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

Haïm Brezis, Petru Mironescu

Research output: Contribution to journalArticlepeer-review

31 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)1355-1376
Number of pages22
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume35
Issue number5
DOIs
StatePublished - Aug 2018

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematical Physics
  • Applied Mathematics

Keywords

  • Gagliardo–Nirenberg inequalities
  • Interpolation inequalities
  • Sobolev spaces

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