Abstract
We establish the equivalence between the Sobolev seminorm [Formula: see text] and a quantity obtained when replacing strong [Formula: see text] by weak [Formula: see text] in the Gagliardo seminorm [Formula: see text] computed at [Formula: see text] As corollaries we derive alternative estimates in some exceptional cases (involving [Formula: see text]) where the "anticipated" fractional Sobolev and Gagliardo-Nirenberg inequalities fail.
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