# A Limiting Absorption Principle for Helmholtz systems and time-harmonic
isotropic Maxwell’s equations

Lucrezia Cossetti, Rainer Mandel1 1Karlsruhe Institute of Technology,
Institute for Analysis, Englerstraße 2, 76131 Karlsruhe, Germany
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this work we investigate the $L^{p}-L^{q}$-mapping properties of the
resolvent associated with the time-harmonic isotropic Maxwell operator. As
spectral parameters close to the spectrum are also covered by our analysis, we
obtain an $L^{p}-L^{q}$-type Limiting Absorption Principle for this operator.
Our analysis relies on new results for Helmholtz systems with zero order non-
Hermitian perturbations. Moreover, we provide an improved version of the
Limiting Absorption Principle for Hermitian (self-adjoint) Helmholtz systems.

###### 2000 Mathematics Subject Classification:

35Q61, 35B45, 35J05

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

##  1\. Introduction

The propagation of electromagnetic waves in continuous three-dimensional media
is governed by the Maxwell’s equations. They consist of four equations, two
vectorial and two scalar ones in the unknowns $\mathcal{D}$ and $\mathcal{E}$
(the electric fields) and $\mathcal{B}$ and $\mathcal{H}$ (the magnetic
fields) for a given current density $\mathcal{J}$. Assuming the absence of
electric charges their macroscopic formulation reads as follows

$\partial_{t}\mathcal{D}-\nabla\times\mathcal{H}=-\mathcal{J},\qquad\partial_{t}\mathcal{B}+\nabla\times\mathcal{E}=0,\qquad\nabla\cdot\mathcal{D}=\nabla\cdot\mathcal{B}=0,$
(1)

with
$\mathcal{D},\mathcal{H},\mathcal{B},\mathcal{E},\mathcal{J}\colon\mathbb{R}\times\mathbb{R}^{3}\to\mathbb{C}^{3}.$
Notice that the restriction to the case of no electric charges ($\rho=0$)
influencing the propagation of the electromagnetic waves or, which is the
same, the divergence-freeness of $\mathcal{D}$ and $\mathcal{B},$ implies
$\nabla\cdot\mathcal{J}=0.$ Constitutive relations that specify the
connections between the electric displacement $\mathcal{D}$ and the electric
field $\mathcal{E}$ and between the magnetic flux density $\mathcal{B}$ and
the magnetic field $\mathcal{H}$ are necessary for meaningful applications of
this model. In general, these relations need not be simple, but in the
physically realistic scenario where ferro-electric and ferro-magnetic
materials are discarded and where the fields are weak enough, the material
laws may be assumed to obey the following linear relations:

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

Here $\varepsilon$ and $\mu$ embody the permittivity respectively the
permeability of the medium. In general anisotropic materials, where the
interaction of fields and matter not only depends on the position in the
material but also on the direction of the fields, these quantities are
mathematically represented as tensors. In this paper we will be exclusively
concerned with the case of isotropic (i.e. direction-independent) media where
$\varepsilon$ and $\mu$ are scalar-valued functions on $\mathbb{R}^{3}.$ For a
more detailed description of Maxwell’s equations we refer the reader to [27,
34].

We will focus on monochromatic waves only, i.e., electromagnetic fields
$\mathcal{E},\mathcal{D},\mathcal{B},\mathcal{H},\mathcal{J}$ that are
periodic functions of time with the same frequency
$\omega\in\mathbb{R}\setminus\\{0\\},$ more specifically
$\mathcal{E}(x,t):=e^{i\omega t}E(x)$, $\mathcal{D}(x,t):=e^{i\omega t}D(x)$,
$\mathcal{B}(x,t):=e^{i\omega t}B(x)$, $\mathcal{H}(x,t):=e^{i\omega t}H(x)$,
$\mathcal{J}(x,t):=e^{i\omega t}J(x)$ for vector fields
$E,D,B,H,J\colon\mathbb{R}^{3}\to\mathbb{C}^{3}.$ This gives rise to the
following time-harmonic analogue of Maxwell’s equations (1) once the linear
constitutive relations from (2) are imposed:

$i\omega\varepsilon E-\nabla\times H=-J,\qquad i\omega\mu H+\nabla\times E=0.$
(3)

In this paper we are interested in the following slightly more general model

$i\zeta\varepsilon E-\nabla\times H=-J_{\textup{e}},\qquad i\zeta\mu
H+\nabla\times E=J_{\textup{m}},$ (4)

where $\zeta\in\mathbb{C}$ and where both electric and magnetic current
densities $J_{\textup{e}}$ and $J_{\textup{m}}$ are included. Allowing for
spectral parameters $\zeta\in\mathbb{C}\setminus\mathbb{R}$ reflects the so-
called Ohm’s law for conducting media, which asserts that the current $J$
induced by the electric field $E$ can be described (in linear approximation)
by $J=\sigma E+J_{\textup{e}}$, where
$\sigma\colon\mathbb{R}^{3}\to\mathbb{R}$ represents the conductivity and
$J_{\textup{e}}$ is the external current density. Thus, plugging in Ohm’s law
into (3) one gets that the first equation in (3) can be rewritten as

$i(\omega\varepsilon-i\sigma)E-\nabla\times H=-J_{\textup{e}},$

which motivates the interest in the model (4).

The main purpose of this paper is to prove an $L^{p}$-type Limiting Absorption
Principle for the time-harmonic Maxwell’s equations (4). Roughly speaking,
proving a Limiting Absorption Principle means proving existence and continuity
of the resolvent operator up to the essential spectrum. In the context of the
Maxwell system (4) this translates into studying the boundedness of solutions
$(E_{\zeta},H_{\zeta})$ of (4) with $\operatorname{Im}(\zeta)\neq 0$ and
characterizing their limits as $\operatorname{Im}(\zeta)\to 0^{\pm}.$ In this
paper we shall prove the following result.

######  Theorem 1.

Let $\omega\in\mathbb{R}\setminus\\{0\\}$ and assume that $1\leq
p,\tilde{p},q\leq\infty$ satisfy

$\frac{2}{3}<\frac{1}{p}<1,\qquad\frac{1}{6}<\frac{1}{q}<\frac{1}{3},\qquad\frac{1}{2}\leq\frac{1}{p}-\frac{1}{q}\leq\frac{2}{3},\qquad
0\leq\frac{1}{\tilde{p}}-\frac{1}{q}\leq\frac{1}{3}.$ (5)

Moreover assume that there are $\varepsilon_{\infty},\mu_{\infty}>0$ such that

  1. (A1)

$\varepsilon,\mu\in W^{1,\infty}(\mathbb{R}^{3})$ are uniformly positive,

  2. (A2)

$|\nabla(\varepsilon\mu)|+|\varepsilon\mu-\varepsilon_{\infty}\mu_{\infty}|+|\nabla\varepsilon|^{2}+|\nabla\mu|^{2}+|D^{2}\varepsilon|+|D^{2}\mu|\in
L^{\frac{3}{2}}(\mathbb{R}^{3})+L^{2}(\mathbb{R}^{3})$,

  3. (A3)

$|\nabla(\varepsilon\mu)|+|\varepsilon\mu-\varepsilon_{\infty}\mu_{\infty}|\in
L^{\frac{q_{1}}{q_{1}-2}}(\mathbb{R}^{3})+L^{\frac{q_{2}}{q_{2}-2}}(\mathbb{R}^{3})$
where $q\in[q_{1},q_{2}]$ and $(p,\tilde{p},q_{1}),(p,\tilde{p},q_{2})$
satisfy (5).

Then for all divergence-free vector fields $J_{\textup{e}},J_{\textup{m}}\in
L^{p}(\mathbb{R}^{3};\mathbb{C}^{3})\cap
L^{\tilde{p}}(\mathbb{R}^{3};\mathbb{C}^{3})$, there are weak solutions
$(E_{\omega}^{\pm},H_{\omega}^{\pm})\in
L^{q}(\mathbb{R}^{3};\mathbb{C}^{6})\cap
H^{1}_{\operatorname{loc}}(\mathbb{R}^{3};\mathbb{C}^{6})$ of the time-
harmonic Maxwell system (4) with $\zeta=\omega$ satisfying

$\|(E_{\omega}^{\pm},H_{\omega}^{\pm})\|_{q}\leq
C(\omega)\big{(}\|(J_{\textup{e}},J_{\textup{m}})\|_{p}+\|(J_{\textup{e}},J_{\textup{m}})\|_{\tilde{p}}\big{)}$
(6)

where $\omega\mapsto C(\omega)$ is continuous on $\mathbb{R}\setminus\\{0\\}$.
Moreover the following holds:

  * (i)

We have $(E_{\zeta},H_{\zeta})\to(E_{\omega}^{\pm},H_{\omega}^{\pm})$ in
$L^{q}(\mathbb{R}^{3};\mathbb{C}^{6})\cap
H^{1}_{\operatorname{loc}}(\mathbb{R}^{3};\mathbb{C}^{6})$ as
$\zeta\to\omega\pm i0$ where $(E_{\zeta},H_{\zeta})\in
H^{1}(\mathbb{R}^{3};\mathbb{C}^{6})$ is the unique weak solution solution of
(4) with divergence-free vector fields
$J_{\textup{e}}^{\zeta},J_{\textup{m}}^{\zeta}\in
L^{p}(\mathbb{R}^{3};\mathbb{C}^{3})\cap
L^{\tilde{p}}(\mathbb{R}^{3};\mathbb{C}^{3})\cap
L^{2}(\mathbb{R}^{3};\mathbb{C}^{6})$ converging to
$J_{\textup{e}},J_{\textup{m}}$ in $L^{p}(\mathbb{R}^{3};\mathbb{C}^{3})\cap
L^{\tilde{p}}(\mathbb{R}^{3};\mathbb{C}^{3})$, respectively.

  * (ii)

The function
$u_{\omega}^{\pm}:=(\varepsilon^{\frac{1}{2}}E_{\omega}^{\pm},\mu^{\frac{1}{2}}H_{\omega}^{\pm})\in
L^{q}(\mathbb{R}^{3};\mathbb{C}^{6})$ solves the Helmholtz system

$(\Delta+\omega^{2}\varepsilon_{\infty}\mu_{\infty})u_{\omega}^{\pm}+\mathcal{V}(\omega)u_{\omega}^{\pm}=\mathcal{L}_{1}(\omega)\tilde{J}+\mathcal{L}_{2}\tilde{J}$

where $\mathcal{V}(\omega),\mathcal{L}_{1}(\omega),\mathcal{L}_{2},\tilde{J}$
are defined at the beginning of Section 3. More precisely, $u_{\omega}^{\pm}$
satisfies the integral equation (45).

  * (iii)

If additionally $J_{\textup{e}},J_{\textup{m}}\in
L^{q}(\mathbb{R}^{3};\mathbb{C}^{3})$, then
$(E_{\omega}^{\pm},H_{\omega}^{\pm})\in
W^{1,q}(\mathbb{R}^{3};\mathbb{C}^{6})$.

######  Remark 2.

  * (a)

We shall not provide explicit values for the constants $C(\omega)$, but
content ourselves with proving estimates that are uniform on compact subsets
of $\mathbb{R}\setminus\\{0\\}$. This is indeed sufficient for the existence
of a map $\omega\mapsto C(\omega)$ that is continuous on
$\mathbb{R}\setminus\\{0\\}$.

  * (b)

The convergence in $H^{1}_{\operatorname{loc}}(\mathbb{R}^{3};\mathbb{C}^{6})$
is stated for simplicity. By standard elliptic regularity theory, convergence
holds in $W^{2,r}_{\operatorname{loc}}(\mathbb{R}^{3};\mathbb{C}^{6})$ where
$r\geq 1$ depends on the local regularity of
$\varepsilon,\mu,J_{\textup{e}},J_{\textup{m}}$.

  * (c)

If the currents $J_{\textup{e}}^{\zeta},J_{\textup{m}}^{\zeta}$ also converge
in $L^{q}(\mathbb{R}^{3};\mathbb{C}^{3})$ then one finds
$(E_{\zeta},H_{\zeta})\to(E_{\omega}^{\pm},H_{\omega}^{\pm})$ in
$W^{1,q}(\mathbb{R}^{3};\mathbb{C}^{6})$.

In the context of Limiting Absorption Principles of time-harmonic Maxwell’s
equations only few results are available. Picard, Weck and Witsch proved a
Limiting Absorption Principle in weighted $L^{2}$-spaces (similar to [2]) for
time-harmonic Maxwell’s equations in an exterior domain with boundary
conditions $\nu\wedge E=0$, see [44, Theorem 2.10]. Since this result is based
on Fredholm’s Alternative, the frequencies
$\omega\in\mathbb{R}\setminus\\{0\\}$ are assumed not to belong to a discrete
(possibly empty) set of eigenvalues. As in Agmon’s fundamental paper [2] about
the perturbed Helmholtz equation, the permittivity $\varepsilon$ and
permeability $\mu$ are assumed to be isotropic and to decay to some positive
constants at infinity faster than $|x|^{-1}$. Despite some quantitative
differences, this is similar to our assumptions (A1),(A2),(A3). The Limiting
Absorption Principle in weighted $L^{2}$-spaces and applications to Strichartz
estimates for Maxwell’s Equations in isotropic media were recently proved by
D’Ancona and Schnaubelt [11].

In the anisotropic case, still in weighted $L^{2}$-spaces, related results
were obtained by Pauly [43, Theorem 3.5]. We note that these results also
apply to discontinuous $\varepsilon,\mu$, which indicates that (A1) may be
relaxed. For results in the $L^{p}-L^{q}$-setting as in this paper we refer to
[48, 37]. In a recent work [41, Theorem 2.1] by Nguyen and Sil, in an
$L^{2}$-framework, the Limiting Absorption Principle is studied in the case of
anisotropic sign-changing coefficients on $\mathbb{R}^{3}$ that are used to
describe metamaterials. We also mention [40, Lemma 4-Lemma 6] where the author
studied existence, uniqueness and stability of solutions to the anisotropic
Maxwell system when the coefficients are constant outside a bounded set and
piecewise $C^{1}(\mathbb{R}^{3})$.

As far as we can see, our contribution is the first dealing with
$L^{p}$-estimates for time-harmonic Maxwell’s equations. All the
aforementioned results relate to the three-dimensional Maxwell operator. A
Limiting Absorption Principle in periodic 1D waveguides can be found in the
recent preprint by De Nittis, Moscolari, Richard and Tiedra de Aldecoa [12].
Further relevant tools for Limiting Absorption Principles such as Carleman
inequalities or Unique Continuation results can be found in [42, 15].

In view of the Limiting Absorption Principle from Theorem 1 and in particular
of the resolvent-type estimate (6), it must be expected that embedded
eigenvalues of the Maxwell operator do not exist under the assumptions of
Theorem 1. In the classical Fredholm theoretical approaches from [2, 44] this
is even a necessary condition for the Limiting Absorption Principle to hold.
Nonetheless, the proof of our Theorem 1 does not allow to derive the absence
of embedding eigenvalues directly. For this reason we prove the absence of
embedded eigenvalues separately. This is object of the following result. As we
shall see in Section 3, this heavily relies on Carleman estimates by Koch and
Tataru [33].

######  Theorem 3.

Assume (A1),(A2) for some $\varepsilon_{\infty},\mu_{\infty}>0$,
$\zeta\in\mathbb{C}$ and let $(E,H)\in
H^{1}_{\operatorname{loc}}(\mathbb{R}^{3};\mathbb{C}^{6})$ be a weak solution
of the homogeneous ($J_{\textup{e}}=J_{\textup{m}}=0$) time-harmonic Maxwell
system (4) that satisfies $(1+|x|)^{\tau_{1}-\frac{1}{2}}(|E|+|H|)\in
L^{2}(\mathbb{R}^{3})$ for some $\tau_{1}>0$. Then $E\equiv H\equiv 0$.

In relation with non-existence of eigenvalues of the Maxwell operator, we
should mention the remarkable contribution of Eidus [14]. Here the author
proves that for sufficiently smooth, real-valued, symmetric and positive
definite matrices $\varepsilon$ and $\mu$ there are no nontrivial solutions of
the homogeneous ($J_{\textup{e}}=J_{\textup{m}}=0$) time-harmonic Maxwell
system (4) with $\zeta\in\mathbb{R}\setminus\\{0\\}$ imposing that the matrix-
valued coefficients $\varepsilon$ and $\mu$ are short-range perturbations of
the identity and satisfy the repulsivity condition

$\langle\partial_{r}(r\varepsilon)u,u\rangle_{L^{2}}\geq\gamma\|u\|_{L^{2}}^{2},\qquad
r=|x|,$ (7)

for some positive $\gamma$ (and analogous condition for $\mu$). We stress that
in [14] the necessity of condition (7) for the absence of bound states is not
discussed, as a matter of fact in [14] condition (7) is only introduced as a
sufficient condition. We refer the reader to [14, Theorem 4.2] for details. In
the same paper an analogous result in the isotropic case is proved, which
shares some similarity with Theorem 3. In this case it turns out that the
nonexistence of eigenvalues follows without the need of the repulsive
condition (7) stated above (see [14, Theorem 4.4]). In the proofs of Theorem 1
and 3 we will use that for any given solution $(E,H)$ of the Maxwell system
(4) the function
$(\tilde{E},\tilde{H}):=(\varepsilon^{\frac{1}{2}}E,\mu^{\frac{1}{2}}H)$
solves a linear Helmholtz system with complex-valued zeroth order
perturbations. The difficulty arises from the fact this perturbation is non-
Hermitian in general. The tools that we will need in the analysis of this
particular system are inspired from the theory for Helmholtz systems with
Hermitian perturbations that we will develop first. For the sake of simplicity
we restrict our attention to the case $n\geq 3$.

######  Theorem 4.

Let $n,m\in\mathbb{N},n\geq 3,$ $\zeta\in\mathbb{C}\setminus\mathbb{R}.$
Assume $V=\overline{V}^{T}\in
L^{\frac{n}{2}}(\mathbb{R}^{n};\mathbb{C}^{m\times
m})+L^{\frac{n+1}{2}}(\mathbb{R}^{n};\mathbb{C}^{m\times m})$ and that $1\leq
p,q\leq\infty$ satisfy

$\frac{n+1}{2n}<\frac{1}{p}\leq
1,\qquad\frac{(n-1)^{2}}{2n(n+1)}<\frac{1}{q}<\frac{n-1}{2n},\qquad\frac{2}{n+1}\leq\frac{1}{p}-\frac{1}{q}\leq\frac{2}{n}.$

Then $R(\zeta):=(\Delta I_{m}+V(x)+\zeta I_{m})^{-1}\colon
L^{p}(\mathbb{R}^{n};\mathbb{C}^{m})\to L^{q}(\mathbb{R}^{n};\mathbb{C}^{m})$
exists as a bounded linear operator and extends by pointwise convergence to
the positive half-axis via

$R(\lambda\pm i0)f:=\lim_{\zeta\to\lambda\pm
i0}R(\zeta)f,\qquad\text{in}\;L^{q}(\mathbb{R}^{n};\mathbb{C}^{m}),$ (8)

for any $f\in L^{p}(\mathbb{R}^{n};\mathbb{C}^{m})$ and $\lambda>0$.

######  Remark 5.

  * (a)

We will actually prove a slightly stronger result than Theorem 4. We will show
that all conclusions mentioned in this theorem are true assuming
$V=\overline{V}^{T}\in L^{\frac{n}{2}}(\mathbb{R}^{n};\mathbb{C}^{m\times
m})+L^{\tilde{\kappa}}(\mathbb{R}^{n};\mathbb{C}^{m\times m})$ with
$\frac{n}{2}\leq\tilde{\kappa}\leq\frac{n+1}{2}$ where the condition
$\frac{(n-1)^{2}}{2n(n+1)}<\frac{1}{q}<\frac{n-1}{2n}$ is replaced by the
weaker one
$\frac{n+1}{2n}-\frac{1}{\tilde{\kappa}}<\frac{1}{q}<\frac{n-1}{2n}$. In other
words, Theorem 4 corresponds to the special case
$\tilde{\kappa}=\frac{n+1}{2}$ of this stronger result.

  * (b)

We assume $\zeta\in\mathbb{C}\setminus\mathbb{R}$ in order to guarantee the
existence of the resolvent $R(\zeta)$. Notice that sign or smallness
assumptions on $V$ may ensure the existence of well-defined resolvents on
given parts of the real line.

  * (c)

The limit in (8) is a pointwise limit and it is natural to ask whether this
convergence also holds in the uniform operator topology. Ideas related to this
question can be found in [24, p.46].

  * (d)

The two-dimensional case $n=2$ can in principle be discussed using the same
techniques. We expect that the same statements hold for $V=\overline{V}^{T}\in
L^{\kappa}(\mathbb{R}^{2};\mathbb{C}^{m\times
m})+L^{\frac{3}{2}}(\mathbb{R}^{2};\mathbb{C}^{m\times m})$ for some
$\kappa>1$.

To explain to what extent Theorem 4 improves earlier results, we first provide
a short summary of the available literature about the scalar case $m=1$.
Goldberg and Schlag [20] were the first to go beyond the Hilbert space
framework in which, since Agmon’s work [2], the Limiting Absorption Principles
for self-adjoint Schrödinger operators were studied. They proved an
$L^{p}$-type Limiting Absorption Principle that inspired our Theorem 4. For
$n=3$ they showed

$\sup_{0<\delta<1,\,\lambda\geq\lambda_{0}}\|R(\lambda+i\delta)\|_{\frac{4}{3}\to
4}\leq C(\lambda_{0},V)\lambda^{-\frac{1}{4}},\qquad\lambda_{0}>0,$ (9)

provided that $V\in L^{\frac{3}{2}}(\mathbb{R}^{3})\cap
L^{p}(\mathbb{R}^{3}),$ $p>\frac{3}{2}.$ In [20, Proposition 1.3] it is even
stated that the assumption on $V$ can be considerably weakened to $V\in
L^{\frac{3}{2}+\varepsilon}(\mathbb{R}^{3})+L^{2-\varepsilon}(\mathbb{R}^{3}),\varepsilon>0$,
provided that embedded eigenvalues for Schrödinger operators with such
pertubations do not exist. The latter was meanwhile proved by Koch and Tataru
[33, Theorem 3]. Huang, Yao and Zheng [24] generalized the result from [20] to
the higher-dimensional case by proving that the estimate

$\sup_{0<\delta<1,\lambda\geq\lambda_{0}}\|R(\lambda+i\delta)\|_{\frac{2(n+1)}{n+3}\to\frac{2(n+1)}{n-1}}\leq
C(\lambda_{0},V)\lambda^{-\frac{1}{n+1}},\qquad\lambda_{0}>0,$

holds for all potentials $V\in L^{\frac{n}{2}}(\mathbb{R}^{n})\cap
L^{p}(\mathbb{R}^{n}),$ $p>\frac{n}{2}$ and all $n\in\mathbb{N},n\geq 3$. The
most recent result in this direction is due to Ionescu and Schlag [26] where a
new Limiting Absorption Principle was proved for a much larger class of
potentials than the ones covered by the aforementioned results. As our Theorem
4, this result covers all $V\in
L^{\frac{n}{2}}(\mathbb{R}^{n})+L^{\frac{n+1}{2}}(\mathbb{R}^{n})$ as one can
check from (1.19) in [26]. In their Theorem 1.3 (d), the resolvent estimate

$\sup_{\lambda\in I,\,0<\delta\leq 1}\|R(\lambda\pm i\delta)\|_{X\to
X^{*}}\leq C(I,V)$

is proved where $I\subset\mathbb{R}\setminus\\{0\\}$ is a compact set that
does not intersect the set of nonzero eigenvalues. For the precise definition
of the Banach space $X$ we refer to [26, p.400]. Notice that these estimates
are self-dual in the sense that they bound the operator norms of the
resolvents acting between some Banach space and the corresponding dual space.
In this respect, our result from Theorem 4 is more general than [26]. Compared
to [24, 20], Theorem 4 requires for less integrability of the potential
(including endpoint cases) and therefore improves the known results. Being
given [33, Theorem 3], the generalization to Helmholtz systems ($m\geq 2$) is
rather trivial.

