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

Abstract

Let $\mathbb{S}^{n-1}$ denote unit sphere in $\mathbb{R}^{n}$ equipped with the normalized Lebesgue measure. Let $\Phi \in L^{s}(\mathbb{S}^{n-1})$ be a homogeneous function of degree zero such that $\int _{\mathbb{S}^{n-1}}\Phi (y^{\prime})d \sigma (y^{\prime})=0$ , where $y^{\prime}=y/|y|$ for any $y\neq 0$ . The commutators of variable Marcinkiewicz fractional integral operator is defined as $$ [b,\mu _{\Phi}]^{m}_{\beta }(f)(x )= \left ( \int \limits _{0} ^{ \infty }\left |\int \limits _{|x -y | \leq s} \frac{\Phi (x -y )[b(x )-b(y )]^{m}}{|x -y |^{n-1-\beta (x )}}f(y )dy \right |^{2} \frac{ds}{s^{3}}\right )^{\frac{1}{2}}. $$ In this paper, we obtain the boundedness of the commutators of the variable Marcinkiewicz fractional integral operator on grand variable Herz spaces ${\dot{K} ^{\alpha (\cdot ), q),\theta}_{ p(\cdot )}(\mathbb{R}^{n})}$ .