Characterizations of Lipschitz functions via the commutators of maximal function in total Morrey spaces

We give necessary and sufficient conditions for the boundedness of the maximal commutators (Formula presented.) and the commutators of the maximal operator (Formula presented.) in total Morrey spaces (Formula presented.) when (Formula presented.) belongs to Lipschitz spaces (Formula presented.), whereby some new characterizations for certain subclasses of Lipschitz spaces (Formula presented.) are obtained. © 2024 John Wiley & Sons Ltd.