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.