| In this paper, the authors prove that the parameterized area integral $\mu_{\Omega,S}^\rho$\\ and the parameterized Littlewood-Paley $g^{\ast}_{\delta}$-function $\mu_{\Omega,\delta}^{\ast,\rho}$ are bounded on two-weight grand homogeneous variable Herz-Morrey spaces $M\dot{K}_{p),\theta,q(\cdot)}^{\alpha(\cdot),\lambda}(\omega_1,\omega_2)$, where $\theta>0$, $\lambda\in(2,\infty)$, $q(\cdot)\in{\mathbb{B}(\mathbb{R}^n)}$, $\alpha(\cdot)\in{L^{\infty}(\mathbb{R}^n)}$, $\omega_1\in A_p{_{\omega_1}}$ for $p_{\omega_1}\in[1,\infty]$ and $\omega_2$ is a weight. Furthermore, the authors prove that the commutators $[b,\mu_{\Omega,S}^{\rho}]$ which is formed by $b\in\mathrm{BMO}(\mathbb{R}^n)$ and the $\mu_{\Omega,S}^{\rho}$, and the $[b,\mu_{\Omega,\delta}^{*,\rho}]$ generated by $b\in\mathrm{BMO}(\mathbb{R}^n)$ and the $\mu_{\Omega,\delta}^{\ast,\rho}$ are bounded on $M\dot{K}_{p),\theta,q(\cdot)}^{\alpha(\cdot),\lambda}(\omega_1,\omega_2)$, respectively. |
