$L^p$ Boundedness of Commutators of Riesz Transforms associated to Schrödinger Operator

Abstract

In this paper we consider $L^p$ boundedness of some commutators of Riesz transforms associated to Schrödinger operator $P=-\Delta +V(x)$ on ${\mathbb{R}}^n,n\ge 3$. We assume that $V(x)$ is non-zero, nonnegative, and belongs to $B_q$ for some $q\ge n/2$. 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 $b\in \mathbf{BMO}({\mathbb{R}}^n)$. Note that the kernel of $T_j(j=1,2,3)$ has no smoothness.

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