$\displaystyle 1+\sum_{l=2}^{\ell}\kappa_{l}(\varepsilon|\xi|)^{l-1},$
$\displaystyle\gamma_{\varepsilon}^{\ell}(\xi)$ $\displaystyle:=$
$\displaystyle\xi\cdot\bigg{(}\sum_{n=1}^{\ell}\Big{(}\kappa_{n}\bar{\boldsymbol{a}}+\sum_{l=1}^{n-1}\kappa_{l}\bar{\boldsymbol{b}}^{n+1-l}\odot(i\tfrac{\xi}{|\xi|})^{n-l}\Big{)}(\varepsilon|\xi|)^{n-1}\bigg{)}\xi,$

with
$\gamma_{\varepsilon}^{\ell}/\beta_{\varepsilon}^{\ell}=\mu_{\varepsilon}^{\operatorname{(III)},\ell}$.
Arguing as in Step 1, and using that $\beta_{\varepsilon}^{\ell}(\xi)\geq 1$
and $\gamma_{\varepsilon}^{\ell}(\xi)\geq\lambda|\xi|^{2}$, we find that any
ancient solution of (2.36) satisfies

$\|D\bar{u}_{\varepsilon}^{\operatorname{(III)},\ell;t}\|_{L^{2}(\mathbb{R}^{d})}\,\lesssim\,\int_{0}^{t}\|(\beta_{\varepsilon}^{\ell})^{\frac{1}{2}}\hat{f}\|_{L^{2}(\mathbb{R}^{d})}.$

Inserting the upper bound

$\beta_{\varepsilon}^{\ell}(\xi)\,\leq\,\sum_{l=1}^{\ell}(\varepsilon
C|\xi|)^{l-1}\mathds{1}_{l\text{
odd}}\,\leq\,C^{\ell}\langle\varepsilon\xi\rangle^{2\lfloor\frac{\ell-1}{2}\rfloor},$

we are led to the claimed a priori estimate on the energy norm.

Step 5\. Proof of (iii): comparison of modified equations.
We analyze the differences
$\bar{v}^{(\star),\ell}_{\varepsilon}:=\bar{u}_{\varepsilon}^{(\star),\ell}-\bar{U}_{\varepsilon}^{\ell}$,
and we start with $(\star)=\text{(I)}$. By definition, it satisfies the
equation

$\partial_{t}^{2}\bar{v}^{\operatorname{(I)},\ell}_{\varepsilon}-\nabla\cdot\Big{(}\bar{\boldsymbol{a}}+\sum_{k=2}^{\ell}\bar{\boldsymbol{b}}^{k}\odot(\varepsilon\nabla)^{k-1}\Big{)}\nabla\bar{v}^{\operatorname{(I)},\ell}_{\varepsilon}=(\chi(\varepsilon^{\alpha}\nabla)-1)f,$

so that (2.32) yields for all $r\geq 0$,

$\|\langle
D\rangle^{r}\bar{v}_{\varepsilon}^{\operatorname{(I)},\ell;t}\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\,\lesssim\,\|\langle
D\rangle^{r}(\chi(\varepsilon^{\alpha}\nabla)-1)f\|_{\operatorname{L}^{1}((0,t);\dot{H}^{-1}(\mathbb{R}^{d}))}.$

By the properties of the cut-off function $\chi$, we find for all
$\xi\in\mathbb{R}^{d}$ and $k\geq 0$,

$|\chi(\varepsilon^{\alpha}\xi)-1|\,\leq\,\mathds{1}_{|\varepsilon^{\alpha}\xi|\geq\frac{1}{2}}\,\leq\,(2\varepsilon^{\alpha}|\xi|)^{k}.$
(2.37)

Choosing $k={\lfloor\ell/\alpha\rfloor}+1$, we then get by Plancherel’s
formula,

$\|\langle
D\rangle^{r}\bar{v}_{\varepsilon}^{\operatorname{(I)},\ell;t}\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\,\lesssim\,(\varepsilon
C)^{\ell}\|\langle
D\rangle^{K_{\alpha}\ell+r}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}$

with $K_{\alpha}\leq 1/\alpha$, as claimed.

We turn to the case $(\star)=\operatorname{(II)}$. By definition, the
difference $\bar{v}_{\varepsilon}^{\operatorname{(II)},\ell}$ satisfies the
following equation,

$\partial_{t}^{2}\bar{v}_{\varepsilon}^{\operatorname{(II)},\ell}-\nabla\cdot\Big{(}\bar{\boldsymbol{a}}+\sum_{k=2}^{\ell}\bar{\boldsymbol{b}}^{k}\odot(\varepsilon\nabla)^{k-1}+\kappa_{\ell}(\varepsilon|\nabla|)^{\ell}\Big{)}\nabla\bar{v}_{\varepsilon}^{\operatorname{(II)},\ell}\,=\,\kappa_{\ell}(\varepsilon|\nabla|)^{\ell}\triangle\bar{U}_{\varepsilon}^{\ell}.$

Hence, combining (2.32) and the a priori estimate of item (ii), together with
the bound $\kappa_{\ell}\leq C^{\ell}$, we get

$\displaystyle\|\langle
D\rangle^{r}v_{\varepsilon}^{\operatorname{(II)},\ell;t}\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}$
$\displaystyle\leq$ $\displaystyle C^{\ell}\|\langle
D\rangle^{r}(\varepsilon|\nabla|)^{\ell}\nabla\bar{U}_{\varepsilon}^{\ell}\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}$
$\displaystyle\lesssim$ $\displaystyle(\varepsilon C)^{\ell}\|\langle
D\rangle^{\ell+r}\nabla\bar{U}_{\varepsilon}^{\ell}\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}$
$\displaystyle\lesssim$ $\displaystyle(\varepsilon C)^{\ell}\langle
t\rangle\|\langle
D\rangle^{\ell+r}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}.$

It remains to treat the case $(\star)=\operatorname{(III)}$. Starting from
(2.18) and (2.24), and recalling $\kappa_{1}=1$, we get the following equation
for the corresponding difference,

$\partial_{t}^{2}\Big{(}1+\sum_{l=2}^{\ell}\kappa_{l}(\varepsilon|\nabla|)^{l-1}\Big{)}\bar{v}_{\varepsilon}^{\operatorname{(III)},\ell}\\\
-\nabla\cdot\bigg{(}\sum_{n=1}^{\ell}\Big{(}\kappa_{n}\bar{\boldsymbol{a}}+\sum_{l=1}^{n-1}\kappa_{l}\bar{\boldsymbol{b}}^{n+1-l}\odot(\tfrac{\nabla}{|\nabla|})^{n-l}\Big{)}(\varepsilon|\nabla|)^{n-1}\bigg{)}\nabla\bar{v}_{\varepsilon}^{\operatorname{(III)},\ell}\\\
\,=\,\sum_{l=2}^{\ell}\kappa_{l}(\varepsilon|\nabla|)^{l-1}(f-\partial_{t}^{2}\bar{U}_{\varepsilon}^{\ell})\\\
+\nabla\cdot\bigg{(}\sum_{n=2}^{\ell}\Big{(}\kappa_{n}\bar{\boldsymbol{a}}+\sum_{l=2}^{n-1}\kappa_{l}\bar{\boldsymbol{b}}^{n+1-l}\odot(\tfrac{\nabla}{|\nabla|})^{n-l}\Big{)}(\varepsilon|\nabla|)^{n-1}\bigg{)}\nabla\bar{U}_{\varepsilon}^{\ell},$

and thus, further using the equation for $\bar{U}_{\varepsilon}^{\ell}$ in the
right-hand side, we get after reorganizing the terms,

$\partial_{t}^{2}\Big{(}1+\sum_{l=2}^{\ell}\kappa_{l}(\varepsilon|\nabla|)^{l-1}\Big{)}\bar{v}_{\varepsilon}^{\operatorname{(III)},\ell}\\\
-\nabla\cdot\bigg{(}\sum_{n=1}^{\ell}\Big{(}\kappa_{n}\bar{\boldsymbol{a}}+\sum_{l=1}^{n-1}\kappa_{l}\bar{\boldsymbol{b}}^{n+1-l}\odot(\tfrac{\nabla}{|\nabla|})^{n-l}\Big{)}(\varepsilon|\nabla|)^{n-1}\bigg{)}\nabla\bar{v}_{\varepsilon}^{\operatorname{(III)},\ell}\\\
\,=\,-\nabla\cdot\Big{(}\sum_{l=2}^{\ell}\sum_{k=\ell+2-l}^{\ell}\kappa_{l}{\boldsymbol{b}}^{k}\odot(\varepsilon\nabla)^{k-1}(\varepsilon|\nabla|)^{l-1}\Big{)}\nabla\bar{U}_{\varepsilon}^{\ell}.$
(2.38)

Combining (2.32) and the a priori estimate of item (ii), together with the
bounds $|{\boldsymbol{b}}^{k}|\leq C^{k}$ and $\kappa_{\ell}\leq C^{\ell}$, we
get

$\displaystyle\|\langle
D\rangle^{r}\bar{v}_{\varepsilon}^{\operatorname{(III)},\ell}\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}$
$\displaystyle\leq$ $\displaystyle(\varepsilon C)^{\ell}\|\langle
D\rangle^{2\ell+r-2}\nabla\bar{U}_{\varepsilon}^{\ell}\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}$
$\displaystyle\leq$ $\displaystyle(\varepsilon C)^{\ell}\langle
t\rangle\|\langle
D\rangle^{2\ell+r-2}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))},$

and the conclusion follows.∎

###  2.6. Proof of Theorem 1

Let $\boldsymbol{a}$ be $Q$-periodic. We split the proof into three steps. We
first establish (1.6) for the energy norm, before turning to the
$\operatorname{L}^{2}$-estimate, which requires some additional care. We start
by assuming that $\operatorname{supp}\hat{f}\subset\mathbb{R}\times B_{R}$ for
some $R\geq 1$, and then conclude with the general case in the last step.

Step 1\. Proof of (1.6) for the energy norm in case
$\operatorname{supp}\hat{f}\subset\mathbb{R}\times B_{R}$ with $\varepsilon
R\ll 1$.
For simplicity, we start by assuming momentarily that the corrector estimates
in Lemma 2.8 hold uniformly in the sense of

$\displaystyle\|(\psi^{n},\sigma^{n})\|_{{W^{1,\infty}(Q)}}$
$\displaystyle\leq$ $\displaystyle C^{n},$
$\displaystyle\text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\|(\zeta^{n,m},\tau^{n,m})\|_{{W^{1,\infty}(Q)}}$ $\displaystyle\leq$
$\displaystyle C^{n+m+1},\qquad\text{for all $n,m\geq 0$.}$ (2.39)

As $\operatorname{supp}\hat{f}\subset\mathbb{R}\times B_{R}$ with $\varepsilon
R\ll 1$, we can consider the solution $\bar{U}_{\varepsilon}^{\ell}$ of the
formal effective equation (2.18) as given by Lemma 2.10(i). From Proposition
2.9 and Lemma B.1, using the assumed uniform corrector estimate (2.6), we then
obtain

$\|D(u_{\varepsilon}^{t}-S_{\varepsilon}^{\ell}[\bar{U}_{\varepsilon}^{\ell;t},f^{t}])\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\\\
\,\leq\,(\varepsilon C)^{\ell}\|\langle
D\rangle^{2\ell}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}+(\varepsilon
C)^{\ell}\|\langle
D\rangle^{2\ell}D\bar{U}_{\varepsilon}^{\ell}\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))},$

and thus, combining this with the a priori bounds of Lemma 2.10(i),

$\|D(u_{\varepsilon}^{t}-S_{\varepsilon}^{\ell}[\bar{U}_{\varepsilon}^{\ell;t},f^{t}])\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\\\
\,\leq\,(\varepsilon C)^{\ell}\langle t\rangle\|\langle
D\rangle^{2\ell}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}.$
(2.40)

It remains to replace $\bar{U}_{\varepsilon}^{\ell}$ by
$\bar{u}_{\varepsilon}^{(\star),\ell}$ in the left-hand side for
$(\star)=\text{(I)}$, (II), or (III). For that purpose, recalling the
definition of the spectral two-scale expansion, cf. (2.14), and using the
assumed uniform corrector estimates (2.6), we note that

$\|D(S_{\varepsilon}^{\ell}[\bar{u}_{\varepsilon}^{(\star),\ell},f]-S_{\varepsilon}^{\ell}[\bar{U}_{\varepsilon}^{\ell},f])\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\,\leq\,C^{\ell}\|\langle\nabla\rangle^{\ell}D(\bar{u}_{\varepsilon}^{(\star),\ell}-\bar{U}_{\varepsilon}^{\ell})\|_{\operatorname{L}^{2}(\mathbb{R}^{d})},$

hence, by Lemma 2.10(iii),

$\|D(S_{\varepsilon}^{\ell}[\bar{u}_{\varepsilon}^{(\star),\ell;t},f^{t}]-S_{\varepsilon}^{\ell}[\bar{U}_{\varepsilon}^{\ell;t},f^{t}])\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\,\leq\,(\varepsilon
C)^{\ell}\langle t\rangle\|\langle
D\rangle^{C\ell}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}.$

Combined with (2.40), this proves the claim (1.6) for the energy norm in case
$\operatorname{supp}\hat{f}\subset\mathbb{R}\times B_{R}$ with $\varepsilon
R\ll 1$, provided that (2.6) holds.

It remains to treat the case when the uniform boundedness assumption (2.6) for
correctors is not satisfied. In that case, we rather appeal to the Sobolev
embedding to estimate products with correctors: for any periodic corrector or
corrector gradient
$\varphi\in\\{\psi^{n},\nabla\psi^{n},\sigma^{n},\nabla\sigma^{n},\zeta^{n,m},\nabla\zeta^{n,m},\tau^{n,m},\nabla\tau^{n,m}\\}_{n,m}$,
we can estimate, for any function $g$,

$\displaystyle\|\varphi(\tfrac{\cdot}{\varepsilon})g\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}^{d}}\Big{(}\fint_{B_{\varepsilon}(x)}|\varphi(\tfrac{\cdot}{\varepsilon})|^{2}\Big{)}\Big{(}\sup_{B_{\varepsilon}(x)}|g|^{2}\Big{)}\,dx$
(2.41) $\displaystyle\lesssim$
$\displaystyle\|\varphi\|_{\operatorname{L}^{2}(Q)}^{2}\int_{\mathbb{R}^{d}}\Big{(}\sup_{B_{\varepsilon}(x)}|g|^{2}\Big{)}\,dx$
$\displaystyle\lesssim$
$\displaystyle\|\varphi\|_{\operatorname{L}^{2}(Q)}^{2}\|g\|_{H^{a}(\mathbb{R}^{d})}^{2},$

provided $a>\frac{d}{2}$. Up to a fixed loss of derivatives in the estimates,
we may then appeal to the $\operatorname{L}^{2}$ corrector estimates in Lemma
2.8, and the above proof of (1.6) for the energy norm is adapted directly.

Step 2\. Proof of (1.6) for the $\operatorname{L}^{2}$-norm in case
$\operatorname{supp}\hat{f}\subset B_{R}$ with $\varepsilon R\ll 1$.
As in Step 1, we aim to apply Proposition 2.9 and Lemma B.1, together with
corrector estimates. However, the following terms are a priori problematic in
the right-hand side of the equation for the spectral two-scale expansion given
by Proposition 2.9,

$T_{\varepsilon}^{\ell}\,:=\,-\varepsilon^{\ell}\sigma^{\ell}_{i_{1}\ldots
i_{\ell}}(\tfrac{\cdot}{\varepsilon}):\gamma_{\ell}(\varepsilon\nabla)\nabla^{2}\nabla^{\ell}_{i_{1}\ldots
i_{\ell}}\bar{U}_{\varepsilon}^{\ell}\\\
+\sum_{n=1}^{\ell}\sum_{k=\ell+2-n}^{\ell+1}\varepsilon^{n+k-2}\psi^{n}_{i_{1}\ldots
i_{n}}(\tfrac{\cdot}{\varepsilon})\bar{\boldsymbol{b}}^{k-1}_{i_{n+1}\ldots
i_{n+k-2}}:\gamma_{\ell}(\varepsilon\nabla)\nabla^{2}\nabla_{i_{1}\ldots
i_{n+k-2}}^{n+k-2}\bar{U}_{\varepsilon}^{\ell}.$ (2.42)

Indeed, these terms are not total derivatives and involve
$\bar{U}_{\varepsilon}^{\ell}$ itself: when applying Lemma B.1 to estimate the
$\operatorname{L}^{2}$-norm of the two-scale expansion error, these terms
would therefore contribute like

$(\varepsilon C)^{\ell}\langle
t\rangle^{2}\|{\langle{D}\rangle^{C\ell}}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}$

with a prefactor $\langle t\rangle^{2}$ instead of $\langle t\rangle$. In
order to improve on this, we shall reformulate $T_{\varepsilon}^{\ell}$ as a
total time-derivative up to terms that depend only locally on $f$. Using the
short-hand notation

$\bar{\mathcal{L}}_{\varepsilon}^{\ell}\,:=\,-\nabla\cdot\Big{(}\bar{\boldsymbol{a}}+\sum_{k=2}^{\ell}\bar{\boldsymbol{b}}^{k}\odot(\varepsilon\nabla)^{k-1}\Big{)}\nabla,$

the effective equation for $\bar{U}_{\varepsilon}^{\ell}$ entails

$\bar{\mathcal{L}}_{\varepsilon}^{\ell}\nabla^{2}\bar{U}_{\varepsilon}^{\ell}=-\partial_{t}^{2}\nabla^{2}\bar{U}_{\varepsilon}^{\ell}+\nabla^{2}f.$

As in the proof of Lemma 2.10(i), cf. (2.28), the assumption $\varepsilon R\ll
1$ precisely ensures that the operator
$\bar{\mathcal{L}}_{\varepsilon}^{\ell}:\operatorname{L}^{2}(\mathbb{R}^{d})\to\dot{H}^{-2}(\mathbb{R}^{d})$
can be inverted when restricted to functions with spatial Fourier transform
supported in $B_{R}$. As by definition both $\bar{U}_{\varepsilon}^{\ell}$ and
$f$ have spatial Fourier transform supported in $B_{R}$, we may then write

$\nabla^{2}\bar{U}_{\varepsilon}^{\ell}=-\partial_{t}^{2}(\bar{\mathcal{L}}_{\varepsilon}^{\ell})^{-1}\nabla^{2}\bar{U}_{\varepsilon}^{\ell}+(\bar{\mathcal{L}}_{\varepsilon}^{\ell})^{-1}\nabla^{2}f.$
(2.43)

By uniform ellipticity (2.28), we have for any function $g$ with
$\operatorname{supp}\hat{g}\subset B_{R}$,

$\|(\bar{\mathcal{L}}_{\varepsilon}^{\ell})^{-1}\nabla^{2}g\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\,\lesssim\,\|g\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}.$
(2.44)

Now using (2.43) to reformulate $T_{\varepsilon}^{\ell}$, cf. (2.42), we get

$T_{\varepsilon}^{\ell}\,=\,\varepsilon^{\ell}\partial_{t}^{2}\Big{(}\sigma^{\ell}_{i_{1}\ldots
i_{\ell}}(\tfrac{\cdot}{\varepsilon}):\gamma_{\ell}(\varepsilon\nabla)\nabla^{\ell}_{i_{1}\ldots
i_{\ell}}(\bar{\mathcal{L}}_{\varepsilon}^{\ell})^{-1}\nabla^{2}\bar{U}_{\varepsilon}^{\ell}\Big{)}\\\
-\sum_{n=1}^{\ell}\sum_{k=\ell+2-n}^{\ell+1}\varepsilon^{n+k-2}\partial_{t}^{2}\Big{(}\psi^{n}_{i_{1}\ldots
i_{n}}(\tfrac{\cdot}{\varepsilon})\bar{\boldsymbol{b}}^{k-1}_{i_{n+1}\ldots
i_{n+k-2}}:\gamma_{\ell}(\varepsilon\nabla)\nabla_{i_{1}\ldots
i_{n+k-2}}^{n+k-2}(\bar{\mathcal{L}}_{\varepsilon}^{\ell})^{-1}\nabla^{2}\bar{U}_{\varepsilon}^{\ell}\Big{)}\\\
-\varepsilon^{\ell}\sigma^{\ell}_{i_{1}\ldots
i_{\ell}}(\tfrac{\cdot}{\varepsilon}):\gamma_{\ell}(\varepsilon\nabla)\nabla^{\ell}_{i_{1}\ldots
i_{\ell}}(\bar{\mathcal{L}}_{\varepsilon}^{\ell})^{-1}\nabla^{2}f\\\
+\sum_{n=1}^{\ell}\sum_{k=\ell+2-n}^{\ell+1}\varepsilon^{n+k-2}\psi^{n}_{i_{1}\ldots
i_{n}}(\tfrac{\cdot}{\varepsilon})\bar{\boldsymbol{b}}^{k-1}_{i_{n+1}\ldots
i_{n+k-2}}:\gamma_{\ell}(\varepsilon\nabla)\nabla_{i_{1}\ldots
i_{n+k-2}}^{n+k-2}(\bar{\mathcal{L}}_{\varepsilon}^{\ell})^{-1}\nabla^{2}f.$

Using this to replace the corresponding terms in the equation for the two-
scale expansion error in Proposition 2.9, appealing to Lemma B.1 to estimate
its $\operatorname{L}^{2}$-norm, using the corrector estimates of Lemma 2.8,
using the Sobolev embedding to estimate products with correctors as in (2.41),
and using (2.44), we get for $a>\frac{d}{2}$,

$\|u_{\varepsilon}^{t}-S_{\varepsilon}^{\ell}[\bar{U}_{\varepsilon}^{\ell;t},f^{t}]\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\\\
\,\leq\,(\varepsilon C)^{\ell}\langle t\rangle\|\langle
D\rangle^{2\ell+a}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}+(\varepsilon
C)^{\ell}\|\langle
D\rangle^{2\ell+a}D\bar{U}_{\varepsilon}^{\ell}\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))},$

and thus, by the a priori estimate of Lemma 2.10(i),

$\|u_{\varepsilon}^{t}-S_{\varepsilon}^{\ell}[\bar{U}_{\varepsilon}^{\ell;t},f^{t}]\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\,\leq\,(\varepsilon
C)^{\ell}\langle t\rangle\|\langle
D\rangle^{2\ell+a}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}.$
(2.45)

It remains to argue as in Step 1 to replace $\bar{U}_{\varepsilon}^{\ell}$ by
$\bar{u}_{\varepsilon}^{(\star),\ell}$ in the left-hand side. For that
purpose, recalling the definition of the spectral two-scale expansion, cf.
(2.14), and using again the corrector estimates of Lemma 2.8 together with the
Sobolev embedding as in (2.41), we note that for $a>\frac{d}{2}$,

$\|S_{\varepsilon}^{\ell}[\bar{u}_{\varepsilon}^{(\star),\ell},f]-S_{\varepsilon}^{\ell}[\bar{U}_{\varepsilon}^{\ell},f]\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\,\leq\,C^{\ell}\|\langle\nabla\rangle^{\ell+a}(\bar{u}_{\varepsilon}^{(\star),\ell}-\bar{U}_{\varepsilon}^{\ell})\|_{\operatorname{L}^{2}(\mathbb{R}^{d})},$

hence, by Lemma 2.10(iii),

$\|S_{\varepsilon}^{\ell}[\bar{u}_{\varepsilon}^{(\star),\ell;t},f^{t}]-S_{\varepsilon}^{\ell}[\bar{U}_{\varepsilon}^{\ell;t},f^{t}]\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\,\leq\,(\varepsilon
C)^{\ell}\langle t\rangle\|\langle
D\rangle^{C\ell}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}.$

Combined with (2.45), this proves the claim (1.6) for the
$\operatorname{L}^{2}$-norm in case $\operatorname{supp}\hat{f}\subset B_{R}$
with $\varepsilon R\ll 1$.

