Abstract
In this paper, rearrangement invariant space is considered and the norm generated by it of the Hardy classes of analytic functions inside and outside the unit ball, respectively. Using the shift operator, the subspace is distinguished. It is proved that infinitely differentiable and compactly supported functions are dense in this subspace. Some properties of functions from Hardy classes are studied. The classical Lebesgue spaces, the grand-Lebesgue spaces, the Orlicz spaces, and many other spaces are rearrangement invariant spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
C. Bennett and R. Sharpley, Interpolation of Operators (Academic, New York, 1988).
D. V. Cruz-Vrible and A. Fiorenza, Variable Lebesgue Spaces (Springer, Basel, 2013).
D. R. Adams, Morrey Spaces (Springer, Switzherland, 2016).
V. Kokilashvili, A. Meskhi, H. Rafeiro, and S. Samko, Integral Operators in Non-Standard Function Spaces, Vol. 1: Variable Exponent Lebesgue and Amalgam Spaces (Springer, Switzerland, 2016).
V. Kokilashvili, A. Meskhi, H. Rafeiro, and S. Samko, Integral Operators in Non-Standard Function Spaces, Vol. 2: Variable Exponent Hölder, Morrey–Campanato and Grand Spaces (Springer, Switzerland, 2016).
R. E. Castillo and H. Rafeiro, An Introductory Course in Lebesgue Spaces (Springer, Switzerland, 2016).
M. M. Reo and Z. D. Ren, Applications of Orlichz Spaces (Marcel Dekker, New York, 2002).
B. T. Bilalov and Z. G. Guseynov, ‘‘Basicity criterion for perturbed systems of exponents in Lebesgue spaces with variable summability,’’ Dokl. Math. 83, 93–96 (2011).
B. T. Bilalov, T. B. Gasymov, and A. A. Guliyeva, ‘‘On solvability of Riemann boundary value problem in Morrey–Hardy classes,’’ Turk. J. Math. 40, 1085–1101 (2016).
B. T. Bilalov and Z. G. Guseynov, ‘‘Basicity of a system of exponents with a piece-wise linear phase in variable spaces,’’ Mediterr. J. Math. 9, 487–498 (2012).
B. T. Bilalov and A. A. Guliyeva, ‘‘On basicity of exponential systems in Morrey-type spaces,’’ Int. J. Math. 25 (6), 1–10 (2014).
B. T. Bilalov, ‘‘The basis property of a perturbed system of exponentials in Morrey-type spaces,’’ Sib. Math. J. 60, 249–271 (2019).
D. M. Israfilov and N. P. Tozman, ‘‘Approximation in Morrey–Smirnov classes,’’ Azerb. J. Math. 1, 99–113 (2011).
I. I. Sharapudinov, ‘‘On Direct and inverse theorems of approximation theory in variable Lebesgue and Sobolev spaces,’’ Azerb. J. Math. 4 (1), 55–72 (2014).
B. T. Bilalov, A. A. Huseynli, and S. R. El-Shabrawy, ‘‘Basis properties of trigonometric systems in weighted Morrey spaces,’’ Azerb. J. Math. 9, 200–226 (2019).
J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces (Springer, Berlin, 1979), Vol. 2.
S. G. Krein, Ju. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators (Nauka, Moscow, 1978) [in Russian].
R. Lecniewicz, On Hardy–Orlicz Spaces 1, Ann. Soc. Math. Polon., 1 (1977).
B. T. Bilalov and A. E. Guliyeva, ‘‘Banach Hardy spaces, Cauchy formula and Riesz theorem,’’ Azerb. J. Math. 10, 157–174 (2020).
ACKNOWLEDGMENTS
The author would like to express her deep gratitude to corresponding member of the National Academy of Sciences of Azerbaijan, Professor Bilal T. Bilalov for his attention to this work.
Author information
Authors and Affiliations
Corresponding author
Additional information
(Submitted by T. K. Yuldashev)
Rights and permissions
About this article
Cite this article
Alili, V.G. Banach Hardy Classes and Some Properties. Lobachevskii J Math 43, 284–292 (2022). https://doi.org/10.1134/S1995080222050031
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080222050031