On Musielak-Orlicz spaces isometric to $L_2$ or $L_\infty$

It is proved that a Musielak-Orlicz space $L_\Phi$ of real valued functions which is isometric to a Hilbert space coincides with $L_2$ up to a weight, that is $\Phi(u,t)= c(t)u^2$. Moreover it is shown that any surjective isometry between $L_\Phi$ and $L_\infty$ is a weighted composition operator and a criterion for $L_\Phi$ to be isometric to $L_\infty$ is presented.