# Strichartz estimates for Maxwell equations in media: the fully anisotropic
case

Robert Schippa<EMAIL_ADDRESS>and  Roland Schnaubelt
<EMAIL_ADDRESS>Department of Mathematics, Karlsruhe Institute of
Technology, Englerstrasse 2, 76131 Karlsruhe, Germany

###### Abstract.

We prove Strichartz estimates for Maxwell equations in media in the fully
anisotropic case with Hölder-continuous coefficients. To this end, we use the
FBI transform to conjugate the problem to phase space. After reducing to a
scalar estimate by means of a matrix symmetrizer, we show oscillatory integral
estimates for a variable-coefficient Fourier extension operator. The
characteristic surface has conical singularities for any non-vanishing time
frequency. Combined with energy estimates, we improve the local well-posedness
for certain fully anisotropic quasilinear Maxwell equations.

###### Key words and phrases:

Maxwell equations, Strichartz estimates, quasilinear wave equation, rough
coefficients, half wave equation, FBI transform

###### 2020 Mathematics Subject Classification:

Primary: 35L45, 35B65, Secondary: 35Q61.

##  1\. Introduction and main results

Maxwell equations are the fundamental principles of electro-magnetism. In
media they relate _displacement and electric field_
$(\mathcal{D},\mathcal{E}):\mathbb{R}\times\mathbb{R}^{3}\to\mathbb{R}^{3}\times\mathbb{R}^{3}$
and _magnetic and magnetizing field_
$(\mathcal{B},\mathcal{H}):\mathbb{R}\times\mathbb{R}^{3}\to\mathbb{R}^{3}\times\mathbb{R}^{3}$
via

(1)
$\left\\{\begin{array}[]{rlrl}\partial_{t}\mathcal{D}\\!\\!\\!\\!&=\nabla\times\mathcal{H}-\mathcal{J}_{e},&\quad\nabla\cdot\mathcal{D}\\!\\!\\!\\!&=\rho_{e},\quad(t,x^{\prime})\in\mathbb{R}\times\mathbb{R}^{3},\\\
\partial_{t}\mathcal{B}\\!\\!\\!\\!&=-\nabla\times\mathcal{E}-\mathcal{J}_{m},&\quad\nabla\cdot\mathcal{B}\\!\\!\\!\\!&=\rho_{m},\end{array}\right.$

where
$(\rho_{e},\rho_{m}):\mathbb{R}\times\mathbb{R}^{3}\to\mathbb{R}\times\mathbb{R}$
denote the _electric and magnetic charges_ and
$(\mathcal{J}_{e},\mathcal{J}_{m}):\mathbb{R}\times\mathbb{R}^{3}\to\mathbb{R}^{3}\times\mathbb{R}^{3}$
the _electric and magnetic current_. (Space-time coordinates are written as
$x=(t,x^{\prime})=(x_{0},x_{1},\ldots,x_{d})\in\mathbb{R}\times\mathbb{R}^{d}$
and the dual variables in Fourier space as
$\xi=(\tau,\xi^{\prime})=(\xi_{0},\xi_{1},\ldots,\xi_{d})\in\mathbb{R}\times\mathbb{R}^{d}$.)
We refer to the physics’ literature for further explanations [4, 9]. We remark
that magnetic charges and currents are hypothetical, but included in the
analysis to highlight symmetry between electric and magnetic field.

To obtain a closed system in (1), one has to link the fields via material
laws. In the linear case, we consider the pointwise laws

(2)
$\mathcal{D}(x)=\varepsilon(x)\mathcal{E}(x),\quad\mathcal{B}(x)=\mu(x)\mathcal{H}(x)$

for the permittivity and permeability $\varepsilon,\mu\in
C^{s}(\mathbb{R}\times\mathbb{R}^{3};\mathbb{R}^{3\times 3})$ with $0<s\leq
1$, whose values are supposed to be symmetric and uniformly elliptic, i.e.,

(3) $\exists\,\lambda,\Lambda>0:\,\forall v\in\mathbb{R}^{3},\,\forall
x\in\mathbb{R}^{4}:\;\quad\lambda|v|^{2}\leq
A^{ij}(x)v_{i}v_{j}\leq\Lambda|v|^{2},\;\;A\in\\{\varepsilon,\mu\\}.$

In the above display sum convention is in use.

In the previous work [15], the first author considered the partially
anisotropic case with
$\varepsilon(x)=\text{diag}(\varepsilon_{1}(x),\varepsilon_{2}(x),\varepsilon_{2}(x))$
and $\mu(x)\equiv\mu_{0}$. In [15], the Maxwell system was diagonalized to a
system of half-wave equations, for which the sharp Strichartz range for the
three dimensional wave equation could be recovered in the isotropic case
$\varepsilon_{1}(x)=\varepsilon_{2}(x)$. In the partially anisotropic case,
$\varepsilon_{1}(x)\neq\varepsilon_{2}(x)$, it remains an open question
whether solutions to Maxwell equations satisfy Strichartz estimates as for
(half-)wave equations in the $C^{2}$-case. However, for Lipschitz coefficients
the first author recovered Strichartz estimates with derivative loss as for
wave equations with Lipschitz coefficients (cf. [22]).

In two space dimensions [17] we established sharp Strichartz estimates in the
fully anisotropic case

$\varepsilon(x)=\begin{pmatrix}\varepsilon^{11}(x)&\varepsilon^{12}(x)\\\
\varepsilon^{21}(x)&\varepsilon^{22}(x)\end{pmatrix},\quad\mu(x)=\mu_{0}(x)$

via diagonalization without imposing additional assumptions on $\varepsilon$.
In three dimensions the fully anisotropic case (see (5)), Strichartz estimates
have not been studied before. In this case the diagonalization procedure
introduces singularities in the conjugation matrices, and we opt for a
different approach. The fully anisotropic case appears in the study of biaxial
crystals (cf. [5, 8]). Moreover, when treating nonlinear material laws
$(\varepsilon(\mathcal{E}),\mu(\mathcal{H}))$ one needs Strichatz estimates
for differentiated fields, see Section 5. However, these fields only fullfil a
Maxwell system with modified coefficients, which are matrix-valued even if the
original $(\varepsilon(\mathcal{E}),\mu(\mathcal{H}))$ are scalar and thus
isotropic.

We work under the following assumption on $\varepsilon$ and $\mu$, which can
be described as uniform anisotropic, possibly off-diagonal material law.

######  Assumption 1.

Let
$\varepsilon,\mu:\mathbb{R}\times\mathbb{R}^{3}\to\mathbb{R}_{\text{sym}}^{3\times
3}$ satisfy (3) and suppose that there is $\Phi\in
C^{1}(\mathbb{R}\times\mathbb{R}^{3};\mathbb{R}^{3\times 3})$ with111In $\Phi$
we deviate from the usual matrix index notation and consider $\varphi_{i}$ as
column vectors.

$\Phi=\begin{pmatrix}\varphi_{1}&\varphi_{2}&\varphi_{3}\end{pmatrix}=\begin{pmatrix}\varphi_{11}&\varphi_{21}&\varphi_{31}\\\
\varphi_{12}&\varphi_{22}&\varphi_{32}\\\
\varphi_{13}&\varphi_{23}&\varphi_{33}\end{pmatrix},\quad\Phi^{t}(x)\Phi(x)=1_{3\times
3},$

and $\varepsilon^{d},\mu^{d}\in
C(\mathbb{R}\times\mathbb{R}^{3};\mathbb{R}^{3\times 3})$ with

$\varepsilon^{d}(x)=\text{diag}(\varepsilon^{1}(x),\varepsilon^{2}(x),\varepsilon^{3}(x))\text{
\ and \ }\mu^{d}(x)=\text{diag}(\mu^{1}(x),\mu^{2}(x),\mu^{3}(x))$

such that

$\varepsilon^{d}=\Phi^{t}\varepsilon\Phi,\quad\mu^{d}=\Phi^{t}\mu\Phi,$

and $\varepsilon^{d}$, $\mu^{d}$ satisfy

$\exists c>0:\,\forall x\in\mathbb{R}^{4}:\,\forall i\neq
j:\,\big{|}\frac{\varepsilon^{i}(x)}{\mu^{i}(x)}-\frac{\varepsilon^{j}(x)}{\mu^{j}(x)}\big{|}\geq
c.$

We let

(4) $P(x,D)=\begin{pmatrix}-\partial_{t}(\varepsilon\cdot)&\nabla\times\\\
\nabla\times&\partial_{t}(\mu\cdot)\end{pmatrix}$

such that (1) is concisely written as

$P(x,D)\begin{pmatrix}\mathcal{E}\\\
\mathcal{H}\end{pmatrix}=\begin{pmatrix}\mathcal{J}_{e}\\\
-\mathcal{J}_{m}\end{pmatrix},\quad\nabla\cdot(\varepsilon\mathcal{E})=\rho_{e},\;\nabla\cdot(\mu\mathcal{H})=\rho_{m}.$

We write $u=(\mathcal{E},\mathcal{H})$ and $\rho_{em}=(\rho_{e},\rho_{m})$.

As noted above in the isotropic and to some extent in the partially
anisotropic case, by [16] the Maxwell system possesses Strichartz estimates as
the scalar wave equation. However, in the fully anisotropic case already for
constant coefficients
$\varepsilon=\text{diag}(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3})$,
$\mu=\text{diag}(\mu_{1},\mu_{2},\mu_{3})$ satisfying

(5)
$\frac{\varepsilon_{1}}{\mu_{1}}\neq\frac{\varepsilon_{2}}{\mu_{2}}\neq\frac{\varepsilon_{3}}{\mu_{3}}\neq\frac{\varepsilon_{1}}{\mu_{1}},$

Liess [11] showed that the dispersive properties in the charge-free case
$\nabla\cdot\mathcal{D}=\nabla\cdot\mathcal{B}=0$ are only the same as for
wave equations in two dimensions, namely

(6)
$\|S_{1}^{\prime}(\mathcal{E},\mathcal{H})(t)\|_{L^{\infty}(\mathbb{R}^{3})}\lesssim(1+|t|)^{-\frac{1}{2}}\|(\mathcal{E},\mathcal{H})(0)\|_{L^{1}(\mathbb{R}^{3})}.$

Here we use a homogeneous dyadic decomposition
$(S^{\prime}_{\lambda})_{\lambda\in 2^{\mathbb{Z}}}$ of spatial frequencies,
and $(S_{\lambda})_{\lambda\in 2^{\mathbb{Z}}}$ denotes a decomposition of
space-time frequencies, see (18).

This surprising behavior is connected to the shape of the characteristic
surface depending on the eigenvalues of $\varepsilon$ and $\mu$, which has
been discussed in [16, Section 2.3]. In [12] the first author and R. Mandel
proved the existence of solutions to the time-harmonic Maxwell equations in
the fully anisotropic case under the assumption that $\varepsilon$ and $\mu$
are commuting. To the best of the authors’ knowledge, this was the first time
the condition (5) appeared explicitly in the literature. In this case the
characteristic surface of Maxwell equations ceases to be regular as the union
of two ellipsoids (as in the (partially) anisotropic case). Instead it becomes
the singular Fresnel surface, see [12] for a detailed quantitative analysis.
The properties of the Fresnel surface are recalled in Section 3. We further
note that the characteristic surface can have more singularities if
$\varepsilon$ and $\mu$ do not commute, [3].

The decreased dispersive effect for the Fresnel surface forces us to consider
the wave admissibility condition in two dimensions:

(7) $\frac{2}{p}+\frac{1}{q}\leq\frac{1}{2},\quad 2\leq p,q\leq\infty.$

By (6) and the interpolation argument of Keel–Tao [7], for $(p,q)$ satisfying
(7) the solutions $(\mathcal{E},\mathcal{H})$ to homogeneous Maxwell equations
with constant coefficients fulfill the Strichartz estimate

$\||D^{\prime}|^{-\rho}(\mathcal{E},\mathcal{H})\|_{L_{t}^{p}(\mathbb{R},L_{x^{\prime}}^{q}(\mathbb{R}^{3}))}\lesssim\|(\mathcal{E},\mathcal{H})(0)\|_{L^{2}(\mathbb{R}^{3})}$

in the charge-free case. We give the details and local-in-time estimates in
the case of non-vanishing charges in Section 2. In the following we denote
space-time Lebesgue spaces by $L_{t}^{p}L_{x^{\prime}}^{q}=L^{p}L^{q}$ for the
sake of brevity, where $L^{p}=L^{p}L^{p}$. In the above display the derivative
loss is denoted by $\rho$ which satisfies the scaling condition

(8) $\rho=3\big{(}\frac{1}{2}-\frac{1}{q}\big{)}-\frac{1}{p}.$

Furthermore, we use fractional derivatives given as Fourier multipliers by

$(|D|^{\alpha}f)\widehat{(}\xi)=|\xi|^{\alpha}\hat{f}(\xi),\;(|D^{\prime}|^{\alpha}f)\widehat{(}\xi)=|\xi^{\prime}|^{\alpha}\hat{f}(\xi),\;(\langle
D\rangle^{\alpha}f)\widehat{(}\xi)=\langle\xi\rangle^{\alpha}\hat{f}(\xi),\quad\text{etc.},$

where $\langle x\rangle=(1+|x|^{2})^{1/2}$.

We first show the following Strichartz estimates for rough coefficients in the
diagonal case.

######  Theorem 1.1.

Let $T>0$ and $\delta>0$, and suppose that $\varepsilon,\mu\in
C^{s}(\mathbb{R}\times\mathbb{R}^{3};\mathbb{R}_{\text{sym}}^{3\times 3})$
with $0<s\leq 1$ and
$\|\partial(\varepsilon,\mu)\|_{L_{T}^{2}L_{x^{\prime}}^{\infty}}<\infty$
satisfy Assumption 1 with $\Phi\equiv 1$. Let $P$ be as in (4). Then the
estimate

$\displaystyle\|\langle
D^{\prime}\rangle^{-\rho+\frac{s-2}{2p}-\delta}u\|_{L^{p}(0,T;L^{q}(\mathbb{R}^{3}))}$
$\displaystyle\lesssim\|u_{0}\|_{L^{2}_{x^{\prime}}}+\|Pu\|_{L^{1}_{T}L_{x^{\prime}}^{2}}+\|\langle
D^{\prime}\rangle^{-\frac{1}{2}+\frac{s-2}{8}}\rho_{em}(0)\|_{L^{2}_{x^{\prime}}}$
(9) $\displaystyle\quad+\|\langle
D^{\prime}\rangle^{-\frac{1}{2}+\frac{s-2}{8}}\partial_{t}\rho_{em}\|_{L^{1}_{T}L^{2}_{x^{\prime}}}$

holds provided that $p,q$ satisfy (7), $q\neq 2$, and $\rho$ is given by (8).
The implicit constant depends on $\delta$, $T$,
$\|(\varepsilon,\mu)\|_{C^{s}}$, and
$\|\partial(\varepsilon,\mu)\|_{L_{T}^{2}L_{x^{\prime}}^{\infty}}$.

We stress that both the different admissibility conditions and the appearance
of the charges highlight the effect of the system character of the Maxwell
equations, compared to the case of scalar wave equations treated in [21, 22,
23].

To establish (1.1), we first show

(10)
$\||D|^{-\rho+\frac{s-2}{4}}u\|_{L^{p}L^{q}}\lesssim_{\|(\varepsilon,\mu)\|_{C^{s}}}\|u\|_{L^{2}}+\||D|^{\frac{s-2}{2}}Pu\|_{L^{2}}+\||D|^{-\frac{1}{2}+\frac{s-2}{4}}\rho_{em}\|_{L^{2}}$

(with a modification at the endpoint $q=\infty$) if also
$\frac{2}{p}+\frac{1}{q}=\frac{1}{2}$ and $q>p$ via phase space analysis, see
Theorem 3.2. The core step in the argument is an estimate of a Fourier
extension operator on the Fresnel surface $S$, where we can reduce to scalar
problem using a symmetrizer. The singularities of $S$ are handled by a dyadic
scaling around them on annuli in Fourier space. Here we also use ideas from
[12] in the time-harmonic case with constant coefficients. The degeneracy at
$|\xi^{\prime}|\gg|\xi_{0}|$ of the main symbol of the Maxwell system is
treated using the charges in a separate microlocal argument.

We note that the derivative loss $\rho+\frac{2-s}{4}$ in (10) is the same as
in [21] for scalar wave equations. We actually improve the loss in (1.1), but
so far we cannot reach the sharp regularity loss established in [22, 23] for
the wave equation or [17, 15] in the 2D or isotropic Maxwell case. In contrast
to these works up to now we cannot use deeper properties of the Hamilton flow
of the problem because of the strong system character in the fully isotropic
case. In these papers also the case $s\in[1,2]$ and different norms on the
right-hand side of (1.1) have been treated. We plan to tackle these issues in
future work.

Starting from (10) we reduce the regularity loss by passing through short-time
Strichartz estimates. In Proposition 3.8 we show the endpoint estimate for
$p=4$ and $q=\infty$ on finite-time intervals

$\displaystyle\|\langle
D^{\prime}\rangle^{-\rho+\frac{s-2}{8}-\delta}u\|_{L_{T}^{4}L^{\infty}_{x^{\prime}}}$
$\displaystyle\lesssim\|u_{0}\|_{L^{2}_{x^{\prime}}}+\|Pu\|_{L^{1}_{T}L^{2}_{x^{\prime}}}+\|\langle
D^{\prime}\rangle^{\frac{s-2}{8}-\frac{1}{2}}\rho_{em}(0)\|_{L^{2}_{x^{\prime}}}$
$\displaystyle\quad+\|\langle
D^{\prime}\rangle^{\frac{s-2}{8}-\frac{1}{2}}\partial_{t}\rho_{em}\|_{L^{1}_{T}L^{2}_{x^{\prime}}}$

with implicit constant depending on $\|(\varepsilon,\mu)\|_{C^{1}}$, $T$ and
$\delta$, where we set
$L^{p}_{T}L^{q}_{x^{\prime}}=L^{p}(0,T;L^{q}_{x^{\prime}})$. The different
form of the charge term stems from an argument involving Duhamel’s formula,
see (3.3) and also [15, 17]. Theorem 1.1 then follows by interpolating the
above display with the standard energy estimate

$\|u\|_{L^{\infty}_{T}L^{2}_{x^{\prime}}}\lesssim_{\|(\varepsilon,\mu)\|_{C^{1}},T}\|u_{0}\|_{L^{2}}+\|Pu\|_{L^{1}_{T}L_{x^{\prime}}^{2}}.$

We note that (10) is shown for $q>6$ and that our method of proof breaks down
for $(p,q)$ closer to the energy point $(p,q)=(\infty,2)$. Broadly speaking,
our arguments show that we can prove Strichartz estimates for characteristic
surfaces $S=\\{\tilde{q}(\xi_{0})=0\\}$ with isolated degeneracies
$\tilde{q}(\xi_{0})=0$, $\nabla\tilde{q}(\xi_{0})=0$,
$\det(\partial^{2}\tilde{q}(\xi_{0}))\neq 0$.

Second, we establish the following variant in the possibly non-diagonal case.

######  Theorem 1.2.

Let $T>0$, $\delta>0$, and suppose that $\varepsilon,\mu\in
C^{1}(\mathbb{R}\times\mathbb{R}^{3};\mathbb{R}_{\text{sym}}^{3\times 3})$
satisfy Assumption 1. Let $P$ be as in (4). Then the estimate

$\displaystyle\|\langle
D^{\prime}\rangle^{-\rho-\frac{1}{2p}-\delta}u\|_{L^{p}(0,T;L^{q}(\mathbb{R}^{3}))}$
$\displaystyle\lesssim\|u_{0}\|_{L^{2}}+\|Pu\|_{L^{1}_{T}L_{x^{\prime}}^{2}}$
$\displaystyle\quad+\|\langle
D^{\prime}\rangle^{-5/8}\rho_{em}(0)\|_{L^{2}_{x^{\prime}}}+\|\langle
D^{\prime}\rangle^{-5/8}\partial_{t}\rho_{em}\|_{L^{1}_{T}L^{2}_{x^{\prime}}}$

holds provided that $(p,q)$ satisfy (7), $q\neq 2$, and $\rho$ is given by
(8). The implicit constant depends on $\delta$, $T$, and
$\|(\varepsilon,\mu)\|_{C^{1}}$.

To show this theorem, one uses the orthogonality transformation $\Phi$ to pass
to a Maxwell-type system with the diagonal coefficents $\varepsilon^{d}$ and
$\mu^{d}$ from Assumption 1, but with modified differential operators in
$x^{\prime}$ depending on $\Phi$. This system can be treated similar to the
system arising in the proof of Theorem 1.1.

We believe that the global existence and regularity of eigenvectors as stated
in Assumption 1 is non-trivial. Hence, we devote Section 6 to a discussion of
existence and regularity of eigenvectors for parameter-dependent matrices. It
turns out that if the eigenvalues are uniformly separated and the parameter
domain is simply connected, the eigenvectors are as regular as the eigenvalues
and admit global parametrisations. The results of Section 6 imply the
following proposition, which leads to a corollary to Theorem 1.2 stated below.

######  Proposition 1.3.

Suppose that $\mu\equiv 1$ and $\varepsilon\in
C^{1}(\mathbb{R}\times\mathbb{R}^{3};\mathbb{R}^{3\times 3}_{\text{sym}})$
satisfies (3) and

$\exists c>0:\,\forall
x\in\mathbb{R}^{4}:\,|\varepsilon_{i}(x)-\varepsilon_{j}(x)|\geq c>0,\quad
i\neq j,$

where $(\varepsilon_{i}(x))_{i=1,2,3}$ are the eigenvalues of
$\varepsilon(x)$. Then $(\varepsilon,\mu)$ fulfill Assumption 1.

######  Corollary 1.4.

Let $\varepsilon$ and $\mu$ be like in Proposition 1.3. Then the Strichartz
estimates (1.1) hold true under the assumptions of Theorem 1.2.

In Section 5 we apply the Strichartz estimates from Theorem 1.1 to improve the
local well-posedness theory of quasilinear Maxwell equations

(11)
$\left\\{\begin{array}[]{rlrlrl}\partial_{t}\mathcal{D}\\!\\!\\!\\!&=\nabla\times\mathcal{H},&\quad\nabla\cdot\mathcal{D}\\!\\!\\!\\!&=0,&\quad\mathcal{D}(0)\\!\\!\\!\\!&=\mathcal{D}_{0}\in
H^{s}(\mathbb{R}^{3};\mathbb{R}^{3}),\\\
\partial_{t}\mathcal{B}\\!\\!\\!\\!&=-\nabla\times\mathcal{E},&\quad\nabla\cdot\mathcal{B}\\!\\!\\!\\!&=0,&\quad\mathcal{B}(0)\\!\\!\\!\\!&=\mathcal{B}_{0}\in
H^{s}(\mathbb{R}^{3};\mathbb{R}^{3}),\end{array}\right.$

with small initial fields and such that $\varepsilon(\mathcal{E})$ has
uniformly separated eigenvalues. For sake of simplicity, we suppose that
$\mu=1_{3\times 3}$. Contrary to the isotropic Kerr case analyzed in [17, 15],
we cannot automatically deduce energy estimates by symmetrization. It turns
out that this requires additional symmetries of the permittivity. We shall
rewrite $\mathcal{D}=\varepsilon(\mathcal{E})\mathcal{E}$ to
$\mathcal{E}=\psi(\mathcal{D})\mathcal{D}$, which is possible for small fields
under mild assumptions on $\varepsilon$ by invoking the implicit function
theorem. In this form, we can phrase the condition for the existence of a
symmetrizer as

(12)
$\varepsilon^{ijk}\big{(}\frac{\partial}{\partial\mathcal{D}_{j}}\psi(\mathcal{D})_{k\ell}\big{)}\mathcal{D}^{\ell}=0$

for $i\in\\{1,2,3\\}$ and summation over $j,k\in\\{1,2,3\\}$, where
$\varepsilon^{ijk}$ denotes the Levi-Civita symbol. (Such a condition was also
used in [10].) We refer to the paragraph before Proposition 5.3 for discussion
of examples like

$\varepsilon=\text{diag}(\varepsilon_{0}^{1},\varepsilon_{0}^{2},\varepsilon_{0}^{3})+\text{diag}(\alpha_{1}|\mathcal{E}_{1}|^{2},\alpha_{2}|\mathcal{E}_{2}|^{2},\alpha_{3}|\mathcal{E}_{3}|^{2}).$

It seems plausible that permittivities obtained like this can model biaxial
crystals with nonlinear electric response.

