Musielak--Orlicz Spaces that are Isomorphic to Subspaces of $L_1$

Abstract

We prove that $\frac{1}{n!}\sum_{\pi\in\mathfrak{S}_n} \left( \sum\limits_{i=1}^n | {x_ia_{i,\pi(i)}}^2 \right)^{\frac{1}{2}}$ is equivalent to a Musielak--Orlicz norm $\|{x}\|_{\Sigma M_i}$. We also provide the converse result, i.e., given the Orlicz functions, we provide a formula for the choice of the matrix that generates the corresponding Musielak--Orlicz norm. As a consequence, we obtain the embedding of 2-concave Musielak--Orlicz spaces into $L_1$.