330345

# Fredholm theory connected with a Douglis-Nirenberg system of differential
equations over $\mathbb{R}^{n}$

M. Faierman  School of Mathematics and Statistics, The University of New South
Wales, UNSW Sydney, NSW 2052, Australia<EMAIL_ADDRESS>
(Date: 27/04/2016)

###### Abstract.

We consider a spectral problem over $\mathbb{R}^{n}$ for a Douglis-Nirenberg
system of differential operators under limited smoothness assumptions and
under the assumption of parameter-ellipticity in a closed sector $\mathcal{L}$
in the complex plane with vertex at the origin. We pose the problem in an
$L_{p}$ Sobolev-Bessel potential space setting, $1<p<\infty$, and denote by
$A_{p}$ the operator induced in this setting by the spectral problem. We then
derive results pertaining to the Fredholm theory for $A_{p}$ for values of the
spectral parameter $\lambda$ lying in $\mathcal{L}$ as well as results
pertaining to the invariance of the Fredholm domain of $A_{p}$ with $p$.

###### Key words and phrases:

Parameter-ellipticity, Douglis-Nirenberg system, Fredholm properties.

###### 1991 Mathematics Subject Classification:

35J45, 47A53

##  1\. Introduction

The Fredholm properties of elliptic pseudodifferential operators, as well as
systems of such operators, defined over $\mathbb{R}^{n}$ has been the subject
of investigation by various authors over the past few decades. We refer for
example to [10], [15], [17], [19], and to the references listed therein for
further details. Of particular interest to us are the works dealing with
Douglis-Nirenberg systems (cf. [10], [17], [18], [19], and [22]) as well as
those dealing with parameter-elliptic operators (cf. [17]), since, as far as
we are aware from an inspection of the literature, there are no works dealing
with parameter-elliptic Douglis-Nirenberg systems over $\mathbb{R}^{n}$ in the
classical $L_{p}$ Sobolev-Bessel potential space setting, $1<p<\infty$, when
the diagonal operators are not all of the same order.

In light of what has just been said, let us mention at this point the paper
[13] of Kozhevnikov wherein a Douglis-Nirenberg system of pseudodifferential
operators acting over a compact manifold without boundary is considered. By
posing the problem in a classical setting (as mentioned above) and by
introducing the so-called Kohzevnikov conditions, the author was able to
establish a priori estimates for solutions as well as various spectral
results. Problems similar to those considered in [13] were also investigated
by Denk, Mennicken, and Volevich [9], and by introducing conditions, which by
the use of the Newton polygon they show to be equivalent to those of
Kozhevnikov, they also establish a priori estimates for solutions as well as
various spectral results.

Motivated by what has been said above, the object of this paper is to derive
information concerning the Fredholm properties of the operator $A_{p}$ induced
in a $L_{p}$ Sobolev-Bessel potential space setting, $1<p<\infty$, by a
spectral problem over $\mathbb{R}^{n}$ for a Douglis-Nirenberg system of
differential operators under limited smoothness assumptions as well as under
the assumption of parameter-ellipticity in a closed sector $\mathcal{L}$ of
the complex plane with vertex at the origin. And fundamental to our work will
be the uniform version of the Kozhevnikov conditions given in Definition 2.3
below which will enable us to derive information pertaining to that part of
the Fredholm domain of $A_{p}$ lying in $\mathcal{L}$.

Turning to the problem under consideration here, let $N\in\mathbb{N}$ with
$N>1$ and let $\\{s_{j}\\}_{1}^{N}$ and $\\{t_{j}\\}_{1}^{N}$ denote sequences
of integers satisfying $s_{1}\geq s_{2}\geq\cdots\geq s_{N}\geq 0$, $t_{1}\geq
t_{2}\geq\cdots\geq t_{N}\geq 0$, and put $m_{j}=s_{j}+t_{j}$ for
$j=1,\ldots,N$. We suppose that
$m_{1}=m_{2}=\cdots=m_{k_{1}}>m_{k_{1}+1}=\cdots=m_{k_{d-1}}>m_{k_{d-1}+1}=\cdots=m_{k_{d}}>0$,
where $k_{d}=N$, and let $\tilde{I}_{r}$ denote the
$(k_{r}-k_{r-1})\times(k_{r}-k_{r-1})$ identity matrix for $r=1,\ldots,d$,
where $k_{0}=0$. We will also use the notation $I_{\ell}$ to denote the
$\ell\times\ell$ unit matrix for $\ell\in\mathbb{N}$. Then we shall be
concerned here with the spectral problem

$A(x,D)u(x)-\lambda u(x)=f(x)\quad\textrm{in}\quad\mathbb{R}^{n},$ (1.1)

where $\,u(x)=(u_{1}(x),\ldots,$ $u_{N}(x))^{T}$, and
$f(x)=\left(f_{1}(x),\ldots,f_{N}(x)\right)^{T}$ are $N\times 1$ matrix
functions defined in $\mathbb{R}^{n},\;^{T}$ denotes transpose, $A(x,D)$ is an
$N\times N$ matrix operator whose entries $A_{jk}(x,D)$ are linear
differential operators defined on $\mathbb{R}^{n}$ of order not exceeding
$s_{j}+t_{k}$ and defined to be $0$ if $s_{j}+t_{k}<0$. Our assumptions
concerning the spectral problem (1.1) will be made precise in Section 2.

In Section 2 we make precise the concept of parameter-ellipticity for the
spectral problem (1.1) and the conditions under which the problem is dealt
with. In Section 3 we pose the spectral problem in an $L_{p}$ Sobolev-Bessel
potential space setting, $1<p<\infty$, and obtain estimates for solutions for
various values of the spectral parameter. Finally in Section 4 we we fix our
attention upon the operator $A_{p}$ induced in the $L_{p}$ Sobolev-Bessel
potential space setting just cited by the spectral problem (1.1), and using
results from Section 3, we derive information concerning the Fredholm
properties of $A_{p}$ for various values of the spectral parameter lying in
$\mathcal{L}$.

##  2\. Preliminaries

In this section we are going to introduce some terminology, definitions, and
assumptions concerning the spectral problem (1.1), which we require for our
work.

Accordingly, we let $x=(x_{1},\ldots,x_{n})=(x^{\prime},x_{n})$ denote a
generic point in $\mathbb{R}^{n}$ and use the notation
$D_{j}=-i\partial/\partial x_{j}$, $D=(D_{1},\ldots,D_{n})$,
$D^{\alpha}=D_{1}^{\alpha_{1}}\cdots
D_{n}^{\alpha_{n}}=D^{\prime\,\alpha^{\prime}}D_{n}^{\alpha_{n}}$, and
$\xi^{\alpha}=\xi_{1}^{\alpha_{1}}\ldots\xi_{n}^{\alpha_{n}}$ for
$\xi=(\xi_{1},\ldots,\xi_{n})$ = $(\xi^{\prime},\xi_{n})\in\mathbb{R}^{n}$,
where $\alpha=(\alpha_{1},\ldots,\alpha_{n})=(\alpha^{\prime},\alpha_{n})$ is
a multi-index whose length $\sum_{j=1}^{n}\alpha_{j}$ is denoted by
$|\alpha|$. Differentiation with respect to another variable, say
$y\in\mathbb{R}^{n}$, instead of $x$ will be indicated by replacing $D$ and
$D^{\alpha}$ by $D_{y}$ and $D_{y}^{\alpha}$, respectively. For $1<p<\infty$,
$s\in\mathbb{N}_{0}=\mathbb{N}\cup\\{\,0\,\\}$, and $G$ an open set in
$\mathbb{R}^{n}$, we let $W_{p}^{s}(G)$ denote the Sobolev space of order $s$
related to $L_{p}(G)$ and denote the norm in this space by
$\|\cdot\|_{s,p,G}$, where $\|u\|_{s,p,G}=\left(\sum_{|\alpha|\leq
s}\int_{G}|D^{\alpha}u(x)|^{p}\,dx\right)^{1/p}$ for $u\in W^{s}_{p}(G)$. In
addition we shall use norms depending upon a parameter
$\lambda\in\mathbb{C}\backslash\\{\,0\,\\}$, namely for $1\leq j\leq N$, we
let

