Fourier transform of $|u|^2*u$

Question

Given the Fourier transform of $u$ is $\widehat{u}$, then what can I say about the Fourier transform of $|u|^2u$? Can I represent it by $\widehat{u}$?

Answer 1

Using the convolution theorem for the function $f(x)=|u(x)|^2u(x)$ $$\hat{f}=\widehat{|u|^2u}=\widehat{u \bar{u} u}=\hat{u} *\hat{\bar{u}}*\hat{u} .$$ Using further properties we have

$$\hat{f}(\xi)=\hat{u}(\xi) *\overline{ \hat{u}(-\xi)} * \hat{u}(\xi). $$