On non-effective weights in Orlicz spaces

Given a weight w in Ω ⊂ ∝N, |Ω| < ∞ and a Young function φ, we consider the weighted modular ∫Ω ω(f(x))w(x)dx and the resulting weighted Orlicz space Lω(w). For a Young function Ω ∉ Δ2(∞) we present a necessary and sufficient conditions in order that Lω(w) = Lω(XΩ) up to the equivalence of norms. We find a necessary and sufficient condition for ω in order that there exists an unbounded weight w such that the above equality of spaces holds. By way of applications we simplify criteria from [5] for continuity of the composition operator from Lω into itself when ω Δ2(∞) and obtain necessary and sufficient condition in order that the composition operator maps Lω. continuously onto Lω.