Hardy and BMO spaces on Weyl chambers

Abstract

Abstract Let W be a finite reflection group associated with a root system R in ℝ d {\mathbb{R}^{d}} . Let C + {C_{+}} denote a positive Weyl chamber distinguished by a choice of R + {R_{+}} , a set of positive roots. We define and investigate Hardy and BMO spaces on C + {C_{+}} in the framework of boundary conditions given by a homomorphism η ∈ Hom ⁡ ( W , ℤ ^ 2 ) {\eta\in\operatorname{Hom}(W,\widehat{\mathbb{Z}}_{2})} which attaches the ± {\pm} signs to the facets of C + {C_{+}} . Specialized to orthogonal root systems, atomic decompositions in H η 1 {H^{1}_{\eta}} and h η 1 {h^{1}_{\eta}} are obtained and the duality problem is also treated.