We finally turn our attention to the last integral along $\Gamma_{\omega}$. We note that in that case, for each $\theta \in [-\varepsilon, \varepsilon]$, one has

$$ \begin{aligned} \mathrm{Re} \left( \Lambda_r \left( \sqrt{\frac{\omega}{\zeta}} + i\theta \right) \right) &\le \frac{n}{|\alpha_r|} \left( -\frac{2}{3\sqrt{\zeta}}\omega^{3/2} - \frac{\omega^2}{\zeta^2} \left( \frac{\zeta}{4} - |\alpha_r|C_0\sqrt{\frac{\omega}{\zeta}} \right) - \sqrt{\omega} \left( \sqrt{\zeta} - \frac{3}{2}\sqrt{\omega} \right) \theta^2 \right) \\ &\quad - \frac{n}{|\alpha_r|} \left( \frac{\zeta}{4} - |\alpha_r|C_0\varepsilon \right) \theta^4 \\ &\le \frac{n}{|\alpha_r|} \left( -\frac{2}{3\sqrt{\zeta}}\omega^{3/2} - \frac{3\sqrt{\zeta}}{4}\sqrt{\omega}\theta^2 \right), \end{aligned} $$

thanks to our assumption (4.29). As a consequence, we get

$$ \begin{aligned} \left| \frac{1}{2\pi i} \int_{\Gamma_{\omega}} e^{\Lambda_r(\tau)} \tau \Phi_{r,j}(\tau) d\tau \right| &\le C e^{-\frac{2n}{3|\alpha_r|\sqrt{\zeta}}\omega^{3/2}-c|j|} \int_{-\varepsilon}^{\varepsilon} (\sqrt{\omega} + |\theta|) e^{-\frac{3\sqrt{\zeta}}{4|\alpha_r|}n\sqrt{\omega}\theta^2} d\theta \\ &\le C e^{-\frac{2n}{3|\alpha_r|\sqrt{\zeta}}\omega^{3/2}-c|j|} \left( \frac{\omega^{1/4}}{n^{1/2}} + \frac{1}{n\omega^{1/2}} \right) \\ &\le C \frac{e^{-c|j|}}{n^{2/3}} \left( \frac{|1-j_0|-n|\alpha_r|}{n^{1/3}} \right)^{-1/2} \exp \left( -c \left( \frac{j_0-n|\alpha_r|}{n^{1/3}} \right)^{3/2} \right). \end{aligned} $$

We now move to the case where $\omega_{\varepsilon} \le \omega \le 1 - |\alpha_r|$. We follow the same strategy and use the contours $\Gamma_{\pm}$, $\Gamma_{\omega_{\varepsilon}}$ and $\Gamma_{\omega_{\varepsilon}}^{\omega_{\varepsilon}}$ independently of $\omega$. We readily notice that with our careful choice of $\omega_{\varepsilon}$ all the previous computations remain valid and thus we also have in that case

$$ \begin{aligned} |\mathcal{R}_{1,r}^n(j, j_0)| &\le \left| \frac{1}{2\pi i} \int_{\Gamma_{\pm}} e^{\Lambda_r(\tau)} \tau \Phi_{r,j}(\tau) d\tau \right| + \left| \frac{1}{2\pi i} \int_{\Gamma_{\omega_{\varepsilon}}^{\omega_{\varepsilon}} \cup \Gamma_{\omega_{\varepsilon}}^{\omega_{\varepsilon}}} e^{\Lambda_r(\tau)} \tau \Phi_{r,j}(\tau) d\tau \right| \\ &\quad + \left| \frac{1}{2\pi i} \int_{\Gamma_{\omega_{\varepsilon}}} e^{\Lambda_r(\tau)} \tau \Phi_{r,j}(\tau) d\tau \right| \\ &\le C \left[ e^{-c j_0 - c|j|} + e^{-cn-c|j|} + e^{-\frac{n}{|\alpha_r|}(\omega - \frac{\omega_{\varepsilon}}{\zeta})\sqrt{\frac{\omega_{\varepsilon}}{\zeta}} - c|j|} \left( \frac{\omega_{\varepsilon}^{1/4}}{n^{1/2}} + \frac{1}{n\omega_{\varepsilon}^{1/2}} \right) \right] \\ &\le C \frac{e^{-c|j|}}{n^{2/3}} \left( \frac{j_0 - n|\alpha_r|}{n^{1/3}} \right)^{-1/2} \exp \left( -c \left( \frac{j_0 - n|\alpha_r|}{n^{1/3}} \right)^{3/2} \right), \end{aligned} $$

since $\omega_{\varepsilon} \le \omega \le 1 - |\alpha_r|$. This concludes the proof of the lemma. $\square$

**Case (III) – Oscillatory tail.** We finally move to the last regime (III) where $-|\alpha_r| < \omega \le -n^{-2/3}|\alpha_r|$. The analysis is further split into two parts.

**Lemma 4.6.** There exist constants $\omega_*> 0$ and $C, c > 0$ such that for each $1 \le j_0 \le n$ and $|j - j_0| \le n$, on has

$$ |\mathcal{R}_{1,r}^n(j, j_0)| \le C \frac{e^{-c|j|}}{n^{5/8}} \left( \frac{1}{n^{1/8}} \left( \frac{|j_0 - n|\alpha_r|}{n^{1/2}} \right)^{-1/2} + \left( \frac{|j_0 - n|\alpha_r|}{n^{1/2}} \right)^{1/4} \right) e^{-c \left( \frac{|j_0 - n|\alpha_r|}{n^{1/2}} \right)^2}, $$

as long as $-\omega_* < \omega \le -n^{-2/3}|\alpha_r|$.
Proof. We let $\omega_* \in (0, |\alpha_r|/3)$ be fixed as follows:

$$ \omega_* \le \frac{1 - \alpha_r^2}{3|\alpha_r|} (-1 + \sqrt{1 + 4\varepsilon})^2, \quad |\alpha_r|C_0\sqrt{\omega_*} \le \frac{1}{8} \left( \frac{1 - \alpha_r^2}{3|\alpha_r|} \right)^{3/2}, \quad \sqrt{\omega_*} \le \frac{1}{3} \left( \frac{1 - \alpha_r^2}{3|\alpha_r|} \right)^{1/2}. \quad (4.30) $$

As a consequence, for each $|\omega| \in [n^{-2/3}|\alpha_r|, \omega_*]$, one has

$$ \frac{1 - \alpha_r^2}{3|\alpha_r|} = \frac{2|\alpha_r|}{3} \frac{1 - \alpha_r^2}{2\alpha_r^2} < (|\alpha_r| - \omega_*) \frac{1 - \alpha_r^2}{2\alpha_r^2} \le \frac{j_0}{n} \frac{1 - \alpha_r^2}{2\alpha_r^2} = \zeta. $$

Defining:

$$ \chi_{\omega} := \frac{|\omega|}{2\zeta} + \sqrt{\frac{|\omega|}{\zeta}} > 0, $$

the above conditions imply in particular that

$$ \chi_{\omega} \le \frac{\varepsilon}{4}, \quad |\alpha_r|C_0\sqrt{\frac{|\omega|}{\zeta}} \le \frac{\zeta}{8}, \quad \sqrt{\frac{|\omega|}{\zeta}} \le \frac{1}{3}, \quad \text{with } |\omega| \in [n^{-2/3}, \omega_*]. $$

We then introduce the following contours which are illustrated in Figure 4.4 (compare again with the choice made in [9] that is entirely similar):

* two horizontal contours $\Gamma_{\pm}$ defined as

$$ \Gamma_{-} = \{t - i\varepsilon \mid t \in [-\eta, 0]\}, \quad \Gamma_{+} = \{-t + i\varepsilon \mid t \in [0, \eta]\}; $$

* two horizontal contours $\Gamma_{\wedge}^{\omega}$ defined as

$$ \Gamma_{>}^{\omega} = \{t - i\varepsilon \mid t \in [0, \sqrt{\frac{|\omega|}{\zeta}}]\}, \quad \Gamma_{<}^{\omega} = \{\sqrt{\frac{|\omega|}{\zeta}} - t + i\varepsilon \mid t \in [0, \sqrt{\frac{|\omega|}{\zeta}}]\}; $$

* two vertical contours $\Gamma_{\pm}^{v}$ defined as

$$ \Gamma_{-}^{v} = \left\{ \sqrt{\frac{|\omega|}{\zeta}} + i\theta \mid \theta \in \left[-\varepsilon, -\chi_{\omega} - \sqrt{\frac{|\omega|}{\zeta}}\right] \right\}, \quad \Gamma_{+}^{v} = \left\{ \sqrt{\frac{|\omega|}{\zeta}} + i\theta \mid \theta \in \left[\chi_{\omega} + \sqrt{\frac{|\omega|}{\zeta}}, \varepsilon\right] \right\}; $$

* two oblique contours $\Gamma_{\pm}^{o}$

$$ \Gamma_{-}^{o} = \left\{ -\frac{|\omega|}{2\zeta} - i\sqrt{\frac{|\omega|}{\zeta}} + t e^{\frac{3i\pi}{4}} \mid t \in \left[-\sqrt{2}\chi_{\omega}, \sqrt{2}\sqrt{\frac{|\omega|}{\zeta}}\right] \right\}, $$

$$ \Gamma_{+}^{o} = \left\{ -\frac{|\omega|}{2\zeta} + i\sqrt{\frac{|\omega|}{\zeta}} + t e^{\frac{i\pi}{4}} \mid t \in \left[-\sqrt{2}\sqrt{\frac{|\omega|}{\zeta}}, \sqrt{2}\chi_{\omega}\right] \right\}. $$
<figure><img src="image_17.png" /><figcaption>Figure 4.4: In blue: the contour $\Gamma_{\text{in}}$ within $\overline{\mathbf{B}}_{\varepsilon}(0)$ and its decomposition into eight parts ($\Gamma_{-}$, $\Gamma_{>}^{\omega}$, $\Gamma_{-}^{v}$, $\Gamma_{-}^{0}$, $\Gamma_{+}^{0}$, $\Gamma_{v}^{+}$, $\Gamma_{>}^{\omega}$ and $\Gamma_{+}$) in the regime where $-\omega_{*} < \omega \le -n^{-2/3}|\alpha_{r}|$. The two red dots correspond to the approximate saddles of the phase $|\omega|\tau + \zeta/3\tau^{3} - \zeta/4\tau^{4}$ in the complex plane and the black bullets represent the end points of the contours.</figcaption></figure>

Using Cauchy's formula, we write

$$ \mathcal{R}_r^n(j, j_0) = \frac{1}{2\pi i} \int_{\Gamma_- \cup \Gamma_+} e^{\Lambda_r(\tau)} \tau \Phi_{r,j}(\tau) d\tau + \frac{1}{2\pi i} \int_{\Gamma_>^\omega \cup \Gamma_-^\omega} e^{\Lambda_r(\tau)} \tau \Phi_{r,j}(\tau) d\tau, \\ + \frac{1}{2\pi i} \int_{\Gamma_-^v \cup \Gamma_+^v} e^{\Lambda_r(\tau)} \tau \Phi_{r,j}(\tau) d\tau + \frac{1}{2\pi i} \int_{\Gamma_-^o \cup \Gamma_+^o} e^{\Lambda_r(\tau)} \tau \Phi_{r,j}(\tau) d\tau. $$

As in the preceding case, we have that the uniform bound

$$ \left| \frac{1}{2\pi i} \int_{\Gamma_{-} \cup \Gamma_{+}} e^{\Lambda_r(\tau)} \tau \Phi_{r,j}(\tau) d\tau \right| \le C e^{-c j_0 - c|j|}, $$

and remark that since $|\omega| \in [n^{-2/3}|\alpha_r|, \omega_*]$, an exponential bound in $j_0$ leads to an exponential bound in both $j_0$ and $n$.

**Bounds along $\Gamma_{>}^{\omega}$.** For each $\tau = t - i\varepsilon \in \Gamma_{>}^{\omega}$ with $t \in [0, \sqrt{|\omega|/\zeta}]$,

$$ \begin{aligned} \operatorname{Re}(\Lambda_r (t - i\varepsilon)) &\le \frac{n}{|\alpha_r|} \left[ |\omega|t + \frac{\zeta}{3}(t^3 - 3\varepsilon^2 t) - \frac{\zeta}{4}(t^4 + \varepsilon^4 - 6\varepsilon^2 t^2) + |\alpha_r|C_0(t^5 + \varepsilon^5) \right] \\ &\le \frac{n}{|\alpha_r|} \left[ t \left( \frac{4}{3}|\omega| - \zeta\varepsilon^2 \left( 1 - \frac{3}{2}\sqrt{|\omega|/\zeta} \right) \right) - t^4 \left( \frac{\zeta}{4} - |\alpha_r|C_0t \right) - \varepsilon^4 \left( \frac{\zeta}{4} - |\alpha_r|C_0\varepsilon \right) \right] \\ &\le \frac{n}{|\alpha_r|} \left[ t \left( \frac{4}{3}|\omega| - \frac{\zeta}{2}\varepsilon^2 \right) - \frac{\zeta}{8}\varepsilon^4 \right] \le -\frac{n}{|\alpha_r|} \left[ t \frac{\zeta}{6}\varepsilon^2 + \frac{\zeta}{8}\varepsilon^4 \right]. \end{aligned} $$
Thus, we obtain

$$ \left| \frac{1}{2\pi i} \int_{\Gamma_{>}} e^{\Lambda_r(\tau)} \tau \Phi_{r,j}(\tau) d\tau \right| \le C e^{-n \frac{\zeta \epsilon^4}{8|\alpha_r|} - c|j|} \int_0^{\sqrt{|\omega|/\zeta}} e^{-n \frac{\zeta}{6|\alpha_r|} t} dt \le C e^{-cn-c|j|}. $$

By symmetry, a similar estimate holds along $\Gamma_{<}^{\omega}$. As for the bounds along $\Gamma_{\pm}$, we remark that since $|\omega| \in [n^{-2/3}|\alpha_r|, \omega_*]$, an exponential bound in $n$ leads to an exponential bound in both $j_0$ and $n$.

**Bounds along $\Gamma_{\pm}^v$.** For each $\theta \in [-\epsilon, -\chi_\omega - \sqrt{|\omega|/\zeta}]$, we have that

$$ \begin{aligned} \mathrm{Re} \left( \Lambda_r \left( \sqrt{\frac{|\omega|}{\zeta}} + i\theta \right) \right) &\le \frac{n}{|\alpha_r|} \left( \frac{4}{3\sqrt{\zeta}} |\omega|^{3/2} - \sqrt{|\omega|} \left( \sqrt{\zeta} - \frac{3}{2}\sqrt{|\omega|} \right) \theta^2 - \left( \frac{\zeta}{4} - |\alpha_r|C_0 \right) \theta^4 \right) \\ &\quad - \frac{n}{|\alpha_r|} \frac{|\omega|^2}{\zeta^2} \left( \frac{\zeta}{4} - |\alpha_r|C_0\sqrt{\frac{|\omega|}{\zeta}} \right). \end{aligned} $$

Thanks to our careful choice for $\epsilon$ and $\omega_*$, we get that

$$ \mathrm{Re} \left( \Lambda_r \left( \sqrt{\frac{|\omega|}{\zeta}} + i\theta \right) \right) \le \frac{n}{|\alpha_r|} \left( \frac{4}{3\sqrt{\zeta}} |\omega|^{3/2} - \frac{\sqrt{\zeta}}{2} \sqrt{|\omega|} \theta^2 \right) - \frac{n|\omega|^2}{8|\alpha_r|\zeta}, \quad \theta \in [-\epsilon, -\chi_\omega - \sqrt{|\omega|/\zeta}]. $$

As a consequence, we obtain

$$ \left| \frac{1}{2\pi i} \int_{\Gamma_{-}^{v}} e^{\Lambda_r(\tau)} \tau \Phi_{r,j}(\tau) d\tau \right| \le C e^{\frac{4}{3\sqrt{\zeta}|\alpha_r|} n|\omega|^{3/2} - \frac{n|\omega|^2}{8|\alpha_r|\zeta} - c|j|} \int_{-\epsilon}^{-\chi_\omega - \sqrt{|\omega|/\zeta}} (\sqrt{|\omega|} + |\theta|) e^{-\frac{\sqrt{\zeta}}{2|\alpha_r|} \sqrt{|\omega|} n\theta^2} d\theta. $$

The last integral on the right-hand side should be estimated carefully since there is an exponentially growing factor in front of the integral. We use Lemma A.2 in Appendix A and obtain the bound:

$$ \int_{-\epsilon}^{-\chi_\omega - \sqrt{|\omega|/\zeta}} e^{-\frac{\sqrt{\zeta}}{2|\alpha_r|} \sqrt{|\omega|} n\theta^2} d\theta \le \int_{2\sqrt{|\omega|/\zeta}}^{+\infty} e^{-\frac{\sqrt{\zeta}}{2|\alpha_r|} \sqrt{|\omega|} n\theta^2} d\theta \le \frac{C}{n|\omega|} e^{-\frac{2}{\sqrt{\zeta}|\alpha_r|} n|\omega|^{3/2}}, $$

and the analogous bound:

$$ \int_{-\epsilon}^{-\chi_\omega - \sqrt{|\omega|/\zeta}} |\theta| e^{-\frac{\sqrt{\zeta}}{2|\alpha_r|} \sqrt{|\omega|} n\theta^2} d\theta \le \int_{2\sqrt{|\omega|/\zeta}}^{+\infty} \theta e^{-\frac{\sqrt{\zeta}}{2|\alpha_r|} \sqrt{|\omega|} n\theta^2} d\theta \le \frac{C}{n\sqrt{|\omega|}} e^{-\frac{2}{\sqrt{\zeta}|\alpha_r|} n|\omega|^{3/2}}, $$

This leads to the final estimate:

$$ \left| \frac{1}{2\pi i} \int_{\Gamma_{-}^{v}} e^{\Lambda_r(\tau)} \tau \Phi_{r,j}(\tau) d\tau \right| \le C \frac{e^{-\frac{2}{3\sqrt{\zeta}|\alpha_r|} n|\omega|^{3/2} - \frac{n|\omega|^2}{8|\alpha_r|\zeta} - c|j|}}{n\sqrt{|\omega|}}, $$

since we have $|\omega| \le \omega_*$. A similar estimate holds along $\Gamma_{+}^v$.
**Bounds along $\Gamma_{\pm}^o$.** We finally handle the contributions along the oblique contours $\Gamma_{\pm}^o$. We focus first on $\Gamma_{-}^o$. For each $\tau \in \Gamma_{-}^o$, we have the parametrization

$$ \tau(t) = -\frac{|\omega|}{2\zeta} - i\sqrt{\frac{|\omega|}{\zeta}} + te^{3i\pi/4}, \quad t \in [-\sqrt{2}\chi_{\omega}, \sqrt{2}\sqrt{\frac{|\omega|}{\zeta}}], $$

and we compute that

$$ |\omega|\tau(t) + \frac{\zeta}{3}\tau(t)^3 - \frac{\zeta}{4}\tau(t)^4 = \sum_{k=0}^{4} p_k(|\omega|)t^k, $$

where each $p_k$ depends on $|\omega|$ only and is a complex valued function whose expansions as $|\omega| \to 0$ is given by

$$ \mathrm{Re}(p_0(|\omega|)) = -\frac{1}{4\zeta}|\omega|^2 + \mathcal{O}(|\omega|^3), \quad \mathrm{Im}(p_0(|\omega|)) = -\frac{2}{3\sqrt{\zeta}}|\omega|^{3/2} + \mathcal{O}(|\omega|^{5/2}), \quad p_1(|\omega|) = \mathcal{O}(|\omega|^2), $$

$$ \mathrm{Re}(p_2(|\omega|)) = -\sqrt{\zeta|\omega|} + \mathcal{O}(|\omega|^{3/2}), \quad \mathrm{Im}(p_2(|\omega|)) = \mathcal{O}(|\omega|), $$

$$ p_3(|\omega|) = \frac{\zeta}{3}e^{i\pi/4} + \mathcal{O}(|\omega|^{1/2}), \quad p_4(|\omega|) = \frac{\zeta}{4}. $$

As a consequence, we get that for each $t \in [-\sqrt{2}\chi_{\omega}, \sqrt{2}\sqrt{\frac{|\omega|}{\zeta}}]$

$$ \sum_{k=1}^{4} \mathrm{Re}(p_k(|\omega|))t^k = -\sqrt{\zeta|\omega|}t^2 + \frac{\zeta}{3\sqrt{2}}t^3 + \underbrace{\mathrm{Re}(p_1(|\omega|))}_{\mathcal{O}(|\omega|^2)} t + \underbrace{\left(\mathrm{Re}(p_2(|\omega|)) + \sqrt{\zeta|\omega|}\right)}_{\mathcal{O}(|\omega|^{3/2})} t^2 \\ + \underbrace{\left(\mathrm{Re}(p_3(|\omega|)) - \frac{\zeta}{3\sqrt{2}}\right)}_{\mathcal{O}(|\omega|^{1/2})} t^3 + \frac{\zeta}{4}t^4, $$

together with

$$ \mathrm{Re}(p_0(|\omega|)) + \frac{j_0}{n}|\alpha_r|\mathrm{Re}(\tau(t)^5\Psi_r(\tau(t))) \leq -\frac{1}{4\zeta}|\omega|^2 + \underbrace{\left(\mathrm{Re}(p_0(|\omega|)) + \frac{1}{4\zeta}|\omega|^2\right)}_{\mathcal{O}(|\omega|^3)} + |\alpha_r|C_0|\tau(t)|^5. $$

Upon decreasing further $\omega_*$ if necessary, we get the existence of a constant $c > 0$ independent of $|\omega|$ and $n$, such that

$$ \sum_{k=1}^{4} \mathrm{Re}(p_k(|\omega|))t^k \leq -c|\alpha_r|\sqrt{|\omega|}t^2, \quad \mathrm{Re}(p_0(|\omega|)) + \frac{j_0}{n}|\alpha_r|\mathrm{Re}(\tau(t)^5\Psi_r(\tau(t))) \leq -c|\alpha_r||\omega|^2, $$

for all $t \in [-\sqrt{2}\chi_{\omega}, \sqrt{2}\sqrt{\frac{|\omega|}{\zeta}}]$. As a consequence, we have

$$ \left| \frac{1}{2\pi i} \int_{\Gamma_{-}^{o}} e^{\Lambda_{r}(\tau)} \tau \Phi_{r,j}(\tau) d\tau \right| \leq C \sqrt{|\omega|} e^{-cn|\omega|^2 - c|j|} \int_{-\sqrt{2}\chi_{\omega}}^{\sqrt{2}\sqrt{\frac{|\omega|}{\zeta}}} e^{-cn\sqrt{|\omega|}t^2} dt. $$
Next, we remark that

$$ \int_{-\sqrt{2}\chi_{\omega}}^{\sqrt{2}\sqrt{\frac{|\omega|}{\zeta}}} e^{-cn\sqrt{|\omega|}t^2} dt \le \frac{C}{n^{1/2}|\omega|^{1/4}}, \quad n^{-2/3}|\alpha_r| \le |\omega| \le \omega_*, $$

such that

$$ \left| \frac{1}{2\pi i} \int_{\Gamma_0^o} e^{\Lambda_r(\tau)} \tau^5 \Phi_{r,j}(\tau) d\tau \right| \le C \frac{|\omega|^{1/4}}{n^{1/2}} e^{-cn|\omega|^2-c|j|}, \quad n^{-2/3}|\alpha_r| \le |\omega| \le \omega_* $$

And a similar estimate holds along $\Gamma_+^o$.

**Conclusion.** In summary, combining all the bounds for the eight segments and retaining only the “worst” contributions, we have obtained the estimate

$$ |\mathcal{R}_{1,r}^n(j, j_0)| \le C \left( \frac{1}{n\sqrt{|\omega|}} + \frac{|\omega|^{1/4}}{n^{1/2}} \right) e^{-cn|\omega|^2-c|j|}, \quad n^{-2/3}|\alpha_r| \le |\omega| \le \omega_* $$

We may rewrite the above estimate as

$$ |\mathcal{R}_{1,r}^n(j, j_0)| \le C \frac{e^{-c|j|}}{n^{5/8}} \left( \frac{1}{n^{1/8}} \left( \frac{|j_0 - n|\alpha_r|}{n^{1/2}} \right)^{-1/2} + \left( \frac{|j_0 - n|\alpha_r|}{n^{1/2}} \right)^{1/4} \right) e^{-c \left( \frac{|j_0 - n|\alpha_r|}{n^{1/2}} \right)^2}, $$

valid in the range $n^{-2/3}|\alpha_r| \le |\omega| \le \omega_*$. $\square$

We should now deal with the final regime $\omega \le -\omega_*$, with $\omega_* > 0$ being given by Lemma 4.6.

**Lemma 4.7.** Let $\omega_* > 0$ be given by Lemma 4.6. Then there exist constants $C, c > 0$ such that for each $1 \le j_0 \le n$ and $|j - j_0| \le n$, there holds:

$$ |\mathcal{R}_{1,r}^n(j, j_0)| \le C e^{-cn-cj_0-c|j|}, $$

as long as $-|\alpha_r| < \omega \le -\omega_*$.

*Proof.* We simply take the vertical contour $\Gamma_{in} = \Gamma_{-\eta}$, and recall that, thanks to conditions (4.11) and (4.12) on $\varepsilon$, we have

$$ \mathrm{Re} (\Lambda_r(-\eta + i\theta)) \le -\frac{|\alpha_r|}{4}\eta, \quad \theta \in [-\varepsilon, \varepsilon]. $$

We readily obtain the desired bound since $-|\alpha_r| < \omega \le -\omega_*$. $\square$

Combining Lemma 4.4, Lemma 4.5, Lemma 4.6 and Lemma 4.7, we have proved Proposition 4.1 for $\mathcal{R}_{1,r}^n(j, j_0)$. Indeed, in the range $n^{-2/3}|\alpha_r| \le |\omega| \le \omega_*$, we can further bound

$$ \frac{1}{n^{1/8}} \left( \frac{|j_0 - n|\alpha_r|}{n^{1/2}} \right)^{-1/2} \le n^{1/12 - 1/8} \le 1, $$

and

$$ \left( \frac{|j_0 - n|\alpha_r|}{n^{1/2}} \right)^{1/4} e^{-c \left( \frac{|j_0 - n|\alpha_r|}{n^{1/2}} \right)^2} \le \tilde{C} e^{-\tilde{c} \left( \frac{|j_0 - n|\alpha_r|}{n^{1/2}} \right)^2}, $$

with a smaller constant $\tilde{c} > 0$ and a suitable constant $\tilde{C}$.
Regarding the second remainder term $\mathcal{R}_{2,r}^n(j, j_0)$, we first observe that for each $\tau \in \overline{\mathbf{B}_\varepsilon(0)}$ one has

$$ \frac{\exp(j_0 \tau^5 \Psi_r(\tau)) - 1 - j_0 \tau^5 \Psi_r(0)}{\tau} = \frac{\exp(j_0 \tau^5 \Psi_r(\tau)) - 1 - j_0 \tau^5 \Psi_r(\tau)}{\tau} + j_0 \tau^4 (\Psi_r(\tau) - \Psi_r(0)), $$

such that

$$ \left| \frac{\exp(j_0 \tau^5 \Psi_r(\tau)) - 1 - j_0 \tau^5 \Psi_r(0)}{\tau} \right| \le C \left[ (j_0 |\tau|^4) + j_0 |\tau|^4 \right] |\tau|. $$

All the previous lemmas can be easily adapted to obtain similar estimates for $\mathcal{R}_{2,r}^n(j, j_0)$ as the ones we derived for $\mathcal{R}_{1,r}^n(j, j_0)$. For example, regarding the uniform bound in Lemma 4.4, the map $g(\theta)$ now reads

$$ g(\theta) := e^{-n \frac{\xi}{4} \theta^4} \left( \frac{\exp(j_0 \mathbf{i} \theta^5 \Psi_r(\mathbf{i}\theta)) - 1 - j_0 \mathbf{i} \theta^5 \Psi_r(0)}{\theta} \right), \quad \theta \in [-\varepsilon, \varepsilon], $$

and we note that

$$ |g'(\theta)| \le C (1 + n \theta^4 + (n \theta^4)^2) e^{-cn\theta^4}, $$

since $1 \le j_0 \le n$, and we observe that $\|g'\|_{L^1([0, \varepsilon])} \le Cn^{-1/4}$. As a consequence, we naturally retrieve the estimate of Lemma 4.4 for $\mathcal{R}_{2,r}^n(j, j_0)$. We let the other cases to the interested reader. $\square$

### 4.3.4 Final decomposition of the temporal Green's function. Proof of Theorem 4.1

Let us recall that, for $j_0 \ge 1$, we have first decomposed (see (4.9)):

$$ \mathcal{G}^n(j, j_0) = \overline{\mathcal{G}}_r^n(j - j_0) \mathbf{1}_{j \ge 1} + \tilde{\mathcal{G}}^n(j, j_0), $$

and the reduced Green's function $\tilde{\mathcal{G}}^n(j, j_0)$ has been decomposed into:

$$ \tilde{\mathcal{G}}^n(j, j_0) = \frac{1}{2\pi\mathbf{i}} \int_{\Gamma_{\text{out}}} e^{n\tau} \tilde{\mathcal{G}}_j^{j_0}(e^{\tau}) e^{\tau} d\tau + \frac{1}{2\pi\mathbf{i}} \int_{\Gamma_{\text{in}}} e^{n\tau} \tilde{\mathcal{G}}_j^{j_0}(e^{\tau}) e^{\tau} d\tau, $$

with suitable contours $\Gamma_{\text{out}}$ and $\Gamma_{\text{in}}$. The part on $\Gamma_{\text{out}}$ satisfies the exponential bound (4.14) and the part on $\Gamma_{\text{in}}$ has been further decomposed in (4.20).

Coming back to our decomposition (4.20) of part of the (reduced) temporal Green's function $\tilde{\mathcal{G}}^n(j, j_0)$, and recalling the initial decomposition (4.9), we can gather Lemma 4.2 (together with the exponential decay of $\mathcal{H}_j$), Lemma 4.3 and Proposition 4.1 to obtain for $j \in \mathbb{Z}$ and $1 \le j_0 \le n$:

$$ \begin{aligned} |\mathcal{G}^n(j, j_0) - \mathcal{H}_j \mathbf{A}_r(-j_0 + n |\alpha_r|, n)| &\le |\overline{\mathcal{G}}_r^n(j - j_0)| \mathbf{1}_{j \ge 1} + |\tilde{\mathcal{G}}^n(j, j_0) - \mathcal{H}_j \mathbf{A}_r(-j_0 + n |\alpha_r|, n)| \\ &\le C \mathbf{M}_r (c, j - j_0 + n |\alpha_r|, n) \mathbf{1}_{j \ge 1} \\ &\quad + C e^{-c|j|} \mathbf{M}_r (c, -j_0 + n |\alpha_r|, n) + C e^{-cn} e^{-c|j|} e^{-c|j_0|}, \end{aligned} $$

where we have used Corollary A.1 in Appendix A to derive the bound of the free Green's function $\overline{\mathcal{G}}_r$ (and the definition (4.6b) of the function $\mathbf{M}_r$). This shows the validity of the bound (4.8) in Theorem 4.1 for the case $1 \le j_0 \le n$.
Let us finally consider the regime $1 \le n \le j_0 - 1$ and show that the estimate (4.8) in Theorem 4.1 is still valid. We apply Lemma 4.1 and obtain:

$$ egin{aligned} |\mathcal{G}^n(j, j_0) - \mathcal{H}_j \mathbf{A}_r(-j_0 + n |\alpha_r|, n)| &= |\overline{\mathcal{G}}_r^n(j - j_0) - \mathcal{H}_j \mathbf{A}_r(-j_0 + n |\alpha_r|, n)| \\ &\le |\overline{\mathcal{G}}_r^n(j - j_0)| + |\mathcal{H}_j| |\mathbf{A}_r(-j_0 + n |\alpha_r|, n)| \\ &\le C \mathbf{M}_r(c, j - j_0 + n |\alpha_r|, n) + C e^{-c|j|} |\mathbf{A}_r(-j_0 + n |\alpha_r|, n)|, \end{aligned} $$

where we have again used Corollary A.1 in Appendix A to derive the bound of the free Green's function $\overline{\mathcal{G}}_r$. It only remains to show that the final term on the right-hand side is exponentially small with respect to both $j_0$ and $n$ and the proof will be complete. We apply Corollary A.6 in Appendix A to estimate the term with the function $\mathbf{A}_r$ on the right-hand side in the considered regime $1 \le n \le j_0 - 1$. We get:

$$ |\mathbf{A}_r(-j_0 + n |\alpha_r|, n)| \le C \exp \left( -c \frac{|j_0 - n| |\alpha_r||^{4/3}}{n^{1/3}} \right) \le C e^{-c j_0}, $$

where the final estimate comes from the fact that we now consider the regime $1 \le n \le j_0 - 1$. Since now $j_0$ dominates $n$, we get the exponential estimate:

$$ |\mathbf{A}_r(-j_0 + n |\alpha_r|, n)| \le C e^{-cn} e^{-cj_0}, $$

and we have thus proved the validity of the bound (4.8) in Theorem 4.1 for the case $1 \le n \le j_0 - 1$.

The proof of Theorem 4.1 is entirely similar in the case $j_0 \le 0$ except that all involved functions are now associated with the left state $u_\ell$ of the shock rather than with $u_r$. We leave the details to the interested reader.

