# Optimality of the rearrangement inequality with applications to Lorentz-type sequence spaces

## TL;DR

This paper characterizes sequences that preserve the order of sums in rearrangement inequalities and applies these results to Lorentz-type sequence spaces, resolving an open problem and showing Garling sequence spaces lack symmetric bases.

## Contribution

It provides a complete characterization of sequences for which the rearrangement inequality holds in Lorentz-type spaces and settles an open problem about the structure of Garling sequence spaces.

## Key findings

- Characterization of sequences preserving the rearrangement inequality.
- Resolution of an open problem on Lorentz-type sequence spaces.
- Proof that Garling sequence spaces have no symmetric basis.

## Abstract

We characterize the sequences $(w_i)_{i=1}^\infty$ of non-negative numbers for which \[ \sum_{i=1}^\infty a_i w_i \quad \text{ is of the same order as } \quad \sup_n \sum_{i=1}^n a_i w_{1+n-i} \] when $(a_i)_{i=1}^\infty$ runs over all non-increasing sequences of non-negative numbers. As a by-product of our work we settle a problem raised in [F. Albiac, Jose L. Ansorena and B. Wallis; arXiv:1703.07772[math.FA]] and prove that Garling sequences spaces have no symmetric basis.