Musielak--Orlicz Spaces that are Isomorphic to Subspaces of L1

Abstract

We prove that $\frac{1}{n!}\sum _{\pi \in {\mathfrak{S}}_n}{\left(\sum _{i=1}^n|{x_i{a}_{i,\pi (i)}}^2\right)}^{\frac{1}{2}}$ is equivalent to a Musielak--Orlicz norm $\Vert x{\Vert }_{\mathrm{\Sigma }{M}_i}$. 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 $L_1$.