We show the following theorem, which improves the local well-posedness via
Strichartz estimates by $1/9$ derivatives compared to energy arguments. It is
the first result of this kind for cases of fully anisotropic Maxwell systems.
This improvement is smaller as in [23] or [15, 17]. (For instance, for scalar
quasilinear wave equations one gains $1/3$ derivatives compared to energy
arguments by the results in [23].) Strichartz estimates with smaller
regularity loss would imply better results, but it is unclear what can be
achieved in the fully anisotropic Maxwell case.

######  Theorem 1.5.

Let $\varepsilon_{i}\in C^{\infty}(\mathbb{R};\mathbb{R})$ and let
$\varepsilon(\mathcal{E})=\mathrm{diag}(\varepsilon_{1}(\mathcal{E}_{1}),\varepsilon_{2}(\mathcal{E}_{2}),\varepsilon_{3}(\mathcal{E}_{3}))$
be uniformly positive definite with uniformly separated eigenvalues, i.e.,
there are $c,\delta>0$ such that for any $\mathcal{E}\in\mathbb{R}^{3}$ with
$|\mathcal{E}|\leq\delta$ we have

$|\varepsilon_{i}(\mathcal{E})-\varepsilon_{j}(\mathcal{E})|\geq c>0.$

Write $\mathcal{D}=\varepsilon(\mathcal{E})\mathcal{E}$ as
$\mathcal{E}=\psi(\mathcal{D})\mathcal{D}$ for
$\|\mathcal{D}\|_{L^{\infty}}\leq\delta$ and assume that $\psi$ satisfies
(12). Then there is $\delta^{\prime}>0$ such that (11) is locally well-posed
for $s>2+\frac{7}{18}$ for initial data
$\|u_{0}\|_{H^{s}}\leq\delta^{\prime}$.

For the proof we use Strichartz estimates to improve on Sobolev embedding.
Under the above symmetry assumptions, we find energy estimates

$\|(\mathcal{E},\mathcal{H})(t)\|_{H^{s}}\lesssim
e^{c(A)\int_{0}^{t}\|\partial(\mathcal{E},\mathcal{H})(s)\|_{L_{x^{\prime}}^{\infty}}ds}\|(\mathcal{E},\mathcal{H})(0)\|_{H^{s}}$

with
$A=\|(\mathcal{E},\mathcal{H})\|_{L_{T}^{\infty}L_{x^{\prime}}^{\infty}}$. An
estimate of
$\|\partial(\mathcal{E},\mathcal{H})\|_{L_{T}^{1}L_{x^{\prime}}^{\infty}}$
based on Sobolev embedding would lead to a regularity of $s>5/2$. We want to
use non-trivial Strichartz estimates for
$\|\partial(\mathcal{E},\mathcal{H})\|_{L_{T}^{4}L_{x^{\prime}}^{\infty}}$
provided that the coefficients are in $C^{\alpha}_{x}$. The Hölder norm is
again controlled by a spatial Sobolev norm and we can close the argument for
$s>3/2+\alpha$, if the derivative loss in the Strichartz estimates _for
$\partial(\mathcal{E},\mathcal{H})$_ is controlled in the regularity $H^{s}$.
This leads to the condition $s>\frac{3}{2}+\frac{8}{9}$. In the next step, we
control the $L^{2}$-norm of differences of solutions
$v=(\mathcal{E}^{1},\mathcal{H}^{1})-(\mathcal{E}^{2},\mathcal{H}^{2})$ by

$\|v(t)\|_{L^{2}}\lesssim e^{c(A)\int_{0}^{t}B(s)ds}\|v(0)\|_{L^{2}}$

with
$A=\|(\mathcal{E}^{1},\mathcal{H}^{1})\|_{L^{\infty}_{T}L^{\infty}_{x^{\prime}}}+\|(\mathcal{E}^{2},\mathcal{H}^{2})\|_{L^{\infty}_{T}L_{x^{\prime}}^{\infty}}$
and
$B(s)=\|(\mathcal{E}^{1},\mathcal{H}^{1})(s)\|_{L^{\infty}_{x^{\prime}}}+\|(\mathcal{E}^{2},\mathcal{H}^{2})(s)\|_{L_{x^{\prime}}^{\infty}}$.
By the same argument as above, we obtain Lipschitz continuous dependence in
$L^{2}$ for initial data in regularity $H^{s}$ if $s>\frac{3}{2}+\frac{8}{9}$.
To find continuous dependence to hold in $H^{s}$, we use the frequency
envelope approach due to Tao [20]. (See also [6] for an exposition, and [17]
for a previous application in the context of Maxwell equations.) This does not
yield uniform continuous dependence, which cannot be expected for a
quasilinear hyperbolic problem.

_Outline of the paper._ In Section 2 we revisit Strichartz estimates in the
constant-coefficient case for fully anisotropic Maxwell equations. In Section
3 we prove Strichartz estimates for diagonal variable coefficients in the
uniformly fully anisotropic case as stated in Theorem 1.1. In Section 4 we
reduce the more general case of Assumption 1 to the case of diagonal
permittivity and permeability and prove Theorem 1.2. In Section 5 we argue how
the Strichartz estimates for rough coefficients improve the local well-
posedness theory for quasilinear Maxwell equations. In Section 6 we show the
global existence and regularity of eigenvectors in case of separated
eigenvalues of parameter-dependent matrices in simply connected domains.

##  2\. Strichartz estimates in the constant-coefficient case

In this section we revisit the Strichartz estimates in the fully anisotropic
case for constant coefficients. For the remainder of the section, let

(13)
$\varepsilon=\text{diag}(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}),\quad\mu=\text{diag}(\mu_{1},\mu_{2},\mu_{3})\quad\text{with
\ }\varepsilon_{i},\ \mu_{j}\in\mathbb{R}_{>0}.$

By the symmetries of Maxwell equations (cf. [12]), we can reduce the general
case of positive definite $\varepsilon,\mu\in\mathbb{R}^{3\times
3}_{\text{sym}}$ to (13) provided that $\varepsilon$ and $\mu$ commute. We
supplement the linear Maxwell equations

(14)
$\left\\{\begin{array}[]{rlrl}\partial_{t}\mathcal{D}\\!\\!\\!\\!&=\nabla\times\mathcal{H},&\quad\nabla\cdot\mathcal{D}\\!\\!\\!\\!&=\rho_{e},\\\
\partial_{t}\mathcal{B}\\!\\!\\!\\!&=-\nabla\times\mathcal{E},&\quad\nabla\cdot\mathcal{B}\\!\\!\\!\\!&=\rho_{m}\end{array}\right.$

on $\mathbb{R}\times\mathbb{R}^{3}$ with the constant-coefficient material
laws

(15)
$\mathcal{D}=\varepsilon\mathcal{E},\quad\mathcal{B}=\mu\mathcal{H},\quad\text{where
\eqref{eq:DiagonalConstantCoefficients} is true.}$

In the charge-free case, dispersive time-decay of solutions to (14) was proved
by Liess [11]. Here we deduce Strichartz estimates also allowing for charges.
The first observation is that non-trivial charges in (14) inhibit global-in-
time Strichartz estimates

$\||D^{\prime}|^{-\rho}u\|_{L^{p}(\mathbb{R};L^{q}_{x^{\prime}})}\lesssim\|u(0)\|_{L^{2}_{x^{\prime}}}$

with $u=(\mathcal{E},\mathcal{H})$,
$\rho=3\big{(}\frac{1}{2}-\frac{1}{q}\big{)}-\frac{1}{p}$, and
$\frac{2}{p}+\frac{1}{q}\leq\frac{1}{2}$.

Indeed, given nonzero
$(\rho_{e},\rho_{m})\in\dot{H}^{\gamma}\times\dot{H}^{\gamma}$, let
$(\Phi_{1},\Phi_{2})$ solve the elliptic equations

(16)
$\nabla\cdot(\varepsilon\nabla\Phi_{1})=\rho_{e},\qquad\nabla\cdot(\mu\nabla\Phi_{2})=\rho_{m}.$

Then (14) possesses the stationary solution $(\mathcal{E},\mathcal{H})(0)=u$
with $\mathcal{E}(0)=\nabla\Phi_{1}$ and $\mathcal{H}(0)=\nabla\Phi_{2}$. We
have $(\Phi_{1},\Phi_{2})\in\dot{H}^{\gamma+2}$ and
$u(0)\in\dot{H}^{\gamma+1}$. However, the solution does not decay, and hence
global-in-time Strichartz estimates for (14) are not possible. To obtain such
estimates in the charge-free case, we use the following dispersive estimate
due to Liess [11, Theorem 1.3].

######  Proposition 2.1.

Let $u=(\mathcal{E},\mathcal{H})$ solve (14) with (15) and
$\rho_{e}=\rho_{m}=0$. We then have

(17)
$\|S_{1}^{\prime}u(t)\|_{L^{\infty}_{x^{\prime}}}\lesssim(1+|t|)^{-\frac{1}{2}}\|u(0)\|_{L^{1}_{x^{\prime}}}.$

To define the needed dyadic frequency decomposition, let $\chi\in
C^{\infty}_{c}(\mathbb{R};\mathbb{R}_{\geq 0})$ be a radially decreasing with
$\chi(x)=1$ for $|x|\leq 1$ and $\chi(x)=0$ for $|x|\geq 2$. We set

(18)
$\begin{split}(S^{\prime}_{\lambda}f)\widehat{(}\xi)&=(\chi(\|\xi^{\prime}\|/\lambda)-\chi(\|\xi^{\prime}\|/2\lambda))\hat{f}(\xi),\\\
(S_{\lambda}f)\widehat{(}\xi)&=(\chi(\|\xi\|/\lambda)-\chi(\|\xi\|/2\lambda))\hat{f}(\xi)\end{split}$

for $\lambda\in 2^{\mathbb{Z}}$. Moreover, we write

$\displaystyle S_{0}^{\prime}$ $\displaystyle=1-\sum_{\lambda\in
2^{\mathbb{N}_{0}}}S^{\prime}_{\lambda},\quad S_{0}=1-\sum_{\lambda\in
2^{\mathbb{N}_{0}}}S_{\lambda},\qquad S_{\geq
1}^{\prime}=1-S^{\prime}_{0},\quad S_{\geq 1}=1-S_{0},$ $\displaystyle
S^{\prime}_{\sim M}$
$\displaystyle=\sum_{\lambda=M/8}^{8M}S^{\prime}_{\lambda},\quad S_{\sim
M}=\sum_{\lambda=M/8}^{8M}S_{\lambda}.$

By Littlewood-Paley decomposition, rescaling, and the Keel–Tao interpolation
argument [7, Theorem 1.2], we find the desired global estimates.

######  Theorem 2.2.

Let $u=(\mathcal{E},\mathcal{H})$ be a solution to (14) with (15) and
$\rho_{e}=\rho_{m}=0$. Then the global Strichartz estimates

(19)
$\||D^{\prime}|^{-\rho}u\|_{L^{p}(\mathbb{R};L^{q}_{x^{\prime}})}\lesssim\|u(0)\|_{L^{2}_{x^{\prime}}}$

hold with $\rho$ given by (8) and $(p,q)$ satisfying (7) and $q<\infty$.

######  Remark 2.3.

For $q=\infty$, estimate (19) is true for the homogeneous Besov space:

$\|u\|_{L_{t}^{p}(\mathbb{R};\dot{B}^{-\rho}_{q,2}(\mathbb{R}^{3}))}\lesssim\|u(0)\|_{L^{2}_{x^{\prime}}}.$

To lighten the discussion in the following, we suppose $q<\infty$. See also
Remark 3.3.

We next show local-in-time estimates involving charges.

######  Theorem 2.4.

Let $\varepsilon$, $\mu$, $\rho$, $p$, and $q$ be as in Theorem 2.2. We then
have

(20) $\|\langle
D^{\prime}\rangle^{-\rho}u\|_{L^{p}_{T}L^{q}_{x^{\prime}}}\lesssim
T^{\frac{1}{p}}(\|u(0)\|_{L^{2}}+\|\rho_{em}\|_{H^{\frac{1}{p}-1}_{x^{\prime}}}).$

###### Proof.

We begin with estimating the low frequencies

$\|\langle
D^{\prime}\rangle^{-\rho}S^{\prime}_{0}u\|_{L^{p}_{T}L^{q}_{x^{\prime}}}\lesssim
T^{\frac{1}{p}}\|u(0)\|_{L^{2}_{x^{\prime}}}$

by Hölder in time and Bernstein’s inequality.

For the estimate of the high frequencies, we split the initial data into
stationary and dispersive components. Let $(\Phi_{1},\Phi_{2})$ be the
solutions to the elliptic equations in (16). Since we confine to high
frequencies, we can replace the homogeneous with inhomogeneous norms. We have
the stationary solution $u_{\text{stat}}(t)=S^{\prime}_{\geq
1}(\nabla\Phi_{1},\nabla\Phi_{2})$ and we write

$S^{\prime}_{\geq 1}(\mathcal{E},\mathcal{H})(0)=S^{\prime}_{\geq
1}(\nabla\Phi_{1},\nabla\Phi_{2})+S^{\prime}_{\geq
1}(\mathcal{E},\mathcal{H})_{\text{disp}}(0).$

We denote the solution emanating from the last term by $u_{\text{disp}}$,
which is clearly charge free. Taking the Fourier transform of
$\nabla\cdot(\varepsilon\mathcal{E}(0))=\nabla\cdot(\varepsilon\nabla\Phi_{1})$,
we obtain

$\frac{i\xi_{k}\varepsilon^{km}\hat{\mathcal{E}}_{m}(0,\xi)}{\xi_{i}\xi_{j}\varepsilon^{ij}}=\hat{\Phi}_{1}(\xi).$

Plancherel’s theorem thus yields
$\|\nabla\Phi_{1}\|_{L^{2}}\lesssim\|\mathcal{E}(0)\|_{L^{2}}$ and likewise
$\|\nabla\Phi_{2}\|_{L^{2}}\lesssim\|\mathcal{H}(0)\|_{L^{2}}$. Hence
$\|u_{\text{disp}}(0)\|_{L^{2}}\lesssim\|(\mathcal{E},\mathcal{H})(0)\|_{L^{2}}$.

By linearity, we have $S^{\prime}_{\geq 1}u=u_{\text{disp}}+u_{\text{stat}}$
and Theorem 2.2 implies

$\displaystyle\||D^{\prime}|^{-\rho}S^{\prime}_{\geq
1}u\|_{L^{p}_{T}L^{q}_{x^{\prime}})}$
$\displaystyle\leq\||D^{\prime}|^{-\rho}S^{\prime}_{\geq
1}u_{\text{disp}}\|_{L^{p}_{T}L^{q}_{x^{\prime}}}+\||D^{\prime}|^{-\rho}S^{\prime}_{\geq
1}u_{\text{stat}}\|_{L^{p}_{T}L^{q}_{x^{\prime}}}$
$\displaystyle\lesssim\|S^{\prime}_{\geq
1}u(0)\|_{L^{2}_{x^{\prime}}}+T^{\frac{1}{p}}\|(\rho_{e},\rho_{m})\|_{H^{\frac{1}{p}-1}_{x^{\prime}}}.$

Inequality (20) follows. ∎

We turn to the inhomogeneous problem

(21)
$\left\\{\begin{array}[]{rlrlrl}\partial_{t}\mathcal{D}\\!\\!\\!\\!&=\nabla\times\mathcal{H}-\mathcal{J}_{e},&\quad\nabla\cdot\mathcal{D}\\!\\!\\!\\!&=\rho_{e},&\quad\mathcal{D}(0)\\!\\!\\!\\!&=\mathcal{D}_{0},\\\
\partial_{t}\mathcal{B}\\!\\!\\!\\!&=-\nabla\times\mathcal{E}-\mathcal{J}_{m},&\quad\nabla\cdot\mathcal{B}\\!\\!\\!\\!&=\rho_{m},&\quad\mathcal{B}(0)\\!\\!\\!\\!&=\mathcal{B}_{0}.\end{array}\right.$

We write $u=(\mathcal{E},\mathcal{H})$ for solutions to (21) as well as
$\mathcal{J}=(\mathcal{J}_{e},\mathcal{J}_{m})$ and
$\nabla\cdot\mathcal{J}=(\nabla\cdot\mathcal{J}_{e},\nabla\cdot\mathcal{J}_{m})$.

######  Theorem 2.5.

Let $u=(\mathcal{E},\mathcal{H})$ be a solution to (21), $(\rho,p,q)$ and
$(\tilde{\rho},\tilde{p},\tilde{q})$ satisfy (7) and (8) with
$q,\tilde{q}<\infty$. For $T<\infty$, then the estimates

(22) $\begin{split}\|\langle
D^{\prime}\rangle^{-\rho}u\|_{L^{p}([0,T],L^{q}(\mathbb{R}^{3}))}&\lesssim
T^{\frac{1}{p}}(\|u(0)\|_{L^{2}}+\|\rho_{em}(0)\|_{H^{\frac{1}{p}-1}})\\\
&\quad+T^{\frac{1}{p}}\big{(}\|\mathcal{J}\|_{L^{1}([0,T],L^{2})}+\|\nabla\cdot\mathcal{J}\|_{L^{1}([0,T],H^{\frac{1}{p}-1})}\big{)}.\end{split}$

hold. If $\rho_{em}=0$ and $\nabla\cdot\mathcal{J}=0$, we have the global
estimates

(23)
$\||D^{\prime}|^{-\rho}u\|_{L^{p}(\mathbb{R};L^{q}(\mathbb{R}^{3}))}\lesssim\|u(0)\|_{L^{2}(\mathbb{R}^{3})}+\||D^{\prime}|^{\tilde{\rho}}\mathcal{J}\|_{L^{\tilde{p}^{\prime}}L^{\tilde{q}^{\prime}}}.$

###### Proof.

We denote the propagator for the free solution by $U(t)$, so that we can write

$(\mathcal{D},\mathcal{B})(t)=U(t)(\mathcal{D},\mathcal{B})(0)-\int_{0}^{t}U(t-s)\mathcal{J}(s)ds$

by Duhamel’s formula. Since $\|\langle
D^{\prime}\rangle^{s}u\|_{L^{2}}\sim\|\langle
D^{\prime}\rangle^{s}(\mathcal{D},\mathcal{B})\|_{L^{2}}$, we can apply
Theorem 2.4 with initial time $s$ and Minkowski’s inequality to find (22) to
hold. Inequality (23) is a consequence of the dispersive estimate (17) and
Keel-Tao interpolation [7, Theorem 1.2]. ∎

##  3\. Proof of Strichartz estimates in the fully anisotropic case with
diagonal material coefficients

In the following we consider diagonal permittivity and permeability

(24)
$\varepsilon=\text{diag}(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}),\quad\mu=\text{diag}(\mu_{1},\mu_{2},\mu_{3}),\;$

which are supposed to satisfy the uniform ellipticity condition

(25) $\exists\lambda,\Lambda>0:\forall
x\in\mathbb{R}^{4}:\,\lambda\leq\nu_{i}(x)\leq\Lambda\text{ for }i=1,2,3\text{
and }\nu\in\\{\varepsilon,\mu\\}.$

Furthermore, we require the following separability condition, which guarantees
uniformity of the curvature bounds for the characteristic surfaces (cf.
Assumption 1):

(26) $\exists c>0:\forall x\in\mathbb{R}^{4}:\forall i\neq
j:\,\big{|}\frac{\varepsilon_{i}(x)}{\mu_{i}(x)}-\frac{\varepsilon_{j}(x)}{\mu_{j}(x)}\big{|}\geq
c.$

In the following we show Strichartz estimates for Maxwell equations with the
rough material laws as above

(29)
$\displaystyle\left\\{\begin{array}[]{rlrl}\partial_{t}\mathcal{D}\\!\\!\\!\\!&=\nabla\times\mathcal{H}-\mathcal{J}_{e},&\quad\nabla\cdot\mathcal{D}\\!\\!\\!\\!&=\rho_{e},\qquad
x\in\mathbb{R}^{4},\\\
\partial_{t}\mathcal{B}\\!\\!\\!\\!&=-\nabla\times\mathcal{E}-\mathcal{J}_{m},&\quad\nabla\cdot\mathcal{B}\\!\\!\\!\\!&=\rho_{m}.\end{array}\right.$

We use the pointwise material laws (2). Setting

$P(x,D)=\begin{pmatrix}-\partial_{t}(\varepsilon\cdot)&\nabla\times\\\
\nabla\times&\partial_{t}(\mu\cdot)\end{pmatrix},\quad
u=\begin{pmatrix}\mathcal{E}\\\
\mathcal{H}\end{pmatrix},\quad\rho_{em}=\begin{pmatrix}\rho_{e}\\\
\rho_{m}\end{pmatrix},$

the Maxwell system (29) becomes

$P(x,D)u=\begin{pmatrix}\mathcal{J}_{e}\\\
-\mathcal{J}_{m}\end{pmatrix},\quad\nabla\cdot\mathcal{D}=\rho_{e},\quad\nabla\cdot\mathcal{B}=\rho_{m}.$

We first show the basic energy estimate which will be used later in an
interpolation argument.

######  Proposition 3.1.

Let $T>0$, $\varepsilon$, $\mu\in
C(\mathbb{R}\times\mathbb{R}^{3};\mathbb{R}^{3\times 3})$ satisfy (25) with
$\partial_{t}(\varepsilon,\mu)\in L^{1}_{T}L^{\infty}_{x^{\prime}}$. We then
have the estimate

(30)
$\|u\|_{L^{\infty}_{T}L^{2}_{x^{\prime}}}\lesssim\|u_{0}\|_{L^{2}}+\|Pu\|_{L_{T}^{1}L^{2}}.$

###### Proof.

We introduce an equivalent norm on $L^{2}(\mathbb{R}^{3})$ by

$\|u\|^{2}_{E^{0}}=\Big{\langle}u,\begin{pmatrix}\varepsilon&0\\\
0&\mu\end{pmatrix}u\Big{\rangle}.$

For homogeneous solutions we compute

$\begin{split}\partial_{t}\big{(}\int_{\mathbb{R}^{3}}\mathcal{E}.\mathcal{D}+\mathcal{H}.\mathcal{B}\big{)}&=\int_{\mathbb{R}^{3}}(\partial_{t}\mathcal{E}).\mathcal{D}+\mathcal{E}.(\partial_{t}\mathcal{D})+(\partial_{t}\mathcal{H}).\mathcal{B}+\mathcal{H}.(\partial_{t}\mathcal{B})\\\
&=:I+II+III+IV.\end{split}$

We find

$II+IV=\int_{\mathbb{R}^{3}}\mathcal{E}.(\nabla\times\mathcal{H})-(\nabla\times\mathcal{E}).\mathcal{H}=0$

by the symmetry of the curl-operator. Note that

$\begin{split}(\partial_{t}\varepsilon)\mathcal{E}+\varepsilon\partial_{t}\mathcal{E}=\nabla\times\mathcal{H}&\iff\partial_{t}\mathcal{E}=\varepsilon^{-1}\nabla\times\mathcal{H}-\varepsilon^{-1}(\partial_{t}\varepsilon)\mathcal{E},\\\
(\partial_{t}\mu)\mathcal{H}+\mu\partial_{t}\mathcal{H}=-\nabla\times\mathcal{E}&\iff\partial_{t}\mathcal{H}=-\mu^{-1}(\nabla\times\mathcal{E})-\mu^{-1}(\partial_{t}\mu)\mathcal{H},\end{split}$

which yields

$\begin{split}I+III&=\int(\varepsilon^{-1}(\nabla\times\mathcal{H})).\mathcal{D}-\mu^{-1}(\nabla\times\mathcal{E}).\mathcal{B}-\varepsilon^{-1}((\partial_{t}\varepsilon)\mathcal{E}).\mathcal{D}-\mu^{-1}((\partial_{t}\mu)\mathcal{H}).\mathcal{B}\\\
&=-\int(\partial_{t}\varepsilon)\mathcal{E}.\mathcal{E}+\mathcal{H}.(\partial_{t}\mu)\mathcal{H}.\end{split}$

We obtain

$|I+III|\lesssim\|\partial_{t}(\varepsilon,\mu)\|_{L^{\infty}_{x^{\prime}}}\|u\|^{2}_{E^{0}},$

which leads to the estimate

$\partial_{t}\|u\|^{2}_{E^{0}}\lesssim\|(\partial_{t}\varepsilon,\partial_{t}\mu)\|_{L_{x^{\prime}}^{\infty}}\|u\|^{2}_{E^{0}}.$

Grønwall’s lemma and the equivalence of the norms imply

(31) $\|u(t)\|^{2}_{L^{2}}\lesssim
e^{C\|(\partial_{t}\varepsilon,\partial_{t}\mu)\|_{L_{t}^{1}L_{x^{\prime}}^{\infty}}}\|u(0)\|^{2}_{L^{2}}.$

For the proof of the inhomogeneous estimate we consider the evolution of the
variables $(\mathcal{D},\mathcal{B})$. By (25), for vanishing currents
$\mathcal{J}_{em}=0$ we have

$\|(\mathcal{D},\mathcal{B})(t)\|_{L^{2}}\lesssim
e^{C\int_{0}^{t}\|\partial_{t}(\varepsilon,\mu)\|_{L^{\infty}_{x^{\prime}}}ds}\|(\mathcal{D},\mathcal{B})(0)\|_{L^{2}}.$

Let $U(t,s)$ be the propagator of the homogeneous Maxwell equations

$\left\\{\begin{array}[]{cl}\partial_{t}(\mathcal{D},\mathcal{B})\\!\\!\\!\\!&=Q(\mathcal{D},\mathcal{B}),\\\
(\mathcal{D},\mathcal{B})(s)\\!\\!\\!\\!&=(\mathcal{D},\mathcal{B})_{0}\in
L^{2},\end{array}\right.$

The claim now follows from Duhamel’s formula, writing

$(\mathcal{D},\mathcal{B})(t)=U(t,0)(\mathcal{D},\mathcal{B})_{0}-\int_{0}^{t}U(t,s)(\mathcal{J}_{e},\mathcal{J}_{m})(s)ds$

and applying Minkowski’s inequality and (31). ∎

We next show the following local-in-time Strichartz estimates.

