Where Sobolev interacts with Gagliardo-Nirenberg

Journal of Functional Analysis. Publié en ligne le 12/03/2019International audienceWe investigate the validity of the fractional Gagliardo-Nirenberg-Sobolev inequality(1) $\displaystyle \|f\|_{W^{r,q}(\Omega)}\lesssim\| f\|_{W^{s_1,p_1}(\Omega)}^\theta\|f\|_{W^{s_2,p_2}(\Omega)}^{1-\theta},\ \forall\, f\in W^{s_1, p_1}(\Omega)\cap W^{s_2, p_2}(\Omega)$.Here, $s_1, s_2, r$ are non-negative numbers (not necessarily integers), $1\le p_1, p_2,q\le \infty$, and we assume, for some $\theta\in (0,1)$, the standard relations(2) $\displaystyle r$<$s:=\theta s_1+(1-\theta)s_2$ and $1/q=(\theta/p_1+(1-\theta)/p_2)-(s-r)/N$.Formally, estimate (1) is obtained by combining the ``pure'' fractional Gagliardo-Nirenberg style interpolation inequality(3) $\displaystyle \|f\|_{W^{s, p}(\Omega)}\lesssim\| f\|_{W^{s_1,p_1}(\Omega)}^\theta\|f\|_{W^{s_2,p_2}(\Omega)}^{1-\theta}$ (with $1/p:=\theta/p_1+(1-\theta)/p_2$)with the fractional Sobolev style embedding(4) $\displaystyle W^{s, p}(\Omega)\hookrightarrow W^{r,q}(\Omega), \ 0\le r$<$s,\, 1\le p$<$q\le\infty,\, 1/q=1/p-(s-r)/N, \, p(s-r)\le N$.Estimates (3) and (4) are true ``most of the time'', but not always; the exact range of validity of (3) and (4) has been known. Combining these results, we infer that (1) is valid ``most of the time''. However, the validity of (1) when (3) and/or (4) fail was unclear. The goal of this paper is to characterize the values of $s_1, s_2, r, p_1, p_2, q, \theta, N$ such that (1) holds