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

We investigate the validity of the Gagliardo-Nirenberg type inequality vertical bar vertical bar f vertical bar vertical bar(s,p)(W)((Omega)) less than or similar to vertical bar vertical bar f vertical bar vertical bar(s1,p1)(theta)(W)((Omega))vertical bar vertical bar f vertical bar vertical bar(s2,p2)(1-theta)(W)((Omega)), (1) with Omega subset of R-N. Here, 0 <= s(1) <= s <= s(2) are non negative numbers (not necessarily integers), 1 <= p(1), p(2) <= infinity, and we assume the standard relations s = theta s(1) + (1 - theta)s(2), 1/p = theta/p(1) + (1 - theta)/p(2) for some theta is an element of(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. (C) 2017 Elsevier Masson SAS. All rights reserved.