Necessary and sufficient conditions for boundedness of commutators of fractional maximal function in variable Lebesgue spaces on stratified groups

In this paper, the main aim is to consider the mapping properties of the maximal or nonlinear commutator for the fractional maximal operator with the symbols belong to the Lipschitz spaces on variable Lebesgue spaces in the context of stratified Lie group, with the help of which some new characterizations to the Lipschitz spaces and nonnegative Lipschitz functions are obtained in the stratified groups context. Meanwhile, Some equivalent relations between the Lipschitz norm and the variable Lebesgue norm are also given.