Step 3\. Approximation argument: proof for general $f$.
For $R\geq 1$, consider the truncated impulse

$f_{R}\,:=\,\chi(\tfrac{1}{R}\nabla)f,$

and let $\bar{u}_{\varepsilon,R}^{(\star),\ell}$ be the solution of the
modified effective equations given by Lemma 2.10(ii) with impulse $f$ replaced
by $f_{R}$. Recalling the definition of the spectral two-scale expansion, cf.
(2.14), and using again corrector estimates, we then note that for
$a>\frac{d}{2}$,

$\|D(S_{\varepsilon}^{\ell}[\bar{u}_{\varepsilon,R}^{(\star),\ell},f_{R}]-S_{\varepsilon}^{\ell}[\bar{u}_{\varepsilon}^{(\star),\ell},f])\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\\\
\,\leq\,C^{\ell}\|\langle\nabla\rangle^{\ell+a}D(\bar{u}_{\varepsilon}^{(\star),\ell}-\bar{u}_{\varepsilon,R}^{(\star),\ell})\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}+C^{\ell}\|\langle
D\rangle^{\ell+a}(f-f_{R})\|_{\operatorname{L}^{2}(\mathbb{R}^{d})},$

and thus, by linearity and by the a priori estimates of Lemma 2.10(ii),

$\|D(S_{\varepsilon}^{\ell}[\bar{u}_{\varepsilon,R}^{(\star),\ell;t},f_{R}^{t}]-S_{\varepsilon}^{\ell}[\bar{u}_{\varepsilon}^{(\star),\ell;t},f^{t}])\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\,\leq\,C^{\ell}\|{\langle
D\rangle^{C\ell}}(f-f_{R})\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}.$

By definition of $f_{R}$, as in (2.37), the right-hand side can now be
estimated as follows, for any $k\geq 0$,

$\|D(S_{\varepsilon}^{\ell}[\bar{u}_{\varepsilon,R}^{(\star),\ell;t},f_{R}^{t}]-S_{\varepsilon}^{\ell}[\bar{u}_{\varepsilon}^{(\star),\ell;t},f^{t}])\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\,\leq\,R^{-k}C^{\ell}\|\langle
D\rangle^{C\ell+k}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}.$

Combining this with the results of Steps 1 and 2, the conclusion follows for
instance with the choice $R=\varepsilon^{-1/2}$ and $k=2\ell$. ∎

###  2.7. Proof of Corollary 1

The bound (1.7) is obtained along the same line as Theorem 1 (using a
straightforward adaptation of Lemma B.1), and it remains to deduce (1.8). For
that purpose, we note that (1.8) would actually follow from (1.7) together
with the bound

$\int_{-\infty}^{t}(t-s)\|\langle
D\rangle^{C\ell}f^{s}\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\,ds\,\leq\,{C_{f}^{\ell}},$
(2.46)

in favor of which we presently argue. Recall that we assume here
$f^{t}(x)=f_{1}(t)f_{2}(x)$, where $f_{1}$ has a smooth and compactly
supported Fourier transform on $\mathbb{R}$ and where $f_{2}$ has a compactly
supported Fourier transform on $\mathbb{R}^{d}$. The assumption on $f_{2}$
entails
$\|\langle\nabla\rangle^{C\ell}f_{2}\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\leq{C_{f_{2}}^{\ell}}$,
while the assumption on $f_{1}$ yields for $s\leq 0$,

$\displaystyle\|\langle\partial_{t}\rangle^{C\ell}f_{1}\|_{\operatorname{L}^{1}((-\infty,s))}$
$\displaystyle\lesssim$ $\displaystyle\langle
s\rangle^{-2}\|\langle\cdot\rangle^{3}\langle\partial_{t}\rangle^{C\ell}f_{1}\|_{\operatorname{L}^{\infty}((-\infty,s))}$
$\displaystyle\leq$ $\displaystyle\langle
s\rangle^{-2}C^{\ell}\|\langle\cdot\rangle^{C\ell}\langle\partial\rangle^{3}\hat{f}_{1}\|_{\operatorname{L}^{\infty}((-\infty,s))}$
$\displaystyle\leq$ $\displaystyle\langle s\rangle^{-2}{C_{f_{1}}^{\ell}},$

where $\hat{f}_{1}$ stands for the temporal Fourier transform of $f_{1}$. The
claim (2.46) follows. ∎

###  2.8. Proof of Theorem 3

We briefly explain how the above proof of Theorem 1 is adapted to the random
setting. As for Theorem 1, we may assume
$\operatorname{supp}\hat{f}\subset\mathbb{R}\times B_{R}$ with $\varepsilon
R\ll 1$, and the general conclusion follows by approximation. As explained in
Appendix A, the only difference with respect to the periodic setting is that
only a finite number $\ell_{*}=\lceil\frac{\beta\wedge d}{2}\rceil$ of
correctors can be defined with stationary gradient, and the highest-order
corrector has a nontrivial sublinear growth. Using Proposition 2.9 in
combination with Lemma B.1 to estimate the energy norm of the two-scale
expansion error of order $\ell\leq\ell_{*}$, and using the corrector estimates
of Appendix A and the Sobolev embedding, we find for $a>\frac{d}{2}$,

$\|D(u_{\varepsilon}^{t}-S_{\varepsilon}^{\ell}[\bar{U}_{\varepsilon}^{\ell;t},f^{t}])\|_{\operatorname{L}^{q}(\Omega;\operatorname{L}^{2}(\mathbb{R}^{d}))}\\\
\,\lesssim\,\varepsilon^{\ell}\|\mu_{\ell}^{*}(\tfrac{\cdot}{\varepsilon})\gamma(\varepsilon\nabla)\langle
D\rangle^{2\ell+a}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}\\\
+\varepsilon^{\ell}\|\mu_{\ell}^{*}(\tfrac{\cdot}{\varepsilon})\gamma(\varepsilon\nabla)\langle
D\rangle^{2\ell+a}D\bar{U}_{\varepsilon}^{\ell}\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))},$

where the weight $\mu_{\ell}^{*}$ originates in the growth of correctors and
is defined in (A.1). A novelty with respect to the proof of Theorem 1 is that
we now need weighted energy estimates for $\bar{U}_{\varepsilon}^{\ell}$ with
sublinear weight $\mu_{\ell}^{*}$. For that purpose, as the homogenized
equation (2.18) has constant coefficients, we note that
$\bar{U}_{\varepsilon}^{\ell}$ displays ballistic transport, and therefore

$\displaystyle\|\mu_{\ell}^{*}(\tfrac{\cdot}{\varepsilon})\gamma(\varepsilon\nabla)\langle
D\rangle^{2\ell+a}D\bar{U}_{\varepsilon}^{\ell;t}\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}$
$\displaystyle\lesssim$
$\displaystyle\mu_{\ell}^{*}(\tfrac{1}{\varepsilon}\langle
t\rangle)\,\|\langle\cdot\rangle\gamma(\varepsilon\nabla)\langle
D\rangle^{C}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}$
$\displaystyle\lesssim$
$\displaystyle\mu_{\ell}^{*}(\tfrac{1}{\varepsilon}\langle
t\rangle)\,\|\langle\cdot\rangle\langle
D\rangle^{C}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}.$

This is easily obtained by interpolation, using that the weight $x$
corresponds to a derivative in Fourier space and using the Duhamel formula for
$\bar{U}_{\varepsilon}^{\ell}$; see e.g. [5, Proof of Proposition 3] for the
details. The above then becomes

$\|D(u_{\varepsilon}^{t}-S_{\varepsilon}^{\ell}[\bar{U}_{\varepsilon}^{\ell;t},f^{t}])\|_{\operatorname{L}^{q}(\Omega;\operatorname{L}^{2}(\mathbb{R}^{d}))}\,\lesssim\,\varepsilon^{\ell}\langle
t\rangle\,\mu_{\ell}^{*}(\tfrac{1}{\varepsilon}\langle
t\rangle)\,\|\langle\cdot\rangle\langle
D\rangle^{C}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}.$

Note that the error estimate for $\ell=\ell_{*}-1$ is occasionally better than
the one for $\ell=\ell_{*}$. Optimizing between the results for
$\ell=\ell_{*}$ and for $\ell=\ell_{*}-1$, we easily conclude

$\|D(u_{\varepsilon}^{t}-S_{\varepsilon}^{\ell_{*}}[\bar{U}_{\varepsilon}^{\ell_{*};t},f^{t}])\|_{\operatorname{L}^{q}(\Omega;\operatorname{L}^{2}(\mathbb{R}^{d}))}\,\lesssim\,\varepsilon^{\ell_{*}}\langle
t\rangle\big{(}\mu_{\ell_{*}}^{*}(\tfrac{1}{\varepsilon}\langle
t\rangle)\wedge\tfrac{1}{\varepsilon}\big{)}\,\|\langle\cdot\rangle\langle
D\rangle^{C}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}.$

As in the proof of Theorem 1, we can replace
$\bar{U}_{\varepsilon}^{\ell_{*}}$ in the left-hand side by the solution of
any well-posed modification of the formal homogenized equation, and we can
derive a similar estimate for the $\operatorname{L}^{2}$-norm. ∎

##  3\. Geometric approach and hyperbolic two-scale expansion

This section is devoted to the definition of hyperbolic correctors and to the
proof of Theorem 2, including the well-posedness of the formal homogenized
equation (1.10). This essentially constitutes a rewriting of [3, 24, 1] and is
needed to rigorously relate those works to the spectral two-scale expansion,
cf. Section 4.

###  3.1. Definition of hyperbolic correctors

We start with the definition of the hyperbolic correctors
$\\{\phi^{n,m}\\}_{n,m}$ and of the homogenized tensors
$\\{\bar{\boldsymbol{a}}^{n,m}\\}_{n,m}$, as motivated in Section 1.5.

######  Definition 3.1 (Hyperbolic correctors).

In the periodic setting, the hyperbolic correctors $\\{\phi^{n,m}\\}_{n,m\geq
0}$, homogenized tensors $\\{\bar{\boldsymbol{a}}^{n,m}\\}_{n\geq 1,m\geq 0}$,
and fluxes $\\{q^{n,m}\\}_{n,m\geq 0}$ are inductively defined as follows:

  1. $\bullet$

We set $\phi^{0,0}:=1$ and $\phi^{0,m}:=0$ for $m\geq 1$, while for $n\geq 1$
and $m\geq 0$ we define $\phi^{n,m}:=(\phi^{n,m}_{j_{1}\ldots j_{n}})_{1\leq
j_{1},\ldots,j_{n}\leq d}$ with $\phi_{j_{1}\ldots j_{n}}^{n,m}\in
H^{1}_{\operatorname{per}}(Q)$ the periodic scalar field that has vanishing
average $\mathbb{E}\big{[}\phi_{j_{1}\ldots j_{n}}^{n,m}\big{]}=0$ and
satisfies

$\quad-\nabla\cdot\boldsymbol{a}\nabla\phi_{j_{1}\ldots
j_{n}}^{n,m}\,=\,\nabla\cdot\big{(}\boldsymbol{a}\phi_{j_{1}\ldots
j_{n-1}}^{n-1,m}e_{j_{n}}\big{)}+e_{j_{n}}\cdot q^{n-1,m}_{j_{1}\ldots
j_{n-1}}.$

  2. $\bullet$

For $n\geq 1$ and $m\geq 0$, we define
$\bar{\boldsymbol{a}}^{n,m}:=(\bar{\boldsymbol{a}}^{n,m}_{j_{1}\ldots
j_{n-1}})_{1\leq j_{1},\ldots,j_{n-1}\leq d}$ as the matrix-valued $(n-1)$th-
order tensor given by

$\quad\bar{\boldsymbol{a}}^{n,m}_{j_{1}\ldots
j_{n-1}}e_{j}\,:=\,\mathbb{E}\left[\boldsymbol{a}\big{(}\nabla\phi_{j_{1}\ldots
j_{n-1}j}^{n,m}+\phi_{j_{1}\ldots j_{n-1}}^{n-1,m}e_{j}\big{)}\right].$

  3. $\bullet$

For $n,m\geq 0$, we define $q^{n,m}:=(q^{n,m}_{j_{1}\ldots j_{n}})_{1\leq
j_{1},\ldots,j_{n}\leq d}$ with $q^{n,m}_{j_{1}\ldots
j_{n}}\in\operatorname{L}^{2}_{\operatorname{per}}(Q)^{d}$ the periodic vector
field given by

$q^{n,m}_{j_{1}\ldots j_{n}}\,:=\,\boldsymbol{a}\big{(}\nabla\phi_{j_{1}\ldots
j_{n}}^{n,m}+\phi_{j_{1}\ldots
j_{n-1}}^{n-1,m}e_{j_{n}}\big{)}-\phi_{j_{1}\ldots
j_{n}j}^{n+1,m-2}e_{j}-\bar{\boldsymbol{a}}_{j_{1}\ldots
j_{n-1}}^{n,m}e_{j_{n}},$

where the definition of $\bar{\boldsymbol{a}}^{n,m}$ ensures
$\mathbb{E}\left[q^{n,m}\right]=0$.

In particular, there holds $\phi^{n,m}=0$, $\bar{\boldsymbol{a}}^{n,m}=0$, and
$q^{n,m}=0$ whenever $m$ is an odd integer.333This is natural as time
derivatives should always come in even numbers in view of equation (1.2). ∎

For all $n$, we note that $\phi^{n,0}$ coincides with the elliptic corrector
of order $n$, cf. [6]. As is common in the elliptic setting, it is useful to
further introduce suitable flux correctors, which indeed allow us to refine
error estimates and slightly improve on the result of [3]. More precisely,
flux correctors are designed to allow a direct optimal exploitation of
cancellations due to fluxes having vanishing average
$\mathbb{E}\big{[}q^{n,m}\big{]}=0$. We start with the definition of flux
correctors for $m\geq 1$.

######  Definition 3.2 (Hyperbolic flux correctors).

For $n\geq 0$ and $m\geq 1$, we define the hyperbolic flux corrector
$\sigma^{n,m}:=(\sigma^{n,m}_{j_{1}\ldots j_{n}})_{1\leq
j_{1},\ldots,j_{n}\leq d}$ by

$\sigma^{n,m}_{j_{1}\ldots j_{n}}=(\nabla\Phi^{n,m}_{j_{1}\ldots
j_{n}})^{\prime},$

where $\Phi^{n,m}_{j_{1}\ldots j_{n}}\in H^{1}_{\operatorname{per}}(Q)^{d}$ is
the periodic vector field that satisfies $\mathbb{E}\big{[}\Phi_{j_{1}\ldots
j_{n}}^{n,m}\big{]}=0$ and

$-\triangle\Phi^{n,m}_{j_{1}\ldots j_{n}}\,=\,q^{n,m}_{j_{1}\ldots
j_{n}}.\qed$

In case $m=0$, as correctors $\phi^{n,0}$ coincide with elliptic correctors,
they display more structure than general hyperbolic correctors. We recall how
this structure can be exploited to construct a suitable flux corrector
$\sigma^{n,0}$ that is skew-symmetric: the following lemma is essentially
borrowed from [15].

######  Lemma 3.3 (Elliptic correctors and flux correctors).

Up to symmetrization of indices, the elliptic correctors
$\\{\phi^{n,0}\\}_{n\geq 0}$ and homogenized tensors
$\\{\bar{\boldsymbol{a}}^{n,0}\\}_{n\geq 1}$ coincide with the modified
families $\\{\tilde{\phi}^{n,0}\\}_{n\geq 0}$ and
$\\{\tilde{\boldsymbol{a}}^{n,0}\\}_{n\geq 1}$ defined via the following cell
problems:

  1. $\bullet$

We set $\tilde{\phi}^{0,0}:=1$ and for $n\geq 1$ we define
$\tilde{\phi}^{n,0}:=(\tilde{\phi}^{n,0}_{j_{1}\ldots j_{n}})_{1\leq
j_{1},\ldots j_{n},\leq d}$ with $\tilde{\phi}_{j_{1}\ldots j_{n}}^{n,0}\in
H^{1}_{\operatorname{per}}(Q)$ the periodic scalar field that has vanishing
average $\mathbb{E}\big{[}\tilde{\phi}_{j_{1}\ldots j_{n}}^{n,0}\big{]}=0$ and
satisfies

$\quad-\nabla\cdot\boldsymbol{a}\nabla\tilde{\phi}_{j_{1}\ldots
j_{n}}^{n,0}\,=\,\nabla\cdot\big{(}\boldsymbol{a}\tilde{\phi}_{j_{1}\ldots
j_{n-1}}^{n-1,0}e_{j_{n}}\big{)}+e_{j_{n}}\cdot\tilde{q}^{n-1,0}_{j_{1}\ldots
j_{n-1}}.$

  2. $\bullet$

For $n\geq 1$, we define
$\tilde{\boldsymbol{a}}^{n,0}:=(\tilde{\boldsymbol{a}}^{n,0}_{j_{1}\ldots
j_{n-1}})_{1\leq j_{1},\ldots,j_{n-1}\leq d}$ with
$\tilde{\boldsymbol{a}}^{n,0}_{j_{1}\ldots j_{n-1}}$ the matrix given by

$\quad\tilde{\boldsymbol{a}}^{n,0}_{j_{1}\ldots
j_{n-1}}e_{j}\,:=\,\mathbb{E}\left[\boldsymbol{a}\big{(}\nabla\tilde{\phi}_{j_{1}\ldots
j_{n-1}j}^{n,0}+\tilde{\phi}_{j_{1}\ldots
j_{n-1}}^{n-1,0}e_{j}\big{)}\right].$

  3. $\bullet$

For $n\geq 0$, we define $\tilde{q}^{n,0}:=(\tilde{q}^{n,0}_{j_{1}\ldots
j_{n}})_{1\leq j_{1},\ldots,j_{n}\leq d}$ with $\tilde{q}^{n,0}_{j_{1}\ldots
j_{n}}\in\operatorname{L}^{2}_{\operatorname{per}}(Q)^{d}$ the periodic vector
field given by

$\quad\tilde{q}_{j_{1}\ldots
j_{n}}^{n,0}:=\boldsymbol{a}\big{(}\nabla\tilde{\phi}_{j_{1}\ldots
j_{n}}^{n,0}+\tilde{\phi}_{j_{1}\ldots
j_{n-1}}^{n-1,0}e_{j_{n}}\big{)}-\tilde{\boldsymbol{a}}^{n,0}_{j_{1}\ldots
j_{n-1}}e_{j_{n}}-\sigma_{j_{1}\ldots j_{n-1}}^{n-1,0}e_{j_{n}},$

where the definition of $\tilde{\boldsymbol{a}}^{n,0}$ and $\sigma^{n-1,0}$
ensures $\mathbb{E}\left[\tilde{q}^{n,0}\right]=0$ and
$\nabla\cdot\tilde{q}^{n,0}_{j_{1}\ldots j_{n}}=0$.