## 4.4 Derivative of the temporal Green's functions

As it will be made clear in the last chapter of this article, it will also be necessary to obtain large time decaying bounds for the family of operators $(\mathcal{L}^n(\mathrm{Id} - \mathbf{S}))_{n \in \mathbb{N}}$ where $\mathbf{S} : \ell^q(\mathbb{Z}; \mathbb{R}) \to \ell^q(\mathbb{Z}; \mathbb{R})$ is the shift operator defined as $(\mathbf{S}\mathbf{h})_j := h_{j+1}$ for all $j \in \mathbb{Z}$ for any sequence $\mathbf{h} = (h_j)_{j \in \mathbb{Z}} \in \ell^q(\mathbb{Z}; \mathbb{R})$. By definition of the operator $\mathcal{L}^n$, for any $\mathbf{h} \in \ell^q(\mathbb{Z}; \mathbb{R})$, we have the decomposition

$$ \forall (n, j) \in \mathbb{N} \times \mathbb{Z}, \quad (\mathcal{L}^n(\mathrm{Id} - \mathbf{S})\mathbf{h})_j = \sum_{j_0 \in \mathbb{Z}} (\mathcal{G}^n(j, j_0) - \mathcal{G}^n(j, j_0 - 1)) h_{j_0}. $$

This motivates the definition of the following quantity

$$ \forall (n, j, j_0) \in \mathbb{N} \times \mathbb{Z} \times \mathbb{Z}, \quad \mathcal{D}^n(j, j_0) := \mathcal{G}^n(j, j_0) - \mathcal{G}^n(j, j_0 - 1). \quad (4.31) $$

The above quantity is thus a discrete spatial derivative of $\mathcal{G}^n(j, j_0)$ with respect to its second argument, and we shall refer to it simply as the derivative of the temporal Green's function. We will now follow the same strategy as presented in the previous sections of this chapter. That is, we shall decompose the derivative of the temporal Green's function into several contributions and derive bounds for each such contributions which are meant to be sufficiently sharp in order to obtain large time decaying bounds for the family of operators $(\mathcal{L}^n(\mathrm{Id} - \mathbf{S}))_{n \in \mathbb{N}}$.
Before stating our main result, we recall that the temporal Green's function satisfies $\mathcal{G}^n(j, j_0) = 0$ whenever $|j - j_0| > n$, since the Lax-Wendroff scheme has a finite stencil. As an elementary consequence, we have that for all $n \in \mathbb{N}$ and $(j, j_0) \in \mathbb{Z}^2$:

$$j - j_0 > n \text{ or } j - j_0 < -n - 1 \Rightarrow \mathcal{D}^n(j, j_0) = 0.$$

As a consequence, throughout this section, we shall only consider $(n, j, j_0) \in \mathbb{N} \times \mathbb{Z} \times \mathbb{Z}$ that satisfies $-n - 1 \le j - j_0 \le n$. Furthermore, a direct application of Lemma 4.1 gives the following result.

**Lemma 4.8.** Let $j_0 \in \mathbb{Z}$ and $n \in \mathbb{N}$. If $j_0 \ge 2$ and $n \le j_0 - 2$, then there holds:

$$\forall j \in \mathbb{Z}, \quad \mathcal{D}^n(j, j_0) = \overline{\mathcal{G}}_r^n(j - j_0) - \overline{\mathcal{G}}_r^n(j - j_0 + 1).$$

If $j_0 \le 0$ and $n \le |j_0|$, then there holds:

$$\forall j \in \mathbb{Z}, \quad \mathcal{D}^n(j, j_0) = \overline{\mathcal{G}}_l^n(j - j_0) - \overline{\mathcal{G}}_l^n(j - j_0 + 1).$$

Finally, we introduce two functions $\mathbf{K}_\ell$ and $\mathbf{K}_r$ on $\mathbb{R}^{+*} \times \mathbb{R} \times \mathbb{R}^{+*}$ as follows:

$$\mathbf{K}_{\ell}(c, x, y) := \begin{cases} \frac{1}{y^{7/12}} \exp(-c|x|^{3/2}/y^{1/2}), & \text{if } x \ge 0, \\ \frac{1}{y^{7/12}}, & \text{if } -y^{1/3} \le x \le 0, \end{cases} \quad (4.32a)$$

$$\frac{1}{y^{5/8}} \exp(-c x^2/y), \quad \text{if } x \le -y^{1/3},$$

$$\frac{1}{y^{7/12}} \exp(-c|x|^{3/2}/y^{1/2}), \quad \text{if } x \le 0,$$

$$\mathbf{K}_r(c, x, y) := \begin{cases} \frac{1}{y^{7/12}}, & \text{if } 0 \le x \le y^{1/3}, \\ \frac{1}{y^{5/8}} \exp(-c x^2/y), & \text{if } y^{1/3} \le x, \end{cases} \quad (4.32b)$$

for all $(c, x, y) \in \mathbb{R}^{+*} \times \mathbb{R} \times \mathbb{R}^{+*}$. Once again, we crucially note that both $\mathbf{K}_\ell$ and $\mathbf{K}_r$ are non-increasing with respect to their first argument.

We can now state our main result regarding the derivative of the Green's function.

**Theorem 4.2** (Pointwise bounds on the derivative of the Green's function). Let the weak solution (2.2) satisfy the Rankine-Hugoniot condition (2.3) and the entropy inequalities (2.4). Let the parameter $\lambda$ satisfy the CFL condition (2.12) and let Assumption 1 be satisfied. Then there exist some positive constants $C$ and $c$ such that for each $n \ge 1$ and $(j, j_0) \in \mathbb{Z}^2$ with $-n - 1 \le j - j_0 \le n$, the derivative $\mathcal{D}^n(j, j_0)$ of the Green's function enjoys the following pointwise bounds:

* for any $j_0 \le 0$:

$$|\mathcal{D}^n(j, j_0)| \le C \mathbf{K}_\ell(c, j_0 - j + n \alpha_\ell, n) \mathbf{1}_{j \le 0} + C e^{-c|j|} \mathbf{M}_\ell(c, j_0 + \alpha_\ell n, n) + C e^{-cn - c|j - j_0|}; \quad (4.33)$$
• and for any $j_0 \ge 1$:

$$|\mathcal{D}^n(j, j_0)| \le C \mathbf{K}_r(c, j-j_0+n |\alpha_r|, n) \mathbf{1}_{j \ge 1} + C e^{-c|j|} \mathbf{M}_r(c, -j_0 + |\alpha_r|n, n) + C e^{-cn-c|j-j_0|}. \quad (4.34)$$

*Proof.* Throughout the proof we assume that $n \ge 1$ and $(j, j_0) \in \mathbb{Z}^2$ with $-n-1 \le j-j_0 \le n$ and we only consider $j_0 \ge 1$ since the analysis for $j_0 \le 0$ follows similar lines.

We first focus on the regime $2 \le j_0 \le n+1$. Using the expression for $\mathcal{G}^n(j, j_0)$, we may instead write

$$\mathcal{D}^n(j, j_0) = \mathbf{1}_{j \ge 1} \left[ \underbrace{\overline{\mathcal{G}}_r^n(j-j_0) - \overline{\mathcal{G}}_r^n(j-j_0+1)}_{:=\mathcal{H}_r^n(j-j_0)} \right] + \underbrace{\frac{1}{2\pi i} \int_{\Gamma} e^{n\tau} [\tilde{\mathcal{G}}_j^{j_0}(e^{\tau}) - \tilde{\mathcal{G}}_j^{j_0-1}(e^{\tau})] e^{\tau} d\tau}_{:=\tilde{\mathcal{G}}^n(j, j_0)}. \quad (4.35)$$

We first handle the second term $\tilde{\mathcal{D}}^n(j, j_0)$. We let $\varepsilon \in (0, \varepsilon_*)$ be fixed as in previous section, that is satisfying conditions (4.11)-(4.12)-(4.13), and $0 < \eta < \min(\eta_\varepsilon, \varepsilon^5)$. From Proposition 3.3, since the map $\tau \mapsto \tilde{\mathcal{G}}_j^{j_0}(e^\tau) - \tilde{\mathcal{G}}_j^{j_0-1}(e^\tau)$ has a holomorphic extension to $\mathbf{B}_{\varepsilon_0}(0)$, we can decompose the contour $\Gamma$ into $\Gamma_{\text{out}} = \{-\eta + i\theta \mid \varepsilon \le \theta \le \pi\}$ and $\Gamma_{\text{in}}$, where $\Gamma_{\text{in}}$ can be any path joining $-\eta - i\varepsilon$ to $-\eta + i\varepsilon$ which remains in $\mathbf{B}_{\varepsilon}(0)$ (see Figure 4.1). As a consequence, we have

$$\left| \frac{1}{2\pi i} \int_{\Gamma_{\text{out}}} e^{n\tau} \left[ \tilde{\mathcal{G}}_j^{j_0}(e^{\tau}) - \tilde{\mathcal{G}}_j^{j_0-1}(e^{\tau}) \right] e^{\tau} d\tau \right| \le C e^{-\eta n} e^{-c|j|-c|j_0|},$$

together with

$$\frac{1}{2\pi i} \int_{\Gamma_{\text{in}}} e^{n\tau} \left[ \tilde{\mathcal{G}}_j^{j_0}(e^{\tau}) - \tilde{\mathcal{G}}_j^{j_0-1}(e^{\tau}) \right] e^{\tau} d\tau = \frac{\mathcal{H}_j}{\alpha_r} \times \frac{1}{2\pi i} \int_{\Gamma_{\text{in}}} e^{n\tau - j_0 \varphi_r(\tau) + j_0 \tau^5 \Psi_r(\tau)} d\tau \\ + \frac{1}{2\pi i} \int_{\Gamma_{\text{in}}} e^{n\tau - j_0 \varphi_r(\tau) + j_0 \tau^5 \Psi_r(\tau)} \tau \Theta_{r,j}(\tau) d\tau,$$

where the sequence $(\mathcal{H}_j)_{j \in \mathbb{Z}}$ is defined in (3.22) and the sequence $(\Theta_{r,j})_{j \in \mathbb{Z}}$ of bounded holomorphic functions is given by Proposition 3.3. The first integral is similar to $\tilde{\mathcal{G}}_{2,r}^n(j, j_0)$ (with definition (4.15)) and enjoys a similar bound, while the second integral is similar to $\tilde{\mathcal{R}}_{1,r}^n(j, j_0)$ (with definition (4.16)) and enjoys a similar bound. As a consequence, for $2 \le j_0 \le n+1$, we have the estimate

$$\left| \tilde{\mathcal{D}}^n(j, j_0) \right| \le C e^{-c|j|} \mathbf{M}_r (c, -j_0 + |\alpha_r|n, n) + C e^{-cn-c|j_0-c|j}, \quad (4.36)$$

for some $C > 0$ and $c > 0$ which are independent of $n, j$ and $j_0$.

Let us focus now on the first term $\mathcal{H}_r^n(j-j_0)$ of (4.35) which can also be written as

$$\mathcal{H}_r^n(j-j_0) = \frac{1}{2\pi i} \int_{\Gamma} e^{n\tau} \left[ \overline{\mathcal{G}}_{r,j-j_0}(e^{\tau}) - \overline{\mathcal{G}}_{r,j-j_0+1}(e^{\tau}) \right] e^{\tau} d\tau.$$

From the expression (3.27) of Proposition 3.2 of the free spatial Green's function there exists $\kappa_\mathcal{O} > 0$ depending on the set $\mathcal{O}$ defined in Chapter 3, such that for $j_0 \le j$ one has

$$| [\overline{\mathcal{G}}_{r,j-j_0}(e^{\tau}) - \overline{\mathcal{G}}_{r,j-j_0+1}(e^{\tau})] e^{\tau} | \le C e^{-c|j-j_0|},$$

for all $\tau \in \mathbb{C}$ with $\text{Re}(\tau) \ge -\kappa_\mathcal{O}$. As a consequence, upon taking $\Gamma = \{-\eta + i\theta \mid -\pi \le \theta \le \pi\}$ with $0 < \eta < \kappa_\mathcal{O}$, we readily get in that case that

$$|\mathcal{H}_r^n(j-j_0)| \le C e^{-\eta n - c|j-j_0|}.$$
On the other hand, for $1 \le j \le j_0 \le n+1$, one has from the expression (3.27) that for all $\tau \in \overline{\mathbf{B}_{\varepsilon_0}(0)}$

$$[\overline{\mathcal{G}}_{r,j-j_0}(e^{\tau}) - \overline{\mathcal{G}}_{r,j-j_0+1}(e^{\tau})] e^{\tau} = \tau \Xi_r(\tau) \exp((j-j_0)\varphi_r(\tau) - (j-j_0)\tau^5\Psi_r(\tau)),$$

for some bounded holomorphic function $\Xi_r$, with the same $\varepsilon_0$ as the one given by Proposition 3.3. We can then once again let $\varepsilon \in (0, \varepsilon_*)$ be fixed as in the previous section and set $0 < \eta < \min(\eta_\varepsilon, \varepsilon^5)$. Using Corollary 3.3, we also have

$$| [\overline{\mathcal{G}}_{r,j-j_0}(e^{\tau}) - \overline{\mathcal{G}}_{r,j-j_0+1}(e^{\tau})] e^{\tau} | \le C e^{-c|j-j_0|},$$

for all $\tau = -\eta + \mathbf{i}\theta$ with $\varepsilon \le \theta \le \pi$. As a consequence, with our now usual notation:

$$\Gamma_{\text{out}} = \{-\eta + \mathbf{i}\theta \mid \varepsilon \le \theta \le \pi\},$$

we get

$$\left| \frac{1}{2\pi \mathbf{i}} \int_{\Gamma_{\text{out}}} e^{n\tau} [\overline{\mathcal{G}}_{r,j-j_0}(e^{\tau}) - \overline{\mathcal{G}}_{r,j-j_0+1}(e^{\tau})] e^{\tau} \, d\tau \right| \le C e^{-\eta n - c|j-j_0|}.$$

On the other hand, if $\Gamma_{\text{in}}$ denotes any path joining $-\eta - \mathbf{i}\varepsilon$ to $-\eta + \mathbf{i}\varepsilon$ which remains in $\overline{\mathbf{B}_{\varepsilon}(0)}$, one has

$$\frac{1}{2\pi \mathbf{i}} \int_{\Gamma_{\text{in}}} e^{n\tau} [\overline{\mathcal{G}}_{r,j-j_0}(e^{\tau}) - \overline{\mathcal{G}}_{r,j-j_0-1}(e^{\tau})] e^{\tau} d\tau = \frac{1}{2\pi \mathbf{i}} \int_{\Gamma_{\text{in}}} e^{n\tau+(j-j_0)\varphi_r(\tau)-(j-j_0)\tau^5\Psi_r(\tau)} \tau \Xi_r(\tau) d\tau.$$

Then applying Lemma 4.4, Lemma 4.5, Lemma 4.6 and Lemma 4.7 with $\Phi_{r,j}$ replaced by $\Xi_r(\tau)$ and $-j_0$ by $j-j_0$, we readily obtain that

$$\left| \frac{1}{2\pi \mathbf{i}} \int_{\Gamma_{\text{in}}} e^{n\tau} [\overline{\mathcal{G}}_{r,j-j_0}(e^{\tau}) - \overline{\mathcal{G}}_{r,j-j_0+1}(e^{\tau})] e^{\tau} d\tau \right| \le C \mathbf{K}_r(c, j-j_0+n|\alpha_r|, n),$$

where the definition of $\mathbf{K}_r$ is given in (4.32b). Combining the above bound with the estimate (4.36) proves the bound (4.34) of $\mathcal{D}^n(j, j_0)$ in the range $2 \le j_0 \le n+1$.

For $j_0 \ge 2$ and $n \le j_0 - 2$, we use Lemma 4.8 which gives that

$$\forall j \in \mathbb{Z}, \quad \mathcal{D}^n(j, j_0) = \overline{\mathcal{G}}_r^n(j - j_0) - \overline{\mathcal{G}}_r^n(j - j_0 + 1).$$

We can then proceed along similar lines as above, and get that

$$|\mathcal{D}^n(j, j_0)| \le C \mathbf{K}_r(c, j - j_0 + n |\alpha_r|, n) + C e^{-\eta n - c|j - j_0|}.$$

Let us now turn to the case $j_0 = 1$. Using the expressions of $\mathcal{G}^n(j, 1)$ and $\mathcal{G}^n(j, 0)$, we have

$$\mathcal{D}^n(j, 1) = \mathbf{1}_{j \ge 1} \overline{\mathcal{G}}_r^n(j-1) - \mathbf{1}_{j \le 0} \overline{\mathcal{G}}_\ell^n(j) + \underbrace{\frac{1}{2\pi \mathbf{i}} \int_{\Gamma} e^{n\tau} [\tilde{\mathcal{G}}_j^1(e^{\tau}) - \tilde{\mathcal{G}}_j^0(e^{\tau})] e^{\tau} d\tau}_{:= \tilde{\mathcal{D}}^n(j, 1)}.$$

First, inspecting the contribution stemming from $\tilde{\mathcal{D}}^n(j, 1)$ and decomposing $\Gamma$ with $\Gamma_{\text{out}}$ and $\Gamma_{\text{in}}$, we observe that from (3.32) of Proposition 3.3, we have:

$$\left| \frac{1}{2\pi \mathbf{i}} \int_{\Gamma_{\text{out}}} e^{n\tau} [\tilde{\mathcal{G}}_j^1(e^{\tau}) - \tilde{\mathcal{G}}_j^0(e^{\tau})] e^{\tau} d\tau \right| \le C e^{-\eta n} e^{-c|j|},$$
together with

$$ \frac{1}{2\pi i} \int_{\Gamma_{\text{in}}} e^{n\tau} [\tilde{\mathcal{G}}_j^1(e^{\tau}) - \tilde{\mathcal{G}}_j^0(e^{\tau})] e^{\tau} d\tau = \left(\frac{\mathcal{H}_j}{\alpha_r} + \gamma_j^r - \gamma_j^\ell\right) \times \frac{1}{2\pi i} \int_{\Gamma_{\text{in}}} e^{n\tau} d\tau + \frac{1}{2\pi i} \int_{\Gamma_{\text{in}}} e^{n\tau} \tau \Theta_{1,j}(\tau) d\tau. $$

As a consequence, upon taking $\Gamma_{\text{in}} = \{-\eta + i\theta \mid |\theta| \le \varepsilon\}$, we get

$$ |\tilde{\mathcal{G}}^n(j, 1)| \le C e^{-cn-c|j|}. $$

Upon noticing that $M_r(c, -1 + |\alpha_r|n, n) = O(e^{-cn})$, estimate (4.36) also holds true for $j_0 = 1$. Coming back to the first two terms in the expression of $\mathcal{L}^n(j, 1)$, we simply note, using Corollary A.1, that

$$ \forall j \ge 1, \quad |\mathcal{G}_r^n(j-1)| \le C e^{-cn}, \quad \text{and} \quad \forall j \le 0, \quad |\mathcal{G}_\ell^n(j)| \le C e^{-cn}. $$

But since one has $-n \le j \le n + 1$, the above exponential bound in $n$ leads to an exponential bound in both $j$ and $n$:

$$ |1_{j \ge 1} \mathcal{G}_r^n(j-1) - 1_{j \le 0} \mathcal{G}_\ell^n(j)| \le C e^{-cn-c|j|}. $$

To conclude, we note that $K_r(c, j - 1 + n |\alpha_r|, n) = O(e^{-cn-c|j|})$ which implies that estimate (4.34) also holds true for $j_0 = 1$. $\square$

## 4.5 Linear estimates

In this final section, we derive large time decaying bounds for the semigroup $(\mathcal{L}^n)_{n \in \mathbb{N}}$ and the family of operators $(\mathcal{L}^n(\text{Id} - \mathbf{S}))_{n \in \mathbb{N}}$ acting on algebraically weighted spaces which are crucial for our forthcoming nonlinear stability analysis.

We first recall some notation for the polynomially weighted spaces of sequences. Given a real number $\gamma \ge 0$ and $q \in [1, +\infty]$, if we define the weight sequence $\omega_\gamma := (1 + |j|^{\gamma})_{j \in \mathbb{Z}}$, we recall our definition of algebraically weighted $\ell^q$ spaces:

$$ \ell_{\gamma}^{q}(\mathbb{Z}; \mathbb{R}) = \{ \mathbf{h} \in \ell^{q}(\mathbb{Z}; \mathbb{R}) \mid \omega_{\gamma} \mathbf{h} \in \ell^{q}(\mathbb{Z}; \mathbb{R}) \}, $$

where $\omega_\gamma \mathbf{h}$ stands for the sequence $((1 + |j|^{\gamma}) h_j)_{j \in \mathbb{Z}}$. For any sequence $\mathbf{h} \in \ell_{\gamma}^q(\mathbb{Z}; \mathbb{R})$, the norm of $\mathbf{h}$ is defined as $\|\mathbf{h}\|_{\ell_{\gamma}^q} := \|\omega_\gamma \mathbf{h}\|_{\ell^q}$.

The main result of this Chapter reads as follows (the reader will observe that this statement is a refinement of Theorem 2.4 that was made more simple for the reader's convenience).

**Theorem 4.3.** Let the weak solution (2.2) satisfy the Rankine-Hugoniot relation (2.3) and the entropy inequalities (2.4). Let the parameter $\lambda$ satisfy the CFL condition (2.12) and let Assumption 1 be satisfied. For any $\gamma_2 \ge \gamma_1 \ge 0$, there exists $C_{\mathcal{L}}(\gamma_1, \gamma_2) > 0$ such that we have the following estimates on the semigroup $(\mathcal{L}^n)_{n \in \mathbb{N}}$:

$$ \forall n \in \mathbb{N}, \quad \|\mathcal{L}^n \mathbf{h}\|_{\ell_{\gamma_1}^1} \le \frac{C_{\mathcal{L}}(\gamma_1, \gamma_2)}{(1+n)^{\gamma_2-\gamma_1-1/8}} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}, \quad \text{for } \mathbf{h} \in \ell_{\gamma_2}^1(\mathbb{Z}; \mathbb{R}) \text{ with } \sum_{j \in \mathbb{Z}} h_j = 0, \qquad (4.37a) $$

$$ \forall n \in \mathbb{N}, \quad \|\mathcal{L}^n \mathbf{h}\|_{\ell_{\gamma_1}^{\infty}} \le \frac{C_{\mathcal{L}}(\gamma_1, \gamma_2)}{(1+n)^{\gamma_2-\gamma_1+\min(1/3, \gamma_1)}} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}, \quad \text{for } \mathbf{h} \in \ell_{\gamma_2}^1(\mathbb{Z}; \mathbb{R}) \text{ with } \sum_{j \in \mathbb{Z}} h_j = 0, \qquad (4.37b) $$
and the following estimates on the family of operators $(\mathcal{L}^n(\mathrm{Id} - \mathbf{S}))_{n \in \mathbb{N}}$:

$$ \forall n \in \mathbb{N}, \quad \|\mathcal{L}^n(\mathrm{Id} - \mathbf{S})\mathbf{h}\|_{\ell_{\gamma_1}^1} \le \frac{C_{\mathcal{L}}(\gamma_1, \gamma_2)}{(1+n)^{\gamma_2-\gamma_1+1/8}} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}, \quad \text{for } \mathbf{h} \in \ell_{\gamma_2}^1(\mathbb{Z}; \mathbb{R}), \quad (4.38a) $$

$$ \forall n \in \mathbb{N}, \quad \|\mathcal{L}^n(\mathrm{Id} - \mathbf{S})\mathbf{h}\|_{\ell_{\gamma_1}^{\infty}} \le \frac{C_{\mathcal{L}}(\gamma_1, \gamma_2)}{(1+n)^{\gamma_2-\gamma_1+1/3+\min(1/4, \gamma_1)}} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}, \quad \text{for } \mathbf{h} \in \ell_{\gamma_2}^1(\mathbb{Z}; \mathbb{R}), \quad (4.38b) $$

$$ \forall n \in \mathbb{N}, \quad \|\mathcal{L}^n(\mathrm{Id} - \mathbf{S})\mathbf{h}\|_{\ell_{\gamma_1}^{\infty}} \le \frac{C_{\mathcal{L}}(\gamma_1, \gamma_2)}{(1+n)^{\gamma_2-\gamma_1+\min(1/8, \gamma_1-1/8)}} \|\mathbf{h}\|_{\ell_{\gamma_2}^{\infty}}, \quad \text{for } \mathbf{h} \in \ell_{\gamma_2}^{\infty}(\mathbb{Z}; \mathbb{R}). \quad (4.38c) $$

### 4.5.1 Proof of the estimates (4.37a) and (4.37b) on the semigroup $(\mathcal{L}^n)_{n \in \mathbb{N}}$

Before proceeding with the proof of the estimates (4.37a) and (4.37b) on the semigroup $(\mathcal{L}^n)_{n \in \mathbb{N}}$ of Theorem 4.3, let us comment on the strategy that we shall follow. For $\gamma \ge 0$ and $\mathbf{h} \in \ell_{\gamma}^q(\mathbb{Z}; \mathbb{R})$ we recall that the action of the semigroup $\mathcal{L}^n$ on $\mathbf{h}$ is given for each $n \in \mathbb{N}$ by

$$ \forall j \in \mathbb{Z}, \quad (\mathcal{L}^n \mathbf{h})_j = \sum_{j_0 \in \mathbb{Z}} \mathcal{G}^n(j, j_0) h_{j_0}, $$

where $\mathcal{G}^n(\cdot, j_0)$ is the temporal Green's function solution of (4.1). The bounds (4.37a)-(4.37b) are trivial for $n=0$ since we have $\gamma_1 \le \gamma_2$ and therefore $\|\mathbf{h}\|_{\ell_{\gamma_1}^1} \le \|\mathbf{h}\|_{\ell_{\gamma_2}^1}$ so we assume from now on $n \in \mathbb{N}^*$. Motivated by the estimates obtained on the temporal Green's function in Theorem 4.1 (for $n \in \mathbb{N}^*$), we introduce a family of operators $(\mathcal{L}_{\mathrm{act}}^n)_{n \in \mathbb{N}^*}$ acting on a given sequence $\mathbf{h} \in \ell_{\gamma}^q(\mathbb{Z}; \mathbb{R})$ as follows:

$$ \forall (n, j) \in \mathbb{N}^* \times \mathbb{Z}, \quad (\mathcal{L}_{\mathrm{act}}^n \mathbf{h})_j := \mathcal{H}_j \sum_{j_0 \ge 1} \mathbf{1}_{|j-j_0| \le n} \mathbf{A}_r(-j_0 + n|\alpha_r|, n) h_{j_0} \\ + \mathcal{H}_j \sum_{j_0 \le 0} \mathbf{1}_{|j-j_0| \le n} \mathbf{A}_\ell(j_0 + n\alpha_\ell, n) h_{j_0}. $$

It is important to observe that $\mathcal{L}_{\mathrm{act}}^n$ does not stand for the $n$-th power of $\mathcal{L}_{\mathrm{act}}^1$, while $\mathcal{L}^n$ is the $n$-th power of $\mathcal{L}$. However, we hope that this will not create any confusion for the reader and we keep the notation as such to bear in mind that all operators are considered at the discrete time $n$.

We recall that the Green's function satisfies $\mathcal{G}^n(j, j_0) = 0$ for $|j - j_0| > n$. With the previous definition for $\mathcal{L}_{\mathrm{act}}^n$, for all $(n, j) \in \mathbb{N}^* \times \mathbb{Z}$, one can therefore decompose

$$ (\mathcal{L}^n \mathbf{h})_j = (\mathcal{L}_{\mathrm{act}}^n \mathbf{h})_j + \sum_{j_0 \ge 1} (\mathcal{G}^n(j, j_0) - \mathbf{1}_{|j-j_0| \le n} \mathcal{H}_j \mathbf{A}_r(-j_0 + n|\alpha_r|, n)) h_{j_0} \\ + \sum_{j_0 \le 0} (\mathcal{G}^n(j, j_0) - \mathbf{1}_{|j-j_0| \le n} \mathcal{H}_j \mathbf{A}_\ell(j_0 + n\alpha_\ell, n)) h_{j_0}. \quad (4.39) $$

We now introduce some notation. For $(n, j) \in \mathbb{N}^* \times \mathbb{Z}$ and $\mathbf{h} \in \ell_{\gamma_2}^1(\mathbb{Z}; \mathbb{R})$, we define:

$$ \mathcal{G}_{r,1}(n, j, \mathbf{h}) := \mathbf{1}_{j \ge 1} \sum_{j_0 \ge 1} \mathbf{1}_{|j-j_0| \le n} \mathbf{M}_r(c, j-j_0+n|\alpha_r|, n) |h_{j_0}|, $$

$$ \mathcal{G}_{\ell,1}(n, j, \mathbf{h}) := \mathbf{1}_{j \le 0} \sum_{j_0 \le 0} \mathbf{1}_{|j-j_0| \le n} \mathbf{M}_{\ell}(c, j_0 - j + n \alpha_{\ell}, n) |h_{j_0}|, $$
and

$$
\mathfrak{S}_{r,2}(n, j, \mathbf{h}) := e^{-c|j|} \sum_{j_0 \ge 1} \mathbf{M}_r (c, -j_0 + n |\alpha_r|, n) |h_{j_0}|,
$$

$$
\mathfrak{S}_{\ell,2}(n, j, \mathbf{h}) := e^{-c|j|} \sum_{j_0 \le 0} \mathbf{M}_{\ell} (c, j_0 + n \alpha_{\ell}, n) |h_{j_0}|,
$$

where we recall that the functions $\mathbf{M}_\ell$ and $\mathbf{M}_r$ are defined in (4.6). We also define:

$$
\mathfrak{S}_{\exp}(n, j, \mathbf{h}) := e^{-cn-c|j|} \sum_{j_0 \in \mathbb{Z}} e^{-c|j_0|} |h_{j_0}|.
$$

Using the estimates derived in Theorem 4.1 and applying the triangle inequality, we remark that for
some suitable positive constants $C$ and $c$ that do not depend on $n$, $j$ nor $\mathbf{h}$, we have

$$
\left| \sum_{j_0 \ge 1} \left( \mathcal{G}^n(j, j_0) - \mathbb{1}_{|j-j_0| \le n} \mathcal{H}_j \mathbf{A}_r(-j_0 + n |\alpha_r|, n) \right) h_{j_0} \right| \\
\le C \mathfrak{S}_{r,1}(n, j, \mathbf{h}) + C \mathfrak{S}_{r,2}(n, j, \mathbf{h}) + C \mathfrak{S}_{\exp}(n, j, \mathbf{h}),
$$

together with

$$
\left| \sum_{j_0 \le 0} \left( \mathcal{G}^n(j, j_0) - \mathbb{1}_{|j-j_0| \le n} \mathcal{H}_j \mathbf{A}_\ell(j_0 + n \alpha_\ell, n) \right) h_{j_0} \right| \\
\le C \mathfrak{S}_{\ell,1}(n, j, \mathbf{h}) + C \mathfrak{S}_{\ell,2}(n, j, \mathbf{h}) + C \mathfrak{S}_{\exp}(n, j, \mathbf{h}).
$$

Combining those two inequalities with the triangle inequality in (4.39), we have for all $(n, j) \in \mathbb{N}^* \times \mathbb{Z}$:

$$
\left| (\mathcal{L}^n \mathbf{h})_j \right| \le \left| (\mathcal{L}_{\text{act}}^n \mathbf{h})_j \right| + C \left( \mathfrak{S}_{r,1}(n,j,\mathbf{h}) + \mathfrak{S}_{\ell,1}(n,j,\mathbf{h}) + \mathfrak{S}_{r,2}(n,j,\mathbf{h}) + \mathfrak{S}_{\ell,2}(n,j,\mathbf{h}) + \mathfrak{S}_{\exp}(n,j,\mathbf{h}) \right). \quad (4.40)
$$

In order to prove the estimates (4.37a) and (4.37b), one simply needs to prove estimates for each of the
terms appearing in the right-hand side of the above inequality. From now on, we shall proceed term by
term.

We start by estimating $(\mathfrak{S}_{r,1}(n, j, \mathbf{h}))_{j \in \mathbb{Z}}$ and $(\mathfrak{S}_{\ell,1}(n, j, \mathbf{h}))_{j \in \mathbb{Z}}$.

