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

We study the fractional maximal commutators Mb,η$$ {M}_{b,\eta } $$ and the commutators [b,Mη]$$ \left[b,{M}_{\eta}\right] $$ of the fractional maximal operator with b∈BMO(X)$$ b\in BMO(X) $$ in the variable Lebesgue spaces Lp(·)(X)$$ {L}^{p\left(\cdotp \right)}(X) $$ over bounded quasi‐metric measure spaces. We give necessary and sufficient conditions for the boundedness of the operators Mb,η$$ {M}_{b,\eta } $$ and [b,Mη]$$ \left[b,{M}_{\eta}\right] $$ on the spaces Lp(·)(X)$$ {L}^{p\left(\cdotp \right)}(X) $$ when b∈BMO(X)$$ b\in BMO(X) $$. Furthermore, we obtain some new characterizations for certain subspaces of BMO(X)$$ BMO(X) $$.