$|||u|||^{(j)}_{s,p,G}=\|u\|_{s,p,G}+|\lambda|^{s/m_{j}}\|u\|_{0,p,G}\quad\textrm{for}\quad
u\in W^{s}_{p}(G).$

In the sequel we shall at times deal with the Bessel-potential space
$H^{s}_{p}(G)$ for $0\geq s\in\mathbb{Z}$ and equipped with either its
ordinary norm $\|\,\cdot\,\|_{s,p,G}$ or its parameter dependent norm
$|||\,\cdot\,|||^{(j)}_{s,p,G},1\leq j\leq N$. Here, for $u\in H^{s}_{p}(G)$,
$\|\,u\,\|_{s,p,G}=\|\,F^{-1}\langle\,\xi\,\rangle^{s}Fu\,\|_{0,p,\mathbb{R}^{n}}$
(resp.
$|||u|||^{(j)}_{s,p,G}=\|\,F^{-1}\langle\xi,\lambda\rangle^{s}_{j}Fu\|_{0,p,\mathbb{R}^{n}}$)
if $G=\mathbb{R}^{n}$, while $\|\,u\,\|_{s,p,G}=$ inf
$\|\,v\,\|_{s,p,\mathbb{R}^{n}}$ (resp.
$|||u|||^{(j)}_{s,p,G}=\textrm{inf}\,|||v|||^{(j)}_{s,p,\mathbb{R}^{n}}$)
otherwise, where the infimum is taken over all $v\in
H^{s}_{p}(\mathbb{R}^{n})$ for which $u=v_{\bigl{|}G}$, $F$ denotes the
Fourier transformation in $\mathbb{R}^{n}(x\to\xi)$,
$\langle\,\xi\,\rangle=(1+|\xi|^{2})^{1/2}$,
$\lambda\in\mathbb{C}\setminus\\{\,0\,\\}$, and
$\langle\xi,\lambda\rangle_{j}=\left(|\xi|^{2}+|\lambda|^{\frac{2}{m_{j}}}\right)^{\frac{1}{2}}$
(see [11, Section 1], [21, p. 177]). Analogous definitions also hold for
$s>0$. However when $s\geq 0$, then $W^{s}_{p}(G)$ and $H^{s}_{p}(G)$ coincide
algebraically and their norms, both ordinary and parameter dependent, are
equivalent. Hence in the sequel, when dealing with the space $H^{s}_{p}(G)$
for $s\geq 0$, we shall suppose that it is equipped with either the ordinary
or parameter dependent norm of $W^{s}_{p}(G)$. Lastly, let
$\mathbb{R}_{+}=\\{\,t\in\mathbb{R}|t>0\,\\}$,
$\mathbb{R}_{-}=\\{\,t\in\mathbb{R}\,\bigl{|}\,t<0\,\\}$.

Next for $\ell\in\mathbb{N}_{0}$, we will use the usual notation
$C^{\ell}(\mathbb{R}^{n})$ to denote the vector space consisting of all those
functions $\phi$ which, together with their partial derivatives of order up to
$\ell$, are continuous on $\mathbb{R}^{n}$. In addition we let
$C^{\ell}(\overline{\mathbb{R}^{n}})$ denote the subspace of
$C^{\ell}(\mathbb{R}^{n})$ consisting of all those functions $\phi\in
C^{\ell}(\mathbb{R}^{n})$ for which $\phi$ as well as its partial derivative
of order up to $\ell$ are bounded and uniformly continuous on
$\mathbb{R}^{n}$. Lastly for $\ell\in\mathbb{N}$ we let
$C^{\ell,0}(\overline{\mathbb{R}^{n}})$ denote the subspace of
$C^{\ell}(\overline{\mathbb{R}^{n}})$ for which $|D^{\alpha}\phi(x)|\to 0$ as
$|x|\to\infty$ for $1\leq|\alpha|\leq\ell$, while for $\ell=0$ we let
$C^{0,0}(\overline{\mathbb{R}^{n}})$ denote the subspace of
$C^{0}(\overline{\mathbb{R}^{n}})$ consisting of those functions $\phi$ for
which $\omega_{\phi}(x)\to 0$ as $|x|\to\infty$, where $\omega_{\phi}(x)=$ sup
$|\phi(x)-\phi(y)|$ for $x,y\in\mathbb{R}^{n}$, and where for each $x$ the
supremum is taken over those values of $y$ for which $|x-y|\leq 1$. Note that
for $\ell\geq 1,C^{\ell,0}(\overline{\mathbb{R}^{n}})\subset
C^{0,0}(\overline{\mathbb{R}^{n}})$.

Turning now to the spectral problem (1.1), (1.2), let us write

$A_{jk}(x,D)=\sum_{|\alpha|\leq
s_{j}+t_{k}}a^{jk}_{\alpha}(x)D^{\alpha}\quad\textrm{for}\quad
x\in\mathbb{R}^{n}\quad\textrm{and}\quad 1\leq j,k\leq N.$

Then as pointed out in [3] there is no loss of generality in making the
following assumptions.

######  Assumption 2.1.

It will henceforth be supposed that $t_{j}>0$ and $s_{j}\geq 0$ for
$j=1,\ldots,N$.

######  Definition 2.2.

We say that the spectral problem (1.1) is minimally smooth if for each pair
$j,k$, $a_{\alpha}^{jk}\in C^{s_{j}}(\overline{\mathbb{R}^{n}})$ for
$|\alpha|\leq s_{j}+t_{k}$ if $s_{j}>0$, while if $s_{j}=0$, then
$a_{\alpha}^{jk}\in L_{\infty}(\mathbb{R}^{n})$ for $|\alpha|<t_{k}$ and
$a_{\alpha}^{jk}\in C^{0}(\overline{\mathbb{R}^{n}})$ for $|\alpha|=t_{k}$.

For $x,\xi\in\mathbb{R}^{n}$ let

$\mathring{{A}}(x,\xi)=\left(\mathring{{A}}_{jk}(x,\xi)\right)_{j,k=1}^{N},$

where $\mathring{{A}}_{jk}(x,\xi)$ consists of those terms in $A_{jk}(x,\xi)$
which are just of order $s_{j}+t_{k}$. Then in the sequel we shall also
require the following notation. For $x,\xi\in\mathbb{R}^{n}$ let

$\mathcal{A}^{(r)}_{11}(x,\xi)=\left(\mathring{{A}}_{jk}(x,\xi)\right)_{j,k=1}^{k_{r}}\quad\mathrm{for}\quad
1\leq r\leq d.$

In addition we let $\tilde{I}_{r,0}=$
diag$(0\cdot\tilde{I}_{1},\ldots,0\cdot\tilde{I}_{r-1},\tilde{I}_{r})\
\textrm{for}\ r=2,\ldots,d$ and $\tilde{I}_{1,0}=\tilde{I}_{1}$.

