Fractional type Marcinkiewicz integral and its commutator on grand variable Herz-Morrey spaces (Q6592856)

In this manuscript, the authors study the boundedness of the fractional type Marcinkiewicz integral and its commutator on grand variable Herz-Morrey spaces. Throughout the paper, the authors prove that 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 \(b\in\mathrm{BMO}(\mathbb{R}^n)\) and \(\mathcal{M}_{\alpha,\rho,m}\) are bounded on the weighted Lebesgue spaces \(L_\omega^p(\mathbb{R}^n)\). Let \(\alpha(\cdot)\) and \(q(\cdot)\) satisfy the \(\log\) decay at both the infinity and origin, in the setting of these conditions, 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.