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 \(M_{\alpha }\) on total Morrey spaces \(L^{p,\lambda ,\mu }(\mathbb {R}^n)\), respectively. Also we give necessary and sufficient conditions for the boundedness of the fractional maximal commutator operator \(M_{b,\alpha }\) and commutator of fractional maximal operator \([b,M_{\alpha }]\) on \(L^{p,\lambda ,\mu }(\mathbb {R}^n)\) when b belongs to \(BMO(\mathbb {R}^n)\) spaces, whereby some new characterizations for certain subclasses of \(BMO(\mathbb {R}^n)\) spaces are obtained.