Hardy space of exact forms on ℝℕ

We show that the Hardy space of divergence-free vector fields on R 3 R³ has a divergence-free atomic decomposition, and thus we characterize its dual as a variant of B M O BMO . Using the duality result we prove a “div-curl" type theorem: for b b in L l o c 2 ( R 3 , R 3 ) L²loc(R³, R³) , sup ∫ b ⋅ ( ∇ u × ∇ v ) d x ∫ b· (∇ u× ∇ v)\ dx is equivalent to a B M O BMO -type norm of b b , where the supremum is taken over all u , v ∈ W 1 , 2 ( R 3 ) u, v∈ W1,2(R³) with ‖