The dual of the Hardy space associated to the Dunkl-Schr\"odinger operator with reverse H\"older class potential

TL;DR

This paper characterizes the dual space of a Dunkl-associated Hardy space linked to a Schr"odinger operator with reverse H"older potential, extending classical harmonic analysis results to Dunkl settings.

Contribution

It provides a new characterization of the BMO space dual to the Dunkl Hardy space, incorporating potential-dependent atoms and establishing maximal function boundedness.

Findings

Dual space of Dunkl Hardy space identified as a subspace of Dunkl BMO.

Atomic decomposition of Hardy space with potential-dependent cancellation conditions.

Boundedness of the uncentered maximal function on Dunkl BMO established.

Abstract

Let $\mathcal{L}_k = -\Delta_k + V$ be a Schr\"odinger operator associated with the Dunkl Laplacian $\Delta_k$, where $V$ is the non-negative potential function belonging to the reverse H\"older class $RH_k^q(\mathbb{R}^n)$ 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-Schr\"odinger operator. The dual space $BMO(\mathcal{L}_k)$ is a subspace of the $BMO_k$ 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 $BMO(\mathcal{L}_k)$.