Microlocal approach to definition of product of distributions

Question

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 $u\in C^s$, $s \ll 1$?

Answer 1

Too long for a comment. For $u$ in $C^s$, $s\in (0,1)$, you can indeed define $u^2$ and then the distribution-derivative of $u^2$, which belongs to $B^{s-1}_{\infty,\infty}$. Now that does not define the product $uu'$, unless you decide to define that product as $\frac12(u^2)'$. Assuming for instance that $u$ is also compactly supported, you can mollify everything and consider $u_\epsilon=u\ast \rho_\epsilon$ and you will have trouble at proving that $$ u u'_\epsilon $$ has a (weak) limit in the distribution sense when $s<1/2$.

About your question, I believe that the answer is negative. One reason is that you may consider only real-valued functions defined on the real line: in that case the wave-front-set is trivially deduced from the singular support and is $$ \text{singsupp} u\times \mathbb R^*. $$ So microlocal analysis is of no help in that situation, whereas your problem of defining $uu'$ in that simple situation remains the same.