On basicity of exponential and trigonometric systems in grand Lebesgue spaces

Year 2022, Volume: 51 Issue: 6, 1577 - 1587, 01.12.2022

Migdad Ismailov Yusuf Zeren , Kader Şimşir Acar , Ilahe F. Aliyarova

https://doi.org/10.15672/hujms.1076849

Cited By: 2

https://izlik.org/JA53MA47WZ

Abstract

Basis properties of exponential and trigonometric systems in grand Lebesgue spaces $ L_{p)} (-\pi,\pi) $ are studied. Based on a shift operator, we consider the subspace $G_{p)} (-\pi,\pi)$ of the space $ L_{p)} (-\pi,\pi) $, where continuous functions are dense, and the boundedness of the singular operator in this subspace is proved. We establish the basicity of exponential system $ \{{e^{int}}\}_{n\in Z}$ for $G_{p)} (-\pi,\pi)$ and the basicity of trigonometric systems $ \{{\sin{nt}}\}_{n\in N }$ and $ \{{\cos{nt}}\}_{n\in N_0}$ for $G_{p)} (0,\pi)$.

Keywords

grand Lebesgue space , exponential system , minimality , density , basis

References

• [1] D.R. Adams, Morrey spaces, Springer Int. Publ. Switzerland, 2016.
• [2] 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 (50), 1085-1101, 2016.
• [3] B.T. Bilalov, T.B. Gasymov and G.V. Maharramova, On one method of investigation of the basis properties of the discontinuous differential operators J. Contemporary Appl. Math. 6 (1), 7481, 2016.
• [4] B.T. Bilalov and Z.G. Guseynov, Basicity of a system of exponents with a piecewise linear phase in variable spaces, Mediterr. J. Math. 9 (3), 487-498, 2012.
• [5] B.T. Bilalov, A.A. Huseynli and S.R. El-Shabrawy, Basis properties of trigonometric systems in weighted Morrey spaces, Azerbaijan J. Math. 9 (2) 200-226, 2019.
• [6] B.T. Bilalov and A.A. Quliyeva, On basicity of exponential systems in Morrey-type spaces, Int. J. Math. 25 (6), 1-10, 2014.
• [7] 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), 2117-2130, 2021.
• [8] 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), 129- 148, 2021.
• [9] B.T. Bilalov and F.Sh. Seyidova, Basicity of a system of exponents with a piecewise linear phase in Morrey-type spaces, Turk. J. Math. 43 (4), 18501866, 2019.
• [10] C. Capone and A. Fiorenza, On small Lebesgue spaces, J. Funct. Spaces Appl. 3, 7389, 2005.
• [11] R.E. Castilo and H. Rafeiro, An Introductory Course in Lebesgue Spaces, Springer Int. Publ. Switzerland, 2016.
• [12] D. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Springer, 2013.
• [13] D. Cruz-Uribe, G. Di Fratta and A. Fiorenza, Modular inequalities for the maximal operator in variable Lebesgue spaces, Nonlin. Anal. 177(part A), 299311, 2018.
• [14] N. Danelia and V. Kokilashvili, Approximation by trigonometric polynomials in subspace of weighted grand Lebesgue spaces, Bulletin of the Georgian National Academy of Sciences 7, 11-15, 2013.
• [15] L. Diening, and S. Samko, Hardy inequality in variable exponent Lebesgue spaces, Fractional Calculus and Appl. Anal. 10 (1), 118, 2007.
• [16] L. Donofrio, C. Sbordone and R. Schiattarella, Grand Sobolev spaces and their application in geometric function theory and PDEs, J. Fixed Point Theory Appl. 13, 309-340, 2013.
• [17] D.E. Edmunds and A. Meskhi, Potential-type operators in $L^{p(x)}$ spaces, Zeitschrift für Analysis and ihre Anwendungen 21 (3), 681690, 2002.
• [18] D.E. Edmunds and H. Triebel, Function spaces, entropy numbers, differential operators, Cambridge Tracts in Mathematics, Cambridge Univ. Press, Cambridge, 1996.
• [19] A. Fiorenza, Duality and reflexivity in grand Lebesgue spaces, Collect. Math. 51(2), 131148, 2000.
• [20] A. Fiorenza and G.E. Karadzhov, Grand and small Lebesgue spaces and their analogs. Z. Anal. Anwend. 23 (4), 657681, 2004.
• [21] L. Greco, A remark on the equality $det Df=Det Df $, Differential Integral Equations 6, 10891100, 1993.
• [22] B. Gupta, A. Fiorenza and P. Jain. The maximal theorem in weighted grand Lebesgue spaces, Studia Math. 188 (2), 123133, 2008.
• [23] D.M. Israfilov and N.P. Tozman, Approximation by polynomials in MorreySmirnov classes, East J. Approx. 14 (3), 255269, 2008.
• [24] T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses, Arch. Rational Mech. Anal. 119, 129143, 1992.
• [25] T. Iwaniec and C. Sbordone, Riesz transforms and elliptic PDEs with VMO coefcients, J. Anal. Math. 74, 183-212, 1998.
• [26] V. Kokilashvili, Boundedness criterion for the Cauchy singular integral operator in weighted grand Lebesgue spaces and application to the Riemann problem, Proc. A.Razmadze Math. Inst. 151, 129133, 2009.
• [27] V.M. Kokilashvili, A. Meskhi, H. Rafeiro and S. Samko , Integral Operators in Non- Standart Function Spaces, v. 2, Variable exponent Holder, Morrey-Campanato and Grand spaces, Birkhauser, 2016.
• [28] C.B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (4), 126-166, 1938.
• [29] A. Nekvinda, Hardy-Littlewood maximal operator on $L^{p(x)} (R)$ Math, Inequal. Appl. 7, 255265, 2004.
• [30] H. Rafeiro and A. Vargas, On the compactness in grand spaces, Georgian Math. J. 22 (1), 141-152, 2015.
• [31] S.G. Samko, Convolution and potential type operators in $L^{p(x)} (R^n)$, Integral Transform. Spec. Funct. 7 (3-4), 261284, 1996.
• [32] I.I. Sharapudinov, On the topology of the space $L^{p(.)}(0,1)$, Mat. Zametki 26 (4), 613- 632 (in Russian), 1979.
• [33] I.I. Sharapudinov, On direct and inverse theorems of approximation theory in variable Lebesgue and Sobolev spaces, Azerbaijan J. Math. 4 (1), 5572, 2014.
• [34] Y. Zeren, M.I. Ismailov and C. Karacam, Korovkin-type theorems and their statistical versions in grand Lebesgue spaces, Turk. J. Math. 44, 1027-1041, 2020.
• [35] Y. Zeren, M. Ismailov and F. Sirin, On basicity of the system of eigenfunctions of one discontinuous spectral problem for second order differential equation for grand Lebesgue space, Turk. J. Math. 44 (5), 1995-1611, 2020.
• [36] C.T. Zorko, Morrey spaces, Proc. Amer. Math. Soc. 1986, 98 (4), 586-592.