Characterizations for the fractional maximal operator and its commutators on total Morrey spaces

We shall give a characterization for the strong and weak type Adams type boundedness of the fractional maximal operator Mα on total Morrey spaces Lp,λ,μ(Rn), respectively. Also we give necessary and sufficient conditions for the boundedness of the fractional maximal commutator operator Mb,α and commutator of fractional maximal operator [b,Mα] on Lp,λ,μ(Rn) when b belongs to BMO(Rn) spaces, whereby some new characterizations for certain subclasses of BMO(Rn) spaces are obtained. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024.