Abstract
When the weight \(\mu \) is more general than normal, the complete characterizations in terms of the symbol g and weights for the conditions of the boundedness and compactness of \(T_g: H^{\infty }_\nu \rightarrow H^{\infty }_\mu \) and \(S_g: H^{\infty }_\nu \rightarrow H^{\infty }_\mu \) are still unknown. Smith et al. firstly gave the sufficient and necessary conditions for the boundedness of Volterra type operators on Banach spaces of bounded analytic functions when the symbol functions are univalent. In this paper, continuing their lines of investigations, we give the complete characterizations of the conditions for the boundedness and compactness of Volterra type operators \(T_g\) and \(S_g\) between Bloch type spaces \({\mathcal {B}}^\infty _\nu \) and weighted Banach spaces \(H^{\infty }_\nu \) with more general weights, which generalize their works.
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16 December 2022
A Correction to this paper has been published: https://doi.org/10.1007/s00020-022-02721-4
References
Aleman, A., Cima, J.: An integral operator on \(H^p\) and Hardy’s inequality. J. Anal. Math. 85, 157–176 (2001)
Aleman, A., Siskakis, A.: An integral operator on \(H^p\). Complex Var. Theory Appl. 28(2), 149–158 (1995)
Aleman, A., Siskakis, A.: Integration operators on Bergman spaces. Indiana Univ. Math. J. 46(2), 337–356 (1997)
Anderson, A., Jovovic, M., Smith, W.: Some integral operators acting on \(H^\infty \). Integral Equ. Oper. Theory 80(2), 275–291 (2014)
Anderson, A.: Some closed range integral operators on spaces of analytic functions. Integral Equ. Oper. Theory 69, 87–99 (2011)
Basallote, M., Contreras, M., Hernández-Mancera, C., Martín, M., Paúl, P.: Volterra operators and semigroups in weighted Banach spaces of analytic functions. Collect. Math. 65(2), 233–249 (2014)
Bierstedt, K., Bonet, J., Galbis, A.: Weighted spaces of holomorphic functions on balanced domains. Mich. Math. J. 40, 271–297 (1993)
Bierstedt, K., Bonet, J., Taskinen, J.: Associated weights and spaces of holomorphic functions. Studia Math. 127, 70–79 (1998)
Constantin, O., Peláez, J.: Integral operators, embedding theorems and a Littlewood–Paley formula on weighted Fock spaces. J. Geom. Anal. 26(2), 1109–1154 (2016)
Contreras, M., Hernandez-Diaz, A.: Weighted composition operators in weighted Banach spaces of analytic functions. J. Aust. Math. Soc. Ser. A 69(1), 41–60 (2000)
Contreras, M., Peláez, J., Pommerenke, C., Rättyä, J.: Integral operators mapping into the space of bounded analytic functions. J. Funct. Anal. 271(10), 2899–2943 (2016)
Eklund, T., Lindström, M., Pirasteh, M., Sanatpour, A., Wikman, N.: Generalized Volterra operators mapping between Banach spaces of analytic functions. Monatsh. Math. (2018). https://doi.org/10.1007/s00605-018-1216-5
Galanopoulos, P., Girela, D., Peláez, J.: Multipliers and integration operators on Dirichlet spaces. Trans. Am. Math. Soc. 363(4), 1855–1886 (2011)
Girela, D., Peláez, J.: Carleson measures, multipliers and integration operators for spaces of Dirichlet type. J. Funct. Anal. 241(1), 334–358 (2006)
Hardy, G., Littlewood, J.: Some properties of fractional integrals. II. Math. Z. 34(1), 403–439 (1932)
Laitila, J., Miihkinen, S., Nieminen, P.: Essential norms and weak compactness of integration operators. Arch. Math. 97(1), 39–48 (2011)
Lin, Q., Liu, J., Wu, Y.: Volterra type operators on \(S^p({\mathbb{D}})\) spaces. J. Math. Anal. Appl. 461, 1100–1114 (2018)
Lusky, W.: Growth Conditions for Harmonic and Holomorphic Functions, Functional Analysis (Trier, 1994), pp. 281–291. de Gruyter, Berlin (1996)
Manhas, J., Zhao, R.: Products of weighted composition operators and differentiation operators between Banach spaces of analytic functions. Acta Sci. Math. (Szeged) 80, 665–679 (2014)
Mengestie, T.: Product of Volterra type integral and composition operators on weighted Fock spaces. J. Geom. Anal. 24(2), 740–755 (2014)
Miihkinen, S.: Strict singularity of a Volterra-type integral operator on \(H^p\). Proc. Am. Math. Soc. 145(1), 165–175 (2017)
Montes-Rodríguez, A.: Weighted composition operators on weighted Banach spaces of analytic functions. J. Lond. Math. Soc. 61(2), 872–884 (2000)
Pommerenke, Ch.: Boundary Behaviour of Conformal Maps, Grundlehren der Mathematischen Wissenschaften 299. Springer-Verlag, Berlin (1992)
Pommerenke, Ch.: Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation (German). Comment. Math. Helv. 52(4), 591–602 (1977)
Shields, A., Williams, D.: Bounded projections and the growth of harmonic conjugates in the unit disc. Mich. Math. J. 29(1), 3–25 (1982)
Smith, W., Stolyarov, D., Volberg, A.: Uniform approximation of Bloch functions and the boundedness of the integration operator on \(H^\infty \). Adv. Math. 314, 185–202 (2017)
Wolf, E.: Composition followed by differentiation between weighted Banach spaces of holomorphic functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 105(2), 315–322 (2011)
Xiao, J.: The \(Q_p\) Carleson measure problem. Adv. Math. 217(5), 2075–2088 (2008)
Zorboska, N.: Intrinsic operators from holomorphic function spaces to growth spaces. Integral Equ. Oper. Theory 87(4), 581–600 (2017)
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This work was supported by NNSF of China (Grant No. 11801094).
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Lin, Q. Volterra Type Operators Between Bloch Type Spaces and Weighted Banach Spaces. Integr. Equ. Oper. Theory 91, 13 (2019). https://doi.org/10.1007/s00020-019-2512-8
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DOI: https://doi.org/10.1007/s00020-019-2512-8