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

We investigate the validity of the Gagliardo–Nirenberg type inequality 1 \|f\|ₖ^ₒ, ₏ () ≲\|f\|ₖ^ₒ_{₁, p₁ () }^ \|f\|ₖ^ₒ_{₂, p₂ () }^1−, with R^N. Here, 0 s₁ s s₂ are non negative numbers (not necessarily integers), 1 p₁, p, p₂, and we assume the standard relations s = s₁ + (1−) s₂, \: 1/ p = / p₁ + (1−) / p₂ for some (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.