Concerning Limiting Absorption Principles for Helmholtz equations ($m=1$) in
other settings and under different assumptions we would like to mention the
papers [9, 8] (Morrey-Campanato spaces) and [47] for dissipative Helmholtz
operators, [39] (sign-changing coefficients), [36, 6, 45] (periodic
potentials) and [7, 38] (critical potentials).

As anticipated (see again Proposition 1.3 in [20]), in the proof of a Limiting
Absorption Principle, excluding embedded eigenvalues usually represents the
discriminating step where the hypotheses on the potential come into play. The
non-existence of positive eigenvalues for the Schrödinger operator has a long
history. In 1959, Kato [29] proved that the Schrödinger operator $-\Delta+V$
in $L^{2}(\mathbb{R}^{n})$ has no embedded eigenvalues if $V$ is continuous
and such that $V(x)=o(|x|^{-1})$ as $|x|\to\infty.$ Later Simon [49] improved
Kato’s result allowing also for long-range potentials. More precisely he
considered potentials V which admit the decomposition $V=V_{1}+V_{2},$ with
$V_{1}(x)=o(|x|^{-1}),V_{2}(x)=o(1),$ and
$\omega_{0}:=\limsup_{|x|\to\infty}x\cdot\nabla V_{2}(x)<\infty.$ Under these
conditions he proved, in three dimensions, the absence of eigenvalues above
the positive threshold $\omega_{0}.$ This result was later improved by Agmon
in [1], where he lowered the threshold to $\omega_{0}/2$ covering also all
dimensions. Similar results were also proved by Froese et al. in [19] (see
also [17, 18, 10, 5] for related results). Using a different approach based on
Carleman estimates Ionescu-Jerison in [25, Theorem 2.5] showed that for all
$\varepsilon>0$ there are $V\in L^{\frac{n+1}{2}+\varepsilon}(\mathbb{R}^{n})$
such that the scalar Schrödinger operator $\Delta+V$ has embedded eigenvalues
with rapidly decaying eigenfunctions. Thus, the exponent $\frac{n+1}{2}$ in
our assumption is optimal. Notice that, as far as asymptotic decay conditions
are investigated, a higher exponent in the Lebesgue space allows to cover a
wider class of perturbations. On the other hand, the optimality of the
exponent $\frac{n}{2}$ is not entirely clear, even though it is known that
standard properties of Schrödinger operators like semi-boundedness need not
hold for potentials with lower integrability. In [30],[32, Theorem 1.a)] it is
shown that 0 can be an embedded eigenvalue when potentials in the class
$L^{\kappa}_{\operatorname{loc}}(\mathbb{R}^{n})\cap L^{1}(\mathbb{R}^{n})$
with $\kappa<\frac{n}{2}$ are considered, see also [28, Remark 6.5]. Up to the
authors’ knowledge, a counterexample for non-zero eigenvalues is not known.

As customary, a basic tool for ruling out embedded eigenvalues is a suitable
Carleman estimate. In our case, due to the weak and almost optimal conditions
$V=\overline{V}^{T}\in L^{\frac{n}{2}}(\mathbb{R}^{n};\mathbb{C}^{m\times
m})+L^{\frac{n+1}{2}}(\mathbb{R}^{n};\mathbb{C}^{m\times m}),$ we need to use
the fine Carleman estimate for scalar Schrödinger operators provided by Koch
and Tataru in [33], which allows to cover this wide class of potentials. We
stress that the possibility to use a scalar Carleman estimate in our vector-
valued setting only works because the chosen weight in the Carleman bound in
[33, Proposition 4] does not depend on the solution itself. Indeed, this fact
ultimately permits to sum up the estimates obtained for the components and to
get an estimate for the full vector field. Analogue results for Helmholtz
systems with first order perturbations cannot be obtained in this way since
the weights in the corresponding Carleman estimates from [33] (see Theorem 8
and Theorem 11) depend on the solution itself. Hence, it is not guaranteed
that one Carleman weight works for all components, which is why systems with
first order perturbations appear to be more difficult.

The rest of the paper is organized as follows: Section 2 is devoted to the
proof of the Limiting Absorption Principle for Helmholtz systems with
Hermitian coefficients stated in Theorem 4. The aforementioned relation
between Maxwell’s equations (4) and Helmholtz systems will be discussed in
Section 3. Here we also provide the proof of Theorem 3 about the absence of
eigenvalues for the Maxwell system (4). Finally, in Section 4, we prove the
most involved result of the paper, namely the Limiting Absorption Principle
for Maxwell’s equations (4) from Theorem 1.

We conclude this introduction with the main notations used in this paper.

### Notations

* For $Z\in\\{\mathbb{R},\mathbb{C},\mathbb{R}^{m},\mathbb{C}^{m},\mathbb{R}^{m\times m},\mathbb{C}^{m\times m}\\}$ we shall shortly write $\|\cdot\|_{p}:=\|\cdot\|_{L^{p}(\mathbb{R}^{n};Z)}$.
* $\mathcal{B}(X,Y)$ denotes the Banach spaces of bounded linear operators between Banach spaces $X,Y$ equipped with the standard operator norm.
* We write $V\in L^{[p_{1},p_{2}]}(\mathbb{R}^{n};Z):=L^{p_{1}}(\mathbb{R}^{n};Z)+L^{p_{2}}(\mathbb{R}^{n};Z)$ if $V$ can be decomposed as $V=V_{1}+V_{2},$ with $V_{1}\in L^{p_{1}}(\mathbb{R}^{n};Z)$ and $V_{2}\in L^{p_{2}}(\mathbb{R}^{n};Z).$
* $\chi_{B}$ represents the indicator of a measurable subset $B\subset\mathbb{R}^{n}.$
* $\zeta\to\lambda\pm i0$ means $\zeta\to\lambda$ with $\pm\operatorname{Im}(\zeta)\searrow 0$.
* $I_{m}$ denotes the identity matrix in $\mathbb{R}^{m\times m},$ $m\in\mathbb{N}.$
* The notation $I$ is used for the identity operator in some function space
* We use the notation $\lesssim$ where we want to indicate that we have an inequality $\leq$ up to a constant factor which does not depend on the relevant parameters.
* $H^{1}(\operatorname{curl};\mathbb{R}^{3}):=\\{u\in L^{2}(\mathbb{R}^{3};\mathbb{C}^{3})\colon\operatorname{curl}u\in L^{2}(\mathbb{R}^{3};\mathbb{C}^{3})\\}.$
* We adopt the following definition for the Fourier transform

$\widehat{f}(\xi):=\mathcal{F}(f)(\xi):=\frac{1}{(2\pi)^{n/2}}\int_{\mathbb{R}^{n}}e^{-ix\xi}f(x)\,\mathrm{d}x.$

##  2\. The LAP for Helmholtz systems – Proof of Theorem 4

This section is concerned with the proof of Theorem 4 that relies on a well-
known perturbative argument based on Fredholm operator theory. This strategy
has its origin in the pioneering work by Agmon [2] where it was used to
establish a Limiting Absorption Principle for Schrödinger operators acting
between weighted $L^{2}$-spaces. Since then, this technique has permeated many
works in the subject. We refer to [50, 46, 13] for some remarkable earlier
contributions.

We summarize Agmon’s approach as follows. Consider a reference operator
$H_{0}$ and let $H$ be a suitable perturbation of $H_{0}$, let
$\zeta\in\mathbb{C}$. The first step is to prove the existence of a right
inverse $R_{0}(\zeta)$ for the operator $H_{0}+\zeta$ satisfying an estimate
of the form

$\|R_{0}(\zeta)f\|_{X_{1}}\leq C(\zeta)\|f\|_{X_{2}},$ (10)

where $X_{1},X_{2}$ are Banach spaces. In Agmon’s paper, for spectral
parameters $\zeta:=\lambda>0$ and the Laplacian $H_{0}=\Delta$ such right
inverses are constructed via the classical Limiting Absorption Principle for
Helmholtz equations, namely by investigating the mapping properties of the
resolvents $R_{0}(\zeta)$ as $\zeta\to\lambda\pm i0,\lambda>0$, see Theorem
4.1 [2]. This is a nontrivial task given that every such $\lambda$ belongs to
the essential spectrum of the (negative) Laplacian and therefore no such
limits can exist when $X_{1}=X_{2}=L^{2}(\mathbb{R}^{n}).$ In [2, 3] this was
circumvented by introducing suitable and, as a matter of fact, optimal
weighted $L^{2}-$spaces such that the operators $R_{0}(\zeta)$ converge in
$\mathcal{L}(X_{2},X_{1})$ as $\zeta\to\lambda\pm i0$ (with different limits).
In order to extend the estimate (10) to the perturbed operator $H$ one assumes
that $V:=H-H_{0}$ is a relatively compact perturbation of $H_{0}$, meaning
that the linear operator $K(\zeta):=-R_{0}(\zeta)V$ is compact on $X_{1}$. In
view of the formula

$H+\zeta=(H_{0}+\zeta)(I-K(\zeta))$

a right inverse $R(\zeta)$ for the operator $H+\zeta$ is given by

$R(\zeta):=(I-K(\zeta))^{-1}R_{0}(\zeta)$

as soon as $I-K(\zeta)\colon X_{1}\to X_{1}$ is bijective. By Fredholm theory,
it suffices to verify injectivity, which is the most delicate part of the
argument. Once this is achieved, one obtains the desired estimate

$\|R(\zeta)f\|_{X_{1}}\leq C(\zeta)\|(I-K(\zeta))^{-1}\|_{X_{1}\to
X_{1}}\|f\|_{X_{2}}.$ (11)

We stress that a good control of the right hand side with respect to $\zeta$
will be of central interest in the following.

In our context the reference operator $H_{0}$ and its perturbation $H$ are the
free and the perturbed matrix-valued Schrödinger operators, namely

$H_{0}:=\Delta,\qquad H:=\Delta+V(x),$

where $V=\overline{V}^{T}\in
L^{[\frac{n}{2},\frac{n+1}{2}]}(\mathbb{R}^{n};\mathbb{C}^{m\times m})$ and
$m\in\mathbb{N}$. Here, the Laplacian $\Delta$ acts as a diagonal operator on
each of the $m$ components and $\zeta\in\mathbb{C}\setminus\mathbb{R}_{\geq
0}$ or $\zeta=\lambda\pm i0,\lambda>0$ as we explain below.

According to the general strategy described above, to get an analogue of
estimate (11) under our assumptions, we need to accomplish the following three
steps:

  1. Step 1: Provide $L^{p}-L^{q}$ estimates for $R_{0}(\zeta)$.

  2. Step 2: Show that the linear operator $K(\zeta)=-R_{0}(\zeta)V\colon L^{q}(\mathbb{R}^{n};\mathbb{C}^{m})\to L^{q}(\mathbb{R}^{n};\mathbb{C}^{m})$ is compact.

  3. Step 3: Prove the injectivity of the Fredholm operator $I-K(\zeta)\colon L^{q}(\mathbb{R}^{n};\mathbb{C}^{m})\to L^{q}(\mathbb{R}^{n};\mathbb{C}^{m})$.

We will see that Step 1 is essentially available in the literature. Only minor
modifications will be needed to pass from the scalar to the vector-valued
framework. To accomplish Step 2, which is rather standard, we will use the
local compactness of Sobolev embeddings. So the main difficulty is to achieve
Step 3. It will be accomplished with the aid of Carleman estimates by Koch and
Tataru [33] and by exploiting the fact that $V$ is Hermitian.
Our results from Theorem 4 even provide the uniform bounds in
$\mathbb{C}\setminus\mathbb{R}_{\geq 0}$

$C(\zeta):=\|R_{0}(\zeta)\|_{p\to
q}\lesssim|\zeta|^{\frac{n}{2}(\frac{1}{p}-\frac{1}{q}-\frac{2}{n})}\qquad\text{and}\qquad\|(I-K(\zeta))^{-1}\|_{q\to
q}\lesssim 1$

as well as continuity properties of $\zeta\mapsto K(\zeta)$ and
$\zeta\mapsto(I-K(\zeta))^{-1}$ needed for the proof of (8). The following
subsections are devoted to the proof of the aforementioned facts.

###  2.1. $L^{p}-L^{q}$ estimates for $R_{0}(\zeta)$

In the scalar case, optimal $L^{p}-L^{q}$ resolvent estimates for $n\geq 3$
are originally due to Kenig, Ruiz and Sogge [31, Theorem 2.3] in the selfdual
case $q=p^{\prime}$ and to Gutiérrez in [23, Theorem 6] in the general case.
For the precise asymptotics with respect to $\zeta$, which results from
rescaling, we refer to [35, p.1419].

######  Theorem 6 (Kenig-Ruiz-Sogge, Gutiérrez).

Let $m=1,n\in\mathbb{N},n\geq 3$ and assume
$\zeta\in\mathbb{C}\setminus\mathbb{R}_{\geq 0}$. Then, for $1\leq
p,q\leq\infty$ such that

$\frac{n+1}{2n}<\frac{1}{p}\leq 1,\qquad
0\leq\frac{1}{q}<\frac{n-1}{2n},\qquad\frac{2}{n+1}\leq\frac{1}{p}-\frac{1}{q}\leq\frac{2}{n},$
(12)

$R_{0}(\zeta)$ is a bounded linear operator from $L^{p}(\mathbb{R}^{n})$ to
$L^{q}(\mathbb{R}^{n};\mathbb{C})$ satisfying

$\|R_{0}(\zeta)f\|_{q}\lesssim|\zeta|^{\frac{n}{2}(\frac{1}{p}-\frac{1}{q}-\frac{2}{n})}\|f\|_{p}.$
(13)

Moreover, there are bounded linear operators $R_{0}(\lambda\pm
i0):L^{p}(\mathbb{R}^{n})\to L^{q}(\mathbb{R}^{n};\mathbb{C})$ such that
$R_{0}(\zeta)f\to R_{0}(\lambda\pm i0)f$ as $\zeta\to\lambda\pm i0$ for all
$f\in L^{p}(\mathbb{R}^{n})$ and

$\|R_{0}(\lambda\pm
i0)f\|_{q}\lesssim\lambda^{\frac{n}{2}(\frac{1}{p}-\frac{1}{q}-\frac{2}{n})}\|f\|_{p}\qquad(\lambda>0).$
(14)

###### Proof.

The estimate (13) is available in the literature mentioned above. The
existence of a bounded linear operator $R_{0}(\lambda\pm i0)$ with
$R_{0}(\zeta)f\rightharpoonup R_{0}(\lambda\pm i0)f$ as $\zeta\to\lambda\pm
i0$ follows from the uniform boundedness of the functions $R_{0}(\zeta)f$ in
$L^{q}(\mathbb{R}^{n})$ for $\zeta$ near $\lambda$ (see (13)) and the
continuity of Cauchy type integrals as in [2, Theorem 4.1]. We indicate how to
prove that this convergence in fact holds in the strong sense. By density of
test functions and (13) it suffices to prove
$R_{0}(\zeta)f-R_{0}(\tilde{\zeta})f\to 0$ for test functions $f\in
C_{c}^{\infty}(\mathbb{R}^{n})$ as
$\zeta,\tilde{\zeta}\to\zeta_{0}\in\mathbb{C}\setminus\\{0\\},\operatorname{Im}(\zeta)\operatorname{Im}(\tilde{\zeta})>0$.
For simplicity we only consider the case
$\operatorname{Im}(\zeta),\operatorname{Im}(\tilde{\zeta})>0.$ Here we can use
$R_{0}(\zeta)f-R_{0}(\tilde{\zeta})f=(G_{\zeta}-G_{\tilde{\zeta}})\ast f$
where, according to [24, p.46], we have for $\zeta=\mu^{2}\neq
0,\operatorname{Re}(\mu),\operatorname{Im}(\mu)>0$ and
$\tilde{\zeta}=\tilde{\mu}^{2}$ sufficiently close to $\zeta$ with
$\operatorname{Re}(\tilde{\mu}),\operatorname{Im}(\tilde{\mu})>0$,

$\displaystyle|G_{\zeta}(z)-G_{\tilde{\zeta}}(z)|\lesssim\begin{cases}|\mu-\tilde{\mu}||z|^{3-n}&,\text{if
}|z|\leq|\mu|^{-1}\\\
|\mu-\tilde{\mu}||\mu|^{\frac{n-3}{2}}|z|^{\frac{3-n}{2}}&,\text{if
}|\mu|^{-1}\leq|z|\leq|\mu-\tilde{\mu}|^{-1}\\\
|\mu|^{\frac{n-3}{2}}|z|^{\frac{1-n}{2}}&,\text{if
}|z|\geq|\mu-\tilde{\mu}|^{-1}.\end{cases}$

So Young’s convolution inequality implies in view of $q>\frac{2n}{n-1}$

$\displaystyle\|R_{0}(\zeta)f-R_{0}(\tilde{\zeta})f\|_{q}$
$\displaystyle\lesssim|\mu-\tilde{\mu}|\,\||z|^{3-n}\chi_{|z|\leq|\mu|^{-1}}\|_{1}\|f\|_{q}$
$\displaystyle+|\mu-\tilde{\mu}|\,\||z|^{\frac{3-n}{2}}\chi_{|\mu|^{-1}\leq|z|\leq|\mu-\tilde{\mu}|^{-1}}\|_{q}\|f\|_{1}$
$\displaystyle+\||z|^{\frac{1-n}{2}}\chi_{|z|\geq|\mu-\tilde{\mu}|^{-1}}\|_{q}\|f\|_{1}$
$\displaystyle\lesssim|\mu-\tilde{\mu}|+|\mu-\tilde{\mu}|\cdot|\mu-\tilde{\mu}|^{\frac{n-3}{2}-\frac{n}{q}}+|\mu-\tilde{\mu}|^{\frac{n-1}{2}-\frac{n}{q}}$
$\displaystyle=|\mu-\tilde{\mu}|+|\mu-\tilde{\mu}|^{\frac{n-1}{2}-\frac{n}{q}}.$

Hence, $(R_{0}(\zeta)f)$ is a Cauchy sequence in $L^{q}$ and thus converges.
Since the limit must coincide with the weak limit, we get the conclusion.
$\Box$

The conditions (12) on $(p,q)$ are optimal for the uniform estimates (14), cf.
[35, p.1419]. For any fixed $\zeta\in\mathbb{C}\setminus\mathbb{R}_{\geq 0}$,
however, the estimate (13) actually holds for a larger range of exponents,
which is due to the improved properties of the Fourier symbol
$1/(|\xi|^{2}-\zeta)$ and related Bessel potential estimates. We refer to [35]
for more details about sharp $L^{p}-L^{q}$ resolvent estimates of the form
(13).
Theorem 6 extends in an obvious way to the system case that we shall need in
the following.

######  Corollary 7 (Step 1).

Let $m,n\in\mathbb{N},n\geq 3$ and assume
$\zeta\in\mathbb{C}\setminus\mathbb{R}_{\geq 0}$. Then, for $1\leq
p,q\leq\infty$ as in (12), $R_{0}(\zeta)$ is a bounded linear operator from
$L^{p}(\mathbb{R}^{n};\mathbb{C}^{m})$ to
$L^{q}(\mathbb{R}^{n};\mathbb{C}^{m})$ satisfying

$\|R_{0}(\zeta)f\|_{q}\lesssim|\zeta|^{\frac{n}{2}(\frac{1}{p}-\frac{1}{q}-\frac{2}{n})}\|f\|_{p}.$

Moreover, there are bounded linear operators $R_{0}(\lambda\pm
i0):L^{p}(\mathbb{R}^{n};\mathbb{C}^{m})\to
L^{q}(\mathbb{R}^{n};\mathbb{C}^{m})$ such that $R_{0}(\zeta)f\to
R_{0}(\lambda\pm i0)f$ as $\zeta\to\lambda\pm i0$ for all $f\in
L^{p}(\mathbb{R}^{n};\mathbb{C}^{m})$. Furthermore, (14) holds.

###  2.2. Compactness of $K(\zeta)$

We first proceed in greater generality by proving the boundedness and
compactness of $K(\zeta)$ as an operator from
$L^{q_{1}}(\mathbb{R}^{n};\mathbb{C}^{m})$ to
$L^{q_{2}}(\mathbb{R}^{n};\mathbb{C}^{m})$ for suitable $q_{1},q_{2}$, as we
will use this more general result later. The proof of Step 2 then follows from
the particular choice $q_{1}=q_{2}=q$, see Corollary 9 below. In order to
simplify the notation in the proofs, we will write
$L^{s}:=L^{s}(\mathbb{R}^{n};\mathbb{C}^{m}),L^{s}_{\operatorname{loc}}:=L^{s}_{\operatorname{loc}}(\mathbb{R}^{n};\mathbb{C}^{m})$,
etc.

######  Proposition 8.

Let $n,m\in\mathbb{N},n\geq 3$ and suppose that $V\in
L^{[\kappa,\tilde{\kappa}]}(\mathbb{R}^{n};\mathbb{C}^{m\times m})$ where
$1\leq\kappa\leq\tilde{\kappa}<\infty$. Then, for
$\zeta\in\mathbb{C}\setminus\mathbb{R}_{\geq 0}$ or $\zeta=\lambda\pm
i0,\lambda>0$, the operator $K(\zeta)=-R_{0}(\zeta)V\colon
L^{q_{1}}(\mathbb{R}^{n};\mathbb{C}^{m})\to
L^{q_{2}}(\mathbb{R}^{n};\mathbb{C}^{m})$ is compact provided that the
following conditions hold for $q_{1},q_{2}\in[1,\infty]$:

$\frac{n+1}{2n}-\frac{1}{\tilde{\kappa}}<\frac{1}{q_{1}}\leq
1-\frac{1}{\kappa},\qquad
0\leq\frac{1}{q_{2}}<\frac{n-1}{2n},\qquad\frac{2}{n+1}-\frac{1}{\tilde{\kappa}}\leq\frac{1}{q_{1}}-\frac{1}{q_{2}}\leq\frac{2}{n}-\frac{1}{\kappa}.$
(15)

Moreover,

$\|K(\zeta)\|_{q_{1}\to
q_{2}}\lesssim\inf_{V=V_{1}+V_{2}}\left[|\zeta|^{\frac{n}{2}(\frac{1}{q_{1}}-\frac{1}{q_{2}}+\frac{1}{\kappa}-\frac{2}{n})}\|V_{1}\|_{\kappa}+|\zeta|^{\frac{n}{2}(\frac{1}{q_{1}}-\frac{1}{q_{2}}+\frac{1}{\tilde{\kappa}}-\frac{2}{n})}\|V_{2}\|_{\tilde{\kappa}}\right].$
(16)

Furthermore, $u_{j}\rightharpoonup u,\zeta_{j}\to\zeta$ implies
$K(\zeta_{j})u_{j}\to K(\zeta)u$. In particular, the operators $K(\zeta)$
depend continuously on $\zeta\in\mathbb{C}\setminus\mathbb{R}_{\geq 0}$ in the
uniform operator topology and we have $K(\zeta)\to K(\lambda\pm i0)$ as
$\zeta\to\lambda\pm i0$.

