Boundedness Characterization of Maximal Commutators on Orlicz Spaces in the Dunkl Setting

Abstract

On the real line, the Dunkl operators Dν( f )(x) := d f (x) dx +(2ν+1) f (x)− f (−x) 2x , ∀x∈R, ∀ν≥− 1 2 are differential-difference operators associated with the reflection group Z2 on R, and on the Rd the Dunkl operators { Dk,j }d j=1 are the differential-difference operators associated with the reflection group Zd 2 on R d. In this paper, in the setting R we show that b ∈ BMO(R,dmν) if and only if the maximal commutator Mb,ν is bounded on Orlicz spaces LΦ(R,dmν). Also in the setting Rd we show that b∈ BMO(R,hk(x)dx) if and only if the maximal commutator Mb,k is bounded on Orlicz spaces LΦ(R,hk(x)dx). AMS subject classifications: 42B20, 42B25, 42B35