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

摘要

We study the fractional maximal commutators Mb,η$$ {M}_{b,eta } $$ and the commutators b,Mη$$ leftb,{M}_{eta}right $$ of the fractional maximal operator with b∈BMO(X)$$ bin BMO(X) $$ in the variable Lebesgue spaces Lp(·)(X)$$ {L}amp;amp;amp;#x0005E;{pleft(cdotp right)}(X) $$ 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$$ leftb,{M}_{eta}right $$ on the spaces Lp(·)(X)$$ {L}amp;amp;amp;#x0005E;{pleft(cdotp right)}(X) $$ when b∈BMO(X)$$ bin BMO(X) $$. Furthermore, we obtain some new characterizations for certain subspaces of BMO(X)$$ BMO(X) $$.