**Lemma 4.9.** For any $\gamma_2 \ge \gamma_1 \ge 0$, there exists $C > 0$, such that for all $n \ge 1$ and $\mathbf{h} \in \ell_{\gamma_2}^1(\mathbb{Z}; \mathbb{R})$, one has:

$$
\left\| (\mathfrak{S}_{r,1}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^1} + \left\| (\mathfrak{S}_{\ell,1}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^1} \le \frac{C}{n^{\gamma_2 - \gamma_1 - 1/8}} \|\mathbf{h}\|_{\ell_{\gamma_2}^1},
$$

$$
\left\| (\mathfrak{S}_{r,1}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} + \left\| (\mathfrak{S}_{\ell,1}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} \le \frac{C}{n^{\gamma_2 - \gamma_1 + 1/3}} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}.
$$

*Proof.* We only prove the estimates for $(\mathfrak{S}_{r,1}(n, j, \mathbf{h}))_{j \in \mathbb{Z}}$, the other case is handled similarly. For $\mathbf{h} \in$
$\ell_{\gamma_2}^1(\mathbb{Z}; \mathbb{R})$ and $n \in \mathbb{N}^*$, we have that

$$
\begin{aligned}
\|(\mathfrak{S}_{r,1}(n,j,\mathbf{h}))_{j \in \mathbb{Z}}\|_{\ell_{\gamma_1}^1} &= \sum_{j \in \mathbb{Z}} (1 + |j|^{\gamma_1}) \mathbf{1}_{j \ge 1} \left( \sum_{j_0 \ge 1} \mathbf{1}_{|j-j_0| \le n} \mathbf{M}_r (c, j-j_0+n|\alpha_r|, n) |h_{j_0}| \right) \\
&= \underbrace{\sum_{j \ge 1} (1 + |j|^{\gamma_1}) \left( \sum_{j_0=1}^j \mathbf{1}_{|j-j_0| \le n} \mathbf{M}_r (c, j-j_0+n|\alpha_r|, n) |h_{j_0}| \right)}_{=: \mathcal{I}_1^n} \\
&\quad + \underbrace{\sum_{j \ge 1} (1 + |j|^{\gamma_1}) \left( \sum_{j_0 \ge j+1} \mathbf{1}_{|j-j_0| \le n} \mathbf{M}_r (c, j-j_0+n|\alpha_r|, n) |h_{j_0}| \right)}_{=: \mathcal{I}_2^n}.
\end{aligned}
$$

From the definition (4.6b) of $\mathbf{M}_r$, we note that for $1 \le j_0 \le j$ with $|j - j_0| \le n$, one has

$$ \mathbf{M}_r (c, j - j_0 + n |\alpha_r|, n) \le C e^{-cn}. $$

Furthermore, Peetre's inequality implies that for $|j - j_0| \le n$, there holds:

$$ (1 + |j|^{\gamma_1}) \le C (1 + |j_0|^{\gamma_1}) (1 + n^{\gamma_1}). $$

We thus obtain the following estimate for the first contribution $\mathcal{I}_1^n$:

$$
\begin{aligned}
\mathcal{I}_1^n &\le C (1 + n^{\gamma_1}) e^{-cn} \sum_{j \ge 1} \sum_{j_0=1}^j \mathbf{1}_{|j-j_0| \le n} (1 + |j_0|)^{\gamma_1} |h_{j_0}| \\
&= C (1 + n^{\gamma_1}) e^{-cn} \sum_{j_0 \ge 1} \left( \sum_{j \ge j_0} \mathbf{1}_{|j-j_0| \le n} \right) (1 + |j_0|^{\gamma_1}) |h_{j_0}| \\
&\le C n (1 + n^{\gamma_1}) e^{-cn} \|\mathbf{h}\|_{\ell_{\gamma_1}^1} \le C e^{-cn} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}.
\end{aligned}
$$

On the other hand, for the second contribution $\mathcal{I}_2^n$, we decompose the sum into two parts:

$$
\begin{aligned}
\mathcal{I}_2^n = & \sum_{j \ge 1} (1 + |j|^{\gamma_1}) \left( \sum_{j_0 \ge j+1} \mathbf{1}_{j_0 < \frac{n|\alpha_r|}{2}} \mathbf{1}_{|j-j_0| \le n} \mathbf{M}_r (c, j-j_0+n|\alpha_r|, n) |h_{j_0}| \right) \\
& + \sum_{j \ge 1} (1 + |j|^{\gamma_1}) \left( \sum_{j_0 \ge j+1} \mathbf{1}_{j_0 \ge \frac{n|\alpha_r|}{2}} \mathbf{1}_{|j-j_0| \le n} \mathbf{M}_r (c, j-j_0+n|\alpha_r|, n) |h_{j_0}| \right).
\end{aligned}
$$

In both sums, since we have $1 \le j \le j_0$, we can always use the inequality:

$$ (1 + |j|^{\gamma_1}) \le (1 + |j_0|^{\gamma_1}). $$

Furthermore, we notice once again that for $j \ge 1$ and $j_0 \ge j+1$ with $1 \le j_0 < \frac{n|\alpha_r|}{2}$, one has an exponential bound:

$$ \mathbf{M}_r (c, j - j_0 + n |\alpha_r|, n) \le C e^{-cn}. $$
As a consequence, one gets

$$
\begin{align*}
\sum_{j \ge 1} (1 + |j|^{\gamma_1}) \left( \sum_{j_0 \ge j+1} \mathbf{1}_{j_0 < \frac{n|\alpha_r|}{2}} \mathbf{1}_{|j-j_0| \le n} \mathbf{M}_r(c, j-j_0+n|\alpha_r|, n) |h_{j_0}| \right) & \\
&\le C e^{-cn} \sum_{j \ge 1} \sum_{j_0 \ge j+1} \mathbf{1}_{j_0 < \frac{n|\alpha_r|}{2}} \mathbf{1}_{|j-j_0| \le n} (1+|j_0|^{\gamma_1}) |h_{j_0}| & \\
&\le C e^{-cn} \sum_{j_0 \in \mathbb{Z}} \left( \sum_{j=1}^{j_0} \mathbf{1}_{|j-j_0| \le n} \right) (1+|j_0|^{\gamma_1}) |h_{j_0}| \le C n e^{-cn} \|\mathbf{h}\|_{\ell_{\gamma_1}^1} & \le C e^{-cn} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}. &
\end{align*}
$$

For the remaining contribution, we have

$$
\begin{align*}
\sum_{j \ge 1} (1 + |j|^{\gamma_1}) \left( \sum_{j_0 \ge j+1} \mathbf{1}_{j_0 \ge \frac{n|\alpha_r|}{2}} \mathbf{1}_{|j-j_0| \le n} \mathbf{M}_r(c, j-j_0+n|\alpha_r|, n) |h_{j_0}| \right) & \\
&\le \sum_{j_0 \ge 1} \mathbf{1}_{j_0 \ge \frac{n|\alpha_r|}{2}} \left( \sum_{j=1}^{j_0} \mathbf{M}_r(c, j-j_0+n|\alpha_r|, n) \right) (1+|j_0|^{\gamma_1}) |h_{j_0}| & \\
&\le C \|\mathbf{M}_r(c, \cdot + n|\alpha_r|, n)\|_{\ell^1(\mathbb{Z})} \sum_{j_0 \ge 1} \mathbf{1}_{j_0 \ge \frac{n|\alpha_r|}{2}} \frac{(1+|j_0|^{\gamma_2})}{(1+|j_0|^{\gamma_2-\gamma_1})} |h_{j_0}| & \\
&\le \frac{C}{n^{\gamma_2-\gamma_1}} \|\mathbf{M}_r(c, \cdot + n|\alpha_r|, n)\|_{\ell^1(\mathbb{Z})} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}. &
\end{align*}
$$

In order to conclude, we just use the following estimate that can be obtained from the definition (4.6b)
and a mere comparison between a series and an integral:

$$
\forall n \in \mathbb{N}^*, \quad \left\| (\mathbf{M}_r(c, j + n |\alpha_r|, n))_{j \in \mathbb{Z}} \right\|_{\ell^1(\mathbb{Z})} \le C n^{1/8}, \quad (4.41)
$$

where, of course, all constants are uniform with respect to $n \in \mathbb{N}^*$. The proof of the $\ell_{\gamma_1}^1$ estimate for
$\mathfrak{S}_{r,1}(n, \cdot, \mathbf{h})$ is now complete.

We now turn our attention to the second estimate in $\ell_{\gamma_1}^\infty$. By definition of the norm in $\ell_{\gamma_1}^\infty$, we have

$$
\begin{align*}
\| (\mathfrak{S}_{r,1}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \|_{\ell_{\gamma_1}^{\infty}}
&= \sup_{j \in \mathbb{Z}} (1 + |j|^{\gamma_1}) \mathbf{1}_{j \ge 1} \left( \sum_{j_0 \ge 1} \mathbf{1}_{|j-j_0| \le n} \mathbf{M}_r(c, j-j_0+n|\alpha_r|, n) |h_{j_0}| \right) \\
&\le \sup_{j \ge 1} (1 + |j|^{\gamma_1}) \left( \sum_{j_0=1}^{j} \mathbf{1}_{|j-j_0| \le n} \mathbf{M}_r(c, j-j_0+n|\alpha_r|, n) |h_{j_0}| \right) \\
&\quad + \sup_{j \ge 1} (1 + |j|^{\gamma_1}) \left( \sum_{j_0 \ge j+1} \mathbf{1}_{j_0 < \frac{n|\alpha_r|}{2}} \mathbf{1}_{|j-j_0| \le n} \mathbf{M}_r(c, j-j_0+n|\alpha_r|, n) |h_{j_0}| \right) \\
&\quad + \sup_{j \ge 1} (1 + |j|^{\gamma_1}) \left( \sum_{j_0 \ge j+1} \mathbf{1}_{j_0 \ge \frac{n|\alpha_r|}{2}} \mathbf{1}_{|j-j_0| \le n} \mathbf{M}_r(c, j-j_0+n|\alpha_r|, n) |h_{j_0}| \right) \\
&=: \mathcal{J}_1^n + \mathcal{J}_2^n + \mathcal{J}_3^n.
\end{align*}
$$
Performing similar computations and estimates as above, one gets first:

$$ \mathcal{J}_1^n + \mathcal{J}_2^n \leq C e^{-cn} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}, $$

which is an even better estimate than the one we are aiming at. On the other hand, we have

$$ \begin{aligned} \mathcal{J}_3^n &\leq C \|\mathbf{M}_r(c, \cdot + n |\alpha_r|, n)\|_{\ell^\infty(\mathbb{Z})} \sum_{j_0 \geq \frac{n|\alpha_r|}{2}} (1 + |j_0|^{\gamma_1}) |h_{j_0}| \\ &\leq \frac{C}{n^{\gamma_2 - \gamma_1}} \|\mathbf{M}_r(c, \cdot + n |\alpha_r|, n)\|_{\ell^\infty(\mathbb{Z})} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}, \end{aligned} $$

and it simply remains to use the property:

$$ \left\| (\mathbf{M}_r(c, j + n |\alpha_r|, n))_{j \in \mathbb{Z}} \right\|_{\ell^\infty(\mathbb{Z})} \leq \frac{1}{n^{1/3}}, \quad (4.42) $$

that can be easily deduced from the definition (4.6b). This concludes the proof of Lemma 4.9. $\square$

Next, we estimate $(\mathfrak{S}_{r,2}(n, j, \mathbf{h}))_{j \in \mathbb{Z}}$ and $(\mathfrak{S}_{\ell,2}(n, j, \mathbf{h}))_{j \in \mathbb{Z}}$.

**Lemma 4.10.** For any $\gamma_2 \geq \gamma_1 \geq 0$, there exists a constant $C > 0$ such that for all $n \geq 1$ and $\mathbf{h} \in \ell_{\gamma_2}^1(\mathbb{Z}; \mathbb{R})$, one has:

$$ \left\| (\mathfrak{S}_{r,2}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^1} + \left\| (\mathfrak{S}_{\ell,2}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^\infty} \leq \frac{C}{n^{\gamma_2+1/3}} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}, $$

and therefore:

$$ \left\| (\mathfrak{S}_{r,2}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^\infty} + \left\| (\mathfrak{S}_{\ell,2}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^\infty} \leq \frac{C}{n^{\gamma_2+1/3}} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}. $$

*Proof.* For $\mathbf{h} \in \ell_{\gamma_2}^1(\mathbb{Z}; \mathbb{R})$, we have that

$$ \begin{aligned} \left\| (\mathfrak{S}_{r,2}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^1} &= \left( \sum_{j \in \mathbb{Z}} (1 + |j|^{\gamma_1}) e^{-c|j|} \right) \left( \sum_{j_0 \geq 1} \mathbf{M}_r(c, -j_0 + n |\alpha_r|, n) |h_{j_0}| \right) \\ &\leq C \sum_{j_0=1}^{\frac{n|\alpha_r|}{2}} \mathbf{M}_r(c, -j_0 + n |\alpha_r|, n) |h_{j_0}| + C \sum_{j_0 \geq \frac{n|\alpha_r|}{2}} \mathbf{M}_r(c, -j_0 + n |\alpha_r|, n) |h_{j_0}|. \end{aligned} $$

We then either use the exponential bound given by the definition (4.6b) or the global bound (4.42), which yields:

$$ \|\mathfrak{S}_{r,2}(n, \cdot, \mathbf{h})\|_{\ell_{\gamma_1}^1} \leq C e^{-cn} \|\mathbf{h}\|_{\ell^1} + \frac{C}{n^{1/3}} \sum_{j_0 \geq \frac{n|\alpha_r|}{2}} \frac{(1 + |j_0|^{\gamma_2})}{n^{\gamma_2}} |h_{j_0}| \leq \frac{C}{n^{\gamma_2+1/3}} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}. $$

The $\ell_{\gamma_1}^\infty$ estimate is straightforward by just observing that the $\ell_{\gamma_1}^\infty$ norm is always smaller than the $\ell_{\gamma_1}^1$ norm. The proof of Lemma 4.10 is thus complete. $\square$

We now handle the most delicate estimates on the family of operators $(\mathcal{L}_{\text{act}}^n)_{n \in \mathbb{N}^*}$.
**Lemma 4.11.** For any $\gamma_2 \ge \gamma_1 \ge 0$, there exists a constant $C > 0$ such that the family of operators $(\mathcal{L}_{\text{act}}^n)_{n \in \mathbb{N}^*}$ satisfies the following estimate:

$$ \forall n \in \mathbb{N}^*, \quad \|\mathcal{L}_{\text{act}}^n \mathbf{h}\|_{\ell_{\gamma_1}^1} \le \frac{C}{n^{\gamma_2}} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}, \quad \text{for any } \mathbf{h} \in \ell_{\gamma_2}^1(\mathbb{Z}; \mathbb{R}) \text{ with } \sum_{j \in \mathbb{Z}} h_j = 0, $$

and therefore:

$$ \forall n \in \mathbb{N}^*, \quad \|\mathcal{L}_{\text{act}}^n \mathbf{h}\|_{\ell_{\gamma_1}^\infty} \le \frac{C}{n^{\gamma_2}} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}, \quad \text{for any } \mathbf{h} \in \ell_{\gamma_2}^1(\mathbb{Z}; \mathbb{R}) \text{ with } \sum_{j \in \mathbb{Z}} h_j = 0. $$

*Proof.* We consider first $\mathbf{h} \in \ell_{\gamma_2}^1(\mathbb{Z}; \mathbb{R})$ without making any assumption on its mass. The first step of the proof consists in writing:

$$ \begin{aligned} \sum_{j_0 \ge 1} \mathbf{1}_{|j-j_0| \le n} \mathbf{A}_r(-j_0 + n |\alpha_r|, n) h_{j_0} &+ \sum_{j_0 \le 0} \mathbf{1}_{|j-j_0| \le n} \mathbf{A}_\ell(j_0 + n \alpha_\ell, n) h_{j_0} \\ & = \sum_{j_0 \ge 1} \mathbf{A}_r(-j_0 + n |\alpha_r|, n) h_{j_0} + \sum_{j_0 \le 0} \mathbf{A}_\ell(j_0 + n \alpha_\ell, n) h_{j_0} \tag{4.43} \\ & \quad - \sum_{j_0 \ge 1} \mathbf{1}_{|j-j_0| > n} \mathbf{A}_r(-j_0 + n |\alpha_r|, n) h_{j_0} - \sum_{j_0 \le 0} \mathbf{1}_{|j-j_0| > n} \mathbf{A}_\ell(j_0 + n \alpha_\ell, n) h_{j_0}, \end{aligned} $$

where the two infinite sums on the right-hand side converge since $\mathbf{A}_r(-j_0+n|\alpha_r|, n)$ is uniformly bounded (see Corollary A.6 in Appendix A) and $\mathbf{h}$ is integrable. Since $|\alpha_r| \in (0, 1)$, we can take $\beta \in (|\alpha_r|, 1)$. As a consequence, we can use the exponential decay of $(\mathcal{H}_j)_{j \in \mathbb{Z}}$ and further decompose:

$$ \left| \mathcal{H}_j \sum_{j_0 \ge 1} \mathbf{1}_{|j-j_0| > n} \mathbf{A}_r(-j_0 + n |\alpha_r|, n) h_{j_0} \right| \le C \sum_{j_0 \ge 1} \mathbf{1}_{|j-j_0| > n} |\mathbf{A}_r(-j_0 + n |\alpha_r|, n)| |h_{j_0}| e^{-c|j|} \\ = C \sum_{j_0=1}^{n\beta} \mathbf{1}_{|j-j_0| > n} |\mathbf{A}_r(-j_0 + n |\alpha_r|, n)| |h_{j_0}| e^{-c|j|} \\ + C \sum_{j_0 > n\beta} \mathbf{1}_{|j-j_0| > n} |\mathbf{A}_r(-j_0 + n |\alpha_r|, n)| |h_{j_0}| e^{-c|j|}. $$

In the first sum, since $n < |j - j_0|$ and $1 \le j_0 \le n\beta$, we get that $(1 - \beta)n < |j|$, and thus

$$ \sum_{j_0=1}^{n\beta} \mathbf{1}_{|j-j_0| > n} |\mathbf{A}_r(-j_0 + n |\alpha_r|, n)| |h_{j_0}| e^{-c|j|} \le \sum_{j_0=1}^{n\beta} \mathbf{1}_{|j-j_0| > n} |\mathbf{A}_r(-j_0 + n |\alpha_r|, n)| |h_{j_0}| e^{-\frac{c}{2}|j|} e^{-\frac{c(1-\beta)}{2}n} \\ \le C n e^{-\frac{c}{2}(1-\beta)n} e^{-\frac{c}{2}|j|} \|\mathbf{h}\|_{\ell^\infty(\mathbb{Z})}, $$

where we have used the property from Corollary A.6 that the quantity $|\mathbf{A}_r(-j_0 + n |\alpha_r|, n)|$ is uniformly bounded and we have also used the fact that the sum with respect to $j_0$ gathers at most $n$ terms (since $\beta \ge 1$). We now consider the sum with respect to $j_0 > n\beta$. Still using Corollary A.6 in Appendix A, but this time for $j_0 > n\beta$ with $\beta \in (|\alpha_r|, 1)$, we have that

$$ |\mathbf{A}_r(-j_0 + n |\alpha_r|, n)| \le C \exp \left( -c \frac{|j_0 - n |\alpha_r||^{4/3}}{n^{1/3}} \right), $$
and thus

$$ \sum_{j_0 > n\beta} |\mathbf{A}_r(-j_0 + n|\alpha_r|, n)| \le C e^{-cn}. $$

As a consequence, we have

$$ \sum_{j_0 > n\beta} \mathbb{1}_{|j-j_0| > n} |\mathbf{A}_r(-j_0 + n|\alpha_r|, n)| |h_{j_0}| e^{-c|j|} \le C e^{-cn-c|j|} \|\mathbf{h}\|_{\ell^{\infty}(\mathbb{Z})}. $$

Summing up, we have proved the following estimate

$$ \sum_{j \in \mathbb{Z}} (1 + |j|)^{\gamma_1} \left| \mathcal{H}_j \sum_{j_0 \ge 1} \mathbb{1}_{|j-j_0| > n} \mathbf{A}_r(-j_0 + n|\alpha_r|, n) h_{j_0} \right| \le C e^{-cn} \|\mathbf{h}\|_{\ell^{\infty}(\mathbb{Z})} \le C e^{-cn} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}, $$

and similarly, we also have

$$ \sum_{j \in \mathbb{Z}} (1 + |j|)^{\gamma_1} \left| \mathcal{H}_j \sum_{j_0 \le 0} \mathbb{1}_{|j-j_0| > n} \mathbf{A}_\ell(j_0 + n \alpha_\ell, n) h_{j_0} \right| \le C e^{-cn} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}. $$

We now go back to the decomposition (4.43) and consider the two series in the right-hand side. We assume from now on that the sequence $\mathbf{h}$ has zero mass, that is:

$$ \sum_{j_0 \in \mathbb{Z}} h_{j_0} = 0. $$

We can therefore write:

$$ \sum_{j_0 \ge 1} \mathbf{A}_r(-j_0 + n|\alpha_r|, n) h_{j_0} + \sum_{j_0 \le 0} \mathbf{A}_\ell(j_0 + n\alpha_\ell, n) h_{j_0} \\ = \sum_{j_0 \ge 1} (\mathbf{A}_r(-j_0 + n|\alpha_r|, n) - 1) h_{j_0} + \sum_{j_0 \le 0} (\mathbf{A}_\ell(j_0 + n\alpha_\ell, n) - 1) h_{j_0}, $$

and we shall study each sum separately. For the first contribution, we further split the sum into two parts:

$$ \sum_{j_0 \ge 1} (\mathbf{A}_r(-j_0 + n|\alpha_r|, n) - 1) h_{j_0} = \sum_{j_0=1}^{\frac{n|\alpha_r|}{2}} (\mathbf{A}_r(-j_0 + n|\alpha_r|, n) - 1) h_{j_0} \\ + \sum_{j_0 > \frac{n|\alpha_r|}{2}} (\mathbf{A}_r(-j_0 + n|\alpha_r|, n) - 1) h_{j_0}. $$

Once again, we will rely on Corollary A.6 in Appendix A. On the one hand, we use the fact for $1 \le j_0 \le \frac{n|\alpha_r|}{2}$, one has an exponential bound:

$$ |\mathbf{A}_r(-j_0 + n|\alpha_r|, n) - 1| \le C e^{-cn}, $$
and on the other hand, that the factor $\mathbf{A}_r(-j_0 + n|\alpha_r|, n)$ is uniformly bounded with respect to $j_0 \in \mathbb{Z}$ and $n \in \mathbb{N}^*$. As a consequence, we get

$$ \left| \sum_{j_0 \ge 1} (\mathbf{A}_r(-j_0 + n|\alpha_r|, n) - 1) h_{j_0} \right| \le C e^{-cn} \|\mathbf{h}\|_{\ell^1} + C \sum_{j_0 > \frac{n|\alpha_r|}{2}} |h_{j_0}| \\ \le C e^{-cn} \|\mathbf{h}\|_{\ell^1} + C \sum_{j_0 > \frac{n|\alpha_r|}{2}} \frac{1 + |j_0|^{\gamma_2}}{1 + |(n|\alpha_r|/2)|^{\gamma_2}|} |h_{j_0}| \le \frac{C}{n^{\gamma_2}} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}. $$

And similarly, we also have

$$ \left| \sum_{j_0 \le 0} (\mathbf{A}_\ell(j_0 + n \alpha_\ell, n) - 1) h_{j_0} \right| \le \frac{C}{n^{\gamma_2}} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}. $$

Summing up, we have obtained

$$ \sum_{j \in \mathbb{Z}} (1 + |j|)^{\gamma_1} \left| \mathcal{H}_j \left( \sum_{j_0 \ge 1} \mathbf{A}_r(-j_0 + n|\alpha_r|, n) h_{j_0} + \sum_{j_0 \le 0} \mathbf{A}_\ell(j_0 + n \alpha_\ell, n) h_{j_0} \right) \right| \le \frac{C}{n^{\gamma_2}} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}. $$

Going back to the decomposition (4.43), this concludes the proof of Lemma 4.11. $\square$

The very last contribution to handle is the one coming from the exponential terms $(\mathcal{S}_{\exp}(n, j, \mathbf{h}))_{j \in \mathbb{Z}}$, and we have the following result whose proof is trivial.

**Lemma 4.12.** For any $\gamma_2 \ge \gamma_1 \ge 0$, there exists a constant $C > 0$ such that for all $n \ge 1$ and $\mathbf{h} \in \ell_{\gamma_2}^1(\mathbb{Z}; \mathbb{R})$, one has

$$ \| (\mathcal{S}_{\exp}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \|_{\ell_{\gamma_1}^1} \le C e^{-cn} \| \mathbf{h} \|_{\ell_{\gamma_2}^1}, $$

and therefore:

$$ \| (\mathcal{S}_{\exp}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \|_{\ell_{\gamma_1}^{\infty}} \le C e^{-cn} \| \mathbf{h} \|_{\ell_{\gamma_2}^1}. $$

Going back to the inequality (4.40), the proof of the estimates (4.37a) and (4.37b) on the semigroup $(\mathcal{L}^n)_{n \in \mathbb{N}}$ of Theorem 4.3 is then a direct consequence of Lemma 4.9, Lemma 4.10, Lemma 4.11 and Lemma 4.12. Indeed, for proving (4.37a), we consider $\mathbf{h} \in \ell_{\gamma_2}^1(\mathbb{Z}; \mathbb{R})$ with mass zero and combine those four preliminary results to obtain:

$$ \| \mathcal{L}^n \mathbf{h} \|_{\ell_{\gamma_1}^1} \le C \left( \frac{1}{n^{\gamma_2}} + \frac{1}{n^{\gamma_2 - \gamma_1 - 1/8}} + \frac{1}{n^{\gamma_2 + 1/3}} + e^{-cn} \right) \| \mathbf{h} \|_{\ell_{\gamma_2}^1} \le \frac{C}{n^{\gamma_2 - \gamma_1 - 1/8}} \| \mathbf{h} \|_{\ell_{\gamma_2}^1}, $$

and the proof of (4.37b) is entirely similar.

### 4.5.2 Proof of the estimates (4.38a), (4.38b) and (4.38c) on the family $(\mathcal{L}^n(\mathrm{Id} - \mathbf{S}))_{n \in \mathbb{N}}$

The starting point of the proof of the estimates (4.38a), (4.38b) and (4.38c) of Theorem 4.3 follows similar lines as in the previous subsection for proving (4.37a) and (4.37b) on the semigroup $(\mathcal{L}^n)_{n \in \mathbb{N}}$. For $\gamma \ge 0$
and $\mathbf{h} \in \ell_{\gamma}^{q}(\mathbb{Z}; \mathbb{R})$ ($q = 1$ or $q = +\infty$) we recall that the action of the family of operators $(\mathcal{L}^n(\mathrm{Id} - \mathbf{S}))_{n \in \mathbb{N}}$ on $\mathbf{h}$ is given for each $n \in \mathbb{N}$ by:

$$ \forall (n, j) \in \mathbb{N} \times \mathbb{Z}, \quad (\mathcal{L}^n(\mathrm{Id} - \mathbf{S})\mathbf{h})_j = \sum_{j_0 \in \mathbb{Z}} (\mathcal{G}^n(j, j_0) - \mathcal{G}^n(j, j_0 - 1)) h_{j_0} = \sum_{j_0 \in \mathbb{Z}} \mathcal{D}^n(j, j_0) h_{j_0}, $$

by definition (4.31) of $\mathcal{D}^n(j, j_0)$. Restricting to $n \ge 1$ (the case $n = 0$ is trivial) and using the estimates obtained on the derivative of the temporal Green's function in Theorem 4.2, we obtain that

$$ \begin{aligned} |(\mathcal{L}^n(\mathrm{Id} - \mathbf{S})\mathbf{h})_j| & \le C \sum_{j_0 \ge 1} \mathbf{1}_{|j-j_0| \le n+1} \left[ \mathbf{1}_{j \ge 1} \mathbf{K}_r(c, j-j_0+n|\alpha_r|, n) + e^{-c|j|} \mathbf{M}_r(c, -j_0+|\alpha_r|n, n) \right] |h_{j_0}| \\ & \quad + C \sum_{j_0 \le 0} \mathbf{1}_{|j-j_0| \le n+1} \left[ \mathbf{1}_{j \le 0} \mathbf{K}_\ell(c, j_0-j+n\alpha_\ell, n) + e^{-c|j|} \mathbf{M}_\ell(c, j_0+n\alpha_\ell, n) \right] |h_{j_0}| \\ & \quad + C e^{-cn} \sum_{j_0 \in \mathbb{Z}} \mathbf{1}_{|j-j_0| \le n+1} e^{-c|j-j_0|} |h_{j_0}|. \end{aligned} $$

This motivates the definition of the following quantities for $(n, j) \in \mathbb{N}^* \times \mathbb{Z}$:

$$ \mathfrak{D}_{r,1}(n, j, \mathbf{h}) := \mathbf{1}_{j \ge 1} \sum_{j_0 \ge 1} \mathbf{1}_{|j-j_0| \le n+1} \mathbf{K}_r(c, j-j_0+n|\alpha_r|, n) |h_{j_0}|, $$

$$ \mathfrak{D}_{\ell,1}(n, j, \mathbf{h}) := \mathbf{1}_{j \le 0} \sum_{j_0 \le 0} \mathbf{1}_{|j-j_0| \le n+1} \mathbf{K}_{\ell}(c, j_0 - j + n \alpha_{\ell}, n) |h_{j_0}|, $$

and

$$ \mathfrak{D}_{r,2}(n, j, \mathbf{h}) := e^{-c|j|} \sum_{j_0 \ge 1} \mathbf{M}_r(c, -j_0 + n|\alpha_r|, n) |h_{j_0}|, $$

$$ \mathfrak{D}_{\ell,2}(n, j, \mathbf{h}) := e^{-c|j|} \sum_{j_0 \le 0} \mathbf{M}_{\ell}(c, j_0 + n \alpha_{\ell}, n) |h_{j_0}|, $$

together with:

$$ \mathfrak{D}_{\exp}(n, j, \mathbf{h}) := e^{-cn} \sum_{j_0 \in \mathbb{Z}} \mathbf{1}_{|j-j_0| \le n+1} e^{-c|j-j_0|} |h_{j_0}|. $$

With these notations at hand, we readily obtain that for all $(n, j) \in \mathbb{N}^* \times \mathbb{Z}$, we have:

$$ |(\mathcal{L}^n(\mathrm{Id} - \mathbf{S})\mathbf{h})_j| \le C (\mathfrak{D}_{r,1}(n, j, \mathbf{h}) + \mathfrak{D}_{\ell,1}(n, j, \mathbf{h}) + \mathfrak{D}_{r,2}(n, j, \mathbf{h}) + \mathfrak{D}_{\ell,2}(n, j, \mathbf{h}) + \mathfrak{D}_{\exp}(n, j, \mathbf{h})) . $$

We will mainly focus our efforts on the terms $\mathfrak{D}_{r,1}(n, j, \mathbf{h})$, $\mathfrak{D}_{\ell,1}(n, j, \mathbf{h})$ and $\mathfrak{D}_{\exp}(n, j, \mathbf{h})$. Indeed, we note that $\mathfrak{D}_{r,2}(n, j, \mathbf{h})$ and $\mathfrak{D}_{\ell,2}(n, j, \mathbf{h})$ are identical to $\mathfrak{S}_{r,2}(n, j, \mathbf{h})$ and $\mathfrak{S}_{\ell,2}(n, j, \mathbf{h})$ in the previous subsection and thus enjoy the same estimates as the ones derived in Lemma 4.10. We shall need however an additional estimate to the one proved in Lemma 4.10 but the proof of it will be very similar to what has already been done. Let us start indeed with the estimates for $\mathfrak{D}_{r,2}(n, j, \mathbf{h})$ and $\mathfrak{D}_{\ell,2}(n, j, \mathbf{h})$.

**Lemma 4.13.** For any $\gamma_2 \ge \gamma_1 \ge 0$, there exists a constant $C > 0$ such that for all $n \ge 1$ and $\mathbf{h} \in \ell_{\gamma_2}^1(\mathbb{Z}; \mathbb{R})$, one has

$$ \left\| (\mathfrak{D}_{r,2}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^1} + \left\| (\mathfrak{D}_{\ell,2}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^1} \le \frac{C}{n^{\gamma_2+1/3}} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}, $$

$$ \left\| (\mathfrak{D}_{r,2}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} + \left\| (\mathfrak{D}_{\ell,2}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} \le \frac{C}{n^{\gamma_2+1/3}} \|\mathbf{h}\|_{\ell_{\gamma_2}^{\infty}} $$
and for $\mathbf{h} \in \ell_{\gamma_2}^{\infty}(\mathbb{Z}; \mathbb{R})$, one has

$$ \left\| (\mathfrak{D}_{r,2}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} + \left\| (\mathfrak{D}_{\ell,2}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} \le \frac{C}{n^{\gamma_2 - 1/8}} \|\mathbf{h}\|_{\ell_{\gamma_2}^{\infty}}. $$

*Proof.* The first two estimates in Lemma 4.13 have already been proved in Lemma 4.10 so we switch directly to the last estimate where the novelty is that now the sequence $\mathbf{h}$ is assumed to belong to the larger space $\ell_{\gamma_2}^{\infty}(\mathbb{Z}; \mathbb{R})$. We thus consider $n \in \mathbb{N}^*$ and $\mathbf{h} \in \ell_{\gamma_2}^{\infty}(\mathbb{Z}; \mathbb{R})$. From the definition of $\mathfrak{D}_{r,2}(n, j, \mathbf{h})$, we have

$$ \begin{aligned} \left\| (\mathfrak{D}_{r,2}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} &= \left( \sup_{j \in \mathbb{Z}} (1 + |j|^{\gamma_1}) e^{-c|j|} \right) \left( \sum_{j_0 \ge 1} \mathbf{M}_r(c, -j_0 + n |\alpha_r|, n) |h_{j_0}| \right) \\ &\le C \sum_{j_0=1}^{\frac{n|\alpha_r|}{2}} \mathbf{M}_r(c, -j_0 + n |\alpha_r|, n) |h_{j_0}| + C \sum_{j_0 \ge \frac{n|\alpha_r|}{2}} \mathbf{M}_r(c, -j_0 + n |\alpha_r|, n) |h_{j_0}| \\ &\le C \|\mathbf{h}\|_{\ell^{\infty}} \sum_{j_0=1}^{\frac{n|\alpha_r|}{2}} \mathbf{M}_r(c, -j_0 + n |\alpha_r|, n) + C \sum_{j_0 \ge \frac{n|\alpha_r|}{2}} \mathbf{M}_r(c, -j_0 + n |\alpha_r|, n) |h_{j_0}|. \end{aligned} $$

By summing the definition (4.6b) and comparing the corresponding series with an integral, we get the exponential estimate:

$$ \sum_{j_0=1}^{\frac{n|\alpha_r|}{2}} \mathbf{M}_r(c, -j_0 + n |\alpha_r|, n) \le C e^{-cn}. $$

For the second sum, we have:

$$ \sum_{j_0 \ge \frac{n|\alpha_r|}{2}} \mathbf{M}_r(c, -j_0 + n |\alpha_r|, n) |h_{j_0}| \le C \sum_{j_0 \ge \frac{n|\alpha_r|}{2}} \mathbf{M}_r(c, -j_0 + n |\alpha_r|, n) \frac{(1 + |j_0|^{\gamma_2})}{n^{\gamma_2}} |h_{j_0}| \\ + \frac{C}{n^{\gamma_2}} \|\mathbf{h}\|_{\ell_{\gamma_2}^{\infty}} \sum_{j_0 \in \mathbb{Z}} \mathbf{M}_r(c, -j_0 + n |\alpha_r|, n), $$

and we simply use (4.41) to conclude.

<figure><img src="image_18.png" /></figure>

Now, we handle the contributions from $\mathfrak{D}_{r,1}(n, j, \mathbf{h})$ and $\mathfrak{D}_{\ell,1}(n, j, \mathbf{h})$.

