On Disjointly Homogeneous Orlicz–Lorentz Spaces

A characterization of disjointly homogeneous Orlicz–Lorentz function spaces $\Lambda_{\varphi,w}$ is obtained. It is used to find necessary and sufficient conditions for an analog of the classical Dunford–Pettis theorem about the equi-integrability of weakly compact sets in $L_1$ to hold in the space $\Lambda_{\varphi,w}$. It is also shown that there exists an Orlicz function $\Phi$ with the upper Matuszewska–Orlicz index equal to $1$ for which such an analog in the space $\Lambda_{\Phi,w}$ does not hold. This answers a recent question of Leśnik, Maligranda, and Tomaszewski.