Why $\|u\|_{\tau}\leq C[u]_{W^{s,p}}^a\|u\|_{L^q}^{1-a}$ not correct for $p=1$?

Question

In an article I have the following lemma :

Let $d\geq 1$, $p>1$, $q\geq 1$, $\tau>0$, $s\in(0,1)$ and $a\in(0,1]$ s.t. $$\frac{1}{\tau}=a\left(\frac{1}{p}-\frac{s}{d}\right)+\frac{1-a}{q}.$$

Then $$\|u\|_{L^\tau(\mathbb R^d)}\leq C[u]_{W^{s,p}(\mathbb R^d)}^a\|u\|_{L^q(\mathbb R^d)}^{1-a},\quad u\in \mathcal C^1_0(\mathbb R^d),$$ for some positive constant independent of $u$.

I recall that $\mathcal C^1_0(\mathbb R^d)$ is the space of $\mathcal C^1$ function over $\mathbb R^d$ that are compactly supported and that $$[u]_{W^{s,p}(\mathbb R^d)}^p=\iint_{\mathbb R^{2d}}\frac{|u(x)-u(y)|^p}{|x-y|^{d+sp}}dxdy.$$

My question is : why doesn't it work for $p=1$ ? I let you the proof below.

Answer 1

The issue with $p = 1$ is that some definitions of fractional Sobolev spaces that were equivalent when $p > 1$ (by the Gagliardo seminorm that you gave, by interpolation between functional spaces, by trace theory, by harmonic analysis (Littlewood-Paley decomeposition)) are not anymore equivalent when $p = 1$.

So, one should be careful about the case $p=1$ and check whether the Sobolev inequality holds for that definition.

With the Gagliardo seminorm as in the question, the Sobolev inequality holds (see for example Augusto C. Ponce, Elliptic PDEs, measures and capacities. From the Poisson equation to nonlinear Thomas-Fermi problems, EMS Tracts in Mathematics 23. Zürich: European Mathematical Society (EMS) (2016), Proposition 15.5 and thus the lemma also holds for $p = 1$.