Geodesic
Gagliardo–Nirenberg inequalities and non-inequalities: The full story
Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 5, pp. 1355-1376

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We investigate the validity of the Gagliardo–Nirenberg type inequality

‖f‖Ws,p(Ω)≲‖f‖Ws1,p1(Ω)θ‖f‖Ws2,p2(Ω)1−θ,
with Ω⊂RN. Here, 0≤s1≤s≤s2 are non negative numbers (not necessarily integers), 1≤p1,p,p2≤∞, and we assume the standard relations
s=θs1+(1−θ)s2,1/p=θ/p1+(1−θ)/p2 for some θ∈(0,1).

By the seminal contributions of E. Gagliardo and L. Nirenberg, (1) holds when s1,s2,s are integers. It turns out that (1) holds for “most” of values of s1,…,p2, but not for all of them. We present an explicit condition on s1,s2,p1,p2 which allows to decide whether (1) holds or fails.

DOI : 10.1016/j.anihpc.2017.11.007
Keywords: Sobolev spaces, Gagliardo–Nirenberg inequalities, Interpolation inequalities 
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     title = {Gagliardo{\textendash}Nirenberg inequalities and non-inequalities: {The} full story},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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Brezis, Haïm; Mironescu, Petru. Gagliardo–Nirenberg inequalities and non-inequalities: The full story. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 5, pp. 1355-1376. doi: 10.1016/j.anihpc.2017.11.007

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