Ultrasymmetric Orlicz spaces

It is proved that ultrasymmetric reflexive Orlicz spaces can be described exactly as all those Orlicz spaces which can be written as some Lorentz spaces. This description is an answer to the problem posed by Pustylnik in [E. Pustylnik, Ultrasymmetric spaces, J. London Math. Soc. (2) 68 (1) (2003) 165–182]. On the other hand, the Lorentz–Orlicz spaces with non-trivial indices of their fundamental functions are ultrasymmetric.