Marcinkiewicz integral operators associated with the Dunkl setting and their commutators on some spaces

In this paper, we introduce a Marcinkiewicz integral operator Md whose kernels satisfy the size and the smoothness conditions associated with the Dunkl metric d. In terms of sharp maximal estimates for M_d and its commutators {Md,b} generated by M_d with b ∈ BMO_d ( R ^N ), we show that the M_d and the commutator {Md,b} are bounded on spaces Lᵖ ( R^N ), Morrey spaces L_dp,q ( R^N ) related to the Dunkl metric d and generalized Morrey spaces L_dφ,p ( RN ) associated with the Dunkl metric d, respectively.