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Journal Articles Annales de l'Institut Henri Poincaré C, Analyse non linéaire Year : 2018

Gagliardo-Nirenberg inequalities and non-inequalities: the full story

Abstract

We investigate the validity of the Gagliardo-Nirenberg type inequality\begin{equation*}(1)\ \|f\|_{W^{s,p}(\Omega)}\lesssim\| f\|_{W^{s_1,p_1}(\Omega)}^\theta\|f\|_{W^{s_2,p_2}(\Omega)}^{1-\theta},\end{equation*}with $\Omega\subset{\mathbb R}^N$.Here, $0\le s_1\le s\le s_2$ are non negative numbers (not necessarily integers), $1\le p_1, p, p_2\le \infty$, and we assume the standard relations\begin{equation*}\ s=\theta s_1+(1-\theta)s_2,\ 1/p=\theta/p_1+(1-\theta)/p_2\text{ for some }\theta\in (0,1).\end{equation*}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 $s_1,\ldots, p_2$, 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.
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Dates and versions

hal-01626613 , version 1 (31-10-2017)

Identifiers

  • HAL Id : hal-01626613 , version 1

Cite

Haïm Brezis, Petru Mironescu. Gagliardo-Nirenberg inequalities and non-inequalities: the full story. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2018, 35 (5), pp.1355-1376. ⟨hal-01626613⟩
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