Musielak-Orlicz Spaces that are Isomorphic to Subspaces of L_1

TL;DR

This paper establishes a connection between Musielak-Orlicz spaces and subspaces of L_1, providing equivalence results and explicit constructions for embeddings, advancing the understanding of the structure of these function spaces.

Contribution

It introduces a new equivalence between a specific average involving permutations and Musielak-Orlicz norms, and provides a formula for matrices generating these norms, enabling embeddings into L_1.

Findings

Equivalence of a permutation-averaged sum to Musielak-Orlicz norms

Explicit formula for matrices generating Musielak-Orlicz norms

Embedding of 2-concave Musielak-Orlicz spaces into L_1

Abstract

In this note we prove that $\frac{1}{n!} \sum_{\pi} (\sum_{i=1}^n |x_i a_{i,\pi(i)} |^2)^{1/2}$ is equivalent to a Musielak-Orlicz norm $\norm{x}_{\sum M_i}$. We also obtain the inverse 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.