A surprising formula for Sobolev norms

Abstract

We establish the equivalence between the Sobolev seminorm k∇ukLp and a quantity obtained when replacing strong Lp by weak Lp in the Gagliardo seminorm |u|Ws,p computed at s = 1. As corollaries we derive alternative estimates in some exceptional cases (involving W1,1) where the “anticipated” fractional Sobolev and Gagliardo–Nirenberg inequalities fail.