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

TL;DR

This paper establishes the boundedness of certain commutators of Riesz transforms related to Schrödinger operators on L^p spaces, under specific potential conditions, extending understanding of their operator behavior without smooth kernels.

Contribution

It proves L^p boundedness of commutators of Riesz transforms associated with Schrödinger operators with potentials in B_q, a result not previously established for kernels lacking smoothness.

Findings

Boundedness of [b, T_j] on L^p for p in a specific interval

Applicable to potentials V in B_q with q ≥ n/2

Extends operator theory to non-smooth kernel cases

Abstract

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.