######  Definition 2.3.

Let $\mathcal{L}$ be a closed sector in the complex plane with vertex at the
origin. Then the spectral problem (1.1) will be called parameter-elliptic in
$\mathcal{L}$ if it is minimally smooth and for each $r,1\leq r\leq d$,

$\bigl{|}\,\textrm{det}\left(\mathcal{A}^{(r)}_{11}(x,\xi)-\lambda\,\tilde{I}_{r,0}\right)\bigr{|}\geq\kappa_{r}|\xi|^{N_{r-1}}$
(2.1)

for $x$, $\xi\in\mathbb{R}^{n}$, and $\lambda\in\mathcal{L}$ with
$\langle\,\xi,\lambda\,\rangle_{k_{r}}=1$, where the $\kappa_{r}$ denote
positive constants and $N_{r}=\sum_{j=1}^{k_{r}}m_{j}$ for $r\geq 1$ and
$N_{0}=0$.

######  Remark 2.4.

It follows from the arguments of [5, Proposition 2.2] that when (2.1) holds,
then $N_{r}$ is even if $r=1$ and if $r>1$ and $n>2$.

######  Definition 2.5.

We say that the spectral problem (1.1) is weakly smooth if it is minimally
smooth and in addition $a^{jk}_{\alpha}\in C^{0,0}(\overline{\mathbb{R}^{n}})$
if $|\alpha|\leq t_{k}$ and to $C^{|\alpha|-t_{k},0}(\overline{R^{n}})$
otherwise, $1\leq j,k\leq N$.

##  3\. Some estimates

In this section we are going to establish some a priori estimates for
solutions of the spectral problem (1.1), which will be used in the sequel.

Accordingly, let us introduce some further notation. For $G$ an open set in
$\mathbb{R}^{n}$ and $\tau=t$ or $s$ we let
$W^{(\tau)}_{p}(G)=\prod_{k=1}^{N}W^{\tau_{k}}_{p}(G)$ and equip
$W^{(\tau)}_{p}(G)$ with either its ordinary norm,
$\|\,u\,\|_{(\tau),p,G}=\sum_{k=1}^{N}\|\,u_{k}\,\|_{\tau_{k},p,G}$ or its
parameter dependent norm
$|||u|||_{(\tau),p,G}=\sum_{k=1}^{N}|||u_{k}|||^{(k)}_{\tau_{k},p,G}$ for
$u=\left(u_{1},\ldots,u_{N}\right)^{T}\in W^{(\tau)}_{p}(G)$. The subspace
$\prod_{k=1}^{N}\textrm{\r{W}}^{\tau_{k}}_{p}(G)$ is denoted by
$\textrm{\r{W}}^{(\tau)}_{p}(G)$. In addition we let
$H^{(-s)}_{p}(G)=\prod_{k=1}^{N}H^{-s_{k}}_{p}(G)$ and equip this space with
either its ordinary norm $\|\cdot\|_{(-s),p,G}$ or its parameter dependent
norm $|||\cdot|||_{(-s),p,G}$, which are defined in an analogous manner to the
way they were defined for $W^{(\tau)}_{p}(G)$.

######  Proposition 3.1.

Suppose that the spectral problem (1.1) is minimally smooth. Suppose also that
$\lambda\in\mathbb{C}$ with $|\lambda|\geq\lambda^{\sharp}$ for some
$\lambda^{\sharp}\in\mathbb{R}_{+}$, that $u\in W^{(t)}_{p}(\mathbb{R}^{n})$,
and that $f$ is defined by (1.1). Then $f\in H^{(-s)}_{p}(\mathbb{R}^{n})$ and
$|||f|||_{(-s),p,\mathbb{R}^{n}}\leq C|||u|||_{(t),p,\mathbb{R}^{n}}$, where
the constant $C$ does not depend upon $u$ and $\lambda$.

###### Proof.

Let $\langle\,\cdot,\cdot\,\rangle_{\mathbb{R}^{n}}$ denote the pairing
between $H^{(-s)}_{p}(\mathbb{R}^{n})$ and its adjoint space
$W^{(s)}_{p^{\prime}}(\mathbb{R}^{n})$, where $p^{\prime}=p/(p-1)$ and both
spaces are equipped with their parameter dependent norms (see [11, Section 1]
and [21, Theorem 2.6.1, p. 198]). Then for $u\in W^{(t)}_{p}(\mathbb{R}^{n})$
and $\zeta=\left(\zeta_{1},\ldots,\zeta_{N}\right)^{T}\in
C^{\infty}_{0}(\mathbb{R}^{n})^{N}$,

$\langle\left(A(x,D)u-\lambda\,u\right),\zeta\rangle_{\mathbb{R}^{n}}=\sum_{j=1}^{N}\left(\sum_{k=1}^{N}A_{jk}(x,D)u_{k}-\lambda\,\delta_{jk}u_{k}\right)(\overline{\zeta_{j}}),$

where $\delta_{jk}$ denotes the Kronecker delta, and for each $j$

$\left(\sum_{k=1}^{N}A_{jk}(x,D)u_{k}-\lambda\,\delta_{jk}u_{k}\right)(\overline{\zeta_{j}})$

denotes the value of the distribution
$\sum_{k=1}^{N}A_{jk}(x,D)u_{k}-\lambda\,\delta_{jk}u_{k}$ at
$\overline{\zeta_{j}}$.

Let us fix our attention upon a particular pair $j,k$. Then

$\displaystyle\left(A_{jk}(x,D)u_{k}-\lambda\,\delta_{jk}u_{k}\right)(\overline{\zeta_{j}})$
$\displaystyle=\sum_{\begin{subarray}{c}t_{k}\leq|\alpha|\leq s_{j}+t_{k}\\\
|\beta|=t_{k}\end{subarray}}\left(D^{\beta}u_{k},D^{\alpha-\beta}\overline{a^{jk}_{\alpha}(x)}\zeta_{j}\right)_{\mathbb{R}^{n}}$
$\displaystyle\quad+\left(\sum_{|\alpha|<t_{k}}a^{jk}_{\alpha}(x)D^{\alpha}u_{k},\zeta_{j}\right)_{\mathbb{R}^{n}}-\lambda\,\delta_{jk}\left(u_{k},\zeta_{j}\right)_{\mathbb{R}^{n}},$

where $\left(\cdot,\cdot\right)_{\mathbb{R}^{n}}$ denotes the pairing between
$L_{p}(\mathbb{R}^{n})$ and its adjoint space
$L_{p^{\prime}}(\mathbb{R}^{n})$. Hence

$\displaystyle\bigl{|}\left(A_{jk}(x,D)u_{k}-\lambda\,\delta_{jk}u_{k}\right)(\overline{\zeta_{j}})\bigr{|}$
$\displaystyle\leq
C\Bigl{(}\|\,u_{k}\,\|_{t_{k},p,\mathbb{R}^{n}}\|\,\zeta_{j}\,\|_{s_{j},p^{\prime},\mathbb{R}^{n}}$
$\displaystyle\quad+|\lambda|^{t_{k}/m_{k}}|\lambda|^{s_{j}/m_{j}}\|\,u_{k}\,\|_{0,p,\mathbb{R}^{n}}\|\,\zeta_{j}\,\|_{0,p^{\prime},\mathbb{R}^{n}}\Bigr{)}$
$\displaystyle\leq
C|||u_{k}|||^{(k)}_{t_{k},p,\mathbb{R}^{n}}|||\zeta_{j}|||^{(j)}_{s_{j},p^{\prime},\mathbb{R}^{n}},$

where the constant $C$ does not depend upon $u_{k},\zeta_{j}$, and $\lambda$.
It now follows from the foregoing results that

$\bigl{|}\langle\left(A(x,D)-\lambda\,I_{N}\right)u,\zeta\rangle_{\Omega}\bigr{|}\leq
C|||u|||_{(t),p,\mathbb{R}^{n}}|||\zeta|||_{(s),p^{\prime},\mathbb{R}^{n}},$

and the assertion of the proposition follows immediately from this last
result. ∎

We now turn to the main results of this section.

