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

Abstract With b belonging to a new B M O θ ( ρ ) $BMO_{\theta}(\rho )$ space, L = − △ + V $L=-\triangle +V$ is a Schrödinger operator on 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 I β L ${\mathcal{I}}_{\beta}^{L}$ . We investigate the boundedness of I β L ${\mathcal{I}}_{\beta}^{L}$ and [ b , I β L ] $[b,{\mathcal{I}}_{\beta}^{L}]$ , which are its commutators with b θ ( ρ ) $b_{\theta}(\rho )$ on vanishing generalized mixed Morrey spaces V M p → , φ α , V $VM_{\vec{p},\varphi}^{\alpha ,V}$ related to Schrödinger operation and generalized mixed Morrey spaces M p → , φ α , V $M_{\vec{p},\varphi}^{\alpha ,V}$ . The boundedness of the operator I β L ${\mathcal{I}}_{\beta}^{L}$ is ensured by finding sufficient conditions on the pair ( φ 1 , φ 2 ) $(\varphi _{1},\varphi _{2})$ , which goes from M p → , φ 1 α , V $M_{\vec{p},\varphi _{1}}^{\alpha ,V}$ to M q → , φ 2 α , V $M_{\vec{q},\varphi _{2}}^{\alpha ,V}$ , and from V M p → , φ 1 α , V $VM_{\vec{p},\varphi _{1}}^{\alpha ,V}$ to V M q → , φ 2 α , V $VM_{\vec{q},\varphi _{2}}^{\alpha ,V}$ , ∑ i = 1 n 1 p i − ∑ i = 1 n 1 q i = β $\sum \limits _{i=1}^{n}\frac{1}{p_{i}}-\sum \limits _{i=1}^{n}\frac{1}{q_{i}}=\beta $ . When b belongs to B M O θ ( ρ ) $BMO_{\theta}(\rho )$ and ( φ 1 , φ 2 ) $(\varphi _{1},\varphi _{2})$ satisfies some conditions, we also show that the commutator operator [ b , I β L ] $[b,{\mathcal{I}}_{\beta}^{L}]$ is bounded from M p → , φ 1 α , V $M_{\vec{p},\varphi _{1}}^{\alpha ,V}$ to M q → , φ 2 α , V $M_{\vec{q},\varphi _{2}}^{\alpha ,V}$ and from V M p → , φ 1 α , V $VM_{\vec{p},\varphi _{1}}^{\alpha ,V}$ to V M q → , φ 2 α , V $VM_{\vec{q},\varphi _{2}}^{\alpha ,V}$ .