**Lemma 4.14.** For any $\gamma_2 \ge \gamma_1 \ge 0$, there exists $C > 0$, such that for all $n \ge 1$, one has

$$ \left\| (\mathfrak{D}_{r,1}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{1}} + \left\| (\mathfrak{D}_{\ell,1}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{1}} \le \frac{C}{n^{\gamma_2 - \gamma_1 + 1/8}} \|\mathbf{h}\|_{\ell_{\gamma_2}^{1}}, \text{ for } \mathbf{h} \in \ell_{\gamma_2}^{1}, $$

$$ \left\| (\mathfrak{D}_{r,1}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} + \left\| (\mathfrak{D}_{\ell,1}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} \le \frac{C}{n^{\gamma_2 - \gamma_1 + 7/12}} \|\mathbf{h}\|_{\ell_{\gamma_2}^{1}}, \text{ for } \mathbf{h} \in \ell_{\gamma_2}^{1}, $$

$$ \left\| (\mathfrak{D}_{r,1}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} + \left\| (\mathfrak{D}_{\ell,1}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} \le \frac{C}{n^{\gamma_2 - \gamma_1 + 1/8}} \|\mathbf{h}\|_{\ell_{\gamma_2}^{\infty}}, \text{ for } \mathbf{h} \in \ell_{\gamma_2}^{\infty}. $$
Proof. For $h \in \ell_{\gamma_2}^1$, we have that

$$
\begin{align*}
\left\| (\mathfrak{D}_{r,1}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^1}
&= \sum_{j \in \mathbb{Z}} (1 + |j|)^{\gamma_1} \mathbf{1}_{j \ge 1} \sum_{j_0 \ge 1} \mathbf{1}_{|j-j_0| \le n+1} \mathbf{K}_r(c, j-j_0+n|\alpha_r|, n) |h_{j_0}| \\
&= \sum_{j \in \mathbb{Z}} (1 + |j|)^{\gamma_1} \mathbf{1}_{j \ge 1} \sum_{j_0=1}^j \mathbf{1}_{|j-j_0| \le n+1} \mathbf{K}_r(c, j-j_0+n|\alpha_r|, n) |h_{j_0}| \\
&\quad + \sum_{j \in \mathbb{Z}} (1 + |j|)^{\gamma_1} \mathbf{1}_{j \ge 1} \sum_{j_0 \ge j+1} \mathbf{1}_{j_0 < \frac{n|\alpha_r|}{2}} \mathbf{1}_{|j-j_0| \le n+1} \mathbf{K}_r(c, j-j_0+n|\alpha_r|, n) |h_{j_0}| \\
&\quad + \sum_{j \in \mathbb{Z}} (1 + |j|)^{\gamma_1} \mathbf{1}_{j \ge 1} \sum_{j_0 \ge j+1} \mathbf{1}_{j_0 \ge \frac{n|\alpha_r|}{2}} \mathbf{1}_{|j-j_0| \le n+1} \mathbf{K}_r(c, j-j_0+n|\alpha_r|, n) |h_{j_0}|.
\end{align*}
$$

Now from the definition (4.32b) of $\mathbf{K}_r$, we infer the following facts

• for $1 \le j_0 \le j$ with $|j - j_0| \le n + 1$, one has $\mathbf{K}_r(c, j - j_0 + n |\alpha_r|, n) \le C e^{-cn}$,

• for $j \ge 1$ and $j_0 \ge j + 1$ with $1 \le j_0 < \frac{n|\alpha_r|}{2}$, one has $\mathbf{K}_r(c, j - j_0 + n |\alpha_r|, n) \le C e^{-cn}$.

As a consequence, one readily derives, as in the previous cases, that

$$
\begin{align*}
\sum_{j \in \mathbb{Z}} (1 + |j|)^{\gamma_1} \mathbf{1}_{j \ge 1} \sum_{j_0=1}^{j} \mathbf{1}_{|j-j_0| \le n+1} \mathbf{K}_r(c, j-j_0+n|\alpha_r|, n) |h_{j_0}| \\
\qquad + \sum_{j \in \mathbb{Z}} (1 + |j|)^{\gamma_1} \mathbf{1}_{j \ge 1} \sum_{j_0 \ge j+1} \mathbf{1}_{j_0 < \frac{n|\alpha_r|}{2}} \mathbf{1}_{|j-j_0| \le n+1} \mathbf{K}_r(c, j-j_0+n|\alpha_r|, n) |h_{j_0}| \\
\le C e^{-cn} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}.
\end{align*}
$$

For the last contribution, inverting the sums gives

$$
\begin{align*}
\sum_{j \in \mathbb{Z}} (1 + |j|)^{\gamma_1} \mathbf{1}_{j \ge 1} \sum_{j_0 \ge j+1} \mathbf{1}_{j_0 \ge \frac{n|\alpha_r|}{2}} \mathbf{1}_{|j-j_0| \le n+1} \mathbf{K}_r(c, j-j_0+n|\alpha_r|, n) |h_{j_0}| \\
\le C \left\| (\mathbf{K}_r(j+n|\alpha_r|, n))_{j \in \mathbb{Z}} \right\|_{\ell^1(\mathbb{Z})} \sum_{j_0 \ge 1} \mathbf{1}_{j_0 \ge \frac{n|\alpha_r|}{2}} (1+|j_0|)^{\gamma_1} |h_{j_0}| &\le \frac{C}{n^{\gamma_2-\gamma_1+1/8}} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}, \\
\text{since } \left\| (\mathbf{K}_r(j+n|\alpha_r|, n))_{j \in \mathbb{Z}} \right\|_{\ell^1(\mathbb{Z})} &\le C/n^{1/8}.
\end{align*}
$$

Finally, the proof of the second estimate is similar to the one for the estimate of $(\mathfrak{S}_{r,1}(n, j, \mathbf{h}))_{j \in \mathbb{Z}}$ and $(\mathfrak{S}_{\ell,1}(n, j, \mathbf{h}))_{j \in \mathbb{Z}}$ in Lemma 4.9, this time using that $\|(\mathbf{K}_r(j + n |\alpha_r|, n))_{j \in \mathbb{Z}}\|_{\ell^\infty(\mathbb{Z})} \le C/n^{7/12}$.

For the last estimate, we first note that, for $\mathbf{h} \in \ell_{\gamma_2}^{\infty}$, we have that

$$
\begin{align*}
\left\| (\mathfrak{D}_{r,1}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}}
&= \sup_{j \in \mathbb{Z}} (1 + |j|)^{\gamma_1} \mathbf{1}_{j \ge 1} \sum_{j_0 \ge 1} \mathbf{1}_{|j-j_0| \le n+1} \mathbf{K}_r(c, j-j_0+n|\alpha_r|, n) |h_{j_0}| \\
&= \sup_{j \in \mathbb{Z}} (1 + |j|)^{\gamma_1} \mathbf{1}_{j \ge 1} \sum_{j_0=1}^{j} \mathbf{1}_{|j-j_0| \le n+1} \mathbf{K}_r(c, j-j_0+n|\alpha_r|, n) |h_{j_0}| \\
&\quad + \sup_{j \in \mathbb{Z}} (1 + |j|)^{\gamma_1} \mathbf{1}_{j \ge 1} \sum_{j_0 \ge j+1} \mathbf{1}_{j_0 < \frac{n|\alpha_r|}{2}} \mathbf{1}_{|j-j_0| \le n+1} \mathbf{K}_r(c, j-j_0+n|\alpha_r|, n) |h_{j_0}| \\
&\quad + \sup_{j \in \mathbb{Z}} (1 + |j|)^{\gamma_1} \mathbf{1}_{j \ge 1} \sum_{j_0 \ge j+1} \mathbf{1}_{j_0 \ge \frac{n|\alpha_r|}{2}} \mathbf{1}_{|j-j_0| \le n+1} \mathbf{K}_r(c, j-j_0+n|\alpha_r|, n) |h_{j_0}|.
\end{align*}
$$
The first two terms contribute to

$$
\begin{align*}
\sup_{j \in \mathbb{Z}} (1 + |j|)^{\gamma_1} \mathbf{1}_{j \ge 1} \sum_{j_0=1}^{j} \mathbf{1}_{|j-j_0| \le n+1} \mathbf{K}_r(c, j-j_0+n|\alpha_r|, n) |h_{j_0}| & \\
& \quad + \sup_{j \in \mathbb{Z}} (1 + |j|)^{\gamma_1} \mathbf{1}_{j \ge 1} \sum_{j_0 \ge j+1} \mathbf{1}_{j_0 < \frac{n|\alpha_r|}{2}} \mathbf{1}_{|j-j_0| \le n+1} \mathbf{K}_r(c, j-j_0+n|\alpha_r|, n) |h_{j_0}| & \\
& \le C e^{-cn} \|\mathbf{h}\|_{\ell_{\gamma_2}^{\infty}}, &
\end{align*}
$$

while for the third term, we note that for each $j \ge 1$

$$
\begin{align*}
(1 + |j|)^{\gamma_1} \sum_{j_0 \ge j+1} \mathbf{1}_{j_0 \ge \frac{n|\alpha_r|}{2}} \mathbf{1}_{|j-j_0| \le n+1} \mathbf{K}_r(c, j-j_0+n|\alpha_r|, n) |h_{j_0}|
& \le C \sum_{j_0 \ge 1} \mathbf{1}_{j_0 \ge \frac{n|\alpha_r|}{2}} |\mathcal{H}_r^n(c, j-j_0)| \frac{(1+|j_0|)^{\gamma_2}}{(1+|j_0|)^{\gamma_2-\gamma_1}} |h_{j_0}| \\
& \le \frac{C}{n^{\gamma_2-\gamma_1+1/8}} \|\mathbf{h}\|_{\ell_{\gamma_2}^{\infty}},
\end{align*}
$$

$$
\text{since } \|(\mathbf{K}_r(j + n|\alpha_r|, n))_{j \in \mathbb{Z}}\|_{\ell^1(\mathbb{Z})} \le C/n^{1/8}.
$$

<figure><img src="image_19.png" /></figure>

The very last contribution to handle is the one coming from the exponential terms $(\mathfrak{D}_{\exp}(n, j, \mathbf{h}))_{j \in \mathbb{Z}}$.
And we have the following result whose proof is trivial and let to the interested reader.

**Lemma 4.15.** For any $\gamma_2 \ge \gamma_1 \ge 0$, there exists a constant $C > 0$ such that for all $n \ge 1$, one has:

$$
\|(\mathfrak{D}_{\exp}(n, j, \mathbf{h}))_{j \in \mathbb{Z}}\|_{\ell_{\gamma_1}^1} \le C e^{-cn} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}, \quad \text{for any } \mathbf{h} \in \ell_{\gamma_2}^1(\mathbb{Z}; \mathbb{R}),
$$

$$
\| (\mathfrak{D}_{\exp}(n, j, \mathbf{h}))_{j \in \mathbb{Z}} \|_{\ell_{\gamma_1}^{\infty}} \le C e^{-cn} \| \mathbf{h} \|_{\ell_{\gamma_2}^{\infty}}, \quad \text{for any } \mathbf{h} \in \ell_{\gamma_2}^{\infty}(\mathbb{Z}; \mathbb{R}).
$$

We can now conclude the proof of Theorem 4.3. For the first estimate (4.38a), we use the estimates
provided by Lemma 4.14, Lemma 4.13 and Lemma 4.15. We thus have

$$
\begin{align*}
\| \mathcal{L}^n (\mathrm{Id} - \mathbf{S}) \mathbf{h} \|_{\ell_{\gamma_1}^1} &\le C \left( \left\| (\mathfrak{D}_{r,1}(n,j,\mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^1} + \left\| (\mathfrak{D}_{\ell,1}(n,j,\mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^1} \right) \\
&\quad + C \left( \left\| (\mathfrak{D}_{r,2}(n,j,\mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^1} + \left\| (\mathfrak{D}_{\ell,2}(n,j,\mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^1} + \left\| (\mathfrak{D}_{\exp}(n,j,\mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^1} \right) \\
&\le C \left( \frac{1}{n^{\gamma_2-\gamma_1+1/8}} + \frac{1}{n^{\gamma_2+1/3}} + e^{-cn} \right) \|\mathbf{h}\|_{\ell_{\gamma_2}^1} \le \frac{C}{n^{\gamma_2-\gamma_1+1/8}} \|\mathbf{h}\|_{\ell_{\gamma_2}^1}.
\end{align*}
$$
For the second estimate (4.38b), we have

$$
\begin{aligned}
\|\mathcal{L}^n(\mathrm{Id} - \mathbf{S})\mathbf{h}\|_{\ell_{\gamma_1}^{\infty}} &\le C \left( \left\| (\mathfrak{D}_{r,1}(n,j,\mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} + \left\| (\mathfrak{D}_{\ell,1}(n,j,\mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} \right) \\
&\quad + C \left( \left\| (\mathfrak{D}_{r,2}(n,j,\mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} + \left\| (\mathfrak{D}_{\ell,2}(n,j,\mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} + \left\| (\mathfrak{D}_{\exp}(n,j,\mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} \right) \\
&\le C \left( \left\| (\mathfrak{D}_{r,1}(n,j,\mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} + \left\| (\mathfrak{D}_{\ell,1}(n,j,\mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} \right) \\
&\quad + C \left( \left\| (\mathfrak{D}_{r,2}(n,j,\mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{1}} + \left\| (\mathfrak{D}_{\ell,2}(n,j,\mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{1}} + \left\| (\mathfrak{D}_{\exp}(n,j,\mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{1}} \right) \\
&\le C \left( \frac{1}{n^{\gamma_2-\gamma_1+7/12}} + \frac{1}{n^{\gamma_2+1/3}} + e^{-cn} \right) \|\mathbf{h}\|_{\ell_{\gamma_2}^{1}} \le \frac{C}{n^{\gamma_2-\gamma_1+1/3+\min(1/4,\gamma_1)}} \|\mathbf{h}\|_{\ell_{\gamma_2}^{1}}.
\end{aligned}
$$

Finally, for the third estimate (4.38c), we use once more the estimates provided by Lemma 4.14, Lemma 4.13 and Lemma 4.15 to get:

$$
\begin{aligned}
\|\mathcal{L}^n(\mathrm{Id} - \mathbf{S})\mathbf{h}\|_{\ell_{\gamma_1}^{\infty}} &\le C \left( \left\| (\mathfrak{D}_{r,1}(n,j,\mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} + \left\| (\mathfrak{D}_{\ell,1}(n,j,\mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} \right) \\
&\quad + C \left( \left\| (\mathfrak{D}_{r,2}(n,j,\mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} + \left\| (\mathfrak{D}_{\ell,2}(n,j,\mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} + \left\| (\mathfrak{D}_{\exp}(n,j,\mathbf{h}))_{j \in \mathbb{Z}} \right\|_{\ell_{\gamma_1}^{\infty}} \right) \\
&\le C \left( \frac{1}{n^{\gamma_2-\gamma_1+1/8}} + \frac{1}{n^{\gamma_2-1/8}} + e^{-cn} \right) \|\mathbf{h}\|_{\ell_{\gamma_2}^{\infty}} \le \frac{C}{n^{\gamma_2-\gamma_1+\min(1/8,\gamma_1-1/8)}} \|\mathbf{h}\|_{\ell_{\gamma_2}^{\infty}}.
\end{aligned}
$$

This completes the proof of Theorem 4.3.
# Chapter 5

## Nonlinear orbital stability

In this last chapter, we finally prove the nonlinear orbital stability of the family of stationary discrete profiles {$v^\theta, \theta \in (-\underline{\theta}, \underline{\theta})$} given by Theorem 2.1 in some well-chosen algebraically weighted $\ell^p$ spaces. More precisely, given an initial sequence $h = (h_j)_{j \in \mathbb{Z}}$ in $\ell^1(\mathbb{Z}; \mathbb{R})$ satisfying the mass condition

$$ \sum_{j \in \mathbb{Z}} h_j = \theta, \quad \theta \in (-\underline{\theta}, \underline{\theta}), $$

we let $u^n = (u_j^n)_{j \in \mathbb{Z}, n \in \mathbb{N}}$ denote the corresponding solution to the Lax-Wendroff scheme (2.5)-(2.6) starting from the initial condition $u^0 = \bar{u} + h$, where we recall that $\bar{u} = v^0$ is the element of the family {$v^\theta, \theta \in (-\underline{\theta}, \underline{\theta})$} given by the explicit expression (2.7). By mass conservation, we automatically have:

$$ \forall n \in \mathbb{N}, \quad \sum_{j \in \mathbb{Z}} u_j^n - \bar{u}_j = \sum_{j \in \mathbb{Z}} u_j^0 - \bar{u}_j = \sum_{j \in \mathbb{Z}} h_j = \theta. $$

As a consequence, for each $n \in \mathbb{N}$, it is natural to define the perturbation $p^n = (p_j^n)_{j \in \mathbb{Z}}$ as

$$ p^n := u^n - v^\theta, $$

where in particular the mass conservation property of $u^n$ implies that for any $n \in \mathbb{N}$, the sequence $p^n$ has zero mass:

$$ \forall n \in \mathbb{N}, \quad \sum_{j \in \mathbb{Z}} p_j^n = 0. $$

From the recurrence formula (2.5) for $u^n = v^\theta + p^n$, we deduce that the sequence $p^n$ satisfies the following equation:

$$ \forall n \in \mathbb{N}, \quad p^{n+1} = \mathcal{L}^\theta p^n + (\mathrm{Id} - \mathbf{S}) \mathcal{N}^\theta(\mathbf{p}^n), \qquad (5.1) $$

where the bounded linear operator $\mathcal{L}^\theta : \ell^q(\mathbb{Z}; \mathbb{C}) \to \ell^q(\mathbb{Z}; \mathbb{C})$ is defined as the linearization of the Lax-Wendroff scheme around the stationary discrete shock profile $v^\theta$, that is:

$$ \forall j \in \mathbb{Z}, \quad (\mathcal{L}^\theta \mathbf{p})_j := p_j - \lambda \left[ \partial_u \mathcal{F}_\lambda(v_j^\theta, v_{j+1}^\theta) p_j + \partial_v \mathcal{F}_\lambda(v_j^\theta, v_{j+1}^\theta) p_{j+1} \right] \\ + \lambda \left[ \partial_u \mathcal{F}_\lambda(v_{j-1}^\theta, v_j^\theta) p_{j-1} + \partial_v \mathcal{F}_\lambda(v_{j-1}^\theta, v_j^\theta) p_j \right], \qquad (5.2) $$
and the nonlinear operator $\mathcal{N}^{\theta}$ is defined by:

$$ \forall j \in \mathbb{Z}, \quad \left( \mathcal{N}^{\theta}(\mathbf{p}) \right)_j := \lambda \mathcal{F}_{\lambda}(v_{j-1}^{\theta} + p_{j-1}, v_j^{\theta} + p_j) - \lambda \mathcal{F}_{\lambda}(v_{j-1}^{\theta}, v_j^{\theta}) \\ - \lambda \left[ \partial_u \mathcal{F}_{\lambda}(v_{j-1}^{\theta}, v_j^{\theta}) p_{j-1} + \partial_v \mathcal{F}_{\lambda}(v_{j-1}^{\theta}, v_j^{\theta}) p_j \right]. \quad (5.3) $$

We also recall that $\mathbf{S}$ is the shift operator defined as $(\mathbf{S}\mathbf{p})_j := p_{j+1}$ for all $j \in \mathbb{Z}$. Actually, we shall rewrite the recurrence relation in time (5.1) more conveniently as

$$ \forall n \in \mathbb{N}, \quad \mathbf{p}^{n+1} = \mathcal{L} \mathbf{p}^n + (\mathcal{L}^{\theta} - \mathcal{L}) \mathbf{p}^n + (\mathrm{Id} - \mathbf{S}) \mathcal{N}^{\theta}(\mathbf{p}^n), $$

where the operator $\mathcal{L} = \mathcal{L}^0$, given by (2.11), is the linearization of (2.5)-(2.6) around the discrete shock profile $\bar{\mathbf{u}} = \mathbf{v}^0$ given by (2.7). Finally, an application of Duhamel's formula gives the final expression for the perturbation $\mathbf{p}^n$ at any time $n \in \mathbb{N}$:

$$ \forall n \in \mathbb{N}, \quad \mathbf{p}^n = \mathcal{L}^n \mathbf{p}^0 + \sum_{m=0}^{n-1} \mathcal{L}^{n-1-m} (\mathcal{L}^{\theta} - \mathcal{L}) \mathbf{p}^m + \sum_{m=0}^{n-1} \mathcal{L}^{n-1-m} (\mathrm{Id} - \mathbf{S}) \mathcal{N}^{\theta}(\mathbf{p}^m). \quad (5.4) $$

As already emphasized in Chapter 2, our nonlinear orbital stability result (that is, Theorem 2.5) holds true in algebraically weighted spaces. Recalling our definition from the previous chapter, for $\gamma \in \mathbb{R}^+$, we define the weight $\omega_{\gamma} = (1 + |j|^{\gamma})_{j \in \mathbb{Z}}$. Then, for $p \in [1, +\infty]$, we introduce the weighted space:

$$ \ell_{\gamma}^{p}(\mathbb{Z}; \mathbb{R}) = \{ \mathbf{h} \in \ell^{p}(\mathbb{Z}; \mathbb{R}) \mid \omega_{\gamma} \mathbf{h} \in \ell^{p}(\mathbb{Z}; \mathbb{R}) \}, $$

with $\omega_{\gamma} \mathbf{h} = ((1 + |j|^{\gamma}) h_j)_{j \in \mathbb{Z}}$ and for $\mathbf{h} \in \ell_{\gamma}^{p}$, we denote $\|\mathbf{h}\|_{\ell_{\gamma}^{p}} = \|\omega_{\gamma} \mathbf{h}\|_{\ell^{p}}$ the norm of $\mathbf{h}$ in the space $\ell_{\gamma}^{p}$. Our strategy will be to bound each term appearing in the right-hand side of (5.4) using the semi-group estimates obtained for $(\mathcal{L}^n)_{n \in \mathbb{N}}$ and $(\mathcal{L}^n(\mathrm{Id} - \mathbf{S}))_{n \in \mathbb{N}}$ in Theorem 4.3. This is exactly the same strategy as in [7] except that the exponents in Theorem 4.3 are different so the whole process should be carried out in order to determine the correct constraints for $\beta$ and $\sigma$ in Theorem 2.5.

We shall need three technical lemmas which we now state. The first lemma provides semi-group estimates for the family of operators $(\mathcal{L}^n (\mathcal{L}^\theta - \mathcal{L}))_{n \in \mathbb{N}}$.

**Lemma 5.1.** For any $(\nu_1, \nu_2) \in [0, +\infty)^2$, there exists a constant $C_{\mathcal{L}}(\nu_1, \nu_2) > 0$ such that for all sequence $\mathbf{p} \in \ell^{\infty}(\mathbb{Z}; \mathbb{R})$ one has $(\mathcal{L}^\theta - \mathcal{L}) \mathbf{p} \in \cap_{\gamma \ge 0} \ell_{\gamma}^1$ and:

$$ \forall n \in \mathbb{N}, \quad \left\| \mathcal{L}^n (\mathcal{L}^\theta - \mathcal{L}) \mathbf{p} \right\|_{\ell_{\nu_1}^1} \le \frac{C_{\mathcal{L}}(\nu_1, \nu_2)}{(n+1)^{\nu_2}} |\theta| \|\mathbf{p}\|_{\ell^{\infty}}, $$

for all $\theta \in (-\underline{\theta}, \underline{\theta})$.

The proof of the above lemma can be found in [7, Proposition 2], and it uses crucially the exponential localisation of the family of discrete shock profiles $\mathbf{v}^\theta$ as stated in Theorem 2.1 and the key remark that for a bounded sequence $\mathbf{p} \in \ell^{\infty}(\mathbb{Z}; \mathbb{R})$ one has

$$ \sum_{j \in \mathbb{Z}} \left( (\mathcal{L}^{\theta} - \mathcal{L}) \mathbf{p} \right)_j = 0, $$

such that one can rely on estimate (4.37a) of Theorem 4.3 with $(\gamma_1, \gamma_2) = (\nu_1, \nu_1 + \nu_2 + \frac{1}{8})$.

The second lemma establishes algebraically weighted bounds for the nonlinear operator $\mathcal{N}^{\theta}$, and a proof is given in [7, Lemma 3.1].
**Lemma 5.2.** Let $(\mathcal{N}^\theta)_{\theta \in (-\underline{\theta}, \underline{\theta})}$ be the family of nonlinear operators defined by (5.3). Let $\gamma_1 \ge 0$ and $\rho > 0$ be given. There exists a constant $C_{\mathcal{N}}(\gamma_1, \rho) > 0$ such that the following holds.

(i) If $\mathbf{p} \in \ell_{\gamma_1}^1(\mathbb{Z}; \mathbb{R})$ with $\|\mathbf{p}\|_{\ell^\infty} \le \rho$, then for all $\theta \in (-\underline{\theta}, \underline{\theta})$, one has $\mathcal{N}^\theta(\mathbf{p}) \in \ell_{2\gamma_1}^1(\mathbb{Z}; \mathbb{R})$ and

$$ \forall \theta \in (-\underline{\theta}, \underline{\theta}), \quad \left\| \mathcal{N}^\theta(\mathbf{p}) \right\|_{\ell_{2\gamma_1}^1} \le C_{\mathcal{N}}(\gamma_1, \rho) \|\mathbf{p}\|_{\ell_{\gamma_1}^1} \|\mathbf{p}\|_{\ell_{\gamma_1}^\infty}. $$

(ii) If $\mathbf{p} \in \ell_{\gamma_1}^\infty(\mathbb{Z}; \mathbb{R})$ with $\|\mathbf{p}\|_{\ell^\infty} \le \rho$, then for all $\theta \in (-\underline{\theta}, \underline{\theta})$ one has $\mathcal{N}^\theta(\mathbf{p}) \in \ell_{2\gamma_1}^\infty(\mathbb{Z}; \mathbb{R})$ and

$$ \forall \theta \in (-\underline{\theta}, \underline{\theta}), \quad \left\| \mathcal{N}^\theta(\mathbf{p}) \right\|_{\ell_{2\gamma_1}^\infty} \le C_{\mathcal{N}}(\gamma_1, \rho) \|\mathbf{p}\|_{\ell_{\gamma_1}^\infty}^2. $$

Finally, we shall also need the following estimates whose proof can be found in [7, Lemma 3.2]. These are discrete versions of [31, Lemma 2.3]. Here and in all this Chapter, we let $\lfloor x \rfloor$ denote the integer part of a real number $x$.

**Lemma 5.3.** Let $a, b$ and $c$ be positive real numbers. Then there exists a constant $C(a, b, c) > 0$ such that

$$ \forall n \in \mathbb{N}, \quad \sum_{m=0}^{\lfloor \frac{n+1}{2} \rfloor} \frac{1}{(1+m-n)^a} \frac{1}{(1+m)^b} \le \frac{C(a, b, c)}{(2+n)^c}, $$

whenever $0 < a - c$ if $b = 1$, or $1 - b \le a - c$ if $b \in [0, 1)$ or $0 \le a - c$ if $b > 1$, together with

$$ \forall n \in \mathbb{N}, \quad \sum_{m=\lfloor \frac{n+1}{2} \rfloor+1}^{n} \frac{1}{(1+m-n)^a} \frac{1}{(1+m)^b} \le \frac{C(a, b, c)}{(2+n)^c}, $$

whenever $0 < b - c$ if $a = 1$, or $1 - a \le b - c$ if $a \in [0, 1)$ or $0 \le b - c$ if $a > 1$.

## 5.1 Fixing the constants

The first step towards the proof of Theorem 2.5 is to fix the two constants $C_0 > 0$ and $\epsilon > 0$ that appear in its statement. As a consequence, once for all we fix $\sigma + \beta \ge \frac{5}{12}$ together with $0 \le \sigma < \beta + \frac{1}{8}$, and we set $\gamma := \sigma + \beta + \frac{1}{8}$. We also fix a positive constant<fnref n="1" /> $\varrho > 0$.

Next, using Theorem 2.1(iii), we get the existence of a constant $C_m(\gamma) > 0$ such that

$$ \forall \theta \in (-\underline{\theta}, \underline{\theta}), \quad \|\mathbf{v}^\theta - \bar{\mathbf{u}}\|_{\ell_\gamma^1} \le C_m(\gamma) |\theta|. $$

We shall denote by $C_{\mathcal{L}}(\beta, \gamma) > 0$ the constant from Theorem 4.3 with $(\gamma_1, \gamma_2) = (\beta, \gamma)$. We can thus set

$$ C_0 := (1 + C_{\mathcal{L}}(\beta, \gamma))(1 + C_m(\gamma)). \qquad (5.5) $$

For any real number $x$, we use the standard notation $x_+ := \max(x, 0)$. With the above parameter $\sigma$, we introduce the number:

$$ \nu_2 := \frac{7}{12} + \left( \sigma - \frac{7}{12} \right)_+. \qquad (5.6) $$

<footnote n="1">¹In case the flux $f$ of the conservation law or the numerical flux $\mathcal{F}_\lambda$ would be defined on an open set and not on the whole space $(\mathbb{R} \text{ or } \mathbb{R}^2)$, this parameter $\varrho$ would help controlling the $\ell^\infty$ norm of the numerical solution so that the numerical solution would be defined for all times.</footnote>
Next, we introduce the following two constants

$$C_1 := 2 C_0 C_{\mathcal{L}}(\beta, \underline{\nu}_2) C\left(\underline{\nu}_2, \sigma + \frac{11}{24}, \sigma\right) + 2 C_0^2 C_{\mathcal{L}}(\beta, 2\beta) C_{\mathcal{N}}(\beta, \varrho) C\left(\beta + \frac{1}{8}, 2\sigma + \frac{11}{24}, \sigma\right),$$

$$C_2 := 2 C_0 C_{\mathcal{L}} \left( \beta, \sigma + \frac{25}{24} \right) C \left( \sigma + \frac{25}{24}, \sigma + \frac{11}{24}, \sigma + \frac{11}{24} \right) \\ + C_0^2 C_{\mathcal{L}}(\beta, 2\beta) C_{\mathcal{N}}(\beta, \varrho) C \left( \beta + \frac{7}{12}, 2\sigma + \frac{11}{24}, \sigma + \frac{11}{24} \right) \\ + C_0^2 C_{\mathcal{L}}(\beta, 2\beta) C_{\mathcal{N}}(\beta, \varrho) C \left( \beta + \frac{1}{8}, 2\sigma + \frac{11}{12}, \sigma + \frac{11}{24} \right),$$

where the constants $C_{\mathcal{L}}(\beta, \underline{\nu}_2)$ and $C_{\mathcal{L}}(\beta, \sigma + \frac{25}{24})$ are given by Lemma 5.1 with $(\nu_1, \nu_2) = (\beta, \underline{\nu}_2)$ and $(\nu_1, \nu_2) = (\beta, \sigma + \frac{25}{24})$, $C_{\mathcal{L}}(\beta, 2\beta)$ is given by Theorem 4.3 with $(\gamma_1, \gamma_2) = (\beta, 2\beta)$, $C_{\mathcal{N}}(\beta, \varrho)$ is given by Lemma 5.2 with $(\gamma_1, \rho) = (\beta, \varrho)$, and the four constants $C(\beta + \frac{1}{8}, 2\sigma + \frac{11}{24}, \sigma)$, $C(\sigma + \frac{25}{24}, \sigma + \frac{11}{24}, \sigma + \frac{11}{24})$, $C(\beta + \frac{7}{12}, 2\sigma + \frac{11}{24}, \sigma + \frac{11}{24})$ and $C(\beta + \frac{1}{8}, 2\sigma + \frac{11}{12}, \sigma + \frac{11}{24})$ are given by Lemma 5.3 with either one of the triplets:

$$(\beta + \frac{1}{8}, 2\sigma + \frac{11}{24}, \sigma), (\sigma + \frac{25}{24}, \sigma + \frac{11}{24}, \sigma + \frac{11}{24}), (\beta + \frac{7}{12}, 2\sigma + \frac{11}{24}, \sigma + \frac{11}{24}), (\beta + \frac{1}{8}, 2\sigma + \frac{11}{12}, \sigma + \frac{11}{24}).$$

At last, we choose $\epsilon > 0$ small enough such that

$$0 < \epsilon < \min \left( \theta, \frac{\varrho}{C_0}, \frac{1 + C_m(\gamma)}{C_1}, \frac{1 + C_m(\gamma)}{C_2} \right). \qquad (5.7)$$

## 5.2 Proof of Theorem 2.5

We now complete the proof of Theorem 2.5. We consider an initial perturbation $h \in \ell_{\gamma}^{1}(\mathbb{Z}; \mathbb{R})$, with $\gamma$ previously defined, and we assume that $h$ is small enough so that

$$\|h\|_{\ell_{\gamma}^{1}} < \epsilon.$$

We then define the excess mass:

$$\theta := \sum_{j \in \mathbb{Z}} h_j.$$

We readily remark that necessarily

$$|\theta| = \left| \sum_{j \in \mathbb{Z}} h_j \right| \le \|h\|_{\ell^1} \le \|h\|_{\ell_{\gamma}^1} < \epsilon < \underline{\theta},$$

the last inequality coming from our choice (5.7) for $\epsilon$. We can therefore use below the discrete shock profile $v^{\theta}$ associated with the excess mass $\theta$. We define the initial condition $u^0 := \bar{u} + h$ and since the numerical flux $\mathcal{F}_{\lambda} \in \mathcal{C}^{\infty}(\mathbb{R}^2; \mathbb{R})$ is globally defined, we directly obtain the existence of $(u^n)_{n \in \mathbb{N}}$ solution of (2.5)-(2.6) starting from this $u^0$. There is no need here to control the $\ell^{\infty}$ norm of the numerical solution at each time step in order to remain within the set where the numerical flux is defined. However, the $\ell^{\infty}$
control will be crucial in order to obtain uniform bounds for the quadratic remainder term $\mathcal{N}^\theta$. Thus, we can introduce the sequence of perturbations $(\mathbf{p}^n)_{n \in \mathbb{N}}$ defined as

$$ \mathbf{p}^n := \mathbf{u}^n - \mathbf{v}^\theta, $$

which is given by (5.4), that is

$$ \forall n \in \mathbb{N}, \quad \mathbf{p}^n = \mathcal{L}^n \mathbf{p}^0 + \sum_{m=0}^{n-1} \mathcal{L}^{n-1-m} (\mathcal{L}^\theta - \mathcal{L}) \mathbf{p}^m + \sum_{m=0}^{n-1} \mathcal{L}^{n-1-m} (\mathrm{Id} - \mathbf{S}) \mathcal{N}^\theta(\mathbf{p}^m). $$

We shall now prove by induction that for all $n \in \mathbb{N}$ one has $\mathbf{p}^n \in \ell_\beta^1(\mathbb{Z}; \mathbb{R})$ together with the bounds:

$$ \forall m = 0, \dots, n, \quad \|\mathbf{p}^m\|_{\ell_\beta^1} \le \frac{C_0}{(1+m)^\sigma} \|\mathbf{h}\|_{\ell_\gamma^1}, \quad \|\mathbf{p}^m\|_{\ell_\beta^\infty} \le \frac{C_0}{(1+m)^{\sigma+\frac{11}{24}}} \|\mathbf{h}\|_{\ell_\gamma^1}, \quad \text{and} \quad \|\mathbf{p}^m\|_{\ell^\infty} \le \varrho. \tag{5.8} $$

### 5.2.1 Initialization step

At $n = 0$, we have by definition that

$$ \mathbf{p}^0 = \mathbf{u}^0 - \mathbf{v}^\theta = \bar{\mathbf{u}} - \mathbf{v}^\theta + \mathbf{h}. $$

As a consequence, using $0 \le \beta \le \gamma$ and obvious inequalities between norms, we have the estimates:

$$ \|\mathbf{p}^0\|_{\ell^\infty} \le \|\mathbf{p}^0\|_{\ell_\beta^\infty} \le \|\mathbf{p}^0\|_{\ell_\beta^1} \le \|\mathbf{p}^0\|_{\ell_\gamma^1} \le \left\|\bar{\mathbf{u}} - \mathbf{v}^\theta\right\|_{\ell_\gamma^1} + \|\mathbf{h}\|_{\ell_\gamma^1} \le (1 + C_m(\gamma)) \|\mathbf{h}\|_{\ell_\gamma^1} \le C_0 \|\mathbf{h}\|_{\ell_\gamma^1} \le C_0 \epsilon \le \varrho. \tag{5.9} $$

This means that the induction assumption (5.8) is satisfied for $n = 0$ (the above chain of inequalities encompasses the three estimates of (5.8)). We finally also recall that $\mathbf{p}^0$ has zero mass:

$$ \sum_{j \in \mathbb{Z}} p_j^0 = 0, $$

and this property will be automatically propagated at later times since we consider a conservative scheme.

