Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
approximate identity; Musielak-Orlicz space; density of smooth functions
approximate identity; Musielak-Orlicz space; density of smooth functions
Summary:
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.
References:
[1] Adams, R. A., Fournier, J. J. F.: 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] Benkirane, A., Douieb, J., Val, M. Ould Mohamedhen: 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] Benkirane, A., Val, M. Ould Mohamedhen: Some approximation properties in Musielak-Orlicz-Sobolev spaces. Thai J. Math. 10 (2012), 371-381. MR 3001860 | Zbl 1264.46024
[4] Bennett, C., Sharpley, R.: 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] Cruz-Uribe, D., Fiorenza, A.: Approximate identities in variable $L^{p}$ spaces. Math. Nachr. 280 (2007), 256-270. DOI 10.1002/mana.200410479 | MR 2292148 | Zbl 1178.42022
[6] Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: 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] Hudzik, H.: A generalization of Sobolev spaces. II. Funct. Approximatio, Comment. Math. 3 (1976), 77-85. MR 0467279 | Zbl 0355.46011
[8] Kamińska, A.: 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] Kamińska, A.: 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] Kamińska, A., Hudzik, H.: 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] Kamińska, A., Kubiak, D.: 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] Kováčik, O., Rákosník, J.: On spaces $L^{p(x)}$ and $W^{k,p(x)}$. Czech. Math. J. 41 (1991), 592-618. DOI 10.21136/CMJ.1991.102493 | MR 1134951 | Zbl 0784.46029
[13] Krasnosel'skiĭ, M. A., Rutitskiĭ, J. B.: Convex Functions and Orlicz Spaces. P. Noordhoff, Groningen (1961). MR 0126722 | Zbl 0095.09103
[14] Kufner, A., John, O., Fučík, S.: 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] Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics 1034, Springer, Berlin (1983). DOI 10.1007/BFb0072210 | MR 0724434 | Zbl 0557.46020
[16] Nakano, H.: Modulared Semi-Ordered Linear Spaces. Tokyo Math. Book Series 1, Maruzen, Tokyo (1950). MR 0038565 | Zbl 0041.23401
[17] Orlicz, W.: Über konjugierte Exponentenfolgen. Studia Math. German 3 (1931), 200-211. DOI 10.4064/sm-3-1-200-211 | Zbl 0003.25203
Search
Partner of
