Riesz transforms associated with Schr\"odinger operators acting on weighted Hardy spaces

TL;DR

This paper introduces weighted Hardy spaces linked to Schr"odinger operators, develops their atomic decomposition, and proves the boundedness of associated Riesz transforms between these spaces and classical weighted Hardy spaces.

Contribution

It defines new weighted Hardy spaces for Schr"odinger operators, establishes their atomic decomposition, and demonstrates the boundedness of Riesz transforms on these spaces.

Findings

Weighted Hardy spaces $H^p_L(w)$ are introduced and characterized.

Atomic decomposition theory for these spaces is developed.

Riesz transforms are bounded from $H^p_L(w)$ to classical weighted Hardy spaces $H^p(w)$.

Abstract

Let $L=-\Delta+V$ be a Schr\"odinger operator acting on $L^2(\mathbb R^n)$, $n\ge1$, where $V\not\equiv 0$ is a nonnegative locally integrable function on $\mathbb R^n$. In this article, we will introduce weighted Hardy spaces $H^p_L(w)$ 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 $H^p_L(w)$ to the classical weighted Hardy space $H^p(w)$ when $\frac{n}{n+1}<p<1$ and$w\in A_1\cap RH_{(2/p)'}$.