Symmetric finite representability of $\ell^p$-spaces in rearrangement invariant spaces on $[0,1]$

TL;DR

This paper characterizes the set of p-values for which the classical sequence space ll^p can be finitely represented in a given rearrangement invariant space on [0,1], focusing on symmetric and disjointly supported functions.

Contribution

It extends previous work by identifying all p for which ll^p is finitely represented in separable rearrangement invariant spaces on [0,1], emphasizing symmetric and disjoint support conditions.

Findings

Identifies all p with finite ll^p representability in X.

Characterizes the structure of ll^p embeddings via disjoint, equimeasurable functions.

Builds on prior results for spaces on (0,) to the case on [0,1].

Abstract

For a separable rearrangement invariant space $X$ on $[0,1]$ of fundamental type we identify the set of all $p\in [1,\infty]$ such that $\ell^p$ is finitely represented in $X$ in such a way that the unit basis vectors of $\ell^p$ ($c_0$ if $p=\infty$) correspond to pairwise disjoint and equimeasurable functions. This can be treated as a follow up of a paper by the first-named author related to separable rearrangement invariant spaces on $(0,\infty)$.