One has the following "generalized version" of Hölder's inequality:
$$ \| u v \|_{L^1} \leq \| u \|_{W^{-s , p}} \| v \|_{W^{s , p'}} $$
where $s \geq 0$ and $\frac{1}{p} + \frac{1}{p'} = 1$, $1 < p , p' < \infty$. I was wondering if something like the following is true.
$$ \| u v \|_{L^r} \leq \| u \|_{W^{-s , p}} \| v \|_{W^{s , p'}} $$
where $s \geq 0$ and $\frac{1}{p} + \frac{1}{p'} = \frac{1}{r}$, $r < p , p' < \infty$?
I've tried proving it with no avail. I think the claim is false for $r \geq 2$ and $s > 0$. For a counterexample in the $r = 2$ case one may just consider the singular behaviour of a typical function in $W^{-s,p}$ ($s > 0$) and then take $v \in C_c^{\infty}$. However, it would be great for me if it was true for $r < 2$.
Edit: What I have said in the above is wrong. What we actually have is
$$ | \langle u , v \rangle_{L^2} | \leq \| u \|_{W^{-s , p}} \| v \|_{W^{s , p'}} $$
where $s \geq 0$ and $\frac{1}{p} + \frac{1}{p'} = 1$, $1 < p , p' < \infty$. If we wanted to replace the left hand side with $\| uv \|_{L^1}$ we need to put magnitude signs on the $u$ and $v$. However, this raises another question. Consider $u \in H^{-1}$ and $v \in H^1$. It is a well known fact that $|v| \in H^1$ (more generally, $v \in H^s$ implies $|v| \in H^s$ if and only if $s < \frac{3}{2}$). Therefore
$$ \langle |u| , |v| \rangle_{L^2} \leq \| |u| \|_{ H^{-1} } \| |v| \|_{H^1} \leq \| |u| \|_{ H^{-1} } \| v \|_{H^1} $$
Question: Is it possible to estimate $\| |u| \|_{H^{-1}}$ by $\| u \|_{H^{-1}}$? If such an inequality were true, I could improve the above so-called generalized Hölder inequality.