Czechoslovak Mathematical Journal, Vol. 70, No. 2, pp. 453-471, 2020
Some approximation results in Musielak-Orlicz spaces
Ahmed Youssfi, Youssef Ahmida
Received July 30, 2018. Published online December 17, 2019.
Abstract: We prove the continuity in norm of the translation operator in the Musielak-Orlicz $L_M$ spaces. An application to the convergence in norm of approximate identities is given, whereby we prove density results of the smooth functions in $L_M$, in both the modular and norm topologies. These density results are then applied to obtain basic topological properties.
Keywords: approximate identity; Musielak-Orlicz space; density of smooth functions
References: [1] R. A. Adams, J. J. F. Fournier: Sobolev Spaces. Pure and Applied Mathematics 140, Academic Press, New York (2003). DOI 10.1016/S0079-8169(13)62896-2 | MR 2424078 | Zbl 1098.46001
[2] A. Benkirane, J. Douieb, M. Ould Mohamedhen Val: An approximation theorem in Musielak-Orlicz-Sobolev spaces. Commentat. Math. 51 (2011), 109-120. DOI 10.14708/cm.v51i1.5313 | MR 2849685 | Zbl 1294.46025
[3] A. Benkirane, M. Ould Mohamedhen Val: Some approximation properties in Musielak-Orlicz-Sobolev spaces. Thai J. Math. 10 (2012), 371-381. MR 3001860 | Zbl 1264.46024
[4] C. Bennett, R. Sharpley: Interpolation of Operators. Pure and Applied Mathematics 129, Academic Press, Boston (1988). DOI 10.1016/S0079-8169(13)62909-8 | MR 0928802 | Zbl 0647.46057
[5] D. Cruz-Uribe, A. Fiorenza: Approximate identities in variable $L^p$ spaces. Math. Nachr. 280 (2007), 256-270. DOI 10.1002/mana.200410479 | MR 2292148 | Zbl 1178.42022
[6] L. Diening, P. Harjulehto, P. Hästö, M. Růžička: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics 2017, Springer, Berlin (2011). DOI 10.1007/978-3-642-18363-8 | MR 2790542 | Zbl 1222.46002
[7] H. Hudzik: A generalization of Sobolev spaces. II. Funct. Approximatio, Comment. Math. 3 (1976), 77-85. MR 0467279 | Zbl 0355.46011
[8] A. Kamińska: On some compactness criterion for Orlicz subspace $E_{\Phi}(\Omega)$. Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 22 (1981), 245-255. DOI 10.14708/cm.v22i2.6021 | MR 0641438 | Zbl 0504.46024
[9] A. Kamińska: Some convexity properties of Musielak-Orlicz spaces of Bochner type. Rend. Circ. Mat. Palermo, II. Ser. Suppl. 10 (1985), 63-73. MR 0894273 | Zbl 0609.46015
[10] A. Kamińska, H. Hudzik: Some remarks on convergence in Orlicz space. Commentat. Math. 21 (1980), 81-88. DOI 10.14708/cm.v21i1.5965 | MR 0577673 | Zbl 0436.46022
[11] A. Kamińska, D. Kubiak: The Daugavet property in the Musielak-Orlicz spaces. J. Math. Anal. Appl. 427 (2015), 873-898. DOI 10.1016/j.jmaa.2015.02.035 | MR 3323013 | Zbl 1325.46012
[12] O. Kováčik, J. Rákosník: On spaces $L^{p(x)}$ and $W^{k,p(x)}$. Czech. Math. J. 41 (1991), 592-618. MR 1134951 | Zbl 0784.46029
[13] M. A. Krasnosel'skiĭ, J. B. Rutitskiĭ: Convex Functions and Orlicz Spaces. P. Noordhoff, Groningen (1961). MR 0126722 | Zbl 0095.09103
[14] A. Kufner, O. John, S. Fučík: Function Spaces. Monographs and Textbooks on Mechanics of Solids and Fluids. Mechanics: Analysis, Noordhoff International Publishing, Leyden; Academia, Prague (1977). MR 0482102 | Zbl 0364.46022
[15] J. Musielak: Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics 1034, Springer, Berlin (1983). DOI 10.1007/BFb0072210 | MR 0724434 | Zbl 0557.46020
[16] H. Nakano: Modulared Semi-Ordered Linear Spaces. Tokyo Math. Book Series 1, Maruzen, Tokyo (1950). MR 0038565 | Zbl 0041.23401
[17] W. Orlicz: Über konjugierte Exponentenfolgen. Studia Math. 3 (1931), 200-211. (In German.) DOI 10.4064/sm-3-1-200-211 | Zbl 0003.25203
Affiliations: Ahmed Youssfi, Youssef Ahmida, Sidi Mohamed Ben Abdellah University, National School of Applied Sciences, Laboratory of Engineering, Systems and Applications (LISA), My Abdellah Avenue, Road Imouzer, P.O. Box 72 Fès-Principale, 30 000 Fez, Morocco, e-mail: ahmed.youssfi@gmail.com, ahmed.youssfi@usmba.ac.ma, youssef.ahmida@usmba.ac.ma
