\(L^p\) boundedness of commutators of Riesz transforms associated to Schrödinger operator (Q2474948)

The authors consider the Schrödinger operator on \(\mathbb{R}^{n}\), \(n \geq 3\), \[ P=-\Delta +V(x), \] with a non-negative potential \(V (x)\) for which the reverse Hölder inequality \[ \biggl(\frac{1}{| B|} \int_{B} V^{q}\,dx\biggl)^{\frac{1}{q}} < c \biggl(\frac{1}{| B |}\int_{B} V\,dx\biggl) \] holds with \(q>\frac{n}{2}\) for every ball \(B\) in \(\mathbb{R}^{n}\). Denote \(T_{1}=(-\Delta+V)^{-1}V\), \(T_{2}=(-\Delta+V)^{-\frac{1}{2}}V^{\frac{1}{2}}\) and \(T_{3}=(-\Delta+V)^{-\frac{1}{2}}\nabla\). The authors study the \(L_{p}\)-boundedness of the commutator operators \([b,T_{j}]=bT_{j}-T_{j}b\) (\(j = 1,2,3\)), where \(b\in\text{BMO}(\mathbb{R}^{n})\).