A measure-$L^\infty$ div-curl lemma
Valeria Banica, Nicolas Burq
TL;DR
The paper addresses the div-curl lemma in the limiting conjugate setting where one factor is a Radon measure and the other lies in $L^\infty$, presenting a concise proof based on a non-unique microlocal Hodge decomposition to define a distributional product. It constructs a robust product $(v\cdot w)_H$ via a Hodge-type decomposition and proves its independence from the decomposition, extending the classical product to several function spaces beyond the standard $L^p-L^{p'}$ framework. It then establishes weak continuity: if sequences $(v_n,w_n)$ converge in the distributional sense and are bounded in the appropriate spaces, their products converge to the product of the limits in the distribution sense. The method, which leverages pseudo-differential operators and localization, applies to a broad range of regularity settings, offering a short and flexible route to compensated compactness in measure-valued contexts.
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 $L^p-L^{p'}$ for $1<p<\infty$, and spaces of non conjugated regularity.