Show that the distribution is of the form $C \delta + f$

Question

I'm trying to solve this problem:

Let $ u = p.v.(1/x)$, $\phi$, $\psi \in C^{\infty}_c$. I want to show that the distribution

$(\phi u )* (\psi u)$

is of the form $C \delta + f$ for some constant C and $f \in C^{\infty}$.

I realize that $\widehat{(\phi u )* (\psi u)} = \widehat{\phi u} \,\cdot \widehat{\psi u}$, but how do I proceed from here?

Thanks in advance.

Answer 1

The following is a very rough sketch.

You also have $\widehat{\varphi u} = \widehat \varphi * \widehat u$. You can explicitly calculate $\widehat u$; if you do not have a nice form for it, evaluate it here.

Say $f = (\widehat \varphi * \widehat u)(\widehat \psi * \widehat u)$. Calculate $f'$ and show it's Schwartz. (This is a technical and nontrivial step.) You should be able to show that $f(-\infty) = f(\infty)$. Lastly, show that if a function $g$ has $g(\infty) = g(-\infty) = 0$, and $f'$ is Schwartz, then $f$ itself is Schwartz. This will show that our function $f$ is $C + \omega$, where $\omega$ is Schwartz. Now apply the inverse Fourier transform.