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 the norm $\Vert \cdot \Vert _{M,a}$ satisfies certain Hardy-type inequalities. This includes the embedding of some Lorentz spaces $\mathrm {d}^n(a,p)$. The approach is based on combinatorial averaging techniques, and a new result of independent interest is proved, which relates suitable averages with Orlicz-Lorentz norms.