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

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

2008-02-21

arxiv.org/abs/0802.3128v1

J. Math. Anal. Appl. 341 (2008) 421-432

Mathematics

Classical Analysis and ODEs

14 pages, 0 figures

Scientific paper

In this paper we consider $L^p$ boundedness of some commutators of Riesz transforms associated to Schr\"{o}dinger operator $P=-\Delta+V(x)$ 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 $T_1=(-\Delta+V)^{-1}V,\ T_2=(-\Delta+V)^{-1/2}V^{1/2}$ 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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