Some properties of convolution in symmetric spaces and approximate identity

Abstract

This paper deals with the symmetric space of functions and its subspace where continuous functions are dense is considered. Main properties of convolution which plays a vital role in harmonic analysis, as in other areas of mathematics are established in this space. Following the classical case, it is proved that the convolution can be approximated by linear combinations of shifts in a subspace of the considered space. An approximate identity for the convolution is also considered in that subspace.

Keywords

• symmetric spaces
• convolution
• approximate identity

References

1. Peetre, J., On the theory of $L^{p,\alpha}$ spaces, J. Funct. Anal., 4 (1964), 71-87. https://doi.org/10.1016/0022-1236(69)90022-6
2. Zorko, C. T., Morrey space, Proc. Of the Amer. Math. Society, 98 (4) (1986), 586-592. DOI: https://doi.org/10.1090/S0002-9939-1986-0861756-X
3. Ky, N. X., On approximation by trigonometric polynomials in $L^{p}_{u}$-spaces, Studia Sci. Math. Hungar, 28 (1993), 183-188. DOI:10.1515/gmj-2012-0043
4. Samko, N., Weighted hardy and singular operators in Morrey spaces, Journal of Math. Anal. and Appl., 350(1), (2009), 56-72. DOI:10.1016/j.jmaa.2008.09.021
5. Israfilov, D. M., Tozman, N. P., Approximation in Morrey-Smirnov classes, Azerb. J. of Math., 1(1), (2011), 99-113. DOI:10.3336/gm.40.1.09
6. Bilalov, B. T., Guliyeva, A. A., On basicity of exponential systems in Morrey-type spaces, Inter. J. of Math., 25(6) (2014), 10 pages. DOI:10.1142/S0129167X14500542
7. Bilalov, B. T., The basis property of a perturbed system of exponentials in Morrey-type spaces, Siberian Math. J., 60(2) (2019), 249-271. DOI:https://doi.org/10.33048/smzh.2019.60.206
8. Bilalov, B. T., Seyidova, F.Sh., Basicity of a system of exponents with a piecewise linear phase in Morrey-type spaces, Turk. J. Math., 43 (2019), 1850-1866. DOI:10.1007/s00009-011-0135-7

9. Bilalov, B. T., The basis properties of power systems in Lp, Siberian Math. J., 47(1) (2016), 18-27. DOI:10.1007/s11202-006-0002-0
10. Bilalov, B. T., Guliyeva, F. A., A completeness criterion for a double power system with degenerate coe¢ cients, Siberian Math. J., 54(3) (2013), 419-424. DOI:10.1134/S0037446613030051
11. Huseynli, A. A., Mirzoyev, V. S., Quliyeva, A. A., On Basicity of the Perturbed System of Exponents in Morrey-Lebesgue Space, Azerbaijan Journal of Mathematics, 7 (2) (2017), 191-209.
12. Guliyeva, F. A., Sadigova, S. R., On Some Properties of Convolution in Morrey Type Spaces, Azerbaijan Journal of Mathematics, 8(1) (2018), 140-150.
13. Bilalov, B. T., Huseynli, A. A., El-Shabrawy S. R., Basis Properties of Trigonometric Systems in Weighted Morrey Spaces, Azerbaijan Journal of Mathematics, 9(2) (2019), 166-192.
14. Guliyeva, F. A., Sadigova, S. R., Bases of the perturbed system of exponents in generalized weighted Lebesgue space with a general weight, Afr. Mat., 28(5-6) (2017), 781-791. DOI:10.1007/s13370-017-0488-6
15. Bilalov, B. T., Gasymov, T. B., Guliyeva, A.A., On solvability of Riemann boundary value problem in Morrey-Hardy classes, Turk. J. of Math. 40(50) (2016), 1085-1101. DOI:10.3906/mat-1507-10
16. Burenkov, V. I., Tararykova, T. V., An analog of Young's inequality for convolutions of functions for general Morrey-type spaces, Proceedings of the Steklov Institute of Mathematics, 293(1) (2016), 107-126. DOI:10.1134/S0081543816040088
17. Bennett, C., Sharpley, R., Interpolation of Operators, Academic Press, 1988, 469 p.
18. Krein, S. G., Petunin, Yu. I., Semenov E. M., Interpolation of linear operators, Moscow, Nauka, 1978, 400 p. (in Russian)
19. Edwards, R., Fourier Series, v.2, Moscow, 1985. (in Russian)
20. Krepela, M., Convolution in rearrangement-invariant spaces defined in terms of oscillation and the maximal function, Journal of Analysis and its Applications, 33(4) (2014), 369-383. DOI:10.4171/ZAA/1517
21. Nursultanov, E., Tikhonov, S., Convolution inequalities in Lorentz spaces, J. Fourier Anal. Appl., 17(3) (2011), 486-505. DOI:10.1007/s00041-010-9159-9
22. O'Neil, R., Convolution operators and $L(p; q)$ spaces, Duke Math. J., 30(1) (1963), 129-142. DOI: 10.1215/S0012-7094-63-03015-1
23. Kwok-Pun, H., Approximation in vanishing rearrangement-invariant Morrey spaces and applications, Revista de la Real Academia de Ciencias Exactas, FÌsicas y Naturales. Serie A. Matematicas 113(4) (2019), 2999-3014. DOI: 10.1007/s13398-019-00668-7
24. Kokilashvili, V., Meskhi A., Rafeiro, H., Samko, S., Variable Exponent Lebesgue and Amalgam Spaces, Springer, Integral Operators in Non-Standard Function Spaces, 248(1) (2016). DOI:10.1007/978-3-319-21015-5
25. Kokilashvili, V., Meskhi, A., Rafeir, H., Samko, S., Variable Exponent Hölder, Morrey Campanato and Grand Spaces, Integral Operators in Non-Standard Function Spaces, Springer, 249(2) (2016). DOI:10.1007/978-3-319-21018-6