######  Proposition 3.2.

Suppose that the spectral problem (1.1) is parameter-elliptic in
$\mathcal{L}$. Suppose also that $u\in W^{(t)}_{p}(\mathbb{R}^{n})$ and that
$f$ is defined by (1.1). Then there exists the constant
$\lambda^{\prime}=\lambda^{\prime}(p)>0$ such that for $\lambda\in\mathcal{L}$
with $|\lambda|\geq\lambda^{\prime}$, the a priori estimate

$|||u|||_{(t),p,\mathbb{R}^{n}}\leq C|||f|||_{(-s),p,\mathbb{R}^{n}}$ (3.1)

holds, where the constant $C$ does not depend upon $u$ and $\lambda$.

######  Proposition 3.3.

Suppose that the spectral problem (1.1) is parameter-elliptic in
$\mathcal{L}$. Then there exists the constant $\lambda^{0}=\lambda^{0}(p)>0$
such that for $\lambda\in\mathcal{L}$ with $|\lambda|\geq\lambda^{0}$, the
spectral problem has a unique solution $u\in W^{(t)}_{p}(\mathbb{R}^{n})$ for
every $f\in H^{(-s)}_{p}(\mathbb{R}^{n})$, and the a priori estimate

$|||u|||_{(t),p,\mathbb{R}^{n}}\leq C|||f|||_{(-s),p,\mathbb{R}^{n}}$

holds, where the constant $C$ does not depend upon $f$ and $\lambda$.

The proofs of Propositions 3.2 and 3.3 can be achieved by modifying the
arguments given in the proofs of Theorems 4.1 and 5.1 of [5]. Furthermore,
since the proof of Proposition 3.3 is somewhat similar to that of Proposition
3.2, we will restrict ourselves to the proof of this latter proposition.

Before beginning the proof, let us now present some results which we require
below.

######  Proposition 3.4.

Suppose that the spectral problem (1.1) is parameter-elliptic in $\mathcal{L}$
and that $f\in H^{(-s)}_{p}(\mathbb{R}^{n})$. Suppose also that
$x^{0}\in\mathbb{R}^{n}$. Then there exits the constant
$\lambda_{1}=\lambda_{1}(p)>0$ such that for $\lambda\in\mathcal{L}$ with
$|\lambda|\geq\lambda_{1}$, the equation

$\textrm{\r{A}}(x^{0},D)u(x)-\lambda\,u(x)=f(x)\quad\textrm{for}\quad
x\in\mathbb{R}^{n}$

has a unique solution $u\in W^{(t)}_{p}(\mathbb{R}^{n})$ and the a priori
estimate

$|||u|||_{(t),p,\mathbb{R}^{n}}\leq C_{1}|||f|||_{(-s),p,\mathbb{R}^{n}},$

holds, where the constant $C_{1}$ does not depend upon $x^{0},f,$ and
$\lambda$.

###### Proof.

It follows from Definition 2.3 and [9] that there exists the constant
$\lambda_{0}>0$ such that for $\lambda\in\mathcal{L}$ with
$|\lambda|\geq\lambda_{0},\xi\in\mathbb{R}^{n}$, and $\alpha$ a multi-index
whose entries are either $0$ or $1$,

$\displaystyle\bigl{|}\,\textrm{det}\,\left(\textrm{\r{A}}(x^{0},\xi)-\lambda\,I_{N}\right)\bigr{|}$
$\displaystyle\geq
C_{0}\prod_{j=1}^{N}\langle\,\xi,\lambda\rangle_{j}^{m_{j}}\quad\textrm{and}$
$\displaystyle\bigl{|}\,\xi^{\alpha}D^{\alpha}_{\xi}\tilde{a}_{jk}(x^{0},\xi,\lambda)\,\bigr{|}$
$\displaystyle\leq
C_{0}^{\prime}\langle\,\xi,\lambda\,\rangle^{-s_{k}}_{k}\langle\,\xi,\lambda\rangle^{-t_{j}}_{j},$

where the constants $C_{0}$ and $C_{0}^{\prime}$ do not depend upon
$\alpha,x^{0},\xi,\lambda,j,k$, and where we have written
$\left(\textrm{\r{A}}(x^{0},\xi)-\lambda\,I_{N}\right)^{-1}=\left(\tilde{a}_{jk}(x^{0},\xi,\lambda\right)_{j,k=1}^{N}$.
The assertions of the proposition now follow from the same arguments as those
used in the proof of [8, Proposition 3.2]. ∎

######  Proposition 3.5.

Suppose that the spectral problem (1.1) is parameter-elliptic in
$\mathcal{L}$. Then there exist the constants $r_{0}=r_{0}(p)$ and
$\lambda_{2}=\lambda_{2}(p)$ in $\mathbb{R}_{+}$ such that for each
$x^{0}\in\mathbb{R}^{n}$ one can find a neighbourhood $V$ of this point with
diam $V\leq r_{0}$ for which the a priori estimate

$|||u|||_{(t),p,\mathbb{R}^{n}}\leq
C|||A(x,D)u-\lambda\,u|||_{(-s),p,\mathbb{R}^{n}}$

holds for each $u\in W^{(t)}_{p}(\mathbb{R}^{n})$ and $\lambda\in\mathcal{L}$
for which supp $u\subset V$ and $|\lambda|\geq\lambda_{2}$, where the constant
$C$ does not depend upon $u$ and $\lambda$, and where diam denotes diameter
and supp denotes support.

###### Proof.

To begin with let us choose $r_{0}$ so that for $x,y\in\mathbb{R}^{n}$ with
$|x-y|\leq r_{0}$ we have for each triple $(j,k,\alpha)$ with $1\leq j,k\leq
N$ and $|\alpha|=s_{j}+t_{k},\,|a^{jk}_{\alpha}(x)-a^{jk}_{\alpha}(y)|\leq
1/8C_{1}\delta_{1}$, where $C_{1}$ denotes the constant of Proposition 3.4,
$\delta_{1}=N^{2}$ max
$\bigl{\\{}\frac{(s_{j}+t_{k})!}{s_{j}!}n_{jk}\,\bigr{\\}}_{j,k=1}^{N}$ and
$n_{jk}$ denotes the number of distinct multi-indices $\alpha$ for which
$|\alpha|=s_{j}+t_{k}$.

