Marcinkiewicz integrals with the Dunkl setting and their commutators on some function spaces

Abstract

In this paper, we introduce a Marcinkiewicz integral operator $\mathcal{M}_{d}$ whose kernels satisfy the size and the smoothness conditions associated with the Dunkl metric $d$. Via establishing the sharp maximal estimate for the $\mathcal{M}_{d}$ and its commutators $\mathcal{M}_{d,b}$ generated by $b\in\mathrm{BMO}_{d}(\mathbb{R}^{N})$ and the $\mathcal{M}_{d}$, we show that the $\mathcal{M}_{d}$ and the commutator $\mathcal{M}_{d,b}$
are bounded on spaces $L^{p}(\mathbb{R}^{N})$, Morrey spaces $L^{p,q}_{d}(\mathbb{R}^{N})$ related to the Dunkl metric $d$ and generalized Morrey spaces $\mathcal{L}^{\varphi,p}_{d}(\mathbb{R}^{N})$ associated with the Dunkl metric $d$, respectively.