# Under what conditions does the “third-order version of Plancherel's theorem” hold?

I have read in a few places that the formula $$\int_\mathbb{R} x(t)^3 \, dt \ = \ \int_{\mathbb{R}^2} \hat{x}(f_1)\hat{x}(f_2)\hat{x}(-f_1-f_2) \, d(f_1,f_2)$$ holds (where $$\hat{x}$$ denotes the Fourier transform of $$x$$), but without any description of the class of functions that $$x$$ belongs to.

Straightforward iterated application of the convolution theorem shows that this formula is true at least if $$x \in \mathcal{S}(\mathbb{R})$$. Being a bit more precise, it will (I think) be true at least for all $$x \in W^{2,1}(\mathbb{R})$$. But more generally:

Is it the case that for all $$x \in L^2(\mathbb{R}) \cap L^3(\mathbb{R})$$, the map $$\,(f_1,f_2) \mapsto \hat{x}(f_1)\hat{x}(f_2)\hat{x}(-f_1-f_2)\,$$ [known as the bispectrum of $$x$$] is absolutely integrable on $$\mathbb{R}^2$$ and the above formula holds?

(And if so, is there a reference that contains this fact?)