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

## Abstract

We investigate the validity of the Gagliardo–Nirenberg type inequality

$\|f\|_{W^{s,p}(\mathrm{\Omega })}≲\|f\|_{W^{s_{1},p_{1}}(\mathrm{\Omega })}^{\theta }\|f\|_{W^{s_{2},p_{2}}(\mathrm{\Omega })}^{1−\theta },$

with $\mathrm{\Omega } \subset \mathbb{R}^{N}$. 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

$s = \theta s_{1} + (1−\theta )s_{2},\:1/ p = \theta / p_{1} + (1−\theta )/ p_{2}\text{ for some }\theta \in (0,1).$

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 $s_{1},…,p_{2}$, 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.