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 $\eta\in\mathrm{Hom}(W,\hat{\mathbb{Z}}_2)$ which attaches the $\pm$ signs to the facets of $C_+$. Specialized to orthogonal root systems, atomic decompositions in $H^1_\eta$ and $h^1_\eta$ are obtained and the duality problem is also treated.