Edit. I realized the previous question was trivial, I rewrote it.
Take a test function $f\in \mathscr D’(\mathbb R)$. It is known that $$ \int \overline f\cdot Hf \,dx=0, $$ where $H$ is the Hilbert transform. In particular, since also $\overline f\cdot Hf\in\mathscr D(\mathbb R)$ (it is not obvious, but it’s easy), then $\overline f\cdot Hf$ is the derivative of a test function. I will call it $g$. Formally, $$ g:=\partial_x^{-1} (\overline f \cdot Hf). $$
Now, all of this makes sense for $f\in L^2$, leading to $g\in L^\infty(\mathbb R)$ such that $g$ goes to zero at $\pm\infty$ (this way, $g$ is uniquely defined).
My main question is: what can be said about $g$ assuming $f$ lies in more general spaces? Is this map a standard thing, is there an explicit formula for $g$ in terms of $f$ (that does not involve antiderivatives), or any kind of $L^p-L^q$ estimates for $p\neq 2$? What about what happens if $f$ belongs to some Sobolev space, like $H^s(\mathbb R)$, $s\in \mathbb R$? My main thought is that the low frequencies of $f\cdot Hf$ “vanish in some sense” so that it should be possible to define its antiderivative unambiguously for suitable $f$, but I can’t find anything about the map $f\mapsto g$. If you know any references that say something about this map that would be great.
Final remarks. I know about Cotlar’s identity, which is insightful, but I didn’t find a way to use it. I should also say that I don’t care about complex valued functions, so one can think about $f$ being a real function (so that $f\cdot Hf$ is a real function up to redefining $H$ to be $i$ times the Hilbert transform).