Fractional Maximal Commutators in Orlicz Spaces on Stratified Lie Groups

We study the fractional maximal commutators \(Mb,α, 0 ≤ α < Q\) on any stratified Lie group \(G\) in the Orlicz spaces \(L^Φ(G)\), here \(Q\) is the homogeneous dimension of \(G\). We give necessary and sufficient conditions for the boundedness of the operators \(Mb,α\) on Orlicz spaces \(L^Φ(G)\) when \(b\) belongs to \(BMO(G)\) spaces, by which some new characterizations of \(BMO(G)\) spaces is obtained. As applications we consider the Schrödinger operator \(-Δ_G + V\) on \(G\), where the nonnegative potential \(V\) belongs to the reverse Hölder class \(B_∞(G)\). The \(L^Φ(G) - L^Ψ(G)\) estimates for the operators \(V^γ(-Δ_G + V)-β\) and \(V^γ∇_G(-Δ_G + V)-β\) are obtained.