Fix a rapidly decreasing function $\psi \in \mathcal{S}(\mathbb{R})$ with the properties that $\int_\mathbb{R} \psi = 0$, $\mathrm{Re}(\psi(\cdot))$ is an even function, and $\mathrm{Im}(\psi(\cdot))$ is an odd function.

Given any bounded integrable function $x \colon \mathbb{R} \to \mathbb{R}$, define \begin{align*} \mathcal{F}x(\xi) \ :=& \ \int_\mathbb{R} x(\tau)e^{-2\pi i\xi\tau} \, d\tau \\ \mathcal{W}x(\xi,t) \ :=& \ \int_\mathbb{R} x(\tau)\psi(\xi(\tau-t)) \, d\tau \end{align*} for all $\xi,t \in \mathbb{R}$. (Obviously $\mathcal{F}x$ is the Fourier transform; $\mathcal{W}x$ is a version of the wavelet transform.)

It is known that

$$ \hspace{20mm} \int_\mathbb{R} x(t)^2 \, dt \ = \ \int_\mathbb{R} |\mathcal{F}x(\xi)|^2 \, d\xi \ = \ C_\psi\!\iint_{\mathbb{R}^2} |\xi\,\mathcal{W}x(\xi,t)|^2 \, d\xi dt \hspace{18mm} (1) $$

where $C_\psi$ is a constant that depends only on $\psi$ and not on $x$, namely $$ C_\psi \ = \ \left( \int_\mathbb{R} \frac{\mathcal{F}\psi(r)^2}{|r|} \, dr \right)^{\!-1}\,. $$ It is also known that the first equality in $(1)$ can be extended to the 3rd moment by

$$ \int_\mathbb{R} x(t)^3 \, dt \ = \ \iint_{\mathbb{R}^2} \mathcal{F}x(\xi_1)\mathcal{F}x(\xi_2)\overline{\mathcal{F}x(\xi_1+\xi_2)} \, d\xi_1d\xi_2 $$

where the bar denotes complex conjugate. This naturally leads to the question of whether the second equality in $(1)$ can be extended to the third moment:

Does there exist a function $f_\psi \colon \mathbb{R}^2 \to \mathbb{C}$ such that for every bounded integrable function $x \colon \mathbb{R} \to \mathbb{R}$, $$ \int_\mathbb{R} x(t)^3 \, dt \ = \ \iiint_{\mathbb{R}^3} f_\psi(\xi_1,\xi_2) \mathcal{W}x(\xi_1,t)\mathcal{W}x(\xi_2,t)\overline{\mathcal{W}x(\xi_1+\xi_2,t)} \, d\xi_1d\xi_2dt \, ? $$ (If not, then [a more open-ended question] are there reasonable conditions on $\psi$ under which one can find a function $f_\psi$ such that the above equality "approximately" holds?)

A little bit of numerical experimentation by myself and a friend seems to suggest that the function $$ f_\psi(\xi_1,\xi_2) \ = \ D_\psi(\xi_1+\xi_2) $$ might work for some constant $D_\psi \in \mathbb{C}$.