. Elliptic, 28 (1955), 149-158.
    [2] A. Aleman, A. Siskakis, Integration operators on Bergman spaces, Indiana Univ. Math. J., 46 (1997), 337-356.
    [3] A. Aleman, Hilbert spaces of analytic functions between the Hardy space and the Dirichlet space, Proc. Amer. Math. Soc., 115 (1992), 97-104. doi: 10.1090/S0002-9939-1992-1079693-X
    [4] N. Arcozzi, R. Rochberg, E. Sawyer, Carleson measures for analytic Besov spaces, Rev. Mat. Iberoam., 18 (2002), 443-510.
    [5] G. Bao, Z. Lou, R. Qian, H. Wulan, On multipliers of Dirichlet type spaces, Complex Anal. Oper. Theory, 9 (2015), 1701-1732. doi: 10.1007/s11785-015-0444-0
    [6] D. Blasi, J. Pau, A characterization of Besov type spaces and applications to Hankel type operators, Michigan Math. J., 56 (2008), 401-417.
    [7] O. Constantin, A Volterra-type integration operator on Fock spaces, Pro. Amer. Math. Soc., 140 (2012), 4247-4257. doi: 10.1090/S0002-9939-2012-11541-2
    [8] P. Duren, Theory of $H^p$ Spaces, Academic Press, New York, 1970.
    [9] M. Essen, H. Wulan, J. Xiao, Several function-theoretic characterizations of Möbius invariant ${\mathcal{Q}}_K$ spaces, J. Funct. Anal., 230 (2006), 78-115. doi: 10.1016/j.jfa.2005.07.004
    [10] J. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.
    [11] R. Kerman, E. Sawyer, Carleson measures and multipliers of Dirichlet-type spaces, Trans. Amer. Math. Soc., 309 (1988), 87-98. doi: 10.1090/S0002-9947-1988-0957062-1
    [12] J. Laitila S. Miihkinen, J. Nieminen, Essential norms and weak compactness of integration operators, Arch. Math., 97 (2011), 39-48. doi: 10.1007/s00013-011-0272-z
    [13] S. Li, J. Liu, C. Yuan, Embedding theorems for Dirichlet type spaces, Canad. Math. Bull., 63 (2020), 106-117. doi: 10.4153/S0008439519000201
    [14] P. Li, J. Liu, Z. Lou, Integral operators on analytic Morrey spaces, Sci. China Math., 57 (2014), 1961-1974. doi: 10.1007/s11425-014-4811-5
    [15] Z. Lou, R. Qian, Small Hankel operator on Dirichlet-type spaces and applications, Math. Inequal. Appl., 19 (2016), 209-220.
    [16] C. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938), 126-166. doi: 10.1090/S0002-9947-1938-1501936-8
    [17] J. Pau, R. Zhao, Carleson measures, Riemann-Stieltjes and multiplication operators on a general family of function spaces, Integr. Equ. Oper. Theory, 78 (2014), 483-514. doi: 10.1007/s00020-014-2124-2
    [18] R. Qian, S. Li, Carleson measure and Volterra type operators on weighted BMOA spaces, Georgian Math. J., 27 (2020), 413-424. doi: 10.1515/gmj-2018-0040
    [19] R. Qian, Y. Shi, Inner function in Dirichlet type spaces, J. Math. Anal. Appl., 421 (2015), 1844-1854. doi: 10.1016/j.jmaa.2014.08.011
    [20] R. Qian, Y. Shi, Univalent functions in Dirichlet-type spaces, Filomat, 30 (2016), 1213-1218. doi: 10.2298/FIL1605213Q
    [21] R. Qian, X. Zhu, Embedding of Dirichlet type spaces into tent spaces and Volterra operators, Canad. Math. Bull., (2020), 1-12.
    [22] R. Qian, X. Zhu, Embedding of $Q_p$ spaces into tent spaces and Volterra integral operator, AIMS Math., 6 (2021), 698-711. doi: 10.3934/math.2021042
    [23] R. Rochberg, Z. Wu, A new characterization of Dirichlet type spaces and applications, Illinois J. Math., 37 (1993), 101-122.
    [24] A. Siskakis, R. Zhao, A Volterra type operator on spaces of analytic function, Contemp. Math., 232 (1999), 299-312. doi: 10.1090/conm/232/03406
    [25] D. Stegenga, Multipliers of the Dirichlet space, Illinois J. Math., 24 (1980), 113-139.
    [26] G. Taylor, Multipliers on $D_{\alpha}$, Trans. Amer. Math. Soc., 123 (1966), 229-240.
    [27] M. Tjani, Distance of a Bloch function to the little Bloch space, Bull. Austral. Math. Soc., 74 (2006), 101-119. doi: 10.1017/S0004972700047493
    [28] J. Wang, The Carleson measure problem between analytic Morrey spaces, Canad. Math. Bull., 59 (2016), 878-890. doi: 10.4153/CMB-2016-013-9
    [29] H. Wulan, J. Zhou, ${\mathcal{Q}_K}$ and Morrey type spaces, Ann. Acad. Sci. Fenn. Math., 38 (2013), 193-207. doi: 10.5186/aasfm.2013.3801
    [30] H. Wulan, K. Zhu, Möbius invariant ${\mathcal{Q}_K}$ spaces, Springer, Cham, 2017.
    [31] J. Xiao, The $Q_p$ Carleson measure problem, Adv. Math., 217 (2008), 2075-2088. doi: 10.1016/j.aim.2007.08.015
    [32] J. Xiao, Holomorphic ${\mathcal{Q}}$ Classes, Springer, LNM 1767, Berlin, 2001.
    [33] L. Yang, Integral operator acting on weighted Dirichlet spaces to Morrey Type spaces, Filomat, 33 (2019), 3723-3736. doi: 10.2298/FIL1912723Y
    [34] K. Zhu, Operator Theory in Function Spaces, American Mathematical Society, Providence, RI, 2007.
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2082) PDF downloads(117) Cited by(0)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog