$\||D|^{-\sigma}(S_{\lambda}P-P^{\lambda}S_{\lambda})u\|^{2}_{L^{1}L^{2}}\lesssim\kappa^{2}\|\tilde{S}_{\lambda}u\|^{2}_{L^{\infty}L^{2}}.$

holds. Factoring out the derivatives, we thus have to show

(70)
$\|(S_{\lambda}\varepsilon_{ij}-\varepsilon_{ij}^{\lambda}S_{\lambda})u\|^{2}_{L^{1}L^{2}}\lesssim\kappa^{2}\lambda^{2(\sigma-1)}\|\tilde{S}_{\lambda}u\|^{2}_{L^{\infty}L^{2}}.$

Because $u$ is compactly supported in time, it suffices to take compactly
supported $\varepsilon_{ij}$. Then, we know that

$\|\varepsilon_{ij}\|_{L^{1}L^{\infty}}\lesssim
1,\quad\|\varepsilon_{ij}\|_{\mathcal{X}^{s}}\lesssim\kappa^{\frac{2+s}{2}}$

(using also the boundedness of $\varepsilon_{ij}$), which gives

$\|S_{\lambda}\varepsilon_{ij}\|_{L^{1}L^{\infty}}\lesssim\min(\kappa^{\frac{2+s}{2}}\lambda^{-s},1).$

As $1-\sigma=2s/(2+s)$, we infer

$\displaystyle\|\varepsilon_{ij}\|_{\dot{B}^{1\infty 1}_{1-\sigma}}$
$\displaystyle=\sum_{\lambda}\lambda^{1-\sigma}\|S_{\lambda}\varepsilon_{ij}\|_{L^{1}L^{\infty}}$
$\displaystyle\lesssim\sum_{\lambda\lesssim\kappa^{\frac{2+s}{2s}}}\lambda^{1-\sigma}+\sum_{\lambda\gtrsim\kappa^{\frac{2+s}{2s}}}\lambda^{1-\sigma}\kappa^{\frac{2+s}{2}}\lambda^{-s}\lesssim\kappa.$

At this point, (70) follows from inequality (4.4) of [39].

Next, we further restrict the Fourier support of $\varepsilon^{\lambda}_{ij}$
to $\kappa^{\frac{1}{2}}\lambda^{\frac{2}{2+s}}$. This is possible since we
can control the error in the second term in (69) by means of

$\displaystyle\|\partial_{x}(U_{\kappa^{\frac{1}{2}}\lambda^{\frac{2}{2+s}}}\varepsilon^{\lambda}_{ij})S_{\lambda}u\|_{L^{1}L^{2}}$
$\displaystyle\lesssim\lambda\|U_{\kappa^{\frac{1}{2}}\lambda^{\frac{2}{2+s}}}\varepsilon_{ij}\|_{L^{1}L^{\infty}}\|S_{\lambda}u\|_{L^{\infty}L^{2}}$
$\displaystyle\lesssim\lambda\big{(}\kappa^{\frac{1}{2}}\lambda^{\frac{2}{2+s}}\big{)}^{-s}\|\varepsilon_{ij}\|_{\mathcal{X}^{s}}\|S_{\lambda}u\|_{L^{\infty}L^{2}}$
$\displaystyle\lesssim\kappa\lambda^{\sigma}\|S_{\lambda}u\|_{L^{\infty}L^{2}},$

where the ultimate inequality follows from the $\mathcal{X}^{s}$-bound for
$\varepsilon$ and $U_{\nu}$ was defined in (51). To apply Theorem 1.3, we also
note that

$\|\partial^{2}_{x}(S_{\lesssim\kappa^{\frac{1}{2}}\lambda^{\frac{2}{2+s}}}\varepsilon^{\lambda}_{ij})\|_{L^{1}L^{\infty}}\lesssim\kappa^{\frac{2-s}{2}}\lambda^{\frac{2(2-s)}{2+s}}\|\varepsilon_{ij}\|_{\mathcal{X}^{s}}\lesssim(\kappa\lambda^{\sigma})^{2},$

where $\kappa\lambda^{\sigma}\geq 1$. Now, (69) is a consequence of Theorem
1.3. ∎

Finally, we discuss how the inhomogeneous estimates given in Theorem 1.5 can
be deduced from Theorem 1.1 and the analysis in [38].

###### Proof of Theorem 1.5.

By rescaling it is enough to consider the case

$\kappa=1,\qquad\|\partial^{2}_{x}\varepsilon\|_{\infty}\leq 1.$

Then we can modify the arguments in the proof of Theorem 1.1 to reduce the
desired inequality to $u$ with support in a cube of size $1$, to localize it
to a dyadic estimate at frequency $\lambda\gtrsim 1$ and for the region
$\\{|\xi_{0}|\lesssim|\xi^{\prime}|\\}$, and to truncate the coefficients at
frequency $\lambda^{\frac{1}{2}}$. Since the diagonalization
$P=\mathcal{M}\mathcal{D}\mathcal{N}$ essentially respects frequency
localization and $L^{p}$-properties, we finally reduce to the inequality

(71)
$\lambda^{-\rho}\|S_{\lambda}u\|_{L^{p}L^{q}}\lesssim\|S_{\lambda}u\|_{L^{2}}+\|\tilde{f}^{\lambda}_{1}\|_{L^{2}}+\lambda^{\rho}\|\tilde{f}^{\lambda}_{2}\|_{L^{p^{\prime}}L^{q^{\prime}}}+\lambda^{-\frac{1}{2}}\|S_{\lambda}\rho_{e}\|_{L^{2}},$

where
$\mathcal{D}(x,D)S_{\lambda}u=\tilde{f}^{\lambda}_{1}+\tilde{f}^{\lambda}_{2}$.
The degenerate component of $\mathcal{D}$ is handled as in Paragraph 3.2.4. We
sketch the reduction to the above display. Using Proposition 3.1, we can write
up to error terms of order $O_{L^{2}}(\lambda^{-N})$:444We suppress frequency
localization and slightly enlarged variants between the pseudo-differential
operators to lighten the notation.

$PS_{\lambda}=\mathcal{M}\mathcal{D}\mathcal{N}S_{\lambda}+E_{\lambda}$

with $\|E_{\lambda}\|_{L^{2}\to L^{2}}\lesssim 1$. Let $v$ denote the original
function and $S_{\lambda}\mathcal{N}S_{\lambda}v=\tilde{S}_{\lambda}u$. We let
$\mathcal{M}S_{\lambda}\mathcal{D}S_{\lambda}\mathcal{N}S_{\lambda}v=f_{1}^{\lambda}+f_{2}^{\lambda}+g^{\lambda}$
with $\|g^{\lambda}\|_{L^{2}}\lesssim\|S_{\lambda}v\|_{L^{2}}$. This requires

$\mathcal{M}\tilde{f}_{1}^{\lambda}=f_{1}^{\lambda}+g^{\lambda},\qquad\mathcal{M}\tilde{f}_{2}^{\lambda}=f_{2}^{\lambda}.$

Multiplication with $\mathcal{N}$ gives (cf. Proposition 2.4)

$\tilde{f}_{1}^{\lambda}+E_{1}^{\lambda}\tilde{f}_{1}^{\lambda}=\mathcal{N}f_{1}^{\lambda}+\mathcal{N}g^{\lambda},\quad\tilde{f}_{2}^{\lambda}+E_{2}^{\lambda}\tilde{f}_{2}^{\lambda}=\mathcal{N}f_{2}^{\lambda},$

where we have

$\|E_{1}^{\lambda}\|_{L^{2}\to L^{2}}\lesssim\lambda^{-1}\quad\text{ and
}\quad\|E_{2}^{\lambda}\|_{L^{p^{\prime}}L^{q^{\prime}}\to
L^{p^{\prime}}L^{q^{\prime}}}\lesssim\lambda^{-1}$

by Lemma 2.3. Hence, the equations for $\tilde{f}_{1}^{\lambda}$ and
$\tilde{f}_{2}^{\lambda}$ can be solved in $L^{2}$ and
$L^{p^{\prime}}L^{q^{\prime}}$, respectively, by the Neumann series, yielding
the estimates

$\|\tilde{f}_{1}^{\lambda}\|_{L^{2}}\lesssim\|f_{1}^{\lambda}\|_{L^{2}}+\|S_{\lambda}v\|_{L^{2}},\quad\|\tilde{f}_{2}^{\lambda}\|_{L^{p^{\prime}}L^{q^{\prime}}}\lesssim\|f_{2}^{\lambda}\|_{L^{p^{\prime}}L^{q^{\prime}}}.$

We abbreviate $\tilde{f}_{i}=\tilde{f}_{i}^{\lambda}$ in the following.

As before, let

$v_{\lambda}=T_{\lambda}S_{\lambda}u_{i},\ \ i=2,3,\quad\text{ and }\quad
q(x,\xi)=\xi_{0}-\big{(}\tilde{\varepsilon}^{ij}_{\lambda^{\frac{1}{2}}}(x)\xi_{i}\xi_{j}\big{)}^{1/2}.$

An application of Theorem 2.1 shows that (cf. [38, p. 418])

$v_{\lambda}\in L^{2}_{\Phi},\qquad[\lambda
q+2(\bar{\partial}q)(\partial-i\lambda\xi)]v_{\lambda}-T_{\lambda}\tilde{f}_{2}\in
L^{2}_{\Phi}.$

Now set

$w=\Phi^{1/2}v_{\lambda},\qquad g=\Phi^{1/2}T_{\lambda}\tilde{f}_{2}.$

The properties of the ODE along the Hamilton flow yield

$\lambda^{-1/4}w\in L^{2}(K\cap U),\qquad\lambda^{-1/4}(H_{q}w-g)\in
L^{2}(K\cap U).$

As above, we use the ODE along the gradient flow of $q$ to decompose
$w=w_{1}+w_{2}$, where

$\displaystyle[(q_{x}\partial_{\xi}+q_{\xi}\partial_{x})+\lambda(q-\xi\cdot
q_{\xi})]w_{1}$ $\displaystyle=f,\qquad w_{1}\big{|}_{K}=0,$
$\displaystyle[(q_{x}\partial_{x}+q_{\xi}\partial_{\xi})+\lambda(q-i\xi\cdot
q_{x}]w_{2}$ $\displaystyle=0,\qquad w_{2}\big{|}_{K}=w.$

The function $w_{2}$ can further be decomposed into “good” and “bad” parts,

$w_{2}=w_{2}^{g}+w_{2}^{b},$

whose traces on the cone $K\cap U$ satisfy

$\lambda^{-1/4}H_{q}w_{2}^{g}\in L^{2},\qquad H_{q}w_{2}^{b}=g.$

The map $w_{2}^{g}$ can be treated as $w_{2}$ in Paragraph 4.1.4. In a similar
way $w_{2}^{b}$ can be controlled by $g$, and thus $\tilde{f}_{2}$, as
indicated on p. 419 of [38]. Analogously, we decompose $w_{1}$ as

$w_{1}=w_{1}^{g}+w_{1}^{b},$

and solve

$(q_{x}\partial_{x}+q_{\xi}\partial_{\xi}+\lambda(q-iq_{x}\cdot\xi))w_{1}^{g}\in
L^{2},\qquad w_{1}^{g}\big{|}_{K}=0,$

respectively,

$(q_{x}\partial_{x}+q_{\xi}\partial_{\xi}+\lambda(q-iq_{x}\cdot\xi))w_{1}^{b}=g,\qquad
w_{1}^{b}\big{|}_{K}=0.$

Again, $w_{1}^{g}$ can be treated as in Paragraph 4.1.4. For the estimate of
$w_{1}^{b}$ we refer to [38, pp. 419–422]. The argument applies due to the
Lipschitz equivalence of wave and half-wave symbol in the phase space region
of interest, which was already discussed above. ∎

We next point out how homogeneous Strichartz estimates yield inhomogeneous
Strichartz estimates. For this we invoke the following consequence of the
Christ–Kiselev lemma (cf. [6]).

######  Lemma 5.1 ([11, Lemma 8.1]).

Let $X$ and $Y$ be Banach spaces and for all $s,t\in\mathbb{R}$ let
$K(s,t):X\to Y$ be an operator-valued kernel from $X$ to $Y$. Suppose we have
the estimate

$\|\int_{\mathbb{R}}K(s,t)f(s)ds\|_{L^{q}(\mathbb{R},Y)}\leq
A\|f\|_{L^{p}(\mathbb{R},X)}$

for some $A>0$ and $1\leq p<q\leq\infty$, and $f\in L^{p}(\mathbb{R};X)$.
Then, we have

$\|\int_{s<t}K(s,t)f(s)ds\|_{L^{q}(\mathbb{R},Y)}\leq
C_{p,q}A\|f\|_{L^{p}(\mathbb{R},X)}.$

We are ready for the proof of Corollary 1.6.

###### Proof of Corollary 1.6.

By well-posedness in $L^{2}$ for $C^{1}$-coefficients, let
$(U(t,s))_{t,s\in\mathbb{R}}$ denote the propagator of $P$ in $L^{2}$. So
$U(t,s)u_{0}$ is the solution at time $t$ to

$\left\\{\begin{aligned} Pu&=0,\\\ u(s)&=u_{0}\in L^{2}.\end{aligned}\right.$

We let $Pu(t)=(\partial_{t}1_{3\times 3}-A(t))u(t)$ so that $A(t)$ denotes the
time-dependent generator. The full solution is given by Duhamel’s formula

$u(t)=U(t,0)u_{0}+\int_{0}^{t}U(t,s)f(s)ds.$

Let $\mathcal{T}:L^{2}\to L^{p}([0,T],L^{q})$, $\mathcal{T}f=\langle
D^{\prime}\rangle^{-\rho-\frac{\sigma}{2}}U(t,0)f$. The estimates from Theorem
1.2 applied to homogeneous solutions for $\varepsilon\in C^{s}$, $1\leq s\leq
2$, yield

$\|\mathcal{T}u_{0}\|_{L^{p}(0,T;L^{q})}\lesssim_{\delta}\|U(t,0)u_{0}\|_{L^{2}}\lesssim_{T,\|\varepsilon\|_{C^{1}}}\|u(0)\|_{L^{2}}.$

Indeed, the full derivatives can be replaced by the purely spatial derivatives
as a consequence of microlocal estimates since the coefficients of $P$ belong
to $C^{1}$, and the energy estimate holds true because
$\|\partial_{x}\varepsilon\|_{L^{1}L^{\infty}}\lesssim_{T}\|\varepsilon\|_{C^{1}}$.
We aim to estimate the Duhamel term

$\langle
D^{\prime}\rangle^{-\rho-\frac{\sigma}{2}}\int_{0}^{t}U(t,s)f(s)ds=\langle
D^{\prime}\rangle^{-\rho-\frac{\sigma}{2}}U(t,0)\int_{0}^{t}U(0,s)f(s)ds.$

By invoking Lemma 5.1, it suffices to estimate

(72) $\|\langle
D^{\prime}\rangle^{-\rho-\frac{\sigma}{2}}U(t,0)\int_{0}^{T}U(0,s)f(s)ds\|_{L^{p}L^{q}}\lesssim\|\langle
D^{\prime}\rangle^{\tilde{\rho}+\frac{\sigma}{2}}f\|_{L^{\tilde{p}^{\prime}}L^{\tilde{q}^{\prime}}}.$

To prove boundedness of (72), we use the $\mathcal{T}\mathcal{T}^{*}$-argument
with duality with respect to $L^{2}$. Note that

$\mathcal{T}^{*}F=\int_{0}^{T}U(s,0)^{*}\big{(}\langle
D^{\prime}\rangle^{-\rho-\frac{\sigma}{2}}f(s)\big{)}ds.$

Hence, to derive estimates for $\int_{0}^{T}U(0,s)f(s)ds$, it suffices to show

$\|\langle
D^{\prime}\rangle^{-\tilde{\rho}-\frac{\sigma}{2}}U(0,t)^{*}g\|_{L^{\tilde{p}}(0,T;L^{\tilde{q}})}\lesssim\|g\|_{L^{2}(\mathbb{R}^{2})}.$

Since $L^{2}$-duality respects divergence-free vector fields, $(u_{1},u_{2})$
is divergence-free, and so is $(f_{1},f_{2})$, we can suppose that
$(g_{1},g_{2})$ is divergence-free.

We compute the time-dependent generator for $U(0,t)^{*}$ by

$\partial_{t}\langle U(0,t)^{*}u,v\rangle=\partial_{t}\langle
u,U(0,t)v\rangle=\langle u,-U(0,t)A(t)v\rangle=-\langle
A(t)^{*}U(0,t)^{*}u,v\rangle$

in the general case $\varepsilon=(\varepsilon^{ij})_{i,j=1,2}$. We thus find
for the full operator

$P^{*}=\begin{pmatrix}-\partial_{t}&0&\varepsilon_{11}\partial_{2}-\varepsilon_{12}\partial_{1}\\\
0&-\partial_{t}&-\varepsilon_{22}\partial_{1}+\varepsilon_{12}\partial_{2}\\\
\partial_{2}&-\partial_{1}&-\partial_{t}\end{pmatrix}=-P^{t}+E$

with $\|E\|_{L^{2}\to L^{2}}\lesssim 1$ for Lipschitz coefficients. Hence, it
suffices to prove estimates for $P^{t}$. These are a consequence of the proof
of Theorem 1.2. The only difference happens in the diagonalization:
Transposing (29)-(31) yields

$(m^{-1})^{t}(x,\xi)d(x,\xi)m^{t}(x,\xi)=p^{t}(x,\xi).$

Note

$m^{t}(x,\xi)=\begin{pmatrix}-\xi_{1}^{*}\varepsilon_{22}(x)+\xi_{2}^{*}\varepsilon_{12}(x)&\xi_{1}^{*}\varepsilon_{12}(x)-\xi_{2}^{*}\varepsilon_{11}(x)&0\\\
\xi_{2}^{*}&-\xi_{1}^{*}&1\\\ -\xi_{2}^{*}&\xi_{1}^{*}&1\end{pmatrix},$

and analogously for the operators in (32)–(34). The estimate for $P^{t}$
corresponding of Theorem 1.2 is

$\|\langle
D^{\prime}\rangle^{-\tilde{\rho}-\frac{\sigma}{2}}u\|_{L^{\tilde{p}}L^{\tilde{q}}}\lesssim_{T,\kappa}\|u\|_{L^{2}}+\|P^{t}u\|_{L^{2}}+\||D^{\prime}|^{-\frac{1}{2}-\frac{\sigma}{2}}\rho_{e}^{*}\|_{L^{2}}$

with $\rho_{e}^{*}=\nabla\cdot(\tilde{\varepsilon}^{-1}\tilde{u})$,
$\tilde{u}=(u_{1},u_{2})$, and the adjugate
$\tilde{\varepsilon}^{-1}=\varepsilon\cdot\det(\varepsilon^{-1})$ of
$\varepsilon^{-1}$.

Consequently, for $U(0,t)^{*}u_{0}$ we find

$\begin{split}\|\langle
D^{\prime}\rangle^{-\tilde{\rho}-\frac{\sigma}{2}}U(0,t)^{*}u_{0}\|_{L^{\tilde{p}}L^{\tilde{q}}}&\lesssim\|U(0,t)^{*}u_{0}\|_{L^{2}}+\|(P^{*}-E)U(0,t)^{*}u_{0}\|_{L^{2}}\\\
&\qquad+\||D^{\prime}|^{-\frac{1}{2}-\frac{\sigma}{2}}\nabla\cdot(\tilde{\varepsilon}^{-1}U(0,t)^{*}u_{0})\|_{L^{2}}\\\
&\lesssim\|u_{0}\|_{L^{2}}+\||D^{\prime}|^{-\frac{1}{2}-\frac{\sigma}{2}}\rho_{e}^{*}\|_{L^{2}},\end{split}$

which follows from the energy estimate in $L^{2}$. For low frequencies, the
energy estimate also allow to dominate the second term in the last line by the
first. For high frequencies, we can take advantage that $\tilde{u}_{0}$ is
divergence free, that
$\rho_{e}^{*}(t)=\nabla\cdot(\tilde{\varepsilon}^{-1}\tilde{u}_{0})$, and that
$\varepsilon$ is isotropic. It follows

$\|S_{\gtrsim
1}^{\prime}(|D^{\prime}|^{-\frac{1}{2}-\frac{\sigma}{2}}\nabla\cdot(\tilde{\varepsilon}^{-1}\tilde{u}))\|_{L^{2}}\lesssim_{\|\varepsilon\|_{C^{1}}}\|u(0)\|_{L^{2}}.$

We have proved for divergence-free $(f_{1},f_{2})$ that

$\|\langle
D^{\prime}\rangle^{-\rho-\frac{\sigma}{2}}\int_{0}^{t}U(t,s)f(s)ds\|_{L^{p}(0,T;L^{q})}\lesssim\|\langle
D^{\prime}\rangle^{\tilde{\rho}+\frac{\sigma}{2}}f\|_{L^{\tilde{p}^{\prime}}(0,T;L^{\tilde{q}^{\prime}})},$

where $(\rho,p,q,2)$, $(\tilde{\rho},\tilde{p},\tilde{q},2)$ are Strichartz
pairs. Note that the case $p=\tilde{p}=2$ cannot be covered by Lemma 5.1, but
$p=2$ is not wave-admissible in two dimensions anyway. We finish the proof by
Duhamel’s formula and the triangle inequality. ∎

In the following we derive Strichartz estimates only with spatial derivatives
for $\|\partial_{x}\varepsilon\|_{L^{2}L^{\infty}}\lesssim 1$. This will
become useful in the quasilinear case where we shall control
$\|\nabla_{x^{\prime}}u\|_{L^{4}L^{\infty}}$ and have
$\|\partial_{x}(\varepsilon(u))\|_{L^{2}L^{\infty}}\lesssim_{\|u\|_{L^{\infty}}}\|\nabla_{x^{\prime}}u\|_{L^{2}L^{\infty}}\lesssim_{T}\|\nabla_{x^{\prime}}u\|_{L^{4}L^{\infty}}$.
Also note that

$\|\partial_{x}\varepsilon\|_{L^{2}L^{\infty}}\lesssim\|\varepsilon\|_{\mathcal{X}^{s}}$

by Sobolev’s embedding for $s>\frac{3}{2}$. Actually, this condition is only
needed in our proof of the microlocal estimate, for which we use the following
commutator estimate for the FBI transform, also due to Tataru [37]. Let
$X=L^{2}L^{\infty}$ and $X^{1}=\\{u\in X:\partial_{x}u\in X\\}$.

######  Theorem 5.2 ([37, Theorem 2]).

Assume that $a\in X^{1}C^{\infty}_{c}$. Then,

$\|R_{\lambda,a}\|_{L^{\infty}L^{2}\to
L^{2}_{\Phi}}=\|T_{\lambda}A_{\lambda}-aT_{\lambda}\|_{L^{\infty}L^{2}\to
L^{2}_{\Phi}}\lesssim_{\|a\|_{X^{1}C^{\infty}_{c}}}\lambda^{-\frac{1}{2}}.$

We can now show Corollary 1.7.

###### Proof of Corollary 1.7.

Let $S_{\lambda}^{\ll\tau}$ denote the smooth frequency localization to
regions $\\{|\xi|\sim\lambda\wedge|\xi^{\prime}|\ll\lambda\\}$. In the
following we derive a favorable estimate of $\|\langle
D^{\prime}\rangle^{-\rho-\frac{\sigma}{p}}S_{\lambda}^{\ll\tau}u\|_{L^{p}L^{q}}$
for $\lambda\gg 1$, $(\rho,p,q,2)$ a Strichartz pair, and $p\neq\infty$. As
symbol at unit frequencies we consider

$p(x,\xi)=i\begin{pmatrix}\xi_{0}&0&-\xi_{2}\\\ 0&\xi_{0}&\xi_{1}\\\
-\varepsilon_{11}(x)\xi_{2}+\varepsilon_{21}(x)\xi_{1}&\varepsilon_{22}(x)\xi_{1}-\varepsilon_{12}(x)\xi_{2}&\xi_{0}\end{pmatrix}s_{0}(\xi)$

with $s_{0}\in C^{\infty}_{c}(B(0,2))$, $s_{0}=1$ for
$1\sim|\xi|\sim|\xi_{0}|\gg|\xi^{\prime}|$ such that
$S_{\lambda}^{\ll\tau}=s_{0}(\xi/\lambda)$.

By Hölder’s inequality and Sobolev embedding, we find

$\begin{split}\|\langle
D^{\prime}\rangle^{-\rho-\frac{\sigma}{p}}S_{\lambda}^{\ll\tau}u\|_{L^{p}L^{q}}&\lesssim\lambda^{\frac{1}{2}-\frac{\sigma}{p}}\|S_{\lambda}^{\ll\tau}u\|_{L^{2}}\\\
&=\lambda^{\frac{1}{2}-\frac{\sigma}{p}}\|T^{*}_{\lambda}p^{-1}(x,\xi)T_{\lambda}\,T_{\lambda}^{*}p(x,\xi)T_{\lambda}\,S_{\lambda}^{\ll\tau}u\|_{L^{2}}\\\
&\lesssim\lambda^{\frac{1}{2}-\frac{\sigma}{p}}\|p(x,\xi)T_{\lambda}S_{\lambda}^{\ll\tau}u\|_{L^{2}}\\\
&\lesssim\lambda^{\frac{1}{2}-\frac{\sigma}{p}}\|R_{\lambda,p}\|_{L^{\infty}L^{2}\to
L^{2}}\|S_{\lambda}^{\ll\tau}u\|_{L^{\infty}L^{2}}\\\
&\quad+\lambda^{-\frac{1}{2}-\frac{\sigma}{p}}\big{(}\|P(x,D)S_{\lambda}^{\ll\tau}u\|_{L^{2}}+\|\partial_{x}\varepsilon\|_{L^{2}L^{\infty}}\|S_{\lambda}^{\ll\tau}u\|_{L^{\infty}L^{2}}\big{)}.\end{split}$

