  We establish the equivalence between the Sobolev semi-norm $\|\nabla
u\|_{L^p}$ and a quantity obtained when replacing the strong $L^p$ by a weak
$L^p$ norm in the Gagliardo semi-norm $|u|_{W^{s,p}}$ computed at $s = 1$. As
corollaries we derive alternative estimates in some exceptional cases
(involving $W^{1,1}$) where the "anticipated" fractional Sobolev and
Gagliardo-Nirenberg inequalities fail.