Next let $x^{0}\in\mathbb{R}^{n}$ and let $V\subset\mathbb{R}^{n}$ be a
neighbourhood of $x^{0}$ with diam $V\leq r_{0}$. Also let $u\in
W^{(t)}_{p}(\mathbb{R}^{n})$ such that supp $u\subset V$. Then bearing in mind
the proof of Proposition 3.1 as well as Proposition 3.4, we see that

$|||u|||_{(t),p,V}\leq
C_{1}|||\left(\textrm{\r{A}}(x^{0},D)-\lambda\,I_{N}\right)u|||_{(-s),p,\mathbb{R}^{n}}$

for $\lambda$ and $C_{1}$ satisfying the conditions cited in Proposition 3.4.
Hence it follows from arguments similar to those used in the proof of [8,
Proposition 4.1] that there exists the constant $\lambda_{2}=\lambda_{2}(p)>0$
such that

$|||u|||_{(t),p,V}\leq
C_{1}^{\prime}|||\left(A(x,D)-\lambda\,I_{N}\right)u|||_{(-s),p,\mathbb{R}^{n}}\quad\textrm{for}\quad\lambda\in\mathcal{L}\quad\textrm{with}\quad|\lambda|\geq\lambda_{2},$

where the constant $C_{1}^{\prime}$ has the same properties as the constant
$C_{1}$. On the other hand, if we refer to the proof of Proposition 3.1 for
notation and argue as in that proof, then we obtain for $\zeta\in
C^{\infty}_{0}(\mathbb{R}^{n})^{N}$ and for values of $\lambda$ just cited,

$\langle\,\left(A(x,D)-\lambda\,I_{N}\right)u,\zeta\,\rangle_{\mathbb{R}^{n}}=\langle\,\left(A(x,D)-\lambda,I_{N}\right)u,\chi\,\zeta\rangle_{V},$

where $\chi\in C^{\infty}_{0}(\mathbb{R}^{n})$ such that $\chi(x)=1$ for
$x\in$ supp $u$ and supp $\chi\subset V$. Hence if we take into account [7,
Proposition 3.5, p. 109], [21, Theorems 4.3.2.1, p. 317 and 4.8.2, p. 332],
and the fact that $\chi\,\zeta\in\textrm{\r{W}}^{(s)}_{p^{\prime}}(V)$, it
follows that

$\displaystyle\bigl{|}\langle\,\left(A(x,D)-\lambda\,I_{N}\right)u,\zeta\,\rangle_{\mathbb{R}^{n}}\bigr{|}$
$\displaystyle\leq|||\left(A(x,D)-\lambda\,I_{N}\right)u|||_{(-s),p,V}|||\chi\,\zeta|||_{(s),p^{\prime},V}$
$\displaystyle\leq
C_{0}|||\left(A(x,D)-\lambda\,I_{N}\right)u|||_{(-s),p,V}|||\zeta||_{(s),p^{\prime},\mathbb{R}^{n}},$

where the constant $C_{0}$ does not depend upon $u$ and $\lambda$. This
completes the proof of the Proposition. ∎

Proof of Proposition 3.2. For $t>0$ let $Q(t)$ denote the open cube in
$\mathbb{R}^{n}$ with centre at the origin and with sides parallel to the
coordinate axes and of length $2t$. Also let $\zeta,\chi\in
C^{\infty}_{0}(\mathbb{R}^{n})$ such that $\zeta(x)=1$ for $x\in Q(1/2)$ and
$\zeta(x)=0$ for $x\in\mathbb{R}^{n}\setminus Q(3/4)$, while $\chi(x)=1$ for
$x\in Q(13/16)$ and $\chi(x)=0$ for $x\in\mathbb{R}^{n}\setminus Q(14/16)$.
Then for $d>0$ and $\gamma\in\mathbb{Z}^{n}$, we let
$Q_{\gamma,d}=\bigl{\\{}\,x\in\mathbb{R}^{n}\,\bigl{|}\,|x-d\gamma|\in
Q(d)\,\bigr{\\}}$, $\zeta_{\gamma,d}(x)=\zeta(\left(x-d\gamma)/d\right)$, and
$\chi_{\gamma,d}(x)=\chi\left((x-d\gamma)/d\right)$.

Next let $d_{0}$ denote a constant satisfying $0<d_{0}<r_{0}/4n^{1/2}$ (see
Proposition 3.5). Then we shall henceforth fix $d\leq d_{0}$ and let
$\\{\,\gamma_{j}\,\\}_{1}^{\infty}$ denote an enumeration of the members of
$\mathbb{Z}^{n}$ and put
$Q_{j}=Q_{\gamma_{j},d},\zeta_{j}(x)=\zeta_{\gamma_{j},d}(x)$ and
$\chi_{j}(x)=\chi_{\gamma_{j},d}(x)$. Hence if for $j\geq 1$ we let
$\eta_{j}(x)=\zeta_{j}(x)/\sum_{j=1}^{\infty}\zeta_{j}(x)$, then
$\\{\,Q_{j}\,\\}_{1}^{\infty}$ is an open covering of $\mathbb{R}^{n}$ and
$\bigl{\\{}\,\eta_{j}(x)\,\bigr{\\}}_{1}^{\infty}$ a partition of unity
subordinate to this covering. Note also that for any $x\in\mathbb{R}^{n}$, $x$
can lie in at most $2^{n}$ of the $Q_{j}$.

Bearing in mind Proposition 3.5 let $u\in W^{(t)}_{p}(\mathbb{R}^{n})$ and
$\lambda\in\mathcal{L}$ with $|\lambda|\geq\lambda_{2}$. Then we have

$|||u|||_{(t),p,\mathbb{R}^{n}}\leq
N^{1/p^{\prime}}\left(\sum_{j=1}^{N}|||\sum_{\ell\geq
1}\eta_{\ell}u_{j}|||_{t_{j},p,\mathbb{R}^{n}}^{(j)\,p}\right)^{1/p},$

which leads to the inequality

$|||u|||_{(t),p,\mathbb{R}^{n}}\leq 2^{n+1}N\left(\sum_{\ell\geq
1}|||\eta_{\ell}u|||^{p}_{(t),p,Q_{\ell}}\right)^{1/p},$

and hence it follows from Proposition 3.5 and its proof that

$|||u|||_{(t),p,\mathbb{R}^{n}}\leq C\left(\sum_{\ell\geq
1}|||\bigl{(}A(x,D)-\lambda\,I_{N}\bigr{)}\eta_{\ell}u|||_{(-s),p,Q_{\ell}}^{p}\right)^{1/p},$
(3.2)

where here and for the remainder of this proof $C$ denotes a generic constant
which may vary from inequality to inequality, but in all cases it does not
depend upon $u,\lambda$ and $\ell$ (see below). Note that here we have used
the fact that under our present definitions of $V$ and $\chi$, we can take the
constant $C$ of Proposition 3.5 to be also independent of $x^{0}$.