###### Proof.

We begin with proving boundedness of $K(\zeta)\colon L^{q_{1}}\to L^{q_{2}}$.
Using the decomposition $V=V_{1}+V_{2},$ $V_{1}\in
L^{\kappa}(\mathbb{R}^{n};\mathbb{C}^{m\times m})$ and $V_{2}\in
L^{\tilde{\kappa}}(\mathbb{R}^{n};\mathbb{C}^{m\times m}),$ it follows from
Corollary 7 that

$\|K(\zeta)f\|_{q_{2}}\leq\|R_{0}(\zeta)\|_{p\to
q_{2}}\|V_{1}f\|_{p}+\|R_{0}(\zeta)\|_{\tilde{p}\to
q_{2}}\|V_{2}f\|_{\tilde{p}}$

whenever the tuples $(p,q_{2})$ and $(\widetilde{p},q_{2})$ satisfy the
conditions in (12). In view of (15) and $\kappa\leq\tilde{\kappa}$ these
conditions are satisfied if we choose $p,\tilde{p}$ according to
$\frac{1}{p}=\frac{1}{\kappa}+\frac{1}{q_{1}}$,
$\frac{1}{\tilde{p}}=\frac{1}{\tilde{\kappa}}+\frac{1}{q_{1}}$. So Hölder’s
inequality and Corollary 7 give

$\|K(\zeta)f\|_{q_{2}}\lesssim\left(|\zeta|^{\frac{n}{2}(\frac{1}{q_{1}}-\frac{1}{q_{2}}+\frac{1}{\kappa}-\frac{2}{n})}\|V_{1}\|_{\kappa}+|\zeta|^{\frac{n}{2}(\frac{1}{q_{1}}-\frac{1}{q_{2}}+\frac{1}{\tilde{\kappa}}-\frac{2}{n})}\|V_{2}\|_{\tilde{\kappa}}\right)\|f\|_{q_{1}},$

which proves the claimed boundedness as well as (16).

Next we show that $u_{j}\rightharpoonup u,\zeta_{j}\to\zeta$ implies
$K(\zeta_{j})u_{j}\to K(\zeta)u$. Here,
$\zeta\in\mathbb{C}\setminus\mathbb{R}_{\geq 0}$ or $\zeta=\lambda\pm
i0,\lambda>0$. Notice that this fact and Corollary 7 imply the compactness of
$K(\zeta)$ (choose $\zeta_{j}:=\zeta$) as well as the existence of a
continuous extension of $\zeta\mapsto K(\zeta)$ in
$\mathcal{B}(L^{q_{1}};L^{q_{2}})$ to the closed upper resp. lower complex
half-planes.

As shown in [20, Lemma 3.1], without loss of generality, we can assume $V$ to
be bounded and to have compact support.

We first prove $K(\zeta_{j})u_{j}\to K(\zeta)u$ in
$L^{q_{2}}_{\operatorname{loc}}$. To this end, it suffices to prove the
uniform boundedness of the operators $K(\zeta_{j})\colon L^{q_{1}}\to
W^{2,q_{1}}(B;\mathbb{C}^{m})$ with respect to $j$ for any given bounded ball
$B\subset\mathbb{R}^{n}$. Indeed, the embedding
$W^{2,q_{1}}(B;\mathbb{C}^{m})\hookrightarrow L^{q_{2}}(B;\mathbb{C}^{m})$ is
compact due to
$\frac{1}{q_{1}}-\frac{1}{q_{2}}\leq\frac{2}{n}-\frac{1}{\kappa}<\frac{2}{n}$
and the Rellich-Kondrachov Theorem, which implies $K(\zeta_{j})u_{j}\to v$ in
$L^{q_{2}}(B;\mathbb{C}^{m})$ for some $v$. This and $u_{j}\rightharpoonup u$
implies $v=K(\zeta)u.$ So we conclude $K(\zeta_{j})u_{j}\to K(\zeta)u$ in
$L^{q_{2}}_{\operatorname{loc}}$ once we have proved the uniform boundedness
of $K(\zeta_{j})\colon L^{q_{1}}\to W^{2,q_{1}}(B;\mathbb{C}^{m})$.

To prove this let $f\in L^{q_{1}}$ be arbitrary. Then $w_{j}:=K(\zeta_{j})f$
satisfies the elliptic system

$\Delta
w_{j}=(\Delta+\zeta_{j})w_{j}-\zeta_{j}w_{j}=-Vf-\zeta_{j}w_{j}\quad\text{in
}2B.$

From elliptic interior regularity estimates and the mapping properties of
$K(\zeta_{j})$ stated in Corollary 7 and Hölder’s inequality we obtain
$\|w_{j}\|_{W^{2,q_{1}}(B;\mathbb{C}^{m})}\lesssim\|f\|_{q_{1}},$ which is
what we had to prove.

To conclude it is sufficient to show $\sup_{j}\|\chi_{\mathbb{R}^{n}\setminus
B}K(\zeta_{j})\|_{q_{1}\to q_{2}}\to 0$ as $B\nearrow\mathbb{R}^{n}.$ To this
end we use

$K(\zeta_{j})f(x)=-\int_{\mathbb{R}^{n}}G_{\zeta_{j}}(x-y)V(y)f(y)\,\mathrm{d}y,$

where $G_{\zeta_{j}}(z)$ is the integral kernel of the resolvent operator
$R_{0}(\zeta_{j})$, which is explicitly given in terms of Bessel functions. We
use the bound $\sup_{j}|G_{\zeta_{j}}(z)|\lesssim|z|^{\frac{1-n}{2}}$ for
$|z|\geq 1$, see (2.21),(2.25) in [31]. Recalling that $V$ is assumed to be
bounded and compactly supported we infer for $M:=\operatorname{supp}V$

$|K(\zeta_{j})f(x)|\lesssim\int_{M}|x-y|^{\frac{1-n}{2}}|V(y)||f(y)|\,\mathrm{d}y\lesssim|x|^{\frac{1-n}{2}}\|V\|_{q_{1}^{\prime}}\|f\|_{q_{1}}\quad\text{if
}\operatorname{dist}(x,M)\geq 1.$

This yields for large enough balls $B$

$\|\chi_{\mathbb{R}^{n}\setminus
B}K(\zeta_{j})f\|_{q_{2}}\lesssim\|V\|_{q_{1}^{\prime}}\|f\|_{q_{1}}\Big{(}\int_{\mathbb{R}^{n}\setminus
B}|x|^{\frac{q_{2}(1-n)}{2}}\,\mathrm{d}x\Big{)}^{\frac{1}{q_{2}}}$

and the conclusion follows due to $q_{2}>\frac{2n}{n-1}$. $\Box$

The second step now results from considering the special case $q_{1}=q_{2}=q$
in Proposition 8.

######  Corollary 9 (Step 2).

Let $n,m\in\mathbb{N},$ $n\geq 3$ and assume that $V\in
L^{[\kappa,\tilde{\kappa}]}(\mathbb{R}^{n};\mathbb{C}^{m\times m})$ where
$\frac{n}{2}\leq\kappa\leq\tilde{\kappa}\leq\frac{n+1}{2}$. Then, for
$\zeta\in\mathbb{C}\setminus\mathbb{R}_{\geq 0}$ or $\zeta=\lambda\pm
i0,\lambda>0$, the operator $K(\zeta)=-R_{0}(\zeta)V\colon
L^{q}(\mathbb{R}^{n};\mathbb{C}^{m})\to L^{q}(\mathbb{R}^{n};\mathbb{C}^{m})$
is compact provided that

$\frac{n+1}{2n}-\frac{1}{\tilde{\kappa}}<\frac{1}{q}<\frac{n-1}{2n}.$ (17)

Furthermore, $u_{j}\rightharpoonup u,\zeta_{j}\to\zeta$ implies
$K(\zeta_{j})u_{j}\to K(\zeta)u$. In particular, the operators $K(\zeta)$
depend continuously on $\zeta\in\mathbb{C}\setminus\mathbb{R}_{\geq 0}$ in the
uniform operator topology and we have $K(\zeta)\to K(\lambda\pm i0)$ as
$\zeta\to\lambda\pm i0$.

###  2.3. Injectivity of $I-K(\zeta)$

We now prove the injectivity of the Fredholm operator $I-K(\zeta)\colon
L^{q}(\mathbb{R}^{n};\mathbb{C}^{m})\to L^{q}(\mathbb{R}^{n};\mathbb{C}^{m})$
for $q$ as in (17). So we have to show that

$u-K(\zeta)u=0,\qquad u\in L^{q}(\mathbb{R}^{n};\mathbb{C}^{m})$ (18)

implies $u=0$. As a starting point, using a bootstrapping procedure, we show
that solutions of (18) display both more local integrability and better decay
at infinity.

######  Proposition 10.

Let $n,m\in\mathbb{N},$ $n\geq 3$, $q$ as in (17) and assume $V\in
L^{[\kappa,\tilde{\kappa}]}(\mathbb{R}^{n};\mathbb{C}^{m\times m})$ where
$\frac{n}{2}\leq\kappa\leq\tilde{\kappa}\leq\frac{n+1}{2}.$ Then any solution
$u\in L^{q}(\mathbb{R}^{n};\mathbb{C}^{m})$ of (18) belongs to

$\displaystyle L^{r}(\mathbb{R}^{n};\mathbb{C}^{m})$ $\displaystyle\cap
H^{1}_{\operatorname{loc}}(\mathbb{R}^{n};\mathbb{C}^{m})\text{ for all
}r\in\Big{(}\frac{2n}{n-1},\frac{2n}{n-3}\Big{)}\,\quad\text{when
}\kappa=\frac{n}{2},$ $\displaystyle L^{r}(\mathbb{R}^{n};\mathbb{C}^{m})$
$\displaystyle\cap
H^{1}_{\operatorname{loc}}(\mathbb{R}^{n};\mathbb{C}^{m})\text{ for all
}r\in\Big{(}\frac{2n}{n-1},\infty\Big{]}\quad\qquad\text{when
}\kappa>\frac{n}{2}.$

Moreover, for any given such $r,q$ we have $\|u\|_{r}\lesssim\|u\|_{q}$.

###### Proof.

We write again $L^{s}:=L^{s}(\mathbb{R}^{n};\mathbb{C}^{m})$. As a starting
point we show that any given solution $u\in L^{q}$ of (18) belongs to $L^{r}$
for all $r\in(\frac{2n}{n-1},q),$ in other words $u$ displays a better decay
at infinity. We give a proof of this fact distinguishing between
$\tilde{\kappa}<\frac{n+1}{2}$ and the limiting case
$\tilde{\kappa}=\frac{n+1}{2}.$ Let us first consider
$\tilde{\kappa}<\frac{n+1}{2}$. Define
$\frac{1}{q_{0}}<\frac{1}{q_{1}}<\dots<\frac{1}{q_{j}}<\frac{1}{q_{j+1}}<\dots<\frac{n-1}{2n}$
by

$\frac{1}{q_{0}}:=\frac{1}{q},\qquad\frac{1}{q_{j+1}}:=\min\bigg{\\{}\frac{1}{2}\Big{(}\frac{1}{q_{j}}+\frac{n-1}{2n}\Big{)},\frac{1}{q_{j}}-\frac{2}{n+1}+\frac{1}{\tilde{\kappa}}\bigg{\\}}\qquad(j\in\mathbb{N}_{0}).$
(19)

Since we are assuming $\tilde{\kappa}<\frac{n+1}{2},$ at each iteration we
indeed get a smaller Lebesgue exponent, namely
$\frac{n-1}{2n}>\frac{1}{q_{j+1}}>\frac{1}{q_{j}}$. Then one shows that the
tuple $(q_{j},q_{j+1})$ satisfies the conditions (15) in Proposition 8, thus
$K(\zeta)$ maps $L^{q_{j}}$ to $L^{q_{j+1}}.$

Applying iteratively Proposition 8 to the equation (18) we obtain $u\in
L^{q_{j}}$ for all $j\in\mathbb{N}_{0}.$ From
$\frac{1}{q_{j}}\nearrow\frac{n-1}{2n}$ we infer $u\in L^{r}$ for all
$\frac{2n}{n-1}<r<q$ by interpolation as well as $\|u\|_{r}\lesssim\|u\|_{q}$.

Now we consider the limiting case $\tilde{\kappa}=\frac{n+1}{2}$. Observe
that, according to the previous definition (19) of $q_{j+1},$ in this limiting
situation there would be no decay gain at each iteration because of
$q_{j+1}=q_{j}$. We can circumvent this by choosing a decomposition
$V=V_{1}+V_{2},$ where $V_{1}\in
L^{\frac{n}{2}}(\mathbb{R}^{n};\mathbb{C}^{m\times m})$ and $V_{2}\in
L^{\frac{n+1}{2}}(\mathbb{R}^{n};\mathbb{C}^{m\times m})$ has a small
$L^{\frac{n+1}{2}}-$norm.

We use this observation in order to justify a similar iteration as above, this
time for exponents
$\frac{1}{q_{0}}<\frac{1}{q_{1}}<\dots<\frac{1}{q_{j}}<\frac{1}{q_{j+1}}<\dots<\frac{n-1}{2n}$
given by

$\frac{1}{q_{0}}:=\frac{1}{q},\qquad\frac{1}{q_{j+1}}:=\min\bigg{\\{}\frac{1}{2}\Big{(}\frac{1}{q_{j}}+\frac{n-1}{2n}\Big{)},\frac{1}{q_{j}}-\frac{2}{n+1}+\frac{2}{n}\bigg{\\}}\qquad(j\in\mathbb{N}_{0}).$

We have to show that $u=K(\zeta)u,u\in L^{q_{j}}$ implies $u\in L^{q_{j+1}}$.
Having done this, we conclude from $\frac{1}{q_{j}}\nearrow\frac{n-1}{2n}$ and
interpolation that $u\in L^{r}$ for all $\frac{2n}{n-1}<r<q$ as well as
$\|u\|_{r}\lesssim\|u\|_{q}$, which then finishes the proof of our first
claim.

So assume $u\in L^{q_{j}}$. One can choose $V=V_{1}+V_{2}$ such that

$C_{j}\|V_{2}\|_{\frac{n+1}{2}}<\frac{1}{2}$ (20)

where $C_{j}$ denotes the operator norm of $R_{0}(\zeta)\colon L^{s_{j}}\to
L^{q_{j+1}}$ where $s_{j}$ is defined via
$\frac{1}{s_{j}}-\frac{1}{q_{j+1}}=\frac{2}{n+1}$. Observe that this operator
norm is finite due to Corollary 7 because of
$\frac{n+1}{2n}<\frac{1}{s_{j}}\leq 1.$

We introduce the auxiliary operator $T:=I+R_{0}(\zeta)V_{2}:L^{q_{j+1}}\to
L^{q_{j+1}}$. Using (20) we find that $T$ is bounded and invertible due to
Corollary 14. So $T$ has a bounded inverse $T^{-1}:L^{q_{j+1}}\to
L^{q_{j+1}}$. Since $u$ satisfies (18), we have

$\int_{\mathbb{R}^{n}}u\phi\,\mathrm{d}x=\int_{\mathbb{R}^{n}}K(\zeta)u\phi\,\mathrm{d}x=-\int_{\mathbb{R}^{n}}R_{0}(\zeta)V_{1}u\phi\,\mathrm{d}x-\int_{\mathbb{R}^{n}}R_{0}(\zeta)V_{2}u\phi\,\mathrm{d}x$

for any given $\phi\in C_{c}^{\infty}(\mathbb{R}^{n};\mathbb{C}^{m})$. Thus
one has

$\Big{|}\int_{\mathbb{R}^{n}}Tu\phi\,\mathrm{d}x\Big{|}\leq\Big{|}\int_{\mathbb{R}^{n}}R_{0}(\zeta)V_{1}u\phi\,\mathrm{d}x\Big{|}\leq\|R_{0}(\zeta)V_{1}u\|_{q_{j+1}}\|\phi\|_{q_{j+1}^{\prime}}\lesssim\|u\|_{q_{j}}\|\phi\|_{q_{j+1}^{\prime}}$
(21)

by Proposition 8. By the the dual characterization of the Lebesgue norms
$\|\cdot\|_{q_{j+1}}$ and using density of the test functions in
$L^{q_{j+1}^{\prime}},$ one gets $\|Tu\|_{q_{j+1}}\leq\|u\|_{q_{j}}$. Then the
boundedness of $T^{-1}$ gives $\|u\|_{q_{j+1}}\lesssim\|u\|_{q_{j}}$, which is
what we had to prove.

Next we prove higher integrability of $u$. In the case $\kappa>\frac{n}{2}$
classical Moser iteration implies $u\in L^{\infty}$ and the claim follows by
interpolation. So it remains to prove $u\in L^{r}$ for
$r\in(q,\frac{2n}{n-3})$ in the limiting case $\kappa=\frac{n}{2}$. This can
be proved similarly to the limiting case $\tilde{\kappa}=\frac{n+1}{2}$ above
using the decomposition $V=V_{1}+V_{2}$ with $V_{1}\in
L^{\frac{n+1}{2}}(\mathbb{R}^{n};\mathbb{C}^{m\times m})$ and $V_{2}\in
L^{\frac{n}{2}}(\mathbb{R}^{n};\mathbb{C}^{m\times m})$ which has a small norm
$L^{\frac{n}{2}}-$ norm. For the sake of brevity we omit the details.

Finally, we discuss the $H^{1}_{\operatorname{loc}}$-regularity. In the case
$\kappa=\frac{n}{2}$ solutions $u$ of (18) satisfy $\Delta u=-(\zeta+V)u$ in
$\mathbb{R}^{n}$ in the distributional sense. Since $\zeta+V\in
L^{\frac{n}{2}}_{\operatorname{loc}}(\mathbb{R}^{n};\mathbb{C}^{m\times m})$
and $u\in L^{r}_{\operatorname{loc}}$ for all $r<\frac{2n}{n-3}$, we conclude
$\Delta u\in L^{s}_{\operatorname{loc}}$ for all $s\in[1,\frac{2n}{n+1})$. So
Caldéron-Zygmund estimates yield $u\in W^{2,s}_{\operatorname{loc}}$ for those
exponents and thus $u\in H^{1}_{\operatorname{loc}}$ by Sobolev’s Embedding
Theorem, which finishes the proof for $\kappa=\frac{n}{2}$. In the case
$\kappa>\frac{n}{2}$ one obtains similarly $u\in
W^{2,\kappa}_{\operatorname{loc}}\subset H^{1}_{\operatorname{loc}}$ and the
proof is finished. $\Box$

With these improved integrability and regularity properties of solutions at
hand, we may turn towards the injectivity of $I-K(\zeta)$. As in the paper by
Goldberg and Schlag [20] we will discuss separately the case
$\zeta\in\mathbb{C}\setminus\mathbb{R}_{\geq 0}$ and the more involved
situation $\zeta=\lambda\pm i0,\lambda>0$. Observe that if $u$ is a solution
to (18), then it solves the corresponding eigenvalue equation

$(\Delta+\zeta)u+Vu=0,\qquad u\in L^{q}(\mathbb{R}^{n};\mathbb{C}^{m}).$ (22)

To prove injectivity of $I-K(\zeta)$ one is thus lead to prove the absence of
embedded eigenvalues for the Schrödinger operator $\Delta+V$. However, for
$\zeta=\lambda>0$, the Helmholtz equation (22) may possess nontrivial
solutions: consider for instance $V=0$ and ordinary Herglotz waves $u$ with
pointwise decay rate $|u(x)|\lesssim(1+|x|)^{(1-n)/2}$. As a consequence, some
extra decay condition at infinity coming from the integral representation of
$u$ from (18) has to be used in order to deduce $u\equiv 0$. In [20] the
authors managed to prove that in the case $n=3,m=1$ solutions $u\in
L^{4}(\mathbb{R}^{3};\mathbb{C})$ of (18) even satisfy
$(1+|\cdot|)^{\tau_{1}-\frac{1}{2}}u\in L^{2}(\mathbb{R}^{3};\mathbb{C})$ for
some $\tau_{1}>0$ so that a fundamental result by Ionescu-Jerison [25] on the
absence of embedded eigenvalues allows to conclude $u\equiv 0$. The ideas
presented in [20] are not limited to $n=3$, but carry over to general
dimensions $n\geq 2$ in a straightforward manner. In order to avoid redundancy
we only state the (scalar) results that generalize Lemma 2.3 and Proposition
2.4 from this paper.

######  Proposition 11 (Goldberg-Schlag).

Let $n\in\mathbb{N},n\geq 2$ and assume $f\in L^{p}(\mathbb{R}^{n})$ where
$1\leq p\leq\frac{2(n+1)}{n+3}$. Then we have for $|t|<\frac{1}{2}$ and
$\gamma:=\frac{1}{2}\min\\{1,\frac{n+1}{p}-\frac{n+3}{2}\\}$ the estimate

$\|\hat{f}((1+t)\cdot)\|_{L^{2}(S_{\lambda})}\lesssim|t|^{\gamma}\|f\|_{p}$

provided that $\hat{f}$ vanishes identically on the unit sphere in
$\mathbb{R}^{n}$.

######  Proposition 12 (Goldberg-Schlag).

Let $n\in\mathbb{N},n\geq 2$ and $f\in L^{p}(\mathbb{R}^{n})$ for
$\max\\{1,\frac{2n}{n+4}\\}\leq p\leq\frac{2(n+1)}{n+3},\,(n,p)\neq(4,1)$.
Then we have for all
$\tau_{1}<\frac{1}{2}\min\\{1,\frac{n+1}{p}-\frac{n+3}{2}\\}$ the estimate

$\|(1+|\cdot|)^{\tau_{1}-\frac{1}{2}}R_{0}(\lambda\pm
i0)f\|_{2}\lesssim\|f\|_{p}$

provided that $\hat{f}$ vanishes identically on the unit sphere in
$\mathbb{R}^{n}$.

Clearly, Proposition 12 generalizes to systems simply by considering each
component separately. This is how we will deduce
$(1+|\cdot|)^{\tau_{1}-\frac{1}{2}}u\in L^{2}(\mathbb{R}^{n};\mathbb{C}^{m})$
for some $\tau_{1}>0$, which is crucial for the absence of embedded
eigenvalues stated in Theorem 13 below.

To some extent this result is already contained in the work [33, Theorem 3] by
Koch and Tataru. Even so, for the reader’s convenience, we decided to provide
a the proof.

