En 2004, J. Bourgain, H. Brezis et P. Mironescu have shown that if \(\Gamma \subset \mathbb{R}^n\) is a closed rectifiable curve with tangent vector \(t\) and \(\varphi : \mathbb{R}^n \to \mathbb{R}^n\) is a vector field, then \[ \int_{\Gamma} \varphi \cdot t \le \lvert \Gamma \rvert\, \lVert D \varphi \rVert_{L^n}. \] J. Bourgain and H. Brezis have generalized this to \[ \int_{\mathbb{R}^n} \varphi \cdot f \le \lVert f \rVert_{L^1} \, \lVert D \varphi \rVert_{L^n} \] whenever \(f\) is a divergence-free vector field. These inequalities are surprising since the quantity \(\lVert D \varphi \rVert_{L^n}\) does not control \(\lVert \varphi \rVert_{L^\infty} \). These inequalities have consequences in the theory of regularity of elliptic systems with \(L^1\) data.

In this domain,

• I have given elementary proofs of the circulation integral inequality and the inequality for divergence-free vector-fields,
• I have obtained inequalities when the divergence is replaced by a general higher-order operator,
• I have studied the relationship between functions satisfying this kind of estimates and the space of functions of bounded mean oscillation \(BMO\),
• with H. Brezis, I have studied the corresponding boundary estimates and posed some problems of optimal constants,
• with S. Chanillo, I have obtained corresponding inequalities on stratified homogeneous groups, including for example the Heisenberg group,
• with S. Chanillo and Po Lam Yung, we have given some applications to fluid dynamics and electromagnetism,
• with S. Chanillo and Po Lam Yung, we have proved similar estimates on a symmetric space of noncompact type.