Fractional integral related to Schrödinger operator on vanishing generalized mixed Morrey spaces

With b belonging to a new $BM{O}_{\theta }(\rho )$ $BMO_{\theta}(\rho )$ space, $L=-\mathrm{△}+V$ $L=-\triangle +V$ is a Schrödinger operator on ${\mathbb{R}}^n$ ${\mathbb{R}^{n}}$ with nonnegative potential V belonging to the reverse Hölder class $R{H}_{n/2}$ $RH_{n/2}$ . The fractional integral operator associated with L is denoted by ${\mathcal{I}}_{\beta }^L$ ${\mathcal{I}}_{\beta}^{L}$ . We investigate the boundedness of ${\mathcal{I}}_{\beta }^L$ ${\mathcal{I}}_{\beta}^{L}$ and $[b,{\mathcal{I}}_{\beta }^L]$ $[b,{\mathcal{I}}_{\beta}^{L}]$ , which are its commutators with $b_{\theta }(\rho )$ $b_{\theta}(\rho )$ on vanishing generalized mixed Morrey spaces $V{M}_{\overrightarrow{p},\phi }^{\alpha ,V}$ $VM_{\vec{p},\varphi}^{\alpha ,V}$ related to Schrödinger operation and generalized mixed Morrey spaces $M_{\overrightarrow{p},\phi }^{\alpha ,V}$ $M_{\vec{p},\varphi}^{\alpha ,V}$ . The boundedness of the operator ${\mathcal{I}}_{\beta }^L$ ${\mathcal{I}}_{\beta}^{L}$ is ensured by finding sufficient conditions on the pair $({\phi }_1,{\phi }_2)$ $(\varphi _{1},\varphi _{2})$ , which goes from $M_{\overrightarrow{p},{\phi }_1}^{\alpha ,V}$ $M_{\vec{p},\varphi _{1}}^{\alpha ,V}$ to $M_{\overrightarrow{q},{\phi }_2}^{\alpha ,V}$ $M_{\vec{q},\varphi _{2}}^{\alpha ,V}$ , and from $V{M}_{\overrightarrow{p},{\phi }_1}^{\alpha ,V}$ $VM_{\vec{p},\varphi _{1}}^{\alpha ,V}$ to $V{M}_{\overrightarrow{q},{\phi }_2}^{\alpha ,V}$ $VM_{\vec{q},\varphi _{2}}^{\alpha ,V}$ , $\sum _{i=1}^n\frac{1}{p_i}-\sum _{i=1}^n\frac{1}{q_i}=\beta$ $\sum \limits _{i=1}^{n}\frac{1}{p_{i}}-\sum \limits _{i=1}^{n}\frac{1}{q_{i}}=\beta $ . When b belongs to $BM{O}_{\theta }(\rho )$ $BMO_{\theta}(\rho )$ and $({\phi }_1,{\phi }_2)$ $(\varphi _{1},\varphi _{2})$ satisfies some conditions, we also show that the commutator operator $[b,{\mathcal{I}}_{\beta }^L]$ $[b,{\mathcal{I}}_{\beta}^{L}]$ is bounded from $M_{\overrightarrow{p},{\phi }_1}^{\alpha ,V}$ $M_{\vec{p},\varphi _{1}}^{\alpha ,V}$ to $M_{\overrightarrow{q},{\phi }_2}^{\alpha ,V}$ $M_{\vec{q},\varphi _{2}}^{\alpha ,V}$ and from $V{M}_{\overrightarrow{p},{\phi }_1}^{\alpha ,V}$ $VM_{\vec{p},\varphi _{1}}^{\alpha ,V}$ to $V{M}_{\overrightarrow{q},{\phi }_2}^{\alpha ,V}$ $VM_{\vec{q},\varphi _{2}}^{\alpha ,V}$ .