  4. $\bullet$

We set $\sigma^{0,0}:=0$ and we define the flux corrector
$\sigma^{n,0}:=(\sigma^{n,0}_{j_{1}\ldots j_{n}})_{1\leq
j_{1},\ldots,j_{n}\leq d}$ for $n\geq 1$ with $\sigma^{n,0}_{j_{1}\ldots
j_{n}}\in H^{1}_{\operatorname{per}}(Q)^{d\times d}$ the periodic skew-
symmetric matrix field that has vanishing average
$\mathbb{E}\big{[}\sigma^{n,0}_{j_{1}\ldots j_{n}}\big{]}=0$ and satisfies

$\quad-\triangle\sigma^{n,0}_{j_{1}\ldots
j_{n}}=\nabla\times\tilde{q}^{n,0}_{j_{1}\ldots
j_{n}},\qquad\nabla\cdot\sigma^{n,0}_{j_{1}\ldots
j_{n}}=\tilde{q}_{j_{1}\ldots j_{n}}^{n,0},$

where we use the vectorial notation $(\nabla\times
F)_{ij}:=\nabla_{i}F_{j}-\nabla_{j}F_{i}$ for a vector field $F$.

More precisely, these modified correctors coincide with
$\\{\phi^{n,0}\\}_{n\geq 0}$ in the sense that we have for all $n\geq 0$ and
$\xi\in\mathbb{R}^{d}$,

$\phi^{n,0}\odot\xi^{\otimes
n}\,=\,\tilde{\phi}^{n,0}\odot\xi^{n},\qquad(\bar{\boldsymbol{a}}^{n,0}\odot\xi^{\otimes(n-1)})\xi\,=\,(\tilde{\boldsymbol{a}}^{n,0}\odot\xi^{\otimes(n-1)})\xi.\qed$

###### Proof.

We refer to [17] or [15] for the construction of the skew-symmetric flux
corrector $\sigma^{n,0}$, based on the compatibility condition
$\nabla\cdot\tilde{q}^{n,0}_{j_{1}\ldots j_{n}}=0$, and we now turn to the
equivalence of $\phi^{n,0}$ and $\tilde{\phi}^{n,0}$. Setting

$\hat{q}_{j_{1}\ldots
j_{n}}^{n,0}:=\boldsymbol{a}\big{(}\nabla\phi_{j_{1}\ldots
j_{n}}^{n,0}+\phi_{j_{1}\ldots
j_{n-1}}^{n-1,0}e_{j_{n}}\big{)}-\bar{\boldsymbol{a}}^{n,0}_{j_{1}\ldots
j_{n-1}}e_{j_{n}}-\sigma^{n-1,0}_{j_{1}\ldots j_{n-1}}e_{j_{n}},$ (3.1)

the equation for $\phi^{n,0}$ in Definition 3.1 for $n\geq 1$ can be written
as

$-\nabla\cdot\boldsymbol{a}\nabla\phi_{j_{1}\ldots
j_{n}}^{n,0}\,=\,\nabla\cdot\big{(}\boldsymbol{a}\phi_{j_{1}\ldots
j_{n-1}}^{n-1,0}e_{j_{n}}\big{)}+e_{j_{n}}\cdot\big{(}\hat{q}^{n-1,0}_{j_{1}\ldots
j_{n-1}}+\sigma_{j_{1}\ldots j_{n-2}}^{n-2,0}e_{j_{n-1}}\big{)}.$ (3.2)

Symmetrizing indices $j_{1},\ldots,j_{n}$, the skew-symmetry of
$\sigma^{n-2,0}$ allows us to drop the corresponding right-hand side term, and
we may conclude by induction that $\phi^{n,0}$ coincides with its modified
version $\tilde{\phi}^{n,0}$ up to symmetrization as stated. ∎

An iterative use of the Poincaré inequality on the unit cell $Q$ ensures the
well-posedness of the above objects and further provides the following a
priori estimates.

######  Lemma 3.4.

In the periodic setting, the above quantities
$\\{\phi^{n,m},\sigma^{n,m},\bar{\boldsymbol{a}}^{n,m})_{n,m}$ are uniquely
defined and satisfy for all $n,m\geq 0$,

$\|(\phi^{n,m},\sigma^{n,m})\|_{H^{1}(Q)}+|\bar{\boldsymbol{a}}^{n,m}|\,\leq\,C^{n+m}.\qed$

We conclude this paragraph with some important vanishing property of the
higher-order hyperbolic homogenized coefficients, which extends the
corresponding elliptic result [15, Lemma 2.3]; see also [5, Proposition 1],
[3, Theorem 2.13], and [1, Theorem 3.5]. The proof is postponed to Section
3.5.

######  Proposition 3.5 (Symmetry of homogenized coefficients).

For all $n\geq 1$ and $m\geq 0$, there holds for any $j_{1},\ldots,j_{n+1}$,

$e_{j_{n+1}}\cdot\bar{\boldsymbol{a}}^{n,m}_{j_{1}\ldots
j_{n-1}}e_{j_{n}}\,=\,(-1)^{n+1}e_{j_{1}}\cdot\bar{\boldsymbol{a}}^{n,m}_{j_{n+1}\ldots
j_{3}}e_{j_{2}}.$

In particular, whenever $n$ is even, we have

$\xi\cdot(\bar{\boldsymbol{a}}^{n,m}\odot\xi^{\otimes(n-1)})\xi=0,\qquad\text{for
all $\xi\in\mathbb{R}^{d}$}.\qed$

###  3.2. Geometric two-scale expansion

Given a smooth function $\bar{w}$, we consider its two-scale expansion
associated with the above-defined hyperbolic correctors, as in (1.38),

$H^{\ell}_{\varepsilon}[\bar{w}]\,:=\,\sum_{n=0}^{\ell}\sum_{m=0}^{\ell-n}\varepsilon^{n+m}\phi^{n,m}(\tfrac{\cdot}{\varepsilon})\odot\nabla^{n}\partial_{t}^{m}\bar{w},$
(3.3)

and we show that it is well-adapted to describe the local behavior of the
solution to the hyperbolic equation in the following sense: as explained in
(1.40), the heterogeneous hyperbolic operator applied to
$H^{\ell}_{\varepsilon}[\bar{w}]$ is equivalent to some higher-order effective
operator applied to $\bar{w}$ (up to $O(\varepsilon^{\ell})$ error terms). The
proof is postponed to Section 3.6.

######  Proposition 3.6 (Geometric two-scale expansion).

Let $\ell\geq 1$, $\varepsilon>0$, and let $\bar{w},f$ be smooth functions
satisfying

$\partial_{t}^{2}\bar{w}-\nabla\cdot\Big{(}\sum_{n=1}^{\ell}\sum_{m=0}^{\ell-n}\bar{\boldsymbol{a}}^{n,m}\odot(\varepsilon\nabla)^{n-1}(\varepsilon\partial_{t})^{m}\Big{)}\nabla\bar{w}\,=\,f.$
(3.4)

Then, the associated geometric two-scale expansion of order $\ell$, defined in
(3.3), satisfies the following relation in the distributional sense in
$\mathbb{R}\times\mathbb{R}^{d}$,

$\big{(}\partial_{t}^{2}-\nabla\cdot\boldsymbol{a}(\tfrac{\cdot}{\varepsilon})\nabla\big{)}H^{\ell}_{\varepsilon}[\bar{w}]\\\
=\,f-\varepsilon^{\ell}\sum_{n=0}^{\ell}\nabla\cdot\Big{(}\big{(}\boldsymbol{a}\phi_{j_{1}\ldots
j_{n}}^{n,\ell-n}-\sigma^{n,\ell-n}_{j_{1}\ldots
j_{n}}\big{)}(\tfrac{\cdot}{\varepsilon})\nabla\nabla^{n}_{j_{1}\ldots
j_{n}}\partial_{t}^{\ell-n}\bar{w}\Big{)}\hskip 99.58464pt\\\
+\varepsilon^{\ell}\sum_{n=1}^{\ell}\partial_{t}\Big{(}\phi^{n,\ell-n}(\tfrac{\cdot}{\varepsilon})\odot\nabla^{n}\partial_{t}^{\ell+1-n}\bar{w}-\sigma^{n-1,\ell+1-n}_{j_{1}\ldots
j_{n-1}}(\tfrac{\cdot}{\varepsilon}):\nabla^{2}\nabla^{n-1}_{j_{1}\ldots
j_{n-1}}\partial_{t}^{\ell-n}\bar{w}\Big{)}.\qed$

######  Remark 3.7.

Note that the last two right-hand side terms in the above equation for the
two-scale expansion $H_{\varepsilon}^{\ell}[\bar{w}]$ are total derivatives
(with respect to time or space): this is not a trivial fact for the last term,
as it is based on the possibility of constructing skew-symmetric flux
correctors $\\{\sigma^{n,0}\\}_{n}$. This happens to be crucial when applying
Lemma B.1 in order to deduce an optimal $\operatorname{L}^{2}$ error estimate.
This slightly refines the analysis of [3]. ∎

###  3.3. Homogenized equations and secular growth problem

As motivated in Proposition 3.6, cf. (3.4), we consider the following formal
homogenized equation, for $\ell\geq 1$,

$\left\\{\begin{array}[]{ll}\partial_{t}^{2}\bar{W}^{\ell}_{\varepsilon}-\nabla\cdot\big{(}\sum_{n=1}^{\ell}\sum_{m=0}^{\ell-n}\bar{\boldsymbol{a}}^{n,m}\odot(\varepsilon\nabla)^{n-1}(\varepsilon\partial_{t})^{m}\big{)}\nabla\bar{W}^{\ell}_{\varepsilon}=f,&\text{in
$\mathbb{R}\times\mathbb{R}^{d}$},\\\
\bar{W}_{\varepsilon}^{\ell}=f=0,&\text{for $t<0$}.\end{array}\right.$ (3.5)

However, this equation mixes higher-order space and time derivatives, and its
well-posedness is problematic. To avoid the secular growth problem described
in Section 1.3, we follow [3] and first show that the differential operator in
(3.5) can be rewritten in such a way that it does no longer mix space and time
derivatives. This is achieved by iteratively using the equation to eliminate
time derivatives up to higher-order terms, which is referred to as the
‘criminal method’ in [3]. The proof is postponed to Section 3.7. Note that the
new homogenized coefficients $\\{\bar{\boldsymbol{b}}^{n}\\}_{n}$ in this
reformulation automatically coincide with the coefficients given by the
spectral approach: this can for instance be deduced a posteriori by comparing
the corresponding homogenization results, Theorems 1 and 2; this allows us to
use already here the same notation $\\{\bar{\boldsymbol{b}}^{n}\\}_{n}$ as for
spectral homogenized coefficients. Note that by definition
$\bar{\boldsymbol{b}}^{1}=\bar{\boldsymbol{a}}^{1,0}=\bar{\boldsymbol{a}}$.

######  Lemma 3.8 (Revamped homogenized equation).

Given $\ell\geq 1$ and $\varepsilon>0$, if $\bar{W}_{\varepsilon}^{\ell},f$
are smooth and satisfy the formal homogenized equation (3.5), then we have

$\partial_{t}^{2}\bar{W}_{\varepsilon}^{\ell}-\nabla\cdot\Big{(}\sum_{n=1}^{\ell}\bar{\boldsymbol{b}}^{n}\odot(\varepsilon\nabla)^{n-1}\Big{)}\nabla\bar{W}_{\varepsilon}^{\ell}\,=\,f+\varepsilon^{2}\nabla\cdot\Big{(}\sum_{n=1}^{\ell-2}\bar{\boldsymbol{c}}^{n}\odot(\varepsilon
D)^{n-1}\Big{)}\nabla f+\nabla\cdot E_{\varepsilon}^{\ell},$

where the coefficients
$\\{\bar{\boldsymbol{b}}^{n},\bar{\boldsymbol{c}}^{n}\\}_{n}$ and the
remainder $E_{\varepsilon}^{\ell}$ are as follows:

  1. $\bullet$

We define the matrix-valued symmetric tensor
$\bar{\boldsymbol{b}}^{p}:=(\bar{\boldsymbol{b}}^{p}_{j_{1}\ldots
j_{p-1}})_{1\leq j_{1},\ldots,j_{p-1}\leq d}$ for $p\geq 1$ such that for all
$\xi\in\mathbb{R}^{d}$,

$\xi\cdot(\bar{\boldsymbol{b}}^{p}\odot\xi^{\otimes(p-1)})\xi\,:=\,\sum_{k\geq
1}\,\sum_{(m_{1},\ldots,m_{k})\in I_{k}}\sum_{n_{1},\ldots,n_{k}\geq 1\atop
k+|n|=p+1}\prod_{j=1}^{k}\xi\cdot\big{(}\bar{\boldsymbol{a}}^{n_{j},m_{j}}\odot\xi^{\otimes(n_{j}-1)}\big{)}\xi,$

in terms of the index set

$I_{k}\,:=\,\Big{\\{}m=(m_{1},\dots,m_{k}):m_{j}\geq 0\leavevmode\nobreak\
\forall j,\leavevmode\nobreak\ \sum_{j=1}^{s}m_{j}\geq 2s\leavevmode\nobreak\
\forall s<k,\leavevmode\nobreak\ |m|=2(k-1)\Big{\\}}.$

  2. $\bullet$

We define the matrix-valued symmetric tensor
$\bar{\boldsymbol{c}}^{p}:=(\bar{\boldsymbol{c}}^{p}_{j_{1}\ldots
j_{p-1}})_{0\leq j_{1},\ldots,j_{p-1}\leq d}$ for all $p\geq 1$ such that for
all $\hat{\xi}=(\xi_{0},\xi)\in\mathbb{R}\times\mathbb{R}^{d}$,

$\quad\xi\cdot(\bar{\boldsymbol{c}}^{p}\odot\hat{\xi}^{\otimes(p-1)})\xi:=\sum_{k\geq
1}\sum_{(m_{1},\ldots,m_{k})\in J_{k}}\sum_{n_{1},\ldots,n_{k}\geq
1\atop|n|+|m|=p+k+1}\xi_{0}^{|m|-2k}\prod_{j=1}^{k}\xi\cdot\big{(}\bar{\boldsymbol{a}}^{n_{j},m_{j}}\odot\xi^{\otimes(n_{j}-1)}\big{)}\xi,$

in terms of the index set

$J_{k}\,:=\,\Big{\\{}m=(m_{1},\dots,m_{k}):m_{j}\geq 0\leavevmode\nobreak\
\forall j,\,\sum_{j=1}^{s}m_{j}\geq 2s\leavevmode\nobreak\ \forall s\leq
k\Big{\\}}.$

  3. $\bullet$

For $\varepsilon\ll 1$ small enough, the error term $E_{\varepsilon}^{\ell}$
satisfies pointwise, for all $r\geq 0$,

$|\langle D\rangle^{r}E_{\varepsilon}^{\ell}|\,\leq\,(\varepsilon
C\ell)^{\ell}\Big{(}\big{|}\langle D\rangle^{r+\ell}\langle\varepsilon
CD\rangle^{\ell^{2}}D\bar{W}_{\varepsilon}^{\ell}\big{|}+\big{|}\langle
D\rangle^{r+\ell-1}\langle\varepsilon CD\rangle^{\ell^{2}}f\big{|}\Big{)}.$

In particular, Lemma 3.4 implies for all $p\geq 1$,

$|\bar{{\boldsymbol{b}}}^{p}|+|\bar{{\boldsymbol{c}}}^{p}|\leq C^{p},$ (3.6)

and Proposition 3.5 entails, whenever $p$ is an even integer,

$\xi\cdot(\bar{\boldsymbol{b}}^{p}\odot\xi^{\otimes(p-1)})\xi=0,\qquad\text{for
all $\xi\in\mathbb{R}^{d}$.}\qed$

The above thus leads us to considering the following modified version of the
formal effective equation (3.5),

$\left\\{\begin{array}[]{ll}\partial_{t}^{2}\bar{V}^{\ell}_{\varepsilon}-\nabla\cdot\big{(}\bar{\boldsymbol{a}}+\sum_{k=2}^{\ell}\bar{\boldsymbol{b}}^{k}\odot(\varepsilon\nabla)^{k-1}\big{)}\nabla\bar{V}_{\varepsilon}^{\ell}&\\\
\hskip
85.35826pt=f+\varepsilon^{2}\nabla\cdot\big{(}\sum_{k=1}^{\ell-2}\bar{\boldsymbol{c}}^{k}\odot(\varepsilon
D)^{k-1}\big{)}\nabla f,&\text{in $\mathbb{R}\times\mathbb{R}^{d}$},\\\
\bar{V}_{\varepsilon}^{\ell}=f=0,&\text{for $t<0$}.\end{array}\right.$ (3.7)

As with the spectral approach, the symbol of the operator
$-\nabla\cdot\big{(}\bar{\boldsymbol{a}}+\sum_{k=2}^{\ell}\bar{\boldsymbol{b}}^{k}\odot(\varepsilon\nabla)^{k-1}\big{)}\nabla$
lacks positivity, so this equation is ill-posed in general. To cure this
issue, we argue exactly as in Section 2.3: we can consider several well-posed
higher-order modifications of this equation, either by high-frequency
filtering, by higher-order regularization, or by the Boussinesq trick, and we
denote by
$\bar{v}_{\varepsilon}^{\text{(I)},\ell},\bar{v}_{\varepsilon}^{\text{(II)},\ell},\bar{v}_{\varepsilon}^{\text{(III)},\ell}$
the corresponding solutions, respectively. We refer to Lemma 2.10 for the
details (up to replacing the impulse $f$ in (2.18) by the specific right-hand
side in (3.7)).

Next, we show that the above modification procedure
$\bar{W}_{\varepsilon}^{\ell}\mapsto\bar{V}_{\varepsilon}^{\ell}$ for the
formal effective equation can be inverted: more precisely, the solution
$\bar{v}_{\varepsilon}^{(\star),\ell}$ of any of the well-posed modifications
of (3.7) is an approximate solution of the formal effective equation (3.5) up
to an $O(\varepsilon^{\ell})$ error.

######  Lemma 3.9 (Inversion procedure).

Given $\ell\geq 1$ and $\varepsilon>0$, if
$\bar{v}_{\varepsilon}^{(\star),\ell}$ is the solution of one of the well-
posed modifications of equation (3.7) as given by Lemma 2.10, then we have

$\partial_{t}^{2}\bar{v}^{(\star),\ell}_{\varepsilon}-\nabla\cdot\Big{(}\sum_{n=1}^{\ell}\sum_{m=0}^{\ell-n}\bar{\boldsymbol{a}}^{n,m}\odot(\varepsilon\nabla)^{n-1}(\varepsilon\partial_{t})^{m}\Big{)}\nabla\bar{v}^{(\star),\ell}_{\varepsilon}=f+\nabla\cdot
F_{\varepsilon}^{(\star),\ell},$

where the error $\nabla\cdot F_{\varepsilon}^{(\star),\ell}$ satisfies for all
$r\geq 0$,

$\|\langle
D\rangle^{r}F_{\varepsilon}^{(\star),\ell;t}\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\,\leq\,(\varepsilon
C\ell)^{\ell}\|\langle D\rangle^{r+C\ell}\langle\varepsilon
CD\rangle^{\ell^{2}}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))},$

where the constant $C$ further depends on the choice of $\alpha$ in case
$(\star)=\operatorname{(I)}$. ∎

###  3.4. Proof of Theorem 2

Applying Proposition 3.6 with $\bar{w}=\bar{v}^{(\star),\ell}_{\varepsilon}$,
appealing to Lemma 3.9, and comparing with the solution $u_{\varepsilon}$ of
the heterogeneous wave equation (1.2), we find

$\big{(}\partial_{t}^{2}-\nabla\cdot\boldsymbol{a}(\tfrac{\cdot}{\varepsilon})\nabla\big{)}(u_{\varepsilon}-H^{\ell}_{\varepsilon}[\bar{v}_{\varepsilon}^{(\star),\ell}])\,=\,-\nabla\cdot
F_{\varepsilon}^{(\star),\ell}\\\
+\varepsilon^{\ell}\sum_{n=1}^{\ell+1}\nabla\cdot\Big{(}\big{(}\boldsymbol{a}\phi_{j_{1}\ldots
j_{n-1}}^{n-1,\ell-n+1}-\sigma^{n-1,\ell-n+1}_{j_{1}\ldots
j_{n-1}}\big{)}(\tfrac{\cdot}{\varepsilon})\nabla\nabla^{n-1}_{j_{1}\ldots
j_{n-1}}\partial_{t}^{\ell-n+1}\bar{v}_{\varepsilon}^{(\star),\ell}\Big{)}\\\
-\varepsilon^{\ell}\sum_{n=0}^{\ell}\partial_{t}\Big{(}\big{(}\phi^{n+1,\ell-n-1}_{j_{1}\ldots
j_{n-1}ji}e_{j}\otimes e_{i}-\sigma^{n-1,\ell-n+1}_{j_{1}\ldots
j_{n-1}}\big{)}(\tfrac{\cdot}{\varepsilon}):\nabla^{2}\nabla^{n-1}_{j_{1}\ldots
j_{n-1}}\partial_{t}^{\ell-n}\bar{v}_{\varepsilon}^{(\star),\ell}\Big{)}.$

By the a priori estimates of Lemma B.1, using corrector estimates of Lemma
3.4, and using the Sobolev embedding to estimate products with correctors as
in (2.41), we get for $a>\frac{d}{2}$,

$\|u_{\varepsilon}^{t}-H_{\varepsilon}^{\ell}[\bar{v}_{\varepsilon}^{(\star),\ell;t}]\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}+\|D(u_{\varepsilon}^{t}-H_{\varepsilon}^{\ell}[\bar{v}_{\varepsilon}^{(\star),\ell;t}])\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\\\
\,\lesssim\,\|\langle D\rangle
F_{\varepsilon}^{(\star),\ell}\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}+(\varepsilon
C)^{\ell}\|\langle
D\rangle^{\ell+a+1}D\bar{v}_{\varepsilon}^{(\star),\ell}\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}.$

Combining this bound with the estimate of Lemma 3.9 on the remainder
$F_{\varepsilon}^{\ell}$, and with the a priori estimates of Lemma 2.10(ii)
for $\bar{v}_{\varepsilon}^{(\star),\ell}$, the conclusion follows. ∎

###  3.5. Proof of Proposition 3.5

We split the proof into two steps.

