# Some notes on commutators of the fractional maximal function on variable Lebesgue spaces

## TL;DR

This paper investigates the boundedness of the commutator of the fractional maximal function on variable Lebesgue spaces, providing new characterizations for functions in Lipschitz and BMO spaces.

## Contribution

It establishes necessary and sufficient conditions for boundedness of the commutator on variable Lebesgue spaces, introducing new characterizations for Lipschitz and BMO functions.

## Key findings

- Characterizations of boundedness for the commutator on variable Lebesgue spaces.
- New criteria for Lipschitz and BMO functions related to the commutator.
- Conditions linking the function space properties to the commutator's boundedness.

## Abstract

Let $0<\alpha<n$ and $M_{\alpha}$ be the fractional maximal function. The nonlinear commutator of $M_{\alpha}$ and a locally integrable function $b$ is given by $[b,M_{\alpha}](f)=bM_{\alpha}(f)-M_{\alpha}(bf)$. In this paper, we mainly give some necessary and sufficient conditions for the boundedness of $[b,M_{\alpha}]$ on variable Lebesgue spaces when $b$ belongs to Lipschitz or $BMO(\rn)$ spaces, by which some new characterizations for certain subclasses of Lipschitz and $BMO(\rn)$ spaces are obtained.