Hardy spaces for the Dunkl harmonic oscillator

Abstract

Let $\Delta$ and $L=\Delta -\Vert \mathbf{x}{\Vert }^2$ be the Dunkl Laplacian and the Dunkl harmonic oscillator respectively. We define the Hardy space ${\mathcal{H}}^1$ associated with the Dunkl harmonic oscillator by means of the nontangential maximal function with respect to the semigroup $e^{tL}$. We prove that the space ${\mathcal{H}}^1$ admits characterizations by relevant Riesz transforms and atomic decompositions. The atoms which occur in the atomic decompositions are of local type.

arXiv comment26 pages