Step 1\. Migration process: proof that for all $n,m,p,q\geq 0$,

$\mathbb{E}\left[\big{(}\nabla\phi^{p,q}_{j_{1}\ldots
j_{p}}\cdot\boldsymbol{a}\nabla\phi^{n,m}_{i_{1}\ldots
i_{n}}-\phi^{p-1,q}_{j_{1}\ldots
j_{p-1}}e_{j_{p}}\cdot\boldsymbol{a}\phi^{n-1,m}_{i_{1}\ldots
i_{n-1}}e_{i_{n}}\big{)}\right]\\\
=-\mathbb{E}\left[\big{(}\nabla\phi^{p+1,q}_{j_{1}\ldots
j_{p}i_{n}}\cdot\boldsymbol{a}\nabla\phi^{n-1,m}_{i_{1}\ldots
i_{n-1}}-\phi^{p,q}_{j_{1}\ldots
j_{p}}e_{i_{n}}\cdot\boldsymbol{a}\phi^{n-2,m}_{i_{1}\ldots
i_{n-2}}e_{i_{n-1}}\big{)}\right]\\\
-\mathbb{E}\left[\big{(}\phi^{p,q}_{j_{1}\ldots
j_{p}}\phi^{n,m-2}_{i_{1}\ldots i_{n}}+\phi^{p+1,q-2}_{j_{1}\ldots
j_{p}i_{n}}\phi^{n-1,m}_{i_{1}\ldots i_{n-1}}\big{)}\right],$ (3.8)

and in addition, for all $n,m,q\geq 0$,

$\sum_{k=1}^{n}(-1)^{k}\mathbb{E}\left[\phi^{k,q}_{ji_{n}\ldots
i_{n-k+2}}\phi^{n-k+1,m}_{i_{1}\ldots
i_{n-k+1}}\right]=\sum_{k=1}^{n}(-1)^{k}\mathbb{E}\left[\phi^{k,q+2}_{ji_{n}\ldots
i_{n-k+2}}\phi^{n-k+1,m-2}_{i_{1}\ldots i_{n-k+1}}\right]\\\
+(-1)^{n}\mathbb{E}\left[\nabla\phi^{n,q+2}_{ji_{n}\ldots
i_{2}}\cdot\boldsymbol{a}\nabla\phi^{0,m}e_{i_{1}}+\phi^{n-1,q+2}_{ji_{n}\ldots
i_{3}}e_{i_{2}}\cdot\boldsymbol{a}\phi^{0,m}e_{i_{1}}\right].$ (3.9)

The equation for $\phi^{n,m}$ in Definition 3.1 yields

$\mathbb{E}\left[\nabla\phi^{p,q}_{j_{1}\ldots
j_{p}}\cdot\boldsymbol{a}\nabla\phi^{n,m}_{i_{1}\ldots
i_{n}}\right]\,=\,-\mathbb{E}\left[\nabla\phi^{p,q}_{j_{1}\ldots
j_{p}}\cdot\boldsymbol{a}\phi^{n-1,m}_{i_{1}\ldots i_{n-1}}e_{i_{n}}\right]\\\
+\mathbb{E}\left[\phi^{p,q}_{j_{1}\ldots
j_{p}}e_{i_{n}}\cdot\boldsymbol{a}\big{(}\nabla\phi^{n-1,m}_{i_{1}\ldots
i_{n-1}}+\phi^{n-2,m}_{i_{1}\ldots
i_{n-2}}e_{i_{n-1}}\big{)}\right]-\mathbb{E}\left[\phi^{p,q}_{j_{1}\ldots
j_{p}}\phi^{n,m-2}_{i_{1}\ldots i_{n}}\right],$

while the equation for $\phi^{{p+1},q}$ leads to

$\mathbb{E}\left[\phi^{p,q}_{j_{1}\ldots
j_{p}}e_{i_{n}}\cdot\boldsymbol{a}\nabla\phi^{n-1,m}_{i_{1}\ldots
i_{n-1}}\right]\,=\,-\mathbb{E}\left[\nabla\phi^{p+1,q}_{j_{1}\ldots
j_{p}i_{n}}\cdot\boldsymbol{a}\nabla\phi^{n-1,m}_{i_{1}\ldots
i_{n-1}}\right]\\\ +\mathbb{E}\left[\big{(}\nabla\phi^{p,q}_{j_{1}\ldots
j_{p}}+\phi^{p-1,q}_{j_{1}\ldots
j_{p-1}}e_{j_{p}}\big{)}\cdot\boldsymbol{a}\phi^{n-1,m}_{i_{1}\ldots
i_{n-1}}e_{i_{n}}\right]-\mathbb{E}\left[\phi^{p+1,q-2}_{j_{1}\ldots
j_{p}i_{n}}\phi^{n-1,m}_{i_{1}\ldots i_{n-1}}\right].$

Summing these two identities, the claim (3.8) easily follows. Next, using
(3.8) in form of

$\mathbb{E}\left[\phi^{k,q}_{ji_{n}\ldots
i_{n-k+2}}\phi^{n-k+1,m}_{i_{1}\ldots
i_{n-k+1}}\right]=-\mathbb{E}\left[\phi^{k-1,q+2}_{ji_{n}\ldots
i_{n-k+3}}\phi^{n-k+2,m-2}_{i_{1}\ldots i_{n-k+2}}\right]\\\
-\mathbb{E}\left[\big{(}\nabla\phi^{k-1,q+2}_{ji_{n}\ldots
i_{n-k+3}}\cdot\boldsymbol{a}\nabla\phi^{n-k+2,m}_{i_{1}\ldots
i_{n-k+2}}-\phi^{k-2,q+2}_{ji_{n}\ldots
j_{n-k+4}}e_{j_{n-k+3}}\cdot\boldsymbol{a}\phi^{n-k+1,m}_{i_{1}\ldots
i_{n-k+1}}e_{i_{n-k+2}}\big{)}\right]\\\
-\mathbb{E}\left[\big{(}\nabla\phi^{k,q+2}_{ji_{n}\ldots
i_{n-k+2}}\cdot\boldsymbol{a}\nabla\phi^{n-k+1,m}_{i_{1}\ldots
i_{n-k+1}}-\phi^{k-1,q+2}_{ji_{n}\ldots
i_{n-k+3}}e_{i_{n-k+2}}\cdot\boldsymbol{a}\phi^{n-k,m}_{i_{1}\ldots
i_{n-k}}e_{i_{n-k+1}}\big{)}\right],$

and summing this identity for $1\leq k\leq n$, we find after straightforward
simplifications

$\sum_{k=1}^{n}(-1)^{k}\mathbb{E}\left[\phi^{k,q}_{ji_{n}\ldots
i_{n-k+2}}\phi^{n-k+1,m}_{i_{1}\ldots
i_{n-k+1}}\right]=-\sum_{k=1}^{n}(-1)^{k}\mathbb{E}\left[\phi^{k-1,q+2}_{ji_{n}\ldots
i_{n-k+3}}\phi^{n-k+2,m-2}_{i_{1}\ldots i_{n-k+2}}\right]\\\
-(-1)^{n}\mathbb{E}\left[\big{(}\nabla\phi^{n,q+2}_{ji_{n}\ldots
i_{2}}\cdot\boldsymbol{a}\nabla\phi^{1,m}_{i_{1}}-\phi^{n-1,q+2}_{ji_{n}\ldots
i_{3}}e_{i_{2}}\cdot\boldsymbol{a}\phi^{0,m}e_{i_{1}}\big{)}\right].$

Using the equation for $\phi^{1,m}$, the second claim (3.9) easily follows.

Step 2\. Conclusion.
For $n\geq 1$ and $m\geq 0$, the definition of $\bar{\boldsymbol{a}}^{n,m}$
and the equation for $\phi^{1,0}$ yield

$e_{j}\cdot\bar{\boldsymbol{a}}^{n,m}_{i_{1}\ldots
i_{n-1}}e_{i_{n}}\,=\,-\mathbb{E}\left[\big{(}\nabla\phi^{1,0}_{j}\cdot\boldsymbol{a}\nabla\phi^{n,m}_{i_{1}\ldots
i_{n}}-e_{j}\cdot\boldsymbol{a}\phi^{n-1,m}_{i_{1}\ldots
i_{n-1}}e_{i_{n}}\big{)}\right].$

Iterating identity (3.8) then leads to

$e_{j}\cdot\bar{\boldsymbol{a}}^{n,m}_{i_{1}\ldots
i_{n-1}}e_{i_{n}}\,=\,(-1)^{n}\mathbb{E}\left[\big{(}\nabla\phi^{n,0}_{ji_{n}\ldots
i_{2}}\cdot\boldsymbol{a}\nabla\phi^{1,m}_{i_{1}}-\phi^{n-1,0}_{ji_{n}\ldots
i_{3}}e_{i_{2}}\cdot\boldsymbol{a}\phi^{0,m}e_{i_{1}}\big{)}\right]\\\
-\sum_{k=1}^{n-1}(-1)^{k}\mathbb{E}\left[\phi^{k,0}_{ji_{n}\ldots
i_{n-k+2}}\phi^{n-k+1,m-2}_{i_{1}\ldots i_{n-k+1}}\right],$

hence, using the equation for $\phi^{1,m}$,

$e_{j}\cdot\bar{\boldsymbol{a}}^{n,m}_{i_{1}\ldots
i_{n-1}}e_{i_{n}}\,=\,(-1)^{n+1}\mathbb{E}\left[\big{(}\nabla\phi^{n,0}_{ji_{n}\ldots
i_{2}}+\phi^{n-1,0}_{ji_{n}\ldots
i_{3}}e_{i_{2}}\big{)}\cdot\boldsymbol{a}\phi^{0,m}e_{i_{1}}\right]\\\
-\sum_{k=1}^{n}(-1)^{k}\mathbb{E}\left[\phi^{k,0}_{ji_{n}\ldots
i_{n-k+2}}\phi^{n-k+1,m-2}_{i_{1}\ldots i_{n-k+1}}\right].$ (3.10)

For $m=0$, this already yields the conclusion

$e_{j}\cdot\bar{\boldsymbol{a}}^{n,0}_{i_{1}\ldots
i_{n-1}}e_{i_{n}}\,=\,(-1)^{n+1}\mathbb{E}\left[\big{(}\nabla\phi^{n,0}_{ji_{n}\ldots
i_{2}}+\phi^{n-1,0}_{ji_{n}\ldots
i_{3}}e_{i_{2}}\big{)}\cdot\boldsymbol{a}e_{i_{1}}\right]\,=\,(-1)^{n+1}e_{i_{1}}\cdot\bar{\boldsymbol{a}}^{n,0}_{ji_{n}\ldots
i_{3}}e_{i_{2}}.$

For $m\geq 2$, identity (3.10) rather takes the form

$e_{j}\cdot\bar{\boldsymbol{a}}^{n,m}_{i_{1}\ldots
i_{n-1}}e_{i_{n}}\,=\,-\sum_{k=1}^{n}(-1)^{k}\mathbb{E}\left[\phi^{k,0}_{ji_{n}\ldots
i_{n-k+2}}\phi^{n-k+1,m-2}_{i_{1}\ldots i_{n-k+1}}\right],$ (3.11)

which we shall combine with an iterative use of identity (3.9). More
precisely, in case $m=4m^{\prime}+2$ with an integer $m^{\prime}\geq 0$,
iterating identity (3.9) (starting from $q=0$ with $m$ replaced by
$m-2=4m^{\prime}$), and recalling $\phi^{0,l}=0$ for $l\geq 1$, we find

$\sum_{k=1}^{n}(-1)^{k}\mathbb{E}\left[\phi^{k,0}_{ji_{n}\ldots
i_{n-k+2}}\phi^{n-k+1,m-2}_{i_{1}\ldots
i_{n-k+1}}\right]=\sum_{k=1}^{n}(-1)^{k}\mathbb{E}\left[\phi^{k,2m^{\prime}}_{ji_{n}\ldots
i_{n-k+2}}\phi^{n-k+1,2m^{\prime}}_{i_{1}\ldots i_{n-k+1}}\right],$

and thus, combining this with (3.11), and replacing $k$ by $n-k+1$ in the sum,

$e_{j}\cdot\bar{\boldsymbol{a}}^{n,m}_{i_{1}\ldots
i_{n-1}}e_{i_{n}}\,=\,-\sum_{k=1}^{n}(-1)^{k}\mathbb{E}\left[\phi^{k,2m^{\prime}}_{ji_{n}\ldots
i_{n-k+2}}\phi^{n-k+1,2m^{\prime}}_{i_{1}\ldots
i_{n-k+1}}\right]\,=\,(-1)^{n+1}e_{i_{1}}\cdot\bar{\boldsymbol{a}}^{n,m}_{ji_{n}\ldots
i_{3}}e_{i_{2}}.$

Similarly, in the case $m=4m^{\prime}$ with integer $m^{\prime}\geq 1$, we
find

$\displaystyle e_{j}\cdot\bar{\boldsymbol{a}}^{n,m}_{i_{1}\ldots
i_{n-1}}e_{i_{n}}$ $\displaystyle=$
$\displaystyle-\sum_{k=1}^{n}(-1)^{k}\mathbb{E}\left[\phi^{k,2m^{\prime}}_{ji_{n}\ldots
i_{n-k+2}}\phi^{n-k+1,2m^{\prime}-2}_{i_{1}\ldots i_{n-k+1}}\right]$
$\displaystyle=$
$\displaystyle-\sum_{k=1}^{n}(-1)^{k}\mathbb{E}\left[\phi^{k,2m^{\prime}-2}_{ji_{n}\ldots
i_{n-k+2}}\phi^{n-k+1,2m^{\prime}}_{i_{1}\ldots i_{n-k+1}}\right]$
$\displaystyle=$
$\displaystyle(-1)^{n+1}e_{i_{1}}\cdot\bar{\boldsymbol{a}}^{n,m}_{ji_{n}\ldots
i_{3}}e_{i_{2}},$

and the conclusion follows. ∎

###  3.6. Proof of Proposition 3.6

We focus on the case $\varepsilon=1$ and drop it from all subscripts in the
notation, while the final result is obtained after $\varepsilon$-rescaling.
Since the correctors $\phi_{j_{1}\ldots j_{n}}^{n,m}$ do not depend on time, a
direct calculation yields for all $n,m\geq 0$,

$(\partial_{t}^{2}-\nabla\cdot\boldsymbol{a}\nabla)\big{(}\phi^{n,m}\odot\nabla^{n}\partial_{t}^{m}\bar{w}\big{)}\,=\,\phi^{n,m}\odot\nabla^{n}\partial_{t}^{m+2}\bar{w}\\\
+(-\nabla\cdot\boldsymbol{a}\nabla\phi^{n,m}_{j_{1}\ldots
j_{n}})\nabla^{n}_{j_{1}\ldots
j_{n}}\partial_{t}^{m}\bar{w}-\nabla\cdot(\boldsymbol{a}\phi^{n,m}_{j_{1}\ldots
j_{n}}e_{j_{n+1}})\nabla^{n+1}_{j_{1}\ldots j_{n+1}}\partial_{t}^{m}\bar{w}\\\
-(e_{j_{n+1}}\cdot\boldsymbol{a}\nabla\phi^{n,m}_{j_{1}\ldots
j_{n}})\nabla^{n+1}_{j_{1}\ldots
j_{n+1}}\partial_{t}^{m}\bar{w}-(e_{j_{n+2}}\cdot\boldsymbol{a}\phi^{n,m}_{j_{1}\ldots
j_{n}}e_{j_{n+1}})\nabla^{n+2}_{j_{1}\ldots j_{n+2}}\partial_{t}^{m}\bar{w}.$

Inserting the defining equation for the hyperbolic corrector $\phi^{n,m}$, cf.
Definition 3.1, we get for all $n\geq 1$ and $m\geq 0$,

$(\partial_{t}^{2}-\nabla\cdot\boldsymbol{a}\nabla)\big{(}\phi^{n,m}\odot\nabla^{n}\partial_{t}^{m}\bar{w}\big{)}\,=\,\phi^{n,m}\odot\nabla^{n}\partial_{t}^{m+2}\bar{w}-\phi^{n,m-2}\odot\nabla^{n}\partial_{t}^{m}\bar{w}\\\
+\nabla\cdot(\boldsymbol{a}\phi^{n-1,m}_{j_{1}\ldots
j_{n-1}}e_{j_{n}})\nabla^{n}_{j_{1}\ldots
j_{n}}\partial_{t}^{m}\bar{w}-\nabla\cdot(\boldsymbol{a}\phi^{n,m}_{j_{1}\ldots
j_{n}}e_{j_{n+1}})\nabla^{n+1}_{j_{1}\ldots j_{n+1}}\partial_{t}^{m}\bar{w}\\\
+(e_{j_{n}}\cdot\boldsymbol{a}\nabla\phi_{j_{1}\ldots
j_{n-1}}^{n-1,m})\nabla^{n}_{j_{1}\ldots
j_{n}}\partial_{t}^{m}\bar{w}-(e_{j_{n+1}}\cdot\boldsymbol{a}\nabla\phi^{n,m}_{j_{1}\ldots
j_{n}})\nabla^{n+1}_{j_{1}\ldots j_{n+1}}\partial_{t}^{m}\bar{w}\\\
+(e_{j_{n}}\cdot\boldsymbol{a}\phi_{j_{1}\ldots
j_{n-2}}^{n-2,m}e_{j_{n-1}})\nabla^{n}_{j_{1}\ldots
j_{n}}\partial_{t}^{m}\bar{w}-(e_{j_{n+2}}\cdot\boldsymbol{a}\phi^{n,m}_{j_{1}\ldots
j_{n}}e_{j_{n+1}})\nabla^{n+2}_{j_{1}\ldots j_{n+2}}\partial_{t}^{m}\bar{w}\\\
-\bar{\boldsymbol{a}}^{n-1,m}_{j_{1}\ldots
j_{n-2}}:\nabla^{2}\nabla^{n-2}_{j_{1}\ldots j_{n-2}}\partial_{t}^{m}\bar{w},$

and thus, after summation over $1\leq n\leq\ell$ and $0\leq m\leq\ell-n$,
recalling the definition (3.3) of the geometric two-scale expansion, and using
$\phi^{0,0}=1$ and $\phi^{0,m}=0$ for $m>0$,

