A surprising formula for Sobolev norms

Haim Brezis, Jean Van Schaftingen, Po-Lam Yung

We establish the equivalence between the Sobolev semi-norm $\Vert \nabla u{\Vert }_{L^p}$ and a quantity obtained when replacing the strong $L^p$ by a weak $L^p$ norm in the Gagliardo semi-n…