The dual of the Hardy space associated to the Dunkl-Schrödinger operator with reverse Hölder class potential

Abstract

Let ${\mathcal{L}}_k=-{\Delta }_k+V$ be a Schrödinger operator associated with the Dunkl Laplacian ${\Delta }_k$, where $V$ is the non-negative potential function belonging to the reverse Hölder class $R{H}_k^q({\mathbb{R}}^n)$ with $q>\max \{1,\frac{n+2\gamma }{2}\}$. Here, $2\gamma$ denotes the degree of homogeneity of the weight function $w_k$, which is determined by the normalized root system and the non-negative multiplicity function $k$. In this paper, we investigate the dual space of the Hardy space $H_{\Tilde{\mathcal{L}}_k}^1$ associated with the Dunkl-Schrödinger operator. The dual space $BMO({\mathcal{L}}_k)$ is a subspace of the $BM{O}_k$ space, which is the Dunkl analogue of the classical $BMO(\mathcal{L})$ space. We provide a characterization for the $BMO({\mathcal{L}}_k)$ space. The duality result is obtained via the atomic decomposition of $H_{\Tilde{\mathcal{L}}_k}^1$, where the cancellation condition of atoms depends on the critical radius function associated with the potential $V$. Finally, we establish the boundedness of the uncentered maximal function on the space $BMO({\mathcal{L}}_k)$.

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