Gagliardo–Nirenberg inequalities and non-inequalities: The full story

By Haim Brezis, Petru Mironescu. Annales de l'Institut Henri Poincare. Analyse non linéar, 2017

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Abstract We investigate the validity of the Gagliardo–Nirenberg type inequality (1) ‖ f ‖ W s , p ( Ω ) ≲ ‖ f ‖ W s 1 , p 1 ( Ω ) θ ‖ f ‖ W s 2 , p 2 ( Ω ) 1 − θ , with Ω ⊂ R N . Here, 0 ≤ s 1 ≤ s ≤ s 2 are non negative numbers (not necessarily integers), 1 ≤ p 1 , p , p 2 ≤ ∞ , and we assume the standard relations s = θ s 1 + ( 1 − θ ) s 2 , 1 / p = θ / p 1 + ( 1 − θ ) / p 2 for some θ ∈ ( 0 , 1 ) . By the seminal contributions of E. Gagliardo and L. Nirenberg, (1) holds when s 1 , s 2 , s are integers. It turns out that (1) holds for “most” of values of s 1 , … , p 2 , but not for all of them. We present an explicit condition on s 1 , s 2 , p 1 , p 2 which allows to decide whether (1) holds or fails.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/3c339fdecf3016d209575d6df95665336cb54f85

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Citations in the corpus

Fractional Orlicz-Sobolev embeddings (2020, background): “A touch of recent contributions in this connection is furnished by [7, 8, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 41, 44, 45, 51, 52, 53, 58, 59, 61, 62, 63, 64, 65, 68, 70, 71, 77, 76, 78, 83, 86].”
On the Regularity of Minima of Non-autonomous Functionals (2019, methodology): “2, (a)] (see also [8]) we know that, if f̃ ∈W (R) ∩ L(R) with p, s, t > 1, then there holds ‖f̃‖W 1,p̃(Rn) ≤ c(n, p, s, t)‖f̃‖ s−1 s L2t(Rn)‖f̃‖ 1 s W s,p(Rn) , (3.”