Isomorphisms of Orlicz spaces

In this paper, we provide some isomorphism preserving conditions for (weighted) Orlicz spaces, and as a main result, it is proved that if there exist a bicontinuous linear operator T : L-w1(Phi)(G(1)) -> L-w2(Phi)(G(2)) and a mapping a bar right arrow xi((a), h(a)) from G(1) to C x G(2) with T lambda(a) = xi(a)lambda(h(a)) T for all a is an element of G(1), then G(1) and G(2) are isomorphic, where Phi is a Delta(2)-regular Young function, G(1) and G(2) are locally compact groups and w(1) and w(2) are weight functions. Also, for a class of Young functions Phi, we show that if CV Phi(G(1)) and CV Phi(G(2)) are isometrically isomorphic, then G(1) and G(2) are isomorphic, CV Phi(G(i)) is the space of all convolution operators on the Orlicz space L-Phi(G(i)) for i = 1, 2.