Commutator of the Maximal Function in Total Morrey Spaces for the Dunkl Operator on the Real Line

On the real line, the Dunkl operators D are differential-difference operators associated with the reflection group Z 2 on R. In this paper, in the setting R we study the commutators of the maximal operator associated with the Dunkl operator [b, M ] in the total D -Morrey spaces L p,, (R, dm ).We give necessary and sufficient conditions for the boundedness of the operator [b, M ] on total D -Morrey spaces L p,, (R, dm ) when b belongs to BM O(R, dm ) spaces, whereby some new characterizations for certain subclasses of BM O(R, dm ) spaces are obtained.