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

Abstract

In this note we give a very short proof of the div-curl lemma in the limit conjugate case $\mathcal{M}-L^{\infty }$, 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 $\mathcal{M}-L^{\infty }$, including the classical div-curl lemma spaces $L^p-L^{p^{\prime }}$ for $1<p<\infty$, and spaces of non conjugated regularity.