Weak compactness in nice Musielak-Orlicz spaces

We prove two weak compactness criteria in Musielak-Orlicz spaces for $N$-functions satisfying the $Δ_2$-condition. They extend criteria from Andô for Orlicz spaces to this setting of non-symmetrical Banach function spaces. As consequences, we prove criteria for a sequence in a Musielak-Orlicz space to be weakly convergent, and show that Musielak-Orlicz spaces with the subsequence splitting property are weakly Banach-Saks. The study includes the case of Musielak-Orlicz sequence spaces.

💡 Research Summary

The paper investigates weak compactness in Musielak‑Orlicz spaces (L_{\varphi}(\Omega)) when the generating function (\varphi) is a generalized (N)-function satisfying the (\Delta_{2})-condition. The authors extend two classical criteria originally proved by T. Andô for Orlicz spaces to this broader, non‑symmetric setting.

First, the authors recall the necessary background on Musielak‑Orlicz spaces: a generalized (\Phi)-function (\varphi:\Omega\times