######  Theorem 3.2.

Let $0<s\leq 1$, $\varepsilon,\mu\in C^{s}(\mathbb{R}^{4};\mathbb{R}^{3\times
3})$ be as in (24) satisfying (25) and (26). Suppose that
$\rho=3\big{(}\frac{1}{2}-\frac{1}{q}\big{)}-\frac{1}{p}$, $2\leq
p<q\leq\infty$, and $\frac{2}{p}+\frac{1}{q}=\frac{1}{2}$. We find the
following estimate to hold:

(32)
$\||D|^{-\rho+\frac{s-2}{4}}u\|_{(L^{p}L^{q})_{2}}\lesssim(1+\|(\varepsilon,\mu)\|_{C^{s}})\|u\|_{L^{2}}+\||D|^{\frac{s-2}{2}}Pu\|_{L^{2}}+\||D|^{-\frac{1}{2}+\frac{s-2}{4}}\rho_{em}\|_{L^{2}}.$

######  Remark 3.3.

We consider the Besov norm with mixed coefficients defined by

$\|f\|_{(L^{p}L^{q})_{2}}=\big{(}\sum_{\lambda\in
2^{\mathbb{N}_{0}}\cup\\{0\\}}\|S_{\lambda}f\|_{L^{p}L^{q}}^{2}\big{)}^{\frac{1}{2}}$

on the left hand-side because at the endpoint $q=\infty$ the Littlewood-Paley
square-function estimate fails. For $q<\infty$ we can simplify (32) to

$\||D|^{-\rho+\frac{s-2}{4}}u\|_{L^{p}L^{q}}\lesssim(1+\|(\varepsilon,\mu)\|_{C^{s}})\|u\|_{L^{2}}+\||D|^{\frac{s-2}{2}}Pu\|_{L^{2}}+\||D|^{-\frac{1}{2}+\frac{s-2}{4}}\rho_{em}\|_{L^{2}}.$

We prove Theorem 3.2 by phase space analysis close to the endpoint. To this
end, we apply the FBI transform (cf. [21, 22, 23]) to change into phase space
after

  * •

reducing to high frequencies,

  * •

spatial and dyadic frequency localisation of the solution,

  * •

frequency truncation of $\varepsilon(x)$ and $\mu(x)$.

After these reductions, as in [17, 15] we arrive at the following proposition.

######  Proposition 3.4.

Let $\lambda\gg\lambda_{0}$ with $\lambda_{0}$ depending on the ellipticity of
$\varepsilon$ and $\mu$ and $\|(\varepsilon,\mu)\|_{C^{s}_{x}}$ for $0<s\leq
1$. Let $P_{\lambda}$ denote the operator $P$ with coefficients frequency
truncated at frequencies $<\lambda/8$ and let $u$ be essentially supported in
the unit cube. Under asumptions of Theorem 3.2, estimate (32) follows from

(33)
$\begin{split}\lambda^{-\rho+\frac{s-2}{4}}\|S_{\lambda}u\|_{L^{p}L^{q}}&\lesssim(1+\|(\varepsilon,\mu)\|_{C^{s}})\|S_{\lambda}u\|_{L^{2}}+\lambda^{\frac{s-2}{2}}\|P_{\lambda}S_{\lambda}u\|_{L^{2}}\\\
&\quad+\lambda^{-\frac{1}{2}+\frac{s-2}{4}}\|S_{\lambda}\rho_{em}\|_{L^{2}}.\end{split}$

The estimate corresponds to Strichartz estimates from [21] for wave equations
with rough coefficients _with integrability exponents as in the two-
dimensional case_. The reason that we cannot hope for the Strichartz range of
the wave equation in three dimensions is that the characteristic surface is no
longer regular, but has conical singularities. In [22, 23] Tataru used the
regularity of the Hamilton flow to recover the Euclidean Strichartz estimates
locally-in-time for $C^{2}$-coefficients and coefficients with
$\partial^{2}g\in L_{t}^{1}L_{x^{\prime}}^{\infty}$. In the present context it
is not clear how to take advantage of the Hamilton flow due to the strong
system character. For this reason, we do not recover the derivative loss over
scaling as for scalar wave equations with Hölder-continuous coefficients in
two space dimensions proved in [22]. (Also note that the results in [22] were
shown to be essentially optimal in [18]).

In a first step we show the above reduction result.

###  3.1. Proof of Proposition 3.4

Choosing $\lambda_{0}$ large enough, the uniform ellipticity is preserved
because

$\|\kappa_{\gtrsim\lambda}\|_{L^{\infty}_{x}}\lesssim\lambda^{-s}\|\kappa_{\gtrsim\lambda}\|_{C^{s}_{x}},\quad\kappa\in\\{\varepsilon;\mu\\}.$

Using Sobolev’s embedding, we first estimate low frequencies by

$\||D|^{-\rho+\frac{s-2}{4}}S_{0}u\|_{L^{p}L^{q}}\lesssim\||D|^{\frac{1}{2}+\frac{s-2}{4}}S_{0}u\|_{L^{2}}\lesssim\|S_{0}u\|_{L^{2}}.$

It remains to show the estimate

$\displaystyle\||D|^{-\rho+\frac{s-2}{4}}S_{\geq 1}u\|_{(L^{p}L^{q})_{2}}$
$\displaystyle\lesssim(1+\|(\varepsilon,\mu)\|_{C^{s}})\|u\|_{L^{2}}+\||D|^{\frac{s-2}{2}}Pu\|_{L^{2}}$
$\displaystyle\quad+\||D|^{-\frac{1}{2}+\frac{s-2}{4}}\rho_{em}\|_{L^{2}}$

for high frequencies. To deduce this inequality from (33) by means of the
definition of $(L^{p}L^{q})_{2}$, we have to bound
$\|P_{\lambda}S_{\lambda}u\|_{L^{2}}$ by $\|S_{\lambda}Pu\|_{L^{2}}$ plus
summable error terms. We write
$\tilde{S}_{\lambda}=\sum_{\mu=\lambda/8}^{8\lambda}S_{\mu}$ and
$P_{\sim\lambda}$ for the operator $P$ with coefficients frequency localized
near $\lambda$. We then obtain

$\displaystyle\|\tilde{S}_{\lambda}P_{\lambda}S_{\lambda}u\|_{L^{2}}$
$\displaystyle\leq\|\tilde{S}_{\lambda}PS_{\lambda}u\|_{L^{2}}+\|\tilde{S}_{\lambda}P_{\sim\lambda}S_{\lambda}u\|_{L^{2}}$
$\displaystyle\leq\|S_{\lambda}Pu\|_{L^{2}}+\|\tilde{S}_{\lambda}[P,S_{\lambda}]u\|_{L^{2}}+\|\tilde{S}_{\lambda}P_{\sim\lambda}S_{\lambda}u\|_{L^{2}}.$

Since $P$ is in divergence form, we can factor out the derivatives and
estimate

$\displaystyle\|\tilde{S}_{\lambda}[P,S_{\lambda}]u\|_{L^{2}}$
$\displaystyle\lesssim\lambda\|[(\varepsilon,\mu),S_{\lambda}]u\|_{L^{2}}\lesssim\lambda^{1-s}\|(\varepsilon,\mu)\|_{C^{s}}\|u\|_{L^{2}},$
$\displaystyle\|\tilde{S}_{\lambda}P_{\sim\lambda}S_{\lambda}u\|_{L^{2}}$
$\displaystyle\lesssim\lambda^{1-s}\|\lambda^{s}(\tilde{S}_{\lambda}(\varepsilon,\mu))S_{\lambda}u\|_{L^{2}}\lesssim\lambda^{1-s}\|(\varepsilon,\mu)\|_{C^{s}}\|u\|_{L^{2}}.$

In the last step, we used a well-known kernel estimate. This finishes the
proof of Proposition 3.4. $\hfill\Box$

In the following we tacitly suppose $\varepsilon$ and $\mu$ are truncated to
frequencies $<\lambda/8$ to lighten the notation.

###  3.2. Applying the FBI transform

In this section we prove (33). In the following we use the FBI transform to
reduce to an estimate in phase space. We refer to [17, Section 2] for further
explanations.

For $\lambda\in 2^{\mathbb{Z}}$, the FBI transform of an integrable function
$f:\mathbb{R}^{m}\to\mathbb{C}$ is defined by

$\begin{split}T_{\lambda}f(z)&=C_{m}\lambda^{\frac{3m}{4}}\int_{\mathbb{R}^{m}}e^{-\frac{\lambda}{2}(z-y)^{2}}f(y)dy,\quad
z=x-i\xi\in T^{*}\mathbb{R}^{m}\simeq\mathbb{R}^{2m},\\\
C_{m}&=2^{-\frac{m}{2}}\pi^{-\frac{3m}{4}}.\end{split}$

The FBI transform is an isometric mapping
$T_{\lambda}:L^{2}(\mathbb{R}^{m})\to L^{2}_{\Phi}(T^{*}\mathbb{R}^{m})$,
where $\Phi(z)=e^{-\lambda\xi^{2}}$. But $T_{\lambda}$ is not surjective
because $T_{\lambda}f$ is holomorphic (which motivates to write $z=x-i\xi$).
The adjoint in $L^{2}_{\Phi}$ provides the inversion formula
$T^{*}_{\lambda}T_{\lambda}f=f$ with

$T_{\lambda}^{*}F(y)=C_{m}\lambda^{\frac{3m}{4}}\int_{\mathbb{R}^{2m}}e^{-\frac{\lambda}{2}(\bar{z}-y)^{2}}\Phi(z)F(z)dxd\xi.$

Originally, the FBI transform was used in the context of microlocal analysis
to obtain an expansion of analytic symbols. Tataru reversed the logic in [21,
22] to find expansions of rough symbols $a\in C^{s}_{x}C^{\infty}_{c}$, which
satisfy suitable error bounds. Let $a\in C^{s}_{x}C^{\infty}_{c}$ with
$a(x,\xi)=0$ for $\xi\notin B(0,2)\backslash B(0,1/2)$ and

$\tilde{a}^{s}_{\lambda}=\sum_{|\alpha|+|\beta|<s}(\partial_{\xi}-\lambda\xi)^{\alpha}\frac{\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)}{|\alpha|!|\beta|!(-i\lambda)^{|\alpha|}\lambda^{|\beta|}}(\frac{1}{i}\partial_{x}-\lambda\xi)^{\beta}.$

For $s\leq 1$, we have $\tilde{a}^{s}_{\lambda}=a$. Tataru proved the
following approximation theorem in [22, Theorem 5] and [21, Theorem 2].

######  Theorem 3.5.

Suppose that $a\in C^{s}_{x}C^{\infty}_{c}$. Set
$a_{\lambda}(x,\xi)=a(x,\xi/\lambda)$ and
$R^{s}_{\lambda}=T_{\lambda}a_{\lambda}(x,D)-\tilde{a}^{s}_{\lambda}(x,\xi)T_{\lambda}$.
Then,

(34) $\|R^{s}_{\lambda}\|_{L^{2}\to
L^{2}_{\Phi}}\lesssim\lambda^{-\frac{s}{2}}.$

Moreover, if $\partial a\in L^{2}_{x_{0}}L^{\infty}_{x^{\prime}}$, then

(35) $\|R^{s}_{\lambda}\|_{L^{\infty}_{x_{0}}L^{2}_{x^{\prime}}\to
L^{2}_{\Phi}}\lesssim\lambda^{-\frac{1}{2}}.$

We return to the proof of (33). Write $P_{\lambda}S_{\lambda}u=J_{\lambda}$
and $T_{\lambda}S_{\lambda}u=v_{\lambda}$,
$T_{\lambda}(J_{\lambda}/\lambda)=f_{\lambda}$. An application of Theorem 3.5
yields

$p(x,\xi)v_{\lambda}=f_{\lambda}+g_{\lambda}\quad\text{with \
}\|g_{\lambda}\|_{L^{2}_{\Phi}}\lesssim_{\|(\varepsilon,\mu)\|_{C^{s}}}\lambda^{-\frac{s}{2}}\|S_{\lambda}u\|_{L^{2}}$

and

$p(x,\xi)=\begin{pmatrix}-i\xi_{0}\varepsilon&i\mathcal{C}(\xi^{\prime})\\\
i\mathcal{C}(\xi^{\prime})&i\xi_{0}\mu\end{pmatrix}\in\mathbb{C}^{6\times 6}.$

In the above display we denote
$\mathcal{C}(\xi^{\prime})=(-\varepsilon^{ijk}\xi_{k})_{ij}$ with
$(\varepsilon^{ijk})$ the Levi-Civita symbol such that $(\nabla\times
f)\widehat{(}\xi)=i\mathcal{C}(\xi^{\prime})\hat{f}(\xi)$. The estimate (33)
becomes

$\lambda^{-\rho+\frac{s-2}{4}}\|T_{\lambda}^{*}v_{\lambda}\|_{L^{p}L^{q}}\lesssim(1+\|(\varepsilon,\mu)\|_{C^{s}})\|v_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{\frac{s}{2}}\|f_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{\frac{s-2}{4}-\frac{1}{2}}\|S_{\lambda}\rho_{em}\|_{L^{2}}$

by invoking (34). It thus suffices to prove

(36)
$\|T_{\lambda}^{*}v_{\lambda}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{\frac{2-s}{4}}\|v_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{\frac{2+s}{4}}\|p(x,\xi)v_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{-\frac{1}{2}}\|S_{\lambda}\rho_{em}\|_{L^{2}}).$

By non-stationary phase, we can suppose that
$\text{supp}(v_{\lambda})\subseteq B(0,1)\times\\{1/4\leq|\xi|\leq 4\\}=:U$
(up to a negligible error). In the following we reduce the vector-valued
estimate to a scalar one. For this purpose we adapt arguments from the
analysis of the constant-coefficient time-harmonic case given in [12]. Let

(37)
$\tilde{q}(x,\xi)=-\xi_{0}^{2}q(x,\xi):=-\xi_{0}^{2}(\xi_{0}^{4}-\xi_{0}^{2}q_{0}(x,\xi)+q_{1}(x,\xi)),$

where

$\displaystyle q_{0}(x,\xi)$
$\displaystyle=\xi_{1}^{2}\big{(}\frac{1}{\varepsilon_{2}\mu_{3}}+\frac{1}{\mu_{2}\varepsilon_{3}}\big{)}+\xi_{2}^{2}\big{(}\frac{1}{\varepsilon_{1}\mu_{3}}+\frac{1}{\mu_{1}\varepsilon_{3}}\big{)}+\xi_{3}^{2}\big{(}\frac{1}{\varepsilon_{1}\mu_{2}}+\frac{1}{\varepsilon_{2}\mu_{1}}\big{)},$
$\displaystyle q_{1}(x,\xi)$
$\displaystyle=\frac{1}{\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}\mu_{1}\mu_{2}\mu_{3}}(\varepsilon_{1}\xi_{1}^{2}+\varepsilon_{2}\xi_{2}^{2}+\varepsilon_{3}\xi_{3}^{2})(\mu_{1}\xi_{1}^{2}+\mu_{2}\xi_{2}^{2}+\mu_{3}\xi_{3}^{2}).$

The dependence of $\varepsilon$ and $\mu$ on $x$ is suppressed in the above
display for the sake of brevity.

######  Proposition 3.6.

For the proof of (36) under the assumptions of Proposition 3.4, it suffices to
show the estimate

(38)
$\|T_{\lambda}^{*}w_{\lambda}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{\frac{2-s}{4}}\|w_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{\frac{2+s}{4}}\|q(x,\xi)w_{\lambda}\|_{L^{2}_{\Phi}})$

for $w_{\lambda}:\mathbb{R}^{4}\times\mathbb{R}^{4}\to\mathbb{C}$ with
$\text{supp}(w_{\lambda})\subseteq[-1,1]^{4}\times\\{(\xi_{0},\xi^{\prime})\in\mathbb{R}\times\mathbb{R}^{3}:|\xi_{0}|\sim|\xi^{\prime}|\sim
1\\}$.

###### Proof.

In the first step, we separate the evolution of the $\mathcal{E}$\- and
$\mathcal{H}$-variables by passing to a second order system. We multiply $p$
with the symmetrizer $\sigma$

$\sigma(x,\xi)=\begin{pmatrix}-i\xi_{0}\varepsilon^{-1}&i\varepsilon^{-1}\mathcal{C}(\xi^{\prime})\mu^{-1}\\\
i\mu^{-1}\mathcal{C}(\xi^{\prime})\varepsilon^{-1}&i\xi_{0}\mu^{-1}\end{pmatrix}\in\mathbb{C}^{6\times
6}$

(cf. [12, Proposition 1.2]), obtaining

(39) $\sigma(x,\xi)p(x,\xi)=\begin{pmatrix}M_{E}(x,\xi)-\xi_{0}^{2}&0\\\
0&M_{H}(x,\xi)-\xi_{0}^{2}\end{pmatrix}.$

Here we set

$\displaystyle
M_{E}(x,\xi)=-\varepsilon^{-1}(x)\mathcal{C}(\xi^{\prime})\mu^{-1}(x)\mathcal{C}(\xi^{\prime}),\quad
M_{H}(x,\xi)=-\mu^{-1}(x)\mathcal{C}(\xi^{\prime})\varepsilon^{-1}(x)\mathcal{C}(\xi^{\prime}).$

Moreover, in [12, p. 1835, Eq. (13)] it was pointed out that

$q(x,\xi)=\det(M_{E}(x,\xi^{\prime})-\xi_{0}^{2})=\det(M_{H}(x,\xi^{\prime})-\xi_{0}^{2}).$

Let $v_{\lambda}=(v_{\lambda,1},v_{\lambda,2})$ denote the grouping into
$3$-vectors. By boundedness of the entries of $\sigma(x,\xi)$ for $\xi\in U$,
it suffices to prove

$\|T_{\lambda}^{*}v_{\lambda}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{\frac{2-s}{4}}\|v_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{\frac{2+s}{4}}\|\sigma(x,\xi)p(x,\xi)v_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{-\frac{1}{2}}\|S_{\lambda}\rho_{em}\|_{L^{2}})$

which can be written as

$\displaystyle\|T_{\lambda}^{*}v_{\lambda,1}\|_{L^{p}L^{q}}$
$\displaystyle\lesssim\lambda^{\rho}(\lambda^{\frac{2-s}{4}}\|v_{\lambda,1}\|_{L^{2}_{\Phi}}+\lambda^{\frac{2+s}{4}}\|(M_{E}(x,\xi^{\prime})-\xi_{0}^{2})v_{\lambda,1}\|_{L^{2}_{\Phi}}$
(40)
$\displaystyle\quad+\lambda^{-\frac{1}{2}}\|S_{\lambda}\rho_{e}\|_{L^{2}}),$
$\displaystyle\|T_{\lambda}^{*}v_{\lambda,2}\|_{L^{p}L^{q}}$
$\displaystyle\lesssim\lambda^{\rho}(\lambda^{\frac{2-s}{4}}\|v_{\lambda,2}\|_{L^{2}_{\Phi}}+\lambda^{\frac{2+s}{4}}\|(M_{H}(x,\xi^{\prime})-\xi_{0}^{2})v_{\lambda,2}\|_{L^{2}_{\Phi}}$
(41)
$\displaystyle\quad+\lambda^{-\frac{1}{2}}\|S_{\lambda}\rho_{m}\|_{L^{2}}).$

The estimates will be proved in the regions

  * (1)

$\\{1\sim|\xi_{0}|\gg|\xi^{\prime}|\\}$,

  * (2)

$\\{1\sim|\xi^{\prime}|\gg|\xi_{0}|\\}$, for which we have to consider the
charges,

  * (3)

$\\{1\sim|\xi_{0}|\sim|\xi^{\prime}|\\}$.

The region $\\{1\sim|\xi_{0}|\gg|\xi^{\prime}|\\}$ is easy to handle. Here we
have

$\|(M_{X}(x,\xi^{\prime})-\xi_{0}^{2})v_{\lambda}\|_{L^{2}_{\Phi}}\gtrsim\|v_{\lambda}\|_{L^{2}_{\Phi}},\quad
X\in\\{E;H\\},$

and so the claim follows by Sobolev embedding via

$\|T_{\lambda}^{*}v_{\lambda,i}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho+\frac{1}{2}}\|v_{\lambda,i}\|_{L^{2}_{\Phi}}\lesssim\lambda^{\rho+\frac{1}{2}+\frac{s}{4}}\|(M_{X}(x,\xi^{\prime})-\xi_{0}^{2})v_{\lambda}\|_{L^{2}_{\Phi}}.$

For the second region $\\{|\xi_{0}|\ll|\xi^{\prime}|\sim 1\\}$ the charges
come into play. We decompose
$v_{\lambda,1}=v_{\lambda,1}^{s}+v_{\lambda,1}^{p}$ with

$v_{\lambda,1}^{p}=\frac{(\xi^{\prime}.\varepsilon^{\lambda}v_{\lambda,1})\xi^{\prime}}{|\xi^{\prime}|^{2}_{\varepsilon}},\quad|\xi^{\prime}|^{2}_{\varepsilon}=\xi^{\prime}.\varepsilon^{\lambda}\xi^{\prime},$

again indicating the frequency cutoff of the coefficients temporarily. The
contribution of $v_{\lambda,1}^{p}$ is estimated by Sobolev embedding:

$\|T_{\lambda}^{*}v_{\lambda,1}^{p}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho+\frac{1}{2}}\|v_{\lambda,1}^{p}\|_{L^{2}_{\Phi}}\lesssim\lambda^{\rho+\frac{1}{2}}\|(\frac{\xi^{\prime}}{|\xi^{\prime}|_{\varepsilon}}.\varepsilon^{\lambda}v_{\lambda,1})\|_{L^{2}_{\Phi}}.$

Theorem 3.5 and commutator estimates yield

$\|\frac{\xi^{\prime}}{|\xi^{\prime}|_{\varepsilon}}.(\varepsilon^{\lambda}v_{\lambda,1})\|_{L^{2}_{\Phi}}\lesssim\lambda^{-\frac{s}{2}}\|u_{\lambda}\|_{L^{2}}+\lambda^{-1}\|S_{\lambda}\nabla^{\prime}.(\varepsilon^{\lambda}\mathcal{E})\|_{L^{2}}.$

By expanding
$\varepsilon^{\lambda}=\varepsilon+(\varepsilon-\varepsilon^{\lambda})$, we
find

$\begin{split}\lambda^{-1}\|S_{\lambda}\nabla^{\prime}.(\varepsilon^{\lambda}\mathcal{E})\|_{L^{2}}&\lesssim\lambda^{-1}\|S_{\lambda}\nabla^{\prime}.(\varepsilon\mathcal{E})\|_{L^{2}}+\|(\varepsilon-\varepsilon^{\lambda})\tilde{S}_{\lambda}\mathcal{E}\|_{L^{2}}\\\
&\lesssim\lambda^{-1}\|S_{\lambda}\rho_{e}\|_{L^{2}}+\lambda^{-s}\|\tilde{S}_{\lambda}\mathcal{E}\|_{L^{2}}.\end{split}$

This is an acceptable estimate for $v_{\lambda,1}^{p}$ in terms of $\rho_{e}$
and $\mathcal{E}$.

