Limiting Bourgain-Brezis estimates for systems: theme and variations

J. Bourgain and H. Brezis have obtained in 2002 some new and surprising estimates for systems of linear differential equations, dealing with the endpoint case ${\mathrm{L}}^1$ of singular integral estimates and the critical Sobolev space ${\mathrm{W}}^{1,n}({\mathbf{R}}^n)$. This paper presents an overview of the results, further developments over the last ten years and challenging open problems.