My question may be simple to an expert, but I'm not:

Let's consider $u \in C^{s}(\mathbb{R}^d)$ be a Hölder function sor some $s\in [0,1/2)$ which we may take very close to $0$.

Of course, $u^2 \in C^{s}(\mathbb{R}^d)$ so that $(u^2)' \in B^{s-1}_{\infty,\infty}$, is a well-defined distribution, with some explicit negative regularity.

So it means that in this case one can define $u'u = (u^2)'$ as a distribution. But if one wants to define the product $u'u$ by using some general rule, like defining $vu$ for $u\in C^s$, $v\in C^{s-1}$, when $s>1/2$. But for $s<1/2$, this is not possible this way.

I know that there are critera like if $\{(x,\xi) \vert (x,\xi) \in WF(u), (x,-\xi)\in WF(v)\}=\varnothing$ then $uv$ is well-defined as a distribution. So my question is the following:

Is there a way to apply these microlocal tools to define $u'u$ when $u\in C^s$, $s \ll 1$?