Recall the Coifman–Meyer Theorem as stated in [Grafakos and Oh - The Kato-Ponce Inequality](https://arxiv.org/abs/1303.5144).

**Theorem**: Let $m \in L^{\infty}\left(\mathbf{R}^{2 n}\right)$ be smooth away from the origin and satisfy
$$
\left|\partial_{\xi}^{\alpha} \partial_{\eta}^{\beta} m\right|(\xi, \eta) \leq C(\alpha, \beta)(|\xi|+|\eta|)^{-|\alpha|-|\beta|}
$$
for all $\xi, \eta \in \mathbf{R} \setminus\{0\}$ and $\alpha, \beta \in \mathbf{Z}^{n}$ multi-indices with $|\alpha|,|\beta| \leq 2 n+1$. Then for all $f, g \in \mathcal{S}\left(\mathbf{R}^{n}\right)$,
$$
\left\|\int_{\mathbf{R}^{2 n}} m(\xi, \eta) \widehat{f}(\xi) \widehat{g}(\eta) e^{i\langle\xi+\eta,\rangle} d \xi d \eta\right\|_{L^{r}\left(\mathbf{R}^{n}\right)} \leq C(p, q, r, m)\|f\|_{L^{p}\left(\mathbf{R}^{n}\right)}\|g\|_{L^{q}\left(\mathbf{R}^{n}\right)}
$$
where $\frac{1}{2}<r<\infty, 1<p, q \leq \infty$ satisfy $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$. 

I was wondering if it makes sense to expect the following estimate to hold,
$$
\left\|\int_{\mathbf{R}^{2 n}} m(\xi, \eta) \widehat{f}(\xi) \widehat{g}(\eta) e^{i\langle\xi+\eta,\rangle} d \xi d \eta\right\|_{L^{p}\left(\mathbf{R}^{n}\right)} \leq C(p, m)\|fg\|_{L^{p}\left(\mathbf{R}^{n}\right)}
$$
for some Schwartz functions $f,g$ and exponent $p\geq 1?$


  [1]: https://arxiv.org/pdf/1303.5144.pdf