### 5.2.2 Induction step

We assume that (5.8) is satisfied up to some integer $n \in \mathbb{N}$ and we now show that it propagates to the integer $n + 1$. Using Duhamel's formula at $n + 1$, we have the following expression for $\mathbf{p}^{n+1}$:

$$ \mathbf{p}^{n+1} = \mathcal{L}^{n+1} \mathbf{p}^0 + \sum_{m=0}^{n} \mathcal{L}^{n-m} (\mathcal{L}^\theta - \mathcal{L}) \mathbf{p}^m + \sum_{m=0}^{n} \mathcal{L}^{n-m} (\mathrm{Id} - \mathbf{S}) \mathcal{N}^\theta(\mathbf{p}^m). $$

We shall now bound each term separately.
**The $\ell_\beta^1$ estimate.** Since the initial perturbation $\mathbf{p}^0 \in \ell_\gamma^1$ is of zero mass, we directly have by Theorem 4.3:

$$ \|\mathcal{L}^{n+1} \mathbf{p}^0\|_{\ell_\beta^1} \le \frac{C_{\mathcal{L}}(\beta, \gamma)}{(2+n)^\sigma} \|\mathbf{p}^0\|_{\ell_\gamma^1} \le \frac{C_{\mathcal{L}}(\beta, \gamma)(1+C_m(\gamma))}{(2+n)^\sigma} \|\mathbf{h}\|_{\ell_\gamma^1}, \quad (5.10) $$

where we have used the following consequence of (5.9):

$$ \|\mathbf{p}^0\|_{\ell_\gamma^1} \le (1 + C_m(\gamma)) \|\mathbf{h}\|_{\ell_\gamma^1}. $$

Regarding the second term, we use Lemma 5.1 with $\nu_1 = \beta$ and $\nu_2$ defined in (5.6), the inequality $|\theta| < \epsilon$ and the induction assumption (5.8) to obtain:

$$ \begin{aligned} \|\mathcal{L}^{n-m} (\mathcal{L}^\theta - \mathcal{L}) \mathbf{p}^m\|_{\ell_\beta^1} &\le \frac{C_{\mathcal{L}}(\beta, \nu_2)}{(n-m+1)^{\nu_2}} |\theta| \|\mathbf{p}^m\|_{\ell^\infty} \\ &\le \frac{C_{\mathcal{L}}(\beta, \nu_2)}{(n-m+1)^{\nu_2}} |\theta| \|\mathbf{p}^m\|_{\ell_\beta^\infty} \\ &\le \frac{\epsilon C_0 C_{\mathcal{L}}(\beta, \nu_2)}{(n-m+1)^{\nu_2} (1+m)^{\sigma+\frac{11}{24}}} \|\mathbf{h}\|_{\ell_\gamma^1}. \end{aligned} $$

Finally, using Lemma 5.3 with $a = \nu_2$, $b = \sigma + \frac{11}{24}$ and $c = \sigma$ (the reader can easily verify that we are always in a position to apply the two inequalities provided by Lemma 5.3 with our choice (5.6) of $\nu_2$), we directly obtain that

$$ \begin{aligned} \left\| \sum_{m=0}^{n} \mathcal{L}^{n-m} \left( \mathcal{L}^{\theta} - \mathcal{L} \right) \mathbf{p}^m \right\|_{\ell_{\beta}^{1}} &\le \sum_{m=0}^{n} \frac{\epsilon C_0 C_{\mathcal{L}}(\beta, \nu_2)}{(n-m+1)^{\nu_2} (1+m)^{\sigma}} \|\mathbf{h}\|_{\ell_{\gamma}^{1}} \\ &\le \frac{2 \epsilon C_0 C_{\mathcal{L}}(\beta, \nu_2) C \left( \nu_2, \sigma + \frac{11}{24}, \sigma \right)}{(2+n)^{\sigma}} \|\mathbf{h}\|_{\ell_{\gamma}^{1}}. \end{aligned} \quad (5.11) $$

For the third term that incorporates the nonlinear contributions, we shall use the estimate (4.38a) of Theorem 4.3 and Lemma 5.2 to derive that

$$ \begin{aligned} \left\| \sum_{m=0}^{n} \mathcal{L}^{n-m} (\mathrm{Id} - \mathbf{S}) \mathcal{N}^{\theta}(\mathbf{p}^m) \right\|_{\ell_{\beta}^1} &\le \sum_{m=0}^{n} \frac{C_{\mathcal{L}}(\beta, 2\beta)}{(1+n-m)^{\beta+\frac{1}{8}}} \left\| \mathcal{N}^{\theta}(\mathbf{p}^m) \right\|_{\ell_{2\beta}^1} \\ &\le \sum_{m=0}^{n} \frac{C_{\mathcal{L}}(\beta, 2\beta) C_{\mathcal{N}}(\beta, \varrho)}{(1+n-m)^{\beta+\frac{1}{8}}} \|\mathbf{p}^m\|_{\ell_\beta^1} \|\mathbf{p}^m\|_{\ell_\beta^\infty} \\ &\le \sum_{m=0}^{n} \frac{C_0^2 C_{\mathcal{L}}(\beta, 2\beta) C_{\mathcal{N}}(\beta, \varrho)}{(1+n-m)^{\beta+\frac{1}{8}} (1+m)^{2\sigma+\frac{11}{24}}} \|\mathbf{h}\|_{\ell_\gamma^1}^2 \\ &\le \frac{2\epsilon C_0^2 C_{\mathcal{L}}(\beta, 2\beta) C_{\mathcal{N}}(\beta, \varrho) C \left( \beta + \frac{1}{8}, 2\sigma + \frac{11}{24}, \sigma \right)}{(2+n)^\sigma} \|\mathbf{h}\|_{\ell_\gamma^1}^1. \end{aligned} $$

For the last inequality, we have applied Lemma 5.3 with $a = \beta + \frac{1}{8}$, $b = 2\sigma + \frac{11}{24}$ and $c = \sigma$. We can readily verify that

$$ 0 < \beta + \frac{1}{8} - \sigma = a - c, \quad 0 < \sigma + \frac{11}{24} = b - c, \quad \text{and} \quad b - c + a - 1 = \beta + \sigma + \frac{1}{8} + \frac{11}{24} - 1 = \sigma + \beta - \frac{5}{12} \ge 0, $$
thanks to our assumptions on $\beta + \sigma \ge \frac{5}{12}$ and $0 \le \sigma < \beta + \frac{1}{8}$.

As a consequence (recalling the definition of the constant $C_1$), we have obtained that

$$ \|\mathbf{p}^{n+1}\|_{\ell_{\beta}^{1}} \le \frac{1}{(2+n)^{\sigma}} \left(C_{\mathcal{L}}(\beta, \gamma)(1+C_{m}(\gamma)) + \epsilon C_{1}\right) \|\mathbf{h}\|_{\ell_{\gamma}^{1}} \le \frac{C_{0}}{(2+n)^{\sigma}} \|\mathbf{h}\|_{\ell_{\gamma}^{1}}, $$

thanks to our choice of $C_0$ and the restrictions on $\epsilon$.

**The $\ell_{\beta}^{\infty}$ estimate.** Using once more that the initial perturbation $\mathbf{p}^0 \in \ell_{\gamma}^1$ is of zero mass, we directly find, using the semi-group estimate of $(\mathcal{L}^n)_{n \in \mathbb{N}}$ in $\ell_{\beta}^{\infty}$ that

$$ \|\mathcal{L}^{n+1} \mathbf{p}^0\|_{\ell_{\beta}^{\infty}} \le \frac{C_{\mathcal{L}}(\beta, \gamma)}{(2+n)^{\sigma+\frac{11}{24}}} \|\mathbf{p}^0\|_{\ell_{\gamma}^1} \le \frac{C_{\mathcal{L}}(\beta, \gamma)(1+C_{m}(\gamma))}{(2+n)^{\sigma+\frac{11}{24}}} \|\mathbf{h}\|_{\ell_{\gamma}^1}, $$

where we have used one more time the following consequence of (5.9):

$$ \|\mathbf{p}^0\|_{\ell_{\gamma}^1} \le (1 + C_m(\gamma)) \|\mathbf{h}\|_{\ell_{\gamma}^1}. $$

Regarding the second term, we use Lemma 5.1 with $\nu_1 = \beta$ and $\nu_2 = \sigma + \frac{25}{24}$ to obtain

$$ \begin{aligned} \|\mathcal{L}^{n-m} (\mathcal{L}^{\theta} - \mathcal{L}) \mathbf{p}^m\|_{\ell_{\beta}^{\infty}} &\le \|\mathcal{L}^{n-m} (\mathcal{L}^{\theta} - \mathcal{L}) \mathbf{p}^m\|_{\ell_{\gamma}^1} \\ &\le \frac{C_{\mathcal{L}}(\beta, \sigma + \frac{25}{24})}{(n-m+1)^{\sigma+\frac{25}{24}}} |\theta| \|\mathbf{p}^m\|_{\ell^{\infty}} \\ &\le \frac{\epsilon C_0 C_{\mathcal{L}}(\beta, \sigma + \frac{25}{24})}{(n-m+1)^{\sigma+\frac{25}{24}} (1+m)^{\sigma+\frac{11}{24}}} \|\mathbf{h}\|_{\ell_{\gamma}^1}. \end{aligned} $$

Next, using Lemma 5.3 with $a = \sigma + \frac{25}{24} > 1$ and $b = c = \sigma + \frac{11}{24}$, noticing that $c < a$, we directly obtain that

$$ \begin{aligned} \left\| \sum_{m=0}^{n} \mathcal{L}^{n-m} (\mathcal{L}^{\theta} - \mathcal{L}) \mathbf{p}^m \right\|_{\ell_{\beta}^{\infty}} &\le \sum_{m=0}^{n} \frac{\epsilon C_0 C_{\mathcal{L}}(\beta, \sigma + \frac{25}{24})}{(n-m+1)^{\sigma+\frac{25}{24}} (1+m)^{\sigma+\frac{11}{24}}} \|\mathbf{h}\|_{\ell_{\gamma}^1} \\ &\le \frac{2\epsilon C_0 C_{\mathcal{L}}(\beta, \sigma + \frac{25}{24}) \mathbf{C}(\sigma + \frac{25}{24}, \sigma + \frac{11}{24}, \sigma + \frac{11}{24})}{(2+n)^{\sigma+\frac{11}{24}}} \|\mathbf{h}\|_{\ell_{\gamma}^1}. \end{aligned} $$

For the third term, we shall use the estimate (4.38b) of Theorem 4.3 and Lemma 5.2 to derive that

$$ \begin{aligned} \left\| \sum_{m=0}^{\lfloor \frac{n+1}{2} \rfloor} \mathcal{L}^{n-m} (\mathrm{Id} - \mathbf{S}) \mathcal{N}^{\theta}(\mathbf{p}^m) \right\|_{\ell_{\beta}^{\infty}} &\le \sum_{m=0}^{\lfloor \frac{n+1}{2} \rfloor} \frac{C_{\mathcal{L}}(\beta, 2\beta)}{(1+n-m)^{\beta+\frac{1}{3}+\min(\frac{1}{4}, \beta)}} \|\mathcal{N}^{\theta}(\mathbf{p}^m)\|_{\ell_{2\beta}^1} \\ &\le \sum_{m=0}^{\lfloor \frac{n+1}{2} \rfloor} \frac{C_{\mathcal{L}}(\beta, 2\beta) C_{\mathcal{N}}(\beta, \varrho)}{(1+n-m)^{\beta+\frac{7}{12}}} \|\mathbf{p}^m\|_{\ell_{\beta}^1} \|\mathbf{p}^m\|_{\ell_{\beta}^{\infty}} \\ &\le \sum_{m=0}^{\lfloor \frac{n+1}{2} \rfloor} \frac{C_0^2 C_{\mathcal{L}}(\beta, 2\beta) C_{\mathcal{N}}(\beta, \varrho)}{(1+n-m)^{\beta+\frac{7}{12}} (1+m)^{2\sigma+\frac{11}{24}}} \|\mathbf{h}\|_{\ell_{\gamma}^1}^2 \\ &\le \frac{\epsilon C_0^2 C_{\mathcal{L}}(\beta, 2\beta) C_{\mathcal{N}}(\beta, \varrho) \mathbf{C}(\beta + \frac{7}{12}, 2\sigma + \frac{11}{24}, \sigma + \frac{11}{24})}{(2+n)^{\sigma+\frac{11}{24}}} \|\mathbf{h}\|_{\ell_{\gamma}^1}. \end{aligned} $$
For the last inequality, we have used the first inequality of Lemma 5.3 with $a = \beta + \frac{7}{12}$, $b = 2\sigma + \frac{11}{24}$ and $\sigma + \frac{11}{24}$, noticing that

$$a - c = \beta + \frac{1}{8} - \sigma > 0, \quad \text{and} \quad a - c + b - 1 = \sigma + \beta - \frac{5}{12} \ge 0,$$

since $0 \le \sigma < \beta + \frac{1}{8}$ and $\beta + \sigma \ge \frac{5}{12}$.

For the remaining part of the sum, we use estimate (4.38c) of Theorem 4.3 and Lemma 5.2 to obtain

$$ \begin{aligned} \left\| \sum_{m=\lfloor \frac{n+1}{2} \rfloor + 1}^{n} \mathcal{L}^{n-m} (\mathrm{Id} - \mathbf{S}) \mathcal{N}^{\theta}(\mathbf{p}^m) \right\|_{\ell_{\beta}^{\infty}} & \le \sum_{m=\lfloor \frac{n+1}{2} \rfloor + 1}^{n} \frac{C_{\mathcal{L}}(\beta, 2\beta)}{(1+n-m)^{\beta+\min(\frac{1}{8}, \beta-\frac{1}{8})}} \left\| \mathcal{N}^{\theta}(\mathbf{p}^m) \right\|_{\ell_{2\beta}^{\infty}} \\ & \le \sum_{m=\lfloor \frac{n+1}{2} \rfloor + 1}^{n} \frac{C_{\mathcal{L}}(\beta, 2\beta) C_{\mathcal{N}}(\beta, \varrho)}{(1+n-m)^{\beta+\frac{1}{8}}} \|\mathbf{p}^m\|_{\ell_{\beta}^{\infty}}^2 \\ & \le \sum_{m=\lfloor \frac{n+1}{2} \rfloor + 1}^{n} \frac{C_{\mathcal{L}}(\beta, 2\beta) C_{\mathcal{N}}(\beta, \varrho) C_0^2}{(1+n-m)^{\beta+\frac{1}{8}} (1+m)^{2\sigma+\frac{11}{12}}} \|\mathbf{h}\|_{\ell_{\gamma}^1}^2 \\ & \le \frac{\epsilon C_0^2 C_{\mathcal{L}}(\beta, 2\beta) C_{\mathcal{N}}(\beta, \varrho) C\left(\beta + \frac{1}{8}, 2\sigma + \frac{11}{12}, \sigma + \frac{11}{24}\right)}{(2+n)^{\sigma+\frac{11}{24}}} \|\mathbf{h}\|_{\ell_{\gamma}^1}. \end{aligned} $$

For the last inequality, we have used the second inequality of Lemma 5.3 with $a = \beta + \frac{1}{8}$, $b = 2\sigma + \frac{11}{24}$ and $\sigma + \frac{11}{24}$, noticing that

$$b - c > 0, \quad \text{and} \quad a - c + b - 1 = \sigma + \beta - \frac{5}{12} \ge 0.$$

As a consequence (recalling our definition for the constant $C_2$), we have obtained that

$$\|\mathbf{p}^{n+1}\|_{\ell_{\beta}^{\infty}} \le \frac{1}{(2+n)^{\sigma+\frac{11}{24}}} \left( C_{\mathcal{L}}(\beta, \gamma) (1+C_m(\gamma)) + \epsilon C_2 \right) \|\mathbf{h}\|_{\ell_{\gamma}^1} \le \frac{C_0}{(2+n)^{\sigma+\frac{11}{24}}} \|\mathbf{h}\|_{\ell_{\gamma}^1}.$$

From there, we also deduce that

$$\|\mathbf{p}^{n+1}\|_{\ell^{\infty}} \le \|\mathbf{p}^{n+1}\|_{\ell_{\beta}^{\infty}} \le C_0 \epsilon \le \varrho.$$

This concludes the proof of Theorem 2.5.
# Appendix A

## The exact and approximate Green's functions for the Cauchy problem

In this appendix, we study the Lax-Wendroff scheme when applied to the linear transport equation:

$$ \partial_t v + a \partial_x v = 0, $$

with $a \neq 0$ and the equation is considered on the whole real line $\mathbb{R}$. As we have already seen earlier in this article, in the linear case, the Lax-Wendroff scheme can be recast under the form:

$$ \forall n \in \mathbb{N}, \quad v^{n+1} = \overline{\mathcal{L}} v^n, \qquad (A.1) $$

where for any integer $n \in \mathbb{N}$, $v^n$ denotes the sequence $(v_j^n)_{j \in \mathbb{Z}}$, and $\overline{\mathcal{L}}$ is the discrete convolution operator defined on any real or complex valued sequence $v = (v_j)_{j \in \mathbb{Z}}$ by:

$$ \forall j \in \mathbb{Z}, \quad (\overline{\mathcal{L}} v)_j := v_j - \frac{\alpha}{2} (v_{j+1} - v_{j-1}) + \frac{\alpha^2}{2} (v_{j+1} - 2v_j + v_{j-1}), $$

where, as in the core of this article, $\alpha$ is a short notation for $\lambda a$, $\lambda > 0$ denoting the fixed ratio $\Delta t/\Delta x$ between the time and space steps. The constant $\alpha$ thus has the sign of the transport velocity $a$. In what follows, we shall apply the results below to either $\alpha = \alpha_\ell \in (0,1)$ or $\alpha = \alpha_r \in (-1,0)$, so that the above operator $\overline{\mathcal{L}}$ corresponds to either one of the operators $\mathcal{L}_\ell$ or $\mathcal{L}_r$ defined in (3.25).

In this appendix, we recall or prove several bounds on the *exact* and *approximate* Green's functions for the Lax-Wendroff scheme. The approximate Green's function is defined below in (A.8) and is meant to reproduce the leading qualitative and quantitative features of the exact Green's function of (A.1). The study of the Green's function of (A.1) was the purpose of the article [9] by one of the authors. We shall recall below the main conclusions of [9] since they are useful for our purpose here. Unsurprisingly, many arguments below for studying the approximate Green's function follow what has been already done in [9] but there are also several new regimes that need to be considered and that did not appear in [9]. The bounds that we prove below (see in particular Theorem A.3) are used in this article to study the so-called activation function $A$ in our decomposition of the Green's function for the linearized operator $\mathcal{L}$ in (2.11) (see Theorem 4.1) and various bounds for some remainder terms. This appendix can also be seen as a main building block for a complete justification of the local limit theorem for finite difference approximations of the transport equation that exhibit dispersion and dissipation (we refer to [23] and
[24, 25] for a presentation and some recent advances on the local limit theorem and its connection to probability theory). The local limit theorem in the non-dispersive (or rather parabolic) case is justified in [11] and the complete justification of the local limit theorem in the dispersive case is a work in progress. We refer to [24] for a justification of the leading order term in the local limit theorem for sequences that exhibit dispersion and dissipation as we consider here.

## A.1 The exact and approximate Green's functions. Main results

As we have recalled above, the Lax-Wendroff scheme for the transport equation on the whole real line reads:

$$v_j^{n+1} = v_j^n - \frac{\alpha}{2} (v_{j+1}^n - v_{j+1}^n) + \frac{\alpha^2}{2} (v_{j+1}^n - 2v_j^n + v_{j+1}^n), \quad (A.2)$$

with $\alpha := \lambda a$. The exact Green's function for (A.2) corresponds to the initial condition defined by:

$$\forall j \in \mathbb{Z}, \quad \bar{\mathcal{G}}_j^0 := \begin{cases} 1, & \text{if } j = 0, \\ 0, & \text{otherwise,} \end{cases}$$

which leads to the solution $(\bar{\mathcal{G}}_j^n)_{(j,n) \in \mathbb{Z} \times \mathbb{N}}$ for (A.2). This sequence $(\bar{\mathcal{G}}_j^n)_{(j,n) \in \mathbb{Z} \times \mathbb{N}}$ is studied in [9], see also [16, 17]. Its main features are recalled below.

We restrict from now on to the case $\alpha \in (-1, 1) \setminus \{0\}$ in order to stick to the two particular situations we are interested in, that is, either $\alpha = \alpha_\ell \in (0, 1)$ or $\alpha = \alpha_r \in (-1, 0)$. We compute the so-called amplification factor for (A.2) and obtain (see for instance [9]):

$$\forall \theta \in \mathbb{R}, \quad \hat{F}_{\text{LW}}(\theta) = 1 - 2\alpha^2 \sin^2 \frac{\theta}{2} + i\alpha \sin \theta.$$

As already reported in [9], the expansion of the amplification factor $\hat{F}_{\text{LW}}$ near the frequency 0 reads:

$$\hat{F}_{\text{LW}}(\theta) = \exp \left( i\alpha\theta - i\frac{\alpha(1-\alpha^2)}{6}\theta^3 - \frac{\alpha^2(1-\alpha^2)}{8}\theta^4 + \mathcal{O}(\theta^5) \right), \quad (A.3)$$

and we have furthermore the dissipation property:

$$\forall \theta \in [-\pi, \pi] \setminus \{0\}, \quad |\hat{F}_{\text{LW}}(\theta)| < 1.$$

Since $\hat{F}_{\text{LW}}$ is a trigonometric polynomial, it extends as a holomorphic function with respect to $\theta$ on the whole complex plane $\mathbb{C}$. For later use, we introduce the coefficients:

$$c_3 := \frac{\alpha(1-\alpha^2)}{6} \neq 0, \quad c_4 := \frac{\alpha^2(1-\alpha^2)}{8} > 0, \quad (A.4)$$

in such a way that (A.3) equivalently reads:

$$\hat{F}_{\text{LW}}(\theta) = \exp \left( i\alpha\theta - i c_3 \theta^3 - c_4 \theta^4 + \mathcal{O}(\theta^5) \right),$$

as $\theta$ tends to 0. The main result proved in [9] can be formulated as follows. It uses the notation $c_3$ and $c_4$ for the coefficients in (A.4) that arise in the Taylor's expansion of the amplification factor $\hat{F}_{\text{LW}}$ at 0.
**Theorem A.1.** Assume that the constant $c_3$ in (A.4) is positive, that is $\alpha \in (0, 1)$. Then there exist two constants $C > 0$ and $c > 0$ such that the Green's function $(\mathcal{G}_j^n)_{(j,n)\in\mathbb{Z}\times\mathbb{N}}$ for (A.2) satisfies the uniform bounds:

$$ \forall n \in \mathbb{N}^*, \quad \forall j \in \mathbb{Z}, \quad |\overline{\mathcal{G}}_j^n| \le \frac{C}{n^{1/3}} \exp\left(-c\left(\frac{j - \alpha n}{n^{1/3}}\right)^{3/2}\right), \quad \text{if } j - \alpha n \ge 0, \quad (A.5) $$

and:

$$ \begin{array}{l} \forall n \in \mathbb{N}^*, \quad \forall j \in \mathbb{Z}, \\ |\overline{\mathcal{G}}_j^n - 2 \operatorname{Re} \mathfrak{g}_j^n| \le \frac{C}{n^{1/3}} \exp \left( -c \left( \frac{|j - \alpha n|}{n^{1/3}} \right)^{3/2} \right) + \frac{C}{n^{1/2}} \exp \left( -c \frac{(j - \alpha n)^2}{n} \right), \\ \qquad \text{if } j - \alpha n < 0, \quad (A.6) \end{array} $$

where $\mathfrak{g}_j^n$ is defined for $n \in \mathbb{N}^*$ and $j \in \mathbb{Z}$ as:

$$ \forall (n,j) \in \mathbb{N}^* \times \mathbb{Z}, \quad \mathfrak{g}_j^n := \frac{1}{2\pi} \exp\left(-\frac{c_4 (j - \alpha n)^2}{9c_3^2 n}\right) \exp\left(\mathbf{i} \frac{2|j - \alpha n|^{3/2}}{3\sqrt{3|c_3|n}} - \mathbf{i} \frac{\pi}{4}\right) \\ \times \int_{-\sqrt{\frac{2|j - \alpha n|}{3|c_3|n}}}^{\sqrt{\frac{2|j - \alpha n|}{3|c_3|n}}} e^{-\sqrt{3|c_3| |j - \alpha n|} n u^2} e^{c_3 n e^{-i\pi/4} u^3} du. \quad (A.7) $$

If $c_3$ is negative, the bounds depending on the sign of $j - \alpha n$ should be switched.

In order to be absolutely complete, it is important to note that Theorem A.1 above is not exactly the statement that is proved in [9]. This is because, in the interval between the completion of [9] and the present work, an error was found in [9], which made us modify the statement and the proof of the main result in [9]. The new statement, that is correct, and whose corollaries follow exactly as in [9], is Theorem A.1 above. The error occurred in the proof of the estimate (A.6) and in the definition (A.7) of the approximation $\mathfrak{g}_j^n$ that incorporates the damped oscillations of the Green's function. Rather than reproducing the whole proof of Theorem A.1 (only a tiny part of the proof needs to be corrected), we shall rather give the proof of Theorem A.3 below for the approximate Green's function since the analysis of the approximate Green's function needs to incorporate new regimes that were not considered in [9]. The proof of the above estimate (A.6) corresponds to the bound (A.12) below for the approximate Green's function. The reader will most certainly experiment no difficulty to adapt the arguments below to derive the statement in (A.6), (A.7).

Let us recall an immediate consequence of Theorem A.1, see [9] for an even more precise statement.

**Corollary A.1.** Let the constant $c_3$ in (A.4) be positive, that is, $\alpha \in (0, 1)$. Then there exist two constants $C > 0$ and $c > 0$ such that the Green's function $(\overline{\mathcal{G}}_j^n)_{(j,n)\in\mathbb{Z}\times\mathbb{N}}$ for (A.2) satisfies:

$$ \forall n \in \mathbb{N}^*, \quad |\overline{\mathcal{G}}_j^n| \le C \begin{cases} \frac{1}{n^{1/3}} \exp\left(-c|j - \alpha n|^{3/2}/n^{1/2}\right), & \text{if } j - \alpha n \ge 0, \\ \frac{1}{n^{1/3}}, & \text{if } -n^{1/3} \le j - \alpha n \le 0, \\ \frac{1}{|j - \alpha n|^{1/4} n^{1/4}} \exp\left(-c|j - \alpha n|^2/n\right), & \text{if } j - \alpha n \le -n^{1/3}, \end{cases} $$
together with the $\ell^1$ estimate:

$$ \forall n \in \mathbb{N}, \quad \sum_{j \in \mathbb{Z}} |\mathcal{G}_j^n| \le C (1+n)^{1/8}. $$

If $c_3$ is negative, that is, $\alpha \in (-1, 0)$, the pointwise estimates for the Green's function $(\mathcal{G}_j^n)_{(j,n) \in \mathbb{Z} \times \mathbb{N}}$ read:

$$ \forall n \in \mathbb{N}^*, \quad |\mathcal{G}_j^n| \le C \begin{cases} \frac{1}{n^{1/3}} \exp(-c|j - \alpha n|^{3/2}/n^{1/2}), & \text{if } j - \alpha n \le 0, \\ \frac{1}{n^{1/3}}, & \text{if } 0 \le j - \alpha n \le n^{1/3}, \\ \frac{1}{|j - \alpha n|^{1/4} n^{1/4}} \exp(-c|j - \alpha n|^2/n), & \text{if } j - \alpha n \ge n^{1/3}, \end{cases} $$

and the $\ell^1$ estimate is unchanged.

The analysis in Chapter 4 will use a very slight variation on Theorem A.1 which we state here for convenience. Actually, the statement in Theorem A.2 below is the core of the proof of Theorem A.1 in [9] even though this result was not stated in such generality in [9]. Once again, the proof of Theorem A.2 below follows from the exact same arguments as those we develop below in the proofs of Theorem A.3 and Theorem A.4. The absorption of the remainder term $\theta^5 \Psi(\theta)$ in the integral is made by choosing $\delta$ small enough, just like we did in the proof of Proposition 4.1 in Chapter 4. We therefore feel free to skip the proof of Theorem A.2.

**Theorem A.2.** Let $\tilde{c}_3$ and $\tilde{c}_4$ be two positive real numbers. Let $\Psi$ denote a holomorphic function on some neighborhood of 0 in $\mathbb{C}$. Let $\underline{C}$ be a positive real number. Then there exists some positive real number $\delta_0 > 0$ such that the following property holds: for any $\delta \in (0, \delta_0]$, there exist two constants $C > 0$ and $c > 0$ such that there holds:

$$ \forall (x, y) \in \mathbb{R} \times [\underline{C}^{-1}, +\infty), \quad \left| \int_{-\delta}^{\delta} e^{ix\theta + i\tilde{c}_3 y\theta^3 - \tilde{c}_4 y\theta^4 + y\theta^5} \Psi(\theta) \, d\theta \right| \\ \le C \begin{cases} \frac{1}{y^{1/3}} \exp(-c|x|^{3/2}/y^{1/2}), & \text{if } 0 \le x \le \underline{C}y, \\ \frac{1}{y^{1/3}}, & \text{if } -y^{1/3} \le x \le 0, \\ \frac{1}{|x|^{1/4} y^{1/4}} \exp(-c|x|^2/y), & \text{if } -\underline{C}y \le x \le -y^{1/3}. \end{cases} $$
If now $\tilde{c}_3$ is negative ($\tilde{c}_4$ being kept positive), the result still holds but with estimates that now read:

$$ \forall (x, y) \in \mathbb{R} \times [\underline{\mathbf{C}}^{-1}, +\infty), \quad \left| \int_{-\delta}^{\delta} e^{ix\theta + i\tilde{c}_3 y\theta^3 - \tilde{c}_4 y\theta^4 + y\theta^5} \Psi(\theta) \, d\theta \right| \\ \le C \begin{cases} \frac{1}{y^{1/3}} \exp(-c|x|^{3/2}/y^{1/2}), & \text{if } -\underline{\mathbf{C}}y \le x \le 0, \\ \frac{1}{y^{1/3}}, & \text{if } 0 \le x \le y^{1/3}, \\ \frac{1}{|x|^{1/4} y^{1/4}} \exp(-c|x|^2/y), & \text{if } y^{1/3} \le x \le \underline{\mathbf{C}}y. \end{cases} $$

We now turn to the approximate Green's function for (A.2). Recalling that we have the following formula for the exact Green's function:

$$ \forall (n, j) \in \mathbb{N} \times \mathbb{Z}, \quad \bar{\mathcal{G}}_j^n := \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{-ij\theta} \hat{F}_{\text{LW}}(\theta)^n d\theta = \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{ij\theta} \hat{F}_{\text{LW}}(-\theta)^n d\theta, $$

the expansion (A.3) and the dissipation property suggests the introduction of the approximate Green's function $\mathcal{G}_j^n$ defined by:

$$ \forall (n, j) \in \mathbb{N}^* \times \mathbb{Z}, \quad \mathcal{G}_j^n := \frac{1}{2\pi} \int_{\mathbb{R}} e^{i(j-\alpha n)\theta} e^{i c_3 n \theta^3} e^{-c_4 n \theta^4} d\theta. \quad (A.8) $$

The relevance of $\mathcal{G}_j^n$ for analyzing the behavior of the exact Green's function for (A.2) is illustrated by numerous simulations in [3]. A rigorous justification (by means of sharp analytical bounds) of these numerical observations is, to some extent, the purpose of the so-called local limit theorem and is the content of a future work by the authors. We show below that the bounds for $\mathcal{G}_j^n$ are “consistent” with those given in Theorem A.1 for the exact Green's function $\bar{\mathcal{G}}_j^n$. Let us quickly observe that the above definition (A.8) only makes sense for $n \in \mathbb{N}^*$ so that the integral is absolutely convergent. We have replaced the compact integration interval $[-\pi, \pi]$ by the whole real line $\mathbb{R}$ for convenience since high frequencies will not modify much the behavior of $\mathcal{G}_j^n$.

For technical reasons that will be made more clear below, it is useful to “extend” the approximate Green's function to a continuous setting and we therefore introduce the function of two variables $\mathbf{G}$ that is defined by:

$$ \forall (x, y) \in \mathbb{R} \times \mathbb{R}^{+*}, \quad \mathbf{G}(x, y) := \frac{1}{2\pi} \int_{\mathbb{R}} e^{ix\theta} e^{ic_3 y\theta^3} e^{-c_4 y\theta^4} d\theta. \quad (A.9) $$

From the definition (A.8), we directly get the relation $\mathcal{G}_j^n = \mathbf{G}(j - \alpha n, n)$ so that any detailed information or bounds on $\mathbf{G}$ will give us information or bounds on $\mathcal{G}_j^n$.

