Potential-free $L^1$-estimates for positivity-preserving Riesz transform related to Schrödinger operator in dimension one

Abstract

Let $V\ge 0$ be a locally integrable function on $\mathbb{R}$. Consider the Schrödinger operator $L=-\frac{d^2}{d{x}^2}+V$. We prove that for all $0<a\le 1$, there is a constant $C_a$, independent of $V$, such that the Riesz transform type operator $V^a{L}^{-a}$ is bounded on $L^1(\mathbb{R})$ and $\Vert V^a{L}^{-a}f{\Vert }_{L^1(\mathbb{R})}\le C_a\Vert f{\Vert }_{L^1(\mathbb{R})}$.