On non-effective weights in Orlicz spaces

Given a weight in , and a Young function , we consider the weighted modular ( ( )) ( ) and the resulting weighted Orlicz space ( ). For a Young function ∆ ( ) we present a necessary and sufficient conditions in order that ( ) = ( ) up to the equivalence of norms. We find a necessary and sufficient condition for in order that there exists an unbounded weight such that the above equality of spaces holds. By way of applications we simplify criteria from [CHKM] for continuity of the composition operator from into itself when ∆ ( ) and obtain necessary and sufficient condition in order that the composition operator maps continuously onto .