$(\partial_{t}^{2}-\nabla\cdot\boldsymbol{a}\nabla)H^{\ell}[\bar{w}]\,=\,\partial_{t}^{2}\bar{w}-\sum_{n=1}^{\ell}\sum_{m=0}^{\ell-n}\bar{\boldsymbol{a}}^{n-1,m}_{j_{1}\ldots
j_{n-2}}:\nabla^{2}\nabla^{n-2}_{j_{1}\ldots
j_{n-2}}\partial_{t}^{m}\bar{w}\\\
+\sum_{n=1}^{\ell}\Big{(}\phi^{n,\ell-n-1}\odot\nabla^{n}\partial_{t}^{\ell+1-n}\bar{w}+\phi^{n,\ell-n}\odot\nabla^{n}\partial_{t}^{\ell+2-n}\bar{w}\Big{)}\\\
-\sum_{n=1}^{\ell}\Big{(}\nabla\cdot(\boldsymbol{a}\phi_{j_{1}\ldots
j_{n}}^{n,\ell-n}e_{j_{n+1}})+e_{j_{n+1}}\cdot\boldsymbol{a}\nabla\phi_{j_{1}\ldots
j_{n}}^{n,\ell-n}\Big{)}\nabla^{n+1}_{j_{1}\ldots
j_{n+1}}\partial_{t}^{\ell-n}\bar{w}\\\
-\sum_{n=1}^{\ell}\Big{(}(e_{j_{n+1}}\cdot\boldsymbol{a}\phi^{n-1,\ell-n}_{j_{1}\ldots
j_{n-1}}e_{j_{n}})\nabla^{n+1}_{j_{1}\ldots
j_{n+1}}\partial_{t}^{\ell-n}\bar{w}+(e_{j_{n+2}}\cdot\boldsymbol{a}\phi^{n,\ell-n}_{j_{1}\ldots
j_{n}}e_{j_{n+1}})\nabla^{n+2}_{j_{1}\ldots
j_{n+2}}\partial_{t}^{\ell-n}\bar{w}\Big{)}.$

As $\bar{\boldsymbol{a}}^{0,m}=0$ for all $m\geq 0$, the second right-hand
side term can be rewritten as

$\sum_{n=1}^{\ell}\sum_{m=0}^{\ell-n}\bar{\boldsymbol{a}}_{j_{1}\ldots
j_{n-2}}^{n-1,m}:\nabla^{2}\nabla^{n-2}_{j_{1}\ldots
j_{n-2}}\partial_{t}^{m}\bar{w}\,=\,\sum_{n=1}^{\ell-1}\sum_{m=0}^{\ell-1-n}\bar{\boldsymbol{a}}_{j_{1}\ldots
j_{n-1}}^{n,m}:\nabla^{2}\nabla^{n-1}_{j_{1}\ldots
j_{n-1}}\partial_{t}^{m}\bar{w}\\\
=\sum_{n=1}^{\ell}\sum_{m=0}^{\ell-n}\nabla\cdot(\bar{\boldsymbol{a}}^{n,m}\odot\nabla^{n-1}\partial_{t}^{m})\nabla\bar{w}-\sum_{n=1}^{\ell}\bar{\boldsymbol{a}}_{j_{1}\ldots
j_{n-1}}^{n,\ell-n}:\nabla^{2}\nabla^{n-1}_{j_{1}\ldots
j_{n-1}}\partial_{t}^{\ell-n}\bar{w}.$

Further rearranging the terms, and noting that

$\nabla\cdot(\boldsymbol{a}\phi_{j_{1}\ldots
j_{n}}^{n,\ell-n}e_{j_{n+1}})\nabla^{n+1}_{j_{1}\ldots
j_{n+1}}\partial_{t}^{\ell-n}\bar{w}+(e_{j_{n+2}}\cdot\boldsymbol{a}\phi^{n,\ell-n}_{j_{1}\ldots
j_{n}}e_{j_{n+1}})\nabla^{n+2}_{j_{1}\ldots
j_{n+2}}\partial_{t}^{\ell-n}\bar{w}\\\
\,=\,\nabla\cdot\Big{(}\boldsymbol{a}\phi_{j_{1}\ldots
j_{n}}^{n,\ell-n}\nabla\nabla^{n}_{j_{1}\ldots
j_{n}}\partial_{t}^{\ell-n}\bar{w}\Big{)},$

we are led to

$(\partial_{t}^{2}-\nabla\cdot\boldsymbol{a}\nabla)H^{\ell}[\bar{w}]\,=\,\partial_{t}^{2}\bar{w}-\sum_{n=1}^{\ell}\sum_{m=0}^{\ell-n}\nabla\cdot(\bar{\boldsymbol{a}}^{n,m}\odot\nabla^{n-1}\partial_{t}^{m})\nabla\bar{w}\\\
+\sum_{n=1}^{\ell}\Big{(}\phi^{n,\ell-n-1}\odot\nabla^{n}\partial_{t}^{\ell+1-n}\bar{w}+\phi^{n,\ell-n}\odot\nabla^{n}\partial_{t}^{\ell+2-n}\bar{w}\Big{)}\\\
-\sum_{n=1}^{\ell}\nabla\cdot\Big{(}\boldsymbol{a}\phi_{j_{1}\ldots
j_{n}}^{n,\ell-n}\nabla\nabla^{n}_{j_{1}\ldots
j_{n}}\partial_{t}^{\ell-n}\bar{w}\Big{)}\\\
-\sum_{n=1}^{\ell}\Big{(}\boldsymbol{a}\big{(}\nabla\phi_{j_{1}\ldots
j_{n}}^{n,\ell-n}+\phi^{n-1,\ell-n}_{j_{1}\ldots
j_{n-1}}e_{j_{n}}\big{)}-\bar{\boldsymbol{a}}_{j_{1}\ldots
j_{n-1}}^{n,\ell-n}e_{j_{n}}\Big{)}\cdot\nabla\nabla^{n}_{j_{1}\ldots
j_{n}}\partial_{t}^{\ell-n}\bar{w}.$ (3.12)

The last right-hand side terms can be reformulated in terms of fluxes, cf.
Definition 3.1,

${\sum_{n=1}^{\ell}\Big{(}\boldsymbol{a}\big{(}\nabla\phi_{j_{1}\ldots
j_{n}}^{n,\ell-n}+\phi^{n-1,\ell-n}_{j_{1}\ldots
j_{n-1}}e_{j_{n}}\big{)}-\bar{\boldsymbol{a}}_{j_{1}\ldots
j_{n-1}}^{n,\ell-n}e_{j_{n}}\Big{)}\cdot\nabla\nabla^{n}_{j_{1}\ldots
j_{n}}\partial_{t}^{\ell-n}\bar{w}}\\\
\,=\,\sum_{n=1}^{\ell}q^{n,\ell-n}_{j_{1}\ldots
j_{n}}\cdot\nabla\nabla^{n}_{j_{1}\ldots
j_{n}}\partial_{t}^{\ell-n}\bar{w}+\sum_{n=1}^{\ell-1}\phi^{n+1,\ell-n-2}\odot\nabla^{n+1}\partial_{t}^{\ell-n}\bar{w}.$

Further noting that we can write

$q^{\ell,0}_{j_{1}\ldots j_{\ell}}\cdot\nabla\nabla^{\ell}_{j_{1}\ldots
j_{\ell}}\bar{w}\,=\,\tilde{q}^{\ell,0}_{j_{1}\ldots
j_{\ell}}\cdot\nabla\nabla^{\ell}_{j_{1}\ldots j_{\ell}}\bar{w},$

in terms of the modified fluxes of Lemma 3.3, and then inserting the
definition of hyperbolic flux correctors and using the skew-symmetry of
$\sigma^{\ell,0}_{j_{1}\ldots j_{\ell}}$, cf. Definition 3.2 and Lemma 3.3, we
deduce

${\sum_{n=1}^{\ell}\Big{(}\boldsymbol{a}\big{(}\nabla\phi_{j_{1}\ldots
j_{n}}^{n,\ell-n}+\phi^{n-1,\ell-n}_{j_{1}\ldots
j_{n-1}}e_{j_{n}}\big{)}-\bar{\boldsymbol{a}}_{j_{1}\ldots
j_{n-1}}^{n,\ell-n}e_{j_{n}}\Big{)}\cdot\nabla\nabla^{n}_{j_{1}\ldots
j_{n}}\partial_{t}^{\ell-n}\bar{w}}\\\
\,=\,-\sum_{n=0}^{\ell}\nabla_{j}(\sigma^{n,\ell-n}_{j_{1}\ldots
j_{n}})_{jj_{n+1}}\nabla^{n+1}_{j_{1}\ldots
j_{n+1}}\partial_{t}^{\ell-n}\bar{w}+\sum_{n=0}^{\ell-1}\phi^{n+1,\ell-n-2}\odot\nabla^{n+1}\partial_{t}^{\ell-n}\bar{w}.$

Rearranging the terms and further using the skew-symmetry of
$\sigma^{\ell,0}_{j_{1}\ldots j_{\ell}}$, we get

${\sum_{n=1}^{\ell}\Big{(}\boldsymbol{a}\big{(}\nabla\phi_{j_{1}\ldots
j_{n}}^{n,\ell-n}+\phi^{n-1,\ell-n}_{j_{1}\ldots
j_{n-1}}e_{j_{n}}\big{)}-\bar{\boldsymbol{a}}_{j_{1}\ldots
j_{n-1}}^{n,\ell-n}e_{j_{n}}\Big{)}\cdot\nabla\nabla^{n}_{j_{1}\ldots
j_{n}}\partial_{t}^{\ell-n}\bar{w}}\\\
\,=\,-\sum_{n=0}^{\ell}\nabla\cdot\Big{(}\sigma^{n,\ell-n}_{j_{1}\ldots
j_{n}}\nabla\nabla^{n}_{j_{1}\ldots
j_{n}}\partial_{t}^{\ell-n}\bar{w}\Big{)}+\sum_{n=0}^{\ell-1}\sigma^{n,\ell-n}_{j_{1}\ldots
j_{n}}:\nabla^{2}\nabla^{n}_{j_{1}\ldots j_{n}}\partial_{t}^{\ell-n}\bar{w}\\\
+\sum_{n=0}^{\ell-1}\phi^{n+1,\ell-n-2}\odot\nabla^{n+1}\partial_{t}^{\ell-n}\bar{w}.$

Inserting this into (3.12) yields the conclusion.∎

###  3.7. Proof of Lemma 3.8

We split the proof into three steps.

Step 1\. Proof that for all $p\geq 1$ there holds

$\partial_{t}^{2}\bar{W}^{\ell}_{\varepsilon}-\sum_{k=1}^{p}\varepsilon^{2(k-1)}\sum_{n\in[\ell]^{k}}\sum_{m\in
I^{n}_{k}}\Big{(}\prod_{j=1}^{k}\nabla\cdot\big{(}\bar{\boldsymbol{a}}^{n_{j},m_{j}}\odot(\varepsilon\nabla)^{n_{j}-1}\big{)}\nabla\Big{)}\bar{W}_{\varepsilon}^{\ell}\\\
\,=\,f+\sum_{k=1}^{p-1}\sum_{n\in[\ell]^{k}}\sum_{m\in
J^{n}_{k}}\varepsilon^{|m|}\Big{(}\prod_{j=1}^{k}\nabla\cdot\big{(}\bar{\boldsymbol{a}}^{n_{j},m_{j}}\odot(\varepsilon\nabla)^{n_{j}-1}\big{)}\nabla\Big{)}\partial_{t}^{|m|-2k}f\\\
+\sum_{n\in[\ell]^{p}}\sum_{m\in
J^{n}_{p}}\varepsilon^{|m|}\Big{(}\prod_{j=1}^{p}\nabla\cdot\big{(}\bar{\boldsymbol{a}}^{n_{j},m_{j}}\odot(\varepsilon\nabla)^{n_{j}-1}\big{)}\nabla\Big{)}\partial_{t}^{|m|-2(p-1)}\bar{W}_{\varepsilon}^{\ell},$
(3.13)

where we have defined the following sets of indices, for all $k\geq 1$ and
$n=(n_{1},\ldots,n_{k})\in[\ell]^{k}:=\\{1,\dots,\ell\\}^{k}$,

$\displaystyle I^{n}_{k}$ $\displaystyle=\Big{\\{}m=(m_{1},\dots,m_{k}):0\leq
m_{j}\leq\ell-n_{j}\leavevmode\nobreak\ \forall j,\,\sum_{j=1}^{s}m_{j}\geq
2s\leavevmode\nobreak\ \forall s<k,\,|m|=2(k-1)\Big{\\}},$ $\displaystyle
J^{n}_{k}$ $\displaystyle=\Big{\\{}m=(m_{1},\dots,m_{k}):0\leq m_{j}\leq\ell-
n_{j}\leavevmode\nobreak\ \forall j,\,\sum_{j=1}^{s}m_{j}\geq
2s\leavevmode\nobreak\ \forall s\leq k\Big{\\}}.$

We argue by induction. For $p=1$, the stated identity (3.13) reduces to

$\partial_{t}^{2}\bar{W}_{\varepsilon}^{\ell}-\nabla\cdot\Big{(}\sum_{n=1}^{\ell}\bar{\boldsymbol{a}}^{n,0}\odot(\varepsilon\nabla)^{n-1}\Big{)}\nabla\bar{W}_{\varepsilon}^{\ell}\,=\,f+\nabla\cdot\Big{(}\sum_{n=1}^{\ell}\sum_{m=2}^{\ell-n}\bar{\boldsymbol{a}}^{n,m}\odot(\varepsilon\nabla)^{n-1}(\varepsilon\partial_{t})^{m}\Big{)}\nabla\bar{W}_{\varepsilon}^{\ell},$

which is a simple reformulation of (3.5), keeping in mind that
$\bar{\boldsymbol{a}}^{n,m}=0$ whenever $m$ is odd, cf. Definition 3.1. Next,
we assume that the claim (3.13) holds for some $p\geq 1$ and we prove that it
then also holds at level $p+1$. Let $n\in[\ell]^{p}$ and $m\in J_{p}^{n}$ be
momentarily fixed. As by definition $|m|\geq 2p$, we may use (3.5) in the form

$\partial_{t}^{|m|-2(p-1)}\bar{W}_{\varepsilon}^{\ell}=\partial_{t}^{|m|-2p}\bigg{(}f+\nabla\cdot\Big{(}\sum_{n^{\prime}=1}^{\ell}\sum_{m^{\prime}=0}^{\ell-n^{\prime}}\bar{\boldsymbol{a}}^{n^{\prime},m^{\prime}}\odot(\varepsilon\nabla)^{n^{\prime}-1}(\varepsilon\partial_{t})^{m^{\prime}}\Big{)}\nabla\bar{W}_{\varepsilon}^{\ell}\bigg{)}.$

Note that for $1\leq n^{\prime}\leq\ell$ and $0\leq
m^{\prime}\leq\ell-n^{\prime}$ we have the equivalences

$\displaystyle(m,m^{\prime})\leavevmode\nobreak\ \in\leavevmode\nobreak\
\left\\{\begin{array}[]{lll}I_{p+1}^{(n,n^{\prime})}&\iff&|m|=2p,\,m^{\prime}=0,\\\
J_{p+1}^{(n,n^{\prime})}&\iff&|m|+m^{\prime}\geq 2(p+1).\end{array}\right.$

Using these observations, the last term in (3.13) can be decomposed as

$\sum_{n\in[\ell]^{p}}\sum_{m\in
J^{n}_{p}}\varepsilon^{|m|}\Big{(}\prod_{j=1}^{p}\nabla\cdot\big{(}\bar{\boldsymbol{a}}^{n_{j},m_{j}}\odot(\varepsilon\nabla)^{n_{j}-1}\big{)}\nabla\Big{)}\partial_{t}^{|m|-2(p-1)}\bar{W}_{\varepsilon}^{\ell}\\\
\,=\,\sum_{n\in[\ell]^{p}}\sum_{m\in
J^{n}_{p}}\varepsilon^{|m|}\Big{(}\prod_{j=1}^{p}\nabla\cdot\big{(}\bar{\boldsymbol{a}}^{n_{j},m_{j}}\odot(\varepsilon\nabla)^{n_{j}-1}\big{)}\nabla\Big{)}\partial_{t}^{|m|-2p}f\\\
+\varepsilon^{2p}\sum_{n\in[\ell]^{p+1}}\sum_{m\in{I}^{n}_{p+1}}\Big{(}\prod_{j=1}^{p+1}\nabla\cdot\big{(}\bar{\boldsymbol{a}}^{n_{j},m_{j}}\odot(\varepsilon\nabla)^{n_{j}-1}\big{)}\nabla\Big{)}\bar{W}_{\varepsilon}^{\ell}\\\
+\sum_{n\in[\ell]^{p+1}}\sum_{m\in
J^{n}_{p+1}}\varepsilon^{|m|}\Big{(}\prod_{j=1}^{p+1}\nabla\cdot\big{(}\bar{\boldsymbol{a}}^{n_{j},m_{j}}\odot(\varepsilon\nabla)^{n_{j}-1}\big{)}\nabla\Big{)}\partial_{t}^{|m|-2p}\bar{W}_{\varepsilon}^{\ell}.$

Inserting this expression into the induction assumption (3.13) yields the
conclusion.

Step 2\. Reformulation.
The definition of the coefficients $\\{\bar{\boldsymbol{b}}^{n}\\}_{n}$ in the
statement leads to

$\displaystyle\nabla\cdot\Big{(}\sum_{n=1}^{\ell}\bar{\boldsymbol{b}}^{n}\odot(\varepsilon\nabla)^{n-1}\Big{)}\nabla$
$\displaystyle=$ $\displaystyle\sum_{r=1}^{\ell}\sum_{k=1}^{\lceil
r/2\rceil}\varepsilon^{2(k-1)}\sum_{n\in[\ell]^{k}\atop k+|n|=r+1}\sum_{m\in
I_{k}^{n}}\,\prod_{j=1}^{k}\nabla\cdot\big{(}\bar{\boldsymbol{a}}^{n_{j},m_{j}}\odot(\varepsilon\nabla)^{n_{j}-1}\big{)}\nabla$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{\lceil\ell/2\rceil}\varepsilon^{2(k-1)}\sum_{n\in[\ell]^{k}\atop|n|\leq\ell-k+1}\sum_{m\in
I_{k}^{n}}\,\prod_{j=1}^{k}\nabla\cdot\big{(}\bar{\boldsymbol{a}}^{n_{j},m_{j}}\odot(\varepsilon\nabla)^{n_{j}-1}\big{)}\nabla.$

(Here we have used the following observation: for $m=(m_{1},\ldots,m_{k})$ and
$n=(n_{1},\ldots,n_{k})$ with $m_{j}\geq 0$ and $n_{j}\geq 1$ for all $j$, the
restrictions $|m|=2(k-1)$ and $k+|n|=r+1$ entail $m_{j}+n_{j}\leq r$, thus
showing that the index set $I_{k}$ in the definition of the coefficients
$\\{\bar{\boldsymbol{b}}^{n}\\}_{n}$ is indeed interchangeable with
$I_{k}^{n}$ here.) Choosing $p:=\lceil\frac{\ell}{2}\rceil$ in (3.13) and
inserting the above, we find

$\partial_{t}^{2}\bar{W}_{\varepsilon}^{\ell}-\nabla\cdot\Big{(}\sum_{n=1}^{\ell}\bar{\boldsymbol{b}}^{n}\odot(\varepsilon\nabla)^{n-1}\Big{)}\nabla\bar{W}_{\varepsilon}^{\ell}\,=\,\nabla\cdot
Q_{\varepsilon}^{\ell}\\\ +f+\sum_{k=1}^{p-1}\sum_{n\in[\ell]^{k}}\sum_{m\in
J^{n}_{k}}\varepsilon^{|m|}\Big{(}\prod_{j=1}^{k}\nabla\cdot\big{(}\bar{\boldsymbol{a}}^{n_{j},m_{j}}\odot(\varepsilon\nabla)^{n_{j}-1}\big{)}\nabla\Big{)}\partial_{t}^{|m|-2k}f,$
(3.14)

where the remainder $\nabla\cdot Q_{\varepsilon}^{\ell}$ is given by

$Q_{\varepsilon}^{\ell}\,:=\\\ \sum_{n\in[\ell]^{p}}\sum_{m\in
J^{n}_{p}}\varepsilon^{|m|}\big{(}\bar{\boldsymbol{a}}^{n_{1},m_{1}}\odot(\varepsilon\nabla)^{n_{1}-1}\big{)}\nabla\Big{(}\prod_{j=2}^{p}\nabla\cdot\big{(}\bar{\boldsymbol{a}}^{n_{j},m_{j}}\odot(\varepsilon\nabla)^{n_{j}-1}\big{)}\nabla\Big{)}\partial_{t}^{|m|-2(p-1)}\bar{W}_{\varepsilon}^{\ell}\\\
+\sum_{k=1}^{p}\varepsilon^{2(k-1)}\sum_{n\in[\ell]^{k}\atop|n|\geq\ell-k+2}\sum_{m\in
I^{n}_{k}}\big{(}\bar{\boldsymbol{a}}^{n_{1},m_{1}}\odot(\varepsilon\nabla)^{n_{1}-1}\big{)}\nabla\Big{(}\prod_{j=2}^{k}\nabla\cdot\big{(}\bar{\boldsymbol{a}}^{n_{j},m_{j}}\odot(\varepsilon\nabla)^{n_{j}-1}\big{)}\nabla\Big{)}\bar{W}_{\varepsilon}^{\ell}.$

(We use the standard convention $\prod_{j=l}^{k}=1$ if $l>k$.) In order to
estimate the remainder $Q_{\varepsilon}^{\ell}$, let us first examine the
$\varepsilon$-scaling and the number of derivatives of
$\bar{W}_{\varepsilon}^{\ell}$ appearing in each of the two contributions in
its definition:

  1. —

In the first contribution in the definition of $Q_{\varepsilon}^{\ell}$, the
terms have scaling $\varepsilon^{|n|+|m|-p}$ and involve $|n|+|m|-p+1$ space-
time derivatives of $\bar{W}_{\varepsilon}^{\ell}$, while the condition on
$m,n$ in the sum ensures the lower bound $|n|+|m|-p\geq|m|\geq 2p\geq\ell$ and
the upper bound
$|n|+|m|-p\leq|n|+p\ell-|n|-p=p(\ell-1)=\lceil\frac{\ell}{2}\rceil(\ell-1)\leq\ell^{2}$.