Since $\|R_{\lambda,p}\|_{L^{\infty}L^{2}\to
L^{2}}\lesssim_{\|\partial_{x}\varepsilon\|_{L^{2}L^{\infty}}}\lambda^{-\frac{1}{2}}$
by Theorem 5.2, the first term is acceptable. The third term comes from
recovering divergence form and is clearly admissible. Also note that we
omitted an error term from localizing $T_{\lambda}S_{\lambda}^{\ll\tau}u$ to
$\\{|\xi_{0}|\sim 1,\;|\xi^{\prime}|\ll 1\\}$ as this gains arbitrarily many
derivatives. We turn to the second term. For this purpose let
$P^{\lambda}(x,D)$ denote $P$ with coefficients frequency truncated at
$\lambda$. The coefficients of $P^{\lambda}$ have Lipschitz norm
$\lesssim\lambda^{\frac{1}{2}}$. Note that

$\begin{split}\|P(x,D)S_{\lambda}^{\ll\tau}u\|_{L^{2}}&\leq\|P^{\lambda}(x,D)S_{\lambda}^{\ll\tau}u\|_{L^{2}}+\|P^{\gtrsim\lambda}(x,D)S_{\lambda}^{\ll\tau}u\|_{L^{2}}\\\
&\lesssim\|P^{\lambda}(x,D)S_{\lambda}^{\ll\tau}u\|_{L^{2}}+\|\partial_{x}\varepsilon\|_{L^{2}L^{\infty}}\|u\|_{L^{\infty}L^{2}}.\end{split}$

Moreover, a kernel estimate yields

$\begin{split}\|P^{\lambda}S_{\lambda}^{\ll\tau}u\|_{L^{2}}&\leq\|S_{\lambda}^{\ll\tau}P^{\lambda}u\|_{L^{2}}+\|[P^{\lambda},S_{\lambda}^{\ll\tau}]u\|_{L^{2}}\\\
&\lesssim\|S_{\lambda}^{\ll\tau}P^{\lambda}u\|_{L^{2}}+\lambda^{\frac{1}{2}}\|u\|_{L^{2}}\\\
&\lesssim\|S_{\lambda}^{\ll\tau}Pu\|_{L^{2}}+\|S_{\lambda}^{\ll\tau}P^{\gtrsim\lambda}u\|_{L^{2}}+\lambda^{\frac{1}{2}}\|u\|_{L^{2}}.\end{split}$

For the first term we use Bernstein’s inequality in time

$\|S_{\lambda}^{\ll\tau}Pu\|_{L^{2}}\lesssim\lambda^{\frac{1}{2}}\|S_{\lambda}^{\ll\tau}Pu\|_{L^{1}L^{2}}$

and note that it is still summable in $\lambda$ due to the additional factor
$\lambda^{-\frac{\sigma}{p}}$ and since we only want to estimate $\|\langle
D^{\prime}\rangle^{-\alpha}u\|_{L^{p}L^{q}}$ for
$\alpha>\rho+\frac{\sigma}{p}$ in Corollary 1.7. Likewise, the third term can
be summed. For the second term we note

$\|S_{\lambda}^{\ll\tau}P^{\gtrsim\lambda}u\|_{L^{2}}\lesssim\|\partial_{x}\varepsilon\|_{L^{2}L^{\infty}}\|u\|_{L^{\infty}L^{2}}.$

This handles the part with spatial frequencies much smaller than temporal
ones.

For the dyadic frequency blocks $\\{|\tau|\sim|\xi^{\prime}|\sim\lambda\\}$ we
can use Theorem 1.4 with $P^{\lambda}$ to recover the dyadic estimate (69):

$\begin{split}\lambda^{-\rho-\frac{\sigma}{p}}\|S_{\lambda}S_{\lambda}^{\prime}u\|_{L^{p}L^{q}}&\lesssim\|S_{\lambda}u\|_{L^{\infty}L^{2}}+\lambda^{-\sigma}\|P^{\lambda}S_{\lambda}S^{\prime}_{\lambda}u\|_{L^{1}L^{2}}\\\
&\quad+\lambda^{-\frac{1}{2}-\frac{\sigma}{p}}\|S^{\prime}_{\lambda}\rho_{e}(0)\|_{L^{2}}+\lambda^{-\frac{1}{2}-\frac{\sigma}{p}}\|S^{\prime}_{\lambda}\partial_{t}\rho_{e}\|_{L^{1}L^{2}}\\\
&\lesssim\|u\|_{L^{\infty}L^{2}}+\lambda^{-\sigma}\|S_{\lambda}Pu\|_{L^{1}L^{2}}\\\
&\quad+\lambda^{-\frac{1}{2}-\frac{\sigma}{p}}\|S^{\prime}_{\lambda}\rho_{e}(0)\|_{L^{2}}+\lambda^{-\frac{1}{2}-\frac{\sigma}{p}}\|S^{\prime}_{\lambda}\partial_{t}\rho_{e}\|_{L^{1}L^{2}}.\end{split}$

We used the same commutator considerations as in the proof of Theorem 1.4 and
the fundamental theorem of calculus together with Minkowski’s inequality. By
the energy estimate, we conclude for $\alpha>\rho+\frac{\sigma}{p}$

$\begin{split}\|\langle
D^{\prime}\rangle^{-\alpha}u\|_{L^{p}(0,T;L^{q})}&\lesssim\|u_{0}\|_{L^{2}}+\|Pu\|_{L^{1}L^{2}}\\\
&\quad+\|\langle
D^{\prime}\rangle^{-\frac{1}{2}-\frac{\sigma}{p}}\rho_{e}(0)\|_{L^{2}}+\|\langle
D^{\prime}\rangle^{-\frac{1}{2}-\frac{\sigma}{p}}\partial_{t}\rho_{e}\|_{L^{1}L^{2}},\end{split}$

and the proof is complete. ∎

##  6\. Local well-posedness for a 2d quasilinear Maxwell system

This section is devoted to the proof of Theorem 1.9. Recall that the system
under consideration is given by

(73) $\left\\{\begin{array}[]{cl}\partial_{t}u_{1}&=\partial_{2}u_{3},\qquad
u(0)=u_{0}\in H^{s}(\mathbb{R}^{2};\mathbb{R})^{3},\\\
\partial_{t}u_{2}&=-\partial_{1}u_{3},\qquad\partial_{1}u_{1}+\partial_{2}u_{2}=0,\\\
\partial_{t}u_{3}&=\partial_{2}(\varepsilon^{-1}(u)u_{1})-\partial_{1}(\varepsilon^{-1}(u)u_{2}),\end{array}\right.$

where $\varepsilon^{-1}(u)=\psi(|u_{1}|^{2}+|u_{2}|^{2})$ and
$\psi:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ is smooth, monotone
increasing and satisfies $\psi(0)=1$. We denote $\tilde{u}=(u_{1},u_{2})$.

Without using dispersive properties, energy methods yield local well-posedness
in $H^{s}(\mathbb{R}^{2})$ for $s>2$. To improve on this by Strichartz
estimates, we follow the arguments from Ifrim–Tataru [14]. Let $A=\sup_{0\leq
t^{\prime}\leq t}\|u(t^{\prime})\|_{L^{\infty}_{x^{\prime}}}$ and
$B(t)=\|\nabla_{x^{\prime}}u(t)\|_{L^{\infty}_{x^{\prime}}}$. In the following
we take local existence of smooth solutions for granted, which follows by
classical arguments, e.g., parabolic regularization. We focus on proving
estimates in rough norms. The argument consists of three steps:

  1. 1)

We prove energy estimates

(74) $E^{s}(u(t))\lesssim
e^{c(A)\int_{0}^{t}B(t^{\prime})dt^{\prime}}E^{s}(u(0)),$

for smooth solutions $u$, where $E^{s}(u)\approx_{A}\|u\|_{H^{s}}$ and $s\geq
0$.

  2. 2)

We show $L^{2}$-Lipschitz bounds for differences of solutions $v=u^{1}-u^{2}$

(75) $\|v(t)\|^{2}_{L^{2}}\lesssim
e^{c(A)\int_{0}^{t}B(t^{\prime})dt^{\prime}}\|v(0)\|^{2}_{L^{2}},$

where $u^{1}$ and $u^{2}$ solve (73) and

$\displaystyle A$ $\displaystyle=\sup_{0\leq t^{\prime}\leq
t}\|u^{1}(t^{\prime})\|_{L^{\infty}_{x^{\prime}}}+\sup_{0\leq t^{\prime}\leq
t}\|u^{2}(t^{\prime})\|_{L^{\infty}_{x^{\prime}}},$ $\displaystyle B(t)$
$\displaystyle=\|\nabla_{x^{\prime}}u^{1}(t)\|_{L^{\infty}_{x^{\prime}}}+\|\nabla_{x^{\prime}}u^{2}(t)\|_{L^{\infty}_{x^{\prime}}}.$

  3. 3)

We conclude the proof of Theorem 1.9, using frequency envelopes. These were
introduced by Tao in the context of wave maps [36] and turned out as very
useful to treat quasilinear evolution equations.

We start with energy estimates.

