Embeddings of Orlicz-Lorentz spaces into $L_1$.

In this article, we show that Orlicz-Lorentz spaces $ell^n_{M,a}$, $ninmathbb N$ with Orlicz function $M$ and weight sequence $a$ are uniformly isomorphic to subspaces of $L_1$ if the norm $|cdot|_{M,a}$ satisfies certain Hardy-type inequalities. This includes the embedding of some Lorentz spaces $d^n(a,p)$. Our approach is based on combinatorial averaging techniques and we prove a new result of independent interest that relates suitable averages with Orlicz-Lorentz norms.