Characterizations of Lipschitz functions via the commutators of maximal function in Orlicz spaces on stratified Lie groups

We give necessary and sufficient conditions for the boundedness of the maximal commutators $M_b$, the commutators of the maximal operator $[b,M]$ and the commutators of the sharp maximal operator $[b,M^{\mathrm{♯}}]$ in Orlicz spaces $L^{\mathrm{\Phi }}(\mathbb{G})$ on any stratified Lie group $\mathbb{G}$ when $b$ belongs to Lipschitz spaces ${\dot{\mathrm{\Lambda }}}_{\beta }(\mathbb{G})$. We obtain some new characterizations for certain subclasses of Lipschitz spaces ${\dot{\mathrm{\Lambda }}}_{\beta }(\mathbb{G})$.