Embeddings of Orlicz-Lorentz spaces into $ L_1$

In this article, we show 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 $\|\,\cdot\,\|_{M,a}$ satisfies certain Hardy-type inequalities. This includes the embedding of some Lorentz spaces $\mathrm{d}^n(a,p)$. 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.