As far as I know Zygmund class of Orlicz spaces or "$L \log L$" is defined as an Orlicz space with the Young function $Q(t) = t \sqrt{\ln(1+t)}$ (or something similar to this in different references).
I wanted to ask does anyone know about the (weak) pre-compactness condition of this space? If for example I have a sequence $\{ \rho_n\}$ s.t. $\Vert \rho_n\Vert_{L_Q} \leq C$ (where $\Vert \cdot \Vert_{L_Q}$ denotes the Zygmund norm) can I obtain a (weak) convergent subsequence?
Thank you