When does the associate space equal the topological dual in quasi-Banach function spaces?

Determine necessary and sufficient conditions for a quasi-Banach function space X under which its associate space X' coincides with its topological dual X*, thereby providing a complete characterization of the equality X' = X* in the quasi-Banach setting beyond the known sufficient condition of absolute continuity of the quasinorm.

Background

Theorem ThmACNDual proves that if a quasi-Banach function space X has absolutely continuous quasinorm then X' = X*, and for Banach function spaces the converse also holds. However, in general quasi-Banach spaces the converse fails, and the authors note the lack of a full characterization.

Understanding when X' equals X* is fundamental for duality, weak topologies, and the ergodic results developed later in the paper, especially those relying on the identification of duals via associate spaces.

A complete characterization would clarify the structural requirements (e.g., norm properties, lattice features, rearrangement-invariance, or other conditions) needed to ensure this duality equivalence in the quasi-Banach framework.