of harmonic analysis, and in particular the oscillatory phase lemma. In particular, Proposition \[RhoDescription\] shows that, for all $\varepsilon>0$, $\rho_\tau$ is polynomially small with respect to $\tau>0$ to any arbitrary order in the set $$\Big\{(t,s) \in (0,T) \times {\mathbb{R}}, \hbox{ s.t. } t + \varepsilon < s \underset{|\alpha| \leq \delta + \varepsilon}\inf \{f'(\alpha)\}  \hbox{ or }     s \sup_{|\alpha| \leq \delta + \varepsilon} \{f'(\alpha)\} < t-\varepsilon \Big\}.$$ To give some insights on this result, let us again consider the informal arguments given above and remark that $z_\tau(s)$ in formally solves $$\partial_s^{\tau} z_\tau = A z_\tau,$$ where $\partial_s^{\tau}$ is the operator defined on the Fourier basis by $$\label{Def-ds}
    \forall \mu \in {\mathbb{R}}, \quad \partial_s^\tau \left(e^{i \mu s}\right) = i \frac{g(\mu \tau)}{\tau} e^{i \mu s}.$$ Thus, the kernel $\tilde \rho_\tau$ in formally satisfies the transport-like equation $$\label{Transport-Pseudo}
    \partial_t \tilde \rho_\tau - \partial_s^\tau \tilde \rho_\tau = 0, \quad (t,s) \in {\mathbb{R}}\times {\mathbb{R}},$$ with an initial data $\rho_\tau(t = 0, s) = \delta_{0}(s)$ where $\delta_{0}(s)$ is the Dirac function, since, according to , $\tilde \rho_\tau(t= 0, \cdot)$ is the Fourier transform of the function taking value $1$ identically.

Note that the solution of $(\partial_t - \partial_s) \rho_* = 0$ with initial data $\rho_*(t = 0) 