Hardy and BMO spaces on Weyl chambers

Let $W$ be a finite reflection group associated with root system $R$ in ${\mathbb{R}}^d$. Let $C_{+}$ denote a positive Weyl chamber distinguished by a choice of $R_{+}$, a set of positive roots. We define and investigate Hardy and BMO spaces on $C_{+}$ in the framework of boundary conditions given by a homomorphism $\eta \in \mathrm{H}\mathrm{o}\mathrm{m}(W,{\widehat{\mathbb{Z}}}_2)$ which attaches the $\pm$ signs to the facets of $C_{+}$. Specialized to orthogonal root systems, atomic decompositions in $H_{\eta }^1$ and $h_{\eta }^1$ are obtained and the duality problem is also treated.