In this paper, we introduce grand Herz–Morrey spaces with variable exponent and prove the boundedness of Riesz potential operators in these spaces.

In this paper, we introduce grand weighted Herz–Morrey spaces with a variable exponent and prove the boundedness of fractional integrals on these spaces.

The aim of this paper is to obtain the boundedness of commutators of Hardy operators with rough kernels on grand variable Herz spaces K ˙ q ⁢ ( ⋅ ) a ⁢ ( ⋅ ) , u , θ ⁢ ( ℝ n ) by applying some properties of variable exponent. Moreover, by using the idea of grand variable Herz–Morrey spaces, we will prove the boundedness of Hardy operators on these spaces.

In this article, we prove the boundedness of an intrinsic square function and sublinear operators under appropriate assumptions about the grand variable Herz–Morrey spaces \(M\dot{K} ^{a(\cdot),u),\theta}_{ \beta, q(\cdot)}(\mathbb{R}^n)\), where \(a(\cdot)\) is log-Hölder continuous both at the origin and at infinity.

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 ) .