Riesz transform characterization of weighted Hardy spaces associated to Schr\"{o}dinger operators

TL;DR

This paper characterizes weighted Hardy spaces related to Schrödinger operators and a critical radius function using localized Riesz transforms, providing new insights into their structure and properties.

Contribution

It introduces a novel characterization of weighted Hardy spaces associated with Schrödinger operators via Riesz transforms, extending previous harmonic analysis frameworks.

Findings

Characterization of weighted local Hardy spaces using localized Riesz transforms.

New representation of Hardy spaces associated with Schrödinger operators.

Extension of Hardy space theory to operators with potentials satisfying reverse Hölder inequality.

Abstract

In this paper, we characterize the weighted local Hardy spaces $h^p_\rho(\omega)$ related to the critical radius function $\rho$ and weights $\omega\in A_{1}^{\rho,\,\infty}(\mathbb{R}^{n})$ by localized Riesz transforms $\widehat{R}_j$, in addition, we give a characterization of weighted Hardy spaces $H^{1}_{\cal L}(\omega)$ via Riesz tranforms associated to Schr\"{o}dinger operator $\cal L$, where $\L=-\Delta+V$ is a Schr\"{o}dinger operator on $\mathbb{R}^{n}$ ($n\ge 3$) and $V$ is a nonnegative function satisfying the reverse H\"older inequality.