Boundedness of commutators of variable Marcinkiewicz fractional integral operator in grand variable Herz spaces

Let ${\mathbb{S}}^{n-1}$ $\mathbb{S}^{n-1}$ denote unit sphere in ${\mathbb{R}}^n$ $\mathbb{R}^{n}$ equipped with the normalized Lebesgue measure. Let $\mathrm{\Phi }\in L^s({\mathbb{S}}^{n-1})$ $\Phi \in L^{s}(\mathbb{S}^{n-1})$ be a homogeneous function of degree zero such that ${\int }_{{\mathbb{S}}^{n-1}}\mathrm{\Phi }(y^{\prime })d\sigma (y^{\prime })=0$ $\int _{\mathbb{S}^{n-1}}\Phi (y^{\prime})d \sigma (y^{\prime})=0$ , where $y^{\prime }=y/|y|$ $y^{\prime}=y/|y|$ for any $y\ne 0$ $y\neq 0$ . The commutators of variable Marcinkiewicz fractional integral operator is defined as ${[b,{\mu }_{\mathrm{\Phi }}]}_{\beta }^m(f)(x)={\left(\underset{0}{\overset{\mathrm{\infty }}{\int }}{|\underset{|x-y|\le s}{\int }\frac{\mathrm{\Phi }(x-y){[b(x)-b(y)]}^m}{{|x-y|}^{n-1-\beta (x)}}f(y)dy|}^2\frac{ds}{s^3}\right)}^{\frac{1}{2}}.$ \( [b,\mu _{\Phi}]^{m}_{\beta }(f)(x )= \left ( \int \limits _{0} ^{ \infty }\left |\int \limits _{|x -y | \leq s} \frac{\Phi (x -y )[b(x )-b(y )]^{m}}{|x -y |^{n-1-\beta (x )}}f(y )dy \right |^{2} \frac{ds}{s^{3}}\right )^{\frac{1}{2}}. \)

In this paper, we obtain the boundedness of the commutators of the variable Marcinkiewicz fractional integral operator on grand variable Herz spaces ${\dot{K}}_{p(\cdot )}^{\alpha (\cdot ),q),\theta }({\mathbb{R}}^n)$ ${\dot{K} ^{\alpha (\cdot ), q),\theta}_{ p(\cdot )}(\mathbb{R}^{n})}$ .