Schrödinger operators with reverse Hölder class potentials in the Dunkl setting and their Hardy spaces

Abstract

For a normalized root system $R$ in ${\mathbb{R}}^N$ and a multiplicity function $k\ge 0$ let $\mathbf{N}=N+{\sum }_{\alpha \in R}k(\alpha )$. Let $L=-\Delta +V$, $V\ge 0$, be the Dunkl--Schrödinger operator on ${\mathbb{R}}^N$. Assume that there exists $q>\max (1,\frac{\mathbf{N}}{2})$ such that $V$ belongs to the reverse Hölder class ${\text{RH}}^q(dw)$. We prove the Fefferman--Phong inequality for $L$. As an application, we conclude that the Hardy space $H_L^1$, 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 $V$.

arXiv comment31 pages