We turn to $v_{\lambda,1}^{s}$ noting that
$\xi^{\prime}.(\varepsilon^{\lambda}v_{1,\lambda}^{s})=0$. In the following we
show that
$|(M_{E}(x,\xi)-\xi_{0}^{2})v_{1}^{\lambda}|\gtrsim|v_{1}^{\lambda}|$ for
$\xi^{\prime}.(\varepsilon^{\lambda}v_{1}^{\lambda})=0$ and
$\\{|\xi_{0}|\ll|\xi^{\prime}|\\}$. We mit again the superscript $\lambda$ of
the coefficients. For $\xi_{0}\neq 0$ we have $q(x,\xi)\neq 0$. Let
$w=(M_{E}-\xi_{0}^{2})v_{1}$. We have to show that

$|(M_{E}-\xi_{0}^{2})^{-1}w|\lesssim|w|$

noting that $\xi^{\prime}.(\varepsilon w)=0$. In [12, Section 2.1] the matrix
$(M_{E}-\xi_{0}^{2})$ was inverted by

$(M_{E}-\xi_{0}^{2})^{-1}=q^{-1}(x,\xi)\text{adj}(M_{E}-\xi_{0}^{2})=\frac{-1}{\xi_{0}^{2}(\xi_{0}^{4}-\xi_{0}^{2}q_{0}(x,\xi)+q_{1}(x,\xi))}Z_{\varepsilon,\mu}\frac{\varepsilon}{\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}},$

wher the components of $Z=Z_{\varepsilon,\mu}$ are given by

$\displaystyle\begin{aligned}
Z_{11}(\xi)&=\xi_{1}^{2}(\frac{\xi_{1}^{2}}{\mu_{2}\mu_{3}}+\frac{\xi_{2}^{2}}{\mu_{1}\mu_{3}}+\frac{\xi_{3}^{2}}{\mu_{1}\mu_{2}})-\xi_{0}^{2}(\frac{\varepsilon_{2}}{\mu_{2}}\xi_{1}^{2}+\frac{\varepsilon_{3}}{\mu_{3}}\xi_{1}^{2}+\frac{\varepsilon_{2}}{\mu_{1}}\xi_{2}^{2}+\frac{\varepsilon_{3}}{\mu_{1}}\xi_{3}^{2})+\xi_{0}^{4}\varepsilon_{2}\varepsilon_{3},\\\
Z_{12}(\xi)&=Z_{21}(\xi)=\xi_{1}\xi_{2}(\frac{\xi_{1}^{2}}{\mu_{2}\mu_{3}}+\frac{\xi_{2}^{2}}{\mu_{1}\mu_{3}}+\frac{\xi_{3}^{2}}{\mu_{1}\mu_{2}}-\xi_{0}^{2}\frac{\varepsilon_{3}}{\mu_{3}}),\\\
Z_{13}(\xi)&=Z_{31}(\xi)=\xi_{1}\xi_{3}(\frac{\xi_{1}^{2}}{\mu_{2}\mu_{3}}+\frac{\xi_{2}^{2}}{\mu_{1}\mu_{3}}+\frac{\xi_{3}^{2}}{\mu_{1}\mu_{2}}-\xi_{0}^{2}\frac{\varepsilon_{2}}{\mu_{2}}),\\\
Z_{22}(\xi)&=\xi_{2}^{2}(\frac{\xi_{1}^{2}}{\mu_{2}\mu_{3}}+\frac{\xi_{2}^{2}}{\mu_{1}\mu_{3}}+\frac{\xi_{3}^{2}}{\mu_{1}\mu_{2}})-\xi_{0}^{2}(\frac{\varepsilon_{1}}{\mu_{2}}\xi_{1}^{2}+\frac{\varepsilon_{3}}{\mu_{3}}\xi_{2}^{2}+\frac{\varepsilon_{1}}{\mu_{1}}\xi_{2}^{2}+\frac{\varepsilon_{3}}{\mu_{2}}\xi_{3}^{2})+\xi_{0}^{4}\varepsilon_{1}\varepsilon_{3},\\\
Z_{23}(\xi)&=Z_{32}(\xi)=\xi_{2}\xi_{3}(\frac{\xi_{1}^{2}}{\mu_{2}\mu_{3}}+\frac{\xi_{2}^{2}}{\mu_{1}\mu_{3}}+\frac{\xi_{3}^{2}}{\mu_{1}\mu_{2}}-\xi_{0}^{2}\frac{\varepsilon_{1}}{\mu_{1}}),\\\
Z_{33}(\xi)&=\xi_{3}^{2}(\frac{\xi_{1}^{2}}{\mu_{2}\mu_{3}}+\frac{\xi_{2}^{2}}{\mu_{1}\mu_{3}}+\frac{\xi_{3}^{2}}{\mu_{1}\mu_{2}})-\xi_{0}^{2}(\frac{\varepsilon_{1}}{\mu_{3}}\xi_{1}^{2}+\frac{\varepsilon_{2}}{\mu_{3}}\xi_{2}^{2}+\frac{\varepsilon_{1}}{\mu_{1}}\xi_{3}^{2}+\frac{\varepsilon_{2}}{\mu_{2}}\xi_{3}^{2})+\xi_{0}^{4}\varepsilon_{1}\varepsilon_{2}.\end{aligned}$

It is a crucial observation that $Z_{\varepsilon,\mu}$ will be applied to
vectors $v$ with $\xi^{\prime}.v=0$. This reduces to
$Z_{\varepsilon,\mu}^{\text{eff}}$, having the coefficcients

$\displaystyle\begin{aligned}
Z^{\text{eff}}_{11}(\xi)&=-\xi_{0}^{2}(\frac{\varepsilon_{2}}{\mu_{2}}\xi_{1}^{2}+\frac{\varepsilon_{3}}{\mu_{3}}\xi_{1}^{2}+\frac{\varepsilon_{2}}{\mu_{1}}\xi_{2}^{2}+\frac{\varepsilon_{3}}{\mu_{1}}\xi_{3}^{2})+\xi_{0}^{4}\varepsilon_{2}\varepsilon_{3},\\\
Z^{\text{eff}}_{12}(\xi)&=Z^{\text{eff}}_{21}(\xi)=-\xi_{1}\xi_{2}\xi_{0}^{2}\frac{\varepsilon_{3}}{\mu_{3}},\\\
Z^{\text{eff}}_{13}(\xi)&=Z^{\text{eff}}_{31}(\xi)=-\xi_{0}^{2}\frac{\varepsilon_{2}}{\mu_{2}}\xi_{1}\xi_{3},\\\
Z^{\text{eff}}_{22}(\xi)&=-\xi_{0}^{2}(\frac{\varepsilon_{1}}{\mu_{2}}\xi_{1}^{2}+\frac{\varepsilon_{3}}{\mu_{3}}\xi_{2}^{2}+\frac{\varepsilon_{1}}{\mu_{1}}\xi_{2}^{2}+\frac{\varepsilon_{3}}{\mu_{2}}\xi_{3}^{2})+\xi_{0}^{4}\varepsilon_{1}\varepsilon_{3},\\\
Z^{\text{eff}}_{23}(\xi)&=Z^{\text{eff}}_{32}(\xi)=-\xi_{2}\xi_{3}\xi_{0}^{2}\frac{\varepsilon_{1}}{\mu_{1}},\\\
Z^{\text{eff}}_{33}(\xi)&=-\xi_{0}^{2}(\frac{\varepsilon_{1}}{\mu_{3}}\xi_{1}^{2}+\frac{\varepsilon_{2}}{\mu_{3}}\xi_{2}^{2}+\frac{\varepsilon_{1}}{\mu_{1}}\xi_{3}^{2}+\frac{\varepsilon_{2}}{\mu_{2}}\xi_{3}^{2})+\xi_{0}^{4}\varepsilon_{1}\varepsilon_{2}.\end{aligned}$

We can write $Z^{\text{eff}}=\xi_{0}^{2}\tilde{Z}^{\text{eff}}$ with
$\tilde{Z}^{\text{eff}}$ having bounded entries for $|\xi_{0}|\lesssim 1$.
This gives

$\displaystyle(M_{E}(x,\xi)-\xi_{0}^{2})^{-1}w$
$\displaystyle=q^{-1}(x,\xi)\text{adj}(M_{E}-\xi_{0}^{2})w$
$\displaystyle=\frac{-1}{(\xi_{0}^{4}-\xi_{0}^{2}q_{0}(x,\xi)+q_{1}(x,\xi))}\tilde{Z}^{\text{eff}}_{\varepsilon,\mu}\frac{\varepsilon}{\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}}w.$

For $|\xi_{0}|\ll|\xi^{\prime}|$ or $|\xi_{0}|\gg|\xi^{\prime}|$, we have

$|\xi_{0}^{4}-\xi_{0}^{2}q_{0}(\xi)+q_{1}(\xi)|\sim 1\text{ provided that
}\max(|\xi_{0}|,|\xi^{\prime}|)\sim 1.$

Consequently, for $|\xi_{0}|\ll|\xi^{\prime}|\sim 1$, it follows

$\|(M_{E}(x,\xi)-\xi_{0}^{2})v\|_{L^{2}_{\Phi}(U)}\gtrsim\|v\|_{L^{2}_{\Phi}}.$

This shows estimate (3.2) in the second region. (41) follows by exchanging the
roles of $\varepsilon$ and $\mu$.

We turn to the third region. Again we focus on (3.2), for which we prove

$\|T_{\lambda}^{*}v_{\lambda,1}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{\frac{2-s}{4}}\|v_{\lambda,1}\|_{L^{2}_{\Phi}}+\lambda^{\frac{2+s}{4}}\|(M_{E}-\xi_{0}^{2})v_{\lambda,1}\|_{L^{2}_{\Phi}})$

for $v_{\lambda}$ supported in $B(0,1)\times\\{|\xi_{0}|\sim|\xi^{\prime}|\sim
1\\}$. This inequality is a consequence from (38). Indeed, for any component
$v_{\lambda,1,k}$, $k=1,2,3$, the estimate (38) yields

$\|T_{\lambda}^{*}v_{\lambda,1,k}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{\frac{2-s}{4}}\|v_{\lambda,1,k}\|_{L^{2}_{\Phi}}+\lambda^{\frac{2+s}{4}}\|(Z_{\varepsilon,\mu}\frac{\varepsilon}{\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}}(M_{E}-\xi_{0}^{2})v_{\lambda,1})_{k}\|_{L^{2}_{\Phi}})$

because
$Z_{\varepsilon,\mu}\frac{\varepsilon}{\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}}(M_{E}-\xi_{0}^{2})=q(x,\xi)$.
The assertion easily follows. ∎

It thus suffices to show for scalar $v_{\lambda}\in L^{2}_{\Phi}$ supported in
$\\{\frac{1}{4}\leq|\xi|\leq 4\\}$ that the estimate

(42)
$\|T_{\lambda}^{*}v_{\lambda}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{\frac{2-s}{4}}\|v_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{\frac{2+s}{4}}\|q(x,\xi)v_{\lambda}\|_{L^{2}_{\Phi}})$

holds true. We consider the operator

$W_{\lambda}=T_{\lambda}^{*}\frac{a(x,\xi)}{\lambda^{-\frac{s}{4}}+\lambda^{\frac{s}{4}}|q(x,\xi)|}$

with $a\in C^{\infty}_{c}$ localizing to
$B(0,1)\times\\{\frac{1}{4}\leq|\xi|\leq 4\\}$. Hence it is enough to
establish

$\|W_{\lambda}\|_{L_{\Phi}^{2}\to
L^{p}L^{q}}\lesssim\lambda^{\rho+\frac{1}{2}}.$

By the $TT^{*}$-argument, we can likewise prove the estimate

$\|\tilde{V}_{\lambda}\|_{L^{p^{\prime}}L^{q^{\prime}}\to
L^{p}L^{q}}\lesssim\lambda^{1+2\rho},\quad\tilde{V}_{\lambda}=T_{\lambda}^{*}\frac{a^{2}(x,\xi)\Phi(\xi)}{(\lambda^{-\frac{s}{4}}+\lambda^{\frac{s}{4}}|q(x,\xi)|)^{2}}T_{\lambda}.$

For this estimate it is crucial to understand the curvature properties of
$\\{\xi\in\mathbb{R}^{4}:q(x,\xi)=0\\}$. Since $q(x,\xi)$ is $4$-homogeneous
in $\xi$, we have

$q(x,\xi)=\xi_{0}^{4}q(x,1,\xi^{\prime}/\xi_{0}).$

So it suffices to consider the Fresnel surface
$S=\\{\xi^{\prime}\in\mathbb{R}^{3}:q(x,1,\xi^{\prime})=0\\}$. The Fresnel
surface is classical and has been studied since the 19th century (cf. [2, 8,
5]). We refer to detailed computations laid out in [12] and recall properties
of the Fresnel surface $S$. We can consider $S$ as a union of three components
$S=S_{1}\cup S_{2}\cup S_{3}$ with

  * •

$S_{1}$ describes a regular bounded surface with Gaussian curvature bounded
from below and above,

  * •

$S_{2}$ is a neighbourhood of the Hamilton circles, which is regular, but has
one principal curvature vanishing along the Hamilton circles,

  * •

$S_{3}$ is a neighbourhood of four conical singularities.

We redenote $a^{2}(x,\xi)$ by $a(x,\xi)$ for the sake of brevity, and split
$a=a_{1}+a_{2}+a_{3}$ and the operator
$V_{\lambda}=V_{\lambda,1}+V_{\lambda,2}+V_{\lambda,3}$ according to the above
components. We establish the three estimates separately, which is meant to
highlight a different dispersive behavior. To see that the uniformity of the
curvature bounds follows from the separability condition (26), we revisit the
analysis of the Fresnel surface provided in [12, Section 3].

First, the analysis of Fresnel’s surface for $|\xi_{0}|\sim|\xi^{\prime}|\sim
1$

$S_{\xi_{0}}=\\{\xi^{\prime}\in\mathbb{R}^{3}:p(\xi_{0},\xi^{\prime})=-\xi_{0}^{2}(\xi_{0}^{4}-\xi_{0}^{2}q_{0}(\xi^{\prime})+q_{1}(\xi^{\prime}))=0\\}$

is reduced to the standard form $\mu_{1}=\mu_{2}=\mu_{3}=\xi_{0}=1$ by means
of the substitution

$\lambda_{i}=\frac{\xi_{i}}{\xi_{0}\sqrt{\mu_{i+1}\mu_{i+2}}}\qquad(i=1,2,3).$

Note that the change of coordinates is as smooth as $\mu$. We use cyclic
notation in the following, i.e., $\mu_{4}:=\mu_{1}$ and $\mu_{5}:=\mu_{2}$,
likewise for $\varepsilon_{i}$. In the following let
$\varepsilon_{i}:=\frac{\varepsilon_{i}}{\mu_{i}}$ (the right-hand side refers
to the original quantities).

We shall analyze
$S^{*}=\\{\lambda\in\mathbb{R}^{3}:1-q_{0}^{*}(\lambda)+q_{1}^{*}(\lambda)=0\\}$
with $q_{0}^{*}$ and $q_{1}^{*}$ denoting the reduced polynomials $q_{0}$ and
$q_{1}$ for $\mu_{1}=\mu_{2}=\mu_{3}=1$. We find that the coordinates of the
four singular points are given by

$\lambda_{j}^{2}=\frac{\varepsilon_{j+2}(\varepsilon_{j}-\varepsilon_{j+1})}{\varepsilon_{j}-\varepsilon_{j+2}},\quad\lambda_{j+1}=0,\quad\lambda_{j+2}^{2}=\frac{\varepsilon_{j}(\varepsilon_{j+2}-\varepsilon_{j+1})}{\varepsilon_{j+2}-\varepsilon_{j}}$

for $\varepsilon_{j+1}\in\langle\varepsilon_{j},\varepsilon_{j+2}\rangle$,
which denotes
$\varepsilon_{j+1}\in[\varepsilon_{j},\varepsilon_{j+2}]\cup[\varepsilon_{j+2},\varepsilon_{j}]$.
By uniform ellipticity and the separation condition (26), the singular points
vary as smoothly in $x\in\mathbb{R}^{4}$ as $(\varepsilon,\mu)$.

To analyze the regularity of the curvature in $(\varepsilon,\mu)$, we recall
the parametrization of the regular part of the Fresnel surface given by

$s=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2},\quad\varepsilon_{1}\lambda_{1}^{2}+\varepsilon_{2}\lambda_{2}^{2}+\varepsilon_{3}\lambda_{3}^{2}=\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}t^{-1},$

and

$\alpha(s,t)=\frac{(t-s)\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}}{s^{2}t-(\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3})st+(\varepsilon_{1}\varepsilon_{2}+\varepsilon_{1}\varepsilon_{3}+\varepsilon_{2}\varepsilon_{3})t-\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}}.$

In $(s,t)$ coordinates, we have for the Gaussian curvature

$K(s,t)=\frac{(\alpha(s,t)-\varepsilon_{1})(\alpha(s,t)-\varepsilon_{2})(\alpha(s,t)-\varepsilon_{3})}{\alpha(s,t)(s-\varepsilon_{1})(s-\varepsilon_{2})(s-\varepsilon_{3})}$

and for the mean curvature

$\begin{split}K_{m}(s,t)&=-\frac{1}{2}\big{(}\frac{s}{\sqrt{\alpha}}K(s,t)-\frac{1}{\sqrt{\alpha}}\big{(}\frac{(\alpha-\varepsilon_{1})(\alpha-\varepsilon_{2})}{(s-\varepsilon_{1})(s-\varepsilon_{2})}+\frac{(\alpha-\varepsilon_{2})(\alpha-\varepsilon_{3})}{(s-\varepsilon_{2})(s-\varepsilon_{3})}\\\
&\qquad+\frac{(\alpha-\varepsilon_{1})(\alpha-\varepsilon_{3})}{(s-\varepsilon_{1})(s-\varepsilon_{3})}\big{)}\big{)}.\end{split}$

The limiting case $s=t$ corresponds to singular points. For separated $s$ and
$t$ we have $\alpha\geq d>0$ and the uniformity of the curvature bounds
follows. For the analysis of the curvature close to the singular points, we
rescale a dyadic decomposition away from the singular points to unit distance
(see below), for which the previous arguments apply.

We now conclude the proof of the Strichartz estimates in Theorem 3.2 for sharp
Strichartz exponents close to the endpoint. Note that the following
proposition implies Strichartz estimates for the contribution of
$V_{\lambda,1}$ like in (42) for the wider range
$\frac{1}{p}+\frac{1}{q}\leq\frac{1}{2}$ and $2<p<q\leq\infty$ by Sobolev
embedding. We recall the defintion of $V_{\lambda,i}$ in (46) below.

######  Proposition 3.7.

Let $\rho=3\big{(}\frac{1}{2}-\frac{1}{q}\big{)}-\frac{1}{p}$ and
$2<p<q\leq\infty$. The estimate

(43) $\|V_{\lambda,i}\|_{L^{p^{\prime}}L^{q^{\prime}}\to
L^{p}L^{q}}\lesssim\lambda^{1+2\rho}$

holds true, if

  * •

$i=1$ and $\frac{1}{p}+\frac{1}{q}=\frac{1}{2}$,

  * •

$i\in\\{2,3\\}$ and $\frac{2}{p}+\frac{1}{q}=\frac{1}{2}$.

For $i=1,2$, the assumption $2<p<q\leq\infty$ is not necessary because we can
interpolate with (43) for $p=\infty$ and $q=2$. We show this case. In the
proof we will see that if fails for $i=3$ at the singular points because these
satisfy $\partial_{\xi}q(x,\xi_{x})=0$.

After factoring out the FBI transform in the $x^{\prime}$ variables by virtue
of its $L^{2}$-boundedness, we it remains to prove the boundedness of

(44)
$\lambda^{-1}T_{\lambda,0}^{*}a_{i}(x,\xi)\Phi(\xi)\delta_{q=0}T_{\lambda,0}:L_{t}^{1}L_{x^{\prime},\xi^{\prime}}^{2}\to
L_{t}^{\infty}L_{x^{\prime},\xi^{\prime}}^{2}$

for the restricted FBI transform $T_{\lambda,0}:L^{2}(\mathbb{R}^{n})\to
L^{2}_{\Phi}(\mathbb{R}^{2}\times\mathbb{R}^{n-1})$ given by

$T_{\lambda,0}f(x_{0},\xi_{0},y^{\prime})=2^{-\frac{1}{2}}\pi^{-\frac{3}{4}}\lambda^{\frac{3}{4}}\int
e^{-\frac{\lambda}{2}(z-y_{0})^{2}}f(y_{0},y^{\prime})dy_{0}$

with $\Phi(x_{0},\xi_{0},x^{\prime})=e^{-\lambda|\xi_{0}|^{2}}$. (Compare the
proof of Proposition 3.7 below.) Hence, to establish (44), we have to show the
boundedness of the kernel

$K(y_{0},\bar{y}_{0})=\lambda^{\frac{1}{2}}\int_{\mathbb{R}}dx_{0}e^{-\frac{\lambda}{2}(x_{0}-y_{0})^{2}}e^{-\frac{\lambda}{2}(x_{0}-\bar{y}_{0})^{2}}\int_{\mathbb{R}}d\xi_{0}e^{i\lambda\xi_{0}.(y_{0}-\bar{y}_{0})}a_{0}(x_{0},\xi_{0})\delta_{q=0}.$

It is enough to check the hyperbolicity condition

(45) $\partial_{\xi_{0}}q(x,\xi)\neq 0\quad\text{for \ }a_{i}(x,\xi)\neq
0,\;\;i=1,2.$

Indeed, from (45) follows

$\big{|}\int_{\mathbb{R}}d\xi_{0}e^{i\lambda\xi_{0}.(y_{0}-\bar{y}_{0})}a_{0}(x_{0},\xi_{0})\delta_{q=0}\big{|}\lesssim
1$

and

$\int_{\mathbb{R}}dx_{0}e^{-\frac{\lambda}{2}(x_{0}-y_{0})^{2}}e^{-\frac{\lambda}{2}(x_{0}-\bar{y}_{0})^{2}}\lesssim\lambda^{-\frac{1}{2}},$

yielding (43) for $(p,q)=(\infty,2)$. For the remaining proof of (45), we
consider

$\partial_{\xi_{0}}q=4\xi_{0}^{3}-2\xi_{0}q_{0}=0.$

This implies $q_{0}=2\xi_{0}^{2}$. Since $q(\xi_{0},\xi^{\prime})=0$, we find
$q_{1}=\xi_{0}^{4}$ and so $q_{1}=(\frac{q_{0}}{2})^{2}$. By homogeneity of
$q$, it is enough to consider $\xi_{0}=1$. Then we use again the
parametrisations of [12]:

$\displaystyle s=\xi_{1}^{2}+\xi_{2}^{2}+\xi_{3}^{2},$
$\displaystyle\qquad\varepsilon_{1}\xi_{1}^{2}+\varepsilon_{2}\xi_{2}^{2}+\varepsilon_{3}\xi_{3}^{2}=\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}t^{-1},$
$\displaystyle q_{1}^{*}(\xi)=st^{-1},$ $\displaystyle\qquad
q_{0}^{*}(\xi)=1+st^{-1}.$

Let $b=st^{-1}$. From $q_{1}=(\frac{q_{0}}{2})^{2}$ follows
$b=\big{(}\frac{1+b}{2}\big{)}^{2}$ and hence $(b-1)^{2}=0$. In the regular
part of $S$ we have $s<t$ or $t<s$ corresponding to the two sheets of $S$,
where $t=s$ formally corresponds to the singular points. This ensures
hyperbolicity within $\text{supp}(a_{1})\cup\text{supp}(a_{2})$. However,
within $\text{supp}(a_{3})$ the argument fails. We can thus prove (43) only
close to the endpoint, which yields the corresponding Strichartz estimates.

