Estimates for the commutators of operator Vα∇(−Δ+V)−β $V^{\alpha }\nabla (-\Delta +V)^{-\beta }$

Abstract

Abstract Let a function b belong to the space BMOθ(ρ) $\operatorname{BMO}_{\theta }(\rho )$, which is larger than the space BMO(Rn) $\operatorname{BMO}(\mathbb{R}^{n})$, and let a nonnegative potential V belong to the reverse Hölder class RHs $\mathit{RH}_{s}$ with n/2<s<n $n/2< s< n$, n≥3 $n\geq 3$. Define the commutator [b,Tβ]f=bTβf−Tβ(bf) $[b,T_{\beta }]f=bT_{ \beta }f-T_{\beta }(bf)$, where the operator Tβ=Vα∇L−β $T_{\beta }=V^{\alpha } \nabla \mathcal{L}^{-\beta }$, β−α=12 $\beta -\alpha =\frac{1}{2}$, 12<β≤1 $\frac{1}{2}< \beta \leq 1$, and L=−Δ+V $\mathcal{L}=-\Delta +V$ is the Schrödinger operator. We have obtained the Lp $L^{p}$-boundedness of the commutator [b,Tβ]f $[b,T_{\beta }]f$ and we have proved that the commutator is bounded from the Hardy space HL1(Rn) $H^{1}_{\mathcal{L}}(\mathbb{R}^{n})$ into weak L1(Rn) $L^{1}(\mathbb{R}^{n})$.

Keywords

• Schrödinger operator
• Hardy space
• Reverse Hölder class
• BMO
• Commutator