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

[摘要] 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.

[发布日期] 2020-11-01

[关键词] Banach function space;Vector measure;Orlicz space;Orlicz norm;Luxemburg norm;Strictly monotone norm