**Problem:** In Appendix (A.6) of [Main][1] paper is written 

> $$\lVert K(x; t_0, t_1, t_2, \frac{1}{2\pi}q_1,
 \frac{1}{2\pi}q_2)\rVert_3 \leq \prod_{\nu=1}^{d} \lVert
 p_{R^{\nu}}^{(d=1)}\rVert_3 \leq C
 \prod_{\nu=1}^{d}\frac{1}{\langle R^{\nu}\rangle^{\frac{1}{7}} }$$ 
> where we have applied a known bound for the $\ell^3$-norm of the one
> dimensional propagator, following [ \[11][2], [13\]][3].

Compacted definitions: 

$$
\begin{align*}
&K(x; t_0, t_1, t_2, \frac{1}{2\pi}q_1,
 \frac{1}{2\pi}q_2) =\\
&\quad\quad\quad e^{-ic(t_0 + t_1 + t_2)}\times \prod\limits_{\substack{\nu = 1}}^{d}\int_{0}^{2\pi}\frac{dp}{2\pi}e^{ipx^{\nu}}e^{i(t_0\cos(p) + t_1\cos(p + q_1^{\nu}) + t_2 \cos(p + q_2^{\nu}))}
\end{align*}
$$

and 
$t_0,t_1,t_2 \in \mathbb{R},\; u_1, u_2 \in \mathbb{T}^d = [0,1]^d, x \in \mathbb{Z}^d$. $R^{\nu} = |t_0 + t_1 e^{iq_1^\nu} + t_2 e^{iq_2^\nu}|$, $\langle x \rangle = \sqrt{1+x^2}$ for all $x \in \mathbb{R}$, and the *free propagator* is $p_t(x) = \int_{\mathbb{T}^d}dke^{i2\pi x\cdot k}e^{-i t\omega(k)}$ where $\omega(k)\colon \mathbb{T}^d \to \mathbb{R}$ is the dispersion relation satisfying $(DR1)$ through $(DR4)$ on page $91$ of the main paper.

**Question:** What is the referenced known bound and corresponding proof?

**Investigation:** See $(21)$ of $[11]$, but note that the paper has a typo on the RHS: $e^{it}$ should be $e^{-it}$. The bound they reference on the following page in Claim $4$ Proposition $5$ I suspect is the referenced bound. However, I am stuck on proving this claim and using it to obtain the above inequalities. In their [listed reference][4] is discussed integral limits which may be relevant. Regarding $[13]$, $(2)$ seems most relevant but I have yet to see how this is applied.


  [1]: https://link.springer.com/article/10.1007/s00222-010-0276-5?error=cookies_not_supported&code=29a8fbc7-e18c-43a0-a805-df9017a6c315
  [2]: https://link.springer.com/article/10.1007/BF02181246
  [3]: https://www.sciencedirect.com/science/article/pii/S0377042700004416?via%3Dihub
  [4]: https://www.worldscientific.com/doi/abs/10.1142/S0129055X93000061