###### Proof of Proposition 3.7.

For $i=1,2$, the result is shown by a straight-forward adaptation of Tataru’s
arguments [21]. We shall be brief, but repeat the argument because in the most
difficult case $i=3$ we will elaborate on it. To prove a uniform bound for the
family of operators

(46)
$\lambda^{-1-2\rho}V_{\lambda,i}=\lambda^{-1-2\rho}T_{\lambda}^{*}\frac{a_{i}(x,\xi)\Phi(\xi)}{(\lambda^{-\frac{s}{4}}+\lambda^{\frac{s}{4}}|q(x,\xi)|)^{2}}T_{\lambda}:L^{p^{\prime}}L^{q^{\prime}}\to
L^{p}L^{q},$

we use the integrability of the weight

$\int\frac{1}{(\lambda^{-\frac{s}{4}}+\lambda^{\frac{s}{4}}|q|)^{2}}dq\lesssim
1$

and can reduce to level sets $\delta_{q(x,\xi)=c}$ by foliation. It is enough
to consider $c=0$ as $\\{\xi\in\mathbb{R}^{4}:q(x,\xi)=0\\}$ is a regular
surface within the supports of $a_{i}$, $i=1,2$, and the supports can be
properly foliated.

For the proof of the non-endpoint estimate (43) in case $2<p<q\leq\infty$, we
consider the analytic family of operators

$V_{\lambda,i}^{\theta}=\lambda^{-\theta(2\rho+1)}\frac{e^{(\theta-1)^{2}}}{\Gamma(1-\theta)}T_{\lambda}^{*}a_{i}(x,\xi)\Phi(\xi)q^{-\theta}(x,\xi)T_{\lambda}$

such that $\lambda^{-1-2\rho}V_{\lambda,i}=V_{\lambda,i}^{1}$.

The estimate follows from interpolation between $\lambda$-independent bouns of

$\displaystyle V_{\lambda,i}^{\theta}:L^{2}\to L^{2},\qquad\qquad\quad\
\,\Re\theta=0,$ $\displaystyle
V_{\lambda,i}^{\theta}:L^{p_{1}^{\prime}}L^{1}\to
L^{p_{1}}L^{\infty},\qquad\Re\theta=\theta_{1},$

where $p_{1},\theta_{1}$ are chosen so that the following points are
collinear:

(47)
$\big{(}\frac{1}{2},\frac{1}{2},0\big{)},\qquad\big{(}\frac{1}{p},\frac{1}{q},1\big{)},\qquad\big{(}\frac{1}{p_{1}},0,\theta_{1}\big{)}.$

The $L^{2}$-estimate is immediate from the $L^{2}$-mapping properties of the
FBI transform. For the $L^{p_{1}^{\prime}}L^{1}\to L^{p_{1}}L^{\infty}$ bound,
we compute the kernel $K^{\theta}_{\lambda,i}$ of $V^{\theta}_{\lambda,i}$ as

$\begin{split}K^{\theta}_{\lambda,i}(y,\bar{y})&=\lambda^{6-\theta(2\rho+1)}\int_{\mathbb{R}^{4}}dxe^{-\frac{\lambda}{2}(y-x)^{2}}e^{-\frac{\lambda}{2}(\bar{y}-x)^{2}}\\\
&\quad\times\frac{e^{(\theta-1)^{2}}}{\Gamma(1-\theta)}\int_{\mathbb{R}^{4}}d\xi
e^{i\lambda(y-\bar{y}).\xi}q^{-\theta}(x,\xi)a_{i}(x,\xi).\end{split}$

We have the well-known oscillatory integral estimate (cf. [19, Chapter IX,
Section 1.2.3])

(48)
$\big{|}\frac{e^{(\theta-1)^{2}}}{\Gamma(1-\theta)}\int_{\mathbb{R}^{d}}e^{iy.\xi}a_{i}(x,\xi)q^{-\theta}(x,\xi)d\xi\big{|}\lesssim(1+|y|)^{(\Re\theta-1)-\frac{k_{i}}{2}},$

where $k_{1}=2$ and $k_{2}=1$ are the numbers of principal curvatures of the
regular surface $\\{q(x,\xi)=0\\}$222The surface is regular in $\xi$ within
the support of $a_{i}$, $i=1,2$. bounded from below. Estimate (43) now follows
in the same spirit as in [21]. We give the details only for $i=2$ because the
case $i=1$ is covered in [21, pp. 363-364] verbatim. For $i=2$, we find the
kernel bound

$|K^{\theta}_{\lambda,i}(y,\bar{y})|\lesssim\lambda^{4-\Re\theta(2\rho+1)}(1+\lambda|y-\bar{y}|)^{\Re\theta-\frac{3}{2}}.$

The relations

$\theta-\frac{3}{2}=-\frac{2}{p}-\frac{1}{q},\quad
4-\theta(2\rho+1)=\frac{2}{p}+\frac{6}{q},$

hold for the first two points in (47). So they hold for the third, which gives

(49) $\theta_{1}-\frac{3}{2}=-\frac{2}{p_{1}},\quad
4-\theta_{1}(2\rho+1)=\frac{2}{p_{1}}.$

Consequently, we have proved the kernel estimate

$|K^{\theta}_{\lambda,i}(y,\bar{y})|\lesssim|y-\bar{y}|^{-\frac{2}{p_{1}}},\qquad\Re\theta=\theta_{1}.$

The Hardy–Littlewood–Sobolev inequality then yields (43) for $i=2$.

We turn to the most difficult case $i=3$. Since there are four isolated
singular points $\xi_{x}^{\prime}$, it suffices to estimate the contribution
of each one separately. We make the decomposition $a_{3}(x,\xi)=\sum_{k\geq
0}a_{3,k}(x,\xi)$ where

$a_{3,k}(x,1,\xi^{\prime})=a_{3,0}(x,1,(\xi^{\prime}-\xi^{\prime}_{x})2^{k})$

localizes smoothly to a $2^{-k}$-annulus around $\xi_{x}^{\prime}$. This
cutoff is extended $1$-homogeneously in $\xi_{0}$, i.e.,
$a_{3,k}(x,\xi_{0},\xi^{\prime})=\xi_{0}a_{3,k}(x,1,\xi^{\prime}/\xi_{0})$.
This is possible because $|\xi_{0}|\in[1/4,4]$. The kernel $K_{k}$ of

$A_{\lambda,k}=T_{\lambda}^{*}\frac{a_{3,k}(x,\xi)\Phi(\xi)}{(\lambda^{-\frac{s}{4}}+\lambda^{\frac{s}{4}}|q(x,\xi)|)^{2}}T_{\lambda}$

is given by

$K_{k}(y,\bar{y})=\lambda^{6-(2\rho+1)}\int_{\mathbb{R}^{4}}dxe^{-\frac{\lambda(y-x)^{2}}{2}}e^{-\frac{\lambda(\bar{y}-x)^{2}}{2}}\int_{\mathbb{R}^{4}}d\xi\frac{e^{i\lambda\xi.(y-\bar{y})}a_{3,k}(x,\xi_{0},\xi^{\prime})}{(\lambda^{-\frac{s}{4}}+\lambda^{\frac{s}{4}}|q(x,\xi)|)^{2}}.$

By homogeneity of $q(x,\xi_{0},\xi^{\prime})$ and
$a_{3}(x,\xi_{0},\xi^{\prime})$ in $\xi_{0}$, we find

$\frac{a_{3,k}(x,\xi_{0},\xi^{\prime})a_{0}(\xi_{0})}{(\lambda^{-\frac{s}{4}}+\lambda^{\frac{s}{4}}|q(x,\xi_{0},\xi^{\prime})|)^{2}}=\frac{\xi_{0}a_{3,k}(x,1,\tilde{\xi}^{\prime})a_{0}(\xi_{0})}{\xi_{0}^{8}(\lambda^{-\frac{s}{4}}\xi_{0}^{-4}+\lambda^{\frac{s}{4}}|q(x,1,\tilde{\xi}^{\prime})|)^{2}}$

with $\tilde{\xi}^{\prime}=\xi^{\prime}/\xi_{0}$ and a suitable smooth cutoff
$a_{0}$.

Furthermore, we carry out a Taylor expansion of $q(x,1,\cdot)$ around
$\xi_{x}^{\prime}$ to find

$q(x,1,\xi^{\prime}+\xi_{x}^{\prime})=\frac{\langle\partial^{2}_{\xi^{\prime}\xi^{\prime}}q(x,1,\xi_{x}^{\prime})\xi^{\prime},\xi^{\prime}\rangle}{2}+O(|\xi^{\prime}|^{3}),$

recalling that

$q(x,1,\xi_{x}^{\prime})=0,\quad\nabla_{\xi^{\prime}}q(x,1,\xi_{x}^{\prime})=0.$

We write

$\begin{split}q(x,1,\tilde{\xi}^{\prime})=q(x,1,2^{-k}\xi^{\prime}+\xi_{x}^{\prime})&=2^{-2k}\big{(}\frac{\langle\partial^{2}_{\xi^{\prime}\xi^{\prime}}q(x,1,\xi_{x}^{\prime})\xi^{\prime},\xi^{\prime}\rangle}{2}+O(2^{-k}|\xi^{\prime}|^{3})\big{)}\\\
&=:2^{-2k}q_{k}(x,1,\xi^{\prime}).\end{split}$

For the analysis of the kernel we perform the change of variables
$\tilde{\xi}^{\prime}=\tilde{\tilde{\xi}}^{\prime}+\xi_{x}^{\prime}$ and
$\xi^{\prime}=2^{k}\tilde{\tilde{\xi}}^{\prime}$ to find

$\begin{split}K_{k}(y,\bar{y})&=\lambda^{6-(2\rho+1)}\\!\\!\int_{\mathbb{R}^{4}}\\!\\!dxe^{-\frac{\lambda}{2}(y-x)^{2}}e^{-\frac{\lambda}{2}(\bar{y}-x)^{2}}\\!\\!\int_{\mathbb{R}}\\!d\xi_{0}\frac{e^{i\lambda\xi_{0}.(y_{0}-\bar{y}_{0})}e^{i\lambda\xi_{0}\xi_{x}^{\prime}.(y^{\prime}-\bar{y}^{\prime})}a_{0}(\xi_{0})}{\xi_{0}^{4}}\\\
&\quad\times\int_{\mathbb{R}^{3}}d\xi^{\prime}\frac{2^{-3k}a_{3,0}(x,1,\xi^{\prime})e^{i\lambda
2^{-k}\xi_{0}\xi^{\prime}.(y^{\prime}-\bar{y}^{\prime})}}{(\lambda^{-\frac{s}{4}}\xi_{0}^{-4}+\lambda^{\frac{s}{4}}2^{-2k}|q_{k}(x,1,\xi^{\prime})|)^{2}}\\\
&=\lambda^{6-(2\rho+1)}\\!\\!\int_{\mathbb{R}^{4}}\\!\\!dxe^{-\frac{\lambda}{2}(y-x)^{2}}e^{-\frac{\lambda}{2}(\bar{y}-x)^{2}}\\!\\!\int_{\mathbb{R}}\\!d\xi_{0}\frac{e^{i\lambda\xi_{0}.(y_{0}-\bar{y}_{0})}e^{i\lambda\xi_{0}\xi_{x}^{\prime}.(y^{\prime}-\bar{y}^{\prime})}a_{0}(\xi_{0})}{\xi_{0}^{4}}\\\
&\quad\times\int_{\mathbb{R}^{3}}d\xi^{\prime}\frac{2^{k}a_{3,0}(x,1,\xi^{\prime})e^{i\lambda
2^{-k}\xi_{0}\xi^{\prime}.(y^{\prime}-\bar{y}^{\prime})}}{(\lambda^{-\frac{s}{4}}\xi_{0}^{-4}2^{2k}+\lambda^{\frac{s}{4}}|q_{k}(x,1,\xi^{\prime})|)^{2}}.\end{split}$

The weight
$(2^{2k}\lambda^{-\frac{s}{4}}\xi_{0}^{-4}+\lambda^{\frac{s}{4}}|q_{k}(x,1,\xi^{\prime})|)^{-2}$
is integrable with respect to the level sets, and we have

$\int_{0}^{\infty}\frac{dq}{(2^{2k}\lambda^{-\frac{s}{4}}+q\lambda^{\frac{s}{4}})^{2}}\lesssim
2^{-2k}.$

After foliation over level sets $q(x,\xi)=c$, we are thus left with the
operator having the kernel

$\displaystyle K^{1}_{k}(y,\bar{y})$
$\displaystyle=\lambda^{6-(2\rho+1)}\\!\\!\int_{\mathbb{R}^{4}}dxe^{-\frac{\lambda}{2}(y-x)^{2}}e^{-\frac{\lambda}{2}(\bar{y}-x)^{2}}\\!\\!\int_{\mathbb{R}}d\xi_{0}\frac{e^{i\lambda\xi_{0}.(y-\bar{y})}e^{i\lambda\xi_{0}\xi_{x}^{\prime}.(y^{\prime}-\bar{y}^{\prime})}a_{0}(\xi_{0})}{\xi_{0}^{4}}$
$\displaystyle\quad\times\int_{\mathbb{R}^{3}}d\xi^{\prime}2^{-k}a_{3,0}(x,1,\xi^{\prime})e^{i\lambda
2^{-k}\xi_{0}\xi^{\prime}.(y^{\prime}-\bar{y}^{\prime})}\delta_{\tilde{q}_{k}(x,1,\xi^{\prime})=0}.$

We estimate it by interpolation using the analytic family of operators
$V^{\theta}_{k,\lambda}$ given by the kernels

(50)
$\begin{split}K^{\theta}_{\lambda,k}&=\lambda^{6-\theta(2\rho+1)}\int_{\mathbb{R}^{4}}dxe^{-\frac{\lambda}{2}(y-x)^{2}}e^{-\frac{\lambda}{2}(\bar{y}-x)^{2}}\int_{\mathbb{R}}d\xi_{0}e^{i\lambda\xi_{0}.(y_{0}-\bar{y}_{0})}\frac{e^{i\lambda\xi_{0}\xi_{x}^{\prime}.(y^{\prime}-\bar{y}^{\prime})}a_{0}(\xi_{0})}{\xi_{0}^{4}}\\\
&\quad\times\frac{e^{(\theta-1)^{2}}}{\Gamma(1-\theta)}\int_{\mathbb{R}^{3}}d\xi^{\prime}2^{-k}a_{3,0}(x,1,\xi^{\prime})e^{i\lambda
2^{-k}\xi_{0}\xi^{\prime}.(y^{\prime}-\bar{y}^{\prime})}(q_{k}+i0)^{-\theta}.\end{split}$

We show+ that the corresponding operators uniformly bounded on the spaces

$\begin{split}&V^{\theta}_{\lambda,k}:L^{2}\to
L^{2}\qquad\quad\;\quad\;\Re\theta=0,\\\
&V^{\theta}_{\lambda,k}:L^{p_{1}^{\prime}}L^{1}\to
L^{p_{1}}L^{\infty}\quad\Re\theta=\theta_{1},\end{split}$

where $p_{1}$, $\theta_{1}$ are chosen so that the points

$\big{(}\frac{1}{2},\frac{1}{2},0\big{)},\quad\big{(}\frac{1}{p},\frac{1}{q},1\big{)},\quad\big{(}\frac{1}{p_{1}},0,\theta_{1}\big{)}$

are collinear.

  * •

Estimate of $V^{\theta}_{\lambda,k}:L^{2}\to L^{2},\quad\Re\theta=0$: After
inverting the changes of variables and using the $L^{2}$-boundedness of the
FBI transform, we find

(51) $\|V^{\theta}_{\lambda,k}\|_{L^{2}\to L^{2}}\lesssim 2^{2k}.$

  * •

Estimate of $V^{\theta}_{\lambda,k}:L^{p_{1}^{\prime}}L^{1}\to
L^{p_{1}}L^{\infty}$, $\Re\theta=\theta_{1}$: Inequality (48) implies the
kernel estimates

$|K_{\lambda,k}^{\theta}|\lesssim\lambda^{4-\theta(2\rho+1)}2^{-k}(1+\lambda
2^{-k}|y^{\prime}-\bar{y}^{\prime}|)^{\Re\theta-\frac{3}{2}}.$

By the relations (49) and provided that
$|y_{0}-\bar{y}_{0}|\lesssim|y^{\prime}-\bar{y}^{\prime}|$, this gives

$|K^{\theta}_{\lambda,k}|\lesssim
2^{\frac{2k}{p_{1}}-k}|y-\bar{y}|^{-\frac{2}{p_{1}}}\leq
2^{\frac{2k}{p_{1}}-k}|y_{0}-\bar{y}_{0}|^{-\frac{2}{p_{1}}}.$

Since $p_{1}\geq 4$, we can apply the Hardy–Littlewood–Sobolev inequality On
the other hand, if $|y_{0}-\bar{y}_{0}|\gg|y^{\prime}-\bar{y}^{\prime}|$, we
can integrate by parts in $\xi_{0}$ in (50), which gains factors
$(\lambda|y_{0}-\bar{y}_{0}|)^{-1}$. So we obtain

$\|V^{\theta}_{k,\lambda}\|_{L^{p_{1}^{\prime}}L^{1}\to
L^{p_{1}}L^{\infty}}\lesssim 2^{\frac{2k}{p_{1}}-k}.$

Interpolation yields

$\|V_{k,\lambda}^{\theta}\|_{L^{p^{\prime}}L^{q^{\prime}}\to
L^{p}L^{q}}\lesssim\|V^{\theta}_{k,\lambda}\|_{L^{2}\to
L^{2}}^{\alpha}\|V^{\theta}_{k,\lambda}\|_{L^{p_{1}^{\prime}}L^{1}\to
L^{p_{1}}L^{\infty}}^{1-\alpha}$

with

$\big{(}\frac{1}{p},\frac{1}{q},1\big{)}=\alpha\big{(}\frac{1}{2},\frac{1}{2},0\big{)}+(1-\alpha)\big{(}\frac{1}{p_{1}},0,\theta_{1}\big{)}\quad\Longrightarrow\quad\alpha=\frac{2}{q}.$

These identities imply $\theta_{1}=\frac{1/2}{1/2-1/q}$ and
$(2k/p_{1})(1-2/q)=k(2/p-2/q)$, and thus the operator norm

$\|V_{k,\lambda}^{\theta}\|_{L^{p^{\prime}}L^{q^{\prime}}\to
L^{p}L^{q}}\lesssim
2^{k\big{(}\frac{6}{q}-1\big{)}}2^{\frac{2k}{p_{1}}(1-\alpha)}\lesssim
2^{k\big{(}\frac{3}{q}-\frac{1}{2}\big{)}}.$

This is summable for $q>6$, but since $\frac{2}{p}+\frac{1}{q}=\frac{1}{2}$
and $p<q$ this is always the case under the assumptions. ∎

Note that (51) reflects the lack of hyperbolicity at the singular point. For
this purpose, we only prove the bounds at the level of the scalar equation
close to the endpoint. We remark that in two dimensions, it suffices to prove
the (allowed) endpoint estimate $L_{t}^{4}L_{x}^{\infty}$. We included the
above arguments to show that the proof remains valid for a characteristic
surface $\tilde{S}=\\{q(\xi)=0\\}$ in higher dimensions with the degeneracy
$q(\xi_{0})=0$, $\nabla q(\xi_{0})=0$, and $\det(\partial^{2}q(\xi_{0}))\neq
0$.

###  3.3. Short-time Strichartz estimates

In the estimate (32) proved so far, there is no gain in the regularity of the
homogeneous part $\|u\|_{L^{2}}$ by the Strichartz estimate over Sobolev
embedding. Nonetheless, by passing through short-time Strichartz estimates we
show the claimed improvement (see also Bahouri–Chemin [1]). In the following
we recast the smoothing in the inhomogeneous term $Pu$ as derivative gain by
using frequency dependent time localization.

######  Proposition 3.8.

Let $(\varepsilon,\mu)\in C^{s}_{x}$ for $0<s\leq 1$ and
$\partial(\varepsilon,\mu)\in L_{t}^{2}L_{x^{\prime}}^{\infty}$. Suppose that
the estimate

(52)
$\||D|^{-\rho+\frac{s-2}{4}}u\|_{(L_{t}^{4}L^{\infty}_{x^{\prime}})_{2}}\lesssim_{C}\|u\|_{L_{x}^{2}}+\||D|^{\frac{s-2}{2}}Pu\|_{L^{2}_{x}}+\||D|^{\frac{s-2}{4}-\frac{1}{2}}\rho_{em}\|_{L^{2}_{x}}$

holds true with
$C=C(\|(\varepsilon,\mu)\|_{C^{s}_{x}},\|\partial(\varepsilon,\mu)\|_{L_{t}^{2}L_{x^{\prime}}^{\infty}})$.
Let $\delta>0$ and $T>0$. We then obtain the inquality

(53) $\begin{split}\|\langle
D^{\prime}\rangle^{-\rho+\frac{s-2}{8}-\delta}u\|_{L^{4}(0,T;L^{\infty}_{x^{\prime}})}&\leq
C(\|u_{0}\|_{L^{2}_{x^{\prime}}}+\|Pu\|_{L^{1}_{T}L^{2}_{x^{\prime}}}+\|\langle
D^{\prime}\rangle^{\frac{s-2}{8}-\frac{1}{2}}\rho_{em}(0)\|_{L^{2}_{x^{\prime}}}\\\
&\quad+\|\langle
D^{\prime}\rangle^{\frac{s-2}{8}-\frac{1}{2}}\partial_{t}\rho_{em}\|_{L^{1}_{T}L^{2}_{x^{\prime}}})\end{split}$

with
$C=C(\|(\varepsilon,\mu)\|_{C_{x}^{s}},\|\partial(\varepsilon,\mu)\|_{L_{t}^{2}L_{x^{\prime}}^{\infty}},T,\delta)$.

###### Proof.

An application of (52) with $P_{\lambda}$ (recall this is the Maxwell operator
with coefficients truncated at frequencies $\leq\lambda/8$) on $S_{\lambda}u$
yields

(54)
$\lambda^{-\rho+\frac{s-2}{4}}\|S_{\lambda}u\|_{L_{t}^{4}L_{x^{\prime}}^{\infty}}\lesssim\|S_{\lambda}u\|_{L_{x}^{2}}+\lambda^{\frac{s-2}{2}}\|P_{\lambda}S_{\lambda}u\|_{L^{2}_{t}L_{x^{\prime}}^{2}}+\lambda^{\frac{s-2}{4}-\frac{1}{2}}\|S_{\lambda}\rho_{em}\|_{L^{2}}.$

For the proof of (53), we use Minkowski’s inequality to split

$\begin{split}\|\langle
D^{\prime}&\rangle^{-\rho+\frac{s-2}{8}-\frac{\delta}{2}}u\|_{L_{T}^{4}L_{x^{\prime}}^{\infty}}\\\
&\leq\sum\nolimits_{\lambda}\|\langle
D^{\prime}\rangle^{-\rho+\frac{s-2}{8}-\frac{\delta}{2}}S_{\lambda}u\|_{L_{T}^{4}L_{x^{\prime}}^{\infty}}\\\
&\leq\sum\nolimits_{\lambda}\big{[}\|\langle
D^{\prime}\rangle^{-\rho+\frac{s-2}{8}-\frac{\delta}{2}}S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{T}^{4}L_{x^{\prime}}^{\infty}}+\|\langle
D^{\prime}\rangle^{-\rho+\frac{s-2}{8}-\frac{\delta}{2}}S_{\lambda}^{\ll\tau}u\|_{L_{T}^{4}L_{x^{\prime}}^{\infty}}\big{]},\end{split}$

where $S_{\lambda}^{\ll\tau}$ projects to the space-time frequencies
$\\{\lambda\sim|\tau|\gg|\xi^{\prime}|\\}$. In this region the operator $P$ is
elliptic, which gains one derivative for Lipschitz coefficients. The summand
can then be bounded by Sobolev embedding. We refer to the proof of [17,
Corollary 1.7] for details on this estimate if $\partial(\varepsilon,\mu)\in
L^{2}_{x_{0}}L^{\infty}_{x^{\prime}}$.

