Gagliardo–Nirenberg inequalities and non-inequalities: The full story
We investigate the validity of the Gagliardo–Nirenberg type inequality \tag{1} \|f\|_{W^{s,p}(\mathrm{\Omega })}≲\|f\|_{W^{s_{1},p_{1}}(\mathrm{\Omega })}^{\theta }\|f\|_{W^{s_{2},p_{2}}(\mathrm{\Omega })}^{1−\theta }, with \mathrm{\Omega } \subset \mathbb{R}^{N} . Here, 0 \leq s_{1} \leq s \leq s_{2} are non negative numbers (not necessarily integers), 1 \leq p_{1},p,p_{2} \leq \infty , and we assume the standard relations s = \theta s_{1} + (1−\theta )s_{2},\:1/ p = \theta / p_{1} + (1−\theta )/ p_{2}\text{ for some }\theta \in (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.