# Commutators of potential type operators with Lipschitz symbols on variable Lebesgue spaces with different weights

## TL;DR

This paper establishes conditions under which commutators of potential type operators with Lipschitz symbols are bounded between variable Lebesgue spaces with different weights, extending previous results with more general conditions and symbols.

## Contribution

It introduces a generalized Fefferman-Phong condition for weighted boundedness of commutators on variable Lebesgue spaces, including Lipschitz symbols and Musielak-Orlicz norms.

## Key findings

- Boundedness of commutators under generalized weight conditions
- Extension to Lipschitz symbols and Musielak-Orlicz norms
- Improved conditions for variable Lebesgue spaces

## Abstract

We prove that a generalized Fefferman-Phong type condition on a pair of weights $u$ and $v$ is sufficient for the boundedness of the commutators of potential type operators from $L^{p(\cdot)}_v$ into $L^{q(\cdot)}_u$. We also give an improvement of this result in the sense that we not only consider a variable version of power bump conditions, but also weaker norms related to Musielak-Orlicz functions. We consider a wider class of symbols including Lipschitz symbols and some generalizations.