Hardy space of exact forms on $\mathbb {R}^N$

Abstract:

We show that the Hardy space of divergence-free vector fields on $\mathbb {R}^{3}$ has a divergence-free atomic decomposition, and thus we characterize its dual as a variant of $BMO$. Using the duality result we prove a “div-curl" type theorem: for $b$ in $L^{2}_{loc}(\mathbb {R}^{3}, \mathbb {R}^{3})$, $\sup \int b\cdot (\nabla u\times \nabla v)\ dx$ is equivalent to a $BMO$-type norm of $b$, where the supremum is taken over all $u, v\in W^{1,2}(\mathbb {R}^{3})$ with $\|\nabla u\|_{L^{2}}, \ \|\nabla v\|_{L^{2}}\le 1.$ This theorem is used to obtain some coercivity results for quadratic forms which arise in the linearization of polyconvex variational integrals studied in nonlinear elasticity. In addition, we introduce Hardy spaces of exact forms on $\mathbb {R}^N$, study their atomic decompositions and dual spaces, and establish “div-curl" type theorems on $\mathbb {R}^N$.