A measure-$L^{\mathrm{\infty }}$ div-curl lemma

In this note we give a very short proof of the div-curl lemma in the limit conjugate case $\mathcal{M}-L^{\mathrm{\infty }}$, where $\mathcal{M}$ is the set of Radon measures on ${\mathbb{R}}^d$. The proof follows the classical approach by defining here the product in the sense of distributions via a non unique microlocal Hodge’s decomposition. The result is valid for many other spaces than $\mathcal{M}-L^{\mathrm{\infty }}$, including the classical div-curl lemma spaces $L^p-L^{p^{\mathrm{\prime }}}$ for $1<p<\mathrm{\infty }$, and spaces of non conjugated regularity.