Let us firstly fix our attention upon a particular $\ell\geq 1$. Then
observing that

$\displaystyle|||\left(A(x,D)-\lambda\,I_{N}\right)\eta_{\ell}u-\eta_{\ell}\left(A(x,D)-\lambda\,I_{N}\right)u|||_{(-s),p,Q_{\ell}}$
$\displaystyle\qquad\leq|||\left(A(x,D)-\lambda\,I_{N}\right)\eta_{\ell}u-\eta_{\ell}\left(A(x,D)-\lambda\,I_{N}\right)u|||_{(-s),p,\mathbb{R}^{n}},$

and referring to the proof of Proposition 3.1 for notation, we can argue as in
that proof to show that

$\displaystyle\bigl{|}\langle\,\left(\bigl{(}A(x,D)-\lambda\,I_{N}\bigr{)}\eta_{\ell}u-\eta_{\ell}\bigl{(}A(x,D)-\lambda\,I_{N}\bigr{)}u\right),\zeta\rangle_{\mathbb{R}^{n}}\bigr{|}$
$\displaystyle\qquad\leq
C\sum_{j=1}^{N}\sum_{k=1}^{N}\left(\|\,\chi_{\ell}u_{k}\,\|_{t_{k}-1,p,Q_{\ell}}\|\,\chi_{\ell}\zeta_{j}\,\|_{s_{j},p^{\prime},Q_{\ell}}+\|\,\chi_{\ell}u_{k}\,\|_{t_{k},p,Q_{\ell}}\|\,\chi_{\ell}\zeta_{j}\|_{s_{j}-1,p^{\prime},Q_{\ell}}\right).$

Hence it follows from [4, Proposition 2.2] and the fact that $\chi_{\ell}$ has
support in $Q_{\ell}$ that

$\displaystyle\bigl{|}\langle\,\left(\bigl{(}A(x,D)-\lambda\,I_{N}\bigr{)}\eta_{\ell}u-\eta_{\ell}\left(A(x,D)-\lambda\,I_{N}\right)u\right),\zeta\,\rangle_{\mathbb{R}^{n}}\bigr{|}$
(3.3) $\displaystyle\qquad\leq
C|\lambda|^{-1/m_{1}}|||u|||_{(t),p,Q_{\ell}}|||\zeta|||_{(s),p^{\prime},Q_{\ell}}.$

Thus we conclude from (3.3) that

$\displaystyle|||\left(A(x,D)-\lambda\,I_{N}\right)\eta_{\ell}u|||_{(-s),p,\mathbb{R}^{n}}\leq$
$\displaystyle\qquad|||\eta_{\ell}\left(A(x,D)-\lambda\,I_{N}\right)u|||_{(-s),p,\mathbb{R}^{n}}+C|\lambda|^{-1/m_{1}}|||u|||_{(t),p,Q_{\ell}},$

and hence that

$\displaystyle\sum_{\ell\geq
1}||\left(A(x,D)-\lambda\,I_{N}\right)\eta_{\ell}u|||_{(-s),p,\mathbb{R}^{n}}^{p}$
(3.4) $\displaystyle\qquad\leq 2^{p}\sum_{\ell\geq
1}|||\eta_{\ell}\left(A(x,D)-\lambda\,I_{N}\right)u|||_{(-s),p,\mathbb{R}^{n}}^{p}+2^{p}C^{p}|\lambda|^{-p/m_{1}}\sum_{\ell\geq
1}|||u|||_{(t),p,Q_{\ell}}^{p}.$

We are now going to use (3.2) and (3.4) to complete the proof of the
proposition. To this end, some further preparation is required. Accordingly,
let us recall that we have so far equipped the space
$W^{s_{j}}_{p^{\prime}}(\mathbb{R}^{n}),1\leq j\leq N$, with the parameter
dependent norm $|||\cdot|||_{s_{j},p^{\prime},\mathbb{R}^{n}}$. For our
purposes it will be convenient now to introduce an equivalent norm, namely the
norm $|||\cdot|||_{s_{j},p^{\prime},\mathbb{R}^{n}}^{\prime}$ defined by

$|||v|||^{\prime}_{s_{j},p^{\prime},\mathbb{R}^{n}}=\\\
\left(\|\,v\,\|^{p^{\prime}}_{s_{j},p^{\prime},\mathbb{}{R}^{n}}+\bigl{[}(1+|\lambda|^{s_{j}/m_{j}})^{p^{\prime}}-1\bigr{]}\|\,v\,\|^{p^{\prime}}_{0,p^{\prime},\mathbb{R}^{n}}\right)^{1/p^{\prime}}$

for $v\in W^{s_{j}}_{p^{\prime}}(\mathbb{R}^{n})$. Note that by the
equivalence of these two norms we mean that there are constants $C_{1}$ and
$C_{2}$, not depending upon $v$ and $\lambda$, such that
$|||v|||_{s_{j},p^{\prime},\mathbb{R}^{n}}\leq
C_{1}|||v|||_{s_{j},p^{\prime},\mathbb{R}^{n}}^{\prime}$ and
$|||v|||^{\prime}_{s_{j},p^{\prime},\mathbb{R}^{n}}\leq
C_{2}|||v|||_{s_{j},p^{\prime},\mathbb{R}^{n}}$.

Supposing now that $\lambda$ is fixed and
$W^{s_{j}}_{p^{\prime}}(\mathbb{R}^{n})$ is equipped with the norm
$|||\cdot|||^{\prime}_{s_{j},p^{\prime},\mathbb{R}^{n}}$, let $\nu_{j}$ denote
the number of distinct multi-indices $\alpha$ satisfying $0\leq|\alpha|\leq
s_{j}$. Then to each $v\in W^{s_{j}}_{p^{\prime}}(\mathbb{R}^{n})$ we can
associate the vector $\mathcal{P}_{j,p^{\prime}}v\in
L_{p^{\prime}}(\Omega)^{\nu_{j}}$, where $\mathcal{P}_{j,p^{\prime}}v$ denotes
the $\nu_{j}$ – vector with components
$\bigl{\\{}\,\Lambda_{j,\alpha}D^{\alpha}\,\bigr{\\}}_{|\alpha|\leq s_{j}}$,
where $\Lambda_{j,\alpha}=1$ if $|\alpha|\geq 1$ and equals
$1+|\lambda|^{s_{j}/m_{j}}$ otherwise, and where the components of
$\mathcal{P}_{j,p^{\prime}}v$ are arranged so that if
$\Lambda_{j,\alpha}D^{\alpha}v$ is the $\ell$-th component of this vector and
$\Lambda_{j,\beta}D^{\beta}v$ the $(\ell+1)$-th component, then
$|\alpha|\leq|\beta|$. Hence if we let $\mathcal{H}_{j,p^{\prime}}$ denote the
closed subspace of $L_{p^{\prime}}(\Omega)^{\nu_{j}}$ spanned by the vectors
$\mathcal{P}_{j,p^{\prime}}v$ for $v\in
W^{s_{j}}_{p^{\prime}}(\mathbb{R}^{n})$, then $\mathcal{H}_{j,p^{\prime}}$ is
isometrically isomorphic to $W^{s_{j}}_{p^{\prime}}(\mathbb{R}^{n})$.