For the first term we want to apply (54). Let $\chi\in C^{\infty}_{c}(-2,2)$
be radially decreasing with $\chi(t)=1$ for $t\in[-1,1]$. Note that passing to
$\chi(\lambda^{\frac{2-s}{2}}t-n)u$ does not essentially change the Fourier
support in time because the inverse Fourier transform of
$\chi_{\lambda^{\frac{2-s}{2}}}(t)=\chi(\lambda^{\frac{2-s}{2}}t-n)$ is
essentially supported in a $\lambda^{\frac{2-s}{2}}$-ball. We suppress
dependence on $n$ in the following due to uniformity of the estimates. We will
apply (54) to $S^{\prime}_{\lambda}\chi_{\lambda^{\frac{2-s}{2}}}u$ with
$I_{\lambda}=\text{supp}(\chi_{\lambda^{\frac{2-s}{2}}})$ and
$I_{\lambda}^{*}=\\{\chi_{\lambda^{\frac{2-s}{2}}}=1\\}$. At first, applying
$\partial_{t}$ to $\chi_{\lambda^{\frac{2-s}{2}}}$ in
$[P_{\lambda},\chi_{\lambda^{\frac{2-s}{2}}}]$ we derive

$\lambda^{\frac{s-2}{2}}\|P_{\lambda}\chi_{\lambda^{\frac{2-s}{2}}}S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{x}^{2}}\lesssim\|S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{t}^{2}(I_{\lambda};L^{2}_{x^{\prime}})}+\lambda^{\frac{s-2}{2}}\|P_{\lambda}S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{t}^{2}(I_{\lambda};L^{2}_{x^{\prime}})}.$

This gives

$\begin{split}\lambda^{-\rho+\frac{s-2}{4}-\frac{\delta}{2}}\|S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{t}^{4}(I_{\lambda}^{*};L_{x^{\prime}}^{\infty})}&\lesssim\|S_{\lambda}^{\prime}u\|_{L_{t}^{2}(I_{\lambda};L^{2}_{x^{\prime}})}+\lambda^{\frac{s-2}{2}}\|P_{\lambda}S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{t}^{2}(I_{\lambda};L^{2}_{x^{\prime}})}\\\
&\quad+\lambda^{\frac{s-2}{4}-\frac{1}{2}}\|S^{\prime}_{\lambda}\rho_{em}\|_{L_{t}^{2}(I_{\lambda};L^{2}_{x^{\prime}})}.\end{split}$

An application of Hölder’s inequality on
$\|S_{\lambda}^{\prime}u\|_{L_{t}^{2}(I_{\lambda};L_{x^{\prime}}^{2})}$ yields

$\begin{split}\lambda^{-\rho+\frac{s-2}{4}-\frac{\delta}{2}}\|S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{t}^{4}(I_{\lambda}^{*};L^{\infty}_{x^{\prime}})}&\lesssim\lambda^{\frac{s-2}{4}}\|S^{\prime}_{\lambda}u\|_{L_{t}^{\infty}(I_{\lambda};L^{2}_{x^{\prime}})}+\lambda^{\frac{s-2}{2}}\|P_{\lambda}S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{t}^{2}(I_{\lambda};L^{2}_{x^{\prime}})}\\\
&\quad+\lambda^{\frac{s-2}{4}-\frac{1}{2}}\|S^{\prime}_{\lambda}\rho_{em}\|_{L_{t}^{2}(I_{\lambda};L^{2}_{x^{\prime}})}.\end{split}$

We sum this estimate over the $T\lambda^{\frac{2-s}{2}}$ disjoint intervals
$I_{\lambda}^{*}$ partitioning $I=(0,T)$ in $\ell^{4}$ such that

$\begin{split}\lambda^{-\rho+\frac{s-2}{4}-\frac{\delta}{2}}\|S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{T}^{4}(I;L_{x^{\prime}}^{\infty})}&\lesssim_{T}\lambda^{\frac{2-s}{8}+\frac{s-2}{4}}\|S^{\prime}_{\lambda}u\|_{L_{t}^{\infty}L^{2}_{x^{\prime}}}+\lambda^{\frac{s-2}{2}}\|P_{\lambda}S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{t}^{2}L^{2}_{x^{\prime}}}\\\
&\quad+\lambda^{\frac{s-2}{4}-\frac{1}{2}}\|S^{\prime}_{\lambda}\rho_{em}\|_{L_{x}^{2}}.\end{split}$

Because of $\ell^{2}\hookrightarrow\ell^{4}$ we do not lose powers of
$\lambda$ when summing $P_{\lambda}S_{\lambda}S^{\prime}_{\lambda}u$ and
$S^{\prime}_{\lambda}\rho_{em}$. By commutator arguments using
$\|\partial(\varepsilon,\mu)\|_{L^{2}_{t}L_{x^{\prime}}^{\infty}}\lesssim 1$,
cf. (4.4) in [23], we find

$\|P_{\lambda}S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{t}^{2}L_{x^{\prime}}^{2}}\lesssim\|S_{\lambda}^{\prime}Pu\|_{L_{t}^{2}L_{x^{\prime}}^{2}}+\|S^{\prime}_{\lambda}u\|_{L_{t}^{\infty}L_{x^{\prime}}^{2}}$

and hence,

$\displaystyle\lambda^{-\rho+\frac{s-2}{8}-\frac{\delta}{2}}\|S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{t}^{4}(I;L^{\infty}_{x^{\prime}})}$
$\displaystyle\lesssim_{T}\|S^{\prime}_{\lambda}u\|_{L^{\infty}_{t}L^{2}_{x^{\prime}}}+\lambda^{\frac{3(s-2)}{8}}\|Pu\|_{L^{2}_{t}L^{2}_{x^{\prime}}}$
$\displaystyle\quad+\lambda^{\frac{s-2}{8}-\frac{1}{2}}\|S^{\prime}_{\lambda}\rho_{em}\|_{L_{x}^{2}}.$

Summing over $\lambda$, we find the estimate

(55) $\|\langle
D^{\prime}\rangle^{-\rho+\frac{s-2}{8}-\delta}u\|_{L_{t}^{4}(I;L^{\infty}_{x^{\prime}})}\lesssim_{T,\delta,X}\|u\|_{L_{t}^{\infty}L^{2}_{x^{\prime}}}+\|Pu\|_{L^{2}_{t}L^{2}_{x^{\prime}}}+\|\langle
D^{\prime}\rangle^{-\frac{1}{2}+\frac{s-2}{8}}\rho_{em}\|_{L^{2}_{x}}$

with
$X=\|(\varepsilon,\mu)\|_{C^{s}_{x}}+\|\partial(\varepsilon,\mu)\|_{L_{t}^{2}L_{x^{\prime}}^{\infty}}$.
(Note that we can treat frequencies $\lambda\sim|\tau|\gg|\xi^{\prime}|$ as
above.)

The $L_{t}^{2}L_{x^{\prime}}^{2}$-norms are changed to
$L_{t}^{1}L_{x^{\prime}}^{2}$-norms by the energy estimate and Duhamel’s
formula: An application of (55) to homogeneous solutions gives

$\|\langle
D^{\prime}\rangle^{-\rho+\frac{s-2}{8}-\delta}u\|_{L_{t}^{4}(I;L^{\infty}_{x^{\prime}})}\lesssim_{T,\delta,X}\|u_{0}\|_{L^{2}_{x^{\prime}}}+\|\langle
D^{\prime}\rangle^{-\frac{1}{2}+\frac{s-2}{8}}\rho_{em}(0)\|_{L^{2}_{x^{\prime}}}.$

Now we write the general function $u$ by Duhamel’s formula
$u(t)=U(t,0)u_{0}+\int_{0}^{t}U(t,s)(Pu)(s)ds$ and note that
$\nabla\cdot((Pu)_{1}(s),(Pu)_{2}(s))=(-\partial_{t}\rho_{e}(s),\partial_{t}\rho_{m}(s))$.
We find after applying Minkowski’s inequality

$\displaystyle\|\langle
D^{\prime}\rangle^{-\rho+\frac{s-2}{8}-\delta}u\|_{L_{t}^{4}(I;L^{\infty}_{x^{\prime}})}$
$\displaystyle\lesssim_{T,\delta,X}\|u_{0}\|_{L^{2}_{x^{\prime}}}+\|Pu\|_{L^{1}_{T}L^{2}_{x^{\prime}}}+\|\langle
D^{\prime}\rangle^{-\frac{1}{2}+\frac{s-2}{8}}\rho_{em}(0)\|_{L^{2}_{x^{\prime}}}$
(56) $\displaystyle\quad+\|\langle
D^{\prime}\rangle^{-\frac{1}{2}+\frac{s-2}{8}}\partial_{t}\rho_{em}\|_{L^{1}_{T}L^{2}_{x^{\prime}}}.$

The proof is complete. ∎

Interpolation with the energy estimate from Proposition 3.1

$\|u\|_{L^{\infty}_{T}L^{2}_{x^{\prime}}}\lesssim\|u(0)\|_{L^{2}_{x^{\prime}}}+\|Pu\|_{L^{1}_{T}L^{2}_{x^{\prime}}}$

yields Theorem 1.1 on the sharp line $\frac{2}{p}+\frac{1}{q}=\frac{1}{2}$.
The general case follows from Sobolev embedding. In the next section we
generalize the result in the Lipschitz case to diagonalizable coefficients.

##  4\. Reducing to the case of diagonal material laws

We now prove Theorem 1.2 by transforming permittivity and permeability to
diagonal matrices. In the following we consider $\varepsilon,\mu\in
C^{1}(\mathbb{R}\times\mathbb{R}^{3};\mathbb{R}^{3\times 3}_{\text{sym}})$.

###  4.1. Orthogonality transformations

Suppose there is $\Phi\in C^{1}(\mathbb{R}^{4};\mathbb{R}^{3\times 3})$ with

(57) $\Phi^{t}(x)\Phi(x)=1_{3\times
3},\quad\varepsilon^{d}=\Phi^{t}\varepsilon\Phi,\quad\mu^{d}=\Phi^{t}\mu\Phi.$

Write $\Phi=(\varphi_{1}\;\varphi_{2}\;\varphi_{3})$. We use the
transformations

$\tilde{\mathcal{E}}=\Phi^{-1}\mathcal{E},\quad\tilde{\mathcal{H}}=\Phi^{-1}\mathcal{H},\quad\tilde{\mathcal{J}}_{k}=\Phi^{-1}\mathcal{J}_{k},\;k\in\\{e,m\\}$

to reduce the analysis to the case that $\varepsilon$ and $\mu$ are diagonal
matrices. If $\mathcal{E}$, $\mathcal{H}$, $\mathcal{J}$ satisfy (29), then
$\tilde{\mathcal{E}}$, $\tilde{\mathcal{H}}$, $\mathcal{\tilde{J}}$ satisfy
the system

(58)
$\left\\{\begin{array}[]{cl}\partial_{t}(\varepsilon^{d}\tilde{\mathcal{E}})&=\Phi^{t}\nabla\times(\Phi\tilde{\mathcal{H}})-(\partial_{t}\Phi)\varepsilon^{d}\tilde{\mathcal{E}}-\tilde{\mathcal{J}}_{e},\\\
\partial_{t}(\mu^{d}\tilde{\mathcal{H}})&=-\Phi^{t}\nabla\times(\Phi\tilde{\mathcal{E}})-(\partial_{t}\Phi)\mu^{d}\tilde{\mathcal{H}}-\tilde{\mathcal{J}}_{m},\\\
\rho_{e}&=\nabla\cdot(\Phi\varepsilon^{d}\tilde{\mathcal{E}}),\quad\rho_{m}=\nabla\cdot(\Phi\mu^{d}\tilde{\mathcal{H}}).\end{array}\right.$

We write $\tilde{u}=(\tilde{\mathcal{E}},\tilde{\mathcal{H}})$. The curl
transforms as follows. We denote

$\nabla\times
f=i\mathcal{C}(D)f,\quad\mathcal{C}(\xi^{\prime})=(-\varepsilon^{ijk}\xi_{k})_{ij},\quad\mathcal{C}(\xi^{\prime})v=\xi^{\prime}\times
v.$

The leading order term of $\Phi^{t}(\nabla\times(\Phi\cdot))$ can be written
as

$\begin{split}\begin{pmatrix}\varphi_{1}^{t}\\\ \varphi_{2}^{t}\\\
\varphi_{3}^{t}\end{pmatrix}\begin{pmatrix}\xi^{\prime}\times\varphi_{1}&\xi^{\prime}\times\varphi_{2}&\xi^{\prime}\times\varphi_{3}\end{pmatrix}&=(\varphi_{i}\cdot(\xi^{\prime}\times\varphi_{j}))_{ij}=(\xi^{\prime}\cdot(\varphi_{j}\times\varphi_{i}))_{ij}\\\
&=(-\varepsilon^{ijk}\varphi_{k}(x)\cdot\xi^{\prime})_{ij}=:(\mathcal{C}(\eta))_{ij}.\end{split}$

We let $\eta_{k}(x,\xi)=\varphi_{k}(x)\cdot\xi^{\prime}$ and333It is important
to work with operators in divergence form. We highlight this by the notation
$a(D,x)$. $\eta_{k}(D,x)=\sum_{j}\partial_{j}(\varphi_{kj}(x)\cdot)$.
Consequently,

$\big{(}\Phi^{t}(\nabla\times(\Phi\cdot))\big{)}_{ij}=(-\varepsilon^{ijk}\varphi_{k}(x)\cdot\nabla_{x^{\prime}})_{ij}+b_{ij}(x)$

with $\|b_{ij}\|_{L^{\infty}}\lesssim\|\Phi\|_{C^{1}}$. In the following we
use that $\varepsilon^{d}$ and $\mu^{d}$ are diagonal as in Assumption 1:

(59)
$\varepsilon^{d}(x)=\text{diag}(\varepsilon_{1}^{d}(x),\varepsilon^{d}_{2}(x),\varepsilon^{d}_{3}(x)),\quad\mu^{d}(x)=\text{diag}(\mu^{d}_{1}(x),\mu_{2}^{d}(x),\mu^{d}_{3}(x)).$

For the electric charge we note that

$\rho_{e}=\nabla\cdot(\varepsilon\mathcal{E})=\nabla\cdot(\Phi\varepsilon^{d}\tilde{u}^{1})=\sum_{j,k=1}^{3}\partial_{j}(\varphi_{kj}\varepsilon_{k}^{d}\tilde{u}_{k}^{1})=:\eta(D,x)\cdot\varepsilon^{d}\tilde{u}^{1}.$

For the magnetic charge we find similarly

$\rho_{m}=\eta(D,x)\cdot(\mu^{d}\tilde{u}^{2}).$

We define

(60)
$\tilde{\rho}_{e}=\eta(D,x)\cdot(\varepsilon^{d}\tilde{u}^{1}),\quad\tilde{\rho}_{m}=\eta(D,x)\cdot(\mu^{d}\tilde{u}^{2}).$

It follows

$\begin{pmatrix}-\partial_{t}(\varepsilon^{d}\cdot)&\Phi^{t}(\nabla\times(\Phi(\cdot)))\\\
\Phi^{t}(\nabla\times(\Phi(\cdot)))&\partial_{t}(\mu^{d}\cdot)\end{pmatrix}=\begin{pmatrix}-\partial_{t}(\varepsilon^{d}\cdot)&\mathcal{C}(\eta(D,x))\\\
\mathcal{C}(\eta(D,x))&\partial_{t}(\mu^{d}\cdot)\end{pmatrix}+R(x)$

with $R(x)\in L^{\infty}$. Let

(61)
$\tilde{P}=\begin{pmatrix}-\partial_{t}(\varepsilon^{d}\cdot)&\mathcal{C}(\eta(D,x))\\\
\mathcal{C}(\eta(D,x))&\partial_{t}(\mu^{d}\cdot)\end{pmatrix}.$

We have the following lemma:

######  Lemma 4.1.

Let $\Phi\in C^{1}(\mathbb{R}^{4};\mathbb{R}^{3\times 3})$ satisfy (57) and
$\varepsilon^{d}$, $\mu^{d}$ be like in (59). Set
$\tilde{u}=(\tilde{\mathcal{E}},\tilde{\mathcal{H}})=(\Phi^{-1}\mathcal{E},\Phi^{-1}\mathcal{H})$,
$\tilde{\mathcal{J}}_{k}=\Phi^{-1}\mathcal{J}_{k}$, $k\in\\{e,m\\}$,
$\varepsilon^{d}=\Phi^{t}\varepsilon\Phi$, $\mu^{d}=\Phi^{t}\mu\Phi$, and

$\tilde{P}=\begin{pmatrix}-\partial_{t}(\varepsilon^{d}\cdot)&\mathcal{C}(\eta(D,x))\\\
\mathcal{C}(\eta(D,x))&\partial_{t}(\mu^{d}\cdot)\end{pmatrix};\quad\tilde{\rho}_{em}=(\eta(D,x)(\varepsilon^{d}\tilde{u}^{1}),\eta(D,x)(\mu^{d}\tilde{u}^{2})).$

Suppose that

(62)
$\||D|^{-\rho-\frac{1}{4}}\tilde{u}\|_{(L^{4}L^{\infty})_{2}}\lesssim\|\tilde{u}\|_{L^{2}}+\||D|^{-\frac{1}{2}}\tilde{P}\tilde{u}\|_{L^{2}_{x}}+\||D|^{-\frac{3}{4}}\tilde{\rho}_{em}\|_{L^{2}}.$

Then, for $0<\delta<\frac{1}{4}$, the following estimate holds:

(63)
$\||D|^{-\rho-\frac{1}{4}-\delta}u\|_{(L^{4}L^{\infty})_{2}}\lesssim\|u\|_{L^{2}}+\||D|^{-\frac{1}{2}}Pu\|_{L^{2}}+\||D|^{-\frac{3}{4}}\rho_{em}\|_{L^{2}}.$

The implicit constants depend on $\|(\Phi,\Phi^{t})\|_{C^{1}}$,
$\|(\varepsilon,\mu)\|_{C^{1}}$.

###### Proof.

The low frequencies can be treated like in the proof of Proposition 3.4 by
Sobolev embedding. For $0<\delta<\frac{1}{4}$, we thus obtain

$\|S_{0}|D|^{-\rho-\frac{1}{4}-\delta}u\|_{L^{4}L^{\infty}}\lesssim\|u\|_{L^{2}}.$

For high frequencies $M\gtrsim 1$, we write

$\begin{split}\|S_{M}|D|^{-\rho-\frac{1}{4}-\delta}u\|_{(L^{4}L^{\infty})_{2}}&\sim
M^{-\rho-\frac{1}{4}-\delta}\|S_{M}(\Phi^{t}\tilde{u})\|_{L^{4}L^{\infty}}\\\
&\lesssim M^{-\rho-\frac{1}{4}-\delta}\sum_{1\leq K\leq
M}\|S_{M}S^{\prime}_{K}(\Phi^{t}\tilde{u})\|_{L^{4}L^{\infty}}\\\
&\quad+M^{-\rho-\frac{1}{4}-\delta}\|S_{M}S^{\prime}_{0}(\Phi^{t}\tilde{u})\|_{L^{4}L^{\infty}}.\end{split}$

The second term is estimated

$M^{-\frac{3}{2}-\delta}\|S_{M}S^{\prime}_{0}(\Phi^{t}u)\|_{L^{4}L^{\infty}}\lesssim
M^{-1}\|\Phi^{t}u\|_{L^{2}}$

by Bernstein’s inequality, with easy summation in $M$. Hence, it suffices to
estimate the spatial frequencies, which are greater than $1$. To this end, we
also use paraproduct decompositions both in space and space-time frequencies.
First, we write (with $X=L_{x_{0}}^{4}L^{\infty}_{x^{\prime}}$)

$\displaystyle\|S_{M}S^{\prime}_{K}(\Phi^{t}\tilde{u})\|_{X}$
$\displaystyle\leq\|S_{M}((S^{\prime}_{\sim K}\Phi^{t})(S^{\prime}_{\ll
K}\tilde{u}))\|_{X}+\|S_{M}(S^{\prime}_{K}((S^{\prime}_{\gtrsim
K}\Phi^{t})(S^{\prime}_{\gtrsim K}\tilde{u}))\|_{X}$ (64)
$\displaystyle\quad+\|S_{M}((S^{\prime}_{\ll K}\Phi^{t})(S^{\prime}_{\sim
K}\tilde{u}))\|_{X}.$

Bernstein’s inequality implies for the first term in (4.1)

$\begin{split}\sum_{M\geq 1}&M^{-2\rho-\frac{1}{2}}\big{(}\sum_{1\leq K\leq
M}\|S_{M}(S^{\prime}_{K}\Phi^{t}S^{\prime}_{\ll
K}\tilde{u})\|_{L_{t}^{4}L_{x^{\prime}}^{\infty}}\big{)}^{2}\\\
&\lesssim\sum_{M\geq 1}M^{-2\rho-\frac{1}{2}}M^{\frac{1}{2}}\big{(}\sum_{K\leq
M}K^{\frac{3}{2}}\|S^{\prime}_{K}\Phi^{t}S^{\prime}_{\ll
K}\tilde{u}\|_{L^{2}_{t,x^{\prime}}}\big{)}^{2}\\\ &\lesssim\sum_{M\geq
1}M^{-\frac{5}{2}}\big{(}\sum_{K\leq
M}K^{\frac{1}{2}}\|\nabla_{x^{\prime}}S^{\prime}_{K}\Phi^{t}\|_{L^{\infty}}\big{)}^{2}\|\tilde{u}\|^{2}_{L^{2}}\lesssim\|\Phi^{t}\|_{C^{1}}^{2}\|\tilde{u}\|_{L^{2}}^{2}.\end{split}$

The second term in (4.1) can be estimated likewise:

$\begin{split}\sum_{M\geq 1}&M^{-2\rho-\frac{1}{2}}\big{(}\sum_{1\leq K\leq
M}\|S_{M}S^{\prime}_{K}(S^{\prime}_{\gtrsim K}\Phi^{t}S^{\prime}_{\gtrsim
K}\tilde{u}\big{)}\|_{L_{t}^{4}L_{x^{\prime}}^{\infty}}\big{)}^{2}\\\
&\lesssim\sum_{M\geq 1}M^{-2\rho}\big{(}\sum_{1\leq K\leq
M}K^{\frac{3}{2}}\|S^{\prime}_{\gtrsim
K}\Phi^{t}\|_{L^{\infty}_{x^{\prime}}}\|S^{\prime}_{\gtrsim
K}\tilde{u}\|_{L^{2}_{x}}\big{)}^{2}\lesssim\|\Phi^{t}\|_{C^{1}}^{2}\|\tilde{u}\|_{L^{2}}^{2}.\end{split}$

For the last term in (4.1), we additionally make the paraproduct decomposition
in space-time frequencies

$\displaystyle\|S_{M}(S^{\prime}_{\ll
K}\Phi^{t}S^{\prime}_{K}\tilde{u})\|_{X}$
$\displaystyle\leq\|(S_{M}S^{\prime}_{\ll K}\Phi^{t})(S_{\ll
M}S^{\prime}_{\sim K}\tilde{u})\|_{X}$ (65)
$\displaystyle\quad+\|S_{M}(S_{\gtrsim M}S^{\prime}_{\ll K}\Phi^{t}S_{\gtrsim
M}S^{\prime}_{\sim K}\tilde{u})\|_{X}$ $\displaystyle\quad+\|(S_{\ll
M}S^{\prime}_{\ll K}\Phi^{t})(S_{\sim M}S^{\prime}_{\sim K}\tilde{u})\|_{X}.$

