Boundedness of Commutators Generated by Campanato-Type Functions and Riesz Transforms Associated with Schrödinger Operators

Abstract

Let $\mathcal{L} = −∆ + V$ be a Schrödinger operator on $\boldsymbol{R}^n , n > 3$, where $∆$ is the Laplacian on $\boldsymbol{R}^n$ and $V ≠ 0$ is a nonnegative function satisfying the reverse Hölder's inequality. Let $[b, T]$ be the commutator generated by the Campanato-type function $b ∈ Λ^β_{\mathcal{L}}$ and the Riesz transform associated with Schrödinger operator $T = ∇(−∆+V )^{\frac{1}{2}}$. In the paper, we establish the boundedness of $[b, T]$ on Lebesgue spaces and Campanato-type spaces.