  2. —

In the second contribution in the definition of $Q_{\varepsilon}^{\ell}$, the
terms have scaling $\varepsilon^{|n|+k-2}$ and involve $|n|+k-1$ derivatives,
while the condition on $n,m$ in the sum ensures the lower bound
$|n|+k-2\geq\ell$ and the upper bound $|n|+k-2\leq
k\ell+k-2=\lceil\frac{\ell}{2}\rceil(\ell+1)-2\leq\ell^{2}$.

Hence, in view of Lemma 3.4, we deduce for $\varepsilon\ll 1$, for all $r\geq
0$,

$|\langle D\rangle^{r}Q_{\varepsilon}^{\ell}|\,\lesssim\,(\varepsilon
C\ell)^{\ell}|\langle D\rangle^{r+\ell}\langle\varepsilon
CD\rangle^{\ell^{2}}D\bar{W}_{\varepsilon}^{\ell}|.$

Likewise, the definition of $\\{\bar{\boldsymbol{c}}^{n}\\}_{n}$ in the
statement leads to

$\varepsilon^{2}\nabla\cdot\Big{(}\sum_{n=1}^{\ell-2}\bar{\boldsymbol{c}}^{n}\odot(\varepsilon
D)^{n-1}\Big{)}\nabla\\\
\,=\,\sum_{k=1}^{\lceil\ell/2\rceil-1}\sum_{n\in[\ell]^{k}}\sum_{m\in
J_{k}^{n}\atop|n|+|m|\leq\ell+k-1}\varepsilon^{|m|}\Big{(}\prod_{j=1}^{k}\nabla\cdot\big{(}\bar{\boldsymbol{a}}^{n_{j},m_{j}}\odot(\varepsilon\nabla)^{n_{j}-1}\big{)}\nabla\Big{)}\partial_{t}^{|m|-2k},$

which, inserted into (3.14), yields

$\partial_{t}^{2}\bar{W}_{\varepsilon}^{\ell}-\nabla\cdot\Big{(}\sum_{n=1}^{\ell}\bar{\boldsymbol{b}}^{n}\odot(\varepsilon\nabla)^{n-1}\Big{)}\nabla\bar{W}_{\varepsilon}^{\ell}\,=\,f+\varepsilon^{2}\nabla\cdot\Big{(}\sum_{n=1}^{\ell-2}\bar{\boldsymbol{c}}^{n}\odot(\varepsilon
D)^{n-1}\Big{)}\nabla
f+\nabla\cdot(Q_{\varepsilon}^{\ell}+R_{\varepsilon}^{\ell}),$

where the additional remainder $\nabla\cdot R^{\ell}_{\varepsilon}$ (which is
not zero only for $\ell\geq 3$) is given by

$R^{\ell}_{\varepsilon}\,=\,\sum_{k=1}^{{p-1}}\sum_{n\in[\ell]^{k}}\sum_{m\in
J_{k}^{n}\atop|n|+|m|\geq\ell+k}\varepsilon^{|m|}\big{(}\bar{\boldsymbol{a}}^{n_{1},m_{1}}\odot(\varepsilon\nabla)^{n_{1}-1}\big{)}\\\
\times\nabla\Big{(}\prod_{j=2}^{k}\nabla\cdot\big{(}\bar{\boldsymbol{a}}^{n_{j},m_{j}}\odot(\varepsilon\nabla)^{n_{j}-1}\big{)}\nabla\Big{)}\partial_{t}^{|m|-2k}f.$

Again, in order to estimate this remainder, we first check the
$\varepsilon$-scaling and the number of derivatives of $f$: the terms have
scaling $\varepsilon^{|n|+|m|-k}$ and involve $|n|+|m|-k-1$ derivatives, while
the condition on $n,m$ in the sum ensures the lower bound $|n|+|m|-k\geq\ell$
and the upper bound $|n|+|m|-k\leq
k(\ell-1)\leq(\lceil\frac{\ell}{2}\rceil-1)(\ell-1)\leq\ell^{2}$. Hence, in
view of Lemma 3.4, we deduce for $\varepsilon\ll 1$, for all $r\geq 0$,

$|\langle D\rangle^{r}R_{\varepsilon}^{\ell}|\,\lesssim\,(\varepsilon
C\ell)^{\ell}|\langle D\rangle^{r+\ell-1}\langle\varepsilon
CD\rangle^{\ell^{2}}f|.$

The conclusion with the remainder estimate follows upon setting
$E_{\varepsilon}^{\ell}=Q_{\varepsilon}^{\ell}+R_{\varepsilon}^{\ell}$.

Step 3\. Growth of the revamped coefficients: proof of (3.6).
We focus on $\bar{\boldsymbol{b}}^{p}$, while the estimation of
$\bar{\boldsymbol{c}}^{p}$ is obtained similarly. By definition of
$\bar{\boldsymbol{b}}^{p}$ in the statement, together with Lemma 3.4, we find

$\displaystyle|\bar{{\boldsymbol{b}}}^{p}|\leq\sum_{k\geq
1}\,\sum_{(m_{1},\ldots,m_{k})\in I_{k}}\sum_{n_{1},\ldots,n_{k}\geq 1\atop
k+|n|=p+1}C^{|n|+|m|}.$

For all admissible indices in the above sum, we have $|n|\geq k$, and thus
$p+1=k+|n|\geq 2k$, hence $k\leq\tfrac{p+1}{2}$. Moreover, any $m\in I_{k}$
satisfies $|m|=2(k-1)$, hence $p+1=k+|n|=|n|+|m|-k+2$. The upper bound for $k$
then yields $|n|+|m|\leq\tfrac{3p-1}{2}$, which leads us to the bound

$|\bar{{\boldsymbol{b}}}^{p}|\,\leq\,C^{p}\,\sharp\\!\left\\{r\in\mathbb{N}^{p+1}:\,|r|\leq\tfrac{3p-1}{2}\right\\},$

and the conclusion $|\bar{\boldsymbol{b}}^{p}|\leq C^{p}$ follows from a
simple counting argument with the balls in bins formula. ∎

###  3.8. Proof of Lemma 3.9

Let $\bar{v}_{\varepsilon}^{(\star),\ell}$ be the solution of a well-posed
modification of (3.7). In terms of

$f_{\varepsilon}^{(\star),\ell}(x)\,:=\,\partial_{t}^{2}\bar{v}^{(\star),\ell}_{\varepsilon}-\nabla\cdot\Big{(}\sum_{n=1}^{\ell}\sum_{m=0}^{\ell-n}\bar{\boldsymbol{a}}^{n,m}\odot(\varepsilon\nabla)^{n-1}(\varepsilon\partial_{t})^{m}\Big{)}\nabla\bar{v}^{(\star),\ell}_{\varepsilon},$
(3.15)

we aim to decompose $f_{\varepsilon}^{(\star),\ell}=f+\nabla\cdot
F_{\varepsilon}^{(\star),\ell}$ for some remainder
$F_{\varepsilon}^{(\star),\ell}$ satisfying the claimed estimates. We split
the proof into three steps, separately considering the cases
$(\star)=\text{(I)}$, (II), (III).

Step 1\. High-order filtering: $(\star)=\text{(I)}$.
Applying Lemma 3.8 to (3.15), we find

$\partial_{t}^{2}\bar{v}_{\varepsilon}^{\text{(I)},\ell}-\nabla\cdot\Big{(}\bar{\boldsymbol{a}}+\sum_{k=2}^{\ell}\bar{\boldsymbol{b}}^{k}\odot(\varepsilon\nabla)^{k-1}\Big{)}\nabla\bar{v}_{\varepsilon}^{\text{(I)},\ell}\\\
=f_{\varepsilon}^{\text{(I)},\ell}+\varepsilon^{2}\nabla\cdot\Big{(}\sum_{n=1}^{\ell-2}\bar{\boldsymbol{c}}^{n}\odot(\varepsilon
D)^{n-1}\Big{)}\nabla f_{\varepsilon}^{\text{(I)},\ell}+\nabla\cdot
E_{\varepsilon}^{\text{(I)},\ell},$

where the remainder satisfies for $\varepsilon\ll 1$, for all $r\geq 0$,

$|\langle D\rangle^{r}E_{\varepsilon}^{\text{(I)},\ell}|\,\leq\,(\varepsilon
C\ell)^{\ell}\Big{(}|\langle D\rangle^{r+\ell}\langle\varepsilon
CD\rangle^{\ell^{2}}D\bar{v}_{\varepsilon}^{\text{(I)},\ell}|+|\langle
D\rangle^{r+\ell-1}\langle\varepsilon
CD\rangle^{\ell^{2}}f_{\varepsilon}^{\text{(I)},\ell}|\Big{)}.$ (3.16)

As $\bar{v}_{\varepsilon}^{\text{(I)},\ell}$ is the solution of the well-posed
modification of (3.7) obtained by high-order filtering in the sense of Lemma
2.10, we deduce

$\chi(\varepsilon^{\alpha}\nabla)\bigg{(}f+\varepsilon^{2}\nabla\cdot\Big{(}\sum_{n=1}^{\ell-2}\bar{\boldsymbol{c}}^{n}\odot(\varepsilon
D)^{n-1}\Big{)}\nabla f\bigg{)}\\\
\,=\,f_{\varepsilon}^{\text{(I)},\ell}+\varepsilon^{2}\nabla\cdot\Big{(}\sum_{n=1}^{\ell-2}\bar{\boldsymbol{c}}^{n}\odot(\varepsilon
D)^{n-1}\Big{)}\nabla f_{\varepsilon}^{\text{(I)},\ell}+\nabla\cdot
E_{\varepsilon}^{\text{(I)},\ell},$

or equivalently,

$f_{\varepsilon}^{\text{(I)},\ell}+\varepsilon^{2}\nabla\cdot\Big{(}\sum_{n=1}^{\ell-2}\bar{\boldsymbol{c}}^{n}\odot(\varepsilon
D)^{n-1}\Big{)}\nabla f_{\varepsilon}^{\text{(I)},\ell}\\\
\,=\,f+\varepsilon^{2}\nabla\cdot\Big{(}\sum_{n=1}^{\ell-2}\bar{\boldsymbol{c}}^{n}\odot(\varepsilon
D)^{n-1}\Big{)}\nabla f+\nabla\cdot\hat{E}_{\varepsilon}^{\text{(I)},\ell},$
(3.17)

with modified remainder

$\hat{E}_{\varepsilon}^{\text{(I)},\ell}\,:=\,-E_{\varepsilon}^{\text{(I)},\ell}-(1-\chi(\varepsilon^{\alpha}\nabla))\triangle^{-1}\nabla\bigg{(}f+\varepsilon^{2}\nabla\cdot\Big{(}\sum_{n=1}^{\ell-2}\bar{\boldsymbol{c}}^{n}\odot(\varepsilon
D)^{n-1}\Big{)}\nabla f\bigg{)}.$ (3.18)

In order to estimate $f_{\varepsilon}^{\text{(I)},\ell}$, it remains to invert
the differential operator

$1+\varepsilon^{2}\nabla\cdot\Big{(}\sum_{n=1}^{\ell-2}\bar{\boldsymbol{c}}^{n}\odot(\varepsilon
D)^{n-1}\Big{)}\nabla$ (3.19)

in the left-hand side of (3.17). Although it is not invertible in general, we
may invert it approximately using a Neumann series truncated at order
$O(\varepsilon^{\ell})$. More precisely, we set

$\mathcal{O}^{\ell}_{\varepsilon}\,:=\,1+\sum_{k=1}^{\lfloor\ell/2\rfloor}(-\varepsilon^{2})^{k}\sum_{n\in[\ell-2]^{k}\atop|n|\leq\ell-k}\leavevmode\nobreak\
\prod_{j=1}^{k}\Big{(}\nabla\cdot\big{(}\bar{\boldsymbol{c}}^{n_{j}}\odot(\varepsilon
D)^{n_{j}-1}\big{)}\nabla\Big{)}.$

By a simple counting argument with the balls in bins formula, the number of
terms in this sum defining $\mathcal{O}^{\ell}_{\varepsilon}$ can be bounded
by

$\sum_{k=1}^{\lfloor\ell/2\rfloor}\sharp\\{n\in[\ell-2]^{k}:|n|\leq\ell-k\\}\,\leq\,C^{\ell},$

and therefore, combining this cardinality bound with the bound
$|\bar{\boldsymbol{c}}^{n}|\leq C^{n}$, we easily deduce for any function $g$,
for all $r\geq 0$,

$|\langle D\rangle^{r}\mathcal{O}^{\ell}_{\varepsilon}g|\leq C^{\ell}|\langle
D\rangle^{r}\langle\varepsilon CD\rangle^{\ell}g|.$ (3.20)

Next, we check that $\mathcal{O}_{\varepsilon}^{\ell}$ is indeed an
approximate inverse for the differential operator in (3.19): by definition of
$\mathcal{O}_{\varepsilon}^{\ell}$, working out cancellations, we find for any
function $g$,

$\mathcal{O}_{\varepsilon}^{\ell}\bigg{(}1+\varepsilon^{2}\nabla\cdot\Big{(}\sum_{n=1}^{\ell-2}\bar{\boldsymbol{c}}^{n}\odot(\varepsilon
D)^{n-1}\Big{)}\nabla\bigg{)}g\,=\,g+\nabla\cdot\mathcal{H}_{\varepsilon}^{\ell}g,$

where the remainder is explicitly given by

$\displaystyle\nabla\cdot\mathcal{H}_{\varepsilon}^{\ell}g\,:=\,$
$\displaystyle-\sum_{k=2}^{\lfloor\ell/2\rfloor}(-\varepsilon^{2})^{k}\sum_{n\in[\ell-2]^{k}\atop\ell+1-k\leq|n|\leq
n_{k}+\ell+1-k}\leavevmode\nobreak\
\prod_{j=1}^{k}\Big{(}\nabla\cdot\big{(}\bar{\boldsymbol{c}}^{n_{j}}\odot(\varepsilon
D)^{n_{j}-1}\big{)}\nabla\Big{)}g$
$\displaystyle-(-\varepsilon^{2})^{\lfloor\ell/2\rfloor+1}\sum_{n\in[\ell-2]^{\lfloor\ell/2\rfloor+1}\atop|n|\leq
n_{k}+\ell-\lfloor\ell/2\rfloor}\leavevmode\nobreak\
\prod_{j=1}^{\lfloor\ell/2\rfloor+1}\Big{(}\nabla\cdot\big{(}\bar{\boldsymbol{c}}^{n_{j}}\odot(\varepsilon
D)^{n_{j}-1}\big{)}\nabla\Big{)}g,$

and can be estimated as follows: for all $r\geq 0$, using
$|\bar{\boldsymbol{c}}^{n}|\leq C^{n}$, we have that

$|\langle D\rangle^{r}\mathcal{H}_{\varepsilon}^{\ell}g|\,\leq\,(\varepsilon
C)^{\ell}|\langle D\rangle^{r+2\ell}g|.$ (3.21)

Applying $\mathcal{O}_{\varepsilon}^{\ell}$ to both sides of (3.17), we then
get

$f_{\varepsilon}^{\text{(I)},\ell}\,=\,f+\nabla\cdot
F_{\varepsilon}^{\text{(I)},\ell},\qquad
F_{\varepsilon}^{\text{(I)},\ell}\,:=\,\mathcal{H}_{\varepsilon}^{\ell}(f-f_{\varepsilon}^{\text{(I)},\ell})+\mathcal{O}_{\varepsilon}^{\ell}\hat{E}_{\varepsilon}^{\text{(I)},\ell},$

and the desired remainder estimate follows by combining (3.15), (3.16),
(3.18), (3.20), and (3.21), together with the a priori estimate of Lemma
2.10(ii) for $\bar{v}_{\varepsilon}^{\text{(I)},\ell}$. Note that the
remainder term that is local with respect to $f$ can also be bounded with an
$\operatorname{L}^{1}(0,t)$-norm instead of an
$\operatorname{L}^{\infty}(0,t)$-norm up to loosing a time derivative.

Step 2\. Higher-order regularization: $(\star)=\text{(II)}$.
As $\bar{v}_{\varepsilon}^{\text{(II)},\ell}$ is the solution of the well-
posed modification of (3.7) obtained by higher-order regularization in the
sense of Lemma 2.10, we get instead of (3.17),

$f_{\varepsilon}^{\text{(II)},\ell}+\varepsilon^{2}\nabla\cdot\Big{(}\sum_{n=1}^{\ell-2}\bar{\boldsymbol{c}}^{n}\odot(\varepsilon
D)^{n-1}\Big{)}\nabla f_{\varepsilon}^{\text{(II)},\ell}\\\
\,=\,f+\varepsilon^{2}\nabla\cdot\Big{(}\sum_{n=1}^{\ell-2}\bar{\boldsymbol{c}}^{n}\odot(\varepsilon
D)^{n-1}\Big{)}\nabla f+\nabla\cdot\hat{E}_{\varepsilon}^{\text{(II)},\ell},$

with remainder

$\hat{E}_{\varepsilon}^{\text{(II)},\ell}\,:=\,-E_{\varepsilon}^{\text{(II)},\ell}+\kappa_{\ell}(\varepsilon|\nabla|)^{\ell}\nabla\bar{v}_{\varepsilon}^{\text{(II)},\ell},$

where $E_{\varepsilon}^{\text{(II)},\ell}$ satisfies the corresponding
estimate (3.16). Now applying the same approximate inverse operator
$\mathcal{O}_{\varepsilon}^{\ell}$ to both sides of this identity, we get

$f_{\varepsilon}^{\text{(II)},\ell}\,=\,f+\nabla\cdot
F_{\varepsilon}^{\text{(II)},\ell},\qquad
F_{\varepsilon}^{\text{(II)},\ell}\,:=\,\mathcal{H}_{\varepsilon}^{\ell}(f-f_{\varepsilon}^{\text{(II)},\ell})+\mathcal{O}_{\varepsilon}^{\ell}\hat{E}_{\varepsilon}^{\text{(II)},\ell},$

and the desired remainder estimate follows similarly.

