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

Abstract

On the real line, the Dunkl operators Dν are differential-difference operators associated with the reflection group Z2 on R. In this paper, in the setting R we study the commutators of the maximal operator associated with the Dunkl operator [b, Mν] in the total Dν-Morrey spaces Lp,λ,µ (R, dmν). We give necessary and sufficient conditions for the boundedness of the operator [b, Mν] on total Dν-Morrey spaces Lp,λ,µ (R, dmν) when b belongs to BMO(R, dmν) spaces, whereby some new characterizations for certain subclasses of BMO(R, dmν) spaces are obtained.