Weak compactness in nice Musielak-Orlicz spaces

TL;DR

This paper establishes new weak compactness criteria in Musielak-Orlicz spaces satisfying the elta_2 condition, extending previous results from Orlicz spaces and exploring implications for weak convergence and Banach-Saks properties.

Contribution

It extends weak compactness criteria from Orlicz spaces to Musielak-Orlicz spaces, including sequence spaces, under the elta_2 condition.

Findings

Criteria for weak convergence in Musielak-Orlicz spaces.

Musielak-Orlicz spaces with the subsequence splitting property are weakly Banach-Saks.

Extension of compactness criteria to non-symmetrical Banach function spaces.

Abstract

We prove two weak compactness criteria in Musielak-Orlicz spaces for $N$-functions satisfying the $\Delta_2$-condition. They extend criteria from And\^o 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.