On the embedding of 2-concave Orlicz spaces into $L^1$

TL;DR

This paper characterizes the embedding of 2-concave Orlicz spaces into L^1 by providing explicit formulas for sequences that make certain sums equivalent to given Orlicz norms.

Contribution

It offers a formula for sequences that realize the equivalence between specific sums and 2-concave Orlicz norms, extending previous characterizations.

Findings

Explicit sequence formulas for Orlicz norm equivalence

Extension of previous embedding results

Characterization of 2-concave Orlicz spaces in L^1

Abstract

In [K--S 1] it was shown that $$ \underset {\pi} \to {\text{Ave}} (\sum_{i=1}^{n}|x_i a_{\pi(i)}|^2)^{\frac {1}{2}} $$ is equivalent to an Orlicz norm whose Orlicz function is 2-concave. Here we give a formula for the sequence $a_1, a_2,....,a_n$ so that the above expression is equivalent to a given Orlicz norm.