We thus consider from now on the function $\mathbf{G}$ defined in (A.9). Our main result, that is Theorem A.3 below, makes use of an auxiliary function that, to some extent, captures the oscillating behavior of the
function $\mathbf{G}$ on the side where $x$ has the opposite sign of $c_3$. This auxiliary function $\mathbf{g}$ is defined as follows:

$$ \forall (x, y) \in \mathbb{R} \times \mathbb{R}^{+*}, \quad \mathbf{g}(x, y) := \frac{1}{2\pi} \exp\left(-\frac{c_4 x^2}{9c_3^2 y}\right) \exp\left(\mathbf{i} \frac{2|x|^{3/2}}{3\sqrt{3|c_3|}y} - \mathbf{i} \frac{\pi}{4}\right) \\ \times \int_{-\sqrt{\frac{2|x|}{3|c_3|y}}}^{\sqrt{\frac{2|x|}{3|c_3|y}}} e^{-\sqrt{3|c_3|} |x| y t^2} e^{c_3 y e^{-i\pi/4} t^3} dt. \quad (A.10) $$

Comparing with (A.7), we see that (A.7) corresponds to $x = j - \alpha n$ and $y = n$ in (A.10). This explains why the proof of the above estimate (A.6) is entirely similar to the proof of (A.12) below. Our main result is then the following.

**Theorem A.3.** Let us assume that the coefficient $c_3$ in (A.4) is positive, that is $\alpha \in (0, 1)$. Let $y_{\min} > 0$ and let $\underline{c} > 0$ be given. Then there exist some constants<fnref n="1" /> $C > 0$ and $c > 0$ such that, for any $(x, y) \in \mathbb{R} \times [y_{\min}, +\infty)$, there holds:

$$ |\mathbf{G}(x, y)| \le \begin{cases} \frac{C}{y^{1/4}} \exp(-c x^{4/3} / y^{1/3}), & \text{if } x \ge \underline{c} y, \\ \frac{C}{y^{1/3}} \exp(-c x^{3/2} / y^{1/2}), & \text{if } 0 \le x \le \underline{c} y, \\ \frac{C}{y^{1/3}}, & \text{if } -y^{1/3} \le x \le 0, \\ \frac{C}{y^{1/4}} \exp(-c |x|^{4/3} / y^{1/3}), & \text{if } x \le -\underline{c} y, \end{cases} \quad (A.11) $$

and:

$$ |\mathbf{G}(x, y) - 2 \operatorname{Re} \mathbf{g}(x, y)| \le \frac{C}{y^{1/3}} \exp \left( -c \frac{|x|^{3/2}}{y^{1/2}} \right) + \frac{C}{y^{1/2}} \exp \left( -c \frac{x^2}{y} \right), \quad (A.12) $$

if $-\underline{c} y \le x \le -y^{1/3}$.

In particular, there exist another constant $\tilde{C} \ge C$ and another constant $\tilde{c} \in (0, c]$ such that for any $(x, y) \in \mathbb{R} \times [y_{\min}, +\infty)$, there holds:

$$ \left| \int_{-\infty}^{x} \mathbf{G}(\xi, y) \mathrm{d}\xi \right| \le \begin{cases} \tilde{C} \exp(-\tilde{c} |x|^{4/3} / y^{1/3}), & \text{if } x \le -\underline{c} y, \\ \tilde{C}, & \text{if } -\underline{c} y \le x \le \underline{c} y. \end{cases} \quad (A.13) $$

and:

$$ \left| 1 - \int_{-\infty}^{x} \mathbf{G}(\xi, y) \mathrm{d}\xi \right| \le \tilde{C} \exp(-\tilde{c} x^{4/3} / y^{1/3}), \quad \text{if } x \ge \underline{c} y. \quad (A.14) $$

Let us note that the case where $c_3$ is negative (that is, where $\alpha$ is negative) simply corresponds to changing the sign of $x$ since the definition (A.9) gives:

$$ \mathbf{G}(-x, y) := \frac{1}{2\pi} \int_{\mathbb{R}} e^{i x \theta} e^{i (-c_3) y \theta^3} e^{-c_4 y \theta^4} d\theta. $$

We shall thus feel free to use without proof the analog of Theorem A.3 in the case where $\alpha$ is negative. (This is precisely what we shall do in Chapter 4 where one of the relevant values for $\alpha$ is $\alpha_r \in (-1, 0).$)

<footnote n="1">¹These constants depend only on $y_{\min}$, $\underline{c}$, $c_3$ and $c_4$, or equivalently on $y_{\min}$, $\underline{c}$ and $\alpha$.</footnote>
As follows from [29], see also [16, 17], it is already known that the Green's function $(\overline{\mathcal{G}}_j^n)_{(j,n)\in\mathbb{Z}\times\mathbb{N}}$ of the Lax-Wendroff scheme (A.2) is not uniformly bounded in $\ell^1(\mathbb{Z})$ (where "uniformly" refers to uniformity with respect to $n \in \mathbb{N}$). This is a characteristic feature of numerical schemes that exhibit dispersion. Since the function $G$ is meant to reproduce the leading behavior of the Green's function, there is no hope to show that $G(\cdot, y)$ belongs to $L_x^1$ uniformly with respect to $y > 0$ (or, say, $y \ge y_{\min} > 0$ in order to get rid of potential trouble when $y$ becomes arbitrarily small). However, a crucial point in our analysis of the linearized operator $\mathcal{L}$ around our reference discrete shock profile is to obtain a uniform bound for the activation function $A$. As we shall see later on, this activation function corresponds to the primitive function of $G$ with respect to its first variable. Deriving a bound for this primitive function can not follow from the triangle inequality. We thus need to capture the leading oscillating behavior of $G$ in order to show that its primitive (with respect to its first variable) is uniformly bounded with respect to both variables. This is the meaning of the estimates (A.13) and (A.14) and this is where the estimate (A.12) is crucial.

The proof of Theorem A.3 is split in several paragraphs in order to emphasize the distinctions between the fast decaying side of the Green's function and its oscillating side (as for the Airy function). Most of what follows is an adaptation of the results in [9] even though several regimes did not appear in [9] because the exact Green's function was considered there and this one has compact support for any $n$. Furthermore, it appears that an error occurred in [9] when introducing the analog of the above function $g$ and in proving an error bound between the Green's function and its (presumably) leading behavior. This appendix is therefore the opportunity to correct this error and to generalize some of the results in [9] to a continuous setting. In the final paragraph of this appendix, we connect the result of Theorem A.3 with the analysis of the activation function $A_r$ (or $A_\ell$), see Corollary A.6, and we explain why the bounds in Theorem A.3 give exactly all that is needed in the analysis of Sections 4.3 and 4.5.

## A.2 The uniform bound

It is sometimes useful to consider $y > 0$ in (A.9) and later restrict to $y \ge y_{\min}$ as in Theorem A.3. This is mostly what we shall do in what follows, where the final restriction $y \ge y_{\min}$ will be used to derive bounds as claimed in Theorem A.3. We first derive a uniform bound for $G(x, y)$ with respect to $x \in \mathbb{R}$.

**Proposition A.1.** Let the function $G$ be defined in (A.9) with a nonzero constant $c_3$. Then there exists a constant $C > 0$ such that for any $y > 0$, there holds:

$$ \sup_{x \in \mathbb{R}} |G(x, y)| \le \frac{C}{y^{1/3}}. $$

*Proof.* The proof of Proposition A.1 is a mere adaptation of the proof of [9, Proposition 2.3]. We recall the details for the sake of completeness. From [24, Lemma 3.1] and the so-called van der Corput Lemma, we have:

$$ \left| \int_a^b e^{i f(\theta)} g(\theta) d\theta \right| \le \frac{C_0}{\left( \min_{\theta \in [a,b]} |f^{(3)}(\theta)| \right)^{1/3}} \left( \|g\|_{L^\infty([a,b])} + \|g'\|_{L^1([a,b])} \right), \quad (A.15) $$

as long as $f$ is real valued and the minimum of $|f^{(3)}|$ on the interval $[a, b]$ is positive. The crucial observation here is that the constant $C_0$ in (A.15) does not depend on $a$ nor $b$ (nor $f$ and $g$, of course). The function $g$ could be complex valued.
We apply the above inequality to $f(\theta) := x\theta + c_3 y\theta^3$ and $g(\theta) := \exp(-c_4 y\theta^4)$, so that both norms $\|g\|_{L^\infty([a,b])}$ and $\|g'\|_{L^1([a,b])}$ can be bounded uniformly with respect to $a, b$ and $y$. We thus obtain the bound:

$$ \left| \int_a^b e^{i x \theta} e^{i c_3 y \theta^3} e^{-c_4 y \theta^4} d\theta \right| \le \frac{C}{y^{1/3}}, $$

and it remains to pass to the limit in both $a$ and $b$ to conclude, the constant $C$ being independent of the interval $[a, b]$. $\square$

## A.3 The fast decaying side

We assume from now on that the coefficient $c_3$ in (A.4) is positive, the case where this coefficient is negative being obtained by switching the sign of $x$. Now that the sign of $c_3$ is fixed, we first deal with the case $x \ge 0$.

**Proposition A.2.** Let the function $G$ be defined in (A.9) and let the constant $c_3$ be positive. Then there exist some constants $c_\# > 0$, $C > 0$ and $c > 0$ such that for any $(x, y) \in \mathbb{R} \times \mathbb{R}^{+*}$, there holds:

$$ |\mathbf{G}(x, y)| \le \begin{cases} \frac{C}{y^{1/4} \max(1, x^{1/4})} \exp \left( -c \frac{x^{3/2}}{y^{1/2}} \right), & \text{if } 0 \le x \le \mathbf{c}_\# y, \\ \frac{C}{y^{1/2}} \exp \left( -c \frac{x^{4/3}}{y^{1/3}} \right), & \text{if } x \ge \mathbf{c}_\# y. \end{cases} $$

*Proof.* Integrating the holomorphic function:

$$ z \in \mathbb{C} \mapsto e^{i x z} e^{i c_3 y z^3} e^{-c_4 y z^4} $$

over a large rectangle and passing to the limit, we easily see that the real line $\mathbb{R}$ over which we integrated it to obtain the defining equation (A.9) can be switched to any line $i\mu + \mathbb{R}$ with $\mu \in \mathbb{R}$. In other words, the Cauchy formula yields:

$$ \forall \mu \in \mathbb{R}, \quad \forall (x, y) \in \mathbb{R} \times \mathbb{R}^{+*}, \quad \mathbf{G}(x, y) = \frac{1}{2\pi} \int_{\mathbb{R}} e^{i x (i\mu + \theta)} e^{i c_3 y (i\mu + \theta)^3} e^{-c_4 y (i\mu + \theta)^4} d\theta. \quad (A.16) $$

Expanding all quantities within the integral and applying the triangle inequality, we obtain the bound:

$$ \forall \mu \in \mathbb{R}, \quad \forall (x, y) \in \mathbb{R} \times \mathbb{R}^{+*}, \quad |\mathbf{G}(x, y)| \le \frac{e^{-x \mu + c_3 y \mu^3 - c_4 y \mu^4}}{2\pi} \int_{\mathbb{R}} e^{-3y\mu(c_3 - 2c_4\mu)\theta^2} e^{-c_4 y \theta^4} d\theta. \quad (A.17) $$

This bound is rather crude but it allows to deal with many regimes in $(x, y)$ by optimizing with respect to the free parameter $\mu \in \mathbb{R}$.

Let us now consider $x \ge 0$ and let us choose $\mu = \mu_0 := (x/(3c_3y))^{1/2} \ge 0$. We use the bound (A.17) and use the obvious inequality $\exp(-c_4 y \mu_0^4) \le 1$ to obtain:

$$ |\mathbf{G}(x, y)| \le \frac{1}{2\pi} \exp \left( -\frac{2 x^{3/2}}{3\sqrt{3} c_3 y^{1/2}} \right) \int_{\mathbb{R}} e^{-3y\mu_0(c_3 - 2c_4\mu_0)\theta^2} e^{-c_4 y \theta^4} d\theta. $$
We define once and for all the positive constant $c_{\sharp} := 3c_3^3/(16c_4^2)$. If we consider the regime $0 \le x \le c_{\sharp}y$, then we have $2c_4\mu_0 \le c_3/2$, so that we get:

$$|\mathbf{G}(x, y)| \le \frac{1}{2\pi} \exp\left(-\frac{2x^{3/2}}{3\sqrt{3c_3}y^{1/2}}\right) \int_{\mathbb{R}} e^{-(3/2)c_3 y \mu_0 \theta^2} e^{-c_4 y \theta^4} d\theta.$$

Since $x$ is nonnegative (and therefore $\mu_0$ is nonnegative too), we can either use one exponential term or the other within the integral to derive a bound for this integral. In the case $x = 0$, only the exponential term $\exp(-c_4 y \theta^4)$ makes the integral converge. Recalling the value $\mu_0 = (x/(3c_3y))^{1/2}$, we thus get:

$$\int_{\mathbb{R}} e^{-(3/2)c_3 y \mu_0 \theta^2} e^{-c_4 y \theta^4} d\theta \le \frac{C}{y^{1/4} \max(1, x^{1/4})},$$

with a constant $C$ that is independent of $y$ and $x \in [0, c_{\sharp}y]$. This yields our first bound (see the statement of Proposition A.2):

$$\forall x \in [0, c_{\sharp}y], \quad |\mathbf{G}(x, y)| \le \frac{C}{y^{1/4} \max(1, x^{1/4})} \exp\left(-c \frac{x^{3/2}}{y^{1/2}}\right). \qquad (A.18)$$

We assume from now on $x \ge c_{\sharp}y > 0$ with the above (already fixed) positive constant $c_{\sharp} = 3c_3^3/(16c_4^2)$. We go back to (A.17) and restrict from now on to positive values of the parameter $\mu$. We use Young's inequality:

$$6c_4 y \mu^2 \theta^2 \le c_4 y \theta^4 + 9c_4 y \mu^4,$$

to derive the bound:

$$\forall \mu > 0, \quad |\mathbf{G}(x, y)| \le \frac{C}{\sqrt{y\mu}} \exp(-x\mu + c_3 y \mu^3 + 8c_4 y \mu^4), \qquad (A.19)$$

where the constant $C$ is independent of $x, y$ and $\mu$. Our final choice for $\mu$ will depend on $x$ and $y$ so it is crucial to get constants that do not depend on $\mu$ all along.

The function :

$$f: \mu \in \mathbb{R}^+ \mapsto -x\mu + c_3 y \mu^3 + 8c_4 y \mu^4, \qquad (A.20)$$

is smooth and strictly convex. Since $f(0) = 0$, $f'(0) < 0$ and $f$ tends to $+\infty$ at $+\infty$, it appears that $f$ attains its unique global minimum on $\mathbb{R}^+$ at some $\underline{\mu} > 0$ that is characterized by $f'(\underline{\mu}) = 0$. Multiplying the relation $f'(\underline{\mu}) = 0$ by $\underline{\mu}$, we obtain:

$$f(\underline{\mu}) = -2c_3 y \underline{\mu}^3 - 24c_4 y \underline{\mu}^4 \le -24c_4 y \underline{\mu}^4.$$

Since we have $x/y \ge c_{\sharp}$, it is not hard to show that we can choose some small positive constant $c_{\sharp} > 0$, whose choice only depends on $c_3, c_4$ $c_{\sharp}$, and such that:

$$f'\left(\mathbf{c}_{\sharp} \frac{x^{1/3}}{y^{1/3}}\right) < 0.$$

From the strict convexity of $f$, this means that we have $\underline{\mu} \ge c_{\sharp} (x/y)^{1/3}$ for some constant $c_{\sharp}$ that only depends on $c_3$ and $c_4$ (we recall that $c_{\sharp}$ only depends on $c_3$ and $c_4$).
Going back to (A.19) and using $\underline{\mu} \ge c_{\#} (x/y)^{1/3}$, we thus obtain the bound:

$$|\mathbf{G}(x, y)| \le \frac{C}{\sqrt{y \underline{\mu}}} \exp(f(\underline{\mu})) \le \frac{C}{\sqrt{y \underline{\mu}}} \exp(-24 c_4 y \underline{\mu}^4) \le \frac{C}{y^{1/3} x^{1/6}} \exp\left(-c \frac{x^{4/3}}{y^{1/3}}\right).$$

We simplify a little bit more this last bound by using again the inequality $x \ge c_{\#} y$ and this yields the estimate that we announced in Proposition A.2, namely:

$$|\mathbf{G}(x, y)| \le \frac{C}{y^{1/2}} \exp\left(-c \frac{x^{4/3}}{y^{1/3}}\right).$$

The proof of Proposition A.2 is now complete.

<figure><img src="image_20.png" /></figure>

Let us observe that in Proposition A.2, the parameter $y$ can take arbitrarily small positive values. We now restrict to $y \ge y_{\min} > 0$ in order to prove the result of Theorem A.3. This will be done in two steps.

**Corollary A.2.** Let the function $\mathbf{G}$ be defined in (A.9) and let $c_3$ be positive. Let $y_{\min} > 0$ be given. Then there exist some constants $c_{\#} > 0$, $C > 0$ and $c > 0$ such that for any $(x, y) \in \mathbb{R} \times [y_{\min}, +\infty)$, there holds:

$$|\mathbf{G}(x, y)| \le \begin{cases} \frac{C}{y^{1/3}} \exp\left(-c \frac{x^{3/2}}{y^{1/2}}\right), & \text{if } 0 \le x \le c_{\#} y, \\ \frac{C}{y^{1/4}} \exp\left(-c \frac{x^{4/3}}{y^{1/3}}\right), & \text{if } x \ge c_{\#} y. \end{cases}$$

*Proof.* We fix the constant $c_{\#} > 0$ as the one given in Proposition A.2. Let us first assume $x \ge c_{\#} y$ so that Proposition A.2 gives:

$$|\mathbf{G}(x, y)| \le \frac{C}{y^{1/2}} \exp\left(-c \frac{x^{4/3}}{y^{1/3}}\right).$$

We then use the bound $y^{1/2} \ge y_{\min}^{1/4} y^{1/4}$ to conclude. We can now assume $0 \le x \le c_{\#} y$ so that Proposition A.2 gives:

$$|\mathbf{G}(x, y)| \le \frac{C}{y^{1/4} \max(1, x^{1/4})} \exp\left(-c \frac{x^{3/2}}{y^{1/2}}\right).$$

In particular, we have $\max(1, x^{1/4}) \ge x^{1/4}$ so if $x \ge y^{1/3} > 0$, we get:

$$|\mathbf{G}(x, y)| \le \frac{C}{y^{1/4} x^{1/4}} \exp\left(-c \frac{x^{3/2}}{y^{1/2}}\right) \le \frac{C}{y^{1/3}} \exp\left(-c \frac{x^{3/2}}{y^{1/2}}\right). \quad (A.21)$$

It remains to examine the case $0 \le x \le \min(y^{1/3}, c_{\#} y)$. Proposition A.1 gives the bound:

$$|\mathbf{G}(x, y)| \le \frac{C}{y^{1/3}} \le \frac{C e^c}{y^{1/3}} \exp\left(-c \frac{x^{3/2}}{y^{1/2}}\right),$$

with the same constant $c > 0$ as in (A.21). This gives the expected bound for $|\mathbf{G}(x, y)|$ in the regime $0 \le x \le c_{\#} y$. The proof of Corollary A.2 is complete.

<figure><img src="image_21.png" /></figure>
It appears that allowing us to choose the value of the constant $c_\#$ that separates the two regimes in Corollary A.2 will be convenient in the analysis (while in Corollary A.2 the constant $c_\#$ is given and is therefore not free to be fixed). We thus prove the following result which is a slight adaptation of Corollary A.2.

**Corollary A.3.** Let the function $\mathbf{G}$ be defined in (A.9) and let $c_3$ be positive. Let $y_{\min} > 0$ be given. Then for any constant $\mathbf{c} > 0$, there exist some constants $C > 0$ and $c > 0$ such that for any $(x, y) \in \mathbb{R} \times [y_{\min}, +\infty)$, there holds:

$$|\mathbf{G}(x, y)| \le \begin{cases} \frac{C}{y^{1/3}} \exp \left( -c \frac{x^{3/2}}{y^{1/2}} \right), & \text{if } 0 \le x \le \underline{c} y, \\ \frac{C}{y^{1/4}} \exp \left( -c \frac{x^{4/3}}{y^{1/3}} \right), & \text{if } x \ge \overline{c} y. \end{cases}$$

*Proof.* We let $\mathbf{c} > 0$ be given and assume that there holds $\mathbf{c} \ge c_\#$ where $c_\#$ is given in Corollary A.2. (The case $\mathbf{c} \le c_\#$ is dealt with similarly as below.) Assuming first $x \ge c\mathbf{y}$, we have $x \ge c_\#\mathbf{y}$ and we can therefore use Corollary A.2 and obtain the desired bound for $|\mathbf{G}(x, y)|$. Corollary A.2 also gives the desired bound in the regime $0 \le x \le c_\#\mathbf{y}$ and it therefore only remains to consider the regime $c_\#\mathbf{y} \le x \le c\mathbf{y}$. From Corollary A.2, we already have the bound:

$$|\mathbf{G}(x, y)| \le \frac{C_1}{y^{1/4}} \exp \left( -c_1 \frac{x^{4/3}}{y^{1/3}} \right),$$

for some constants $C_1$ and $c_1$ that are independent of $x$ and $y$. Since we have $c_\#\mathbf{y} \le x$, we can first use a slight part of the exponential term to improve the power of the algebraic factor:

$$|\mathbf{G}(x, y)| \le \frac{C_1}{y^{1/4}} \exp(-c_1 \mathbf{c}_\#^{4/3} y) \le \frac{C_2}{y^{1/3}} \exp \left( -\frac{c_1 \mathbf{c}_\#^{4/3}}{2} y \right),$$

and we now wish to show that the right-hand side of the latter inequality is less than:

$$\frac{C}{y^{1/3}} \exp \left( -c \frac{x^{3/2}}{y^{1/2}} \right)$$

for some appropriate constants $C$ and $c$ (recalling that $x$ belongs to the interval $[c_\#\mathbf{y}, c\mathbf{y}]$). We fix $c_2 := c_1 \mathbf{c}_\#^{4/3} / (2c^{3/2})$ so that we have (we use $x \le c\mathbf{y}$):

$$\exp \left( -\frac{c_1 \mathbf{c}_\#^{4/3}}{2} y \right) \le \exp \left( -c_2 \frac{x^{3/2}}{y^{1/2}} \right),$$

and we therefore obtain:

$$|\mathbf{G}(x, y)| \le \frac{C_2}{y^{1/3}} \exp \left( -c_2 \frac{x^{3/2}}{y^{1/2}} \right),$$

as expected. The proof of Corollary A.3 is complete.

<figure><img src="image_22.png" /></figure>
## A.4 The oscillating side

We still assume that the coefficient $c_3$ in (A.4) is positive but we now consider the case where $x$ is negative. This corresponds to the oscillating side of the Green's function. The analysis is split in four regimes. The case $|x| \le y^{1/3}$ is dealt with straightforwardly with the uniform bound of Proposition A.1. The second (and most difficult) regime corresponds to $x \le -y^{1/3}$ and $|x|/y$ is sufficiently small. The third regime corresponds to the case where $|x|/y$ is sufficiently large and the fourth and last regime corresponds to the case where $|x|$ and $y$ are of comparable sizes. In the end, we collect all bounds and prove the result announced in Theorem A.3. We start with the most difficult case for which $|x|/y$ is small.

**Proposition A.3.** Let the function $\mathbf{G}$ be defined in (A.9) and let $c_3$ be positive. Let the function $\mathbf{g}$ be defined in (A.10). Then there exist some constants $c_\flat > 0$, $C > 0$ and $c > 0$ such that for any $(x, y) \in \mathbb{R} \times \mathbb{R}^{+*}$, there holds:

$$|\mathbf{G}(x, y) - 2 \operatorname{Re} \mathbf{g}(x, y)| \le \frac{C}{y^{1/3} \min(1, y^{1/3})} \exp \left(-c \frac{|x|^{3/2}}{y^{1/2}}\right) + \frac{C}{y^{1/2}} \exp \left(-c \frac{x^2}{y}\right),$$

$$\text{if } -\mathbf{c}_\flat y \le x \le -y^{1/3}.$$

*Proof.* We always consider $x < 0$ and introduce the notation $\omega := |x|/y$. We follow the analysis in [9] with slight modifications since this is precisely at this stage that [9] has some incorrect argument. Rather than integrating the function:

$$z \in \mathbb{C} \mapsto e^{ixz} e^{ic_3 y z^3} e^{-c_4 y z^4}$$

on a horizontal line, we apply once again the Cauchy formula and use the contour depicted in Figure A.1, that is:

* A horizontal half-line from $-\infty + i\sqrt{\omega/(3c_3)}$ to $-2\sqrt{\omega/(3c_3)} - (2c_4/(9c_3^2))\omega + i\sqrt{\omega/(3c_3)}$,

* A segment (with slope $-\pi/4$) from the point $-2\sqrt{\omega/(3c_3)} - (2c_4/(9c_3^2))\omega + i\sqrt{\omega/(3c_3)}$ to the point $-i(\sqrt{\omega/(3c_3)} + (2c_4/(9c_3^2))\omega)$,

* A segment (with slope $\pi/4$) from the point $-i(\sqrt{\omega/(3c_3)} + (2c_4/(9c_3^2))\omega)$ to the point $2\sqrt{\omega/(3c_3)} + (2c_4/(9c_3^2))\omega + i\sqrt{\omega/(3c_3)}$,

* A horizontal half-line from $2\sqrt{\omega/(3c_3)} + (2c_4/(9c_3^2))\omega + i\sqrt{\omega/(3c_3)}$ to $+\infty + i\sqrt{\omega/(3c_3)}$.

The corresponding contributions $\varepsilon_1(x, y)$, $\mathcal{H}_\flat(x, y)$, $\mathcal{H}_\sharp(x, y)$ and $\varepsilon_2(x, y)$ are reported in Figure A.1 and their exact expressions are given below.

In this way, we obtain the decomposition:

$$\mathbf{G}(x, y) = \varepsilon_1(x, y) + \varepsilon_2(x, y) + \mathcal{H}_\flat(x, y) + \mathcal{H}_\sharp(x, y), \quad (A.22)$$

where the four contributions $\varepsilon_1(x, y)$, $\varepsilon_2(x, y)$, $\mathcal{H}_\flat(x, y)$ and $\mathcal{H}_\sharp(x, y)$ have the following expressions:

$$\varepsilon_1(x, y) = \frac{e^{-x \mu + c_3 y \mu^3 - c_4 y \mu^4}}{2\pi} \int_{-\infty}^{-\Xi(\omega)} e^{ix\theta + i c_3 y \theta^3 - i 3 c_3 y \mu^2 \theta + i 4 c_4 y \mu^3 \theta - i 4 c_4 y \mu \theta^3} \times e^{-3 c_3 y \mu \theta^2 + 6 c_4 y \mu^2 \theta^2} e^{-c_4 y \theta^4} d\theta,$$
<figure><img src="image_23.png" /><figcaption>Figure A.1: The integration contour in the case $x \le -y^{1/3}$ and $|x|/y$ small. The two red bullets represent the approximate saddle points of the phase $-i\omega z + i c_3 z^3 - c_4 z^4$ and the black bullets represent the end points of the contours along which we compute the contributions $\varepsilon_1(x, y)$, $\varepsilon_2(x, y)$, $\mathcal{H}_b(x, y)$, $\mathcal{H}_\#(x, y)$.</figcaption></figure>

where the upper bound $\Xi(\omega)$ and the parameter $\mu$ in the expression of $\varepsilon_1(x, y)$ are defined as:

$$ \Xi(\omega) := 2 \sqrt{\frac{\omega}{3c_3}} + \frac{2c_4}{9c_3^2}\omega, \quad \mu := \sqrt{\frac{\omega}{3c_3}}. \qquad (A.23) $$

The contribution $\varepsilon_2(x, y)$ has the following expression (with the same definition for $\Xi(\omega)$ and $\mu$):

$$ \varepsilon_2(x, y) = \frac{e^{-x\mu+c_3y\mu^3-c_4y\mu^4}}{2\pi} \int_{\Xi(\omega)}^{+\infty} e^{ix\theta+i c_3 y \theta^3 - i 3 c_3 y \mu^2 \theta + i 4 c_4 y \mu^3 \theta - i 4 c_4 y \mu \theta^3} \times e^{-3c_3 y \mu \theta^2 + 6 c_4 y \mu^2 \theta^2} e^{-c_4 y \theta^4} d\theta = \overline{\varepsilon_1(x, y)}. \quad (A.24) $$

The contribution $\mathcal{H}_b(x, y)$ is given by:

$$ \mathcal{H}_b(x, y) = \frac{e^{-i\pi/4}}{2\pi} \int_{-\sqrt{\frac{2\omega}{3c_3}} - \frac{2\sqrt{2}c_4}{9c_3^2}\omega}^{\sqrt{\frac{2\omega}{3c_3}}} e^{-i\omega y \Theta(t) + i c_3 y \Theta(t)^3 - c_4 y \Theta(t)^4} dt, \quad (A.25) $$

with:

$$ \forall t \in \left[ -\sqrt{\frac{2\omega}{3c_3}} - \frac{2\sqrt{2}c_4}{9c_3^2}\omega, \sqrt{\frac{2\omega}{3c_3}} \right], \quad \Theta(t) := -\sqrt{\frac{\omega}{3c_3}} - \frac{2c_4}{9c_3^2}\omega + t e^{-i\pi/4}, \quad (A.26) $$

which corresponds to the parametrization of the first slanted segment in Figure A.1 (the one with slope $-\pi/4$). The parametrization of the second segment is entirely similar and we obtain the expression:

$$ \mathcal{H}_\#(x, y) = \overline{\mathcal{H}_b(x, y)}. $$

Let us start with the estimate of $\varepsilon_2(x, y)$ whose expression is given in (A.24). By using the expression for the parameter $\mu$ and recalling that we have $x = -\omega y$, we get:

$$ \varepsilon_2(x, y) = \frac{1}{2\pi} \exp \left( \frac{4\omega^{3/2}y}{3\sqrt{3}c_3} - \frac{c_4}{9c_3^2}\omega^2 y \right) \int_{\Xi(\omega)}^{+\infty} e^{i\dots} \times e^{-\sqrt{3}c_3\overline{\omega}\omega\theta^2 + \frac{2c_4}{c_3}\omega y\theta^2} e^{-c_4y\theta^4} d\theta, $$
where the three dots within the integral stand for a real quantity whose precise expression is useless since we are going to use the triangle inequality so that the modulus of this exponential term will be bounded by 1. Indeed, applying the triangle inequality yields the bound:

$$ \begin{aligned} |\varepsilon_2(x, y)| & \le \frac{1}{2\pi} \exp\left(\frac{4\omega^{3/2}y}{3\sqrt{3}c_3} - \frac{c_4}{9c_3^2}\omega^2 y\right) \int_{\Xi(\omega)}^{+\infty} e^{-\sqrt{3}c_3\omega y\theta^2 + \frac{2c_4}{c_3}\omega y\theta^2} e^{-c_4 y\theta^4} d\theta \\ & \le \frac{1}{2\pi} \exp\left(\frac{4\omega^{3/2}y}{3\sqrt{3}c_3}\right) \int_{\Xi(\omega)}^{+\infty} e^{-\sqrt{3}c_3\omega y\theta^2 + \frac{2c_4}{c_3}\omega y\theta^2} e^{-c_4 y\theta^4} d\theta. \end{aligned} $$

We first restrict $\omega = |x|/y$ by imposing:

$$ \omega \le \frac{3c_3^3}{16c_4^2}, \qquad (A.27) $$

so that we have:

$$ \frac{2c_4}{c_3}\omega \le \frac{1}{2}\sqrt{3c_3\omega}. $$

We thus get:

$$ \begin{aligned} |\varepsilon_2(x, y)| & \le \frac{1}{2\pi} \exp\left(\frac{4\omega^{3/2}y}{3\sqrt{3}c_3}\right) \int_{\Xi(\omega)}^{+\infty} e^{-\frac{\sqrt{3}c_3\omega}{2}y\theta^2} e^{-c_4 y\theta^4} d\theta \\ & \le \frac{1}{2\pi} \exp\left(\frac{4\omega^{3/2}y}{3\sqrt{3}c_3}\right) \int_{\Xi(\omega)}^{+\infty} e^{-\frac{\sqrt{3}c_3\omega}{2}y\theta^2} d\theta \\ & \le \frac{1}{2\pi} \exp\left(\frac{4\omega^{3/2}y}{3\sqrt{3}c_3}\right) \int_{2\sqrt{\frac{\omega}{3c_3}}}^{+\infty} e^{-\frac{\sqrt{3}c_3\omega}{2}y\theta^2} d\theta, \end{aligned} $$

where the final inequality comes from the definition of $\Xi(\omega)$. We then use the inequality:

$$ \forall a > 0, \quad \forall X > 0, \quad \int_X^{+\infty} e^{-a\theta^2} d\theta \le \frac{1}{2aX} e^{-aX^2}, $$

to obtain (for a suitable constant $C$):

$$ |\varepsilon_2(x, y)| \le \frac{C}{\omega y} \exp\left(\frac{4\omega^{3/2}y}{3\sqrt{3}c_3}\right) \exp\left(-\frac{2\omega^{3/2}y}{\sqrt{3}c_3}\right) = \frac{C}{\omega y} \exp\left(-\frac{2\omega^{3/2}y}{3\sqrt{3}c_3}\right), $$

and this gives, going back to $x$ and $y$, the estimate:

$$ |\varepsilon_2(x, y)| \le \frac{C}{|x|} \exp\left(-c \frac{|x|^{3/2}}{y^{1/2}}\right). $$

Restricting to the regime $x \le -y^{1/3}$, we obtain the final estimate:

$$ |\varepsilon_2(x, y)| \le \frac{C}{y^{1/3}} \exp\left(-c \frac{|x|^{3/2}}{y^{1/2}}\right), \qquad (A.28) $$
as long as we have $x \le -y^{1/3}$ and $|x|/y \le 3c_3^3/(16c_4^2)$. Of course a similar estimate holds for $\varepsilon_1(x, y)$ since it is the complex conjugate of $\varepsilon_2(x, y)$.