######  Proposition 6.1.

Let $s\geq 0$. Then, we find (74) to hold. Let $s>11/6$. For $u_{0}\in H^{s}$,
there is a time $T=T(\|u_{0}\|_{H^{s}})$ such that $T$ is lower semicontinuous
and

$\sup_{t\in[0,T]}\|u(t)\|_{H^{s}}\lesssim\|u_{0}\|_{H^{s}}.$

###### Proof.

We consider energy norms

$\|u\|^{2}_{E^{s}}=\langle\langle D^{\prime}\rangle^{s}u,C(u)\langle
D^{\prime}\rangle^{s}u\rangle\approx_{A}\|u\|^{2}_{H^{s}},$

where we define the smooth map $C$ in (78) below. We rewrite (73) as

(76) $\partial_{t}u=\mathcal{A}^{j}(u)\partial_{j}u,$

with the coefficient matrices

$\displaystyle\mathcal{A}^{1}(u)$ $\displaystyle=\begin{pmatrix}0&0&0\\\
0&0&-1\\\
-2\psi^{\prime}(|\tilde{u}|^{2})u_{1}u_{2}&-2\psi^{\prime}(|\tilde{u}|^{2})u_{2}^{2}-\psi(|\tilde{u}|^{2})&0\end{pmatrix},$
$\displaystyle\mathcal{A}^{2}(u)$ $\displaystyle=\begin{pmatrix}0&0&1\\\
0&0&0\\\
2\psi^{\prime}(|\tilde{u}|^{2})u_{1}^{2}+\psi(|\tilde{u}|^{2})&2\psi^{\prime}(|\tilde{u}|^{2})u_{1}u_{2}&0\end{pmatrix}.$

For the time derivative we find

(77) $\displaystyle\frac{d}{dt}\|u\|^{2}_{E^{s}}$
$\displaystyle=\langle\langle
D^{\prime}\rangle^{s}(\mathcal{A}^{j}(u)\partial_{j}u),C(u)\langle
D^{\prime}\rangle^{s}u\rangle+\langle\langle
D^{\prime}\rangle^{s}u,C(u)\langle
D^{\prime}\rangle^{s}(\mathcal{A}^{j}(u)\partial_{j}u)\rangle$
$\displaystyle\quad+\langle\langle
D^{\prime}\rangle^{s}u,C^{\prime}(u)(\partial_{t}u)\langle
D^{\prime}\rangle^{s}u\rangle,$

at time $t$, which is suppressed. Because of (76), the last term is estimated
by

$|\langle D^{\prime}\rangle^{s}u,\tilde{C}(u)(\partial_{t}u)\langle
D^{\prime}\rangle^{s}u\rangle|\lesssim c(A)B\|u\|^{2}_{H^{s}}.$

The first term in (77) can be expressed as

$\displaystyle\langle\langle D^{\prime}\rangle^{s}$
$\displaystyle(\mathcal{A}^{j}(u)\partial_{j}u),C(u)\langle
D^{\prime}\rangle^{s}u\rangle$
$\displaystyle=\langle\mathcal{A}^{j}(u)(\langle
D^{\prime}\rangle^{s}\partial_{j}u)+(\langle
D^{\prime}\rangle^{s}\mathcal{A}^{j}(u)-\mathcal{A}^{j}(u)\langle
D^{\prime}\rangle^{s})\partial_{j}u,C(u)\langle D^{\prime}\rangle^{s}u\rangle$
$\displaystyle=\langle\mathcal{A}^{j}(u)(\langle
D^{\prime}\rangle^{s}\partial_{j}u),C(u)\langle
D^{\prime}\rangle^{s}u\rangle+II.$

Paraproduct/Moser estimates yield

$II\leq\|(\langle
D^{\prime}\rangle^{s}\mathcal{A}^{j}(u)-\mathcal{A}^{j}(u)\langle
D^{\prime}\rangle^{s})\partial_{j}u\|_{L^{2}}\|C(u)\langle
D^{\prime}\rangle^{s}u\|_{L^{2}}\lesssim_{A}B\|u\|^{2}_{H^{s}}.$

Integrating by parts, the other summand becomes

$\begin{split}\langle\mathcal{A}^{j}(u)&\langle
D^{\prime}\rangle^{s}\partial_{j}u,C(u)\langle
D^{\prime}\rangle^{s}u\rangle_{L^{2}}\\\ &=-\langle\langle
D^{\prime}\rangle^{s}u,\partial_{j}(\mathcal{A}_{j}(u)^{*}C(u)\langle
D^{\prime}\rangle^{s}u)\rangle_{L^{2}}\\\ &=-\langle\langle
D^{\prime}\rangle^{s}u,\mathcal{A}^{j}(u)^{*}C(u)\langle
D^{\prime}\rangle^{s}\partial_{j}u\rangle_{L^{2}}+\mathcal{O}_{A}(B\|u\|^{2}_{H^{s}}).\end{split}$

The second summand in (77) can be treated analogously. We seek to cancel the
highest-order terms. This gives the condition

$\mathcal{A}^{j}(u)^{*}C(u)=C(u)^{*}\mathcal{A}^{j}(u).$

Solving the system of equations and setting $C_{33}=1$, we find

(78) $C(u)=\begin{pmatrix}\psi+2\psi^{\prime}\cdot
u_{1}^{2}&2\psi^{\prime}\cdot u_{1}u_{2}&0\\\ 2\psi^{\prime}\cdot
u_{1}u_{2}&\psi+2\psi^{\prime}\cdot u_{2}^{2}&0\\\ 0&0&1\end{pmatrix}.$

We thus conclude

$\frac{d}{dt}\|u(t)\|^{2}_{E^{s}}\lesssim_{A}B(t)\|u(t)\|^{2}_{E^{s}},$

and Grønwall’s lemma implies (74). To check that

$\|u\|^{2}_{H^{s}}\approx_{A}\|u\|^{2}_{E^{s}},$

it is enough to show that $C$ is uniformly elliptic. This follows from Young’s
inequality in the computation

$\displaystyle\Big{\langle}\begin{pmatrix}\xi_{1}\\\ \xi_{2}\end{pmatrix}$
$\displaystyle,\begin{pmatrix}\psi+2\psi^{\prime}\cdot
u_{1}^{2}&2\psi^{\prime}\cdot u_{1}u_{2}\\\ 2\psi^{\prime}\cdot
u_{1}u_{2}&\psi+2\psi^{\prime}\cdot
u_{2}^{2}\end{pmatrix}\begin{pmatrix}\xi_{1}\\\
\xi_{2}\end{pmatrix}\Big{\rangle}$ $\displaystyle\geq(\psi+2\psi^{\prime}\cdot
u_{1}^{2})\xi_{1}^{2}-4\psi^{\prime}\cdot(\frac{u_{1}^{2}\xi_{1}^{2}}{2}+\frac{u_{2}^{2}\xi_{2}^{2}}{2})+(\psi+2\psi^{\prime}\cdot
u_{2}^{2})\xi_{2}^{2}$ $\displaystyle=\psi\cdot|\xi|^{2}.$

The second claim of the proposition is a consequence of Sobolev’s embedding
for $s>2$ . To improve on this, we use Strichartz estimates and show

(79)
$\|\nabla_{x^{\prime}}u\|_{L^{4}(0,T;L^{\infty})}\lesssim\|u_{0}\|_{H^{s}}$

for $s>11/6$. To bootstrap, we require that
$\|\nabla_{x^{\prime}}u\|_{L^{4}(0,T_{0};L^{\infty})}\leq K$ for a fixed
number $K>0$ and a maximally defined time $T_{0}>0$. Take $T\in(0,T_{0})$ with
$\|\partial_{x}\varepsilon\|_{L^{2}(0,T;L^{\infty})}\lesssim_{A}T^{1/4}K\leq
1$ and
$\|\partial_{x}\varepsilon\|_{L^{1}(0,T;L^{\infty})}\lesssim_{A}T^{\frac{3}{4}}K\leq
1$. We thus have uniform constants in the energy inequality (74) and in the
Strichartz estimate

(80) $\|\langle
D^{\prime}\rangle^{-\alpha}w\|_{L^{p}(0,T;L^{q})}\lesssim\|w_{0}\|_{L^{2}}+\|P(x,D)w\|_{L^{1}L^{2}},$

for $\alpha>\rho+\frac{1}{3p}$ from Corollary 1.7 with $\tilde{s}=1$, if
$\partial_{1}w_{1}+\partial_{2}w_{2}=0$. For low frequencies, Bernstein’s
inequality and (74) yield

$\|S^{\prime}_{\lesssim
1}\nabla_{x^{\prime}}u\|_{L^{4}L^{\infty}}\lesssim_{A}T^{\frac{1}{4}}\|u_{0}\|_{L^{2}}.$

For high frequencies, we define the auxiliary function $v=\langle
D^{\prime}\rangle^{s}u$ satisfying $\partial_{1}v_{1}+\partial_{2}v_{2}=0$.
Similar as above, a fixed time estimate gives