Step 3\. Boussinesq trick: $(\star)=\text{(III)}$.
By definition, $\bar{v}_{\varepsilon}^{\text{(III)},\ell}$ is the solution of
the well-posed modification of (3.7) obtained by Boussinesq trick in the sense
of Lemma 2.10, that is,

$\partial_{t}^{2}\Big{(}1+\sum_{l=2}^{\ell}\kappa_{l}(\varepsilon|\nabla|)^{l-1}\Big{)}\bar{v}_{\varepsilon}^{\text{(III)},\ell}\\\
-\nabla\cdot\bigg{(}\sum_{n=1}^{\ell}\Big{(}\kappa_{n}\bar{\boldsymbol{a}}+\sum_{l=1}^{n-1}\kappa_{l}\bar{\boldsymbol{b}}^{n+1-l}\odot(\tfrac{\nabla}{|\nabla|})^{n-l}\Big{)}(\varepsilon|\nabla|)^{n-1}\bigg{)}\nabla\bar{v}_{\varepsilon}^{\text{(III)},\ell}\\\
\,=\,\Big{(}1+\sum_{l=2}^{\ell}\kappa_{l}(\varepsilon|\nabla|)^{l-1}\Big{)}\bigg{(}f+\varepsilon^{2}\nabla\cdot\Big{(}\sum_{k=1}^{\ell-2}\bar{\boldsymbol{c}}^{k}\odot(\varepsilon
D)^{k-1}\Big{)}\nabla f\bigg{)},$ (3.22)

with $\bar{v}_{\varepsilon}^{\text{(III)},\ell}=f=0$ for $t\leq 0$. We recall
that the coefficients $\\{\kappa_{l}\\}_{l}$ are defined in (2.23) and that
well-posedness is indeed ensured by Lemma 2.10(ii). Using the identity

$\sum_{n=1}^{\ell}\Big{(}\kappa_{n}\bar{\boldsymbol{a}}+\sum_{l=1}^{n-1}\kappa_{l}\bar{\boldsymbol{b}}^{n+1-l}\odot(\tfrac{\nabla}{|\nabla|})^{n-l}\Big{)}(\varepsilon|\nabla|)^{n-1}\\\
=\Big{(}1+\sum_{l=2}^{\ell}\kappa_{l}(\varepsilon|\nabla|)^{l-1}\Big{)}\Big{(}\bar{\boldsymbol{a}}+\sum_{k=2}^{\ell}\bar{\boldsymbol{b}}^{k}\odot(\varepsilon\nabla)^{k-1}\Big{)}\\\
-\sum_{n=\ell+1}^{2\ell-1}\Big{(}\sum_{l=n+1-\ell}^{\ell}\kappa_{l}\bar{\boldsymbol{b}}^{n+1-l}\odot(\tfrac{\nabla}{|\nabla|})^{n-l}\Big{)}(\varepsilon|\nabla|)^{n-1},$

the above equation (3.22) can be alternatively written as

$\partial_{t}^{2}\bar{v}_{\varepsilon}^{\text{(III)},\ell}-\nabla\cdot\Big{(}\bar{\boldsymbol{a}}+\sum_{k=2}^{\ell}\bar{\boldsymbol{b}}^{k}\odot(\varepsilon\nabla)^{k-1}\Big{)}\nabla\bar{v}_{\varepsilon}^{\text{(III)},\ell}\,=\,f+\varepsilon^{2}\nabla\cdot\Big{(}\sum_{k=1}^{\ell-2}\bar{\boldsymbol{c}}^{k}\odot(\varepsilon
D)^{k-1}\Big{)}\nabla f-\nabla\cdot G_{\varepsilon}^{\ell},$

in terms of

$G_{\varepsilon}^{\ell}\,:=\,\sum_{n=\ell+1}^{2\ell-1}\Big{(}\sum_{l=n+1-\ell}^{\ell}\kappa_{l}\bar{\boldsymbol{b}}^{n+1-l}\odot(\tfrac{\nabla}{|\nabla|})^{n-l}\Big{)}(\varepsilon|\nabla|)^{n-1}\Big{(}1+\sum_{l=2}^{\ell}\kappa_{l}(\varepsilon|\nabla|)^{l-1}\Big{)}^{-1}\nabla\bar{v}_{\varepsilon}^{\text{(III)},\ell},$

where the inverse operator is obviously well-defined as $\kappa_{l}\geq 0$ for
all $l$. We then get, instead of (3.17),

$f_{\varepsilon}^{\text{(III)},\ell}+\varepsilon^{2}\nabla\cdot\Big{(}\sum_{n=1}^{\ell-2}\bar{\boldsymbol{c}}^{n}\odot(\varepsilon
D)^{n-1}\Big{)}\nabla f_{\varepsilon}^{\text{(III)},\ell}\\\
\,=\,f+\varepsilon^{2}\nabla\cdot\Big{(}\sum_{k=1}^{\ell-2}\bar{\boldsymbol{c}}^{k}\odot(\varepsilon
D)^{k-1}\Big{)}\nabla f+\nabla\cdot\hat{E}_{\varepsilon}^{\text{(III)},\ell},$

with remainder

$\hat{E}_{\varepsilon}^{\text{(III)},\ell}\,:=\,-E_{\varepsilon}^{\text{(III)},\ell}-G_{\varepsilon}^{\ell},$

where $E_{\varepsilon}^{\text{(III)},\ell}$ satisfies the corresponding
estimate (3.16). The conclusion then follows similarly as in the first two
steps.∎

##  4\. Relating spectral to hyperbolic correctors

This last section is devoted to the proof of (1.12). It relies on an
algorithmic procedure to relate spectral and hyperbolic correctors. We split
the proof into two steps.

Step 1\. Reformulation of the hyperbolic two-scale expansion: for
$\varepsilon\ll 1$, we have for all $\ell\geq 1$,

$\bigg{\|}H^{\ell}_{\varepsilon}[\bar{v}_{\varepsilon}^{\text{(I)},\ell;t}]-\sum_{n=0}^{\ell}\varepsilon^{n}\mathring{\psi}_{\ell}^{n}(\tfrac{\cdot}{\varepsilon})\odot\nabla^{n}\bar{u}^{\text{(I)},\ell;t}_{\varepsilon}\\\
-\varepsilon^{3}\sum_{n=0}^{\ell-3}\sum_{2m=0}^{\ell-n-3}\varepsilon^{n+2m}\mathring{\zeta}_{\ell}^{n,2m}(\tfrac{\cdot}{\varepsilon})\odot\nabla^{n+1}\partial_{t}^{2m}f^{t}\bigg{\|}_{H^{1}(\mathbb{R}^{d})}\\\
\,\leq\,(\varepsilon C)^{\ell}\|\langle
D\rangle^{C\ell}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))},$
(4.1)

where for each $n,m$ the correctors $\mathring{\psi}_{\ell}^{n}$ and
$\mathring{\zeta}_{\ell}^{n,m}$ are suitable linear combinations of hyperbolic
correctors $\\{\phi^{n^{\prime},m^{\prime}}\\}_{n^{\prime},m^{\prime}}$ with
coefficients involving $\ell$ and
$\\{\bar{\boldsymbol{b}}^{n^{\prime}},\bar{\boldsymbol{c}}^{n^{\prime}}\\}_{n^{\prime}}$.

Comparing the effective equation in the spectral and in the geometric
approach, cf. (2.18) and (3.7), and using the well-posed modification by high-
order filtering, we get

$\bar{v}_{\varepsilon}^{\text{(I)},\ell}\,=\,\bar{u}_{\varepsilon}^{\text{(I)},\ell}+\varepsilon^{2}\nabla\cdot\Big{(}\sum_{k=1}^{\ell-2}\bar{\boldsymbol{c}}^{k}\odot(\varepsilon
D)^{k-1}\Big{)}\nabla\bar{u}_{\varepsilon}^{\text{(I)},\ell}.$ (4.2)

Inserting this into the definition (3.3) of the geometric two-scale expansion
$H_{\varepsilon}^{\ell}[\bar{v}_{\varepsilon}^{\text{(I)},\ell}]$, and
rearranging terms, we get

$H^{\ell}_{\varepsilon}[\bar{v}_{\varepsilon}^{\text{(I)},\ell}]\,=\,\sum_{n=0}^{\ell}\sum_{m=0}^{\ell-n}\varepsilon^{n+m}z^{n,m}_{\ell}(\tfrac{\cdot}{\varepsilon})\odot\nabla^{n}\partial_{t}^{m}\bar{u}_{\varepsilon}^{\text{(I)},\ell}+R^{\ell}_{\varepsilon,1}[\bar{u}_{\varepsilon}^{\text{(I)},\ell}],$
(4.3)

where for each $n,m$ the corrector $z^{n,m}_{\ell}$ is some linear combination
of hyperbolic correctors
$\\{\phi^{n^{\prime},m^{\prime}}\\}_{n^{\prime},m^{\prime}}$ with coefficients
involving $\ell$ and $\\{\bar{\boldsymbol{c}}^{n^{\prime}}\\}_{n^{\prime}}$,
and where the remainder term is given by

$R^{\ell}_{\varepsilon,1}[\bar{u}_{\varepsilon}^{\text{(I)},\ell}]\,:=\\\
\sum_{n=0}^{\ell}\sum_{m=0}^{\ell-n}\sum_{k=1}^{\ell-2}\mathds{1}_{n+m+k\geq\ell}\,\varepsilon^{n+m+k+1}\phi^{n,m}_{j_{1}\ldots
j_{n}}(\tfrac{\cdot}{\varepsilon})(\bar{\boldsymbol{c}}^{k}\odot
D^{k-1}):\nabla^{2}\nabla_{j_{1}\ldots
j_{n}}^{n}\partial_{t}^{m}\bar{u}_{\varepsilon}^{\text{(I)},\ell}.$

By construction, we note that $z^{0,0}_{\ell}=\phi^{0,0}=1$,
$z^{0,m}_{\ell}=\phi^{0,m}=0$ for all $m\geq 1$, and that $z^{n,m}_{\ell}$
does not depend on $\ell$ provided $n+m<\ell$. We emphasize that the
$z^{n,m}_{\ell}$’s do a priori not have vanishing average. Using Lemma 3.4
together with (3.6), and using the Sobolev embedding in form of (2.41) to
bound products with correctors, the remainder can be estimated as follows, for
$a>\frac{d}{2}$,

$\|R^{\ell}_{\varepsilon,1}[\bar{u}_{\varepsilon}^{\text{(I)},\ell}]\|_{H^{1}(\mathbb{R}^{d})}\,\leq\,(\varepsilon
C)^{\ell}\|\langle
D\rangle^{2\ell+a-1}D\bar{u}_{\varepsilon}^{\text{(I)},\ell}\|_{\operatorname{L}^{2}(\mathbb{R}^{d})},$

and thus, by the a priori estimate of Lemma 2.10(ii),

$\|R^{\ell}_{\varepsilon,1}[\bar{u}_{\varepsilon}^{\text{(I)},\ell;t}]\|_{H^{1}(\mathbb{R}^{d})}\leq(\varepsilon
C)^{\ell}\|\langle
D\rangle^{2\ell+a-{1}}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}.$
(4.4)

Next, using the equation (2.18) for $\bar{u}_{\varepsilon}^{\text{(I)},\ell}$,
we shall proceed to remove the time derivatives appearing in the geometric
two-scale expansion (4.3). For that purpose, arguing by induction and
successively separating terms of order $\varepsilon^{\ell+1}$ or higher, we
show that the equation for $\bar{u}_{\varepsilon}^{\text{(I)},\ell}$ yields
for all $m\geq 1$,

$\partial_{t}^{2m}\bar{u}^{\text{(I)},\ell}_{\varepsilon}-\sum_{1\leq\beta_{1},\ldots,\beta_{m}\leq\ell\atop|\beta|\leq
m+\ell}\varepsilon^{|\beta|-m}\bar{B}^{\beta}(\nabla)\bar{u}^{\text{(I)},\ell}_{\varepsilon}\\\
=\chi(\varepsilon^{\alpha}\nabla)\partial_{t}^{2(m-1)}f+\chi(\varepsilon^{\alpha}\nabla)\sum_{k=1}^{m-1}\sum_{1\leq\beta_{1},\ldots,\beta_{k}\leq\ell\atop|\beta|\leq
k+\ell}\varepsilon^{|\beta|-k}\bar{B}^{\beta}(\nabla)\partial_{t}^{2(m-k-1)}f\\\
+\sum_{k=2}^{m}\sum_{\beta\in
L_{k}}\varepsilon^{|\beta|-k}\bar{B}^{\beta}(\nabla)\partial_{t}^{2(m-k)}\bar{u}^{\text{(I)},\ell}_{\varepsilon},$
(4.5)

in terms of the index sets

$L_{k}\,:=\,\Big{\\{}\beta=(\beta_{1},\dots,\beta_{k}):1\leq\beta_{j}\leq\ell\leavevmode\nobreak\
\forall j,\leavevmode\nobreak\ \sum_{j=1}^{s}\beta_{j}\leq
s+\ell\leavevmode\nobreak\ \forall s<k,\leavevmode\nobreak\
|\beta|>k+\ell\Big{\\}},$

where for all $k\geq 1$ and $\beta=(\beta_{1},\ldots,\beta_{k})$ we use the
short-hand notation

$\bar{B}^{\beta}(\nabla)\,:=\,\prod_{j=1}^{k}\nabla\cdot(\bar{\boldsymbol{b}}^{\beta_{j}}\odot\nabla^{\beta_{j}-1})\nabla.$
(4.6)

We prove identity (4.5) by induction. First, for $m=1$, it simply coincides
with (the filtered version of) equation (2.18) for
$\bar{u}_{\varepsilon}^{\text{(I)},\ell}$. Next, assuming that (4.5) holds for
some $m\geq 1$, applying $\partial_{t}^{2}$ to both sides of the identity, and
using the equation for $\bar{u}_{\varepsilon}^{\text{(I)},\ell}$ to replace
$\partial_{t}^{2}\bar{u}_{\varepsilon}^{\text{(I)},\ell}$ in the left-hand
side, we find

$\partial_{t}^{2(m+1)}\bar{u}^{\ell}_{\varepsilon}-\sum_{1\leq\beta_{1},\ldots,\beta_{m}\leq\ell\atop|\beta|\leq
m+\ell}\varepsilon^{|\beta|-m}\bar{B}^{\beta}(\nabla)\Big{(}\sum_{n=1}^{\ell}\varepsilon^{n-1}\nabla\cdot(\bar{\boldsymbol{b}}^{n}\odot\nabla^{n-1})\nabla\Big{)}\bar{u}^{\ell}_{\varepsilon}\\\
=\chi(\varepsilon^{\alpha}\nabla)\partial_{t}^{2m}f+\chi(\varepsilon^{\alpha}\nabla)\sum_{k=1}^{m}\sum_{1\leq\beta_{1},\ldots,\beta_{k}\leq\ell\atop|\beta|\leq
k+\ell}\varepsilon^{|\beta|-k}\bar{B}^{\beta}(\nabla)\partial_{t}^{2(m-k)}f\\\
+\sum_{k=2}^{m}\sum_{\beta\in
L_{k}}\varepsilon^{|\beta|-k}\bar{B}^{\beta}(\nabla)\partial_{t}^{2(m-k+1)}\bar{u}^{\ell}_{\varepsilon}.$

After suitably splitting powers of $\varepsilon$ in the left-hand side, this
indeed proves identity (4.5) with $m$ replaced by $m+1$.

Now inserting (4.5) into the two-scale expansion (4.3) in order to replace
time derivatives, and again separating terms of order $\varepsilon^{\ell+1}$
or higher, we are led to

$H^{\ell}_{\varepsilon}[\bar{v}_{\varepsilon}^{\text{(I)},\ell}]\,=\,\sum_{n=0}^{\ell}\varepsilon^{n}z^{n,0}_{\ell}(\tfrac{\cdot}{\varepsilon})\odot\nabla^{n}\bar{u}^{\text{(I)},\ell}_{\varepsilon}\\\
+\sum_{n=0}^{\ell}\sum_{2m=2}^{\ell-n}\sum_{1\leq\beta_{1},\ldots,\beta_{m}\leq\ell\atop
n+m+|\beta|\leq\ell}\varepsilon^{n+m+|\beta|}z^{n,2m}_{\ell}(\tfrac{\cdot}{\varepsilon})\odot\bar{B}^{\beta}(\nabla)\nabla^{n}\bar{u}^{\text{(I)},\ell}_{\varepsilon}\\\
+\sum_{n=0}^{\ell}\sum_{2m=2}^{\ell-n}\varepsilon^{n+2m}z^{n,2m}_{\ell}(\tfrac{\cdot}{\varepsilon})\odot\chi(\varepsilon^{\alpha}\nabla)\nabla^{n}\partial_{t}^{2(m-1)}f\\\
+\sum_{n=0}^{\ell}\sum_{2m=2}^{\ell-n}\sum_{j=1}^{m-1}\sum_{1\leq\beta_{1},\ldots,\beta_{j}\leq\ell\atop
n+2m+|\beta|\leq
j+\ell}\varepsilon^{n+2m+|\beta|-j}z^{n,2m}_{\ell}(\tfrac{\cdot}{\varepsilon})\odot\chi(\varepsilon^{\alpha}\nabla)\bar{B}^{\beta}(\nabla)\nabla^{n}\partial_{t}^{2(m-j-1)}f\\\
+R^{\ell}_{\varepsilon,1}[\bar{u}_{\varepsilon}^{\text{(I)},\ell}]+R^{\ell}_{\varepsilon,2}[\bar{u}_{\varepsilon}^{\text{(I)},\ell}]+R^{\ell}_{\varepsilon,3}[f],$
(4.7)

where the last two remainder terms take the form

$\displaystyle
R^{\ell}_{\varepsilon,2}[\bar{u}_{\varepsilon}^{\text{(I)},\ell}]$
$\displaystyle:=$
$\displaystyle\sum_{n=0}^{\ell}\sum_{2m=2}^{\ell-n}\sum_{j=2}^{m}\sum_{\beta\in
L_{j}}\varepsilon^{n+2m+|\beta|-j}z_{\ell}^{n,2m}\left(\tfrac{\cdot}{\varepsilon}\right)\odot\bar{B}^{\beta}(\nabla)\nabla^{n}\partial_{t}^{2(m-j)}\bar{u}^{\text{(I)},\ell}_{\varepsilon}$
$\displaystyle+\sum_{n=0}^{\ell}\sum_{2m=2}^{\ell-n}\sum_{1\leq\beta_{1},\ldots,\beta_{m}\leq\ell\atop|\beta|\leq
m+\ell,n+m+|\beta|>\ell}\varepsilon^{n+m+|\beta|}z_{\ell}^{n,2m}\left(\tfrac{\cdot}{\varepsilon}\right)\odot\bar{B}^{\beta}(\nabla)\nabla^{n}\bar{u}^{\text{(I)},\ell}_{\varepsilon},$
$\displaystyle R^{\ell}_{\varepsilon,3}[f]$ $\displaystyle:=$
$\displaystyle\sum_{n=0}^{\ell}\sum_{2m=2}^{\ell-n}\sum_{j=1}^{m-1}\sum_{1\leq\beta_{1},\ldots,\beta_{j}\leq\ell\atop|\beta|\leq
j+\ell,\leavevmode\nobreak\
n+2m+|\beta|>j+\ell}\varepsilon^{n+2m+|\beta|-j}z_{\ell}^{n,2m}(\tfrac{\cdot}{\varepsilon})$
$\displaystyle\hskip
113.81102pt\odot\chi(\varepsilon^{\alpha}\nabla)\bar{B}^{\beta}(\nabla)\nabla^{n}\partial_{t}^{2(m-j-1)}f.$

Equivalently, in view of (4.6), this can be reformulated as

$H^{\ell}_{\varepsilon}[\bar{v}_{\varepsilon}^{\text{(I)},\ell}]\,=\\\
\sum_{n=0}^{\ell}\varepsilon^{n}\mathring{\psi}_{\ell}^{n}(\tfrac{\cdot}{\varepsilon})\odot\nabla^{n}\bar{u}^{\text{(I)},\ell}_{\varepsilon}+\varepsilon^{3}\sum_{n=0}^{\ell-3}\sum_{2m=0}^{\ell-n-3}\varepsilon^{n+2m}\mathring{\zeta}_{\ell}^{n,2m}(\tfrac{\cdot}{\varepsilon})\odot\chi(\varepsilon^{\alpha}\nabla)\nabla^{n+1}\partial_{t}^{2m}f\\\
+R^{\ell}_{\varepsilon,1}[\bar{u}_{\varepsilon}^{\ell}]+R^{\ell}_{\varepsilon,2}[\bar{u}_{\varepsilon}^{\ell}]+R^{\ell}_{\varepsilon,3}[f],$
(4.8)

where for each $n,m$ the correctors $\mathring{\psi}^{n}_{\ell}$ and
$\mathring{\zeta}^{n,2m}_{\ell}$ are suitable linear combinations of
$\\{z^{n^{\prime},2m^{\prime}}\\}_{n^{\prime},m^{\prime}}$ with coefficients
involving $\ell$ and $\\{\bar{\boldsymbol{b}}^{n^{\prime}}\\}_{n^{\prime}}$.
We emphasize that $\mathring{\psi}^{n}_{\ell}$ and
$\mathring{\zeta}^{n,2m}_{\ell}$ do a priori not have vanishing average.

We turn to the estimation of the last two remainder terms in (4.8). Recalling
the way that $z^{n,2m}$ depends on
$\\{\phi^{n^{\prime},m^{\prime}}\\}_{n^{\prime},m^{\prime}}$ and on
$\\{{\boldsymbol{c}}^{n^{\prime}}\\}_{n^{\prime}}$, using Lemma 3.4 together
with (3.6), and using again the Sobolev embedding in form of (2.41) to bound
products with correctors, we find for $\varepsilon\ll 1$ and $a>\frac{d}{2}$,

$\|R_{\varepsilon,2}^{\ell}[\bar{u}_{\varepsilon}^{\text{(I)},\ell}]\|_{H^{1}(\mathbb{R}^{d})}\,\leq\,(\varepsilon
C)^{\ell}\|\langle
D\rangle^{3\ell+a-1}D\bar{u}_{\varepsilon}^{\ell}\|_{\operatorname{L}^{2}(\mathbb{R}^{d})},$

and thus, further combining this with the a priori estimate of Lemma 2.10(ii),

$\|R_{\varepsilon,2}^{\ell}[\bar{u}_{\varepsilon}^{\text{(I)},\ell;t}]\|_{H^{1}(\mathbb{R}^{d})}\,\leq\,(\varepsilon
C)^{\ell}\|\langle
D\rangle^{3\ell+a-1}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}.$

Similarly, we find for $\varepsilon\ll 1$ and $a>\frac{d}{2}$,

$\|R_{\varepsilon,3}^{\ell}[f]\|_{H^{1}(\mathbb{R}^{d})}\,\leq\,(\varepsilon
C)^{\ell}\|\langle
D\rangle^{2\ell+a-1}f\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}.$

Finally, we recall that the cut-off $\chi(\varepsilon^{\alpha}\nabla)$ can be
removed in the right-hand side of (4.8) up to higher-order errors: for
$a>\frac{d}{2}$,

${\bigg{\|}\varepsilon^{3}\sum_{n=0}^{\ell-3}\sum_{2m=0}^{\ell-n-3}\varepsilon^{n+2m}\mathring{\zeta}_{\ell}^{n,2m}(\tfrac{\cdot}{\varepsilon})\odot(1-\chi(\varepsilon^{\alpha}\nabla))\nabla^{n+1}\partial_{t}^{2m}f\bigg{\|}_{H^{1}(\mathbb{R}^{d})}}\\\
\,\leq\,\varepsilon^{3}C^{\ell}\|(1-\chi(\varepsilon^{\alpha}\nabla))\langle
D\rangle^{\ell+a-1}f\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\,\leq\,(\varepsilon
C)^{\ell}\|\langle
D\rangle^{\ell+{\lceil\frac{\ell}{\alpha}\rceil}+a-1}f\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\\\
{\,\leq\,(\varepsilon C)^{\ell}\|\langle
D\rangle^{\ell+{\lceil\frac{\ell}{\alpha}\rceil}+a}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}}.$

Inserting these bounds together with (4.4) into (4.8), the claim (4.1)
follows.

