Sobolev-Type Theorem for Commutators of Hardy Operators in Grand Herz Spaces

The higher-order commutators of fractional Hardy-type operators of variable order ζ(z) are shown to be bounded from the grand variable Herz spaces \({\dot{K}}_{p\left(\cdot \right)}^{a\left(\cdot \right),\left.u\right),\theta }\left({\mathbb{R}}^{n}\right)\) ${\dot{K}}_{p\left(·\right)}^{a\left(·\right),\left(u\right),\theta }\left({\mathbb{R}}^n\right)$ into the weighted space \({\dot{K}}_{\rho ,q\left(\cdot \right)}^{a\left(\cdot \right),\left.u\right),\theta }\left({\mathbb{R}}^{n}\right)\) ${\dot{K}}_{\rho ,q\left(·\right)}^{a\left(·\right),\left(u\right),\theta }\left({\mathbb{R}}^n\right)$ , where \(\rho ={\left(1+\left|{z}_{1}\right|\right)}^{-\lambda }\) $\rho ={\left(1,+,\left|z_1\right|\right)}^{-\lambda }$ and \(\frac{1}{q\left(z\right)}=\frac{1}{p\left(z\right)}-\frac{\zeta \left(z\right)}{n}\) $\frac{1}{q\left(z\right)}=\frac{1}{p\left(z\right)}-\frac{\zeta \left(z\right)}{n}$ if p(z) is not necessarily constant at infinity.