######  Theorem 13.

Let $n,m\in\mathbb{N},n\geq 3$, assume $V\in
L^{[\frac{n}{2},\frac{n+1}{2}]}(\mathbb{R}^{n};\mathbb{C}^{m\times m})$ and
let $u\in H^{1}_{\operatorname{loc}}(\mathbb{R}^{n};\mathbb{C}^{m})$ be a
solution of (22) for $\zeta\in\mathbb{R}_{>0}$ satisfying
$|x|^{\tau_{1}-\frac{1}{2}}u\in L^{2}(\mathbb{R}^{n};\mathbb{C}^{m})$. Then
$u\equiv 0$.

###### Proof.

The key point is the $L^{p}$-Carleman estimate proved in [33]. More precisely,
introducing the Carleman weight $h_{\epsilon}(t)$ such that
$h_{\epsilon}^{\prime}(t)=\tau_{1}+(\tau
e^{\frac{t}{2}}-\tau_{1})\frac{\tau^{2}}{\tau^{2}+\epsilon e^{t}},$
$\epsilon>0,$ Koch and Tataru proved in [33, Proposition 4]

$\|e^{h_{\epsilon}(\ln|x|)}v\|_{\frac{2n}{n-2}}+\|e^{h_{\epsilon}(\ln|x|)}v\|_{\frac{2(n+1)}{n-1}}\lesssim\inf_{(\Delta+\zeta)v=f+g}\|e^{h_{\epsilon}(\ln|x|)}f\|_{\frac{2n}{n+2}}+\|e^{h_{\epsilon}(\ln|x|)}g\|_{\frac{2(n+1)}{n+3}}$
(23)

for all $v$ supported in $\mathbb{R}^{n}\setminus B_{1}$ such that
$|x|^{\tau_{1}-\frac{1}{2}}v\in L^{2}(\mathbb{R}^{n})$. Notice that (23) is
uniform with respect to $0<\epsilon\leq\epsilon_{0}$ and $\tau\geq\tau_{0}>0$
for some $\varepsilon_{0},\tau_{0}>0.$ Notice that in [33] the estimate (23)
is proved even for more general classes of Helmholtz-type operators also
allowing for long-range perturbations. Moreover, the corresponding estimate
(7) in that paper is formulated with even stronger norms on the left and
weaker norms on the right, as one can check using the embeddings

$\displaystyle\begin{aligned}
L^{\frac{2(n+1)}{n+3}}(\mathbb{R}^{n})+L^{\frac{2n}{n+2}}(\mathbb{R}^{n})&\hookrightarrow
W^{-\frac{1}{n+1},\frac{2(n+1)}{n+3}}(\mathbb{R}^{n}),\\\
L^{\frac{2(n+1)}{n-1}}(\mathbb{R}^{n})\cap
L^{\frac{2n}{n-2}}(\mathbb{R}^{n})&\hookleftarrow
W^{\frac{1}{n+1},\frac{2(n+1)}{n-1}}(\mathbb{R}^{n}).\end{aligned}$ (24)

Step 3.1: Exponential decay. As anticipated we first show that $u$ decays at
infinity faster than $e^{-\tau|x|^{1/2}}$ in some integrated sense. In order
to apply (23) for this purpose, we need to localize the support of $u$ to a
spatial region far from the origin. So we pick a non-negative bump function
$\phi\in C^{\infty}(\mathbb{R}^{n})$ such that $\phi(x)=1$ for $|x|>2R$ and
$\phi(x)=0$ for $|x|\leq R$ for a sufficiently large $R$ to be chosen later
and define $v:=\phi u$. Then $v$ solves

$(\Delta+\zeta)v=\left[(\Delta\phi)u+2\nabla\phi\cdot\nabla
u+V_{1}v\right]+V_{2}v.$

Now we apply the scalar Carleman estimate (23) to each component of the
system. Summing up the resulting estimates one gets

$\displaystyle\begin{aligned}
&\|e^{h_{\epsilon}(\ln|x|)}v\|_{\frac{2n}{n-2}}+\|e^{h_{\epsilon}(\ln|x|)}v\|_{\frac{2(n+1)}{n-1}}\\\
&\lesssim\|e^{h_{\epsilon}(\ln|x|)}\left((\Delta\phi)u+2\nabla\phi\cdot\nabla
u+V_{1}v\right)\|_{\frac{2n}{n+2}}+\|e^{h_{\epsilon}(\ln|x|)}V_{2}v\|_{\frac{2(n+1)}{n+3}}\\\
&\lesssim_{R}\|e^{h_{\epsilon}(\ln|x|)}\left(|u|+|\nabla
u|\right)\chi_{B_{2R}\setminus B_{R}}\|_{\frac{2n}{n+2}}\\\
&+\|e^{h_{\epsilon}(\ln|x|)}V_{1}v\|_{\frac{2n}{n+2}}+\|e^{h_{\epsilon}(\ln|x|)}V_{2}v\|_{\frac{2(n+1)}{n+3}}\\\
&\lesssim\|e^{h_{\epsilon}(\ln|x|)}\left(|u|+|\nabla
u|\right)\chi_{B_{2R}\setminus B_{R}}\|_{\frac{2n}{n+2}}\\\
&+\|V_{1}\chi_{\mathbb{R}^{n}\setminus
B_{R}}\|_{\frac{n}{2}}\|e^{h_{\epsilon}(\ln|x|)}v\|_{\frac{2n}{n-2}}+\|V_{2}\chi_{\mathbb{R}^{n}\setminus
B_{R}}\|_{\frac{n+1}{2}}\|e^{h_{\epsilon}(\ln|x|)}v\|_{\frac{2(n+1)}{n-1}}.\end{aligned}$

Here we used that $\phi$ is supported in $\mathbb{R}^{n}\setminus B_{R}$ and
$|\nabla\phi|,\Delta\phi$ are supported in $\overline{B_{2R}}\setminus B_{R}$.
For $R$ sufficiently large we can absorb the potential-dependent terms on the
right-hand side and get

$\|e^{h_{\epsilon}(\ln|x|)}v\|_{\frac{2n}{n-2}}+\|e^{h_{\epsilon}(\ln|x|)}v\|_{\frac{2(n+1)}{n-1}}\lesssim_{R}\|e^{h_{\epsilon}(\ln|x|)}(|u|+|\nabla
u|)\chi_{B_{2R}\setminus B_{R}}\|_{\frac{2n}{n+2}}.$

Since $h_{\epsilon}(\ln|x|)\nearrow\tau|x|^{1/2}$ as $\epsilon\to 0^{+}$, the
Monotone Convergence Theorem implies

$\|e^{\tau|x|^{1/2}}v\|_{\frac{2n}{n-2}}+\|e^{\tau|x|^{1/2}}v\|_{\frac{2(n+1)}{n-1}}\lesssim_{R}\|e^{\tau|x|^{1/2}}(|u|+|\nabla
u|)\chi_{B_{2R}\setminus B_{R}}\|_{\frac{2n}{n+2}}.$ (25)

Since the right side is finite by Proposition 10 (notice that the presence of
the exponential factor is irrelevant as we are localized in a bounded region),
$v$ and hence $u$ have exponential decay at infinity.

Step 3.2: Compact support. From (25) we even infer that $u$ is supported in
$\overline{B_{2R}}$. Indeed, $|x|^{1/2}\leq\sqrt{2R}$ on $B_{2R}\setminus
B_{R}$ implies

$e^{-\tau\sqrt{2R}}\Big{(}\|e^{\tau|x|^{1/2}}v\|_{\frac{2n}{n-2}}+\|e^{\tau|x|^{1/2}}v\|_{\frac{2(n+1)}{n-1}}\Big{)}\lesssim_{R}\|(|u|+|\nabla
u|)\chi_{B_{2R}\setminus B_{R}}\|_{\frac{2n}{n+2}}.$

Letting $\tau$ go to infinity shows that $v$ and hence $u$ vanishes
identically outside $\overline{B_{2R}}$.

Step 3.3: Triviality. In virtue of the conclusions provided by Step 3.2,
proving the triviality of $u$ reduces to proving the weak unique continuation
property for the differential inequality $|\Delta u|\leq|V||u|.$ At this stage
we only need that our potential $V\in
L^{[\frac{n}{2},\frac{n+1}{2}]}(\mathbb{R}^{n};\mathbb{C}^{m\times m})$
belongs to
$L^{\frac{n}{2}}_{\operatorname{loc}}(\mathbb{R}^{n};\mathbb{C}^{m\times m}),$
as only local properties of the solution $u$ are investigated. So [28, Theorem
6.3] applies and we obtain $u\equiv 0$. $\Box$

Now we are in the position to prove the injectivity of $I-K(\zeta)$.

######  Corollary 14 (Step 3).

Let $n,m\in\mathbb{N},$ $n\geq 3$ and assume $V=\overline{V}^{T}\in
L^{[\kappa,\tilde{\kappa}]}(\mathbb{R}^{n};\mathbb{C}^{m\times m})$ where
$\frac{n}{2}\leq\kappa\leq\tilde{\kappa}\leq\frac{n+1}{2}$. Then, for
$\zeta\in\mathbb{C}\setminus\mathbb{R}_{\geq 0}$ or $\zeta=\lambda\pm
i0,\lambda>0$, the operator $I-K(\zeta)\colon
L^{q}(\mathbb{R}^{n};\mathbb{C}^{m})\to L^{q}(\mathbb{R}^{n};\mathbb{C}^{m})$
is bijective and satisfies

$\|(I-K(\zeta))^{-1}\|_{q\to q}\leq C<\infty\qquad\text{for all
}\zeta\in\mathbb{C}\setminus\mathbb{R}_{\geq 0}$

provided that $q$ satisfies (17). Moreover, $\zeta\mapsto(I-K(\zeta))^{-1}$ is
continuous on $\mathbb{C}\setminus\mathbb{R}_{\geq 0}$ and we have
$(I-K(\zeta))^{-1}\to(I-K(\lambda\pm i0))^{-1}$ in the uniform operator
topology as $\zeta\to\lambda,\operatorname{Im}(\zeta)\to 0^{\pm}$.

###### Proof.

Once again we write $L^{s}:=L^{s}(\mathbb{R}^{n};\mathbb{C}^{m})$. We only
consider the case $\zeta=\lambda+i0,$ $\lambda>0$ since the remaining case
$\zeta\in\mathbb{C}\setminus\mathbb{R}_{\geq 0}$ is straightforward and can be
handled as in equation (3.6) in [20]. We first prove that the operator is
injective, so our aim is to show that any given solution $u$ of (18) must be
trivial. Proposition 10 implies $u\in L^{r}$ whenever
$\frac{n-3}{2n}<\frac{1}{r}<\frac{n-1}{2n}$. So $V\in
L^{[\kappa,\tilde{\kappa}]}(\mathbb{R}^{n};\mathbb{C}^{m\times m})\subset
L^{[\frac{n}{2},\frac{n+1}{2}]}(\mathbb{R}^{n};\mathbb{C}^{m\times m})$
implies $Vu\in L^{[p_{1},p_{2}]}(\mathbb{R}^{n};\mathbb{C}^{m})$ for some
$p_{1},p_{2}$ satisfying

$\frac{2}{n}+\frac{n-3}{2n}<\frac{1}{p_{1}}<\frac{2}{n}+\frac{n-1}{2n},\qquad\frac{2}{n+1}+\frac{n-3}{2n}<\frac{1}{p_{2}}<\frac{2}{n+1}+\frac{n-1}{2n}.$

This implies $Vu\in L^{p}$ where
$\frac{2}{n}+\frac{n-3}{2n}<\frac{1}{p}<\frac{2}{n+1}+\frac{n-1}{2n}$. In
particular, we deduce from $u\in L^{r}$ for
$\frac{n-3}{2n}<\frac{1}{r}<\frac{n-1}{2n}$ the statement

$u\in L^{p^{\prime}},\;g:=Vu\in
L^{p}\qquad\text{whenever}\quad\frac{n+1}{2n}<\frac{1}{p}<\frac{n^{2}+4n-1}{2n(n+1)}.$

So a density argument and $V=\overline{V}^{T}$ imply

$\displaystyle 0=\operatorname{Im}\left(\langle
u,Vu\rangle\right)\stackrel{{\scriptstyle\eqref{eq:ev-
eq}}}{{=}}\operatorname{Im}\left(\langle
K(\lambda+i0)u,g\rangle\right)=-\operatorname{Im}\left(\langle
R_{0}(\lambda+i0)g,g\rangle\right)=-c\sqrt{\lambda}\int_{S^{n-1}}|\widehat{g}(\sqrt{\lambda}\omega)|^{2}\mathrm{d}\sigma(\omega)$
(26)

for some positive number $c>0$, cf. (3.7) in [20]. This implies that $\hat{g}$
vanishes identically on the sphere of radius $\sqrt{\lambda}$ so that
Proposition 12 (choose $\frac{2n(n+1)}{n^{2}+4n-1}<p<\frac{2(n+1)}{n+3}$)
implies

$(1+|\cdot|)^{\tau_{1}-\frac{1}{2}}u=(1+|\cdot|)^{\tau_{1}-\frac{1}{2}}K(\lambda+i0)u=-(1+|\cdot|)^{\tau_{1}-\frac{1}{2}}R_{0}(\lambda+i0)g\in
L^{2}$

provided that $0<\tau_{1}<\frac{1}{2}\min\\{1,\frac{n+1}{p}-\frac{n+3}{2}\\}$.
In view of $u\in H^{1}_{\operatorname{loc}}$, which is a consequence of
Proposition 10, we see that the hypotheses of Theorem 13 are satisfied and we
conclude $u\equiv 0$, which proves the injectivity of $I-K(\zeta)$ and hence
its invertibility.

To prove the continuity of $\zeta\mapsto(I-K(\zeta))^{-1}$ from
$\mathbb{C}\setminus\mathbb{R}_{\geq 0}$ to the space
$\mathcal{B}(L^{q};L^{q})$ we use the identity

$~{}(I-K(\zeta_{1}))^{-1}-(I-K(\zeta_{2}))^{-1}=(I-K(\zeta_{1}))^{-1}(K(\zeta_{1})-K(\zeta_{2}))(I-K(\zeta_{2}))^{-1}$
(27)

for $\zeta_{1},\zeta_{2}\in\mathbb{C}\setminus\mathbb{R}_{\geq 0}$. Hence, for
all $\zeta_{1},\zeta_{2}$ belonging to $A\setminus\mathbb{R}$ and compact sets
$A\subset\mathbb{C}\setminus\mathbb{R}_{\geq 0}$ we have

$\|(I-K(\zeta_{1}))^{-1}-(I-K(\zeta_{2}))^{-1}\|_{q\to
q}\lesssim\|K(\zeta_{1})-K(\zeta_{2})\|_{q\to q}.$

Here we used that the operator norms of
$(I-K(\zeta_{1}))^{-1},(I-K(\zeta_{2}))^{-1}$ are uniformly bounded on $A$.
Hence, the continuity of $K$ and (27) imply the continuity of
$\zeta\mapsto(I-K(\zeta))^{-1}$. In the same way, the existence of a
continuous extension of $\zeta\mapsto K(\zeta)$ to the positive half-axis in
the operator norm topology provided by Corollary 9 implies the existence of a
continuous extension of $\zeta\mapsto(I-K(\zeta))^{-1}$ in the operator norm
topology. This and Theorem 6 finally imply that $\zeta\mapsto
R(\zeta)=(I-K(\zeta))^{-1}R_{0}(\zeta)$ is pointwise convergent as
$\zeta\to\lambda\pm i0$, $\lambda>0$ as claimed in Theorem 4.

Finally, using a decomposition of $V$ as in Proposition 10 with small
$L^{\frac{n}{2}}$-part, the bound (16) from Proposition 8 implies

$\|K(\zeta)\|_{q\to q}\leq\varepsilon+C_{\varepsilon}|\zeta|^{-\frac{1}{n+1}}$

for all $\varepsilon>0,$ $\zeta\in\mathbb{C}\setminus\mathbb{R}_{\geq 0}$ and
some $C_{\varepsilon}>0$. In particular, the norms of these operators tend to
zero as $|\zeta|\to\infty$. Thus, one can choose sufficiently large $R$ and
sufficiently small $\varepsilon$ such that $\|K(\zeta)\|_{q\to q}<\frac{1}{2}$
provided that $\zeta\in\mathbb{C}\setminus\mathbb{R}_{\geq 0}$ and
$|\zeta|\geq R.$ So the Neumann series expansion for $|\zeta|\geq R$ and the
uniform boundedness for $|\zeta|<R$ show that $\|(I-K(\zeta))^{-1}\|\leq
C<\infty$ for all $\zeta\in\mathbb{C}\setminus\mathbb{R}_{\geq 0}$ and the
claim is proved. $\Box$

######  Remark 15.

For a better understanding of the forthcoming sections, we stress here that
the hypothesis of $V$ being Hermitian is crucial in the proof of the
injectivity, thus invertibility, of $I-K(\zeta).$ Indeed, the vanishing of
$\hat{g}$ on the sphere of radius $\sqrt{\lambda},$ which is a fundamental
step in the proof, strongly relies on the identity $\operatorname{Im}(\langle
u,Vu\rangle)=0$, see (26).

##  3\. Absence of eigenvalues for Maxwell’s equations – Proof of Theorem 3

In this section we prove Theorem 3. To this end we rewrite the time-harmonic
Maxwell system (4) as a linear Helmholtz system without first order terms so
that the result essentially follows from Theorem 13. To write down this
Helmholtz system we introduce some notation. For any given $\varepsilon,\mu$
as in (A1),(A2) and $\zeta\in\mathbb{C}$ we define the $6\times 6$ complex-
valued block matrices/operators

$\mathcal{V}(\zeta):=\begin{pmatrix}V_{1}(\zeta)&-i\zeta v\times\\\ i\zeta
v\times&V_{2}(\zeta)\\\
\end{pmatrix},\quad\mathcal{L}_{1}(\zeta):=\begin{pmatrix}i\zeta(\varepsilon\mu)^{\frac{1}{2}}I_{3}&-\frac{1}{2}\nabla(\log\varepsilon)\times\\\
-\frac{1}{2}\nabla(\log\mu)\times&-i\zeta(\varepsilon\mu)^{\frac{1}{2}}I_{3}\\\
\end{pmatrix},\quad\mathcal{L}_{2}:=\begin{pmatrix}0&-\nabla\times\\\
-\nabla\times&0\\\ \end{pmatrix}.$ (28)

Here,
$v:=2\nabla((\varepsilon\mu)^{\frac{1}{2}}):\mathbb{R}^{3}\to\mathbb{R}^{3}$
and $V_{1}(\zeta),V_{2}(\zeta):\mathbb{R}^{3}\to\mathbb{C}^{3\times 3}$ are
given by

$\begin{split}&V_{1}(\zeta):=-\varepsilon^{-\frac{1}{2}}\Delta(\varepsilon^{\frac{1}{2}})I_{3}+\nabla\nabla^{T}(\log\varepsilon)-\zeta^{2}(\varepsilon_{\infty}\mu_{\infty}-\varepsilon\mu)I_{3},\\\
&V_{2}(\zeta):=-\mu^{-\frac{1}{2}}\Delta(\mu^{\frac{1}{2}})I_{3}+\nabla\nabla^{T}(\log\mu)-\zeta^{2}(\varepsilon_{\infty}\mu_{\infty}-\varepsilon\mu)I_{3},\end{split}$
(29)

In the following Lemma we show that solutions of the time-harmonic Maxwell
system give rise to solutions of a $6\times 6$ Helmholtz system with complex-
valued coefficients given by (28),(29). The proof, which is purely
computational, can be found in Appendix A.

######  Lemma 16.

Assume (A1), $\zeta\in\mathbb{C}$ and let $J_{e},J_{m}\in
L^{2}_{\operatorname{loc}}(\mathbb{R}^{3};\mathbb{C}^{3})$ be divergence-free.
Then every weak solution $(E,H)\in
H^{1}_{\operatorname{loc}}(\mathbb{R}^{3};\mathbb{C}^{6})$ of the time-
harmonic Maxwell system (4) satisfies

$(\Delta+\zeta^{2}\varepsilon_{\infty}\mu_{\infty})\begin{pmatrix}\tilde{E}\\\
\tilde{H}\\\ \end{pmatrix}+\mathcal{V}(\zeta)\begin{pmatrix}\tilde{E}\\\
\tilde{H}\\\
\end{pmatrix}=\mathcal{L}_{1}(\zeta)\begin{pmatrix}\tilde{J}_{e}\\\
\tilde{J}_{m}\\\ \end{pmatrix}+\mathcal{L}_{2}\begin{pmatrix}\tilde{J}_{e}\\\
\tilde{J}_{m}\\\ \end{pmatrix}\quad\text{in }\mathbb{R}^{3}$ (30)

in the weak sense where
$(\tilde{E},\tilde{H}):=(\varepsilon^{\frac{1}{2}}E,\mu^{\frac{1}{2}}H)$ and
$(\tilde{J}_{e},\tilde{J}_{m}):=(\mu^{\frac{1}{2}}J_{e},\varepsilon^{\frac{1}{2}}J_{m})$.

######  Remark 17.

Lemma 16 allows to deduce further properties of solutions of time-harmonic
Maxwell’s equations from the corresponding theory for elliptic PDEs. For
instance, one may deduce local regularity properties as we will do in
Proposition 19. We refer to [4, Section 3] for other approaches to regularity
results for time-harmonic Maxwell’s equations. Further features such as
Harnack inequalities or maximum principles can be proved as well. Our
assumptions on the data $\varepsilon,\mu$ may however be far from optimal.

Proof of Theorem 3: Let $(E,H)\in
H^{1}_{\operatorname{loc}}(\mathbb{R}^{3};\mathbb{C}^{6})$ be a weak solution
of the homogeneous ($J_{\textup{e}}=J_{\textup{m}}=0$) time-harmonic Maxwell
system (4) for $\zeta\in\mathbb{C}$ and assume
$(1+|x|)^{\tau_{1}-\frac{1}{2}}(|E|+|H|)\in L^{2}(\mathbb{R}^{3})$ for some
$\tau_{1}>0$. From (4) we infer $(1+|x|)^{\tau_{1}-\frac{1}{2}}(|\nabla\times
E|+|\nabla\times H|)\in L^{2}(\mathbb{R}^{3})$ as well as

$\int_{\mathbb{R}^{3}}\frac{1}{\mu}(\nabla\times
E)\cdot(\nabla\times\phi)\,\mathrm{d}x=\zeta^{2}\int_{\mathbb{R}^{3}}\varepsilon
E\phi\qquad\text{for all }\phi\in
C_{c}^{\infty}(\mathbb{R}^{3};\mathbb{C}^{3}).$

By density of test functions and $E\in
H^{1}_{\operatorname{loc}}(\mathbb{R}^{3};\mathbb{C}^{3})$ we obtain
($\phi=\chi\bar{E}$)

$\int_{\mathbb{R}^{3}}\frac{1}{\mu}|\nabla\times
E|^{2}\chi\,\mathrm{d}x+\int_{\mathbb{R}^{3}}\frac{1}{\mu}(\nabla\times
E)\cdot(\nabla\chi\times\bar{E})\,\mathrm{d}x=\zeta^{2}\int_{\mathbb{R}^{3}}\varepsilon|E|^{2}\chi\,\mathrm{d}x\qquad\text{for
all }\chi\in C_{c}^{\infty}(\mathbb{R}^{3}).$ (31)

We choose $\chi=\chi^{*}(\cdot/R)$ where $\chi^{*}\in
C_{c}^{\infty}(\mathbb{R}^{3})$ is a real-valued radially nonincreasing
nonnegative function that is identically one near the origin so that
$\chi^{*}(\cdot/R)\nearrow 1$ as $R\to\infty$. Using
$|\nabla\chi^{*}(z)|\lesssim(1+|z|)^{-1}$ we get for $R\geq 1$

$\displaystyle\left|\int_{\mathbb{R}^{3}}\frac{1}{\mu}(\nabla\times
E)\cdot(\nabla\chi\times\bar{E})\,\mathrm{d}x\right|$ $\displaystyle\lesssim
R^{-1}\int_{\mathbb{R}^{3}}|\nabla\times E||\nabla\chi^{*}(x/R)||E|\,dx$
$\displaystyle\lesssim R^{-1}\int_{\mathbb{R}^{3}}|\nabla\times
E||E|(1+|x|/R)^{-1}\,dx$ $\displaystyle=\int_{\mathbb{R}^{3}}|\nabla\times
E||E|\cdot(R+|x|)^{-1}\,dx$ $\displaystyle\lesssim
R^{-2\tau_{1}}\int_{\mathbb{R}^{3}}|\nabla\times
E||E|(1+|x|)^{2\tau_{1}-1}\,dx$ $\displaystyle\lesssim
R^{-2\tau_{1}}\qquad(R\to\infty).$

In other words, the second integral in (31) vanishes as $R\to\infty$.

