Hardy-Littlewood-Sobolev inequality in total Morrey spaces

We study some necessary and sufficient conditions for the boundedness of the Riesz potential operator Iα and its commutator on the total Morrey spaces Lp,λ,μ(Rn). We characterize the strong and weak Spanne type and Adams type boundedness of Iα on Lp,λ,μ(Rn), respectively. We also give necessary and sufficient conditions for the boundedness of the commutator of the Riesz potential operator [b,Iα] on Lp,λ,μ(Rn) when b belongs to the spaces BMO(Rn). As applications, we obtain some estimates for the Marcinkiewicz operator and fractional powers of some analytic semigroups on the total Morrey spaces. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.