The first and second term in (4.1) can be estimated using Bernstein’s
inequality. For the first term in (4.1) we find

$\begin{split}\sum_{M\geq 1}&M^{-2\rho-\frac{1}{2}}\big{(}\sum_{1\leq K\leq
M}\|(S_{M}S^{\prime}_{\ll K}\Phi^{t})(S_{\ll M}S^{\prime}_{\sim
K}\tilde{u})\|_{L^{4}L^{\infty}}\big{)}^{2}\\\ &\lesssim\sum_{M\geq
1}M^{-2\rho-\frac{1}{2}}\big{(}\sum_{1\leq K\leq
M}K^{\frac{3}{2}}M^{\frac{1}{4}}\|S_{M}\Phi^{t}\|_{L^{\infty}}\|\tilde{u}\|_{L^{2}}\big{)}^{2}\\\
&\lesssim\sum_{M\geq
1}M^{-3}M^{3}M^{\frac{1}{2}}\|S_{M}\Phi^{t}\|_{L^{\infty}}^{2}\|\tilde{u}\|_{L^{2}}^{2}\lesssim\|\Phi^{t}\|_{C^{1}}^{2}\|\tilde{u}\|_{L^{2}}^{2}.\end{split}$

The estimate of the second term in (4.1) is given by

$\begin{split}\sum_{M\geq 1}&M^{-2\rho-\frac{1}{2}}\big{(}\sum_{1\leq K\leq
M}\|S_{M}(S_{\gtrsim M}S^{\prime}_{\ll K}\Phi^{t}S_{\gtrsim M}S^{\prime}_{\sim
K}\tilde{u}\big{)}\|_{L^{4}L^{\infty}}\big{)}^{2}\\\ &\lesssim\sum_{M\geq
1}M^{-3}M^{\frac{3}{2}+\epsilon}\|\Phi\|_{C^{1}}^{2}\|\tilde{u}\|_{L^{2}}^{2}\lesssim\|\Phi\|_{C^{1}}^{2}\|\tilde{u}\|_{L^{2}}^{2}.\end{split}$

For the third term in (4.1), we find

$\begin{split}\sum_{M\geq 1}&M^{-2\rho-\frac{1}{2}-2\delta}\big{(}\sum_{1\leq
K\leq M}\|(S_{\ll M}S^{\prime}_{\ll K}\Phi^{t})(S_{\sim M}S^{\prime}_{\sim
K}\tilde{u})\|_{L^{4}L^{\infty}}\big{)}^{2}\\\
&\lesssim\|\Phi^{t}\|_{L^{\infty}}^{2}\sum_{M\geq
1}M^{-2\delta}\big{(}\sum_{1\leq K\leq M}\|\langle
D\rangle^{-\rho-\frac{1}{4}}S_{\sim M}S^{\prime}_{\sim
K}\tilde{u}\|_{L^{4}L^{\infty}}\big{)}^{2}\\\
&\lesssim\|\Phi^{t}\|_{L^{\infty}}^{2}(\|\tilde{u}\|_{L^{2}}+\||D|^{-\frac{1}{2}}\tilde{P}\tilde{u}\|_{L^{2}}+\||D|^{-\frac{3}{4}}\tilde{\rho}_{em}\|_{L^{2}})^{2}.\end{split}$

In the last step we used the hypothesis. It remains to show

$\|\tilde{u}\|_{L^{2}}+\||D|^{-\frac{1}{2}}\tilde{P}\tilde{u}\|_{L^{2}}+\||D|^{-\frac{3}{4}}\tilde{\rho}_{em}\|_{L^{2}}\lesssim\|u\|_{L^{2}}+\||D|^{-\frac{1}{2}}Pu\|_{L^{2}}+\||D|^{-\frac{3}{4}}\rho_{em}\|_{L^{2}}.$

We first note that the definitions imply

$\|\tilde{u}\|_{L^{2}}+\||D|^{-\frac{3}{4}}\tilde{\rho}_{em}\|_{L^{2}}\lesssim\|u\|_{L^{2}}+\||D|^{-\frac{3}{4}}\rho_{em}\|_{L^{2}}.$

We turn to the estimate of the second term. By the above we can write
$\tilde{P}\tilde{u}=P^{\prime}\tilde{u}+Ru$ with

$P^{\prime}=\begin{pmatrix}-\partial_{t}(\varepsilon^{d}\cdot)&\Phi^{t}(\nabla\times(\Phi\cdot))\\\
\Phi^{t}\nabla\times(\Phi\cdot)&\partial_{t}(\mu^{d}\cdot)\end{pmatrix},\qquad
R\in L^{\infty}.$

Moreover, $P^{\prime}\tilde{u}=\tilde{R}\tilde{u}+\Phi^{t}\mathcal{J}$, where
we write $\Phi^{t}$ instead of $\text{diag}(\Phi^{t},\Phi^{t}).$ Since
$\tilde{P}$ is in divergence form, the low frequencies satisfy with implicit
constant depending on the coefficients:

$\||D|^{-\frac{1}{2}}S_{0}(\tilde{P}\tilde{u})\|_{L^{2}}\lesssim\|\tilde{u}\|_{L^{2}}.$

For the high frequencies, we plug in the above relations to find

$\begin{split}\||D|^{-\frac{1}{2}}S_{\geq
1}(\tilde{P}\tilde{u})\|_{L^{2}_{x}}&\lesssim\||D|^{-\frac{1}{2}}S_{\geq
1}(\tilde{R}\tilde{u}+Ru+\Phi^{t}Pu)\|_{L^{2}}\\\
&\lesssim(\|\tilde{R}\|_{L^{\infty}}+\|R\|_{L^{\infty}})\|u\|_{L^{2}}+\||D|^{-\frac{1}{2}}S_{\geq
1}(\Phi^{t}Pu)\|_{L^{2}_{x}}.\end{split}$

For the last term we again use a paraproduct decomposition, writing

(66)
$\begin{split}\|S_{M}(\Phi^{t}Pu)\|_{L_{x}^{2}}&\leq\|(S_{M}\Phi^{t})S_{\ll
M}(Pu)\|_{L^{2}_{x}}+\|(S_{\ll M}\Phi^{t})S_{M}(Pu)\|_{L^{2}_{x}}\\\
&\quad+\|\sum_{K\gtrsim M}S_{K}\Phi^{t}S_{K}(Pu)\|_{L^{2}_{x}}\end{split}$

for $M\geq 1$. The first term in (66) is estimated by

$\begin{split}\sum_{M\geq 1}M^{-1}\|S_{M}\Phi^{t}S_{\ll
M}(Pu)\|_{L^{2}}^{2}&\lesssim\sum_{M\geq
1}M^{-1}\|S_{M}\Phi^{t}\|_{L^{\infty}}^{2}\sum_{K\ll
M}\|S_{K}(Pu)\|_{L^{2}}^{2}\\\ &\lesssim\sum_{M\geq
1}M^{-1}\|S_{M}\Phi^{t}\|^{2}_{L^{\infty}}\\!\sum_{1\leq K\ll
M}\\!K\||D|^{-\frac{1}{2}}S_{K}(Pu)\|_{L^{2}}^{2}\\\ &\lesssim\sum_{M\geq
1}\|S_{M}\Phi^{t}\|_{L^{\infty}}^{2}\||D|^{-\frac{1}{2}}Pu\|_{L^{2}}^{2}\\\
&\lesssim\|\Phi^{t}\|_{C^{1}}^{2}\||D|^{-\frac{1}{2}}Pu\|_{L^{2}}^{2}.\end{split}$

Similarly, for the second term in (66) we obtain

$\sum_{M\geq 1}M^{-1}\|S_{\ll
M}\Phi^{t}S_{M}(Pu)\|_{L^{2}}^{2}\lesssim\|\Phi^{t}\|_{C^{1}}^{2}\||D|^{-\frac{1}{2}}Pu\|_{L^{2}}^{2}.$

Finally, we estimate the third term in (66) by

$\displaystyle\sum_{M\geq 1}$ $\displaystyle M^{-1}\sum_{K\geq
M}\|S_{K}\Phi^{t}S_{K}(Pu)\|_{L^{2}}^{2}$ $\displaystyle\lesssim\sum_{M\geq
1}M^{-1}\sum_{K\gtrsim
M}K\|S_{K}\Phi^{t}\|_{L^{\infty}}^{2}K^{-1}\|S_{K}(Pu)\|_{L^{2}}^{2}$
$\displaystyle\lesssim\sum_{M\geq
1}M^{-1}\|\Phi^{t}\|_{C^{1}}^{2}\||D|^{-\frac{1}{2}}Pu\|_{L^{2}}^{2}\lesssim\|\Phi^{t}\|_{C^{1}}^{2}\||D|^{-\frac{1}{2}}Pu\|_{L^{2}}^{2}.\qed$

###  4.2. Proof of Strichartz estimates for the transformed equation

The proof of Theorem 1.2 in the general case revolves around the proof of
analogs of (32), which is

(67)
$\||D|^{-\rho-\frac{1}{4}}\tilde{u}\|_{(L^{p}L^{q})_{2}}\lesssim(1+\|(\varepsilon^{d},\mu^{d})\|_{C^{1}})\|\tilde{u}\|_{L^{2}}+\||D|^{-\frac{1}{2}}\tilde{P}\tilde{u}\|_{L^{2}}+\||D|^{-\frac{3}{4}}\tilde{\rho}_{em}\|_{L^{2}}.$

Using Lemma 4.1, this inequality allows us to complete the proof of Theorem
1.2.

######  Proposition 4.2.

Let $\varepsilon^{d}$, $\mu^{d}\in
C^{1}(\mathbb{R}\times\mathbb{R}^{3};\mathbb{R}^{3\times 3}_{sym})$ be
diagonal matrices, which are uniformly elliptic. Let $\tilde{P}$,
$\tilde{\rho}_{e}$, and $\tilde{\rho}_{m}$ be and in (60) and (61). Then we
find (67) to hold for $2\leq p<q\leq\infty$,
$\rho=3\big{(}\frac{1}{2}-\frac{1}{q}\big{)}-\frac{1}{p}$, and
$\frac{2}{p}+\frac{1}{q}=\frac{1}{2}$.

In the following we apply the analysis of Section 3 with the role of the
partial derivatives $\partial_{k}$ presently played by the differential
operator $\eta_{k}=\varphi_{k}^{\leq\lambda}\cdot\nabla_{x^{\prime}}$. By
dyadic frequency localization and the usual commutator estimates, (67) follows
from

$\lambda^{-\rho-\frac{1}{4}}\|S_{\lambda}\tilde{u}\|_{L^{p}L^{q}}\lesssim\|S_{\lambda}u\|_{L^{2}}+\lambda^{-\frac{1}{2}}\|\tilde{P}_{\lambda}S_{\lambda}u\|_{L^{2}}+\lambda^{-\frac{3}{4}}\|S_{\lambda}\tilde{\rho}_{em}\|_{L^{2}}.$

Above $\tilde{P}_{\lambda}$ denotes the frequency truncated version at
frequencies $<\lambda/8$ of $\tilde{P}$ given by

$\tilde{P}_{\lambda}=\begin{pmatrix}-\partial_{t}(\varepsilon^{d}_{\lambda}\cdot)&\mathcal{C}(\eta_{\leq\lambda}(D,x))\\\
\mathcal{C}(\eta_{\lambda}(D,x))&\partial_{t}(\mu^{d}_{\lambda}\cdot)\end{pmatrix}.$

In $\eta_{\leq\lambda}$ the coefficients of $\varphi$ are also frequency
truncated at frequencies $\lambda/8$. The frequency truncation is suppressed
in the following to lighten the notation.

We apply the FBI transform and let
$v_{\lambda}=T_{\lambda}\tilde{u}_{\lambda}$ and
$T_{\lambda}(\tilde{\mathcal{J}}/\lambda)=f_{\lambda}$. For
$C^{1}$-coefficients an application of Theorem 3.5 yields

$\tilde{p}(x,\xi)v_{\lambda}=f_{\lambda}+g_{\lambda}\quad\text{with \
}\|g_{\lambda}\|_{L^{2}_{\Phi}}\lesssim_{\|(\varepsilon^{d},\mu^{d})\|_{C^{1}}}\lambda^{-1/2}\|S_{\lambda}u\|_{L^{2}}$

and

$\tilde{p}(x,\xi)=\begin{pmatrix}-i\xi_{0}\varepsilon^{d}&i\mathcal{C}(\eta(x,\xi))\\\
i\mathcal{C}(\eta(x,\xi))&i\xi_{0}\mu^{d}\end{pmatrix}\in\mathbb{C}^{6\times
6}.$

The estimate (67) becomes

$\lambda^{-\rho-\frac{1}{4}}\|T_{\lambda}^{*}\tilde{v}_{\lambda}\|_{L^{p}L^{q}}\lesssim(1+\|(\varepsilon^{d},\mu^{d})\|_{C^{1}})\|\tilde{v}_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{\frac{1}{2}}\|\tilde{f}_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{-\frac{3}{4}}\|S_{\lambda}\tilde{\rho}_{em}\|_{L^{2}}.$

We recast this as

(68)
$\|T_{\lambda}^{*}\tilde{v}_{\lambda}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{\frac{1}{4}}\|\tilde{v}_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{\frac{3}{4}}\|\tilde{p}(x,\xi)\tilde{v}_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{-\frac{1}{2}}\|S_{\lambda}\tilde{\rho}_{em}\|_{L^{2}}).$

Next, we reduce (68) to a scalar estimate. For this purpose, corresponding to
(37) we let

$\tilde{q}(x,\xi)=-\xi_{0}^{2}(\xi_{0}^{4}-\xi_{0}^{2}q_{0}(x,\eta^{\prime}(\xi))+q_{1}(x,\eta^{\prime}(\xi))),$

with $q_{0}$ and $q_{1}$ defined like in Section 3 and define the symmetrizer

$\tilde{\sigma}(x,\xi)=\begin{pmatrix}-i\xi_{0}\varepsilon_{d}^{-1}&i\varepsilon_{d}^{-1}\mathcal{C}(\eta(x,\xi))\mu_{d}^{-1}\\\
i\mu_{d}^{-1}\mathcal{C}(\eta(x,\xi))\varepsilon_{d}^{-1}&i\xi_{0}\mu_{d}^{-1}\end{pmatrix}\in\mathbb{C}^{6\times
6},$

obtaining

$\tilde{\sigma}(x,\xi)\tilde{p}(x,\xi)=\begin{pmatrix}M_{E}(x,\eta)-\xi_{0}^{2}&0\\\
0&M_{H}(x,\eta)-\xi_{0}^{2}\end{pmatrix},$

compare (39). We state the announced reduction.

######  Proposition 4.3.

For the proof of (68) under the assumptions of Proposition 4.2, it suffices to
show the estimate

(69)
$\|T_{\lambda}^{*}w_{\lambda}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{1/4}\|w_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{3/4}\|\tilde{q}(x,\xi)w_{\lambda}\|_{L^{2}_{\Phi}}).$

###### Proof.

Clearly, (68) follows from

$\|T_{\lambda}^{*}\tilde{v}_{\lambda}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{1/4}\|\tilde{v}_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{3/4}\|\tilde{\sigma}(x,\xi)\tilde{p}(x,\xi)\tilde{v}_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{-1/2}\|S_{\lambda}\tilde{\rho}_{em}\|_{L^{2}}).$

For this purpose, we show that

(70)
$\|T^{*}_{\lambda}\tilde{v}_{\lambda,1}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{1/4}\|\tilde{v}_{\lambda,1}\|_{L^{2}_{\Phi}}+\lambda^{3/4}\|(M_{E}(x,\eta)-\xi_{0}^{2})\tilde{v}_{\lambda,1}\|_{L^{2}_{\Phi}}+\lambda^{-1/2}\|S_{\lambda}\tilde{\rho}_{e}\|_{L^{2}}),$
(71)
$\|T_{\lambda}^{*}\tilde{v}_{\lambda,2}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{1/4}\|\tilde{v}_{\lambda,2}\|_{L^{2}_{\Phi}}+\lambda^{3/4}\|(M_{H}(x,\eta)-\Xi_{0}^{2})\tilde{v}_{\lambda,2}\|_{L^{2}_{\Phi}}+\lambda^{-1/2}\|S_{\lambda}\tilde{\rho}_{m}\|_{L^{2}}).$

In the following it becomes relevant that $|\eta^{\prime}|\sim|\xi^{\prime}|$.
We prove the estimates in the regions

  * (1)

$\\{|\xi_{0}|\gg|\eta^{\prime}|\\}$,

  * (2)

$\\{|\xi_{0}|\ll|\eta^{\prime}|\\}$, for which we take the generalized charges
into account,

  * (3)

$\\{|\xi_{0}|\sim|\eta^{\prime}|\\}$.

The first region is handled like in the proof of Proposition 3.6 using
$|\eta^{\prime}|\sim|\xi^{\prime}|$. For the estimate in the second region, we
focus on (70) because (71) can be proved in a similar way. We decompose
$\tilde{v}_{\lambda,1}=\tilde{v}^{s}_{\lambda,1}+\tilde{v}_{\lambda,1}^{p}$
with

$\tilde{v}^{p}_{\lambda,1}=\frac{(\eta^{\prime}.\tilde{\varepsilon}^{\lambda}\tilde{v}_{\lambda,1})\eta^{\prime}}{|\eta^{\prime}|^{2}_{\varepsilon^{d}}}.$

The contribution of $\tilde{v}^{p}_{\lambda,1}$ is treated by Sobolev
embedding

$\|T_{\lambda}^{*}\tilde{v}^{p}_{\lambda,1}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho+\frac{1}{2}}\|\tilde{v}^{p}_{\lambda,1}\|_{L^{2}_{\Phi}}\lesssim\lambda^{\rho+\frac{1}{2}}\|(\eta^{\prime}.\varepsilon^{d}_{\lambda}\tilde{v}_{\lambda,1})\|_{L^{2}_{\Phi}}.$

By Theorem 3.5 and a commutator estimate, we find

$\|\eta^{\prime}.(\varepsilon^{d}_{\lambda}v_{\lambda,1})\|_{L^{2}_{\Phi}}\lesssim\lambda^{-\frac{1}{2}}\|v_{\lambda,1}\|_{L^{2}}+\|S_{\lambda}\frac{\eta^{\prime}(x,D)}{\lambda}(\varepsilon^{d}\tilde{\mathcal{E}})\|_{L^{2}}.$

Secondly, it follows like in the proof of Proposition 3.6 by the same
algebraic relations, replacing $\xi^{\prime}$ with $\eta^{\prime}$, that
$|(M_{E}(x,\eta^{\prime})-\xi_{0}^{2})v_{1}^{\lambda}|\gtrsim|v_{1}^{\lambda}|$
for $\eta^{\prime}.(\varepsilon^{d}_{\lambda}\tilde{v}^{\lambda}_{1})=0$ and
$\\{|\xi_{0}|\ll|\xi^{\prime}|\\}$. The details are omitted.

Similarly, by replacing $\eta^{\prime}$ with $\xi^{\prime}$, we argue that the
estimate

$\|T_{\lambda}^{*}v_{\lambda,1}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{1/4}\|v_{\lambda,1}\|_{L^{2}_{\Phi}}+\lambda^{3/4}\|(M_{E}-\xi_{0}^{2})v_{\lambda,1}\|_{L^{2}_{\Phi}})$

holds true provided that

$\|T_{\lambda}^{*}v_{\lambda,1,k}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{1/4}\|v_{\lambda,1,k}\|_{L^{2}_{\Phi}}+\lambda^{3/4}\|(Z_{\varepsilon,\mu}\frac{\varepsilon^{d}}{\varepsilon^{d}_{1}\varepsilon^{d}_{2}\varepsilon^{d}_{3}}(M_{E}-\xi_{0}^{2})v_{\lambda,1})_{k}\|_{L^{2}_{\Phi}}),$

but this is (69). ∎

To prove (69), we modify the arguments of Section 3. We consider the operator

$\tilde{W}_{\lambda}=T_{\lambda}^{*}\frac{a(x,\xi)}{\lambda^{\frac{1}{4}}+\lambda^{\frac{3}{4}}|\tilde{q}(x,\xi)|},$

for which we shall prove

$\|\tilde{W}_{\lambda}\|_{L^{2}\to
L^{p}L^{q}}\lesssim\lambda^{\rho+\frac{1}{2}}.$

By the $TT^{*}$-argument, we can likewise prove the estimate

$\tilde{V}_{\lambda}=T_{\lambda}^{*}\frac{a^{2}(x,\eta)\Phi(\xi)}{(\lambda^{-\frac{1}{4}}+\lambda^{\frac{1}{4}}|\tilde{q}(x,\xi)|)^{2}}T_{\lambda}.$

Again we have to understand the curvature properties of
$\\{\xi\in\mathbb{R}^{4}:\,\tilde{q}(x,\xi)=0\\}$. Note that
$\tilde{q}(x,\xi)$ is $4$-homogeneous in $\xi$, i.e.,

$\tilde{q}(x,\xi_{0},\xi^{\prime})=\xi_{0}^{4}\tilde{q}(x,1,\xi^{\prime}/\xi_{0}).$

This reduces again to the analysis of the surface
$\tilde{S}=\\{\xi^{\prime}\in\mathbb{R}^{3}:\tilde{q}(x,1,\xi^{\prime})=0\\}$.
By the change of variables
$\eta_{k}^{\prime}=\varphi_{k}^{\leq\lambda}(x)\cdot\xi^{\prime}$ we can
reduce to the Fresnel surface
$S=\\{\xi^{\prime}\in\mathbb{R}^{3}:q(x,1,\xi^{\prime})=0\\}$. Like in Section
3 we split replace $a^{2}(x,\xi)$ by $a(x,\xi)$ and split it into
$a=a_{1}+a_{2}+a_{3}$ according to the regions of the Fresnel surface $S$.
Correspondingly, we decompose
$\tilde{V}_{\lambda}=\tilde{V}_{\lambda,1}+\tilde{V}_{\lambda,2}+\tilde{V}_{\lambda,3}$.
We arrive at the following:

######  Proposition 4.4.

Let $\rho=3\big{(}\frac{1}{2}-\frac{1}{q}\big{)}-\frac{1}{p}$ and $2<p\leq
q\leq\infty$. The estimate

$\|\tilde{V}_{\lambda,i}\|_{L^{p^{\prime}}L^{q^{\prime}}\to
L^{p}L^{q}}\lesssim\lambda^{1+2\rho}$

holds true, if

  * •

$i=1$ and $\frac{1}{p}+\frac{1}{q}=\frac{1}{2}$,

  * •

$i\in\\{2,3\\}$ and $\frac{2}{p}+\frac{1}{q}=\frac{1}{2}$.

###### Proof.

For fixed $x\in[-1,1]^{4}$ we observe that the change of variables

(72) $\eta_{k}^{\prime}=\varphi_{k}^{\leq\lambda}(x)\cdot\xi^{\prime}$

is non-degenerate for $\lambda\gg 1$. Indeed,
$|J\eta^{\prime}|=|\det(\varphi_{1}^{\leq\lambda},\varphi_{2}^{\leq\lambda},\varphi_{3}^{\leq\lambda})|\sim
1$.

Hence, for $i=1,2$, a uniform bound for the family of operators

$\lambda^{-1-2\rho}T_{\lambda}^{*}\frac{a_{i}(x,\eta)\Phi(\xi)}{(\lambda^{-\frac{1}{4}}+\lambda^{\frac{1}{4}}|\tilde{q}(x,\xi)|)^{2}}T_{\lambda}:L^{p^{\prime}}L^{q^{\prime}}\to
L^{p}L^{q}$

follows from the integrability of the weight

$\int\frac{1}{(\lambda^{-\frac{1}{4}}+\lambda^{\frac{1}{4}}|\tilde{q}|)^{2}}d\tilde{q}\lesssim
1,$

by which we can reduce to level sets $\delta_{\tilde{q}(x,\xi)=c}$ by
foliation. Note that within $\text{supp}(a_{1})\cup\text{supp}(a_{2})$ the
surface $\\{\xi\in\mathbb{R}^{4}:\tilde{q}(x,\xi)=0\\}$ is a regular surface.
By the regular change of variables (72) the curvature properties are
inherited.

