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 \(\mathcal {M}_{\alpha ,\rho ,m}\) and its higher order commutator \(\mathcal {M}_{\alpha ,\rho ,m,b^l}\) generated by \(b\in \textrm{BMO}({\mathbb {R}}^n)\) and \(\mathcal {M}_{\alpha ,\rho ,m}\) on the weighted Lebesgue spaces \(L_\omega ^p({\mathbb {R}}^n)\). Under assumption that the variable exponents \(\alpha (\cdot )\) and \(q(\cdot )\) satisfy the \(\log \) decay at infinity and origin, the authors show that the \(\mathcal {M}_{\alpha ,\rho ,m}\) and \(\mathcal {M}_{\alpha ,\rho ,m,b^l}\) are bounded on the grand variable Herz spaces \(\dot{K}_{q(\cdot )}^{\alpha (\cdot ),p),\theta }({\mathbb {R}}^n)\) and the grand variable Herz-Morrey spaces \(M\dot{K}_{p),\theta ,q(\cdot )}^{\alpha (\cdot ),\lambda }({\mathbb {R}}^n)\), respectively.