(81) $\|P(x,u,D)v(t)\|_{L^{2}}=\|[P(x,u,D),\langle
D^{\prime}\rangle^{s}]u(t)\|_{L^{2}}\lesssim_{A}\|\nabla
u(t)\|_{L_{x^{\prime}}^{\infty}}\|u(t)\|_{H^{s}},$

where $P(x,u,D)$ has coefficients $\varepsilon(u)^{-1}$. For the Strichartz
pair $(3/4,4,\infty,2)$, inequality (80) implies

$\displaystyle\|S_{\geq 1}^{\prime}\nabla_{x^{\prime}}u\|_{L^{4}L^{\infty}}$
$\displaystyle\lesssim\|\langle
D^{\prime}\rangle^{1-s}v\|_{L^{4}L^{\infty}}\lesssim\|v_{0}\|_{L^{2}}+\|P(x,u,D)v\|_{L^{1}L^{2}},$

since $s>1+\rho+\frac{1}{3p}=\frac{11}{6}$ by our assumption. By virtue of
(81) and (74), we conclude

$\displaystyle\|S_{\geq 1}^{\prime}\nabla_{x^{\prime}}u\|_{L^{4}L^{\infty}}$
$\displaystyle\lesssim_{A}\|u_{0}\|_{H^{s}}+\|\nabla_{x^{\prime}}u\|_{L^{1}L^{\infty}_{x^{\prime}}}\|u\|_{L^{\infty}H^{s}}$
$\displaystyle\lesssim\|u_{0}\|_{H^{s}}+T^{\frac{4}{3}}\|\nabla_{x^{\prime}}u\|_{L^{4}L_{x^{\prime}}^{\infty}}\|u\|_{L^{\infty}H^{s}}$
$\displaystyle\lesssim\|u_{0}\|_{H^{s}}+T^{\frac{4}{3}}\|\nabla_{x^{\prime}}u\|_{L^{4}L_{x^{\prime}}^{\infty}}\|u_{0}\|_{H^{s}}$

on $[0,T]$ also using the equivalence
$\|u(t)\|_{E^{s}}\approx_{A}\|u(t)\|_{H^{s}}$. Starting with a sufficiently
large $K$, we can now fix a small $T_{1}=T_{1}(\|u_{0}\|_{H^{s}})\in(0,T_{0})$
such that
$\|\nabla_{x^{\prime}}u\|_{L^{4}(0,T_{1};L^{\infty})}\lesssim\|u_{0}\|_{H^{s}}<K$.
∎

We turn to the $L^{2}$-bound for differences.

######  Proposition 6.2.

Let $u^{1}$ and $u^{2}$ be two solutions to (73) with finite $A$ and $B$, and
set $v=u^{1}-u^{2}$. Then, we find (75) to hold. Moreover, if $s>11/6$, there
is a time $T=T(\|u^{i}(0)\|_{H^{s}})$ such that $T$ is lower semicontinuous
and

(82)
$\sup_{t\in[0,T]}\|v(t)\|_{L^{2}}\lesssim_{\|u^{i}(0)\|_{H^{s}}}\|v(0)\|_{L^{2}}.$

###### Proof.

We observe that $v$ solves the equation

$\displaystyle\partial_{t}v$
$\displaystyle=\mathcal{A}^{j}(u^{1})\partial_{j}v+[\mathcal{A}^{1}(u^{1})-\mathcal{A}^{1}(u^{2})]\partial_{1}u^{2}+[\mathcal{A}^{2}(u^{1})-\mathcal{A}^{2}(u^{2})]\partial_{2}u^{2}$
$\displaystyle=\mathcal{A}^{j}(u^{1})\partial_{j}v+\mathcal{B}^{j}(u^{1},u^{2})(v,\partial_{j}u^{2}).$

To prove the claim, we work with the equivalent norm

$\|v\|_{2,u^{1}}^{2}=\langle v,C(u^{1})v\rangle_{2}$

with $C$ from the previous proof. For fixed $t$, we calculate

$\displaystyle\frac{1}{2}\frac{d}{dt}\|v\|^{2}_{2,u^{1}}$
$\displaystyle=\langle\mathcal{A}^{j}(u^{1})\partial_{j}v,C(u^{1})v\rangle+\langle
v,C(u^{1})\mathcal{A}^{j}(u^{1})\partial_{j}v\rangle$
$\displaystyle\quad+\langle\mathcal{B}^{j}(u^{1},u^{2})(v,\partial_{j}u^{2}),C(u^{1})v\rangle+\langle
v,C(u^{1})\mathcal{B}^{j}(u^{1},u^{2})(v,\partial_{j}u^{2})\rangle$
$\displaystyle\quad+\mathcal{O}_{A}(B\|v\|_{2}^{2}),$

where the error accounts for the contribution of the time derivative of
$C(u^{1})$. Furthermore,

$\displaystyle|\langle\mathcal{B}^{j}(u^{1},u^{2})(v,\partial_{j}u^{2}),C(u^{1})v\rangle|$
$\displaystyle\leq\|\mathcal{B}^{j}(u^{1},u^{2})(v,\partial_{j}u^{2})\|_{2}\,\|C(u^{1})v\|_{2}\lesssim_{A}B\|v\|_{2}^{2}.$

The key estimate

$|\langle\mathcal{A}^{j}(u^{1})\partial_{j}v,C(u^{1})v\rangle+\langle
v,C(u^{1})\mathcal{A}^{j}(u^{1})\partial_{j}v\rangle|\lesssim_{A}B\|v\|_{2}^{2}$

is carried out as in the proof of Proposition 6.1, employing our choice of
$C$. Estimate (79) now implies (82). ∎

In the third step, we show continuous dependence by means of the frequency
envelope argument detailed in Ifrim–Tataru [14]. The envelopes represent the
dyadically localized Sobolev energy. We use the following tailored version,
where $P_{k}$ is the standard Littlewood–Paley projector at frequency $2^{k}$.

######  Definition 6.3.

$(c_{k})_{k\geq 0}\in\ell^{2}$ is called a _frequency envelope_ for a function
$u$ in $H^{s}$ if it has the following properties:

  * a)

Energy bound:

$\|P_{k}u\|_{H^{s}}\leq c_{k}.$

  * b)

Slowly varying: There is $\delta>0$ such that for all $j,k\in\mathbb{N}$

$\frac{c_{k}}{c_{j}}\lesssim 2^{-\delta|j-k|}.$

The envelopes are called _sharp_ if they also
$\|u\|^{2}_{H^{s}}\approx\sum_{k}c_{k}^{2}$ for a family of functions.

As noted in [14] such envelopes always exist. The idea is to show that for a
solution not only the $H^{s}$-norm is propagated, but also the frequency
envelope for a time depending on the control parameters. This allows to infer
continuous dependence. We give the details.

_Regularization._ Let $u_{0}\in H^{s}(\mathbb{R}^{2})$ and $(c_{k})_{k\geq 0}$
be a sharp frequency envelope for $u_{0}$ in $H^{s}$. The regularized initial
data $u_{0}^{n}=P_{\leq n}u_{0}$ for $n\in\mathbb{N}$ have the following
properties.

  * i)

Uniform bounds:

$\|P_{k}u_{0}^{n}\|_{H^{s}}\lesssim c_{k},$

  * ii)

High frequency bounds:

$\|u_{0}^{n}\|_{H^{s+j}}\lesssim 2^{jn}c_{n},$

  * iii)

Difference bounds:

$\|u_{0}^{n+1}-u_{0}^{n}\|_{L^{2}}\lesssim 2^{-sn}c_{n},$

  * iv)

Limit as $n\to\infty$:

$u_{0}=\lim_{n\to\infty}u_{0}^{n}\text{ in }H^{s}.$

The regularized initial data give rise to a family of smooth solutions.

Uniform bounds: Propositions 6.1 and 6.2 now yield a time interval of length
$T=T(\|u_{0}\|_{H^{s}})$ on which the solutions exist, and also $L^{2}$-bounds
their differences:

  * i)

High frequency bounds:

$\|u^{n}\|_{C([0,T],H^{s+j})}\lesssim 2^{nj}c_{n},$

  * ii)

Difference bounds:

$\|u^{n+1}-u^{n}\|_{C([0,T],L^{2})}\lesssim 2^{-sn}c_{n}.$

Interpolation implies

$\|u^{n+1}-u^{n}\|_{C([0,T],H^{m})}\lesssim c_{n}2^{-(s-m)n}$

for $m\geq 0$. By the $L^{2}$-bound, we have

$\|u-u^{n}\|_{C([0,T],L^{2})}\lesssim 2^{-sn}.$

This gives convergence in $L^{2}$ since

$u-u^{n}=\sum_{j=n}^{\infty}(u^{j+1}-u^{j}).$

The frequency localization of the summands at $2^{j+1}$ and the error bound
show that

$\|u-u^{n}\|_{C([0,T],H^{s})}\lesssim c_{\geq n}=\big{(}\sum_{j\geq
n}c_{j}^{2}\big{)}^{1/2},$

which gives convergence in $C([0,T],H^{s})$. We can now show Theorem 1.9.