For the more involved case of neighbourhoods of the conical singularities, the
additional dyadic decomposition is carried out in $\eta$. After rescaling to
unit distance of the singularity, one can then proceed as in Proposition 3.7.
∎

###  4.3. Conclusion of the proof of Theorem 1.2

With the estimates for $\tilde{V}_{\lambda,i}$ at disposal, the propositions
of the previuous subsection yield the Strichartz estimates

$\||D|^{-\rho-\frac{1}{4}}\tilde{u}\|_{(L^{4}L^{\infty})_{2}}\lesssim(1+\|\partial(\varepsilon^{d},\mu^{d})\|_{L^{\infty}})\|\tilde{u}\|_{L^{2}}+\||D|^{-\frac{1}{2}}\tilde{P}\tilde{u}\|_{L^{2}}+\||D|^{-\frac{3}{4}}\tilde{\rho}_{em}\|_{L^{2}}.$

These estimates transpire to the original quantities by the considerations in
Section 4.1, leading to

$\||D|^{-\rho-\frac{1}{4}}u\|_{L_{t}^{4}L^{\infty}_{x^{\prime}}}\lesssim(1+\|\partial(\varepsilon,\mu)\|_{L^{\infty}})\|u\|_{L^{2}}+\||D|^{-\frac{1}{2}}Pu\|_{L^{2}}+\||D|^{-\frac{3}{4}}\rho_{em}\|_{L^{2}}.$

We can now conclude like in Section 3.3. Proposition 3.8 implies

$\begin{split}\|\langle
D^{\prime}\rangle^{-\rho-\frac{1}{8}-\delta}u\|_{L_{t}^{4}(0,T;L^{\infty}_{x^{\prime}})}&\leq
C(\|u_{0}\|_{L^{2}}+\|Pu\|_{L_{T}^{1}L^{2}_{x^{\prime}}}\\\ &\qquad+\|\langle
D^{\prime}\rangle^{-\frac{5}{8}}\rho_{em}(0)\|_{L^{2}_{x^{\prime}}}+\|\langle
D^{\prime}\rangle^{-\frac{5}{8}}\partial_{t}\rho_{em}\|_{L^{1}_{t}L^{2}_{x^{\prime}}})\end{split}$

with $C=C(X,T,\delta)$, $X=\|\partial(\varepsilon,\mu)\|_{L^{\infty}}$.
Interpolation with Proposition 3.1 yields Theorem 1.1 for
$\frac{2}{p}+\frac{1}{q}=\frac{1}{2}$. The general case follows from Sobolev
embedding. $\hfill\Box$

##  5\. Application to quasilinear Maxwell equations

With the Strichartz estimates at disposal, we can improve local well-posedness
results for quasilinear Maxwell equations in the fully anisotropic case. This
section is devoted to the proof of Theorem 1.5. By local well-posedness we
mean existence, uniqueness, and continuous dependence of the data-to-solution
mapping. We focus on proving estimates in rough norms for smooth solutions,
whose existence we take for granted. The data-to-solution mapping can then be
extended by continuity to rough initial data via standard arguments. We also
refer to our previous work [17].

We consider

(73)
$\left\\{\begin{array}[]{rlrlrl}\partial_{t}\mathcal{D}\\!\\!\\!\\!&=\nabla\times\mathcal{H},&\quad\nabla\cdot\mathcal{D}\\!\\!\\!\\!&=0,&\quad\mathcal{D}(0)\\!\\!\\!\\!&=\mathcal{D}_{0}\in
H^{s}(\mathbb{R}^{3};\mathbb{R}^{3}),\\\
\partial_{t}\mathcal{B}\\!\\!\\!\\!&=-\nabla\times\mathcal{E},&\quad\nabla\cdot\mathcal{B}\\!\\!\\!\\!&=0,&\quad\mathcal{B}(0)\\!\\!\\!\\!&=\mathcal{B}_{0}\in
H^{s}(\mathbb{R}^{3};\mathbb{R}^{3}),\end{array}\right.$

with fields

(74) $\|\mathcal{E}\|_{L^{\infty}}+\|\mathcal{B}\|_{L^{\infty}}\leq\delta,$

$\delta$ to be specified later, and constitutive relations

$\varepsilon=\varepsilon(\mathcal{E})\in\mathbb{R}^{3\times
3}_{\text{sym}},\quad\mu=1_{3\times 3}$

such that $\varepsilon(\mathcal{E})$ is uniformly elliptic and has uniformly
separated eigenvalues for $|\mathcal{E}|\leq\delta$. We use this smallness
condition to guarantee these two crucial properties of the permittivities, it
is not used in the wellposedness analysis given below.

We let $u=(\mathcal{D},\mathcal{B})$ in the following because these are the
variables of which the time-derivative is given. The proof of the theorem
follows along the general scheme of [15, 17]: Set $A=\sup_{0\leq
t^{\prime}\leq t}\|u(t^{\prime})\|_{L^{\infty}_{x^{\prime}}}$ and
$B(t)=\|\nabla_{x^{\prime}}u(t)\|_{L^{\infty}_{x^{\prime}}}$.

  * •

We show energy estimates to hold for $s\geq 0$:

(75) $E^{s}(u(t))\lesssim E^{s}(u(0))e^{C\int_{0}^{t}B(s)ds}$

with $E^{s}(u)\approx_{A}\|u\|_{H^{s}}$.

  * •

For differences of solutions we prove $L^{2}$-bounds depending on the
$H^{s}$-norm $v=u^{1}-u^{2}$ of the initial data:

$\|v(t)\|_{L^{2}}\lesssim\|v_{0}\|_{L^{2}}\text{ for }0\leq t\leq
T=T(\|u^{i}(0)\|_{H^{s}}).$

  * •

We use the frequency envelope approach due to Tao [20] (see also [6]) to show
continuous dependence (but no locally uniform continuous dependence).

To improve on the threshold $s>\frac{5}{2}$ dictated by the energy method, we
want to estimate $\|\partial_{x}u\|_{L_{T}^{1}L_{x^{\prime}}^{\infty}}$ by
Strichartz estimates. This is carried out via a bootstrap argument. For $u$ a
solution in $H_{x^{\prime}}^{\frac{3}{2}+\alpha}$, the coefficients of the
quasilinear Maxwell equation are $\alpha$ Hölder-continuous and in the scope
of Theorem 1.1. If the provided Strichartz estimates can control
$\|\partial_{x}u\|_{L_{T}^{1}L_{x^{\prime}}^{\infty}}$ in terms of the
$H_{x^{\prime}}^{\frac{3}{2}+\alpha}$-norm, we can close the iteration. To
find $\alpha$, we equate the derivative loss $\rho+1$ of the Strichartz
estimates with $\frac{3}{2}+\alpha$, obtaining

$3\big{(}\frac{1}{2}-\frac{1}{q}\big{)}-\frac{1}{p}+\frac{2-\alpha}{2p}+1=\frac{3}{2}+\alpha,\quad
q=\infty,\;p=4.$

This gives $\alpha=\frac{8}{9}$ and suggests that the argument closes for
initial data with regularity $s>\frac{5}{2}-\frac{1}{9}$.

One new ingredient compared to previous works is to find the energy norm for
(73), which is carried out in detail below. It turns out that in the fully
anisotropic case introducing a suitable energy norm requires more care than in
the isotropic case considered in [17, 15]. We recall the following fact from
[15].

######  Proposition 5.1 (Existence of energy estimates).

Suppose that there is symmetric $C(u)$ such that

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

and $C(u)$ is uniformly elliptic. Then, we have

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

and (75) holds, for $s\geq 0$.

To this end, we rewrite Maxwell equations in conservative form
$\partial_{t}u=\mathcal{A}^{j}(u)\partial_{j}u$. However, to state (73) in
conservative form, we have to recast
$\mathcal{E}=\psi(\mathcal{D})\mathcal{D}$, for which we can formulate a
necessary and sufficient condition on the existence of symmetrizers. Note that
$\psi(\mathcal{D})$ is symmetric for symmetric $\varepsilon(\mathcal{E})$.

######  Proposition 5.2.

There exists

(77) $C(u)=\begin{pmatrix}C^{1}(u)&0\\\ 0&1_{3\times
3}\end{pmatrix}\in\mathbb{R}^{6\times 6}_{sym}$

with $C^{1}(u)\in\mathbb{R}^{3\times 3}$ that satisfies (76) if and only if

(78)
$\varepsilon^{ijk}\partial_{j}\psi(\mathcal{D})_{k\ell}\mathcal{D}_{\ell}=0$

for $i\in\\{1,2,3\\}$ with summation over $k,j\in\\{1,2,3\\}$.

###### Proof.

We write for $j=1,2,3$

$\mathcal{A}^{j}(u)=\begin{pmatrix}0&A_{1}^{j}(u)\\\
A_{2}^{j}(u)&0\end{pmatrix};\quad
A_{1}^{j}(x),A_{2}^{j}(x)\in\mathbb{R}^{3\times 3}$

with $(A_{1}^{j})_{mn}=-\varepsilon_{jmn}$. For the special form (77) the
equation (76) translates to

(79) $(A_{1}^{j})^{t}C^{1}=A^{j}_{2}.$

We compute $A^{j}_{2}$ from

$-(\nabla\times(\psi(\mathcal{D})\mathcal{D}))_{i}=(A_{2}^{j}(\mathcal{D})\partial_{j}\mathcal{D})_{i},$

where we have

$\begin{split}(\nabla\times(\psi(\mathcal{D})\mathcal{D}))_{i}&=\varepsilon^{ijk}\partial_{j}(\psi(\mathcal{D})\mathcal{D})_{k}\\\
&=\varepsilon^{ijk}\partial_{j}(\psi(\mathcal{D})_{km}\mathcal{D}_{m})\\\
&=\varepsilon^{ijk}[\psi(\mathcal{D})_{km}\partial_{j}\mathcal{D}_{m}]+\varepsilon^{ijk}\partial_{\ell}\psi(\mathcal{D})_{km}(\partial_{j}\mathcal{D}_{\ell})\mathcal{D}_{m}.\end{split}$

We define the matrices

$\mathcal{B}^{j}_{im}=-\varepsilon^{ijk}\psi(\mathcal{D})_{km},\quad\mathcal{C}^{j}_{im}=-\varepsilon^{ijk}\partial_{m}\psi(\mathcal{D})_{kl}\mathcal{D}_{l}$

and let $A_{2}^{j}=\mathcal{B}^{j}+\mathcal{C}^{j}$. We obtain

$\displaystyle\mathcal{B}^{1}$ $\displaystyle=\begin{pmatrix}0&0&0\\\
\psi(\mathcal{D})_{31}&\psi(\mathcal{D})_{32}&\psi(\mathcal{D})_{33}\\\
-\psi(\mathcal{D})_{21}&-\psi(\mathcal{D})_{22}&-\psi(\mathcal{D})_{23}\end{pmatrix},$
$\displaystyle\mathcal{B}^{2}$
$\displaystyle=\begin{pmatrix}-\psi(\mathcal{D})_{31}&-\psi(\mathcal{D})_{32}&-\psi(\mathcal{D})_{33}\\\
0&0&0\\\
\psi(\mathcal{D})_{11}&\psi(\mathcal{D})_{12}&\psi(\mathcal{D})_{13}\end{pmatrix},$
$\displaystyle\mathcal{B}^{3}$
$\displaystyle=\begin{pmatrix}\psi(\mathcal{D})_{21}&\psi(\mathcal{D})_{22}&\psi(\mathcal{D})_{23}\\\
-\psi(\mathcal{D})_{11}&-\psi(\mathcal{D})_{12}&-\psi(\mathcal{D})_{13}\\\
0&0&0\end{pmatrix}.$

Moreover,

$\begin{split}\mathcal{C}^{1}&=\begin{pmatrix}0&0&0\\\
\partial_{1}\psi(\mathcal{D})_{3\ell}\mathcal{D}_{\ell}&\partial_{2}\psi(\mathcal{D})_{3\ell}\mathcal{D}_{\ell}&\partial_{3}\psi(\mathcal{D})_{3\ell}\mathcal{D}_{\ell}\\\
-\partial_{1}\psi(\mathcal{D})_{2\ell}\mathcal{D}_{\ell}&-\partial_{2}\psi(\mathcal{D})_{2\ell}\mathcal{D}_{\ell}&-\partial_{3}\psi(\mathcal{D})_{2\ell}\mathcal{D}_{\ell}\end{pmatrix},\\\
\mathcal{C}^{2}&=\begin{pmatrix}-\partial_{1}\psi(\mathcal{D})_{3\ell}\mathcal{D}_{\ell}&-\partial_{2}\psi(\mathcal{D})_{3\ell}\mathcal{D}_{\ell}&-\partial_{3}\psi(\mathcal{D})_{3\ell}\mathcal{D}_{\ell}\\\
0&0&0\\\
\partial_{1}\psi(\mathcal{D})_{1\ell}\mathcal{D}_{\ell}&\partial_{2}\psi(\mathcal{D})_{1\ell}\mathcal{D}_{\ell}&\partial_{3}\psi(\mathcal{D})_{1\ell}\mathcal{D}_{\ell}\end{pmatrix},\\\
\mathcal{C}^{3}&=\begin{pmatrix}\partial_{1}\psi(\mathcal{D})_{2\ell}\mathcal{D}_{\ell}&\partial_{2}\psi(\mathcal{D})_{2\ell}\mathcal{D}_{\ell}&\partial_{3}\psi(\mathcal{D})_{2\ell}\mathcal{D}_{\ell}\\\
-\partial_{1}\psi(\mathcal{D}_{1\ell}\mathcal{D}_{\ell}&-\partial_{2}\psi(\mathcal{D})_{1\ell}\mathcal{D}_{\ell}&-\partial_{3}\psi(\mathcal{D})_{1\ell}\mathcal{D}_{\ell}\\\
0&0&0\end{pmatrix}.\end{split}$

Note that

$A_{1}^{1}=\begin{pmatrix}0&0&0\\\ 0&0&-1\\\
0&1&0\end{pmatrix},\;A_{1}^{2}=\begin{pmatrix}0&0&1\\\ 0&0&0\\\
-1&0&0\end{pmatrix},\;A_{1}^{3}=\begin{pmatrix}0&-1&0\\\ 1&0&0\\\
0&0&0\end{pmatrix}.$

This yields

$\begin{split}(A_{1}^{1})^{t}C^{1}&=\begin{pmatrix}0&0&0\\\
C^{1}_{31}&C^{1}_{32}&C^{1}_{33}\\\
-C^{1}_{21}&-C^{1}_{22}&-C^{1}_{23}\end{pmatrix},\;(A_{1}^{2})^{t}C^{1}=\begin{pmatrix}-C^{1}_{31}&-C^{1}_{32}&-C^{1}_{33}\\\
0&0&0\\\ C^{1}_{11}&C^{1}_{12}&C^{1}_{13}\end{pmatrix},\\\
(A_{1}^{3})^{t}C^{1}&=\begin{pmatrix}C^{1}_{21}&C^{1}_{22}&C^{1}_{23}\\\
-C^{1}_{11}&-C^{1}_{12}&-C^{1}_{13}\\\ 0&0&0\end{pmatrix}.\end{split}$

A comparison of the coefficients in (79) yields (78) by symmetry of
$\psi(\mathcal{D})$. For instance, we find

$\begin{split}((A^{1}_{1})^{t}C^{1})_{21}&=C^{1}_{31}=\psi(\mathcal{D})_{31}+\partial_{1}\psi(\mathcal{D})_{3\ell}\mathcal{D}_{\ell},\\\
((A^{2}_{1})^{t}C^{1})_{31}&=C^{1}_{13}=\psi(\mathcal{D})_{13}+\partial_{3}\psi(\mathcal{D})_{1\ell}\mathcal{D}_{\ell}.\end{split}$

This gives
$\varepsilon^{1jk}\partial_{j}\psi(\mathcal{D})_{k\ell}\mathcal{D}_{\ell}=0$.
The other conditions are found likewise. ∎

We remark that this assumption can be difficult to verify starting with
$\varepsilon=\varepsilon(\mathcal{E})$. It turns out that the seemingly
natural ansatz

$\varepsilon(\mathcal{E})=\text{diag}(\varepsilon_{0}^{1},\varepsilon_{0}^{2},\varepsilon_{0}^{3})+\text{diag}(\alpha_{1},\alpha_{2},\alpha_{3})|\mathcal{E}|^{2}$

for
$\varepsilon_{0}^{1}\neq\varepsilon_{0}^{2}\neq\varepsilon_{0}^{3}\neq\varepsilon_{0}^{1}$
fails if $\alpha_{i}\neq\alpha_{j}$ for some $i\neq j$. An admissible choice
of $\varepsilon$ is

(80)
$\varepsilon(\mathcal{E})=\text{diag}(\varepsilon_{0}^{1},\varepsilon_{0}^{2},\varepsilon_{0}^{3})+\text{diag}(\alpha_{1}|\mathcal{E}_{1}|^{2},\alpha_{2}|\mathcal{E}_{2}|^{2},\alpha_{3}|\mathcal{E}_{3}|^{2})$

with $\varepsilon_{0}^{i}>0$. In this case, we have
$\mathcal{D}_{i}=(\varepsilon_{0}^{i}+\alpha_{i}|\mathcal{E}_{i}|^{2})\mathcal{E}_{i}$,
which admits inversion as

$\mathcal{E}_{i}=\psi_{i}(\mathcal{D}_{i})\mathcal{D}_{i}\text{ and
}\psi_{i}(0)=(\varepsilon_{0}^{i})^{-1}.$

This allows for verification of (78). Instead of starting with
$\varepsilon=\varepsilon(\mathcal{E})$, one can make the ansatz

(81)
$\psi(\mathcal{D})_{ij}=\varepsilon^{0}_{ij}+\alpha_{ij}\mathcal{D}_{i}\mathcal{D}_{j},$

for which it is easy to see that (78) requires symmetry of
$(\varepsilon^{0}_{ij})_{ij}$ and $(\alpha_{ij})_{ij}$. Then, by the implicit
function theorem, for small fields

(82) $\|\mathcal{E}\|_{L^{\infty}}+\|\mathcal{D}\|_{L^{\infty}}\leq\delta$

we can rewrite $\mathcal{D}=\varepsilon(\mathcal{E})\mathcal{E}$ with
$\varepsilon(\mathcal{E})=\psi(\mathcal{D}(\mathcal{E}))^{-1}$. Clearly,
$\varepsilon(\mathcal{E})$ is symmetric as an inverse of a symmetric matrix,
and the uniform anisotropy and ellipticity are true for small
$\|\mathcal{E}\|_{L^{\infty}}$. Taking $\delta$ smaller than in (82), if
necessary, specifies $\delta$ in (74).

By the assumptions of Theorem 1.5, the energy estimate

(83)
$\|u(t)\|_{H^{s}}\lesssim_{\delta}\|u(0)\|_{H^{s}}e^{C\int_{0}^{t}B(s)ds}$

is true for $\|u\|_{L_{t,x^{\prime}}^{\infty}}\leq\delta$. In the next step we
use Strichartz estimates to show a priori estimates for $s>2+\frac{7}{18}$.
Note that by Sobolev embedding (83) yields a priori estimates for
$s>\frac{5}{2}$.

######  Proposition 5.3.

Under the assumptions of Theorem 1.5, the a priori estimate

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

holds for $s>2+\frac{7}{18}$, $T=T(\|u_{0}\|_{H^{s}})$ and
$\|u_{0}\|_{H^{s}}\leq\delta$ for some $\delta>0$.

###### Proof.

Suppose that the smooth solution $u=(\mathcal{D},\mathcal{H})$ exists for
$[0,T_{0}]$. Let $0<T\leq T_{0}$. The proof follows from bootstrapping the
energy estimate (83)

$E^{s}(u(t))\lesssim_{A}e^{C\int_{0}^{T}B(s)ds}E^{s}(u(0))$

and the Strichartz estimate

(84)
$\|\partial_{x^{\prime}}u\|_{L_{T}^{4}L_{x^{\prime}}^{\infty}}\lesssim\|\langle
D^{\prime}\rangle^{s}u\|_{L_{T}^{\infty}L_{x}^{2}}$

for $s\geq s_{0}>2+\frac{7}{18}$. The constant in the Strichartz estimate has
to be uniform provided that $T$,
$\|\partial_{x}u\|_{L_{T}^{4}L_{x^{\prime}}^{\infty}}$,
$\|u\|_{C^{\frac{8}{9}+\sigma}_{x}}$ are bounded and $s\geq s_{0}$.

We have to establish (84), for which we make use of commutator arguments.
These only apply after changing Maxwell equations to non-divergence form. Let

$\varepsilon(\mathcal{E})=\text{diag}(\varphi_{1}(\mathcal{E}_{1}),\varphi_{2}(\mathcal{E}_{2}),\varphi_{3}(\mathcal{E}_{3}))$

and denote

$\tilde{\varepsilon}(\mathcal{E})=\text{diag}(\tilde{\varepsilon}_{1}(\mathcal{E}_{1}),\tilde{\varepsilon}_{2}(\mathcal{E}_{2}),\tilde{\varepsilon}_{3}(\mathcal{E}_{3}))\quad\text{with
\
}\tilde{\varepsilon}_{i}(\mathcal{E})=\varphi_{i}(\mathcal{E}_{i})+\varphi_{i}^{\prime}(\mathcal{E}_{i})\mathcal{E}_{i}.$

Note that
$\partial_{x}(\varepsilon(\mathcal{E})\mathcal{E})=\tilde{\varepsilon}(\mathcal{E})\partial_{x}\mathcal{E}$.

For the proof of (84) we have to show

$\|\partial_{x^{\prime}}(\mathcal{E},\mathcal{H})\|_{L_{T}^{4}L_{x^{\prime}}^{\infty}}\lesssim\|(\mathcal{E},\mathcal{H})\|_{L_{T}^{\infty}H^{s}_{x^{\prime}}}\quad\text{for
\ }s\geq s_{0}>\frac{3}{2}+\frac{8}{9}.$

Via a continuity argument, we shall control

$\|\langle
D^{\prime}\rangle^{1+\kappa}(\mathcal{E},\mathcal{H})\|_{L_{T}^{4+\beta}L_{x^{\prime}}^{q}}\lesssim\|(\mathcal{E},\mathcal{H})\|_{L_{T}^{\infty}H^{s}_{x^{\prime}}}\quad\text{for
\ }s\geq s_{0}>\frac{3}{2}+\frac{8}{9},$

$q$ large enough, and $(4+\beta,q)$ being a sharp Strichartz pair. The
slightly larger exponent in $L_{t}^{p}$ allows us to bring powers of $T$ into
play. The positive number $\kappa$ is chosen such that
$\|f\|_{L^{\infty}(\mathbb{R}^{3})}\lesssim\|\langle
D^{\prime}\rangle^{\kappa}f\|_{L^{q}(\mathbb{R}^{3})}$ by Sobolev embedding.

We consider the Maxwell system

$\left\\{\begin{array}[]{rlrl}\partial_{t}(\tilde{\varepsilon}\mathcal{E})\\!\\!\\!\\!&=\nabla\times\mathcal{H},&\quad\nabla\cdot(\tilde{\varepsilon}\mathcal{E})\\!\\!\\!\\!&=\rho_{e},\\\
\partial_{t}\mathcal{H}\\!\\!\\!\\!&=-\nabla\times\mathcal{E},&\quad\nabla\cdot\mathcal{H}\\!\\!\\!\\!&=0,\end{array}\right.$

and denote

$\tilde{P}=\begin{pmatrix}-\partial_{t}(\tilde{\varepsilon}\cdot)&\nabla\times\\\
\nabla\times&\partial_{t}\end{pmatrix},\quad
v=(v_{1},v_{2}):\mathbb{R}\times\mathbb{R}^{3}\to\mathbb{R}^{3}\times\mathbb{R}^{3}.$

We can choose $\kappa,\beta,\sigma>0$, $\gamma\in(0,\frac{1}{9})$, and
$q<\infty$ such that $W^{\kappa,q}_{x^{\prime}}\hookrightarrow
L^{\infty}_{x^{\prime}}$, $\frac{3}{2}+\frac{8}{9}+\gamma+\sigma\leq s_{0}$
and such that the proof of Theorem 1.1, see (55), yields