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

We study the fractional maximal commutators Mb, and the commutators[b, M] of the fractional maximal operator with b ∈ BMO(X) in the variable Lebesgue spaces Lp(·)(X) over bounded quasi-metric measure spaces. We give necessary and sufficient conditions for the boundedness of the operators Mb, and [b, M] on the spaces Lp(·)(X) when b ∈ BMO(X). Furthermore, we obtain some new characterizations for certain subspaces of BMO(X).