Hardy and BMO spaces on Weyl chambers

Abstract

Let W be a finite reflection group associated with a root system R in

. 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.