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

Abstract

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