BMO estimate for the higher order commutators of Marcinkiewicz integral operator on grand Herz-Morrey spaces

Abstract

Let $\mathbb{S}^{n-1}$ denote the unit sphere in $\mathbb{R}^n$ with the normalized Lebesgue measure. Let $\Phi\in L^{r}(\mathbb{S}^{n-1})$ is a homogeneous function of degree zero and $b$ is a locally integrable function on $\mathbb{R}^n$. In this paper we define the higher order commutators of Marcinkiewicz integral $[b,\mu_{\Phi}]^m$ and prove the boundedness of $[b,\mu_{\Phi}]^m$ under some proper assumptions on grand variable Herz-Morrey spaces $M\dot{K}^{\alpha(.),\beta}_{u,v(.)}(\mathbb{R}^n)$.