Turning next to the space $W^{(s)}_{p^{\prime}}(\mathbb{R}^{n})$, we have so
far equipped this space with the parameter dependent norm
$|||\cdot|||_{(s),p^{\prime},\mathbb{R}^{n}}$. Let us also equip
$W^{(s)}_{p^{\prime}}(\mathbb{R}^{n})$ with the norm
$|||\cdot|||_{(s),p^{\prime},\mathbb{R}^{n}}^{\prime}$, which is equivalent to
$|||\cdot|||_{(s),p^{\prime},\mathbb{R}^{n}}$, defined by
$|||v|||^{\prime\,p^{\prime}}_{(s),p^{\prime}\,\mathbb{R}^{n}}=$
$\sum_{j=1}^{N}\left(|||v_{j}|||_{s_{j},p^{\prime},\mathbb{R}^{n}}^{\prime}\right)^{p^{\prime}}$
for $v=(v_{1},\ldots,v_{N})^{T}\in W^{(s)}_{p^{\prime}}(\mathbb{R}^{n})$. Note
that the norms $|||\cdot|||_{(s),p^{\prime},\mathbb{R}^{n}}$ and
$|||\cdot|||^{\prime}_{(s),p^{\prime}\mathbb{R}^{n}}$ induce equivalent norms
on the adjoint space of $W^{(s)}_{p^{\prime}}(\mathbb{R}^{n})$,
$H^{(-s)}_{p}(\mathbb{R}^{n})$, and hence if we denote by
$|||\cdot|||^{\prime}_{(-s),p,\mathbb{R}^{n}}$ the norm induced on
$H^{(-s)}_{p}(\mathbb{R}^{n})$ by the norm
$|||\cdot|||^{\prime}_{(s),p^{\prime},\mathbb{R}^{n}}$, then
$|||\cdot|||^{\prime}_{(-s),p,\mathbb{R}^{n}}$ and
$|||\cdot|||_{(-s),p,\mathbb{R}^{n}}$ are equivalent norms on this space. Note
also that when $W^{(s)}_{p^{\prime}}(\mathbb{R}^{n})$ is equipped with the
norm $|||\cdot|||^{\prime}_{(s),p^{\prime},\mathbb{R}^{n}}$, then
$W^{(s)}_{p^{\prime}}(\mathbb{R}^{n})$ is isometrically isomorphic to
$\mathcal{H}_{p^{\prime}}$, the closed subspace of
$L_{p^{\prime}}(\mathbb{R}^{n})^{\sum_{j=1}^{N}\nu_{j}}$ defined by
$\mathcal{H}_{p^{\prime}}=\prod_{j=1}^{N}\mathcal{H}_{j,p^{\prime}}$. Hence it
follows from [1, Theorem 3.8, p. 49] that for each $u\in
H^{(-s)}_{p}(\mathbb{R}^{n})$, there is a $f\in
L_{p}(\mathbb{R}^{n})^{\sum_{j=1}^{N}\nu_{j}}$ such that
$|||u|||^{\prime}_{(-s),p,\mathbb{R}^{n}}=\|\,f\,\|$, where $\|\cdot\|$
denotes the norm in $L_{p}(\mathbb{R}^{n})^{\sum_{j=1}^{N}\nu_{j}}$.

Fixing our attention upon (3.4) and referring to the proof of Proposition 3.1
for notation, we have for $\ell\geq 1$,

$\displaystyle|||\eta_{\ell}\left(A(x,D)-\lambda\,I_{N}\right)u|||_{(-s),p,\mathbb{R}^{n}}$
$\displaystyle\leq
C|||\eta_{\ell}\left(A(x,D)-\lambda\,I_{N}\right)u|||^{\prime}_{(-s),p,\mathbb{R}^{n}}$
$\displaystyle=C\,\sup\,\bigl{|}\langle\,\left(A(x,D)-\lambda\,I_{N}\right)u,\eta_{\ell}\zeta\,\rangle_{\mathbb{R}^{n}}\bigr{|}$
$\displaystyle=C\sup\,\bigl{|}\sum_{j=1}^{N}\sum_{k=1}^{\nu_{j}}\left(f_{j,k},\Lambda_{j,\alpha}D^{\alpha}\eta_{\ell}\zeta_{j}\right)_{\mathbb{R}^{n}}\bigr{|}$

for some $f\in L_{p}(\mathbb{R}^{n})^{\sum_{j=1}^{N}\nu_{j}}$, where the
supremum is taken over the set $\bigl{\\{}\,\zeta\in
C^{\infty}_{0}(\mathbb{R}^{n})^{N}\,\bigl{|}\,\break|||\zeta|||^{\prime}_{0,p,\mathbb{R}^{n}}=1\,\bigr{\\}}$,
and where we have written $f=(f_{1},\ldots,f_{N})^{T}$,
$f_{j}=(f_{j,1},\ldots,f_{j,\nu_{j}})^{T}$, and
$\Lambda_{j,\alpha}D^{\alpha}\eta_{\ell}\zeta_{j}$ denotes the $k$-th
component of $\mathcal{P}_{j,p^{\prime}}\eta_{\ell}\zeta_{j}$. Hence it
follows that

$\displaystyle\sum_{\ell\geq
1}|||\eta_{\ell}\left(A(x,D)-\lambda\,I_{N}\right)u|||^{p}_{(-s),p,\mathbb{R}^{n}}$
$\displaystyle\leq C^{p}\sum_{\ell\geq
1}\sum_{j=1}^{N}\sum_{k=1}^{\nu_{j}}\|\,f_{j,k}\,\|^{p}_{0,p,Q_{\ell}}$ (3.5)
$\displaystyle\leq
C^{\prime}|||\left(A(x,D)-\lambda\,I_{N}\right)|||^{p}_{(-s),p,\mathbb{R}^{n}},$

where the constant $C^{\prime}$ has the same properties as the constant $C$.
Hence it follows from equations (3.2), (3.4), and (3.5) that we may choose
$\lambda^{\prime}(p)$ so that the assertion of the proposition holds.
$\square$

To terminate the work of this section, we now present a result which will be
used in the sequel. Here, for $\ell\in\mathbb{N}$, we use the notation
$(t+\ell)$ to denote the multi-index whose entries are
$t_{j}+\ell,j=1,\ldots,N$. The multi-indices $(-s+\ell),(-r_{j}+\ell-1/p)$ are
defined analogously.

######  Proposition 3.6.

Suppose that the spectral problem (1.1) is parameter-elliptic in $\mathcal{L}$
and that $\ell_{0}\in\mathbb{N}$. Suppose also that for each pair $j,k,1\leq
j,k\leq N$, $a^{jk}_{\alpha}\in C^{s_{j}+\ell_{0}}(\overline{\mathbb{R}^{n}})$
for $|\alpha|\leq s_{j}+t_{k}$. Lastly suppose that $u\in
W^{(t)}_{p}(\mathbb{R}^{n}),\lambda\in\mathcal{L}\setminus\\{\,0\,\\}$, and
that $f$ is defined by (1.1). Consequently, if $f\in
H^{(-s+\ell_{0})}_{p}(\mathbb{R}^{n})$, then $u\in
W^{(t+\ell_{0})}_{p}(\mathbb{R}^{n})$.

###### Proof.

Referring to the proof of Proposition 3.2 for terminology, it follows from
(3.1) and from arguments similar to those used in the proof of Proposition 3.1
that for $\ell\geq 1$,

$\displaystyle\|\,\eta_{\ell}u\,\|_{(t),p,Q_{\ell}}$ $\displaystyle\leq
C\bigl{(}\|\,\bigl{(}A(x,D)-\lambda\,I_{N}\bigr{)}\eta_{\ell}u\,\|_{(-s),p,Q_{\ell}}$
(3.6) $\displaystyle\quad+\|\,\eta_{\ell}u\,\|_{(t-1),p,Q_{\ell}}\bigr{)},$

where here and for the remainder of this proof, $C$ denotes a generic constant
which may vary from inequality to inequality, but in all cases it does not
depend upon $\ell,u$, and $\lambda$. By employing arguments similar to those
used in the proof of Proposition 3.1, it is not difficult to verify that
$\left(A(x,D)-\lambda\,I_{N}\right)\eta_{\ell}u\in H^{(-s+1)}_{p}(Q_{\ell})$.
Hence for $\ell\geq 1$ we can apply the differential quotient method to (3.6)
as in [14, Proof of Theorem 3.1, p. 123] (see also [2]) to deduce that
$\eta_{\ell}u\in W^{(t+1)}_{p}(Q_{\ell})$ and

