When and where the Orlicz and Luxemburg (quasi-) norms are equivalent?

[EN] We study the equivalence between the Orlicz and Luxemburg (quasi-) norms in the context of the generalized Orlicz spaces associated to an N-function Phi and a (quasi-) Banach function space X over a positive finite measure mu. We show that the Orlicz and the Luxemburg spaces do not coincide in general, and also that under mild requirements (sigma-Fatou property, strictly monotone renorming) the coincidence holds. We use as a technical tool the classes L-omega(Phi)(m), L-Phi(m) and L-Phi(parallel to m parallel to) of Orlicz spaces of scalar integrable functions with respect to a Banachspace-valued countably additive vector measure m, providing also some new results on these spaces. (C) 2020 Elsevier Inc. All rights reserved.This research has been partially supported by La Junta de Andalucia (Spain) under the grant FQM-133. The fifth author gratefully acknowledges the support of the Ministerio de Ciencia, Innovacion y Universidades (Spain) and FEDER under grant MTM2016-77054-C2-1-P2.Del Campo, R.; Fernández, A.; Mayoral, F.; Naranjo, F.; Sánchez Pérez, EA. (2020). When and where the Orlicz and Luxemburg (quasi-) norms are equivalent?. Journal of Mathematical Analysis and Applications. 491(1):1-18. https://doi.org/10.1016/j.jmaa.2020.1243021184911Del Campo, R., Fernández, A., Ferrando, I., Mayoral, F., & Naranjo, F. (2008). Multiplication operators on spaces of integrable functions with respect to a vector measure. Journal of Mathematical Analysis and Applications, 343(1), 514-524. doi:10.1016/j.jmaa.2008.01.080Del Campo, R., Fernández, A., Mayoral, F., & Naranjo, F. (2016). Reflexivity of function spaces associated to a σ-finite vector measure. Journal of Mathematical Analysis and Applications, 438(1), 339-350. doi:10.1016/j.jmaa.2016.01.076Del Campo, R., Fernández, A., Mayoral, F., & Naranjo, F. (2019). The de la Vallée-Poussin theorem and Orlicz spaces associated to a vector measure. Journal of Mathematical Analysis and Applications, 470(1), 279-291. doi:10.1016/j.jmaa.2018.10.001Delgado, O. (2004). Banach function subspaces of L1 of a vector measure and related Orlicz spaces. Indagationes Mathematicae, 15(4), 485-495. doi:10.1016/s0019-3577(05)00004-2Ferrando, I., & Galaz-Fontes, F. (2009). Multiplication operators on vector measure Orlicz spaces. Indagationes Mathematicae, 20(1), 57-71. doi:10.1016/s0019-3577(09)80003-7Jain, P., Persson, L. E., & Upreti, P. (2007). Inequalities and properties of some generalized Orlicz classes and spaces. Acta Mathematica Hungarica, 117(1-2), 161-174. doi:10.1007/s10474-007-6083-9Kalton, N. J. (1984). Convexity conditions for non-locally convex lattices. Glasgow Mathematical Journal, 25(2), 141-152. doi:10.1017/s0017089500005553Lewis, D. R. (1972). On integrability and summability in vector spaces. Illinois Journal of Mathematics, 16(2). doi:10.1215/ijm/1256052286Moore, L. (1971). Strictly increasing Riesz norms. Pacific Journal of Mathematics, 37(1), 171-180. doi:10.2140/pjm.1971.37.171Roy, S. K., & Chakraborty, N. D. (1986). Orlicz spaces for a family of measures. Analysis Mathematica, 12(3), 229-235. doi:10.1007/bf01907709Schep, A. R. (2009). Products and factors of Banach function spaces. Positivity, 14(2), 301-319. doi:10.1007/s11117-009-0019-