Marcinkiewicz integrals associated with Schrödinger operator and their commutators on vanishing generalized Morrey spaces

Abstract

Abstract Let L = − Δ + V $L=-\Delta+V$ be a Schrödinger operator, where Δ is the Laplacian on R n $\mathbb{R}^{n}$ and the non-negative potential V belongs to the reverse Hölder class RH q $\mathit{RH}_{q}$ for q ≥ n / 2 $q \ge n/2$ . In this paper, we study the boundedness of the Marcinkiewicz integral operators μ j L $\mu_{j}^{L}$ and their commutators [ b , μ j L ] $[b,\mu_{j}^{L}]$ with b ∈ BMO θ ( ρ ) $b \in \mathit{BMO}_{\theta}(\rho)$ on generalized Morrey spaces M p , φ α , V ( R n ) $M_{p,\varphi }^{\alpha,V}(\mathbb{R}^{n})$ associated with Schrödinger operator and vanishing generalized Morrey spaces VM p , φ α , V ( R n ) $\mathit{VM}_{p,\varphi}^{\alpha ,V}(\mathbb{R}^{n})$ associated with Schrödinger operator. We find the sufficient conditions on the pair ( φ 1 , φ 2 ) $(\varphi_{1},\varphi_{2})$ which ensure the boundedness of the operators μ j L $\mu_{j}^{L}$ from one vanishing generalized Morrey space VM p , φ 1 α , V $\mathit{VM}_{p,\varphi_{1}}^{\alpha,V}$ to another VM p , φ 2 α , V $\mathit{VM}_{p,\varphi_{2}}^{\alpha,V}$ , 1 < p < ∞ $1< p<\infty$ and from the space VM 1 , φ 1 α , V $\mathit{VM}_{1,\varphi_{1}}^{\alpha,V}$ to the weak space V W M 1 , φ 2 α , V $VWM_{1,\varphi _{2}}^{\alpha,V}$ . When b belongs to BMO θ ( ρ ) $\mathit{BMO}_{\theta}(\rho)$ and ( φ 1 , φ 2 ) $(\varphi_{1},\varphi _{2})$ satisfies some conditions, we also show that [ b , μ j L ] $[b,\mu_{j}^{L}]$ is bounded from M p , φ 1 α , V $M_{p,\varphi_{1}}^{\alpha,V}$ to M p , φ 2 α , V $M_{p,\varphi _{2}}^{\alpha,V}$ and from VM p , φ 1 α , V $\mathit{VM}_{p,\varphi_{1}}^{\alpha,V}$ to VM p , φ 2 α , V $\mathit{VM}_{p,\varphi_{2}}^{\alpha,V}$ , 1 < p < ∞ $1< p<\infty$ .

Keywords

• Schrödinger operator
• Marcinkiewicz integral
• vanishing generalized Morrey space
• commutator
• BMO