Characterizations of anisotropic Lipschitz functions via thecommutators of anisotropic maximal function in total anisotropicMorrey spaces

We shall give necessary and sufficient conditions for the boundedness of the anisotropic maximal commutators M d b and the commutators of the anisotropic maximal operator [b, M d ] in total Morrey spaces L d p,, (R n ) when b belongs to anisotropic Lipschitz spaces ,d (R n ), whereby some new characterizations for certain subclasses of anisotropic Lipschitz spaces ,d (R n ) are obtained.