# Microlocal approach to definition of product of distributions

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

• I think the theory of paraproducts (e.g., the Coifman-Meyer multiplier theorem) is more relevant here than classical microlocal analysis; one can hope to define various paraproducts at regularities below what one might naively expect if there are suitable cancellations in the associated bilinear symbol. See also the div-curl lemma for another instance of this phenomenon. Oct 27, 2022 at 16:53
• @TerryTao thank you for your comment. In fact I had in mind these kind of div-curl lemma somewhat magial cancellations, but I'm not an expert on this. What would you suggest as a good reference to see these "cancellations in the associated bilinear symbol" in action? Nov 1, 2022 at 16:19
• I don't have it at hand, but perhaps Taylor's "Tools for PDE" covers some of this material. Nov 1, 2022 at 19:00
• Thank you, I'll take a look at Taylor's books! Nov 1, 2022 at 22:47

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.