###### Proof of Theorem 1.9.

Consider a sequence of initial data

$u_{0k}\rightarrow u_{0}\text{ \ in }H^{s}(\mathbb{R}^{2}),\quad
s>\frac{11}{6}.$

Propositions 6.1 and 6.2 show that the solutions $u_{k}$ with inital values
$u_{0,k}$ exist on a common time interval $[0,T]$ with
$T=T(\|u_{0}\|_{H^{s}})$ and that $u_{k}$ coverges to $u$ in $C([0,T],L^{2})$.
For the solutions $u^{n}_{k}$ and $u^{n}$ with regularized initial data
$u^{n}_{0k}$ and $u^{n}$, we have $u^{n}_{0,k}\to u^{n}_{0}$ in $H^{s}$ and
hence

$\displaystyle u^{n}_{k}\to u^{n}\text{ \ in }C([0,T],H^{s})$

as $k\to\infty$ by the above reasoning. We then derive

$\displaystyle\|u_{k}-u\|_{C([0,T],H^{s})}$
$\displaystyle\lesssim\|u^{n}_{k}-u^{n}\|_{C([0,T],H^{s})}+\|u^{n}-u\|_{C([0,T],H^{s})}$
$\displaystyle\quad+\|u^{n}_{k}-u_{k}\|_{C([0,T],H^{s})}$
$\displaystyle\lesssim\|u_{k}^{n}-u^{n}\|_{C([0,T],H^{s})}+c_{\geq
n}+c^{k}_{\geq n}.$

The convergence $u_{0k}\to u_{0}$ in $H^{s}$ allows us to choose a sequence of
frequency envelopes $c^{k}\to c$ in $\ell^{2}$. We can thus take $n$ uniformly
in $k$ such that the second and third term become arbitrarily small, and then
let $k\to\infty$. ∎

##  7\. Sharpness of derivative loss

Next, we connect the Maxwell system with a wave equation to infer the
sharpness of derivative loss in Theorem 1.4 for permittivity coefficients
$\varepsilon^{ij}\in C^{s}$ with $1\leq s\leq 2$. For this purpose, we recall
the counterexamples in Smith–Tataru [32]. Smith and Tataru constructed time-
independent $C^{s}$-metrics $g$ for $0\leq s\leq 2$ and solutions $u$ to the
corresponding wave equations, which exhaust a derivative loss in Strichartz
estimates analogous to that proved in Theorem 1.4. They treated the second-
order hyperbolic operator

$Q(t,x,\partial_{t},\partial_{x})=\partial_{t}^{2}-\partial_{i}g^{ij}(t,x)\partial_{j}$

on $[0,1]\times\mathbb{R}^{n}$ with $g^{ij}\in C^{s}$ for $n\geq 2$ and $0\leq
s\leq 2$, and the Strichartz estimates

$\begin{split}\|u\|_{L^{p}(0,1;L^{q})}&\lesssim\|u\|_{L^{\infty}(0,1;H^{\rho^{\prime}})}+\|\partial_{t}u\|_{L^{\infty}(0,1;H^{\rho^{\prime}-1})}+\|Qu\|_{L_{t}^{1}(0,1;H^{\rho^{\prime}-1})},\end{split}$

where $(\rho,p,q,n)$ is a Strichartz pair and

$\rho^{\prime}=\rho+\frac{\sigma}{p},\quad\sigma=\frac{2-s}{2+s},\quad\delta=\frac{2}{2+s}=\frac{1+\sigma}{2}.$

In [32] it is proved that $\rho^{\prime}$ cannot be lowered. For $2/3\leq
s\leq 2$, the authors at first look at the equation

(83)
$Q_{\lambda}(y,\partial_{t},\partial_{x},\partial_{y})=\partial_{t}^{2}-g_{\lambda}(y)\partial_{x}^{2}-\Delta_{y},$

with $(x,y)\in\mathbb{R}\times\mathbb{R}^{n-1}$ and
$g_{\lambda}(y)=1+\lambda^{2\sigma}|y|^{2}$. Later $g_{\lambda}$ is replaced
by $\tilde{g}_{\lambda}(y)=1+\lambda^{2\sigma-2\delta}a(\lambda^{\delta}|y|)$
for a smooth $a\geq 0$ supported in $[0,2)$ with $a(r)=r^{2}$ for $r\in[0,1]$.
The $C^{s}$ norm of $\tilde{g}_{\lambda}$ is uniformly bounded in $\lambda$.

We consider smooth solutions to Maxwell equations in two spatial dimensions

(84)
$\left\\{\begin{array}[]{cl}\partial_{t}\mathcal{D}&=\nabla_{\perp}\mathcal{H},\qquad\partial_{1}\mathcal{D}_{1}+\partial_{2}\mathcal{D}_{2}=\rho_{e},\\\
\partial_{t}\mathcal{H}&=-\nabla\times\mathcal{E}=\partial_{2}\mathcal{E}_{1}-\partial_{1}\mathcal{E}_{2},\end{array}\right.$

with the rough permittivity $\varepsilon_{\lambda}$ given by

$\varepsilon_{\lambda}^{-1}(x,y)=\begin{pmatrix}1&0\\\
0&1+\lambda^{2\sigma}y^{2}\end{pmatrix}$

with $\lambda\geq 2$. Corresponding to the Strichartz estimate above, we look
at

(85)
$\|(\mathcal{D},\mathcal{H})\|_{L^{p}L^{q}}\lesssim\|(\mathcal{D},\mathcal{H})\|_{L^{\infty}H^{\gamma}}+\|P(\mathcal{D},\mathcal{H})\|_{L^{1}H^{\gamma-\sigma}}+\|\rho_{e}\|_{L^{\infty}H^{\gamma-\frac{1}{2}-\frac{\sigma}{p}}}$

with $P$ as in (6), omitting the time interval $(0,1)$. We want to show that
this estimate can only hold for all $u$ if $\gamma\geq\rho+\frac{\sigma}{p}$,
cf. Theorem 1.4.

The field $\mathcal{H}$ from (84) solves the wave equation

$\displaystyle\partial_{t}^{2}\mathcal{H}$
$\displaystyle=(1+\lambda^{2\sigma}y^{2})\partial_{1}^{2}\mathcal{H}+\partial_{2}^{2}\mathcal{H}=:\Delta_{\varepsilon_{\lambda}^{-1}}\mathcal{H}.$

As in [32] we choose

(86)
$\mathcal{H}(t,x,y)=\mathcal{H}^{\lambda}(t,x,y)=\frac{1}{(\log\lambda)^{2}}\int\beta((\log\lambda)^{-2}r)u^{\lambda}_{r}(t,x,y)dr$

for the function

(87)
$u^{\lambda}_{r}(t,x,y)=e^{ir\lambda(t-x)-\frac{i}{2}\lambda^{\sigma}t-\frac{1}{2}r\lambda^{2\sigma}y^{2}}$

with $0\neq\beta\in C^{\infty}_{c}(\mathbb{R})$ satisfying
$\operatorname{supp}\beta\subseteq[1,2]$ and $\beta\geq 0$. As noted in [32],
the function $\mathcal{H}^{\lambda}$ is essentially supported in
$K_{\lambda}^{t}=\\{|t-x|\leq\lambda^{-1}(\log\lambda)^{-2},|y|\leq\lambda^{-\delta}(\log\lambda)^{-1}\\}$.

To solve (84), we set

(88)
$\mathcal{D}^{\lambda}(t,x,y)=\int_{0}^{t}\nabla_{\perp}\mathcal{H}^{\lambda}(s,x,y)ds+C^{\lambda}(x,y)+\varepsilon_{\lambda}\nabla\psi^{\lambda}(x,y)$

with $C^{\lambda}$ and $\psi^{\lambda}$ to be determined.

