Divergence-free Hardy space on \(\mathbb R^N_+\) (Q1777542)

In [Trans. Am. Math. Soc. 357, No. 4, 1469--1496 (2005; Zbl 1079.42014)], the authors studied the divergence-free \(H^{1}_{\text{div}}({\mathbb R}^N)\) space and proved, in particular, an atomic decomposition for this space. In the present paper, they establish a similar atomic decomposition for a divergence free Hardy space on the upper half space of \({\mathbb R}^N\) which turns to be the subspace of \(H^{1}_{\text{div}}({\mathbb R}^N)\) made of forms supported in \({\mathbb R}^{N}_{+}\). The dual of this Hardy space is also described, and some applications to div-curl lemmata and coercivity properties of polyconvex quadratic forms are given.