From this we conclude as follows. In the case
$\operatorname{Im}(\zeta^{2})\neq 0$ we take the imaginary part of (31) and
get from the Monotone Convergence Theorem
$\int_{\mathbb{R}^{3}}\varepsilon|E|^{2}=0$, hence $E=0$. In the case
$\operatorname{Re}(\zeta^{2})\leq 0$ we take the real part of (31) and obtain
$\int_{\mathbb{R}^{3}}\frac{1}{\mu}|\nabla\times
E|^{2}-\operatorname{Re}(\zeta^{2})\varepsilon|E|^{2}=0$. Again, $E=0$. So we
have $E=0$ in both cases, which then implies $H=0$ because of (4). This proves
the absence of eigenvalues for all
$\zeta\in\mathbb{C}\setminus\mathbb{R}\cup\\{0\\}$.

We now prove the claim for $\zeta\in\mathbb{R}\setminus\\{0\\}$. We deduce
from Lemma 16 that $(\tilde{E},\tilde{H})\in
H^{1}_{\operatorname{loc}}(\mathbb{R}^{3};\mathbb{C}^{6})$ is a weak solution
of the Helmholtz system (30) for $\tilde{J}_{e}=\tilde{J}_{m}=0$. After
decomposing $\tilde{E},\tilde{H}$ and the coefficient matrix
$\mathcal{V}(\zeta)$ into real and imaginary part, we find that
$u:=(\operatorname{Re}(\tilde{E}),\operatorname{Re}(\tilde{H}),\operatorname{Im}(\tilde{E}),\operatorname{Im}(\tilde{H}))$
is a weak solution in
$H^{1}_{\operatorname{loc}}(\mathbb{R}^{3};\mathbb{R}^{12})$ of a real
$(12\times 12)$-Helmholtz system of the form $(\Delta+\lambda)u+Vu=0$ in
$\mathbb{R}^{3}$ where $\lambda=\zeta^{2}\varepsilon_{\infty}\mu_{\infty}$ and
$V\in L^{[\frac{3}{2},2]}(\mathbb{R}^{3};\mathbb{R}^{12\times 12})$. The
latter is a consequence of (A2). Since our assumptions imply
$(1+|x|)^{\tau_{1}-\frac{1}{2}}u\in L^{2}(\mathbb{R}^{3};\mathbb{R}^{12})$ and
$\lambda=\zeta^{2}\varepsilon_{\infty}\mu_{\infty}>0$, Theorem 13 yields
$u\equiv 0$ and thus $E\equiv H\equiv 0.$ This finishes the proof. $\Box$

######  Remark 18.

In the proof of Theorem 3 we used two different approaches to treat the cases
$\zeta\in\mathbb{C}\setminus\mathbb{R}\cup\\{0\\}$ and
$\zeta\in\mathbb{R}\setminus\\{0\\}.$ As a matter of fact, we cannot deduce
the absence of eigenvalues $\zeta\in\mathbb{C}\setminus\mathbb{R}$ by a
reduction to the Helmholtz-type system (30) as in the case
$\zeta\in\mathbb{R}\setminus\\{0\\}$. Indeed, as soon as we allow
$\operatorname{Im}(\lambda)\neq 0$ (recall that
$\lambda=\zeta^{2}\varepsilon_{\infty}\mu_{\infty}$), the function
$u:=(\operatorname{Re}(\tilde{E}),\operatorname{Re}(\tilde{H}),\operatorname{Im}(\tilde{E}),\operatorname{Im}(\tilde{H}))$
satisfies $(\Delta+\operatorname{Re}\lambda)u+Wu+Vu=0,$ where $W$ is a
constant-valued $12\times 12$-matrix given by

$W=\begin{pmatrix}0&\operatorname{Im}(\lambda)I_{6}\\\
-\operatorname{Im}(\lambda)I_{6}&0\end{pmatrix}.$

The lack of decay of $W$ rules out the possibility of applying Theorem 13 and
as a consequence one cannot conclude $u\equiv 0.$ So the difficulty of
treating complex-valued potentials cannot be resolved just by taking the real
and imaginary parts of the equation. On the contrary, as we will see in the
next section, the treatment of complex-valued potentials requires a more
accurate analysis of the problem.

##  4\. The LAP for Maxwell’s equations – Proof of Theorem 1

This section is devoted to the proof of the Limiting Absorption Principle for
the time-harmonic Maxwell system (4) stated in Theorem 1. We shall first give
an overview of the main steps of the proof, the rigorous details are provided
afterwards. We start by considering $(E_{\zeta},H_{\zeta})\in
H^{1}(\mathbb{R}^{3};\mathbb{C}^{6})$, the uniquely determined solutions of
the approximating time-harmonic Maxwell system

$i\zeta\varepsilon E_{\zeta}-\nabla\times
H_{\zeta}=-J_{\textup{e}}^{\zeta},\qquad i\zeta\mu H_{\zeta}+\nabla\times
E_{\zeta}=J_{\textup{m}}^{\zeta},$ (32)

where $\zeta\in\mathbb{C}\setminus\mathbb{R}$ and
$J_{\textup{e}}^{\zeta},J_{\textup{m}}^{\zeta}\in
L^{p}(\mathbb{R}^{3};\mathbb{C}^{3})\cap
L^{\tilde{p}}(\mathbb{R}^{3};\mathbb{C}^{3})\cap
L^{2}(\mathbb{R}^{3};\mathbb{C}^{3})$ are divergence-free currents that
approximate the given divergence-free currents
$J_{\textup{e}},J_{\textup{m}}\in L^{p}(\mathbb{R}^{3};\mathbb{C}^{3})\cap
L^{\tilde{p}}(\mathbb{R}^{3};\mathbb{C}^{3})$ as
$\zeta\to\omega\in\mathbb{R}_{>0}$. The necessity of introducing the
approximating problem (32) with square integrable currents
$(J_{\textup{e}}^{\zeta},J_{\textup{m}}^{\zeta})$ comes from the fact that, up
to our knowledge, the existence and uniqueness of solutions to the time-
harmonic Maxwell system (4) for general divergence-free currents
$(J_{\textup{e}},J_{\textup{m}})\in L^{p}(\mathbb{R}^{3};\mathbb{C}^{6})\cap
L^{\tilde{p}}(\mathbb{R}^{3};\mathbb{C}^{6})$ are not known. In our analysis
we will use the results and notation of the previous sections, notably
$\mathcal{V}(\zeta),\mathcal{L}_{1}(\zeta),\mathcal{L}_{2},V_{1}(\zeta),V_{2}(\zeta)$
and the vector field

$v=2\nabla((\varepsilon\mu)^{\frac{1}{2}})$

from (28) and (29).

In order to prove the Limiting Absorption Principle, the main task is to prove
the convergence of the sequence $(E_{\zeta},H_{\zeta})$ in
$L^{q}(\mathbb{R}^{3};\mathbb{C}^{6})$ as $\zeta\to\omega\pm i0$. Notice that
Sobolev’s Embedding Theorem implies $(E_{\zeta},H_{\zeta})\in
H^{1}(\mathbb{R}^{3};\mathbb{C}^{6})\subset
L^{q}(\mathbb{R}^{3};\mathbb{C}^{6})$ due to $3<q<6$. From Lemma 16 one infers
that the functions

$u_{\zeta}:=(\tilde{E}_{\zeta},\tilde{H}_{\zeta}):=(\varepsilon^{\frac{1}{2}}E_{\zeta},\mu^{\frac{1}{2}}H_{\zeta})$
(33)

are solutions of the Helmholtz system

$(\Delta+\zeta^{2}\varepsilon_{\infty}\mu_{\infty})u_{\zeta}+\mathcal{V}(\zeta)u_{\zeta}=\mathcal{L}_{1}(\zeta)\tilde{J}^{\zeta}+\mathcal{L}_{2}\tilde{J}^{\zeta},$
(34)

where $\mathcal{V}(\zeta),\mathcal{L}_{1}(\zeta)$ and $\mathcal{L}_{2}$ are
defined in (28) and
$\tilde{J}^{\zeta}=(\tilde{J}_{\textup{e}}^{\zeta},\tilde{J}_{\textup{m}}^{\zeta}):=(\mu^{\frac{1}{2}}J_{\textup{e}}^{\zeta},\varepsilon^{\frac{1}{2}}J_{\textup{m}}^{\zeta}).$
Due to the explicit relation between the spectral parameters in the Maxwell
and Helmholtz systems, the limiting case $\zeta=\omega\pm
i0,\omega\in\mathbb{R}\setminus\\{0\\}$ in the Maxwell system corresponds to
the limiting case in the Helmholtz system
$\zeta=\lambda\pm\operatorname{sign}(\omega)i0$,
$\lambda=\omega^{2}\varepsilon_{\infty}\mu_{\infty}>0$. The boundedness
assumption on $\varepsilon,\mu$ from (A1) implies that, as soon as we are able
to provide a uniform bound of $\|u_{\zeta}\|_{q}$ as
$|\operatorname{Im}(\zeta)|\to 0$, a corresponding bound also holds true for
$\|(E_{\zeta},H_{\zeta})\|_{q}.$

In order to prove such bounds for $\|u_{\zeta}\|_{q}$ we need to investigate
the vectorial Helmholtz type operators

$\Delta+\zeta^{2}\varepsilon_{\infty}\mu_{\infty}I_{m}+\mathcal{V}(\zeta),$

that we may rewrite as
$(\Delta+\zeta^{2}\varepsilon_{\infty}\mu_{\infty})(I-\mathcal{K}(\zeta))$
where

$\begin{split}\mathcal{K}(\zeta)&:=-R_{0}\big{(}\zeta^{2}\varepsilon_{\infty}\mu_{\infty}\big{)}\mathcal{V}(\zeta)\qquad\qquad\quad(\zeta\in\mathbb{C}\setminus\mathbb{R})\\\
\mathcal{K}(\omega\pm i0)&:=-R_{0}\big{(}(\omega\pm
i0)^{2}\varepsilon_{\infty}\mu_{\infty}\big{)}\mathcal{V}(\omega)\qquad(\omega\in\mathbb{R}\setminus\\{0\\})\end{split}$
(35)

and $(\omega\pm
i0)^{2}\varepsilon_{\infty}\mu_{\infty}:=\omega^{2}\varepsilon_{\infty}\mu_{\infty}\pm\operatorname{sign}(\omega)i0$.
Since the potential $\mathcal{V}(\zeta)$ is in general not Hermitian (see
(28)), not even for $\zeta=\omega\pm i0,\omega\in\mathbb{R}\setminus\\{0\\}$,
we cannot verify the sufficient condition $\operatorname{Im}(\langle
u,\mathcal{V}(\zeta)u\rangle)=0$ for the bijectivity of
$I-\mathcal{K}(\zeta)$, see the proof of Corollary 14 and Remark 15.
Nevertheless, in Proposition 21 we will quantify the potential lack of
injectivity of the operator $I-\mathcal{K}(\zeta)$ through the following
injectivity estimates

$\|u\|_{q_{1}}+\|u\|_{q_{2}}\lesssim\|(I-\mathcal{K}(\zeta))u\|_{q_{1}}+\|(I-\mathcal{K}(\zeta))u\|_{q_{2}}+\left|\operatorname{Im}(\langle
u,\mathcal{V}(\zeta)u\rangle)\right|^{1/2}$ (36)

for all $u=(u_{e},u_{m})\in L^{q_{1}}(\mathbb{R}^{3};\mathbb{C}^{6})\cap
L^{q_{2}}(\mathbb{R}^{3};\mathbb{C}^{6})$ and
$|\operatorname{Re}(\zeta)|\geq\delta>0$. We remark that it is for proving
(36) that the additional integrability assumptions from (A3) enter the proof
of Theorem 1.

The estimate (36) is valid for any $u=(u_{e},u_{m})\in
L^{q}(\mathbb{R}^{3};\mathbb{C}^{6}),$ no matter whether $u$ is a solution of
the Helmholtz system (34) or not. On the other hand, if we consider the family
$u_{\zeta}$ of solutions to (34) defined in (33), the following representation
formula in terms of the resolvent $R_{0}$ of the Laplacian is available

$(I-\mathcal{K}(\zeta))u_{\zeta}=R_{0}\big{(}\zeta^{2}\varepsilon_{\infty}\mu_{\infty}\big{)}\big{[}\mathcal{L}_{1}(\zeta)\tilde{J}^{\zeta}+\mathcal{L}_{2}\tilde{J}^{\zeta}\big{]}.$
(37)

Thus the injectivity estimate (36),(42) yields

$\displaystyle\begin{aligned}
\|u_{\zeta}\|_{q_{1}}+\|u_{\zeta}\|_{q_{2}}&\lesssim\|R_{0}\big{(}\zeta^{2}\varepsilon_{\infty}\mu_{\infty}\big{)}\big{[}\mathcal{L}_{1}(\zeta)\tilde{J}^{\zeta}+\mathcal{L}_{2}\tilde{J}^{\zeta}\big{]}\|_{q_{1}}+\|R_{0}\big{(}\zeta^{2}\varepsilon_{\infty}\mu_{\infty}\big{)}\big{[}\mathcal{L}_{1}(\zeta)\tilde{J}^{\zeta}+\mathcal{L}_{2}\tilde{J}^{\zeta}\big{]}\|_{q_{2}}\\\
&\phantom{\lesssim}+\left|\operatorname{Im}(\langle
u,\mathcal{V}(\zeta)u\rangle)\right|^{1/2}.\end{aligned}$ (38)

As soon as (38) is proved, the final step will be to bound the right-hand side
of (38) in terms of $\|J\|_{p}+\|J\|_{\tilde{p}}$ and other terms depending on
$\|u_{\zeta}\|_{q_{1}}+\|u_{\zeta}\|_{q_{2}}$ that become negligible as
$\zeta\to\omega\pm i0$. Estimating the first two terms only requires minor
modifications of Corollary 7 and it is explicitly performed in Proposition 27.
On the other hand, in order to estimate the third term, the structure of
Maxwell’s equations (32) comes into play. Indeed, in order to get the claimed
bound, we shall not only make use of the explicit expression of $u_{\zeta}$ in
terms of $(E_{\zeta},H_{\zeta})$ from (33), but also that
$(E_{\zeta},H_{\zeta})$ solves Maxwell’s equations (32). Combining the
previous steps one gets the following uniform estimate

$\|E_{\zeta}\|_{q}+\|H_{\zeta}\|_{q}\lesssim\|J\|_{p}+\|J\|_{\tilde{p}}+o(1)\qquad\text{as
}\zeta\to\omega\pm i0.$ (39)

From (39), in a rather standard way (see Subsection 4.3), one obtains the
Limiting Absorption Principle contained in Theorem 1.

In the following $p,\tilde{p},q,q_{1},q_{2}$ are chosen as in Theorem 1. We
will use the notation from (28) and (29). In the proofs we will write
$L^{s}:=L^{s}(\mathbb{R}^{3};\mathbb{C}^{6})$ and similarly for
$W^{1,s},L^{s}_{\operatorname{loc}}$ etc.

###  4.1. Injectivity estimates

First we recall from Proposition 10 (with
$\kappa=\frac{n}{2}=\frac{3}{2},\tilde{\kappa}=\frac{n+1}{2}=2$) the
regularity and integrability properties of
$L^{q}(\mathbb{R}^{3};\mathbb{C}^{6})$-solutions to
$(I-\mathcal{K}(\zeta))u=0$. Using the definitions (28),(35) and assumption
(A2) we get the following result.

######  Proposition 19.

Assume (A2) and let $q$ satisfy $3<q<6$. Moreover assume
$(I-\mathcal{K}(\zeta))u=0$ for some $u\in
L^{q}(\mathbb{R}^{3};\mathbb{C}^{6})$ where
$\zeta\in\mathbb{C}\setminus\mathbb{R}$ or $\zeta=\omega\pm i0,$
$\omega\in\mathbb{R}\setminus\\{0\\}$. Then any solution $u\in
L^{q}(\mathbb{R}^{3};\mathbb{C}^{6})$ of $(I-\mathcal{K}(\zeta))u=0$ belongs
to $L^{r}(\mathbb{R}^{3};\mathbb{C}^{6})\cap
H^{1}_{\operatorname{loc}}(\mathbb{R}^{3};\mathbb{C}^{6})$ for all
$r\in(3,\infty)$. Moreover, for any given such $r,q$ we have
$\|u\|_{r}\lesssim\|u\|_{q}$.

As in the Helmholtz case these integrability properties are actually better
for $\zeta\in\mathbb{C}\setminus\mathbb{R}$ where we even have $u\in
H^{1}(\mathbb{R}^{3};\mathbb{C}^{6})$. Next we present the crucial scalar
condition ensuring the injectivity of
$I-\mathcal{K}(\zeta)=I+R_{0}(\zeta^{2}\varepsilon_{\infty}\mu_{\infty})[\mathcal{V}(\zeta)\cdot]$.
It comes as no surprise that the condition to guarantee injectivity only
involves $\operatorname{Im}(\langle u,\mathcal{V}(\zeta)u\rangle)$. It can be
seen as an isotropic inhomogeneous variant of the Silver-Müller type radiation
condition, which, roughly speaking, represents the analogue for Maxwell
equations of the well-known Sommerfeld radiation conditions in the context of
Helmholtz equations.

######  Proposition 20.

Let (A2) hold and assume $3<q<6$. Moreover assume $(I-\mathcal{K}(\zeta))u=0$
for some $u=(u_{e},u_{m})\in L^{q}(\mathbb{R}^{3};\mathbb{C}^{6})$ where
$\zeta\in\mathbb{C}$ with $\operatorname{Im}(\zeta^{2})\neq 0$ or
$\zeta=\omega\pm i0,$ $\omega\in\mathbb{R}\setminus\\{0\\}$. Then

$\operatorname{Im}(\langle
u,\mathcal{V}(\zeta)u\rangle)\qquad\Longleftrightarrow\qquad u=0.$ (40)

###### Proof.

We only need to prove the implication “$\Longrightarrow$”, so we assume that
$\operatorname{Im}(\langle u,\mathcal{V}(\zeta)u\rangle)=0$. This implies

$\begin{split}0&=\operatorname{Im}\left(\int_{\mathbb{R}^{3}}\overline{u}\cdot\mathcal{V}(\zeta)u\,\mathrm{d}x\right)\\\
&=\operatorname{Im}\left(\int_{\mathbb{R}^{3}}\overline{\mathcal{K}(\zeta)u}\cdot\mathcal{V}(\zeta)u\,\mathrm{d}x\right)\\\
&\stackrel{{\scriptstyle\eqref{eq:defK_MAxwell}}}{{=}}-\operatorname{Im}\left(\int_{\mathbb{R}^{3}}\overline{R_{0}(\zeta^{2}\varepsilon_{\infty}\mu_{\infty})[\mathcal{V}(\zeta)u]}\cdot\mathcal{V}(\zeta)u\,\mathrm{d}x\right).\end{split}$

For $\operatorname{Im}(\zeta^{2})\neq 0$ we deduce as in equation (3.6) of
[20] that $\mathcal{V}(\zeta)u\equiv 0$ and hence
$u\equiv\mathcal{K}(\zeta)u\equiv 0$ by (35). In the case $\zeta=\omega\pm
i0,\omega\in\mathbb{R}\setminus\\{0\\}$ we get as in the proof of Corollary 14
for $\lambda=\omega^{2}\varepsilon_{\infty}\mu_{\infty}>0$

$\displaystyle 0$
$\displaystyle=\operatorname{Im}\left(\int_{\mathbb{R}^{3}}\overline{\mathcal{K}(\omega\pm
i0)u}\cdot\mathcal{V}(\omega)u\right)$
$\displaystyle=-\operatorname{Im}\left(\int_{\mathbb{R}^{3}}\overline{R_{0}(\lambda\pm\operatorname{sign}(\omega)i0)[\mathcal{V}(\omega)u]}\cdot\mathcal{V}(\omega)u\right)$
$\displaystyle=\mp\operatorname{sign}(\omega)c\int_{S_{\lambda}}|\widehat{\mathcal{V}(\omega)u}|^{2}\,d\sigma_{\lambda}$

for some $c\neq 0$. Hence, $\widehat{\mathcal{V}(\omega)u}=0$ on $S_{\lambda}$
in the $L^{2}$-trace sense. Moreover, $\mathcal{V}(\omega)u\in L^{p}$ for some
(sufficiently large) $p\in[1,\frac{4}{3})$ by assumption (A2) and Proposition
19. So Proposition 12 implies for some $\tau_{1}>0$

$(1+|\cdot|)^{\tau_{1}-\frac{1}{2}}u=(1+|\cdot|)^{\tau_{1}-\frac{1}{2}}\mathcal{K}(\omega\pm
i0)u=-(1+|\cdot|)^{\tau_{1}-\frac{1}{2}}R_{0}(\lambda\pm\operatorname{sign}(\omega)i0)[\mathcal{V}(\omega)u]\in
L^{2}.$

Given that $u$ solves the homogeneous Helmholtz system (22) we deduce from
Theorem 13 $u=0$. $\Box$

From this fact we deduce our injectivity estimates. We take the condition
$\operatorname{Im}(\zeta^{2})\neq 0$ or $\zeta=\omega\pm
i0,\omega\in\mathbb{R}\setminus\\{0\\}$ from the previous proposition into
account by restricting our attention to spectral parameters $\zeta$ with
nontrivial real parts. Recall from Theorem 6 and Proposition 8 that this
implies continuity properties of $\zeta\mapsto\mathcal{K}(\zeta)$.

