Hardy-Littlewood-Sobolev inequality in total Morrey spaces on stratified lie groups

Abstract

We study some necessary and sufficient conditions for the boundedness of the fractional integral operator Iα and its commutator [b, Iα] on the total Morrey spaces Lp,λ,µ(G), where G is a stratified Lie group. We characterize the strong and weak Spanne type and Adams type boundedness of Iα on Lp,λ,µ(G), respectively. We also give necessary and sufficient conditions for the boundedness of the commutator of the fractional integral operator [b, Iα] on Lp,λ,µ(G) when b belongs to the spaces BMO(G).