We now turn to the contribution $\mathcal{H}_b(x, y)$ in (A.25). With the definition (A.26) for $\Theta(t)$, we expand:

$$-i\omega y \Theta(t) + ic_3 y \Theta(t)^3 - c_4 y \Theta(t)^4 = y \sum_{k=0}^{4} p_k(\omega) t^k,$$

where the quantities $p_0(\omega), \dots, p_4(\omega)$ have the following behavior<fnref n="2" /> as $\omega > 0$ tends to zero:

$$\mathrm{Re} \, p_0(\omega) = -\frac{c_4}{9c_3^2} \omega^2 + O(\omega^3), \qquad (\mathrm{A.29a})$$

$$\mathrm{Im} \, p_0(\omega) = \frac{2}{3\sqrt{3}c_3} \omega^{3/2} + O(\omega^{5/2}), \qquad (\mathrm{A.29b})$$

$$p_1(\omega) = O(\omega^2), \qquad (\mathrm{A.29c})$$

$$\mathrm{Re} \, p_2(\omega) = -\sqrt{3}c_3\omega + O(\omega^{3/2}), \qquad (\mathrm{A.29d})$$

$$\mathrm{Im} \, p_2(\omega) = O(\omega), \qquad (\mathrm{A.29e})$$

$$p_3(\omega) = c_3 e^{-i\pi/4} + O(\omega^{1/2}), \qquad (\mathrm{A.29f})$$

$$p_4(\omega) = c_4. \qquad (\mathrm{A.29g})$$

We thus have:

$$\mathcal{H}_b(x, y) = \frac{e^{p_0(\omega) y - i\pi/4}}{2\pi} \int_{-\sqrt{\frac{2\omega}{3c_3}} - \frac{2\sqrt{2}c_4}{9c_3^2}\omega}^{\sqrt{\frac{2\omega}{3c_3}}} \exp\left(\sum_{k=1}^{4} p_k(\omega) y t^k\right) dt.$$

We now use the inequality:

$$\forall z \in \mathbb{C}, \quad |e^z - 1| \le |z| e^{|z|}, \qquad (\mathrm{A.30})$$

with the specific value:

$$z := y \left\{ p_1(\omega) t + (p_2(\omega) + \sqrt{3}c_3\omega) t^2 + (p_3(\omega) - c_3 e^{-i\pi/4}) t^3 + p_4(\omega) t^4 \right\}.$$

On the considered integration interval for $t$, we thus have (see (A.29c), (A.29d), (A.29e), (A.29f), (A.29g)):

$$|z| \le C y (\omega^{5/2} + \omega t^2),$$

with a constant $C$ that is independent of $y$, $\omega$ and $t$ (for the relevant values of $t$). This gives the estimate:

$$\left| \mathcal{H}_b(x, y) - \frac{e^{p_0(\omega) y - i\pi/4}}{2\pi} \int_{-\sqrt{\frac{2\omega}{3c_3}} - \frac{2\sqrt{2}c_4}{9c_3^2}\omega}^{\sqrt{\frac{2\omega}{3c_3}}} \exp\left(-\sqrt{3}c_3\omega y t^2 + ic_3 e^{-i\pi/4} y t^3\right) dt \right| \\ \le C y e^{\mathrm{Re} \, p_0(\omega) y + C\omega^{5/2} y} \int_{-\sqrt{\frac{2\omega}{3c_3}} - \frac{2\sqrt{2}c_4}{9c_3^2}\omega}^{\sqrt{\frac{2\omega}{3c_3}}} (\omega^{5/2} + \omega t^2) e^{-\sqrt{3}c_3\omega y t^2 + \frac{c_3}{\sqrt{2}} y t^3 + C\omega y t^2} dt.$$

<footnote n="2">²Here we see that our choice of contour only uses approximate saddle points because $p_1(\omega)$ is not zero. This does not matter so much since $p_1(\omega)$ will be small enough to produce error terms that can be controlled in our analysis.</footnote>
We first use (A.29a) to estimate the real part of $p_0(\omega)$ and choose $\omega$ small enough so that we have:

$$ \mathrm{Re} \, p_0(\omega) + C \, \omega^{5/2} \leq -c \, \omega^2, $$

for a suitable constant $c > 0$ that does not depend on $\omega$. We thus get the estimate:

$$ \left| \mathcal{H}_b(x, y) - \frac{e^{p_0(\omega)y - i\pi/4}}{2\pi} \int_{-\sqrt{\frac{2\omega}{3c_3}} - \frac{2\sqrt{2}c_4}{9c_3^2}}^{\sqrt{\frac{2\omega}{3c_3}}} \exp\left(-\sqrt{3c_3\omega}y t^2 + i\mathbf{c}_3 e^{-i\pi/4}y t^3\right) dt \right| \\ \leq C y e^{-c\omega^2 y} \int_{-\sqrt{\frac{2\omega}{3c_3}} - \frac{2\sqrt{2}c_4}{9c_3^2}}^{\sqrt{\frac{2\omega}{3c_3}}} \left(\omega^{5/2} + \omega t^2\right) e^{-\sqrt{3c_3\omega}y t^2 + \frac{c_3}{\sqrt{2}}y t^3 + C\omega y t^2} dt. $$

As in [9], let us now observe that when $t$ is nonpositive in the integral, we have $c_3 t^3 \le 0$ (recall that $c_3$ is positive), and when $t$ is positive, on the considered integration interval, we have:

$$ \frac{c_3}{\sqrt{2}} t^3 \le \sqrt{\frac{c_3 \omega}{3}} t^2. $$

If $\omega$ is chosen sufficiently small (and this choice is, of course, independent of $y$), we thus get:

$$ e^{-\sqrt{3c_3\omega}yt^2 + \frac{c_3}{\sqrt{2}}yt^3 + C\omega y t^2} \le e^{-c\sqrt{\omega}y t^2}, $$

for the relevant values of $t$. This gives the estimate:

$$ \left| \mathcal{H}_b(x, y) - \frac{e^{p_0(\omega)y - i\pi/4}}{2\pi} \int_{-\sqrt{\frac{2\omega}{3c_3}} - \frac{2\sqrt{2}c_4}{9c_3^2}}^{\sqrt{\frac{2\omega}{3c_3}}} \exp\left(-\sqrt{3c_3\omega}y t^2 + i\mathbf{c}_3 e^{-i\pi/4}y t^3\right) dt \right| \\ \leq C y e^{-c\omega^2 y} \int_{-\sqrt{\frac{2\omega}{3c_3}} - \frac{2\sqrt{2}c_4}{9c_3^2}}^{\sqrt{\frac{2\omega}{3c_3}}} \left(\omega^{5/2} + \omega t^2\right) e^{-c\sqrt{\omega}y t^2} dt. $$

Integrating now with respect to $t$ (over $\mathbb{R}$ rather than over the above compact interval), we thus have:

$$ \left| \mathcal{H}_b(x, y) - \frac{e^{p_0(\omega)y - i\pi/4}}{2\pi} \int_{-\sqrt{\frac{2\omega}{3c_3}} - \frac{2\sqrt{2}c_4}{9c_3^2}}^{\sqrt{\frac{2\omega}{3c_3}}} \exp\left(-\sqrt{3c_3\omega}y t^2 + i\mathbf{c}_3 e^{-i\pi/4}y t^3\right) dt \right| \\ \leq C y^{1/2} \omega^{9/4} e^{-c\omega^2 y} + \frac{C}{\sqrt{y}} \omega^{1/4} e^{-c\omega^2 y} \leq \frac{C}{\sqrt{y}} (1 + \omega^2 y) e^{-c\omega^2 y}. $$

Up to diminishing the constant $c > 0$ and increasing the constant $C$, we thus get the final estimate (recall $\omega = |x|/y$):

$$ \left| \mathcal{H}_b(x, y) - \frac{e^{p_0(\omega)y - i\pi/4}}{2\pi} \int_{-\sqrt{\frac{2\omega}{3c_3}} - \frac{2\sqrt{2}c_4}{9c_3^2}}^{\sqrt{\frac{2\omega}{3c_3}}} \exp\left(-\sqrt{3c_3\omega}y t^2 + i\mathbf{c}_3 e^{-i\pi/4}y t^3\right) dt \right| \\ \le \frac{C}{y^{1/2}} \exp\left(-c\frac{x^2}{y}\right). \quad (A.31) $$
We now use the inequality (A.30) one more time with the value:

$$z := y \left\{ p_0(\omega) + \frac{c_4}{9c_3^2} \omega^2 - i \frac{2}{3\sqrt{3}c_3} \omega^{3/2} \right\},$$

which gives (use (A.29a) and (A.29b)):

$$\left| e^{p_0(\omega)y} - e^{-\frac{c_4}{9c_3^2} \omega^2 y} e^{\frac{2}{3\sqrt{3}c_3} \omega^{3/2} y} \right| \le C \omega^{5/2} y e^{C \omega^{5/2} y}.$$

We can then follow the same kind of estimate as just above and derive the estimate (combining with (A.31)):

$$\left| \mathcal{H}_b(x, y) - \frac{e^{-\frac{c_4}{9c_3^2} x^2} e^{\frac{2|x|^3/2}{3\sqrt{3}c_3 y^{1/2}} - i\pi/4}}{2\pi} \int_{-\sqrt{\frac{2\omega}{3c_3}} - \frac{2\sqrt{2}c_4}{9c_3^2} \omega}^{\sqrt{\frac{2\omega}{3c_3}}} \exp\left(-\sqrt{3c_3\omega}y t^2 + i c_3 e^{-i\pi/4} y t^3\right) dt \right| \\ \le \frac{C}{y^{1/2}} \exp\left(-c\frac{x^2}{y}\right), \quad (A.32)$$

provided that $\omega$ is sufficiently small.

If we compare with the definition (A.10) of $g$, we see that we have almost made the quantity $g(x, y)$ appear, except for the fact that the integration interval is not exactly the good one since it is not symmetric. This final estimate is not more difficult than the previous ones. Namely, we easily estimate:

$$\left| \frac{e^{-\frac{c_4}{9c_3^2} x^2} e^{\frac{2|x|^3/2}{3\sqrt{3}c_3 y^{1/2}} - i\pi/4}}{2\pi} \int_{-\sqrt{\frac{2\omega}{3c_3}} - \frac{2\sqrt{2}c_4}{9c_3^2} \omega}^{-\sqrt{\frac{2\omega}{3c_3}}} \exp\left(-\sqrt{3c_3\omega}y t^2 + i c_3 e^{-i\pi/4} y t^3\right) dt \right| \\ \le C e^{-c\omega^2 y} \int_{-\sqrt{\frac{2\omega}{3c_3}} - \frac{2\sqrt{2}c_4}{9c_3^2} \omega}^{-\sqrt{\frac{2\omega}{3c_3}}} \exp\left(-\sqrt{3c_3\omega}y t^2 + \frac{c_3}{\sqrt{2}} y t^3\right) dt.$$

Since $t$ is negative on the considered interval, we have:

$$\left| \frac{e^{-\frac{c_4}{9c_3^2} x^2} e^{\frac{2|x|^3/2}{3\sqrt{3}c_3 y^{1/2}} - i\pi/4}}{2\pi} \int_{-\sqrt{\frac{2\omega}{3c_3}} - \frac{2\sqrt{2}c_4}{9c_3^2} \omega}^{-\sqrt{\frac{2\omega}{3c_3}}} \exp\left(-\sqrt{3c_3\omega}y t^2 + i c_3 e^{-i\pi/4} y t^3\right) dt \right| \\ \le C e^{-c\omega^2 y} \int_{-\sqrt{\frac{2\omega}{3c_3}} - \frac{2\sqrt{2}c_4}{9c_3^2} \omega}^{-\sqrt{\frac{2\omega}{3c_3}}} \exp\left(-\sqrt{3c_3\omega}y t^2\right) dt,$$

and we can then bound the final integral by the maximum of the integrated function times the length of the interval. This gives the estimate:

$$\left| \frac{e^{-\frac{c_4}{9c_3^2} x^2} e^{\frac{2|x|^3/2}{3\sqrt{3}c_3 y^{1/2}} - i\pi/4}}{2\pi} \int_{-\sqrt{\frac{2\omega}{3c_3}} - \frac{2\sqrt{2}c_4}{9c_3^2} \omega}^{-\sqrt{\frac{2\omega}{3c_3}}} \exp\left(-\sqrt{3c_3\omega}y t^2 + i c_3 e^{-i\pi/4} y t^3\right) dt \right| \\ \le C \omega e^{-c\omega^2 y} e^{-c\omega^{3/2} y} \le \frac{C}{\omega^{1/2} y} (\omega^{3/2} y) e^{-c\omega^{3/2} y}.$$
We can then again decrease the constant c and increase C to get:

$$ \left| \frac{e^{-\frac{c_4}{9c_3^2}x^2} e^{\frac{2|x|^{3/2}}{3\sqrt{3}c_3 y^{1/2}}} - i\pi/4}{2\pi} \int_{-\sqrt{\frac{2\omega}{3c_3}}-\frac{2\sqrt{2}c_4}{9c_3^2}\omega}^{-\sqrt{\frac{2\omega}{3c_3}}} \exp\left(-\sqrt{3c_3\omega}y t^2 + i c_3 e^{-i\pi/4} y t^3\right) dt \right| \le \frac{C}{\omega^{1/2}y} e^{-c\omega^{3/2}y}. $$

Since we restrict to the case $|x| \ge y^{1/3}$, we have $\omega^{1/2} \ge y^{-1/3}$ and this gives:

$$ \left| \frac{e^{-\frac{c_4}{9c_3^2}x^2} e^{\frac{2|x|^{3/2}}{3\sqrt{3}c_3 y^{1/2}}} - i\pi/4}{2\pi} \int_{-\sqrt{\frac{2\omega}{3c_3}}-\frac{2\sqrt{2}c_4}{9c_3^2}\omega}^{-\sqrt{\frac{2\omega}{3c_3}}} \exp\left(-\sqrt{3c_3\omega}y t^2 + i c_3 e^{-i\pi/4} y t^3\right) dt \right| \le \frac{C}{y^{2/3}} \exp\left(-c\frac{|x|^{3/2}}{y^{1/2}}\right). \quad (A.33) $$

We now combine (A.32) and (A.33) and recall the definition (A.10) to get:

$$ |\mathcal{H}_b(x, y) - \mathfrak{g}(x, y)| \le \frac{C}{y^{2/3}} \exp\left(-c \frac{|x|^{3/2}}{y^{1/2}}\right) + \frac{C}{y^{1/2}} \exp\left(-c \frac{x^2}{y}\right). $$

Taking twice the real part and recalling that the remaining contribution $\mathcal{H}_\#(x, y)$ is the complex conjugate of $\mathcal{H}_b(x, y)$, we get:

$$ |\mathcal{H}_b(x, y) + \mathcal{H}_\#(x, y) - 2 \operatorname{Re} \mathfrak{g}(x, y)| \le \frac{C}{y^{2/3}} \exp\left(-c \frac{|x|^{3/2}}{y^{1/2}}\right) + \frac{C}{y^{1/2}} \exp\left(-c \frac{x^2}{y}\right). $$

It remains<fnref n="3" /> to combine this estimate with (A.28) and the corresponding one for $\varepsilon_1(x, y)$ to obtain the estimate of Proposition A.3 (recall the decomposition (A.22)). We have not kept track all along of the smallness requirement on $\omega = |x|/y$ but the final estimate holds as long as $\omega$ is small enough, and this smallness condition is independent of $y$, which corresponds to the statement of Proposition A.3 for the condition on $x: x \ge -c_b y$ for some $c_b > 0$. □

We now deal with the regime where $x$ is negative and $|x|/y$ is large enough. As in Proposition A.2, this is a regime that had not been considered in [9].

**Proposition A.4.** Let the function $\mathbf{G}$ be defined in (A.9) and let $c_3$ be positive. Let $c_b > 0$ be the constant given in Proposition A.3. Then there exist some constants $C_b > c_b$, $C > 0$ and $c > 0$ such that for any $(x, y) \in \mathbb{R} \times \mathbb{R}^+$ with $x \le -C_b y$, there holds:

$$ |\mathbf{G}(x, y)| \le \frac{C}{y^{1/4}} \exp\left(-c \frac{|x|^{4/3}}{y^{1/3}}\right). $$

<footnote n="3">³The error in [9] occurred here since the author had tried to get rid of the term $t^3$ in the integral by applying similar arguments but an error occurred when estimating the corresponding integrals. Hence the final expressions for the analog of the function $\mathfrak{g}$ and the error bound are not correct.</footnote>
Proof. We go back to the bound (A.17) and only consider from now on negative values for both $x$ and $\mu$. We use Young's inequality:

$$6 c_4 y \mu^2 \theta^2 \leq \frac{c_4}{2} y \theta^4 + 18 c_4 y \mu^4,$$

as well as the obvious inequality $\exp(c_3 y \mu^3) \leq 1$ to obtain:

$$\forall \mu \leq 0, \quad |\mathbf{G}(x, y)| \leq \frac{e^{-x\mu + 17c_4 y\mu^4}}{2\pi} \int_{\mathbb{R}} e^{3c_3 y|\mu|\theta^2} e^{-\frac{c_4}{2}y\theta^4} d\theta.$$

We choose the specific value $\mu = \mu_0 := -(|x|/(68 c_4 y))^{1/3} < 0$ so as to minimize the exponential factor before the integral. We thus get a bound$^4$:

$$|\mathbf{G}(x, y)| \leq C \exp \left( -c \frac{|x|^{4/3}}{y^{1/3}} \right) \int_{0}^{+\infty} e^{3c_3 y |\mu_0| \theta^2} e^{-\frac{c_4}{2} y \theta^4} d\theta.$$

The integral is estimated by cutting at $\theta_0 > 0$ such that $3 c_3 |\mu_0| = (c_4/4) \theta_0^2$, so that we have:

$$\int_{\theta_0}^{+\infty} e^{3c_3 y |\mu_0| \theta^2} e^{-\frac{c_4}{2} y \theta^4} d\theta = \int_{\theta_0}^{+\infty} e^{\frac{c_4}{4} y \theta_0^2 \theta^2} e^{-\frac{c_4}{2} y \theta^4} d\theta \leq \int_{\theta_0}^{+\infty} e^{-\frac{c_4}{4} y \theta^4} d\theta \leq \frac{C}{y^{1/4}},$$

and we also have:

$$\int_{0}^{\theta_0} e^{3c_3 y |\mu_0| \theta^2} e^{-\frac{c_4}{2} y \theta^4} d\theta \leq \int_{0}^{\theta_0} e^{3c_3 y |\mu_0| \theta^2} d\theta \leq \theta_0 e^{3c_3 y |\mu_0| \theta_0^2} \leq C |\mu_0|^{1/2} e^{C y \mu_0^2}.$$

Going back to the definition of $\mu_0$ in terms of $x$ and $y$, we thus get the estimate:

$$|\mathbf{G}(x, y)| \leq \frac{C}{y^{1/4}} \exp \left( -c \frac{|x|^{4/3}}{y^{1/3}} \right) + C \left( \frac{|x|}{y} \right)^{1/6} \exp(C y^{1/3} |x|^{2/3}) \exp \left( -c \frac{|x|^{4/3}}{y^{1/3}} \right),$$

where all constants $C$ and $c$ are independent of $x < 0$ and $y > 0$. It is now not difficult to show that if the constant $C_b$ is chosen large enough and is we restrict to the regime $x \leq -C_b y$, then the decaying exponential term $\exp(-c|x|^{4/3}/y^{1/3})$ can absorb the large term $\exp(C y^{1/3} |x|^{2/3})$. There is of course no loss of generality in assuming that $C_b$ is chosen larger than the constant $c_b$ given in Proposition A.3. Now that $C_b$ is fixed and that we restrict to the regime $x \leq -C_b y$, we get the estimate:

$$\begin{aligned} |\mathbf{G}(x, y)| & \leq \frac{C}{y^{1/4}} \exp \left( -c \frac{|x|^{4/3}}{y^{1/3}} \right) + \frac{C}{y^{1/8}} \left( \frac{|x|}{y^{1/4}} \right)^{1/6} \exp \left( -c \frac{|x|^{4/3}}{y^{1/3}} \right) \\ & \leq \frac{C}{y^{1/4}} \exp \left( -c \frac{|x|^{4/3}}{y^{1/3}} \right) + \frac{C}{y^{1/8}} \exp \left( -c \frac{|x|^{4/3}}{y^{1/3}} \right), \end{aligned}$$

with (new) constants $C$ and $c$ that are still independent of $y > 0$ and $x \leq -C_b y$. We can still spare half of the last decaying exponential term on the right-hand side to gain a decaying factor $\exp(-cy)$ and then use a crude bound:

$$\frac{1}{y^{1/8}} \exp(-cy) \leq \frac{C}{y^{1/4}},$$

<footnote>$^4$The integral is reduced to the domain $\mathbb{R}^+$ by an obvious change of variable, which only modifies the constant in the inequality by a factor 2.</footnote>
so that, eventually, we get the claimed bound:

$$|\mathbf{G}(x, y)| \leq \frac{C}{y^{1/4}} \exp \left( -c \frac{|x|^{4/3}}{y^{1/3}} \right),$$

for $x \leq -\mathbf{C}_b y$. The proof of Proposition A.4 is complete.

<figure><img src="image_24.png" /></figure>

We now turn to the final, intermediate regime where $x$ is negative and $|x|$ and $y$ are comparable.

**Proposition A.5.** Let the function $\mathbf{G}$ be defined in (A.9) and let $c_3$ be positive. Let the constant $\mathbf{c}_b$ be the one given by Proposition A.3 and let the constant $\mathbf{C}_b$ be the one given by Proposition A.4. Then there exist some constants $C > 0$ and $c > 0$ such that for any $(x, y) \in \mathbb{R} \times \mathbb{R}^{+*}$ with $-\mathbf{C}_b y \leq x \leq -\mathbf{c}_b y$, there holds:

$$|\mathbf{G}(x, y)| \leq \frac{C}{\min(1, y^{1/4})} \exp(-cy).$$

*Proof.* We go back to the definition (A.9) for $\mathbf{G}(x, y)$ but we now deform the integration axis $\mathbb{R}$ as depicted on Figure A.2. Namely, we introduce some parameter $\delta > 0$ to be fixed later on and first consider the contribution:

$$\varepsilon(x, y) := \frac{1}{2\pi} \int_{-\infty}^{-\delta} e^{i\mathbf{x}\theta} e^{i\mathbf{c}_3 \mathbf{y}\theta^3} e^{-c_4 \mathbf{y}\theta^4} d\theta + \frac{1}{2\pi} \int_{\delta}^{+\infty} e^{i\mathbf{x}\theta} e^{i\mathbf{c}_3 \mathbf{y}\theta^3} e^{-c_4 \mathbf{y}\theta^4} d\theta. \quad (A.34)$$

The integral $\mathcal{H}_1(x, y)$ corresponds to the integral along the segment from $-\delta$ to $-i\delta$, namely:

$$\mathcal{H}_1(x, y) := \frac{\delta(1 - i)}{2\pi} \int_0^1 e^{i\mathbf{x}(\delta t - i\delta(1-t))} e^{i\mathbf{c}_3 \mathbf{y}(\delta t - i\delta(1-t))^3} e^{-c_4 \mathbf{y}(\delta t - i\delta(1-t))^4} dt, \quad (A.35)$$

while $\mathcal{H}_2(x, y)$ corresponds to the integral along the segment from $-i\delta$ to $\delta$, namely:

$$\mathcal{H}_2(x, y) := \frac{\delta(1 + i)}{2\pi} \int_0^1 e^{i\mathbf{x}(\delta t - i\delta(1-t))} e^{i\mathbf{c}_3 \mathbf{y}(\delta t - i\delta(1-t))^3} e^{-c_4 \mathbf{y}(\delta t - i\delta(1-t))^4} dt. \quad (A.36)$$

<figure><img src="image_25.png" /><figcaption>Figure A.2: The integration contour in the case $-\mathbf{C}_b y \leq x \leq -\mathbf{c}_b y$.</figcaption></figure>

Cauchy's formula allows us to decompose $\mathbf{G}(x, y)$ as:

$$\mathbf{G}(x, y) = \mathcal{H}_1(x, y) + \mathcal{H}_2(x, y) + \varepsilon(x, y),$$

and we now estimate each contribution in $\mathbf{G}(x, y)$ one by one. We first estimate the integral $\mathcal{H}_1(x, y)$ defined in (A.35) and explain how the parameter $\delta > 0$ is fixed. We restrict from on to the regime $-\mathbf{C}_b y \leq x \leq -\mathbf{c}_b y$ where the constant $\mathbf{c}_b$, resp. $\mathbf{C}_b$, is given by Proposition A.3, resp. Proposition A.4.
This implies in particular that $x$ is negative. We expand the polynomial quantities within the exponential terms in (A.35) and use the triangle inequality to obtain:

$$|\mathcal{H}_1(x, y)| \le C \delta \int_0^1 \exp\left(-|x|\delta t - c_3 y \delta^3 t^3 + 3c_3 y \delta^3 (1-t)^2 t\right) \\ \times \exp\left(-c_4 y \delta^4 (1-t)^4 + 6c_4 y \delta^4 (1-t)^2 t^2 - c_4 y \delta^4 t^4\right) dt.$$

Estimating exponentials of negative terms by 1, we have:

$$|\mathcal{H}_1(x, y)| \le C \delta \int_0^1 \exp\left(-|x|\delta t + 3c_3 \delta^3 y t + 6c_4 \delta^4 y t^2 - c_4 \delta^4 y (1-t)^4\right) dt.$$

We fix the constant $\delta > 0$ by imposing:

$$3c_3 \delta^2 \le \frac{c_b}{3}, \quad \text{and} \quad 6c_4 \delta^3 \le \frac{c_b}{3}. \qquad (A.37)$$

Since $\delta$ is fixed and only depends on already fixed parameters, we allow constants below to depend on $\delta$. We use the restrictions (A.37) in the previous estimate for $\mathcal{H}_1(x, y)$ and use furthermore the inequality $|x| \ge c_b y$ to obtain:

$$|\mathcal{H}_1(x, y)| \le C \int_0^1 \exp\left(-y\left(\frac{c_b}{3}\delta t + c_4 \delta^4 (1-t)^4\right)\right) dt.$$

We now observe that (A.37) implies that the function:

$$t \in [0, 1] \mapsto \frac{c_b}{3} \delta t + c_4 \delta^4 (1-t)^4,$$

is increasing. We thus have the exponentially decaying bound:

$$|\mathcal{H}_1(x, y)| \le C \exp(-c_4 \delta^4 y) = C \exp(-cy).$$

By a simple change of variable $t \to 1-t$, we easily find that the integral $\mathcal{H}_2(x, y)$ in (A.36) equals the complex conjugate of $\mathcal{H}_1(x, y)$ so the previous estimate for $\mathcal{H}_1(x, y)$ also applies to $\mathcal{H}_2(x, y)$.

We end the argument with the last remaining term $\varepsilon(x, y)$ defined in (A.34). We recall that $\delta > 0$ has been fixed in the analysis of the contributions $\mathcal{H}_1(x, y)$ and $\mathcal{H}_2(x, y)$. By applying the triangle inequality, we have:

$$|\varepsilon(x, y)| \le \frac{1}{2\pi} \int_{-\infty}^{-\delta} e^{-c_4 y \theta^4} d\theta + \frac{1}{2\pi} \int_{\delta}^{+\infty} e^{-c_4 y \theta^4} d\theta = \frac{1}{\pi} \int_{\delta}^{+\infty} e^{-c_4 y \theta^4} d\theta.$$

We thus get the estimate:

$$|\varepsilon(x, y)| \le \frac{e^{-\frac{c_4}{2} \delta^4 y}}{\pi} \int_{\delta}^{+\infty} e^{-\frac{c_4}{2} y \theta^4} d\theta \le \frac{C}{y^{1/4}} \exp(-cy).$$

Going back to the decomposition of $\mathbf{G}(x, y)$, we collect the estimates of $\mathcal{H}_1(x, y)$, $\mathcal{H}_2(x, y)$ and $\varepsilon(x, y)$ to get:

$$|\mathbf{G}(x, y)| \le C \exp(-cy) + \frac{C}{y^{1/4}} \exp(-cy).$$

The conclusion of Proposition A.5 follows. □
We first collect the results of Proposition A.4 and Proposition A.5 to obtain the following unified estimate, the proof of which is left to the interested reader. The constant $c_b$ in Corollary A.4 below is, of course, the same as the one given by Proposition A.3.

**Corollary A.4.** Let the function $\mathbf{G}$ be defined in (A.9) and let $c_3$ be positive. Let $y_{\min} > 0$ be given. Then there exist some constants $c_b > 0$, $C > 0$ and $c > 0$ such that for any $(x, y) \in \mathbb{R} \times [y_{\min}, +\infty)$, there holds:

$$|\mathbf{G}(x, y)| \le \frac{C}{y^{1/4}} \exp \left( -c \frac{|x|^{4/3}}{y^{1/3}} \right),$$

as long as $x$ satisfies $x \le -c_b y$.

The final Corollary is precisely what we are aiming at, namely at estimates that are analogous to those of Proposition A.3 and Corollary A.4 but with now a transition at $-c_b y$ with an arbitrary constant $c$. The result is the following.

**Corollary A.5.** Let the function $\mathbf{G}$ be defined in (A.9) and let $c_3$ be positive. Let the function $\mathfrak{g}$ be defined in (A.10). Let also $y_{\min} > 0$ be given. Then for any constant $c > 0$, there exist constants $C > 0$ and $c > 0$ such that for any $(x, y) \in \mathbb{R} \times [y_{\min}, +\infty)$, there holds:

$$|\mathbf{G}(x, y) - 2 \operatorname{Re} \mathfrak{g}(x, y)| \le \frac{C}{y^{1/3}} \exp \left( -c \frac{|x|^{3/2}}{y^{1/2}} \right) + \frac{C}{y^{1/2}} \exp \left( -c \frac{x^2}{y} \right), \quad (A.38)$$

if $-c_b y \le x \le -y^{1/3}$, and there holds:

$$|\mathbf{G}(x, y)| \le \frac{C}{y^{1/4}} \exp \left( -c \frac{|x|^{4/3}}{y^{1/3}} \right), \quad (A.39)$$

if $x \le -c_b y$.

*Proof.* The argument is mostly similar to that we used in the proof of Corollary A.3 except that we now need to control also $\mathfrak{g}(x, y)$ in the region where $|x|$ and $y$ are comparable. Let us for instance deal with the case $c \ge c_b$ where $c_b > 0$ is the constant given in Proposition A.3. If $x \le -c_b y$, we have $x \le -c_b y$ and the desired estimate (A.39) for $|\mathbf{G}(x, y)|$ is given by Corollary A.4. Moreover, if $x$ satisfies $-c_b y \le x \le -y^{1/3}$, the estimate (A.38) is given by Proposition A.3 and by using $y \ge y_{\min} > 0$ in order to estimate from below the quantity $\min(1, y^{1/3})$.

It thus remains to control the left hand side of (A.38) in the case where $x$ is negative and $|x|/y \in [c_b, c]$ which we assume from now on. We have:

$$\begin{aligned} |\mathbf{G}(x, y) - 2 \operatorname{Re} \mathfrak{g}(x, y)| & \le |\mathbf{G}(x, y)| + 2 |\mathfrak{g}(x, y)| \\ & \le \frac{C}{y^{1/4}} \exp \left( -c \frac{|x|^{4/3}}{y^{1/3}} \right) + 2 |\mathfrak{g}(x, y)|, \end{aligned}$$

where the second inequality follows from Corollary A.4 since we have $x \le -c_b y$. The quantity $\mathfrak{g}(x, y)$ is estimated directly from the definition (A.10). Applying the triangle inequality, we have:

$$|\mathfrak{g}(x, y)| \le C \exp \left( -c \frac{x^2}{y} \right) \int_{-\sqrt{\frac{2|x|}{3c_3 y}}}^{\sqrt{\frac{2|x|}{3c_3 y}}} e^{-\sqrt{3c_3 |x|} y t^2 + \frac{c_3}{\sqrt{2}} y t^3} dt,$$
and we have already seen that on the considered integration interval, there holds:

$$e^{-\sqrt{3}c_3|x|y t^2 + \frac{c_3}{\sqrt{2}}y t^3} \le e^{-c\sqrt{|x|}y t^2}$$

for some appropriate numerical constant $c > 0$. Since we consider the regime $|x|/y \in [c_b, c]$, we thus have:

$$|g(x, y)| \le C \exp \left( -c \frac{x^2}{y} \right) \int_{\mathbb{R}} e^{-c y_{\min} t^2} dt \le C \exp(-cy),$$

which yields:

$$|G(x, y) - 2 \operatorname{Re} g(x, y)| \le C \exp(-cy).$$

As we have already seen several times before, there is no difficulty at this stage to derive the estimate (A.38) for the regime $|x|/y \in [c_b, c]$ and $y \ge y_{\min}$. This completes the proof of Corollary A.5. □

## A.5 Proof of Theorem A.3

The combination of Proposition A.1, Corollary A.3 and Corollary A.5 already gives the estimates (A.11) and (A.12) of the function $G$ as given in Theorem A.3. It therefore only remains to prove the estimates (A.13) and (A.14) for the primitive function of $G$ with respect to its first variable. We still assume $c_3 > 0$ and start with the first estimate in (A.13). We consider a fixed positive constant $c$ and let $x \le -c y$. We then use the estimate in the fourth case of (A.11) to get:

$$\left| \int_{-\infty}^{x} G(\xi, y) \, d\xi \right| \le \int_{-\infty}^{x} |G(\xi, y)| \, d\xi \le \frac{C}{y^{1/4}} \int_{-\infty}^{x} \exp \left( -c \frac{|\xi|^{4/3}}{y^{1/3}} \right) \, d\xi.$$

We then perform a change of variable to obtain:

$$\left| \int_{-\infty}^{x} G(\xi, y) \, d\xi \right| \le C \int_{-\infty}^{x/y^{1/4}} \exp(-c|\eta|^{4/3}) \, d\eta = C \int_{-\infty}^{x/y^{1/4}} \frac{(4/3)c|\eta|^{1/3}}{(4/3)c|\eta|^{1/3}} \exp(-c|\eta|^{4/3}) \, d\eta.$$

Since we assume $x \le -c y$, we have $|\eta|^{1/3} \ge c^{1/3} y^{1/4}$ in the last integral on the right-hand side, and this gives:

$$\left| \int_{-\infty}^{x} G(\xi, y) \, d\xi \right| \le \frac{C}{y^{1/4}} \int_{-\infty}^{x/y^{1/4}} (4/3) c |\eta|^{1/3} \exp(-c|\eta|^{4/3}) \, d\eta.$$

We thus end up with the estimate:

$$\left| \int_{-\infty}^{x} G(\xi, y) \, d\xi \right| \le \frac{C}{y^{1/4}} \exp \left( -c \frac{|x|^{4/3}}{y^{1/3}} \right) \le C \exp \left( -c \frac{|x|^{4/3}}{y^{1/3}} \right),$$

for $x \le -c y$ and $y \ge y_{\min}$, an estimate from which the first half of (A.13) follows directly. Note in particular that we have the uniform estimate:

$$\left| \int_{-\infty}^{-c y} G(\xi, y) \, d\xi \right| \le C, \qquad (A.40)$$

for any $y \ge y_{\min}$.
From the definition (A.9), we know that for any $y > 0$, $\mathbf{G}(\cdot, y)$ is the inverse Fourier transform of the Schwartz class function:

$$ \theta \in \mathbb{R} \mapsto e^{i c_3 y} e^{\theta^3} e^{-c_4 y^4}. $$

In particular, this means that the (partial) Fourier transform of $\mathbf{G}$ with respect to its first variable is given by:

$$ \mathcal{F}_x(\mathbf{G})(\theta, y) = e^{i c_3 y} e^{\theta^3} e^{-c_4 y^4}. $$

Evaluating at $\theta = 0$, we get the relation:

$$ \int_{\mathbb{R}} \mathbf{G}(x, y) dx = 1, $$

so that proving the bound (A.14) amounts to showing the bound:

$$ \left| \int_{x}^{+\infty} \mathbf{G}(\xi, y) d\xi \right| \leq \tilde{C} \exp(-\tilde{c}y), \quad \text{if } x \geq \underline{c}y. $$

The proof of this bound follows from the exact same argument as above except that we now use the first case in (A.11).