######  Proposition 21.

Assume (A2),(A3). Then, for any given compact subset
$K\subset\\{\zeta\in\mathbb{C}:\operatorname{Re}(\zeta)\neq 0\\}$ we have for
all $\zeta\in K\setminus\mathbb{R}$ and all $u=(u_{e},u_{m})\in
L^{q_{1}}(\mathbb{R}^{3};\mathbb{C}^{6})\cap
L^{q_{2}}(\mathbb{R}^{3};\mathbb{C}^{6})$

$\displaystyle\|u\|_{q_{1}}+\|u\|_{q_{2}}$
$\displaystyle\lesssim\|(I-\mathcal{K}(\zeta))u\|_{q_{1}}+\|(I-\mathcal{K}(\zeta))u\|_{q_{2}}+|\operatorname{Im}(\langle
u,\mathcal{V}(\zeta)u\rangle)|^{1/2}.$

###### Proof.

We argue by contradiction and assume that there are sequences
$(\zeta^{j})\subset K$ and $(u^{j})\subset L^{q_{1}}\cap L^{q_{2}}$ such that
$\|u^{j}\|_{q_{1}}+\|u^{j}\|_{q_{2}}=1$ and

$\displaystyle\begin{aligned}
\|(I-\mathcal{K}(\zeta^{j}))u^{j}\|_{q_{1}}+\|(I-\mathcal{K}(\zeta^{j}))u^{j}\|_{q_{2}}\to
0,\quad\operatorname{Im}(\langle u^{j},\mathcal{V}(\zeta^{j})u^{j}\rangle)\to
0.\end{aligned}$ (41)

We choose subsequences such that $\zeta^{j}\to\zeta^{*}\in K$ and
$u^{j}\rightharpoonup u^{*}$ in $L^{q_{1}}$ and $L^{q_{2}}$. In the case
$\zeta^{*}\in\mathbb{R}\setminus\\{0\\},\operatorname{Im}(\zeta^{j})\to 0^{+}$
we will write $\mathcal{K}(\zeta^{*})$ instead of $\mathcal{K}(\zeta^{*}+i0)$
for notational simplicity. Clearly similar arguments apply in the case
$\operatorname{Im}(\zeta^{j})\to 0^{-}$. The second part of Corollary 9 and
(35) imply $\mathcal{K}(\zeta^{j})u^{j}\to\mathcal{K}(\zeta^{*})u^{*}$ so that
$\|(I-\mathcal{K}(\zeta^{j}))u^{j}\|_{q_{1}}+\|(I-\mathcal{K}(\zeta^{j}))u^{j}\|_{q_{2}}\to
0$ gives $(I-\mathcal{K}(\zeta^{*}))u^{*}=0$ and thus $u^{j}\to u^{*}$ in
$L^{q}_{1}$ and in $L^{q_{2}}$. From the second part of (41) we want to deduce
that the injectivity condition (40) holds for $(u^{*},\zeta^{*})$. So we need
to show

$\operatorname{Im}(\langle
u^{*},\mathcal{V}(\zeta^{*})u^{*}\rangle=\lim_{j\to\infty}\operatorname{Im}(\langle
u^{j},\mathcal{V}(\zeta^{j})u^{j}\rangle=0.$

To verify the first equality we use formula

$\displaystyle\begin{aligned}
&\operatorname{Im}\left(\int_{\mathbb{R}^{3}}\overline{u}\cdot\mathcal{V}(\zeta)u\,\mathrm{d}x\right)\\\
&=\operatorname{Im}\left(\int_{\mathbb{R}^{3}}\overline{u}_{e}V_{1}(\zeta)u_{e}+\overline{u}_{m}V_{2}(\zeta)u_{m}+i\zeta\big{(}\overline{u}_{e}\cdot(v\times
u_{m})-\overline{u}_{m}\cdot(v\times u_{e})\big{)}\,\mathrm{d}x\right)\\\
&=-\operatorname{Im}(\zeta^{2})\int_{\mathbb{R}^{3}}(\varepsilon_{\infty}\mu_{\infty}-\varepsilon\mu)|u|^{2}\,\mathrm{d}x-\operatorname{Im}\left(i\zeta\int_{\mathbb{R}^{3}}v\cdot(u_{m}\times\overline{u}_{e})-v\cdot(u_{e}\times\overline{u}_{m})\,\mathrm{d}x\right)\\\
&=\operatorname{Im}(\zeta^{2})\int_{\mathbb{R}^{3}}(\varepsilon\mu-\varepsilon_{\infty}\mu_{\infty})|u|^{2}\,\mathrm{d}x-\operatorname{Im}\left(i\zeta\int_{\mathbb{R}^{3}}v\cdot
2\operatorname{Re}(u_{m}\times\overline{u}_{e})\,\mathrm{d}x\right)\\\
&=\operatorname{Im}(\zeta^{2})\int_{\mathbb{R}^{3}}(\varepsilon\mu-\varepsilon_{\infty}\mu_{\infty})|u|^{2}\,\mathrm{d}x-2\operatorname{Re}(\zeta)\int_{\mathbb{R}^{3}}v\cdot\operatorname{Re}(u_{m}\times\overline{u}_{e})\,\mathrm{d}x.\end{aligned}$
(42)

We write $|\varepsilon_{\infty}\mu_{\infty}-\varepsilon\mu|+|v|=m_{1}+m_{2}$
where $m_{1}\in L^{\frac{q_{1}}{q_{1}-2}}(\mathbb{R}^{3}),m_{2}\in
L^{\frac{q_{2}}{q_{2}-2}}(\mathbb{R}^{3})$. This is possible due to assumption
(A3).

$\displaystyle\left|\operatorname{Im}((\zeta^{j})^{2})\int_{\mathbb{R}^{3}}(\varepsilon\mu-\varepsilon_{\infty}\mu_{\infty})|u^{j}|^{2}\,\mathrm{d}x-\operatorname{Im}((\zeta^{*})^{2})\int_{\mathbb{R}^{3}}(\varepsilon\mu-\varepsilon_{\infty}\mu_{\infty})|u^{*}|^{2}\,\mathrm{d}x\right|$
$\displaystyle\leq|(\zeta^{j})^{2}-(\zeta^{*})^{2}|\int_{\mathbb{R}^{3}}(m_{1}+m_{2})|u^{j}|^{2}\,\mathrm{d}x+|\zeta^{*}|^{2}\int_{\mathbb{R}^{3}}(m_{1}+m_{2})||u^{j}|^{2}-|u^{*}|^{2}|\,\mathrm{d}x$
$\displaystyle\lesssim|(\zeta^{j})^{2}-(\zeta^{*})^{2}|(\|m_{1}\|_{\frac{q_{1}}{q_{1}-2}}\|u^{j}\|_{q_{1}}^{2}+\|m_{2}\|_{\frac{q_{2}}{q-2}}\|u^{j}\|_{q_{2}}^{2})$
$\displaystyle+\|m_{1}\|_{\frac{q_{1}}{q_{1}-2}}\|u^{j}-u\|_{q_{1}}\||u^{j}|+|u|\|_{q_{1}}+\|m_{2}\|_{\frac{q_{2}}{q_{2}-2}}\|u^{j}-u\|_{q_{2}}\||u^{j}|+|u|\|_{q_{2}}$
$\displaystyle\lesssim(\|m_{1}\|_{\frac{q_{1}}{q_{1}-2}}+\|m_{2}\|_{\frac{q_{2}}{q_{2}-2}})(|(\zeta^{j})^{2}-(\zeta^{*})^{2}|+\|u_{j}-u\|_{q_{1}}+\|u_{j}-u\|_{q_{2}})$
$\displaystyle=o(1)\qquad(j\to\infty).$

Analogous computations yield

$\operatorname{Re}(\zeta^{j})\int_{\mathbb{R}^{3}}v\cdot\operatorname{Re}(u_{m}^{j}\times\overline{u}_{e}^{j})\,\mathrm{d}x\to\operatorname{Re}(\zeta^{*})\int_{\mathbb{R}^{3}}v\cdot\operatorname{Re}(u_{m}^{*}\times\overline{u}_{e}^{*})\,\mathrm{d}x\qquad(j\to\infty).$

So (40) holds and Proposition 20 implies $u^{*}=0$. This however contradicts
$u^{j}\to u^{*}=0$ and $\|u^{j}\|_{q_{1}}+\|u^{j}\|_{q_{2}}=1$. So the
assumption was false, which proves the claim. $\Box$

###  4.2. Bounds for $E_{\zeta},H_{\zeta}$

Proposition 21 makes it possible to bound the $L^{q}$-norm of solutions
$u_{\zeta}:=(u_{e}^{\zeta},u_{m}^{\zeta}):=(\tilde{E}_{\zeta},\tilde{H}_{\zeta})$
of the Helmholtz system (30) with $\zeta\in\mathbb{C}\setminus\mathbb{R}$ in
terms of $J$ as soon as we find suitable bounds for $\operatorname{Im}(\langle
u_{\zeta},\mathcal{V}(\zeta)u_{\zeta}\rangle).$ Those are provided in the next
proposition.

######  Proposition 22.

Let the assumptions (A1),(A2),(A3) hold. Then, for any given
$\zeta\in\mathbb{C}\setminus\mathbb{R}$ the solutions
$u_{\zeta}:=(u_{e}^{\zeta},u_{m}^{\zeta}):=(\tilde{E}_{\zeta},\tilde{H}_{\zeta}):=(\varepsilon^{\frac{1}{2}}E_{\zeta},\mu^{\frac{1}{2}}H_{\zeta})\in
L^{q}(\mathbb{R}^{3};\mathbb{C}^{6})$ of the Helmholtz system (30) satisfy

$\displaystyle\begin{aligned}
\int_{\mathbb{R}^{3}}v\cdot\operatorname{Re}(u_{m}^{\zeta}\times\overline{u_{e}^{\zeta}})\,\mathrm{d}x&=\operatorname{Im}(\zeta)\int_{\mathbb{R}^{3}}(\varepsilon\mu-\varepsilon_{\infty}\mu_{\infty})|u_{\zeta}|^{2}\,\mathrm{d}x\\\
&+\int_{\mathbb{R}^{3}}(\varepsilon\mu-\varepsilon_{\infty}\mu_{\infty})\,\operatorname{Re}\left(\mu^{-\frac{1}{2}}\overline{J_{\textup{m}}^{\zeta}}\cdot
u_{m}^{\zeta}-\varepsilon^{-\frac{1}{2}}J_{\textup{e}}^{\zeta}\cdot\overline{u_{e}^{\zeta}}\,\mathrm{d}x\right).\end{aligned}$

In particular, for $\zeta\in K\setminus\mathbb{R}$ and any compact set
$K\subset\mathbb{C}$,

$\displaystyle\begin{aligned} |\operatorname{Im}(\langle
u_{\zeta},\mathcal{V}(\zeta)u_{\zeta}\rangle)|\lesssim|\operatorname{Im}(\zeta)|(\|u_{\zeta}\|_{q_{1}}+\|u_{\zeta}\|_{q_{2}})^{2}+(\|J_{\textup{e}}^{\zeta}\|_{p}+\|J_{\textup{m}}^{\zeta}\|_{p})(\|u_{\zeta}\|_{q_{1}}+\|u_{\zeta}\|_{q_{2}}).\end{aligned}$
(43)

###### Proof.

We recall from (29) the identity

$v=2\nabla((\varepsilon\mu)^{1/2})=(\varepsilon\mu)^{-\frac{1}{2}}\nabla(\varepsilon\mu)=(\varepsilon\mu)^{-\frac{1}{2}}\nabla(\varepsilon\mu-\varepsilon_{\infty}\mu_{\infty}).$

Then integration by parts gives

$\displaystyle\int_{\mathbb{R}^{3}}v\cdot\operatorname{Re}(u_{m}^{\zeta}\times\overline{u_{e}^{\zeta}})\,\mathrm{d}x$
$\displaystyle=\int_{\mathbb{R}^{3}}\nabla(\varepsilon\mu-\varepsilon_{\infty}\mu_{\infty})\cdot\operatorname{Re}\left(H_{\zeta}\times\overline{E_{\zeta}}\,\mathrm{d}x\right)$
$\displaystyle=-\int_{\mathbb{R}^{3}}(\varepsilon\mu-\varepsilon_{\infty}\mu_{\infty})\,\left[\nabla\cdot\operatorname{Re}\left(H_{\zeta}\times\overline{E_{\zeta}}\right)\right]\,\mathrm{d}x$
$\displaystyle=-\int_{\mathbb{R}^{3}}(\varepsilon\mu-\varepsilon_{\infty}\mu_{\infty})\,\operatorname{Re}\left((\nabla\times
H_{\zeta})\cdot\overline{E_{\zeta}}-(\overline{\nabla\times E_{\zeta}})\cdot
H_{\zeta}\right)\,\mathrm{d}x$
$\displaystyle=-\int_{\mathbb{R}^{3}}(\varepsilon\mu-\varepsilon_{\infty}\mu_{\infty})\,\operatorname{Re}\left((i\zeta\varepsilon
E_{\zeta}+J_{\textup{e}}^{\zeta})\cdot\overline{E}_{\zeta}-\overline{(-i\zeta\mu
H_{\zeta}+J_{\textup{m}}^{\zeta})}\cdot H_{\zeta}\right)\,\mathrm{d}x$
$\displaystyle=-\int_{\mathbb{R}^{3}}(\varepsilon\mu-\varepsilon_{\infty}\mu_{\infty})\,\operatorname{Re}\left(i\zeta\varepsilon|E_{\zeta}|^{2}+J_{\textup{e}}^{\zeta}\cdot\overline{E_{\zeta}}-i\overline{\zeta}\mu|H_{\zeta}|^{2}-\overline{J_{\textup{m}}^{\zeta}}\cdot
H_{\zeta}\right)\,\mathrm{d}x$
$\displaystyle=\operatorname{Im}(\zeta)\int_{\mathbb{R}^{3}}(\varepsilon\mu-\varepsilon_{\infty}\mu_{\infty})\,\left(\varepsilon|E_{\zeta}|^{2}+\mu|H_{\zeta}|^{2}\right)\,\mathrm{d}x+\int_{\mathbb{R}^{3}}(\varepsilon\mu-\varepsilon_{\infty}\mu_{\infty})\,\operatorname{Re}\left(\overline{J_{\textup{m}}^{\zeta}}\cdot
H_{\zeta}-J_{\textup{e}}^{\zeta}\cdot\overline{E_{\zeta}}\,\mathrm{d}x\right)$
$\displaystyle=\operatorname{Im}(\zeta)\int_{\mathbb{R}^{3}}(\varepsilon\mu-\varepsilon_{\infty}\mu_{\infty})|u_{\zeta}|^{2}\,\mathrm{d}x+\int_{\mathbb{R}^{3}}(\varepsilon\mu-\varepsilon_{\infty}\mu_{\infty})\,\operatorname{Re}\left(\mu^{-\frac{1}{2}}\overline{J_{\textup{m}}^{\zeta}}\cdot
u_{m}^{\zeta}-\varepsilon^{-\frac{1}{2}}J_{\textup{e}}^{\zeta}\cdot\overline{u_{e}^{\zeta}}\right)\,\mathrm{d}x.$

So Hölder’s inequality and (A3) imply with $m_{1},m_{2}$ as in the proof of
Proposition 21

$\displaystyle\left|2\operatorname{Re}(\zeta)\int_{\mathbb{R}^{3}}v\cdot\operatorname{Re}(u_{m}^{\zeta}\times\overline{u_{e}^{\zeta}})\,\mathrm{d}x\right|$
$\displaystyle\lesssim|\operatorname{Im}(\zeta)|(\|m_{1}\|_{\frac{q_{1}}{q_{1}-2}}\|u_{\zeta}\|_{q_{1}}^{2}+\|m_{2}\|_{\frac{q_{2}}{q_{2}-2}}\|u_{\zeta}\|_{q_{2}}^{2})$
$\displaystyle+\int_{\mathbb{R}^{3}}(m_{1}+m_{2})(|J_{\textup{m}}^{\zeta}||u_{m}^{\zeta}|+|J_{\textup{e}}^{\zeta}||u_{e}^{\zeta}|)\,\mathrm{d}x$

We have $m_{1}\in L^{\frac{q_{1}}{q_{1}-2}}(\mathbb{R}^{3})\cap
L^{\infty}(\mathbb{R}^{3})$ and $m_{2}\in
L^{\frac{q_{2}}{q_{2}-2}}(\mathbb{R}^{3})\cap L^{\infty}(\mathbb{R}^{3})$ by
(A1), $J_{\textup{m}}^{\zeta},J_{\textup{e}}^{\zeta}\in L^{p}\cap
L^{\tilde{p}}$ and $u_{m}^{\zeta},u_{e}^{\zeta}\in L^{q_{1}}\cap L^{q_{2}}$.
Our assumptions on $p,\tilde{p},q_{1},q_{2}$ from the theorem imply

$\displaystyle\frac{1}{\infty}+\frac{1}{\tilde{p}}+\frac{1}{q_{1}}$
$\displaystyle\leq 1\leq\frac{q_{1}-2}{q_{1}}+\frac{1}{p}+\frac{1}{q_{1}},$
$\displaystyle\frac{1}{\infty}+\frac{1}{\tilde{p}}+\frac{1}{q_{2}}$
$\displaystyle\leq 1\leq\frac{q_{2}-2}{q_{2}}+\frac{1}{p}+\frac{1}{q_{2}}.$

So we can find suitable exponents for Hölder’s inequality and thus obtain

$\displaystyle\left|2\operatorname{Re}(\zeta)\int_{\mathbb{R}^{3}}v\cdot\operatorname{Re}(u_{m}^{\zeta}\times\overline{u_{e}^{\zeta}})\,\mathrm{d}x\right|$
$\displaystyle\lesssim|\operatorname{Im}(\zeta)|(\|u_{\zeta}\|_{q_{1}}+\|u_{\zeta}\|_{q_{2}})^{2}$
$\displaystyle+(\|J_{\textup{e}}^{\zeta}\|_{p}+\|J_{\textup{m}}^{\zeta}\|_{p}+\|J_{\textup{e}}^{\zeta}\|_{\tilde{p}}+\|J_{\textup{m}}^{\zeta}\|_{\tilde{p}})(\|u_{\zeta}\|_{q_{1}}+\|u_{\zeta}\|_{q_{2}}).$

Moreover,
$\operatorname{Im}(\zeta^{2})=2\operatorname{Re}(\zeta)\operatorname{Im}(\zeta)$
gives

$\left|\operatorname{Im}(\zeta^{2})\int_{\mathbb{R}^{3}}(\varepsilon\mu-\varepsilon_{\infty}\mu_{\infty})|u_{\zeta}|^{2}\,\mathrm{d}x\right|\lesssim|\operatorname{Im}(\zeta)|(\|u_{\zeta}\|_{q_{1}}+\|u_{\zeta}\|_{q_{2}})^{2}$

so that (43) is proved in view of (42). $\Box$

Combining this fact and the injectivity estimates from Proposition 21 we
obtain uniform bounds for the solutions $(E_{\zeta},H_{\zeta})$ provided that
$|\operatorname{Im}(\zeta)|$ is sufficiently small.

######  Corollary 23.

Let the assumptions (A1),(A2),(A3) hold and let $K\subset\mathbb{C}$ be
compact. Then, for $|\operatorname{Im}(\zeta)|$ sufficiently small, any
solution $(E_{\zeta},H_{\zeta})$ of the time-harmonic Maxwell system (4) for
$\zeta\in K\setminus\mathbb{R}$ satisfies

$\displaystyle\begin{aligned}
\|E_{\zeta}\|_{q}+\|H_{\zeta}\|_{q}&\lesssim\|R_{0}(\zeta^{2}\varepsilon_{\infty}\mu_{\infty})[\mathcal{L}_{1}(\zeta)\tilde{J}^{\zeta}+\mathcal{L}_{2}\tilde{J}^{\zeta}]\|_{q_{1}}\\\
&+\|R_{0}(\zeta^{2}\varepsilon_{\infty}\mu_{\infty})[\mathcal{L}_{1}(\zeta)\tilde{J}^{\zeta}+\mathcal{L}_{2}\tilde{J}^{\zeta}]\|_{q_{2}}+\|J^{\zeta}\|_{p}+\|J^{\zeta}\|_{\tilde{p}}.\end{aligned}$
(44)

###### Proof.

We define
$u_{\zeta}:=(\tilde{E}_{\zeta},\tilde{H}_{\zeta}):=(\varepsilon^{\frac{1}{2}}E_{\zeta},\mu^{\frac{1}{2}}H_{\zeta})$.
By Lemma 16 these functions solve the Helmholtz system (30) and hence satisfy
the representation formula (37). So Proposition 21 and Proposition 22 give

$\displaystyle\|u_{\zeta}\|_{q_{1}}+\|u_{\zeta}\|_{q_{2}}$
$\displaystyle\lesssim\|(I-\mathcal{K}(\zeta))u_{\zeta}\|_{q_{1}}+\|(I-\mathcal{K}(\zeta))u_{\zeta}\|_{q_{2}}+|\operatorname{Im}(\langle
u_{\zeta},\mathcal{V}(\zeta)u_{\zeta}\rangle)|^{1/2}$
$\displaystyle\lesssim\|R_{0}(\zeta^{2}\varepsilon_{\infty}\mu_{\infty})\big{[}\mathcal{L}_{1}(\zeta)\tilde{J}^{\zeta}+\mathcal{L}_{2}\tilde{J}^{\zeta}\big{]}\|_{q_{1}}+\|R_{0}(\zeta^{2}\varepsilon_{\infty}\mu_{\infty})\big{[}\mathcal{L}_{1}(\zeta)\tilde{J}^{\zeta}+\mathcal{L}_{2}\tilde{J}^{\zeta}\big{]}\|_{q_{2}}$
$\displaystyle+\sqrt{|\operatorname{Im}(\zeta)|}(\|u_{\zeta}\|_{q_{1}}+\|u_{\zeta}\|_{q_{2}})+(\|J_{\textup{e}}^{\zeta}\|_{p}+\|J_{\textup{m}}^{\zeta}\|_{p})^{\frac{1}{2}}(\|u_{\zeta}\|_{q_{1}}+\|u_{\zeta}\|_{q_{2}})^{\frac{1}{2}}.$

This and
$\|u_{\zeta}\|_{q}\lesssim\|u_{\zeta}\|_{q_{1}}+\|u_{\zeta}\|_{q_{2}}$ yields
the corresponding bound for $u_{\zeta}$ provided that
$|\operatorname{Im}(\zeta)|$ is sufficiently small. Assumption (A1) implies
$\|E_{\zeta}\|_{r}+\|H_{\zeta}\|_{r}\lesssim\|u_{\zeta}\|_{r}$ for
$r\in\\{q_{1},q_{2}\\}$ and (44) follows. $\Box$

###  4.3. Proof of the Limiting Absorption Principle

We first prove the existence of the functions $(E_{\zeta},H_{\zeta})$ the
bounds for which we provided above. We recall that it is defined as the unique
solution in $H^{1}(\mathbb{R}^{3};\mathbb{C}^{6})\subset
L^{q}(\mathbb{R}^{3};\mathbb{C}^{6})$ of the time-harmonic Maxwell system (4)
with divergence-free currents $J_{\textup{e}}^{\zeta},J_{\textup{m}}^{\zeta}$
lying in $L^{p}(\mathbb{R}^{3};\mathbb{C}^{3})\cap
L^{\tilde{p}}(\mathbb{R}^{3};\mathbb{C}^{3})\cap
L^{2}(\mathbb{R}^{3};\mathbb{C}^{3})$ that converge to
$J_{\textup{e}},J_{\textup{m}}$, respectively. (The reason for considering
$J_{\textup{e}}^{\zeta},J_{\textup{m}}^{\zeta}$ instead of
$J_{\textup{e}},J_{\textup{m}}$ is because the existence of
$L^{q}(\mathbb{R}^{3};\mathbb{C}^{6})$-solutions $(E_{\zeta},H_{\zeta})$ for
the currents $J_{\textup{e}},J_{\textup{m}}\in
L^{p}(\mathbb{R}^{3};\mathbb{C}^{3})\cap
L^{\tilde{p}}(\mathbb{R}^{3};\mathbb{C}^{3})$ is not clear.) In the next
proposition we first show that divergence-free currents
$J_{\textup{e}},J_{\textup{m}}\in L^{p}(\mathbb{R}^{3};\mathbb{C}^{3})\cap
L^{\tilde{p}}(\mathbb{R}^{3};\mathbb{C}^{3})$ can be approximated by a
sequence of divergence-free currents
$J_{\textup{e}}^{\zeta},J_{\textup{m}}^{\zeta}\in
L^{p}(\mathbb{R}^{3};\mathbb{C}^{3})\cap
L^{\tilde{p}}(\mathbb{R}^{3};\mathbb{C}^{3})\cap
L^{2}(\mathbb{R}^{3};\mathbb{C}^{3})$.

