# Subrearrangement-invariant function spaces

## TL;DR

This paper introduces the concept of subrearrangement-invariance in function spaces, explores its relationship with rearrangement-invariance, and constructs new examples of such spaces with specific properties.

## Contribution

It defines subrearrangement-invariance as a generalization of 1-subsymmetry, proves that all rearrangement-invariant spaces are subrearrangement-invariant, and constructs new spaces inspired by Garling.

## Key findings

- Not all function spaces admit an equivalent subrearrangement-invariant norm.
- Not all subrearrangement-invariant spaces admit an equivalent rearrangement-invariant norm.
- Constructed spaces contain copies of ll_p.

## Abstract

Rearrangement-invariance in function spaces can be viewed as a kind of generalization of 1-symmetry for Schauder bases. We define subrearrangement-invariance in function spaces as an analogous generalization of 1-subsymmetry. It is then shown that every rearrangement-invariant function space is also subrearrangement-invariant. Examples are given to demonstrate that not every function space on $(0,\infty)$ admits an equivalent subrearrangement-invariant norm, and that not every subrearrangement-invariant function space on $(0,\infty)$ admits an equivalent rearrangement-invariant norm. The latter involves constructing a new family of function spaces inspired by D.J.H.\ Garling, and we further study them by showing that they are Banach spaces containing copies of $\ell_p$.