Commutators of fractional maximal operator in variable Lebesgue spaces over bounded quasi-metric measure spaces

We study the fractional maximal commutators (Formula presented.) and the commutators (Formula presented.) of the fractional maximal operator with (Formula presented.) in the variable Lebesgue spaces (Formula presented.) over bounded quasi-metric measure spaces. We give necessary and sufficient conditions for the boundedness of the operators (Formula presented.) and (Formula presented.) on the spaces (Formula presented.) when (Formula presented.). Furthermore, we obtain some new characterizations for certain subspaces of (Formula presented.). © 2022 John Wiley & Sons, Ltd.