Schr\"odinger operators with reverse H\"older class potentials in the Dunkl setting and their Hardy spaces

TL;DR

This paper studies Dunkl Schr"odinger operators with reverse H"older class potentials, proving a Fefferman--Phong inequality and establishing an atomic decomposition for the associated Hardy space, extending harmonic analysis tools in this setting.

Contribution

It introduces a Hardy space framework for Dunkl Schr"odinger operators with reverse H"older potentials, including atomic decomposition results.

Findings

Proves Fefferman--Phong inequality for Dunkl--Schr"odinger operators.

Establishes atomic decomposition of Hardy space $H^1_{L}$ with localized atoms.

Extends harmonic analysis techniques to Dunkl setting with reverse H"older potentials.

Abstract

For a normalized root system $R$ in $\mathbb R^N$ and a multiplicity function $k\geq 0$ let $\mathbf N=N+\sum_{\alpha \in R} k(\alpha)$. Let $L=-\Delta +V$, $V\geq 0$, be the Dunkl--Schr\"odinger operator on $\mathbb R^N$. Assume that there exists $q >\max(1,\frac{\mathbf{N}}{2})$ such that $V$ belongs to the reverse H\"older class $\text{RH}^q(dw)$. We prove the Fefferman--Phong inequality for $L$. As an application, we conclude that the Hardy space $H^1_{L}$, which is originally defined by means of the maximal function associated with the semigroup $e^{tL}$, admits an atomic decomposition with local atoms in the sense of Goldberg, where their localization are adapted to $V$.