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

We investigate the validity of the Gagliardo–Nirenberg type inequality {1} \|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.