2018/08/01 by Petru Mironescu, Haïm Brezis · 166 citations
paper · doi:10.1016/j.anihpc.2017.11.007
published in Annales de l'Institut Henri Poincaré C, Analyse non linéaire 35(5), 1355-1376 (European Mathematical Society - EMS - Publishing House GmbH)
crossref created 2017/11/24 · crossref issued 2018/08/01 · crossref published 2018/08/01 · crossref published-print 2018/08/01 · crossref deposited 2025/07/31 · crossref indexed 2026/08/02
We investigate the validity of the Gagliardo–Nirenberg type inequality ‖f‖Ws,p(Ω )≲‖f‖_W^s1,p1(Ω )θ ‖f‖_W^s2,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 = θ s1 + (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.