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We investigate the validity of the Gagliardo–Nirenberg type inequality
with . Here, are non negative numbers (not necessarily integers), , and we assume the standard relations
By the seminal contributions of E. Gagliardo and L. Nirenberg, (1) holds when are integers. It turns out that (1) holds for “most” of values of , but not for all of them. We present an explicit condition on which allows to decide whether (1) holds or fails.
Cite this article
Petru Mironescu, Haïm Brezis, Gagliardo–Nirenberg inequalities and non-inequalities: The full story. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 5, pp. 1355–1376DOI 10.1016/J.ANIHPC.2017.11.007