Pointwise multipliers of Musielak–Orlicz spaces and factorization

We prove that the space of pointwise multipliers between two distinct Musielak–Orlicz spaces is another Musielak–Orlicz space and the function defining it is given by an appropriately generalized Legendre transform. In particular, we obtain characterization of pointwise multipliers between Nakano spaces. We also discuss factorization problem for Musielak–Orlicz spaces and exhibit some differences between Orlicz and Musielak–Orlicz cases.