######  Proposition 24.

Let $p,\tilde{p}\in(1,\infty)$ and assume $J_{\textup{e}},J_{\textup{m}}\in
L^{p}(\mathbb{R}^{3};\mathbb{C}^{3})\cap
L^{\tilde{p}}(\mathbb{R}^{3};\mathbb{C}^{3})$ to be divergence-free. Then
there are divergence-free vector fields
$J_{\textup{e}}^{\zeta},J_{\textup{m}}^{\zeta}\in
L^{p}(\mathbb{R}^{3};\mathbb{C}^{3})\cap
L^{\tilde{p}}(\mathbb{R}^{3};\mathbb{C}^{3})\cap
L^{2}(\mathbb{R}^{3};\mathbb{C}^{3})$ satisfying
$(J_{\textup{e}}^{\zeta},J_{\textup{m}}^{\zeta})\to(J_{\textup{e}},J_{\textup{m}})$
in $L^{p}(\mathbb{R}^{3};\mathbb{C}^{3})\cap
L^{\tilde{p}}(\mathbb{R}^{3};\mathbb{C}^{3})$ as
$\zeta\to\omega\in\mathbb{R}\setminus\\{0\\}$.

###### Proof.

The map $\Pi\colon f\mapsto f-\mathcal{F}^{-1}(|\xi|^{-2}\xi\xi^{T}\hat{f})$
is a bounded linear operator from $L^{p}\cap L^{\tilde{p}}$ to the divergence-
free vector fields in $L^{p}\cap L^{\tilde{p}}$. This is a consequence of the
$L^{p}\cap L^{\tilde{p}}$-boundedness of Riesz transforms. So for any given
$f\in L^{p}\cap L^{\tilde{p}}$ we can choose $(f_{n})\subset\mathcal{S}$ such
that $f_{n}$ converges to $f$ in $L^{p}\cap L^{\tilde{p}}$. The sequence $(\Pi
f_{n})$ then has the desired properties. $\Box$

Next we show that for $J_{\textup{e}}^{\zeta},J_{\textup{m}}^{\zeta}$ as in
Proposition 24 there are uniquely determined solutions $(E_{\zeta},H_{\zeta})$
in $H^{1}(\mathbb{R}^{3};\mathbb{C}^{6})$. In the proof we will need the
following result for $r=2$.

######  Proposition 25.

Assume (A1) and $\zeta\in\mathbb{C},r\in(1,\infty)$. Then every solution of
(4) satisfies

$\|\nabla E\|_{r}+\|\nabla
H\|_{r}\lesssim(1+|\zeta|)(\|E\|_{r}+\|H\|_{r})+\|J_{\textup{e}}\|_{r}+\|J_{\textup{m}}\|_{r}$

###### Proof.

Since $D=\varepsilon E$ and $B=\mu H$ are divergence-free, we have

$\nabla\cdot E=\varepsilon^{-1}\nabla\varepsilon\cdot E,\qquad\nabla\cdot
H=\mu^{-1}\nabla\mu\cdot H.$

This and [51, Theorem 1.1] imply

$\displaystyle\|\nabla E\|_{r}+\|\nabla H\|_{r}$
$\displaystyle\lesssim\|\nabla\times E\|_{r}+\|\nabla\cdot
E\|_{r}+\|\nabla\times H\|_{r}+\|\nabla\cdot H\|_{r}$
$\displaystyle\lesssim\|-i\zeta\mu
H+J_{\textup{m}}\|_{r}+\|\varepsilon^{-1}\nabla\varepsilon\cdot
E\|_{r}+\|i\zeta\varepsilon E+J_{\textup{e}}\|_{r}+\|\mu^{-1}\nabla\mu\cdot
H\|_{r}$
$\displaystyle\lesssim(1+|\zeta|)(\|E\|_{r}+\|H\|_{r})+\|J_{\textup{e}}\|_{r}+\|J_{\textup{m}}\|_{r}.$

$\Box$

######  Proposition 26.

Assume (A1). Then, for $\zeta\in\mathbb{C}$ with
$\operatorname{Re}(\zeta),\operatorname{Im}(\zeta)\neq 0$, there is a unique
solution $(E_{\zeta},H_{\zeta})\in H^{1}(\mathbb{R}^{3};\mathbb{C}^{6})$ of
(4) for the divergence-free currents given by
$J_{\textup{e}}^{\zeta},J_{\textup{m}}^{\zeta}\in
L^{2}(\mathbb{R}^{3};\mathbb{C}^{3})$.

###### Proof.

The existence and uniqueness of such a solution $(E_{\zeta},H_{\zeta})\in
H(\operatorname{curl};\mathbb{R}^{3})\times
H(\operatorname{curl};\mathbb{R}^{3})$ can be proved as in [16, Section 7.4].
From $\varepsilon,\mu\in W^{1,\infty}(\mathbb{R}^{3})$ by (A1) and Proposition
25 for $r=2$ we obtain $(E_{\zeta},H_{\zeta})\in
H^{1}(\mathbb{R}^{3};\mathbb{C}^{6})$. $\Box$

The preceding propositions ensure that the sequences of solutions
$(E_{\zeta},H_{\zeta})$ we were speaking of really exist in the space
$H^{1}(\mathbb{R}^{3};\mathbb{C}^{6})$ and in particular in
$L^{q}(\mathbb{R}^{3};\mathbb{C}^{6})$ for all $q\in(3,6)$ by Sobolev’s
Embedding Theorem. In Corollary 23 we showed that $(E_{\zeta},H_{\zeta})$
remain bounded once we have bounds for suitable Lebesgue-norms of
$\tilde{J}^{\zeta}$ and
$R_{0}(\zeta^{2}\varepsilon_{\infty}\mu_{\infty})\big{[}\mathcal{L}_{1}(\zeta)\tilde{J}^{\zeta}+\mathcal{L}_{2}\tilde{J}^{\zeta}\big{]}$
which are independent of $\operatorname{Im}(\zeta)$. As mentioned earlier,
this can be achieved rather easily with the aid of Theorem 6 and a suitable
modification of it when first order derivates are involved, see Theorem 30 in
the Appendix.

######  Proposition 27.

Assume (A1) and let
$K\subset\\{\zeta\in\mathbb{C}:\operatorname{Re}(\zeta)\neq 0\\}$ be compact.
Then, for $\zeta\in K\setminus\mathbb{R}$ and
$\tilde{J}^{\zeta}:=(\mu^{\frac{1}{2}}J_{\textup{e}}^{\zeta},\varepsilon^{\frac{1}{2}}J_{\textup{m}}^{\zeta})$
as above, we have

$\|R_{0}(\zeta^{2}\varepsilon_{\infty}\mu_{\infty})\big{[}\mathcal{L}_{1}(\zeta)\tilde{J}+\mathcal{L}_{2}\tilde{J}^{\zeta}\big{]}\|_{q_{1}}+\|R_{0}(\zeta^{2}\varepsilon_{\infty}\mu_{\infty})\big{[}\mathcal{L}_{1}(\zeta)\tilde{J}+\mathcal{L}_{2}\tilde{J}^{\zeta}\big{]}\|_{q_{2}}\lesssim\|J^{\zeta}\|_{p}+\|J^{\zeta}\|_{\tilde{p}}.$

###### Proof.

To bound the term involving $\mathcal{L}_{1}$ we use Theorem 6. Since
$\|\mathcal{L}_{1}(\zeta)\|_{\infty}\lesssim 1+|\zeta|\lesssim 1$ by the
definition of $\mathcal{L}_{1}$ from (29) and assumption (A1) we get

$\displaystyle\|R_{0}(\zeta^{2}\varepsilon_{\infty}\mu_{\infty})\big{[}\mathcal{L}_{1}(\zeta)\tilde{J}^{\zeta}\big{]}\|_{q_{1}}\lesssim\|\mathcal{L}_{1}(\zeta)\tilde{J}^{\zeta}\|_{p}\lesssim\|\tilde{J}^{\zeta}\|_{p}\lesssim\|J^{\zeta}\|_{p}.$

The estimate for the term involving $\mathcal{L}_{2}$ corresponds to the
special case $n=3,m=6$ in Theorem 30:

$\displaystyle\|R_{0}(\zeta^{2}\varepsilon_{\infty}\mu_{\infty})\big{[}\mathcal{L}_{2}\tilde{J}^{\zeta}\big{]}\|_{q_{1}}$
$\displaystyle\lesssim\|\tilde{J}^{\zeta}\|_{p}+\|\tilde{J}^{\zeta}\|_{\tilde{p}}\lesssim\|J^{\zeta}\|_{p}+\|J^{\zeta}\|_{\tilde{p}}.$

Since the same holds for $q_{1}$ replaced by $q_{2}$, this proves the claim.
$\Box$

Now we are in the position to prove the Limiting Absorption Principle for
time-harmonic Maxwell’s equations (4).

Proof of Theorem 1: In order to prove Theorem 1, it suffices to combine the
auxiliary results that we established above. So assume (A1),(A2),(A3) and let
$p,q$ and $J_{\textup{e}},J_{\textup{m}}\in L^{p}\cap L^{\tilde{p}}$ be given
as in the theorem. We prove the existence of the solutions
$(E_{\omega}^{\pm},H_{\omega}^{\pm})$ with the desired properties by proving
the convergence of the solutions $(E_{\zeta},H_{\zeta})$ as outlined in part
(i) of the theorem. To reduce the notation we only consider the limit
$\zeta\to\omega+i0$.

Proof of (i): For $\zeta\in\mathbb{C}\setminus\mathbb{R}$ with
$\operatorname{Im}(\zeta)>0,\operatorname{Re}(\zeta)\neq 0$ let
$J_{\textup{e}}^{\zeta},J_{\textup{m}}^{\zeta}\in L^{p}\cap L^{\tilde{p}}\cap
L^{2}$ be the divergence-free approximating sequence whose existence is
ensured by Proposition 24. Let then $(E_{\zeta},H_{\zeta})$ denote the unique
$H^{1}$-solutions of the corresponding inhomogeneous time-harmonic Maxwell
system (4) from Proposition 26. Corollary 23 yields for small
$|\operatorname{Im}(\zeta)|$

$\displaystyle\|E_{\zeta}\|_{q}+\|H_{\zeta}\|_{q}$
$\displaystyle\lesssim\|R_{0}(\zeta^{2}\varepsilon_{\infty}\mu_{\infty})\big{[}\mathcal{L}_{1}(\zeta)\tilde{J}^{\zeta}+\mathcal{L}_{2}\tilde{J}^{\zeta}\big{]}\|_{q_{1}}+\|R_{0}(\zeta^{2}\varepsilon_{\infty}\mu_{\infty})\big{[}\mathcal{L}_{1}(\zeta)\tilde{J}^{\zeta}+\mathcal{L}_{2}\tilde{J}^{\zeta}\big{]}\|_{q_{2}}+\|J^{\zeta}\|_{p}+\|J^{\zeta}\|_{\tilde{p}}$

where
$\tilde{J}^{\zeta}:=(\mu^{\frac{1}{2}}J_{\textup{e}}^{\zeta},\varepsilon^{\frac{1}{2}}J_{\textup{m}}^{\zeta})$.
So Proposition 27 implies

$\|E_{\zeta}\|_{q}+\|H_{\zeta}\|_{q}\lesssim\|J^{\zeta}\|_{p}+\|J^{\zeta}\|_{\tilde{p}}=\|J\|_{p}+\|J\|_{\tilde{p}}+o(1)\quad\text{as
}\zeta\to\omega+i0,$

which proves that the sequence of approximate solutions
$(E_{\zeta},H_{\zeta})$ is bounded in $L^{q}$. So a subsequence of
$(u_{\zeta})$ defined via
$u_{\zeta}:=(\tilde{E}_{\zeta},\tilde{H}_{\zeta}):=(\varepsilon^{\frac{1}{2}}E_{\zeta},\mu^{\frac{1}{2}}H_{\zeta})$
converges weakly to some
$u_{\omega}^{+}=:(\tilde{E}_{\omega}^{+},\tilde{H}_{\omega}^{+})$ in $L^{q}$.
Defining
$(E_{\omega}^{+},H_{\omega}^{+}):=(\varepsilon^{-\frac{1}{2}}\tilde{E}_{\omega}^{+},\mu^{-\frac{1}{2}}\tilde{H}_{\omega}^{+})$
we thus obtain a weak solution of the time-harmonic Maxwell system (4) (for
$\zeta=\omega$) that satisfies

$\|E_{\omega}^{+}\|_{q}+\|H_{\omega}^{+}\|_{q}\lesssim\|\tilde{E}_{\omega}^{+}\|_{q}+\|\tilde{H}_{\omega}^{+}\|_{q}\lesssim\|J\|_{p}+\|J\|_{\tilde{p}}.$

In the first estimate assumption (A1) is used. This proves the existence of
the solution $(E_{\omega}^{+},H_{\omega}^{+})$ along with the corresponding
norm estimate. To conclude the proof of (i) we need to show that for any given
approximations $J_{\textup{e}}^{\zeta},J_{\textup{m}}^{\zeta}$ as above the
full sequence $(u_{\zeta})$ converges to $u_{\omega}^{+}$.

So let $(\zeta_{j}),(\tilde{\zeta}_{j})$ sequences converging to $\omega+i0$
and let $J^{1,\zeta_{j}},J^{2,\tilde{\zeta}_{j}}$ be divergence-free currents
converging to $J$. Let $u_{\omega}^{1},u_{\omega}^{2}\in L^{q}$ denote the
corresponding weak limits, i.e., $u_{\zeta_{j}}\rightharpoonup
u_{\omega}^{1},u_{\tilde{\zeta}_{j}}\rightharpoonup u_{\omega}^{2}$. We need
to show $u_{\omega}^{1}=u_{\omega}^{2}$. From (37) we infer
$(I-\mathcal{K}(\zeta_{j}))u_{\zeta_{j}}=f^{1}_{\zeta_{j}}$ and
$(I-\mathcal{K}(\tilde{\zeta}_{j}))u_{\tilde{\zeta}_{j}}=f^{2}_{\tilde{\zeta}_{j}}$
where

$\displaystyle f_{\zeta}^{1}$
$\displaystyle:=R_{0}\big{(}\zeta^{2}\varepsilon_{\infty}\mu_{\infty}\big{)}\big{[}\mathcal{L}_{1}(\zeta)\tilde{J}^{1,\zeta}+\mathcal{L}_{2}\tilde{J}^{1,\zeta}\big{]},$
$\displaystyle f_{\zeta}^{2}$
$\displaystyle:=R_{0}\big{(}\zeta^{2}\varepsilon_{\infty}\mu_{\infty}\big{)}\big{[}\mathcal{L}_{1}(\zeta)\tilde{J}^{2,\zeta}+\mathcal{L}_{2}\tilde{J}^{2,\zeta}\big{]},$
$\displaystyle f_{\omega}^{+}$
$\displaystyle:=R_{0}\big{(}(\omega+i0)^{2}\varepsilon_{\infty}\mu_{\infty}\big{)}\big{[}\mathcal{L}_{1}(\omega)\tilde{J}+\mathcal{L}_{2}\tilde{J}\big{]}.$

Using first Proposition 27 and $\tilde{J}^{1,\zeta_{j}}\to\tilde{J}$, then the
uniform boundedness of the resolvents $R_{0}(\zeta_{j})$ and
$\mathcal{L}_{1}(\zeta_{j})\to\mathcal{L}_{1}(\omega)$ in $L^{\infty}$, and
finally the pointwise convergence
$R_{0}(\zeta_{j}^{2}\varepsilon_{\infty}\mu_{\infty})\to
R_{0}((\omega+i0)^{2}\varepsilon_{\infty}\mu_{\infty})$ we infer

$\displaystyle f_{\zeta_{j}}^{1}$
$\displaystyle=R_{0}\big{(}\zeta_{j}^{2}\varepsilon_{\infty}\mu_{\infty}\big{)}\big{[}\mathcal{L}_{1}(\zeta_{j})\tilde{J}^{1,\zeta_{j}}+\mathcal{L}_{2}\tilde{J}^{1,\zeta_{j}}\big{]}$
$\displaystyle=R_{0}\big{(}\zeta_{j}^{2}\varepsilon_{\infty}\mu_{\infty}\big{)}\big{[}\mathcal{L}_{1}(\zeta_{j})\tilde{J}+\mathcal{L}_{2}\tilde{J}\big{]}+o(1)$
$\displaystyle=R_{0}\big{(}\zeta_{j}^{2}\varepsilon_{\infty}\mu_{\infty}\big{)}\big{[}\mathcal{L}_{1}(\omega)\tilde{J}+\mathcal{L}_{2}\tilde{J}\big{]}+o(1)$
$\displaystyle=R_{0}\big{(}(\omega+i0)^{2}\varepsilon_{\infty}\mu_{\infty}\big{)}\big{[}\mathcal{L}_{1}(\omega)\tilde{J}+\mathcal{L}_{2}\tilde{J}\big{]}+o(1)$
$\displaystyle=f_{\omega}^{+}+o(1).$

Since the same argument applies to $f_{\tilde{\zeta}_{j}}^{2}$, we get
$f_{\zeta_{j}}^{1}-f^{2}_{\tilde{\zeta}_{j}}\to
f_{\omega}^{+}-f_{\omega}^{+}=0$. Using the continuity and compactness
properties of $\mathcal{K}$ from Corollary 9 we infer that
$w:=u_{\omega}^{1}-u_{\omega}^{2}$ satisfies

$\displaystyle w$ $\displaystyle=u_{\zeta_{j}}-u_{\tilde{\zeta}_{j}}+o_{w}(1)$
$\displaystyle=\mathcal{K}(\zeta_{j})u_{\zeta_{j}}-\mathcal{K}(\zeta_{j})u_{\tilde{\zeta}_{j}}+f^{1}_{\zeta_{j}}-f^{2}_{\tilde{\zeta}_{j}}+o_{w}(1)$
$\displaystyle=\mathcal{K}(\omega+i0)u_{\omega}^{1}-\mathcal{K}(\omega+i0)u_{\omega}^{2}+o_{w}(1)$
$\displaystyle=\mathcal{K}(\omega+i0)w+o_{w}(1).$

Here, $o_{w}(1)$ stands for a null sequence in the weak topology in $L^{q}$.
The function $(\tilde{w}_{\textup{e}},\tilde{w}_{\textup{m}})$ given by
$(\varepsilon^{1/2}\tilde{w}_{\textup{e}},\mu^{1/2}\tilde{w}_{\textup{m}}):=w$
is a weak solution of the homogeneous time-harmonic Maxwell system (4) for
$\zeta=\omega$ and $\tilde{J}=0$. Repeating the computations in Proposition 22
in the limiting case $\operatorname{Im}(\zeta)=0$ one finds

$\displaystyle
2\omega\int_{\mathbb{R}^{3}}v\cdot\operatorname{Re}(w_{m}\times\overline{w_{e}})\,\mathrm{d}x=0.$

So Proposition 20 gives $w=0$. This proves that all possible weak limits
coincide. Hence, the standard subsequence-of-subsequence argument ensures that
all approximating sequences weakly converge to the same limit as
$\zeta\to\omega+i0$.

To finish the proof of (i) it remains to show that this convergence also holds
in the strong sense and hence, by elliptic regularity theory, in
$H^{1}_{\operatorname{loc}}(\mathbb{R}^{3};\mathbb{C}^{6})$. To this end we
recall from above $(I-\mathcal{K}(\zeta_{j}))u_{\zeta_{j}}=f_{\zeta_{j}}$.
From the definition of $\mathcal{K}$ and the second part of Proposition 8 we
get
$\mathcal{K}(\zeta_{j})u_{\zeta_{j}}\to\mathcal{K}(\omega+i0)u_{\omega}^{+}$
in $L^{q}$. Moreover, we showed above $f_{\zeta_{j}}\to f_{\omega}^{+}$ in
$L^{q}$ as $j\to\infty$. Hence, we conclude that $(u_{\zeta_{j}})$ converges
in $L^{q}$. Since the limit necessarily coincides with the weak limit, we
finally obtain $u_{\zeta_{j}}\to u_{\omega}^{+}$ as $\zeta_{j}\to\omega+i0$.
This proves (i) as well as

$(I-\mathcal{K}(\omega\pm i0))u_{\omega}^{\pm}=R_{0}\big{(}(\omega\pm
i0)^{2}\varepsilon_{\infty}\mu_{\infty}\big{)}\big{[}\mathcal{L}_{1}(\omega)\tilde{J}+\mathcal{L}_{2}\tilde{J}\big{]}.$
(45)

In particular, $u_{\omega}^{+}$ solves the Helmholtz system mentioned in the
theorem.