Below we choose $C^{\lambda}$ with support in

$K_{\lambda}^{0}=\\{|x|\leq\lambda^{-1}(\log\lambda)^{-2},|y|\leq\lambda^{-\delta}(\log\lambda)^{-1}\\}.$

In view of the second line of (84), we compute

$\partial_{t}\mathcal{H}^{\lambda}+\nabla\times(\varepsilon^{-1}_{\lambda}\mathcal{D}^{\lambda})=\partial_{t}\mathcal{H}^{\lambda}(t,x,y)-\int_{0}^{t}\Delta_{\varepsilon_{\lambda}^{-1}}\mathcal{H}^{\lambda}(s,x,y)ds+\nabla\times(\varepsilon_{\lambda}^{-1}C^{\lambda}).$

Inserting the equation

$\partial_{t}^{2}\mathcal{H}^{\lambda}-\Delta_{\varepsilon_{\lambda}^{-1}}\mathcal{H}^{\lambda}=-\tfrac{1}{4}\lambda^{2\sigma}\mathcal{H}^{\lambda}$

from [32], we obtain

$\partial_{t}\mathcal{H}^{\lambda}(t,x,y)-\int_{0}^{t}\Delta_{\varepsilon_{\lambda}^{-1}}\mathcal{H}^{\lambda}(s,x,y)ds=\partial_{t}\mathcal{H}^{\lambda}(0,x,y)-\frac{\lambda^{2\sigma}}{4}\int_{0}^{t}\mathcal{H}^{\lambda}(s,x,y)ds.$

Because of (86) and (87), we can calculate the time integral. Compared to (89)
it gives a lower-order term on the right-hand side (85) as $\lambda\to\infty$.
(Here we use that $2\sigma\leq 1$ since $s\geq 2/3$.) Hence, we seek
$\nabla\times(\varepsilon_{\lambda}^{-1}C^{\lambda})\approx\partial_{t}\mathcal{H}^{\lambda}(0,x,y)$.
We set

$C_{1}^{\lambda}=0,\quad
C_{2}^{\lambda}(x,y)=\varphi(x,y)g_{\lambda}(y)^{-1}\int_{-\infty}^{x}\partial_{t}\mathcal{H}^{\lambda}(0,x^{\prime},y)dx^{\prime},$

with $\varphi$ localized to $B(0,1/2)$, which contains the essential support
of $\partial_{t}\mathcal{H}^{\lambda}(0,x,y)$ for $\lambda\gg 1$. Due to (86)
and (87), one can check that also the term

$r(x,y)=\nabla\times(\varepsilon_{\lambda}^{-1}C^{\lambda})-\partial_{t}\mathcal{H}^{\lambda}(0)$

gives a lower-order term. Using (5) and (6) in [32], we have

$\displaystyle\|\mathcal{H}^{\lambda}\|_{L^{p}L^{q}}$
$\displaystyle\gtrsim\lambda^{-(1+\delta)/q}(\log\lambda)^{-3/q},$ (89)
$\displaystyle\|\mathcal{H}^{\lambda}(t)\|_{H^{\gamma}(\mathbb{R}^{2})}$
$\displaystyle\lesssim\lambda^{\gamma-(1+\delta)/2}(\log\lambda)^{2\gamma-3/2}.$

If we can also prove that

(90)
$\|\rho^{\lambda}_{e}\|_{H^{\gamma-\frac{1}{2}-\frac{\sigma}{p}}}\lesssim\|\mathcal{H}^{\lambda}\|_{H^{\gamma}}\quad\text{
and
}\quad\|\mathcal{D}^{\lambda}\|_{H^{\gamma}}\lesssim\|\mathcal{H}^{\lambda}\|_{H^{\gamma}},$

letting $\lambda\to\infty$ we see that the validity of (85) implies
$\gamma\geq\rho+\frac{\sigma}{p}$. This remains true if we modify
$1+\lambda^{2\sigma}|y|^{2}$ to $\tilde{g}_{\lambda}(y)$ in
$\varepsilon_{\lambda}^{-1}$, since $\mathcal{H}^{\lambda}$ decays fast enough
outside $K_{\lambda}$. (See p. 202 in [32].)

To make the charge as small as possible, in view of (88) we solve
approximately

(91)
$\partial_{1}C_{1}^{\lambda}+\partial_{2}C_{2}^{\lambda}=-(\partial_{1}^{2}+\partial_{2}(1+\lambda^{2\sigma}y^{2})^{-1}\partial_{2})\psi^{\lambda}=-\Delta_{\tilde{\varepsilon}_{\lambda}^{-1}}\psi^{\lambda}.$

For this purpose, we solve (91) in $B(0,1)$ with Dirichlet boundary
conditions. Let $\psi^{\lambda}_{0}$ denote the solution in $B(0,1)$. Let
$\chi:[0,\infty)\rightarrow[0,1]$ be a smooth, monotone decreasing function
with $\chi(r)=1$ for $r\in[0,\frac{1}{2}]$ and $\chi(r)=0$ for
$r\geq\frac{3}{4}$. We set
$\psi^{\lambda}(x,y)=\chi(|(x,y)|)\psi^{\lambda}_{0}(x,y)$.

We can now show (90). We treat $\mathcal{D}^{\lambda}$ by estimating the
contributions of the terms in (88) separately: From (87) follows that the
contribution of $\int_{0}^{t}\nabla_{\perp}\mathcal{H}^{\lambda}ds$ is
acceptable because the anti-derivative in $t$ cancels the derivative loss. The
same argument bounds the contribution of $C^{\lambda}$, so that
$\|C^{\lambda}\|_{H^{\gamma}}\lesssim\|\mathcal{H}^{\lambda}\|_{H^{\gamma}}$.
By elliptic regularity, we thus obtain the bound
$\|\psi^{\lambda}\|_{H^{\gamma+1}}\lesssim\|C^{\lambda}\|_{H^{\gamma}}\lesssim\|\mathcal{H}^{\lambda}\|_{H^{\gamma}}$
since the coefficients in $\Delta_{\tilde{\varepsilon}_{\lambda}^{-1}}$ are
uniformly bounded in $C^{s}$. Here we need the additional assumption $1\leq
s\leq 2$, which is not necessary in [32].

Lastly, we compute for the charge

$\rho_{e}^{\lambda}=\nabla\cdot\mathcal{D}^{\lambda}=\nabla\cdot
C^{\lambda}+\Delta_{\tilde{\varepsilon}_{\lambda}^{-1}}\psi^{\lambda}.$

Hence, we find $\rho_{e}^{\lambda}=0$ in $B(0,1/2)$ and $B(0,3/4)^{c}$. In
$B(0,3/4)\backslash B(0,1/2)$, we find

$\rho^{\lambda}_{e}=2(\varepsilon_{\lambda}\nabla\chi)(\nabla\psi^{\lambda}_{0})+(\partial_{2}g_{\lambda}^{-1})(\partial_{2}\psi^{\lambda}_{0})\chi+\text{l.o.t.}.$

Since
$\|\psi^{\lambda}\|_{H^{\gamma+1}}\lesssim\|\mathcal{H}^{\lambda}\|_{H^{\gamma}}$
and $g_{\lambda}^{-1}\in C^{s}$ with $s\geq 1$, only the second summand in the
display is problematic. First we handle
$\gamma-\frac{1}{2}-\frac{\sigma}{p}\leq 0$, in which case the estimate

$\|\rho_{e}^{\lambda}\|_{H^{\gamma-\frac{1}{2}-\frac{\sigma}{p}}}\lesssim\|\mathcal{H}^{\lambda}\|_{H^{\gamma}}$

is straight-forward. For $\gamma-\frac{1}{2}-\frac{\sigma}{p}>0$, observe that
$\|g_{\lambda}^{-1}\|_{C^{\theta}(K^{0}_{\lambda})}\lesssim\lambda^{2\sigma-(2-\theta)\delta}$.
We refer to [32] for the true coefficient $\tilde{g}_{\lambda}$ with
$\|\tilde{g}^{-1}_{\lambda}\|_{C^{\theta}(B(0,1))}\lesssim\|g_{\lambda}^{-1}\|_{C^{\theta}(K^{0}_{\lambda})}$.
Suppose that $0<\gamma<\rho+\frac{\sigma}{p}$. Hence,
$\|\partial_{2}g_{\lambda}^{-1}\|_{C^{\gamma-\frac{1}{2}-\frac{\sigma}{p}}}\lesssim\lambda^{\alpha}$
with $\alpha=2\sigma-(\frac{3}{2}-\gamma+\frac{\sigma}{p})\delta$.
Furthermore,

$\begin{split}\|(\partial_{2}g_{\lambda}^{-1})(\partial_{2}\psi^{\lambda}_{0})\chi\|_{H^{\gamma-\frac{1}{2}-\frac{\sigma}{p}}}&\lesssim\|\partial_{2}g_{\lambda}^{-1}\|_{C^{\gamma-\frac{1}{2}-\frac{\sigma}{p}}}\|\partial_{2}\psi^{\lambda}_{0}\|_{L^{2}}+\|\partial_{2}\psi_{0}^{\lambda}\|_{H^{\gamma-\frac{1}{2}-\frac{\sigma}{p}}}\\\
&\lesssim\lambda^{\alpha}\lambda^{-\frac{1+\delta}{2}}(\log\lambda)^{-\frac{3}{2}}+\|\mathcal{H}^{\lambda}\|_{H^{\gamma}}.\end{split}$

It remains to check $\alpha<\gamma$, which follows from
$(1-\delta)\gamma>\frac{5}{4}\sigma-\frac{3}{4}$. The latter inequality holds
as $\gamma>0$, $\delta\leq\frac{2}{3}$, and $\sigma\leq\frac{1}{3}$ because of
$s\geq 1$. $\Box$

### Acknowledgement

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) – Project-ID 258734477 – SFB 1173.

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