Riesz transforms associated with Schrödinger operators acting on weighted Hardy spaces

Abstract

Let $L=-\Delta +V$ be a Schrödinger operator acting on $L^2({\mathbb{R}}^n)$, $n\ge 1$, where $V\not\equiv 0$ is a nonnegative locally integrable function on ${\mathbb{R}}^n$. In this article, we will introduce weighted Hardy spaces $H_L^p(w)$ associated with $L$ by means of the area integral function and study their atomic decomposition theory. We also show that the Riesz transform $\nabla L^{-1/2}$ associated with $L$ is bounded from our new space $H_L^p(w)$ to the classical weighted Hardy space $H^p(w)$ when $\frac{n}{n+1}<p<1$ and$w\in A_1\cap R{H}_{(2/p{)}^{\prime }}$.

17 pages