与薛定谔算子相关的里斯变换的交换子的$L^p$有界性
$L^p$ Boundedness of Commutators of Riesz Transforms associated to Schrödinger Operator
摘要：
本文考虑了与Schr dinger算子$P=-Δ+V(x)$相关联的里斯变换的某些算子在$\mathbb{R}^n, n\geq 3$上的$L^p$有界性.我们假设$V(x)$是非零、非负的，并且对于某些$q \geq n/2$属于$B_q$。设$T_1=(-Δ+V)^{-1}V,\ T_2=(-Δ+V)^{-1/2}V^{1/2}$和$T_3=(-Δ+V)^{-1/2}\nabla$。当$p$在一个区间上时，我们得到了$[b,T_j] (j=1,2,3)$是$L^p(\mathbb{R}^n)$上的有界算子，其中$b \in \mathbf{BMO}(\mathbb{R}^n)$ .注意，$T_j (j=1,2,3)$的内核没有平滑性。
英文摘要：
In this paper we consider $L^p$ boundedness of some commutators of Riesz transforms associated to Schrödinger operator $P=-Δ+V(x)$ on $\mathbb{R}^n, n\geq 3$. We assume that $V(x)$ is non-zero, nonnegative, and belongs to $B_q$ for some $q \geq n/2$. Let $T_1=(-Δ+V)^{-1}V,\ T_2=(-Δ+V)^{-1/2}V^{1/2}$ and $T_3=(-Δ+V)^{-1/2}\nabla$. 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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