A remark on the div-curl lemma (Q2895970)

The div-curl lemma in [\textit{R. Coifman} et al., J. Math. Pures Appl., IX. Sér. 72, No. 3, 247--286 (1993; Zbl 0864.42009)] states that, if \(\vec{f}\in (L^p(\mathbb{R}^d))^d\) and \(\vec{g}\in (L^q(\mathbb{R}^d))^d\) where \(1/p+1/q=1\) and if \(\vec{f}\) is divergence-free and \(\vec{g}\) is curl-free, then \(\vec{f}\cdot\vec{g}\) lies in the real Hardy space \(H^1(\mathbb{R}^d)\). To motivate the main result here, the author begins by stating an extension of the div-curl lemma to weak \(L^p\)-spaces. He then introduces the notion of a Calderón-Zygmund pair \((X,Y)\) of Banach spaces on \(\mathbb{R}^d\). He denotes by \(X_0\) the closure of the space \(\mathcal{D}\) of test functions in \(X\). The main result states that, if \((X,Y)\) is such a Calderón-Zygmund pair, if \(\vec{f}\in X_0^d\) and \(\vec{g}\in Y^d\),, then if one of \(\vec{f}\), \(\vec{g}\) is divergence-free and the other is curl-free, \(\vec{f}\cdot \vec{g}\) belongs to the real Hardy spaces \(H^1(\mathbb{R}^d)\).NEWLINENEWLINEA Calderón-Zygmund pair \((X,Y)\) is defined as a pair of Banach spaces such that one has continuous embeddings \(\mathcal{D}(\mathbb{R}^d)\subset X\subset \mathcal{D}'(\mathbb{R}^d)\) and likewise for \(Y\); the dual \(X_0^\ast\) of the closure \(X_0\) of \(\mathcal{D}\) in \(X\) coincides with \(Y\) with equivalent norms, and similarly with the roles of \(X\) and \(Y\) reversed; and, finally, every Calderón-Zygmund operator may be extended to a bounded operator on \(X_0\) and on \(Y_0\).NEWLINENEWLINEThe proof relies on decompositions of divergence-free vector fields in terms of divergence-free wavelets and, in particular, in terms of their multiresolution properties. The author finishes by discussing examples of Calderón-Zygmund pairs, including examples for which the div-curl lemma was previously known as well as new examples.