Fractional Maximal Commutators in Orlicz Spaces on Stratified Lie Groups

We study the fractional maximal commutators \(M_{b,\alpha}, 0 \leq \alpha < Q\) on any stratified Lie group \(G\) in the Orlicz spaces \(L^\Phi(G)\), here \(Q\) is the homogeneous dimension of \(G\). We give necessary and sufficient conditions for the boundedness of the operators \(M_{b,\alpha}\) on Orlicz spaces \(L^\Phi(G)\) 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 \(-\Delta_G + V\) on \(G\), where the nonnegative potential \(V\) belongs to the reverse Hölder class \(B_\infty(G)\). The \(L^\Phi(G) - L^\Psi(G)\) estimates for the operators \(V^\gamma(-\Delta_G + V)^{-\beta}\) and \(V^\gamma\nabla_G(-\Delta_G + V)^{-\beta}\) are obtained.