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

Petru Mironescu; Haïm Brezis

doi:10.1016/j.anihpc.2017.11.007

JournalsaihpcVol. 35, No. 5pp. 1355–1376

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

Petru Mironescu
Université de Lyon, Université Lyon 1, CNRS UMR 5208 Institut Camille Jordan, 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France; Simion Stoilow Institute of Mathematics of the Romanian Academy, Calea Griviţei 21, 010702 Bucureşti, Romania
Haïm Brezis
Department of Mathematics, Rutgers University, Hill Center, Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854, USA; Mathematics Department, Technion - Israel Institute of Technology, Haifa, 32000, Israel

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Abstract

We investigate the validity of the Gagliardo–Nirenberg type inequality

∥ f ∥_{W^{s, p} (Ω)} ≲ ∥ f ∥_{W^{s_{1}, p_{1}} (Ω)}^{θ} ∥ f ∥_{W^{s_{2}, p_{2}} (Ω)}^{1 - θ}, (1)

with $Ω \subset R^{N}$ . Here, $0 \leq s_{1} \leq s \leq s_{2}$ are non negative numbers (not necessarily integers), $1 \leq p_{1}, p, p_{2} \leq \infty$ , 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 \in (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 $s_{1}, \dots, p_{2}$ , 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.

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. 1355–1376

DOI 10.1016/J.ANIHPC.2017.11.007