$L^p$ Boundedness of Commutators of Riesz Transforms associated to Schr\"{o}dinger Operator

In this paper we consider $L^p$ boundedness of some commutators of Riesz transforms associated to Schr"{o}dinger operator $P=-\mathrm{\Delta }+V(x)$ on ${\mathbb{R}}^n,n\ge 3$. We assume that $V(x)$ is non-zero, nonnegative, and belongs to $B_q$ for some $q\ge n\mathrm{/}2$. Let $T_1=(-\Delta+V)^{-1}V,
T_2=(-\Delta+V)^{-1/2}V^{1/2}$and$T_3=(-\Delta+V)^{-1/2}\nabla$. We obtain that $[b,T_j](j=1,2,3)$ are bounded operators on $L^p({\mathbb{R}}^n)$ when $p$ ranges in a interval, where $b\in \mathbf{B}\mathbf{M}\mathbf{O}({\mathbb{R}}^n)$. Note that the kernel of $T_j(j=1,2,3)$ has no smoothness.