Abstract
We introduce and study the variable generalized Hölder spaces of holomorphic functions over the unit disc in the complex plane. These spaces are defined either directly in terms of modulus of continuity or in terms of estimates of derivatives near the boundary. We provide conditions of Zygmund type for imbedding of the former into the latter and vice versa. We study mapping properties of variable order fractional integrals in the frameworks of such spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availibility
The authors confirm that data all generated or analysed during this study are included in this article.
References
Blasco, O., Karapetyants, A., Restrepo, J.: Holomorphic Hölder-type spaces and composition operators. Math. Meth. Appl. Sci. 43, 10005–10026 (2020)
Karapetyants, A., Restrepo, J.: Composition operators on holomorphic variable exponent spaces. Math. Meth. Appl. Sci. 47, 1–12 (2021)
Karapetyants, A., Samko, S.: Generalized Hölder spaces of holomorphic functions in domains in the complex plane. Mediterr. J. Math. 15, 226 (2018)
Karapetyants, A., Samko, S.: Hadamard–Bergman convolution operators. Complex Anal. Oper. Theory 14, 77 (2020). https://doi.org/10.1007/s11785-020-01035-w
Kokilashvili, V., Meskhi, A., Rafeiro, H., Samko, S.: Integral Operators in Non-standard Function Spaces. Vol. I: Variable Exponent Lebesgue and Amalgam Spaces. Operator Theory: Advances and Applications. Birkhäuser, Basel (2016)
Kokilashvili, V., Meskhi, A., Rafeiro, H., Samko, S.: Integral Operators in Non-standard Function Spaces. Vol. II: Variable Exponent Hölder, Morrey–Campanato and Grand Spaces. Operator Theory: Advances and Applications. Birkhäuser, Basel (2016)
Rafeiro, H., Samko, S.: Fractional integrals and derivatives: mapping properties. Fract. Calc. Appl. Anal. 19(3), 580–607 (2016). https://doi.org/10.1515/fca-2016-0032
Samko, S., Vakulov, B.: Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators. J. Math. Anal. Appl. 310(1), 229–246 (2005)
Samko, N., Samko, S., Vakulov, B.: Fractional integrals and hypersingular integrals in variable order Hölder spaces on homogeneous spaces. J. Funct. Spaces 8, Article ID 659456, 30 (2010). https://doi.org/10.1155/2010/659456
Samko, N., Vakulov, B.: Spherical fractional and hypersingular integrals of variable order in generalized Hölder spaces with variable characteristic. Math. Nachr. 284(2–3), 355–369 (2011)
Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball. Springer Verlag, New York (2004)
Zhu, K.: Operator Theory in Function Spaces, vol. 138. AMS Mathematical Surveys and Monographs, Providence (2007)
Acknowledgements
Alexey Karapetyants is supported by the Russian Foundation for Fundamental Research, Project 20-51-46003-a. Stefan Samko is supported by Russian Foundation for Fundamental Research under the Grants 19-01-00223-a and by Russian Foundation for Basic research and TUBITAK under the Grant 20-51-46003-a.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
This work does not have any conflicts of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Karapetyants, A., Samko, S. Variable order fractional integrals in variable generalized Hölder spaces of holomorphic functions. Anal.Math.Phys. 11, 156 (2021). https://doi.org/10.1007/s13324-021-00587-0
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13324-021-00587-0
Keywords
- Hadamard–Bergman convolution operators
- Hölder space of holomorphic functions
- Fractional integro-differentiation