×

Some approximation results in Musielak-Orlicz spaces. (English) Zbl 1524.46046

This paper deals with the Musielak-Orlicz spaces \(L_M\), which have as a particular case the so-called Nakano spaces, or the Lebesgue spaces of variable exponent \(L^{p(\cdot)}(\Omega)\). The authors establish some approximation results on these spaces, both with respect to the modular associated to a \(\varphi\)-function \(M\) (the standard Musielak-Orlicz notation is used), and to convergence in norm. As the authors explain, these results find applications in the existence theory for solutions of partial differential equations involving non-standard growths described in terms of Musielak-Orlicz functions.
In essence, the authors show density results for smooth functions in Musielak spaces, first proving a result on \(M\)-mean continuity of bounded functions compactly supported on \(\Omega\), which they apply to obtain the convergence in norm of approximate identities. Continuity in norm of the translation (Theorem 2.1 and Corollary 2.1) is proved under some reasonable conditions, as well as some results on the density of functions on \(C^\infty_0(\Omega)\) (Theorem 2.2).

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B10 Duality and reflexivity in normed linear and Banach spaces
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adams, R. A.; Fournier, J. J. F., Sobolev Spaces, Pure and Applied Mathematics 140, Academic Press, New York (2003) · Zbl 1098.46001 · doi:10.1016/S0079-8169(13)62896-2
[2] Benkirane, A.; Douieb, J.; Val, M. Ould Mohamedhen, An approximation theorem in Musielak-Orlicz-Sobolev spaces, Commentat. Math. 51 (2011), 109-120 · Zbl 1294.46025 · doi:10.14708/cm.v51i1.5313
[3] Benkirane, A.; Val, M. Ould Mohamedhen, Some approximation properties in Musielak-Orlicz-Sobolev spaces, Thai J. Math. 10 (2012), 371-381 · Zbl 1264.46024
[4] Bennett, C.; Sharpley, R., Interpolation of Operators, Pure and Applied Mathematics 129, Academic Press, Boston (1988) · Zbl 0647.46057 · doi:10.1016/S0079-8169(13)62909-8
[5] Cruz-Uribe, D.; Fiorenza, A., Approximate identities in variable \(L^p\) spaces, Math. Nachr. 280 (2007), 256-270 · Zbl 1178.42022 · doi:10.1002/mana.200410479
[6] Diening, L.; Harjulehto, P.; Hästö, P.; Růžička, M., Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics 2017, Springer, Berlin (2011) · Zbl 1222.46002 · doi:10.1007/978-3-642-18363-8
[7] Hudzik, H., A generalization of Sobolev spaces. II, Funct. Approximatio, Comment. Math. 3 (1976), 77-85 · 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 · Zbl 0504.46024 · doi:10.14708/cm.v22i2.6021
[9] Kamińska, A., Some convexity properties of Musielak-Orlicz spaces of Bochner type, Rend. Circ. Mat. Palermo, II. Ser. Suppl. 10 (1985), 63-73 · Zbl 0609.46015
[10] Kamińska, A.; Hudzik, H., Some remarks on convergence in Orlicz space, Commentat. Math. 21 (1980), 81-88 · Zbl 0436.46022 · doi:10.14708/cm.v21i1.5965
[11] Kamińska, A.; Kubiak, D., The Daugavet property in the Musielak-Orlicz spaces, J. Math. Anal. Appl. 427 (2015), 873-898 · Zbl 1325.46012 · doi:10.1016/j.jmaa.2015.02.035
[12] Kováčik, O.; Rákosník, J., On spaces \(L^{p(x)}\) and \(W^{k,p(x)}\), Czech. Math. J. 41 (1991), 592-618 · Zbl 0784.46029
[13] Krasnosel’skiĭ, M. A.; Rutitskiĭ, J. B., Convex Functions and Orlicz Spaces, P. Noordhoff, Groningen (1961) · 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) · Zbl 0364.46022
[15] Musielak, J., Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics 1034, Springer, Berlin (1983) · Zbl 0557.46020 · doi:10.1007/BFb0072210
[16] Nakano, H., Modulared Semi-Ordered Linear Spaces, Tokyo Math. Book Series 1, Maruzen, Tokyo (1950) · Zbl 0041.23401
[17] Orlicz, W., Über konjugierte Exponentenfolgen, Studia Math. German 3 (1931), 200-211 · Zbl 0003.25203 · doi:10.4064/sm-3-1-200-211
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.