Embeddings of Orlicz–Lorentz spaces into $L_1$

It is shown that the Orlicz–Lorentz spaces $\ell ^n_{M,a}$, $n\in \mathbb {N}$, with Orlicz function $M$ and weight sequence $a$ are uniformly isomorphic to subspaces of $L_1$ if norm $\| \cdot \|_{M,a}$ satisfies certain Hardy-type inequalities. This includes embedding some Lorentz $\mathrm {d}^n(a,p)$. The approach based on combinatorial averaging techniques, a new result independent interest proved, which relates suitable averages norms.