We finally prove part (iii) of the theorem, so we assume that the divergence-
free currents $(J_{\textup{e}},J_{\textup{m}})$ lie in the smaller space
$L^{p}\cap L^{q}\subset L^{p}\cap L^{\tilde{p}}$ (because $p<\tilde{p}<q$).
From above we get a solution $(E,H)\in L^{q}$ of (4) which, according to
Proposition 25, satisfies

$\displaystyle\|\nabla E\|_{q}+\|\nabla
H\|_{q}\lesssim(1+|\zeta|)(\|E\|_{q}+\|H\|_{q})+\|J_{\textup{e}}\|_{q}+\|J_{\textup{m}}\|_{q}\lesssim\|J\|_{p}+\|J\|_{q}.$

Hence we conclude $E,H\in W^{1,q}(\mathbb{R}^{3};\mathbb{C}^{3})$ as claimed.
$\Box$

## Appendix A: Proof of Lemma 16

This appendix is devoted to the proof of Lemma 16 and, in particular, to
showing the validity of (30). For notational simplicity we verify (30) in the
pointwise sense assuming that classical derivatives exist. This carries over
to weak solutions by moving first order derivatives to the test functions. So
let $(E,H)$ denote a weak solution of (4) as assumed. Then
$(\tilde{E},\tilde{H})=(\varepsilon^{-\frac{1}{2}}D,\mu^{-\frac{1}{2}}B)$
where $D,B$ are divergence-free vector fields. The latter follows from the
fact that $J_{e},J_{m}$ are divergence-free. So we have
$\nabla\times\nabla\times D=\nabla(\nabla\cdot D)-\Delta D=-\Delta D$ and
obtain

$\displaystyle\begin{aligned}
\Delta\tilde{E}&=\Delta(\varepsilon^{-\frac{1}{2}})D+2[\nabla(\varepsilon^{-\frac{1}{2}})\cdot\nabla]D+\varepsilon^{-\frac{1}{2}}\Delta
D\\\
&=\left(\frac{3}{4}\varepsilon^{-2}|\nabla\varepsilon|^{2}-\frac{1}{2}\varepsilon^{-1}\Delta\varepsilon\right)\tilde{E}-\varepsilon^{-\frac{3}{2}}[\nabla\varepsilon\cdot\nabla]D-\varepsilon^{-\frac{1}{2}}\nabla\times\nabla\times
D.\end{aligned}$ (46)

The second order term can be simplified with the aid of (4). The vector
calculus identities

$\displaystyle\nabla\times(\psi A)$ $\displaystyle=\nabla\psi\times
A+\psi(\nabla\times A),$ $\displaystyle\nabla\times(A\times C)$
$\displaystyle=A(\nabla\cdot C)-C(\nabla\cdot
A)+(C\cdot\nabla)A-(A\cdot\nabla)C$

for scalar fields $\psi$ and vector fields $A,C$ lead to

$\displaystyle\nabla\times\nabla\times D$
$\displaystyle=\nabla\times\nabla\times(\varepsilon E)$
$\displaystyle=\nabla\times\left(\nabla\varepsilon\times
E+\varepsilon(\nabla\times E)\right)$
$\displaystyle=\nabla\times\left(\nabla\varepsilon\times
E+\varepsilon\left[-i\zeta\mu H+J_{m}\right]\right)$
$\displaystyle=\nabla\times\left(\nabla\varepsilon\times
E\right)-i\zeta\nabla(\varepsilon\mu)\times
H-i\zeta\varepsilon\mu(\nabla\times H)+\nabla\times(\varepsilon J_{m})$
$\displaystyle=\nabla\varepsilon(\nabla\cdot
E)-(\Delta\varepsilon)E+(E\cdot\nabla)\nabla\varepsilon-(\nabla\varepsilon\cdot\nabla)E$
$\displaystyle-i\zeta\nabla(\varepsilon\mu)\times
H-i\zeta\varepsilon\mu(i\zeta\varepsilon
E+J_{e})+\nabla\times(\varepsilon^{\frac{1}{2}}\tilde{J}_{m}).$

To simplify these terms we use that $D=\varepsilon E$ is divergence-free and
thus $\nabla\cdot E=-\varepsilon^{-1}\nabla\varepsilon\cdot E$. Moreover,

$(\nabla\varepsilon\cdot\nabla)E=(\nabla\varepsilon\cdot\nabla)(\varepsilon^{-1}D)=-\varepsilon^{-2}|\nabla\varepsilon|^{2}D+\varepsilon^{-1}(\nabla\varepsilon\cdot\nabla)D.$

This implies

$\displaystyle-\varepsilon^{-\frac{1}{2}}\nabla\times\nabla\times D$
$\displaystyle=-\varepsilon^{-\frac{1}{2}}\cdot\left[-\varepsilon^{-1}\nabla\varepsilon(\nabla\varepsilon\cdot
E)-(\Delta\varepsilon)E+(E\cdot\nabla)\nabla\varepsilon+\varepsilon^{-2}|\nabla\varepsilon|^{2}D-\varepsilon^{-1}(\nabla\varepsilon\cdot\nabla)D\right]$
$\displaystyle+i\zeta(\varepsilon\mu)^{-\frac{1}{2}}\nabla(\varepsilon\mu)\times\tilde{H}-\zeta^{2}\varepsilon\mu\tilde{E}+i\zeta(\varepsilon\mu)^{\frac{1}{2}}\tilde{J}_{e}-\varepsilon^{-\frac{1}{2}}\nabla\times(\varepsilon^{\frac{1}{2}}\tilde{J}_{m})$
$\displaystyle=\varepsilon^{-2}\nabla\varepsilon(\nabla\varepsilon)^{T}\tilde{E}+\varepsilon^{-1}(\Delta\varepsilon)\tilde{E}-\varepsilon^{-1}(\tilde{E}\cdot\nabla)\nabla\varepsilon-\varepsilon^{-2}|\nabla\varepsilon|^{2}\tilde{E}+\varepsilon^{-\frac{3}{2}}(\nabla\varepsilon\cdot\nabla)D$
$\displaystyle+i\zeta(\varepsilon\mu)^{-\frac{1}{2}}\nabla(\varepsilon\mu)\times\tilde{H}-\zeta^{2}\varepsilon\mu\tilde{E}+i\zeta(\varepsilon\mu)^{\frac{1}{2}}\tilde{J}_{e}-\nabla\times\tilde{J}_{m}-\frac{1}{2}\varepsilon^{-1}\nabla\varepsilon\times\tilde{J}_{m}.$

Combining this formula with (46) we find that the first order terms (involving
$D$) cancel and

$\displaystyle\Delta\tilde{E}$
$\displaystyle=\left(\frac{3}{4}\varepsilon^{-2}|\nabla\varepsilon|^{2}-\frac{1}{2}\varepsilon^{-1}\Delta\varepsilon\right)\tilde{E}+\varepsilon^{-2}\nabla\varepsilon(\nabla\varepsilon)^{T}\tilde{E}+\varepsilon^{-1}\Delta\varepsilon\tilde{E}-\varepsilon^{-1}(\tilde{E}\cdot\nabla)\nabla\varepsilon-\varepsilon^{-2}|\nabla\varepsilon|^{2}\tilde{E}$
$\displaystyle+i\zeta(\varepsilon\mu)^{-\frac{1}{2}}\nabla(\varepsilon\mu)\times\tilde{H}-\zeta^{2}\varepsilon\mu\tilde{E}+i\zeta(\varepsilon\mu)^{\frac{1}{2}}\tilde{J}_{e}-\nabla\times\tilde{J}_{m}-\frac{1}{2}\varepsilon^{-1}\nabla\varepsilon\times\tilde{J}_{m}$
$\displaystyle=\left(-\frac{1}{4}\varepsilon^{-2}|\nabla\varepsilon|^{2}+\frac{1}{2}\varepsilon^{-1}\Delta\varepsilon+\varepsilon^{-2}\nabla\varepsilon(\nabla\varepsilon)^{T}-\zeta^{2}\varepsilon\mu-\varepsilon^{-1}\nabla\nabla^{T}\varepsilon\right)\tilde{E}$
$\displaystyle+i\zeta(\varepsilon\mu)^{-\frac{1}{2}}\nabla(\varepsilon\mu)\times\tilde{H}+i\zeta(\varepsilon\mu)^{\frac{1}{2}}\tilde{J}_{e}-\nabla\times\tilde{J}_{m}-\frac{1}{2}\varepsilon^{-1}\nabla\varepsilon\times\tilde{J}_{m}$
$\displaystyle=\Big{[}\varepsilon^{-\frac{1}{2}}\Delta(\varepsilon^{\frac{1}{2}})-\nabla\nabla^{T}(\log\varepsilon)+\zeta^{2}(\varepsilon_{\infty}\mu_{\infty}-\varepsilon\mu)\Big{]}\tilde{E}-\zeta^{2}\varepsilon_{\infty}\mu_{\infty}\tilde{E}$
$\displaystyle+2i\zeta\nabla((\varepsilon\mu)^{\frac{1}{2}})\times\tilde{H}+i\zeta(\varepsilon\mu)^{\frac{1}{2}}\tilde{J}_{e}-\nabla\times\tilde{J}_{m}-\frac{1}{2}\nabla(\log\varepsilon)\times\tilde{J}_{m}.$

This corresponds to the first line in (30). To derive the second line, one
proceeds in an analogous manner and subsequently derives the formulas

$\displaystyle\Delta\tilde{H}$
$\displaystyle=\left(\frac{3}{4}\mu^{-2}|\nabla\mu|^{2}-\frac{1}{2}\mu^{-1}\Delta\mu\right)\tilde{H}-\mu^{-\frac{3}{2}}[\nabla\mu\cdot\nabla]B-\mu^{-\frac{1}{2}}\nabla\times\nabla\times
B,$ $\displaystyle\nabla\times\nabla\times B$
$\displaystyle=\nabla\mu(\nabla\cdot
H)-(\Delta\mu)H+(H\cdot\nabla)\nabla\mu-(\nabla\mu\cdot\nabla)H$
$\displaystyle+i\zeta\nabla(\varepsilon\mu)\times
E+i\zeta\varepsilon\mu(-i\zeta\mu
H+J_{m})+\nabla\times(\mu^{\frac{1}{2}}\tilde{J}_{e}),$
$\displaystyle-\mu^{-\frac{1}{2}}\nabla\times\nabla\times B$
$\displaystyle=\mu^{-2}\nabla\mu(\nabla\mu)^{T}\tilde{H}+\mu^{-1}(\Delta\mu)\tilde{H}-\mu^{-1}(\tilde{H}\cdot\nabla)\nabla\mu-\mu^{-2}|\nabla\mu|^{2}\tilde{H}+\mu^{-\frac{3}{2}}(\nabla\mu\cdot\nabla)B$
$\displaystyle-i\zeta(\varepsilon\mu)^{-\frac{1}{2}}\nabla(\varepsilon\mu)\times\tilde{E}-\zeta^{2}\varepsilon\mu\tilde{H}-i\zeta(\varepsilon\mu)^{\frac{1}{2}}\tilde{J}_{m}-\nabla\times\tilde{J}_{e}-\frac{1}{2}\mu^{-1}\nabla\mu\times\tilde{J}_{e},$
$\displaystyle\Delta\tilde{H}$
$\displaystyle=\Big{[}\mu^{-\frac{1}{2}}\Delta(\mu^{\frac{1}{2}})-\nabla\nabla^{T}(\log\mu)+\zeta^{2}(\varepsilon_{\infty}\mu_{\infty}-\varepsilon\mu)\Big{]}\tilde{H}-\zeta^{2}\varepsilon_{\infty}\mu_{\infty}\tilde{H}$
$\displaystyle-2i\zeta\nabla((\varepsilon\mu)^{\frac{1}{2}})\times\tilde{E}-i\zeta(\varepsilon\mu)^{\frac{1}{2}}\tilde{J}_{m}-\nabla\times\tilde{J}_{e}-\frac{1}{2}\nabla(\log\mu)\times\tilde{J}_{e}.$

$\Box$

## Appendix B: Uniform estimates for $R_{0}(\zeta)\partial_{j}$

This appendix is devoted to the proof of Theorem 30 (see below) that we needed
in the proof of the Limiting Absorption Principle for Maxwell’s equations. In
order to do that we need the following classical result on Fourier
multipliers.

######  Theorem 28 (Mikhlin-Hörmander).

Let $n\in\mathbb{N},1<r<\infty$. For $k:=\lfloor\frac{n}{2}\rfloor+1$ assume
that $m\in C^{k}(\mathbb{R}^{n})$ satisfies $|\partial_{\alpha}m(\xi)|\leq
A|\xi|^{-|\alpha|}$ for all multi-indices $\alpha\in\mathbb{N}_{0}^{n}$ with
$|\alpha|\leq k$. Then

$\|\mathcal{F}^{-1}(m\mathcal{F}f)\|_{r}\leq C_{n,r}A\|f\|_{r}$

###### Proof.

The result is a particular case of Theorem 6.2.7 in [21]. $\Box$

We also need some boundedness properties of Bessel potentials.

######  Theorem 29.

Assume $n\in\mathbb{N},n\geq 2$. Let $\mathcal{J}_{1}$ be the Bessel potential
of order $1$ defined as

$\mathcal{J}_{1}f:=\mathcal{F}^{-1}\left(\frac{1}{\sqrt{1+|\xi|^{2}}}\hat{f}\right),\qquad\text{alternatively},\qquad\mathcal{J}_{1}f:=G\ast
f:=\mathcal{F}^{-1}\left(\frac{1}{\sqrt{1+|\xi|^{2}}}\right)\ast f,$

Assume $1\leq p\leq q\leq\infty$ be such that

$0\leq\frac{1}{p}-\frac{1}{q}\leq\frac{1}{n},\qquad\left(\frac{1}{p},\frac{1}{q}\right)\notin\left\\{\left(1,1-\frac{1}{n}\right),\left(\frac{1}{n},0\right)\right\\}.$

Then $\mathcal{J}_{1}$ is a bounded operator from $L^{p}(\mathbb{R}^{n})$ to
$L^{q}(\mathbb{R}^{n}).$

###### Proof.

From [22, Corollary 1.2.6 (a),(b)] and interpolation we have

$\mathcal{J}_{1}\colon L^{p}(\mathbb{R}^{n})\to
L^{q}(\mathbb{R}^{n}),\qquad\text{if}\;1<p\leq q<\infty,\quad
0\leq\frac{1}{p}-\frac{1}{q}\leq\frac{1}{n}.$

Thus, it remains to study the case $p=1$ and the case $q=\infty.$ Let us start
with the case $p=1.$ Again from [22, Corollary 1.2.6 (b)] we have that
$\mathcal{J}_{1}\colon L^{1}(\mathbb{R}^{n})\to L^{q,\infty}(\mathbb{R}^{n}),$
when $p=1$ and for $\frac{1}{p}-\frac{1}{q}=\frac{1}{n}.$ Moreover, from [22,
Corollary 1.2.6 (a)], we know that $\mathcal{J}_{1}\colon
L^{1}(\mathbb{R}^{n})\to L^{1}(\mathbb{R}^{n}).$ Thus, Marcinkiewicz’s
interpolation theorem gives that $\mathcal{J}_{1}\colon
L^{1}(\mathbb{R}^{n})\to L^{q}(\mathbb{R}^{n}),$ with
$1-\frac{1}{q}<\frac{1}{n}.$ We continue with the case $q=\infty.$ We know
from the proof of [22, Corollary 1.2.6 (b)] that the kernel $G$ satisfies
$|G(x)|\lesssim|x|^{1-n}$ if $|x|\leq 2$ and $|G(x)|\lesssim
e^{-\frac{|x|}{2}}$ if $|x|\geq 2.$ Using Young’s convolution inequality we
thus get

$\|\mathcal{J}_{1}f\|_{\infty}=\|G\ast
f\|_{\infty}\leq\|G\|_{p^{\prime}}\|f\|_{p},\qquad\frac{1}{p}+\frac{1}{p^{\prime}}=1.$

The proof is concluded once one observes that $\|G\|_{p^{\prime}}<\infty$ for
$0\leq\frac{1}{p}<\frac{1}{n}.$ $\Box$

With these results at hands, we are in the position to prove the following
estimates that complement Theorem 6.

######  Theorem 30.

Let $m,n\in\mathbb{N},n\geq 3$ and assume
$\zeta\in\mathbb{C}\setminus\mathbb{R}_{\geq 0}.$ Then, for $1\leq
p,\tilde{p},q\leq\infty$ such that

$\begin{split}1&\geq\frac{1}{p}>\frac{n+1}{2n},\qquad
0\leq\frac{1}{q}<\frac{n-1}{2n},\qquad\frac{2}{n+1}\leq\frac{1}{p}-\frac{1}{q}\leq\frac{2}{n},\\\
0&\leq\frac{1}{\tilde{p}}-\frac{1}{q}\leq\frac{1}{n},\qquad\left(\frac{1}{\tilde{p}},\frac{1}{q}\right)\notin\left\\{\left(1,1-\frac{1}{n}\right),\left(\frac{1}{n},0\right)\right\\},\end{split}$
(47)

$R_{0}(\zeta)\partial_{j}f$ is a bounded linear operator from
$L^{p}(\mathbb{R}^{n};\mathbb{C}^{m})$ to
$L^{q}(\mathbb{R}^{n};\mathbb{C}^{m})$ satisfying

$\|R_{0}(\zeta)\partial_{j}f\|_{q}\lesssim|\zeta|^{\frac{n}{2}(\frac{1}{p}-\frac{1}{q}-\frac{1}{n})}\|f\|_{p}+|\zeta|^{\frac{n}{2}(\frac{1}{\tilde{p}}-\frac{1}{q}-\frac{1}{n})}\|f\|_{\tilde{p}},\qquad
j=1,2,\dots,n.$ (48)

Moreover, there are bounded linear operators $R_{0}(\lambda\pm
i0)\partial_{j}:L^{p}(\mathbb{R}^{n};\mathbb{C}^{m})\to
L^{q}(\mathbb{R}^{n};\mathbb{C}^{m})$ such that $R_{0}(\zeta)\partial_{j}f\to
R_{0}(\lambda\pm i0)\partial_{j}f,$ $j=1,2,\dots,n$ as $\zeta\to\lambda\pm
i0,\lambda\in\mathbb{R}_{>0}$ for all $f\in
L^{p}(\mathbb{R}^{n};\mathbb{C}^{m})$ and

$\|R_{0}(\lambda\pm
i0)\partial_{j}f\|_{q}\lesssim|\lambda|^{\frac{n}{2}(\frac{1}{p}-\frac{1}{q}-\frac{1}{n})}\|f\|_{p}+|\lambda|^{\frac{n}{2}(\frac{1}{\tilde{p}}-\frac{1}{q}-\frac{1}{n})}\|f\|_{\tilde{p}}\qquad(\lambda>0).$

###### Proof.

We first isolate the singularity (in Fourier space) of the Fourier multiplier
$\frac{1}{|\xi|^{2}-\zeta}$. In order to do that we introduce the cut-off
function $\chi\in C^{\infty}_{0}(\mathbb{R}^{n})$ with $\chi(\xi)=1$ whenever
$|\xi|\leq 2$ and we define
$\chi_{\zeta}(\xi):=\chi(|\zeta|^{-\frac{1}{2}}\xi).$ We then write

$R_{0}(\zeta)(\partial_{j}f)=\mathcal{F}^{-1}\left(\frac{\chi_{\zeta}(\xi)(i\xi_{j}\hat{f}(\xi))}{|\xi|^{2}-\zeta}\right)+\mathcal{F}^{-1}\left(\frac{(1-\chi_{\zeta}(\xi))(i\xi_{j}\hat{f}(\xi))}{|\xi|^{2}-\zeta}\right)\qquad
j=1,2,\dots,n.$ (49)

Observe that $\chi_{\zeta}$ is nontrivial in a neighborhood of the sphere of
radius $\zeta,$ on the contrary $1-\chi_{\zeta}$ vanishes in the same
neighborhood. In other words, the singularity of the multiplier affects only
the first term of the right-hand side of (49). The latter can be estimated
with the aid of Theorem 6. More specifically, one has

$\begin{split}\left\|\mathcal{F}^{-1}\left(\frac{\chi_{\zeta}(\xi)(i\xi_{j}\hat{f}(\xi))}{|\xi|^{2}-\zeta}\right)\right\|_{q}&=|\zeta|^{-\frac{1}{2}-\frac{n}{2q}}\left\|\mathcal{F}^{-1}\left(\frac{\chi(\xi)\xi_{j}\mathcal{F}[f(|\zeta|^{-1/2}\cdot)](\xi)}{|\xi|^{2}-\frac{\zeta}{|\zeta|}}\right)\right\|_{q}\\\
&\lesssim|\zeta|^{-\frac{1}{2}-\frac{n}{2q}}\left\|\mathcal{F}^{-1}\left(\chi(\xi)\xi_{j}\mathcal{F}[f(|\zeta|^{-1/2}\cdot)](\xi)\right)\right\|_{p}\\\
&\lesssim|\zeta|^{-\frac{1}{2}-\frac{n}{2q}}\left\|\mathcal{F}^{-1}\left(\chi(\xi)\xi_{j}\right)\ast
f(|\zeta|^{-1/2}\cdot)\right\|_{p}\\\
&\lesssim|\zeta|^{-\frac{1}{2}-\frac{n}{2q}}\|f(|\zeta|^{-1/2}\cdot)\|_{p}\\\
&\lesssim|\zeta|^{\frac{n}{2}(\frac{1}{p}-\frac{1}{q}-\frac{1}{n})}\|f\|_{p}.\end{split}$

Here we used Young’s convolution inequality and that
$\mathcal{F}^{-1}(\chi(\xi)\xi_{j})$ is integrable (being a Schwartz
function).

We now turn to the estimate of the second term in the sum in (49). We shall
use that

$m(\xi):=\frac{(1-\chi(\xi))\xi_{j}\sqrt{1+|\xi|^{2}}}{|\xi|^{2}-\zeta/|\zeta|},\qquad
j=1,2,\dots,n,$

is a $L^{r}(\mathbb{R}^{n})-L^{r}(\mathbb{R}^{n})$ multiplier for
$1<r<\infty,$ due to the simplified version of the Mikhlin-Hörmander Theorem
stated in Theorem 28. Recall that $\chi$ satisfies $1-\chi\equiv 0$ on a
neighborhood of the unit sphere. Notice that, by (47) we have
$0<\frac{1}{q}<1$ or $0<\frac{1}{\tilde{p}}<1.$ In the case $0<\frac{1}{q}<1$
we use the above observation for $r=q$ and Theorem 29 implies

$\begin{split}\mathcal{F}^{-1}\left(\frac{(1-\chi_{\zeta}(\xi))(i\xi_{j}\hat{f}(\xi))}{|\xi|^{2}-\zeta}\right)&=|\zeta|^{-\frac{1}{2}-\frac{n}{2q}}\left\|\mathcal{F}^{-1}\left(\frac{(1-\chi(\xi))\xi_{j}\mathcal{F}[f(|\zeta|^{-1/2}\cdot)](\xi)}{|\xi|^{2}-1}\right)\right\|_{q}\\\
&=|\zeta|^{-\frac{1}{2}-\frac{n}{2q}}\left\|\mathcal{F}^{-1}\left(m(\xi)\mathcal{F}\left(\mathcal{F}^{-1}\left(\frac{1}{\sqrt{|\xi|^{2}+1}}\mathcal{F}[f(|\zeta|^{-1/2}\cdot)](\xi)\right)\right)\right)\right\|_{q}\\\
&=|\zeta|^{-\frac{1}{2}-\frac{n}{2q}}\left\|\mathcal{F}^{-1}\left(m(\xi)\mathcal{F}\left(\mathcal{J}_{1}[f(|\zeta|^{-1/2}\cdot)]\right)\right)\right\|_{q}\\\
&\lesssim|\zeta|^{-\frac{1}{2}-\frac{n}{2q}}\left\|\mathcal{J}_{1}[f(|\zeta|^{-1/2}\cdot)]\right\|_{q}\\\
&\lesssim|\zeta|^{-\frac{1}{2}-\frac{n}{2q}}\|f(|\zeta|^{-1/2}\cdot)\|_{\tilde{p}}\\\
&\leq|\zeta|^{\frac{n}{2}(\frac{1}{\tilde{p}}-\frac{1}{q}-\frac{1}{n})}\|f\|_{\tilde{p}}.\end{split}$

In the complementary case $0<\frac{1}{\tilde{p}}<1,$ we use the Mikhlin-
Hörmander Theorem 28 for $r=\tilde{p}$ and proceed similarly. Plugging the two
previous bounds in (49) gives (48). The final part of Theorem 30 can be proved
as in Theorem 6. $\Box$

## Acknowledgements