# Necessary conditions for the boundedness of linear and bilinear commutators on Banach function spaces

## TL;DR

This paper generalizes the necessary conditions for the boundedness of linear and bilinear commutators, showing that if such commutators are bounded on Banach function spaces, then the symbol function must belong to BMO, extending previous Lebesgue space results.

## Contribution

It extends the known necessity of BMO for commutator boundedness from Lebesgue spaces to a broad class of Banach function spaces under modest assumptions.

## Key findings

- Bounded commutators imply the symbol is in BMO for various Banach spaces.
- Generalization of previous Lebesgue space results to Banach function spaces.
- Establishment of necessary conditions for boundedness of linear and bilinear commutators.

## Abstract

In this article we extend recent results by the first author on the necessity of $BMO$ for the boundedness of commutators on the classical Lebesgue spaces. We generalize these results to a large class of Banach function spaces. We show that with modest assumptions on the underlying spaces and on the operator $T$, if the commutator $[b,T]$ is bounded, then the function $b$ is in $BMO$.