The boundedness of commutators associated with Schrödinger operators on Herz spaces

Abstract

Abstract Let L = − Δ + V $L=-\Delta+V$ be a Schrödinger operators on R n $\mathbb{R}^{n}$ , n ≥ 3 $n\geq3$ , where the nonnegative potential V belongs to the reverse Hölder class B s $B_{s}$ for s > d 2 $s>\frac{d}{2}$ . Let T β = ( − Δ + V ) − β V β $T_{\beta}=(-\Delta +V)^{-\beta}V^{\beta}$ , β > 0 $\beta>0$ , and R L = ∇ ( − Δ + V ) − 1 / 2 $R_{L}=\nabla(-\Delta+V)^{-1/2}$ be the Riesz transform associated to L. We prove that the operator T β $T_{\beta}$ and R L $R_{L}$ are bounded on Herz spaces K ˙ q α , p ( R n ) $\dot {K}_{q}^{\alpha ,p}(\mathbb{R}^{n})$ and K q α , p ( R n ) ${K}_{q}^{\alpha,p}(\mathbb{R}^{n})$ , respectively. Suppose that b ∈ BMO σ ( ρ ) $b\in \operatorname{BMO}_{\sigma}(\rho)$ , which is larger than BMO ( R n ) $\operatorname{BMO}(\mathbb{R}^{n})$ . By a maximal estimate, we obtain the boundedness of commutators [ b , T β ] $[b, T_{\beta}]$ and [ b , R L ] $[b, R_{L}]$ on Herz spaces.

Keywords

• commutator
• Herz space
• Schrödinger operator