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.