Characterizations of commutators of the maximal function in total Morrey spaces on stratified Lie groups

The aim of this paper is to study the maximal commutators \(M_{b}\) and the commutators of the maximal operator [b, M] in the total Morrey spaces \(L^{p,\lambda ,\mu }(\mathbb {G})\) on any stratified Lie group \(\mathbb {G}\) when b belongs to Lipschitz spaces \({\dot{\Lambda }}_{\beta }(\mathbb {G})\). Some new characterizations for certain subclasses of Lipschitz spaces \({\dot{\Lambda }}_{\beta }(\mathbb {G})\) are given.