Embeddings of Orlicz-Lorentz spaces into $L_1$

Abstract

In this article, we show that Orlicz-Lorentz spaces ${\ell }_{M,a}^n$, $n\in \mathbb{N}$ 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 $d^n(a,p)$. 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.

14 pages