The last part of the proof of Theorem A.3 aims at showing the second half of (A.13), that is, at proving a uniform bound for the primitive function of $\mathbf{G}$ in the case $|x| \leq \underline{c}y$. Let us first assume $x \geq 0$ and thus $x \in [0, \underline{c}y]$. We have already seen the relation:

$$ \int_{-\infty}^{x} \mathbf{G}(\xi, y) d\xi = 1 - \int_{x}^{\underline{c}y} \mathbf{G}(\xi, y) d\xi - \int_{\underline{c}y}^{+\infty} \mathbf{G}(\xi, y) d\xi, $$

and we thus already have from the arguments above (namely, the case $x \geq \underline{c}y$):

$$ \left| \int_{-\infty}^{x} \mathbf{G}(\xi, y) d\xi \right| \leq C + \int_{x}^{\underline{c}y} |\mathbf{G}(\xi, y)| d\xi. $$

The final integral on the right-hand side is estimated by using the second case in (A.11), which gives the uniform control (after an obvious change of variable in $\xi$):

$$ \left| \int_{-\infty}^{x} \mathbf{G}(\xi, y) d\xi \right| \leq C. $$

We can actually push a little further this argument and obtain from (A.11) a uniform bound for:

$$ \int_{-\infty}^{x} \mathbf{G}(\xi, y) d\xi $$

as long as $x$ belongs to the interval $[-y^{1/3}, \underline{c}y]$ and not only when $x$ belongs to $[0, \underline{c}y]$ (use now the third case in (A.11)). It thus remains to examine the case $x \in [-\underline{c}y, -y^{1/3}]$ for which we need to take into account the oscillating behavior of the Green's function. This is the only regime where applying the triangle inequality to estimate the primitive function does not (and can not !) work. Let therefore $x \in [-\underline{c}y, -y^{1/3}]$. We decompose:

$$ \int_{-\infty}^{x} \mathbf{G}(\xi, y) d\xi = \int_{-\infty}^{-\underline{c}y} \mathbf{G}(\xi, y) d\xi + \int_{-\underline{c}y}^{x} (\mathbf{G}(\xi, y) - 2 \operatorname{Re} \mathfrak{g}(\xi, y)) d\xi \\ + 2 \operatorname{Re} \int_{-\underline{c}y}^{x} \mathfrak{g}(\xi, y) d\xi. $$
The first term on the right-hand side has already been estimated, see (A.40), and the second term on the right-hand side is estimated by using the bound<fnref n="5" /> (A.12). At this stage, we already have an estimate that reads:

$$ \left| \int_{-\infty}^{x} \mathbf{G}(\xi, y) \, d\xi \right| \le C + 2 \left| \int_{-c_y}^{x} \mathfrak{g}(\xi, y) \, d\xi \right|, \qquad (A.41) $$

and the remaining point of the proof is to derive a uniform estimate for the primitive function of the explicit function $\mathfrak{g}$ whose expression is given in (A.10). The proof relies on integration by parts, as detailed below.

We first introduce the notation:

$$ \beta_0 := \frac{c_4}{9c_3^2} > 0, \quad \beta_1 := \frac{2}{3\sqrt{3c_3}} > 0, $$

so that, after performing a change of variable in the integral of (A.10), we obtain the expression:

$$ \mathfrak{g}(x, y) := \frac{3 \beta_1 e^{-i \pi/4} |x|^{1/2}}{2 \sqrt{2} \pi y^{1/2}} \exp \left( -\beta_0 \frac{x^2}{y} \right) \exp \left( i \beta_1 \frac{|x|^{3/2}}{y^{1/2}} \right) \int_{-1}^{1} e^{-3 \beta_1 \frac{|x|^{3/2}}{y^{1/2}} u^2} e^{\beta_1 (1-i) \frac{|x|^{3/2}}{y^{1/2}} u^3} \, du. \qquad (A.42) $$

Let us define a function $H$ on $\mathbb{R}^+$ as follows (the constant $\beta_1 > 0$ being fixed as above):

$$ \forall w \ge 0, \quad H(w) := \exp(i\beta_1 w) \int_{-1}^{1} e^{-3\beta_1 w u^2} e^{\beta_1 (1-i) w u^3} \, du. \qquad (A.43) $$

With the help of (A.43), we can rewrite (A.42) and obtain the relation:

$$ \forall (x, y) \in \mathbb{R} \times \mathbb{R}^{+*}, \quad y^{1/2} \mathfrak{g}(y^{1/2} x, y) := \frac{\beta_1 e^{-i \pi/4}}{\sqrt{2} \pi} e^{-\beta_0 x^2} \left( \frac{3}{2} y^{1/4} |x|^{1/2} \right) H(y^{1/4} |x|^{3/2}). \qquad (A.44) $$

where the factor $(3/2) y^{1/4} |x|^{1/2}$ equals, up to a sign, the derivative of the function $(x \mapsto y^{1/4} |x|^{3/2})$. We are thus in a very favorable position for applying integration by parts. Before going further, we prove the following Lemma.

**Lemma A.1.** Let the function $H$ be defined on $\mathbb{R}^+$ by (A.43). Then there exists a constant $C > 0$ that only depends on $\beta_1$ and such that:

$$ \forall w \ge 0, \quad \left| \int_{0}^{w} H(w') \, dw' \right| \le C. $$

*Proof.* We consider $w \ge 0$. By applying the Fubini Theorem, we get the relation:

$$ \int_{0}^{w} H(w') \, dw' = \int_{-1}^{1} \int_{0}^{w} \exp \left( \beta_1 w' \left( -3u^2 + u^3 + i(1-u^3) \right) \right) dw' \, du, $$

and it turns out that the continuous function:

$$ u \in [-1, 1] \mapsto -3u^2 + u^3 + i(1 - u^3), $$

<footnote n="5">⁵The whole point of (A.12) was precisely to extract from $\mathbf{G}$ its leading oscillating behavior so that the difference between $\mathbf{G}$ and this “leading order term” would actually become uniformly integrable.</footnote>
does not vanish. Its modulus is thus uniformly bounded from below. We get:

$$ \int_{0}^{w} H(w') dw' = \int_{-1}^{1} \frac{\exp\left(\beta_1 w \left(-3u^2 + u^3 + \mathbf{i}(1-u^3)\right)\right) - 1}{\beta_1 \left(-3u^2 + u^3 + \mathbf{i}(1-u^3)\right)} du, $$

and the triangle inequality (as well as a lower bound for the modulus of the denominator) yields:

$$ \left| \int_{0}^{w} H(w') dw' \right| \le C \int_{-1}^{1} \exp \left( \beta_1 w (-3u^2 + u^3) \right) + 1 du. $$

We have $-3u^2 + u^3 \le 0$ for $u \in [-1, 1]$ and $w \ge 0$ so the uniform bound of Lemma A.1 follows. □

We now go back to our main problem, which is to prove a bound for the primitive function of $\mathfrak{g}$, see (A.41). We use the expression (A.44) and obtain:

$$ \begin{aligned} \int_{-\mathfrak{c}y}^{x} \mathfrak{g}(\xi, y) d\xi &= \int_{-\mathfrak{c}y^{1/2}}^{x/y^{1/2}} y^{1/2} \mathfrak{g}(y^{1/2} w, y) dw \\ &= \frac{\beta_1 e^{-\mathbf{i}\pi/4}}{\sqrt{2}\pi} \int_{|x|/y^{1/2}}^{\mathfrak{c}y^{1/2}} e^{-\beta_0 w^2} \left( \frac{3}{2} y^{1/4} w^{1/2} H(y^{1/4} w^{3/2}) \right) dw \\ &= \frac{\beta_1 e^{-\mathbf{i}\pi/4}}{\sqrt{2}\pi} \int_{|x|/y^{1/2}}^{\mathfrak{c}y^{1/2}} e^{-\beta_0 w^2} \frac{d}{dw} \left( \int_{0}^{y^{1/4} w^{3/2}} H(w') dw' \right) dw. \end{aligned} $$

It then only remains to integrate by parts the final integral and to apply Lemma A.1 in order to derive the uniform bound:

$$ \left| \int_{-\mathfrak{c}y}^{x} \mathfrak{g}(\xi, y) d\xi \right| \le C. $$

This final argument is a prototype application of Abel's transform (in the continuous setting). This completes the proof of Theorem A.3.

## A.6 Consequences

This final paragraph is devoted to applying Theorem A.3 in order to derive suitable bounds for the activation function $\mathfrak{A}$ and other quantities that arise in our decomposition of the Green's function of the operator $\mathcal{L}$ in (2.11). We therefore now go back to the framework of stationary discrete shock profiles and use the index $r$, resp. $\ell$, to refer to the right, resp. left, state of the discrete shock (2.2). The analysis of Section 4.3 uses the following quantities defined for any $j_0 \in \mathbb{N}^*$ and $n \in \mathbb{N}^*$:

$$ \mathfrak{A}_r^n(j_0) := \frac{1}{2\mathbf{i}\pi} \int_{\eta+\mathbf{i}\mathbb{R}} e^{n\tau-j_0\varphi_r(\tau)} \frac{d\tau}{\tau}, $$

where $\eta$ is any positive number (the Cauchy formula shows that the definition is independent of $\eta$) and:

$$ \mathfrak{B}_r^n(j_0) := \frac{1}{2\mathbf{i}\pi} \int_{\mathbf{i}\mathbb{R}} e^{n\tau-j_0\varphi_r(\tau)} d\tau. $$

In both definitions of $\mathfrak{A}_r^n(j_0)$ and $\mathfrak{B}_r^n(j_0)$, the function $\varphi_r$ is defined by:

$$ \forall \tau \in \mathbb{C}, \quad \varphi_r(\tau) := -\frac{1}{\alpha_r} \tau + \frac{1 - \alpha_r^2}{6 \alpha_r^3} \tau^3 - \frac{1 - \alpha_r^2}{8 \alpha_r^3} \tau^4. $$
At last, we recall that $\alpha_r$ belongs to the interval $(-1, 0)$, see (2.12). This is a consequence of Lax shock inequalities and the choice of the CFL parameter.

By using the parametrization $\tau = i|\alpha_r|\theta$ in the definition of $\mathfrak{B}_r^n(j_0)$, we obtain the expression:

$$ \mathfrak{B}_r^n(j_0) := \frac{|\alpha_r|}{2\pi} \int_{\mathbb{R}} e^{-i(j_0+n\alpha_r)\theta} e^{-i j_0 \frac{1-\alpha_r^2}{6}\theta^3} e^{j_0 \alpha_r \frac{1-\alpha_r^2}{8}\theta^4} d\theta, $$

and the integral is convergent since $j_0$ is positive and $\alpha_r$ belongs to the interval $(-1, 0)$. Going back to the definitions (A.4) (with the choice $\alpha = \alpha_r \in (-1, 0)$) and (A.9), we have thus obtained the relation:

$$ \forall j_0 \in \mathbb{N}^*, \quad \forall n \in \mathbb{N}^*, \quad \mathfrak{B}_r^n(j_0) := |\alpha_r| \mathbf{G}_r \left( -j_0 + n |\alpha_r|, \frac{j_0}{|\alpha_r|} \right), \quad (A.45) $$

which is the reason why Theorem A.3 will give us exactly what we need for proving the bounds we need in our analysis. The index $r$ in $\mathbf{G}_r$ refers to the fact that we have made the choice $\alpha = \alpha_r$ when defining the constants $c_3$ and $c_4$ in (A.4).

Let us now turn to the activation function $\mathfrak{A}_r^n(j_0)$. Choosing the parametrization $\tau = |\alpha_r|(\eta + i\theta)$ for any $\eta > 0$, we get:

$$ \mathfrak{A}_r^n(j_0) := \frac{1}{2\pi} \int_{\mathbb{R}} e^{-(j_0+n\alpha_r)(\eta+i\theta)} e^{j_0 \frac{1-\alpha_r^2}{6}(\eta+i\theta)^3} e^{j_0 \alpha_r \frac{1-\alpha_r^2}{8}(\eta+i\theta)^4} \frac{d\theta}{\eta+i\theta}. $$

With the choice $\alpha = \alpha_r$ and the definition (A.4) for the coefficients $c_3$ and $c_4$, we can introduce the function $\mathbf{A}_r$ defined by:

$$ \forall (x, y) \in \mathbb{R} \times \mathbb{R}^{+*}, \quad \mathbf{A}_r(x, y) := \frac{1}{2\pi} \int_{\mathbb{R}} e^{x(\eta+i\theta)} e^{-c_3 y (\eta+i\theta)^3} e^{-c_4 y (\eta+i\theta)^4} \frac{d\theta}{\eta+i\theta}, \quad (A.46) $$

where the definition is independent of the choice of $\eta > 0$ because of Cauchy's formula. With this notation, we have expressed the activation function $\mathfrak{A}_r^n(j_0)$ as:

$$ \mathfrak{A}_r^n(j_0) = \mathbf{A}_r \left( -j_0 + n |\alpha_r|, \frac{j_0}{|\alpha_r|} \right). \quad (A.47) $$

It remains to connect the function $\mathbf{A}_r$ with the primitive function of $\mathbf{G}_r$ with respect to its first variable. From the dominated convergence theorem, we have that $\mathbf{A}_r$ is differentiable with respect to its first variable and (passing to the limit $\eta \to 0$ in the expression of the partial derivative):

$$ \frac{\partial \mathbf{A}_r}{\partial x}(x, y) = \mathbf{G}_r(x, y). $$

Moreover, the factor $\exp(x\eta)$ can be extracted from the integral and we thus have:

$$ \lim_{x \to -\infty} \mathbf{A}_r(x, y) = 0, $$

from which we get the general expression<fnref n="6" />:

$$ \mathbf{A}_r(x, y) = \int_{-\infty}^{x} \mathbf{G}_r(\xi, y) d\xi. $$

<footnote n="6">⁶The integral is convergent because of the bounds that we proved in Theorem A.3.</footnote>
Going back to (A.47), this means that we have expressed the activation function $\mathfrak{A}_r^n(j_0)$ as follows:

$$ \forall j_0 \in \mathbb{N}^*, \quad \forall n \in \mathbb{N}^*, \quad \mathfrak{A}_r^n(j_0) = \int_{-\infty}^{-j_0+n|\alpha_r|} \mathbf{G}_r\left(\xi, \frac{j_0}{|\alpha_r|}\right) d\xi. \quad (A.48) $$

Theorem A.3 can be recast into the following compact form that will be helpful in the analysis of Chapter 4.

**Corollary A.6.** Let the function $\mathbf{A}_r$ be defined in (A.46) with constants $c_3$ and $c_4$ as in (A.4) with the choice $\alpha = \alpha_r \in (-1, 0)$. Then for any constant $\underline{c} > 0$, there exist two positive constants $C$ and $c$ such that for any $n \in \mathbb{N}^*$ and any $j_0 \in \mathbb{N}^*$, there holds:

$$ |\mathbf{A}_r(-j_0 + n|\alpha_r|, n)| \le \begin{cases} C \exp\left(-c \frac{|j_0 - n|\alpha_r|^{4/3}}{n^{1/3}}\right), & \text{if } -j_0 + n|\alpha_r| \le -\underline{c}n, \\ C, & \text{if } -\underline{c}n \le -j_0 + n|\alpha_r| \le \underline{c}n. \end{cases} $$

and:

$$ |1 - \mathbf{A}_r(-j_0 + n|\alpha_r|, n)| \le C \exp\left(-c \frac{|j_0 - n|\alpha_r|^{4/3}}{n^{1/3}}\right), \quad \text{if } -j_0 + n|\alpha_r| \ge \underline{c}n. $$

In particular, there holds:

$$ \sup_{j_0 \in \mathbb{Z}, n \in \mathbb{N}^*} |\mathbf{A}_r(-j_0 + n|\alpha_r|, n)| < +\infty. $$

## A.7 Higher order estimates

Another crucial estimate that was needed in the analysis of Section 4.3 (see the proof of Lemma 4.2) aims at controlling the difference:

$$ \mathbf{A}_r \left( -j_0 + n |\alpha_r|, \frac{j_0}{|\alpha_r|} \right) - \mathbf{A}_r (-j_0 + n |\alpha_r|, n), $$

with the function $\mathbf{A}_r$ defined in (A.46) and $j_0, n \in \mathbb{N}^*$. Unsurprisingly, the most direct way to estimate this difference is to apply the mean value theorem, which gives rise to the partial derivative of $\mathbf{A}_r$ with respect to its second variable. This leads us to introduce the family of correctors:

$$ \forall p \in \mathbb{N}^*, \quad \forall (x, y) \in \mathbb{R} \times \mathbb{R}^{+*}, \quad \mathbf{G}_p(x, y) := \frac{1}{2\pi} \int_{\mathbb{R}} (\dot{\mathbf{i}}\theta)^p e^{i\mathbf{x}\theta} e^{i\mathbf{c}_3 y \theta^3} e^{-i\mathbf{c}_4 y \theta^4} d\theta. \quad (A.49) $$

The case $p = 0$ corresponds to the definition (A.9) of $\mathbf{G}$ and the relevance of the functions $\mathbf{G}_p$ for controlling the above difference between the two values of $\mathbf{A}_r$ will be the purpose of Corollary A.7 below. We aim here at generalizing Theorem A.3 and at obtaining sharp bounds on $\mathbf{G}_p$ for any $p \in \mathbb{N}^*$. Our result is the following.

**Theorem A.4.** Let us assume that the coefficient $c_3$ in (A.4) is positive, that is $\alpha \in (0, 1)$. Let $p \in \mathbb{N}^*$. Let $y_{\min} > 0$ and let $\underline{c} > 0$ be given. Then there exist some constants $C > 0$ and $c > 0$ such that, for any
$(x, y) \in \mathbb{R} \times [y_{\min}, +\infty)$, there holds:

$$ |\mathbf{G}_p(x, y)| \le \begin{cases} \frac{C}{y^{(p+1)/4}} \exp(-c x^{4/3}/y^{1/3}), & \text{if } x \ge \underline{c} y, \\ \frac{C}{y^{1/3+p/4}} \exp(-c x^{3/2}/y^{1/2}), & \text{if } 0 \le x \le \underline{c} y, \\ \frac{C}{y^{1/3+p/4}}, & \text{if } -y^{1/3} \le x \le 0, \\ \frac{C}{|x|^{1/4} y^{(p+1)/4}} \exp(-c x^2/y), & \text{if } -\underline{c} y \le x \le -y^{1/3}, \\ \frac{C}{y^{(p+1)/4}} \exp(-c |x|^{4/3}/y^{1/3}), & \text{if } x \le -\underline{c} y. \end{cases} \quad (A.50) $$

If $c_3$ is negative, the same estimate holds for $\mathbf{G}_p$ with $x$ being switched to $-x$.

Applying Theorem A.4 gives us the desired estimate for the difference between the two evaluations of $\mathbf{A}_r$, namely we have the following Corollary.

**Corollary A.7.** Let the function $\mathbf{A}_r$ be defined in (A.46). There exist two positive constants $C$ and $c$ such that for any $n \in \mathbb{N}^*$ and any $j_0 \in \mathbb{N}^*$ that satisfies $j_0 \in [n |\alpha_r|/2, n]$, there holds:

$$ \left| \mathbf{A}_r \left( -j_0 + n |\alpha_r|, \frac{j_0}{|\alpha_r|} \right) - \mathbf{A}_r \left( -j_0 + n |\alpha_r|, n \right) \right| \le C \begin{cases} \frac{1}{n^{1/3}} \exp \left( -c \left| j_0 - n |\alpha_r| \right|^{3/2} / n^{1/2} \right), & \text{if } j_0 \ge n |\alpha_r|, \\ \frac{1}{n^{1/3}}, & \text{if } 0 \le n |\alpha_r| - j_0 \le n^{1/3}, \\ \frac{1}{|j_0 - n |\alpha_r||^{1/4} n^{1/4}} \exp \left( -c \left| j_0 - n |\alpha_r| \right|^2 / n \right), & \text{if } n |\alpha_r| - j_0 \ge n^{1/3}. \end{cases} $$

*Proof of Theorem A.4.* A very large part of the proof of Theorem A.4 follows that of Theorem A.3. We therefore feel free to refer to the various steps of the proof of Theorem A.3 (that corresponds to the case $p=0$) and to shorten many details.

• Step 1: the uniform estimate. We follow the same argument as in the proof of Proposition A.1 but, keeping similar notation, we now use the choice $f(\theta) := x\theta + c_3 y\theta^3$ and $g(\theta) := (i\theta)^p \exp(-c_4 y\theta^4)$, so that we have the estimates:

$$ \| g \|_{L^{\infty}([a,b])} \le \frac{C}{y^{p/4}}, \quad \| g' \|_{L^1([a,b])} \le \frac{C}{y^{p/4}}, $$

for any $y > 0$, uniformly with respect to the interval $[a, b]$. By applying the same arguments as in the proof of Proposition A.1, we get the uniform estimate:

$$ \forall y > 0, \quad \sup_{x \in \mathbb{R}} |\mathbf{G}_p(x, y)| \le \frac{C}{y^{1/3+p/4}}, \quad (A.51) $$

with a constant $C$ that only depends on $p$.
• Step 2: the fast decaying side. Part I. Changing the integration line $\mathbb{R}$ to $i\mu + \mathbb{R}$ for any $\mu \in \mathbb{R}$ thanks to the Cauchy formula, we then apply the triangle inequality and get a similar bound as the one we had obtained in (A.17), namely:

$$ \forall \mu \in \mathbb{R}, \quad \forall (x, y) \in \mathbb{R} \times \mathbb{R}^{+*}, \\ |\mathbf{G}_p(x, y)| \le C_p e^{-x\mu+c_3 y\mu^3-c_4 y\mu^4} \int_{\mathbb{R}} (|\mu|^p + |\theta|^p) e^{-3y\mu(c_3-2c_4\mu)\theta^2} e^{-c_4 y\theta^4} d\theta, \quad (A.52) $$

where the constant $C_p$ only depends on $p \in \mathbb{N}^*$ that is a given fixed integer. The important property is that $C_p$ does not depend on $x, y$ nor $\mu$.

Let us assume that $c_3$ is positive. We first consider the regime where $x$ is positive and such that the positive parameter $\mu_0$ defined by $\mu_0 := (x/(3c_3y))^{1/2}$ satisfies $2c_4\mu_0 \le c_3/2$, which corresponds to the constraint $0 < x \le c_4y$ for some well-defined positive constant $c_4$ (the same one as in the proof of Proposition A.2). Choosing the parameter $\mu_0$ in (A.52), we follow the same arguments as in the proof of Proposition A.2 and get the bound (compare with (A.18)):

$$ |\mathbf{G}_p(x, y)| \le C \exp \left( -c \frac{x^{3/2}}{y^{1/2}} \right) \int_{\mathbb{R}} \left( \frac{x^{p/2}}{y^{p/2}} + |\theta|^p \right) e^{-c x^{1/2} y^{1/2} \theta^2} d\theta, $$

for suitable constants $C$ and $c$. Integrating with respect to $\theta$, we thus get the bound:

$$ |\mathbf{G}_p(x, y)| \le C \left\{ \frac{1}{y^{1/4+p/3} \max(1, x^{1/4})} + \frac{1}{y^{(p+1)/4} \max(1, x^{(p+1)/4})} \right\} \exp \left( -c \frac{x^{3/2}}{y^{1/2}} \right), \quad (A.53) $$

which holds for any $y > 0$ and $x \in (0, c_4y]$. We now argue similarly as in the proof of Corollary A.2. We consider $y \ge y_{\min} > 0$ and $x \in [0, c_4y]$. For $x \le y^{1/(3(1+p))}$, we use the uniform bound (A.51) and for $x \in [y^{1/(3(1+p))}, c_4y]$, we use (A.53). This combination of (A.51) and (A.53) gives the unified estimate:

$$ \forall y \ge y_{\min}, \quad \forall x \in [0, c_4 y] \quad |\mathbf{G}_p(x, y)| \le \frac{C}{y^{1/3+p/4}} \exp \left( -c \frac{x^{3/2}}{y^{1/2}} \right), \quad (A.54) $$

where $C$ and $c$ are appropriate constants that do not depend on $y$ and $x$.

• Step 3: the fast decaying side. Part II. The constant $c_4$ has been fixed and we consider the regime $y > 0, x \ge c_4y$. We argue as in the proof of Proposition A.2 and first use Young's inequality to obtain:

$$ \forall \mu > 0, \quad |\mathbf{G}_p(x, y)| \le C \left\{ \frac{\mu^{p-1/2}}{y^{1/2}} + \frac{1}{(\mu y)^{(p+1)/2}} \right\} \exp(f(\mu)), \quad (A.55) $$

where $f$ is the convex function defined in (A.20) whose minimum over $\mathbb{R}^+$ is attained at some $\underline{\mu} > 0$. We have already shown the lower bound $\underline{\mu} \ge c_4 (x/y)^{1/3}$ in the proof of Proposition A.2 and the equality:

$$ \underbrace{3 c_3 y \underline{\mu}^2 + 32 c_4 y \underline{\mu}^3}_{\ge 0} = x, $$
directly gives the upper bound $\underline{\mu} \le C_\nu(x/y)^{1/3}$ for yet another constant $C_\nu$. Choosing the optimal parameter $\underline{\mu}$ in (A.55) and following arguments as in the proof of Proposition A.2 for the upper estimate of $f(\underline{\mu})$, we end up with the estimate:

$$|\mathbf{G}_p(x, y)| \le C \left\{ \frac{1}{y^{p/4+9/24}} + \frac{1}{y^{(p+1)/2}} \right\} \exp \left( -c \frac{x^{4/3}}{y^{1/3}} \right),$$

for $x \ge c_\# y$. Using now $y \ge y_{\min}$, we end up with the estimate:

$$\forall y \ge y_{\min}, \forall x \ge c_\# y \quad |\mathbf{G}_p(x, y)| \le \frac{C}{y^{(p+1)/4}} \exp \left( -c \frac{x^{4/3}}{y^{1/3}} \right), \quad (A.56)$$

for suitable constants $C$ and $c$.

It remains to argue as in Corollaries A.2 and A.3 to pass from a given constant $c_\#$ to an arbitrary given constant $c > 0$ given a priori. At his stage, we have already shown the validity of the first three estimates in (A.50).

• Step 4: the oscillating side. Part I. We still assume that $c_3$ is positive and now assume that $x$ is negative. We follow the proof of Proposition A.3 and consider the same contour deformation as the one depicted in Figure A.1. This gives rise to a decomposition:

$$\mathbf{G}_p(x, y) = \varepsilon_1(x, y) + \varepsilon_2(x, y) + \mathcal{H}_\nu(x, y) + \mathcal{H}_\#(x, y),$$

that is entirely similar to the one in (A.22) except that the four integrals now incorporate the contribution of the polynomial factor $(\mathbf{i}\theta)^p$. For instance, we have (keeping the notation $\omega := |x|/y$):

$$\varepsilon_2(x, y) = \frac{1}{2\pi} \exp \left( \frac{4\omega^{3/2}y}{3\sqrt{3c_3}} - \frac{c_4}{9c_3^2}\omega^2 y \right) \int_{\Xi(\omega)}^{+\infty} \left( \mathbf{i}\theta - \sqrt{\frac{\omega}{3c_3}} \right)^p e^{\mathbf{i}\dots} e^{-\sqrt{3c_3\omega}y\theta^2 + \frac{2c_4}{c_3}\omega y\theta^2} e^{-c_4 y\theta^4} d\theta,$$

where, again, the three dots within the integral stand for a real quantity whose precise expression is useless, and $\Xi(\omega)$ stands for the quantity defined in (A.23). Applying the triangle inequality yields the bound:

$$|\varepsilon_2(x, y)| \le C \exp \left( \frac{4\omega^{3/2}y}{3\sqrt{3c_3}} \right) \int_{\Xi(\omega)}^{+\infty} \left( \theta^p + \omega^{p/2} \right) e^{-\sqrt{3c_3\omega}y\theta^2 + \frac{2c_4}{c_3}\omega y\theta^2} e^{-c_4 y\theta^4} d\theta,$$

where the constant $C$ does not depend on $\omega$ and $y$. We restrict again $\omega = |x|/y$ by imposing the condition (A.27) so that we have:

$$\frac{2c_4}{c_3}\omega \le \frac{1}{2}\sqrt{3c_3\omega}.$$

This restriction corresponds to an inequality $\omega \le \omega_0$ for some well-chosen constant $\omega_0 > 0$. This yields the estimate:

$$|\varepsilon_2(x, y)| \le C \exp \left( \frac{4\omega^{3/2}y}{3\sqrt{3c_3}} \right) \int_2^{+\infty} \left( \theta^p + \omega^{p/2} \right) e^{-\frac{\sqrt{3c_3\omega}}{2}y\theta^2} d\theta. \quad (A.57)$$

We now use the following Lemma which follows from integration by parts and an induction argument (the proof is left to the reader).
**Lemma A.2.** Let the sequence $(Q_k)_{k \in \mathbb{N}}$ of real polynomials be defined by:

$$Q_0(Y) := \frac{1}{2}, \\ \forall k \in \mathbb{N}, \quad Q_{k+1}(Y) := \frac{1}{2} Y^{k+1} + (k+1) Q_k(Y).$$

Then for any integer $\nu \in \mathbb{N}$ and for any real numbers $a > 0$ and $X > 0$, there holds:

$$\int_X^{+\infty} \theta^\nu e^{-a\theta^2} d\theta \le \begin{cases} a^{-(\nu+1)/2} Q_{(\nu-1)/2}(aX^2) e^{-aX^2}, & \text{if } \nu \text{ is odd,} \\ a^{-\nu/2-1} X^{-1} Q_{\nu/2}(aX^2) e^{-aX^2}, & \text{if } \nu \text{ is even.} \end{cases}$$

Let us assume for a moment that $p$ is odd. Applying Lemma A.2 in (A.57), we obtain the estimate:

$$|\varepsilon_2(x, y)| \le C \exp \left( -\frac{2 \omega^{3/2} y}{3 \sqrt{3} c_3} \right) \left( \frac{Q(\omega^{3/2} y)}{\omega^{(p+1)/4} y^{(p+1)/2}} + \frac{\omega^{p/2}}{\omega y} \right),$$

where $Q$ is a real polynomial with nonnegative coefficients. Since the exponential term can absorb any polynomial expression of the same argument, we end up with the estimate:

$$|\varepsilon_2(x, y)| \le C \exp \left( -c \omega^{3/2} y \right) \left( \frac{1}{\omega^{(p+1)/4} y^{(p+1)/2}} + \frac{\omega^{p/2}}{\omega y} \right), \quad (A.58)$$

if $p$ is odd. If $p$ is even, applying Lemma A.2 in (A.57) yields the final estimate:

$$|\varepsilon_2(x, y)| \le C \exp \left( -c \omega^{3/2} y \right) \left( \frac{1}{\omega^{p/4+1} y^{p/2+1}} + \frac{\omega^{p/2}}{\omega y} \right). \quad (A.59)$$

Of course, there is a similar estimate for the contribution $\varepsilon_1(x, y)$ that is the complex conjugate of $\varepsilon_2(x, y)$. Let us now turn to the contribution $\mathcal{H}_b(x, y)$ that corresponds to the inclined segment on the left in Figure A.1. Keeping the notation of the proof of Proposition A.3, we have:

$$\mathcal{H}_b(x, y) = \frac{e^{p_0(\omega) y - i \pi / 4}}{2 \pi} \int_{-\sqrt{\frac{2\omega}{3c_3}} - \frac{2\sqrt{2}c_4}{9c_3^2}}^{\sqrt{\frac{2\omega}{3c_3}}} \omega (i\Theta(t))^p \exp \left( y \sum_{k=1}^4 p_k(\omega) t^k \right) dt,$$

where $(t \mapsto \Theta(t))$ parametrizes the segment so that we have a uniform estimate:

$$|\Theta(t)| \le C \sqrt{\omega},$$

for $\omega \le \omega_0$.

Instead of trying to isolate the leading contribution in $\mathcal{H}_b(x, y)$ as we did in the proof of Proposition A.3, we rather apply the triangle inequality and use the behavior (A.29) of $p_0, \dots, p_4$ when $\omega$ is small. Starting from:

$$|\mathcal{H}_b(x, y)| \le C \omega^{p/2} e^{(\text{Re } p_0(\omega)) y} \int_{-\sqrt{\frac{2\omega}{3c_3}} - \frac{2\sqrt{2}c_4}{9c_3^2}}^{\sqrt{\frac{2\omega}{3c_3}}} \exp \left( y \sum_{k=1}^4 (\text{Re } p_k(\omega)) t^k \right) dt,$$