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

Abstract

We study the equivalence between the Orlicz and Luxemburg (quasi-) norms in the context of the generalized Orlicz spaces associated to an N-function Φ and a (quasi-) Banach function space X over a positive finite measure μ. We show that the Orlicz and the Luxemburg spaces do not coincide in general, and also that under mild requirements (σ-Fatou property, strictly monotone renorming) the coincidence holds. We use as a technical tool the classes LΦ w(m), LΦ(m) and LΦ( m ) of Orlicz spaces of scalar integrable functions with respect to a Banach space-valued countably additive vector measure m, providing also some new results on these spaces.