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 L_ω(w). For a Young function Ω ∉ Δ₂(∞) we present a necessary and sufficient conditions in order that L_ω(w) = L_ω(X_Ω) 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 L_ω into itself when ω Δ₂(∞) and obtain necessary and sufficient condition in order that the composition operator maps L_ω. continuously onto L_ω.