Fractional type Marcinkiewicz integral and its commutator on grand variable Herz-Morrey spaces

The aim of this paper is to establish the boundedness of the fractional type Marcinkiewicz integral operator \(\mathcal {M}_{\alpha ,\rho ,m}\) $\mathcal{M}_{\alpha ,\rho ,m}^{}$ and its higher order commutator \(\mathcal {M}_{\alpha ,\rho ,m,b^l}\) $\mathcal{M}_{\alpha ,\rho ,m,b^l}^{}$ generated by \(b\in \textrm{BMO}({\mathbb {R}}^n)\) $b\in \text{BMO}({\mathbb{R}}^n)$ and \(\mathcal {M}_{\alpha ,\rho ,m}\) $\mathcal{M}_{\alpha ,\rho ,m}^{}$ on the weighted Lebesgue spaces \(L_\omega ^p({\mathbb {R}}^n)\) $L_{\omega }^p({\mathbb{R}}^n)$ . Under assumption that the variable exponents \(\alpha (\cdot )\) $\alpha (·)$ and \(q(\cdot )\) $q(·)$ satisfy the \(\log \) $log$ decay at infinity and origin, the authors show that the \(\mathcal {M}_{\alpha ,\rho ,m}\) $\mathcal{M}_{\alpha ,\rho ,m}^{}$ and \(\mathcal {M}_{\alpha ,\rho ,m,b^l}\) $\mathcal{M}_{\alpha ,\rho ,m,b^l}^{}$ are bounded on the grand variable Herz spaces \(\dot{K}_{q(\cdot )}^{\alpha (\cdot ),p),\theta }({\mathbb {R}}^n)\) ${\dot{K}}_{q(·)}^{\alpha (·),p),\theta }({\mathbb{R}}^n)$ and the grand variable Herz-Morrey spaces \(M\dot{K}_{p),\theta ,q(\cdot )}^{\alpha (\cdot ),\lambda }({\mathbb {R}}^n)\) $M{\dot{K}}_{p),\theta ,q(·)}^{\alpha (·),\lambda }({\mathbb{R}}^n)$ , respectively.