Boundedness of Classical Operators in Weighted Spaces of Holomorphic Functions

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/u5398-4279-7225-c Boundedness of Classical Operators in Weighted Spaces of Holomorphic Functions Abanin, A. V. , Korablina, Yu. V. Vladikavkaz Mathematical Journal 2020. Vol. 22. Issue 3. Abstract: We establish some criteria of the boundedness for some classical operators acting from an abstract Banach space of holomorphic functions in a complex domain to a weighted space of the same functions equipped with sup-norm. It is presented a further development of Zorboska’s idea that conditions of the boundedness of weighted composition operators including multiplication and usual composition ones and Volterra operator can be formulated in terms of \(\delta\)-functions norms in the corresponding dual spaces. As a consequence we obtain criteria of the boundedness of the above mentioned operators on generalized Bergman and Fock spaces. In particular cases it is possible to state these criteria in terms of weights defining spaces and functions giving the composition operator. In~comparison with the previous results we essentially extend the class of weighted holomorphic spaces in the unit disc that admits a realization of Zorboska’s method. In~addition, we develop an extension of this approach to weighted spaces of entire functions. In this relation we introduce the class of almost harmonic weights and obtain some estimates of \(\delta\)-functions norms in spaces dual to the generalized Fock spaces giving by almost harmonic weights. Keywords: weighted spaces of holomorphic functions, weighted composition operator, Volterra operator, Bergman spaces, Fock spaces Language: Russian Download the full text For citation: Abanin, A. V. and Korablina, Yu. V. Boundedness of Classical Operators in Weighted Spaces of Holomorphic Functions, Vladikavkaz Math. J., 2020, vol. 22, no. 3, pp.5-17. DOI 10.46698/u5398-4279-7225-c + References 1. Tien, P. T. Translation Operators on Weighted Spaces of Entire Functions, Proceedings of the American Mathematical Society, 2017, vol. 145, no. 2, pp. 805-815. DOI: 10.1090/proc/13254. 2. Abanin, A. V. and Tien, P. T. Invariant Subspaces for Classical Operators on Weighted Spaces of Holomorphic Functions, Integral Equations and Operator Theory, 2017, vol. 89, no. 3, pp. 409-438. DOI: 10.1007/s00020-017-2401-y. 3. Zorboska, N. Intrinsic Operators from Holomorphic Function Spaces to Growth Spaces, Integral Equations and Operator Theory, 2017, vol. 87, no. 4, pp. 581-600. DOI: 10.1007/s00020-017-2361-2. 4. Bierstedt, K. D., Bonet, J. and Taskinen, J. Associated Weights and Spaces of Holomorphic Functions, Studia Mathematica, 1998, vol. 127, no. 2, pp. 137-168. DOI: 10.4064/sm-127-2-137-168. 5. Abanin, A. V. and Tien, P. T. Differentiation and Integration Operators on Weighted Banach Spaces of Holomorphic Functions, Mathematische Nachrichten, 2017, vol. 290, no. 8-9, pp. 1144-1162. DOI: 10.1002/mana.201500405. 6. Zhu, K. Analysis on Fock Spaces. Graduate Texts in Mathematics, 263, New York, Springer, 2012, 346 p. 7. Baladai, R. A. and Khabibullin, B. N. From Integral Estimates of Functions to Uniform and Locally Averaged Ones, Russian Mathematics, 2017, vol. 61, pp. 11-20. DOI: 10.3103/S1066369X17100036. 8. Constantin, O. and Pelaez, J. A. Integral Operators, Embedding Theorems and a Littlewood-Paley Formula on Weighted Fock Spaces, The Journal of Geometric Analysis, 2016, vol. 26, no. 2, pp. 1109-1154. DOI: 10.1007/s12220-015-9585-7. 9. Bonet, J. and Taskinen J. A Note about Volterra Operators on Weighted Banach Spaces of Entire Functions, Mathematische Nachrichten, 2015, vol. 288, no. 11-12, pp. 1216-1225. DOI: 10.1002/mana.201400099. 10. Mengestie, T. and Ueki, S.-I. Integral, Differential and Multiplication Operators on Generalized Fock Spaces, Complex Analysis and Operator Theory, 2019, vol. 13, no. 3, pp. 935-953. DOI: 10.1007/s11785-018-0820-7. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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