A surprising formula for Sobolev norms

Abstract

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-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.

14 pages

References in corpus (2)

• Gamma-convergence of nonlocal perimeter functionals
• Positive solutions of transport equations and classical nonuniqueness of characteristic curves

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