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.

arXiv comment14 pages

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Cited by in corpus (10)

• A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I
• A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II
• Going to Lorentz when fractional Sobolev, Gagliardo and Nirenberg estimates fail
• Families of functionals representing Sobolev norms
• Spaces of Besov-Sobolev type and a problem on nonlinear approximation
• A weak-type expression of the Orlicz modular
• New estimates for a class of non-local approximations of the total variation
• Trace conjunction inequalities
• A new formula for the $L^p$ norm
• A Logarithmic Uncertainty Principle for Functions with Radial Symmetry