Step 2\. Conclusion.
In view of (4.1), the result of Theorem 2 yields

$\bigg{\|}u_{\varepsilon}^{t}-\sum_{n=0}^{\ell}\varepsilon^{n}\mathring{\psi}_{\ell}^{n}(\tfrac{\cdot}{\varepsilon})\odot\nabla^{n}\bar{u}^{\text{(I)},\ell;t}_{\varepsilon}-\varepsilon^{3}\sum_{n=0}^{\ell-3}\sum_{2m=0}^{\ell-n-3}\varepsilon^{n+2m}\mathring{\zeta}_{\ell}^{n,2m}(\tfrac{\cdot}{\varepsilon})\odot\nabla^{n+1}\partial_{t}^{2m}f^{t}\bigg{\|}_{H^{1}(\mathbb{R}^{d})}\\\
\,\lesssim\,(\varepsilon C\ell)^{\ell}\langle t\rangle\|\langle
D\rangle^{C\ell^{2}}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}.$

Now we compare this with the result of Theorem 1: using (2.7) to expand
$\gamma_{\ell}(\varepsilon\nabla)$ in the spectral two-scale expansion (1.4),
discarding terms of order $O(\varepsilon^{\ell})$, and using Lemma 2.10(ii) to
estimate the latter, the result (1.6) of Theorem 1 leads us to

$\bigg{\|}\sum_{n=0}^{\ell}\varepsilon^{n}\widetilde{\psi}_{\ell}^{n}(\tfrac{\cdot}{\varepsilon})\odot\nabla^{n}\bar{u}^{\text{(I)},\ell;t}_{\varepsilon}+\varepsilon^{3}\sum_{n=0}^{\ell-3}\sum_{2m=0}^{\ell-n-3}\varepsilon^{n+2m}\widetilde{\zeta}_{\ell}^{n,2m}(\tfrac{\cdot}{\varepsilon})\odot\nabla^{n+1}\partial_{t}^{2m}f^{t}\bigg{\|}_{H^{1}(\mathbb{R}^{d})}\\\
\,\lesssim\,(\varepsilon C\ell)^{\ell}\langle t\rangle\|\langle
D\rangle^{C\ell^{2}}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))},$
(4.9)

in terms of the corrector differences

$\widetilde{\psi}_{\ell}^{n}=\sum_{k=0}^{n}\gamma_{\ell}^{k}\otimes_{s}\psi^{n-k}-\mathring{\psi}_{\ell}^{n},\qquad\widetilde{\zeta}_{\ell}^{n,2m}=\sum_{k=0}^{n}\gamma_{\ell}^{k}\otimes_{s}\zeta^{n-k,2m}-\mathring{\zeta}_{\ell}^{n,2m},$

where $\otimes_{s}$ stands for symmetric tensor product. We claim that (4.9)
entails $\widetilde{\psi}_{\ell}^{n}=0$ for all $n\leq\ell-1$,
$\widetilde{\zeta}_{\ell}^{n,2m}=0$ for all $n+2m+3\leq\ell-1$, and
$\nabla\widetilde{\psi}_{\ell}^{\ell}=0$ and
$\nabla\widetilde{\zeta}_{\ell}^{n,2m}=0$ for $n+2m+3=\ell$. We argue by
induction and prove:

$\left\\{\begin{array}[]{lll}\text{for all
$j\leq\ell-1$}&:&\text{$\widetilde{\psi}_{\ell}^{j}=0$ and
$\widetilde{\zeta}_{\ell}^{n,2m}=0$ for $n+2m+3=j$},\\\ \text{for
$j=\ell$}&:&\text{$\nabla\widetilde{\psi}_{\ell}^{\ell}=0$ and
$\nabla\widetilde{\zeta}_{\ell}^{n,2m}=0$ for
$n+2m+3=\ell$.}\end{array}\right.$ (4.10)

Assume that this result is known to hold for $j<j_{0}$, given some
$j_{0}\leq\ell-1$. Using the a priori estimates of Lemma 2.10(ii), first note
that we have $\bar{u}_{\varepsilon}^{\operatorname{(I)},\ell}\to\bar{u}$ in
$C^{\infty}_{\operatorname{loc}}(\mathbb{R};H^{\infty}(\mathbb{R}^{d}))$ as
$\varepsilon\downarrow 0$, where $\bar{u}$ is the solution of the standard
homogenized wave equation (1.3). Given $h\in
C_{\operatorname{per}}^{\infty}(Q)$, multiplying the expression in the left-
hand side of (4.9) by $\varepsilon^{-j_{0}}h(\frac{\cdot}{\varepsilon})$, and
passing to the limit $\varepsilon\downarrow 0$ in the
$\operatorname{L}^{2}$-norm, we then get

$\mathbb{E}\big{[}h\widetilde{\psi}_{\ell}^{j_{0}}\big{]}\odot\nabla^{j_{0}}\bar{u}+\sum_{n+2m+3=j_{0}}\mathbb{E}\big{[}h\widetilde{\zeta}_{\ell}^{n,2m}\big{]}\odot\nabla^{n+1}\partial_{t}^{2m}f\,=\,0.$

As $\bar{u}$ satisfies the homogenized wave equation (1.3), applying the wave
operator $\partial_{t}^{2}-\nabla\cdot\bar{\boldsymbol{a}}\nabla$ to this
pointwise identity leads us to

$\mathbb{E}\big{[}h\widetilde{\psi}_{\ell}^{j_{0}}\big{]}\odot\nabla^{j_{0}}f+\sum_{n+2m+3=j_{0}}\mathbb{E}\big{[}h\widetilde{\zeta}_{\ell}^{n,2m}\big{]}\odot\nabla^{n+1}\partial_{t}^{2m}(\partial_{t}^{2}-\nabla\cdot\bar{\boldsymbol{a}}\nabla)f\,=\,0.$

Choosing for instance $f^{t}(x)=\exp(-(t^{2}+|x|^{2}))$ for $t\geq 1$, it is
easily deduced by induction, by a linear independence argument, that
$\mathbb{E}\big{[}h\widetilde{\psi}_{\ell}^{j_{0}}\big{]}=0$ and
$\mathbb{E}\big{[}h\widetilde{\zeta}_{\ell}^{n,2m}\big{]}=0$ for all $n,m$
with $n+2m+3=j_{0}$. As $h\in C^{\infty}_{\operatorname{per}}(Q)$ is
arbitrary, we deduce that the claim (4.10) also holds for $j=j_{0}$. The same
argument can be adapted to the case $j_{0}=\ell$, rather starting from the
estimate on the $H^{1}$-norm in (4.9). This ends the proof of (4.10).

In other words, we have thus proved that, given $\ell\geq 1$, for all
$j\leq\ell-1$ and for all $n,m$ with $n+m+3\leq\ell-1$, we have

$\displaystyle\mathring{\psi}_{\ell}^{j}=\sum_{k=0}^{j}\gamma_{\ell}^{k}\otimes_{s}\psi^{j-k},\qquad\mathring{\zeta}_{\ell}^{n,2m}=\sum_{k=0}^{n}\gamma_{\ell}^{k}\otimes_{s}\zeta^{n-k,2m},$

while for $j=\ell$ and for all $n,m$ with $n+m+3=\ell$,

$\displaystyle\nabla_{i}\mathring{\psi}_{\ell}^{\ell}=\sum_{k=0}^{\ell}\gamma_{\ell}^{k}\otimes_{s}\nabla_{i}\psi^{\ell-k},\qquad\nabla_{i}\mathring{\zeta}_{\ell}^{n,2m}=\sum_{k=0}^{n}\gamma_{\ell}^{k}\otimes_{s}\nabla_{i}\zeta^{n-k,2m}.$

Inserting this back into (4.1), using (2.7) to reconstruct
$\gamma_{\ell}(\varepsilon\nabla)$, discarding terms of order
$O(\varepsilon^{\ell})$, and using Lemma 2.10(ii) to estimate the latter, we
may then recognize the definition (1.4) of the spectral two-scale expansion,
to the effect of

$\|H^{\ell}_{\varepsilon}[\bar{v}_{\varepsilon}^{\text{(I)},\ell;t}]-S_{\varepsilon}^{\ell}[\bar{u}_{\varepsilon}^{\text{(I)},\ell;t},f]\|_{H^{1}(\mathbb{R}^{d})}\,\lesssim\,(\varepsilon
C)^{\ell}\|\langle
D\rangle^{C\ell}f\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}.$

Combined with the result (1.6) of Theorem 1, this yields the conclusion
(1.12).∎

##  Appendix A Correctors in the random setting

This section is devoted to the definition and bounds on spectral and
hyperbolic correctors in the random setting. As in the elliptic case [18, 4,
21, 15], the main difference with the periodic setting is that only a finite
number of correctors can be defined, depending both on space dimension and on
mixing properties of the coefficient field. For simplicity, we focus on the
Gaussian setting of Definition 1.3. The corrector estimates below were first
obtained in [19, 20, 18] for the first corrector in the elliptic setting. A
proof of the present statement follows from applying iteratively the annealed
Calderón-Zygmund estimates of [15, 11]; see in particular a similar argument
in [15, Proof of Proposition 2.2]. Note that the present result corrects
inaccuracies of the corresponding statement given in [5, Proposition C.4] for
spectral correctors.

######  Theorem A.1.

Let $\boldsymbol{a}$ be Gaussian with parameter $\beta>0$ in the sense of
Definition 1.3. We then define $\ell_{*}\,:=\,\lceil\tfrac{\beta\wedge
d}{2}\rceil$ and

$\mu_{n}^{*}(x)\,:=\,\left\\{\begin{array}[]{lll}1&:&n<\ell_{*},\\\
\log(2+|x|)^{\frac{1}{2}}&:&n=\ell_{*},\leavevmode\nobreak\
\beta>d,\leavevmode\nobreak\ \text{$d$ even},\\\ &&\text{or
}n=\ell_{*},\leavevmode\nobreak\ \beta<d,\leavevmode\nobreak\ \beta\in
2\mathbb{N},\\\ \log(2+|x|)&:&n=\ell_{*},\leavevmode\nobreak\
\beta=d,\leavevmode\nobreak\ \text{$d$ even},\\\ \langle
x\rangle^{\frac{1}{2}}&:&n=\ell_{*},\leavevmode\nobreak\
\beta>d,\leavevmode\nobreak\ \text{$d$ odd},\\\ \langle
x\rangle^{\frac{1}{2}}\log(2+|x|)^{\frac{1}{2}}&:&n=\ell_{*},\leavevmode\nobreak\
\beta=d,\leavevmode\nobreak\ \text{$d$ odd},\\\ \langle
x\rangle^{n-\frac{\beta}{2}}&:&n=\ell_{*},\leavevmode\nobreak\
\beta<d,\leavevmode\nobreak\ \beta\notin 2\mathbb{N}.\end{array}\right.$ (A.1)

  1. (i)

_Spectral correctors:_ The correctors $\\{\psi^{n},\sigma^{n}\\}_{0\leq
n<\ell_{*}}$ and $\\{\zeta^{n,m},\tau^{n,m}\\}_{n+2m<\ell_{*}-3}$ can be
uniquely defined by the corrector equations of Section 2.1 as centered
stationary random fields, and in addition the correctors $\psi^{n},\sigma^{n}$
with $n=\ell_{*}$ and $\zeta^{n,m},\tau^{n,m}$ with $n+2m=\ell_{*}-3$ can be
uniquely defined as (non-stationary) random fields with centered stationary
gradient. The homogenized coefficient $\bar{\boldsymbol{b}}^{n}$ is well-
defined for $0\leq n\leq\ell_{*}$. Moreover, the following moment bounds hold
for all $q<\infty$ and $x\in\mathbb{R}^{d}$,

$\displaystyle\qquad\Big{\|}\Big{(}\fint_{B(x)}|(\psi^{n},\sigma^{n})|^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{\operatorname{L}^{q}(\Omega)}$
$\displaystyle\leavevmode\nobreak\ \lesssim_{q}\leavevmode\nobreak\
\mu_{n}^{*}(x),\quad$ $\displaystyle\text{for $0\leq n\leq\ell_{*}$},$
$\displaystyle\qquad\Big{\|}\Big{(}\fint_{B(x)}|(\nabla\psi^{n},\nabla\sigma^{n})|^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{\operatorname{L}^{q}(\Omega)}$
$\displaystyle\leavevmode\nobreak\ \lesssim_{q}\leavevmode\nobreak\ 1,\quad$
$\displaystyle\text{for $0\leq n\leq\ell_{*}$},$
$\displaystyle\quad\Big{\|}\Big{(}\fint_{B(x)}|(\zeta^{n,m},\tau^{n,m})|^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{\operatorname{L}^{q}(\Omega)}$
$\displaystyle\leavevmode\nobreak\ \lesssim_{q}\leavevmode\nobreak\
\mu^{*}_{n+2m+3}(x),\quad$ $\displaystyle\text{for $n+2m\leq\ell_{*}-3$},$
$\displaystyle\qquad\Big{\|}\Big{(}\fint_{B(x)}|(\nabla\zeta^{n,m},\nabla\tau^{n,m})|^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{\operatorname{L}^{q}(\Omega)}$
$\displaystyle\leavevmode\nobreak\ \lesssim_{q}\leavevmode\nobreak\ 1,\quad$
$\displaystyle\text{for $n+2m\leq\ell_{*}-3$}.$

  2. (ii)

_Hyperbolic correctors:_ The correctors
$\\{\phi^{n,m},\sigma^{n,m}\\}_{n+m<\ell_{*}}$ can be uniquely defined by the
corrector equations of Section 3.1 as centered stationary random fields, and
in addition the correctors $\phi^{n,m},\sigma^{n,m}$ with $n+m=\ell_{*}$ can
be uniquely defined as (non-stationary) random fields with centered stationary
gradient. The homogenized coefficient $\bar{\boldsymbol{a}}^{n,m}$ is well-
defined for $n+m\leq\ell_{*}$. Moreover, for $n+m\leq\ell_{*}$, the following
moment bounds hold for all $q<\infty$ and $x\in\mathbb{R}^{d}$,

$\displaystyle\qquad\Big{\|}\Big{(}\fint_{B(x)}|(\phi^{n,m},\sigma^{n,m})|^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{\operatorname{L}^{q}(\Omega)}$
$\displaystyle\leavevmode\nobreak\ \lesssim_{q}\leavevmode\nobreak\
\mu_{n+m}^{*}(x),$
$\displaystyle\qquad\Big{\|}\Big{(}\fint_{B(x)}|(\nabla\phi^{n,m},\nabla\sigma^{n,m})|^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{\operatorname{L}^{q}(\Omega)}$
$\displaystyle\leavevmode\nobreak\ \lesssim_{q}\leavevmode\nobreak\ 1.\qed$

##  Appendix B A priori estimates for the wave equation

We state the following general a priori estimates for linear wave equations,
which are used throughout; a short proof is included for convenience. In
contrast with the situation in the elliptic setting, we emphasize that putting
the impulse in divergence form essentially only brings an improvement when
estimating the $\operatorname{L}^{2}$-norm, and not the energy norm.

######  Lemma B.1 (A priori estimates).

Let $\mathcal{L}$ be a self-adjoint operator on
$\operatorname{L}^{2}(\mathbb{R}^{d})$ satisfying the bound
$-\triangle\leq\mathcal{L}\leq-C_{0}\triangle$ for some constant
$C_{0}<\infty$. Given
$F_{1}\in\operatorname{L}^{1}_{\operatorname{loc}}(\mathbb{R}^{+};\operatorname{L}^{2}(\mathbb{R}^{d}))$
and $F_{2},F_{3}\in
W^{1,1}_{\operatorname{loc}}(\mathbb{R}^{+};\operatorname{L}^{2}(\mathbb{R}^{d}))$
with $F_{2}|_{t=0}=F_{3}|_{t=0}=0$, let $z$ be the solution of the wave
equation

$\left\\{\begin{array}[]{l}(\partial_{t}^{2}+\mathcal{L})z\,=\,F_{1}+\nabla\cdot
F_{2}+\partial_{t}F_{3}\quad\text{in $\mathbb{R}^{d}$},\\\
z|_{t=0}=\partial_{t}z|_{t=0}=0.\end{array}\right.$

Then, for all $t\geq 0$, we have

$\|z^{t}\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\,\lesssim_{C_{0}}\,t\|F_{1}\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}+\|(F_{2},F_{3})\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))},$

and

$\|Dz^{t}\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\,\lesssim_{C_{0}}\,\|(F_{1},\partial_{t}F_{2},\partial_{t}F_{3})\|_{\operatorname{L}^{1}((0,t);\operatorname{L}^{2}(\mathbb{R}^{d}))}.\qed$

###### Proof.

The assumption $-\triangle\leq\mathcal{L}\leq-C_{0}\triangle$ entails that
$\sqrt{\mathcal{L}}$ defines a bounded linear operator
$\operatorname{L}^{2}(\mathbb{R}^{d})\to\dot{H}^{-1}(\mathbb{R}^{d})$ with
bounded inverse. We may therefore define $\tilde{F}_{2}$ as the solution of
$\sqrt{\mathcal{L}}\tilde{F}_{2}=\nabla\cdot F_{2}$, which satisfies
$\|\tilde{F}_{2}\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\lesssim_{C_{0}}\|F_{2}\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}$.
The solution $z$ of the wave equation can then be represented in terms of
Duhamel’s formula,

$z^{t}=\int_{0}^{t}\frac{\sin((t-s)\sqrt{\mathcal{L}})}{\sqrt{\mathcal{L}}}\,(F_{1}^{s}+\partial_{s}F_{3}^{s})\,ds+\int_{0}^{t}\sin((t-s)\sqrt{\mathcal{L}})\,\tilde{F}_{2}^{s}\,ds.$

Integrating by parts in the integral for $F_{3}$, with $F_{3}|_{t=0}=0$, this
can be rewritten as

$z^{t}=\int_{0}^{t}\frac{\sin((t-s)\sqrt{\mathcal{L}})}{\sqrt{\mathcal{L}}}\,F_{1}^{s}\,ds+\int_{0}^{t}\sin((t-s)\sqrt{\mathcal{L}})\,\tilde{F}_{2}^{s}\,ds+\int_{0}^{t}\cos((t-s)\sqrt{\mathcal{L}})\,F_{3}^{s}\,ds,$

and the stated $\operatorname{L}^{2}$-estimate follows from spectral calculus.
For the energy estimate, we rather use energy conservation in form of

$\displaystyle\tfrac{1}{2}\partial_{t}\int_{\mathbb{R}^{d}}\big{(}|\partial_{t}z|^{2}+z\mathcal{L}z+2F_{2}\cdot\nabla
z\big{)}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}(\partial_{t}z)\,(\partial^{2}_{t}+\mathcal{L})z+\partial_{t}\int_{\mathbb{R}^{d}}F_{2}\cdot\nabla
z$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}(\partial_{t}z)\big{(}F_{1}+\partial_{t}F_{3}+\nabla\cdot
F_{2}\big{)}+\partial_{t}\int_{\mathbb{R}^{d}}F_{2}\cdot\nabla z$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}(\partial_{t}z)\big{(}F_{1}+\partial_{t}F_{3}\big{)}+\int_{\mathbb{R}^{d}}\partial_{t}F_{2}\cdot\nabla
z$ $\displaystyle\leq$
$\displaystyle\|Dz\|_{\operatorname{L}^{2}(\mathbb{R}^{d})}\|(F_{1},\partial_{t}F_{2},\partial_{t}F_{3})\|_{\operatorname{L}^{2}(\mathbb{R}^{d})},$

and the conclusion follows. ∎

## Acknowledgements

MD acknowledges financial support from the F.R.S.-FNRS, as well as from the
European Union (ERC, PASTIS, Grant Agreement n∘101075879), and AG from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (Grant Agreement n∘864066).444Views and
opinions expressed are however those of the authors only and do not
necessarily reflect those of the European Union or the European Research
Council Executive Agency. Neither the European Union nor the granting
authority can be held responsible for them.

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