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

OpenAlex · İngilizce

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