Document Type : Research Paper

Authors

1 Department of Non-harmonic Analysis, Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan.

2 Department of Mathematics, Yildiz Technical University, Istanbul, Turkey.

Abstract

In this paper, the concept of a pseudosymmetric space which is a natural generalization of the concept of a symmetric space is defined. All basic concepts such as the Luxemburg representation theorem, the Boyd indices, the fundamental function and its properties, Calderon's theorem, etc., is transferred over the pseudosymmetric case. Examples are given for pseudosymmetric spaces. The quasi-symmetric spaces expand the scope of the application of symmetric space results.

Keywords

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[1] D.R. Adams, Morrey spaces, Switzherland, Springer, 2016.
[2] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, 1988.
[3] B.T. Bilalov, The basis property of a perturbed system of exponentials in Morrey-type spaces, Sib. Math. J., 60(2) (2019), pp. 323-350.
[4] B.T. Bilalov, T.B. Gasymov and A.A. Guliyeva, On solvability of Riemann boundary value problem in Morrey-Hardy classes, Turkish J. Math., 40(50) (2016), pp. 1085-1101.
[5] B.T. Bilalov and A.A. Guliyeva, On basicity of the perturbed systems of exponents in Morrey-Lebesgue space, Internat. J. Math., 25:1450054 (2014), pp. 1-10.
[6] 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(3) (2012), pp. 487-498.
[7] B.T. Bilalov, A.A. Huseynli and S.R. El-Shabrawy, Basis Properties of Trigonometric Systems in Weighted Morrey Spaces, Azerb. J. Math., 9(2) (2019), pp. 200-226.
[8] B.T. Bilalov and S.R. Sadigova, On solvability in the small of higher order elliptic equations in grand-Sobolev spaces, Complex Var. Elliptic Equ., 66(12) (2021), pp. 2117-2130.
[9] B.T. Bilalov and S.R. Sadigova, Interior Schauder-type estimates for higher-order elliptic operators in grand-Sobolev spaces, Sahand Commun. Math. Anal., 18(2) (2021), pp. 129-148.
[10] B.Bilalov and S. Sadigova, On the fredholmness of the Dirichlet problem for a second-order elliptic equation in grand-Sobolev spaces, Ric. Mat. (2021).
[11] B.T. Bilalov and F.Sh. Seyidova, Basicity of a system of exponents with a piecewise linear phase in Morrey-type spaces, Turkish J. Math., 43 (2019), pp. 1850--1866.
[12] L. Caso, R. D'Ambrosio and L. Softova, Generalized Morrey Spaces over Unbounded Domains, Azerb. J. Math., 10(1) (2020), pp. 193-208.
[13] R.E. Castillo and H. Rafeiro, An introductory course in Lebesgue spaces, Springer, 2016.
[14] D.V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue spaces, Birkhauser, Springer, 2013.
[15] P. Harjulehto and P. Hasto, Orlicz spaces generalized Orlicz spaces, Springer, 2019.
[16] D.M. Israfilov and N.P. Tozman, Approximation in Morrey-Smirnov classes, Azerb. J. Math., 1(1) (2011), pp. 99-113.
[17] V. Kokilashvili, A. Meskhi, H. Rafeiro and S. Samko, Integral Operators in Non-Standard Function Spaces, Variable Exponent Lebesgue and Amalgam Spaces, Springer, (1) 2016.
[18] V. Kokilashvili, A. Meskhi, H. Rafeiro and S. Samko, Integral Operators in Non-Standard Function Spaces, Variable Exponent H\"older, Morrey--Campanato and Grand Spaces, Springer, (2) 2016.
[19] S.G. Krein, Ju.I. Petunin and E.M. Semenov, Interpolation of Linear Operators, Nauka, Moscow, 1978.
[20] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, I, Springer-Verlag, 1977.
[21] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, II, Springer-Verlag, 1979.
[22] W.A.J. Luxemberg, Banach function spaces, Ph.D. Thesis, Delft Ins. of Technology, Assen (Netherlands), 1955.
[23] D.K. Palagachev, M.A. Ragusa and L.G. Softova, Regular oblique derivative problem in Morrey spaces, Electron. J. Differential Equations, 2000(39) (2000), pp. 1-17.
[24] D.K. Palagachev and L.G. Softova, Singular integral operators, Morrey spaces and fine regularity of solutions to PDE's, Potential Anal., 20 (2004), pp. 237-263.
[25] M.M. Reo and Z.D. Ren, Applications of Orlichz Spaces, New-York-Basel, 2002.
[26] I.I. Sharapudinov, On Direct And Inverse Theorems of Approximation Theory in Variable Lebesgue and Sobolev Spaces, Azerb. J. Math., 4(1) (2014), pp. 55-72.
[27] L.G. Softova, The Dirichlet problem for elliptic equations with VMO coefficients in generalized Morrey spaces, Operator Theory, 229 (2013), pp. 365-380.
[28] Y. Zeren, M.I. Ismailov and C. Karacam, Korovkin-type theorems and their statistical versions in grand Lebesgue spaces, Turkish J. Math., 44 (2020), pp. 1027-1041.
[29] Y. Zeren, M. Ismailov and Fatih Sirin, On basicity of the system of eigenfunctions of one discontinuous spectral problem for second order differential equation for grand Lebesgue space, Turkish J. Math., 44(5) (2020), pp. 1995-1612.