$\|\,\eta_{\ell}u\,\|_{(t+1),p,Q_{\ell}}\leq
C\left(\|\,\bigl{(}A(x,D)-\lambda\,I_{N}\bigr{)}\eta_{\ell}u\,\|_{(-s+1),p,Q_{\ell}}+\|\,\eta_{\ell}u\,\|_{(t),p,Q_{\ell}}\right).$
(3.7)

We conclude from the foregoing results and from arguments similar to those
used in the proof of Proposition 3.2 that $u\in W^{(t+1)}_{p}(B(r))$ for every
$r>0$ where $B(r)=\bigl{\\{}x\in\mathbb{R}^{n}\,\bigl{|}|x|<r\,\bigr{\\}}$.
Let us now show that $u\in W^{(t+1)}_{p}(\mathbb{R}^{n})$ and that

$\displaystyle\|\,u\,\|_{(t+1),p,\mathbb{R}^{n}}$ $\displaystyle\leq
C\bigl{(}\|\,\bigl{(}A(x,D)-\lambda\,I_{N}\bigr{)}u\,\|_{(-s+1),p,\mathbb{R}^{n}}$
(3.8) $\displaystyle\quad+\|\,u\,\|_{(t),p,\mathbb{R}^{n}}\bigr{)}.$

Accordingly, it is clear that

$\|\,u\,\|_{(t+1),p,\mathbb{R}^{n}}\leq
N^{1/p^{\prime}}\left(\sum_{j=1}^{N}\|\,\sum_{\ell\geq
1}\eta_{\ell}u_{j}\,\|^{p}_{t_{j}+1,p,\mathbb{R}^{n}}\right)^{1/p},$

which leads to the inequality

$\|\,u\,\|_{(t+1),p,\mathbb{R}^{n}}\leq 2^{n+1}N\left(\sum_{\ell\geq
1}\|\,\eta_{\ell}u_{j}\,\|^{p}_{(t+1),p,Q_{\ell}}\right)^{1/p}.$ (3.9)

Furthermore, by appealing to (3.7) and by arguing in a manner similar to that
in the proof of Proposition 3.2, we can also show that

$\displaystyle\sum_{\ell\geq 1}\|\,\eta_{\ell}u\,\|^{p}_{(t+1),p,Q_{\ell}}$
$\displaystyle\leq C\sum_{\ell\geq
1}\left(\|\,\eta_{\ell}\bigl{(}A(x,D)-\lambda\,I_{N}\bigr{)}u\,\|^{p}_{(-s+1),p,Q_{\ell}}\right.$
(3.10) $\displaystyle\quad+\left.\|\,u\,\|^{p}_{(t),p,Q_{\ell}}\right).$

When $s_{N}\geq 1$, then arguments similar to those used in the proof of
Proposition 3.2 show that the expression on the right side of (3.10) is
majorized by

$C^{\prime}\left(\|\,\bigl{(}A(x,D)-\lambda\,I_{N}\bigr{)}u\,\|^{p}_{(-s+1),p,\mathbb{R}^{n}}+\|\,u\,\|^{p}_{(t),p,\mathbb{R}^{n}}\right),$

where the constant $C^{\prime}$ has the same properties as $C$, and hence it
follows from (3.7), (3.9), and (3.10) that $u\in
W^{(t+1)}_{p}(\mathbb{R}^{n})$ and that the inequality (3.8) holds.

Suppose next that for some $r,1<r\leq d$, $s_{j}=0$ for $j>k_{r-1}$ and
$s_{j}>0$ for $j\leq k_{r-1}$. Then

$H^{(-s+1)}_{p}(\mathbb{R}^{n})=H^{(-s+1)_{1}}_{p}(\mathbb{R}^{n})\times
W^{1}_{p}(\mathbb{R}^{n})^{N-k_{r-1}},$

where
$H^{(-s+1)_{1}}_{p}(\mathbb{R}^{n})=\prod_{j=1}^{k_{r-1}}H^{-s_{j}+1}_{p}(\mathbb{R}^{n})$,
and where now
$H^{(-s+1)}_{p}(\mathbb{R}^{n}),H^{(-s+1)_{1}}_{p}(\mathbb{}{R}^{n})$ and its
adjoint space
$W^{(s-1)_{1}}_{p^{\prime}}(\mathbb{R}^{n})=\prod_{j=1}^{k_{r-1}}W^{s_{j}-1}_{p^{\prime}}(\mathbb{R}^{n})$,
and $W^{1}_{p}(\mathbb{R}^{n})^{N-k_{r-1}}$ are equipped with their ordinary
norms,
$\|\,\cdot\,\|_{(-s+1),p,\mathbb{R}^{n}},\|\,\cdot\,\|_{(-s+1)_{1},p,\mathbb{R}^{n}},\|\,\cdot\,\|_{(s-1)_{1},p^{\prime},\mathbb{R}^{n}}$,
and $\|\,\cdot\,\|_{(1)_{1},p,\mathbb{R}^{n}}$, respectively, which are
defined in a manner analogous to the way $W^{(\tau)}_{p}(G)$ and
$H^{(-s)}_{p}(G)$ were defined at the beginning of this section. For our
purposes it will be convenient to impose equivalent norms on
$W^{(s-1)_{1}}_{p^{\prime}}(\mathbb{R}^{n})$ and
$W^{1}_{p}(\mathbb{R}^{n})^{N-k_{r-1}}$ namely
$\|\,\cdot\,\|^{\prime}_{(s-1)_{1},p^{\prime},\mathbb{R}^{n}}$ and
$\|\,\cdot\,\|^{\prime}_{(1)_{1},p,\mathbb{R}^{n}}$, respectively, where

$\|\,u\,\|^{\prime}_{(s-1)_{1},p^{\prime},\mathbb{R}^{n}}=\left(\sum_{j=1}^{k_{r-1}}\|\,u_{j}\,\|^{p}_{s_{j}-1,p^{\prime},\mathbb{R}^{n}}\right)^{1/p}$

for $u\in W^{(s-1)_{1}}_{p^{\prime}}(\mathbb{R}^{n})$ and

$\|\,u\,\|^{\prime}_{(1)_{1},p,\mathbb{R}^{n}}=\left(\sum_{j=k_{r-1}+1}^{N}\|\,u_{j}\,\|^{p}_{1,p,\mathbb{R}^{n}}\right)^{1/p}$

for $u\in W^{1}_{p}(\mathbb{R}^{n})^{N-k_{r-1}}$. Note that the norms
$\|\,\cdot\,\|_{(s-1)_{1},p^{\prime},\mathbb{R}^{n}}$ and
$\|\,\cdot\,\|^{\prime}_{(s-1)_{1},p^{\prime},\mathbb{R}^{n}}$ induce
equivalent norms on the adjoint space of
$W^{(s-1)_{1}}_{p^{\prime}}(\mathbb{R}^{n})$,
$H^{(-s+1)_{1}}_{p}(\mathbb{R}^{n})$, and hence if we denote by
$\|\,\cdot\,\|^{\prime}_{(-s+1)_{1},p,\mathbb{R}^{n}}$ the norm induced on
$H^{(-s+1)_{1}}_{p}(\mathbb{R}^{n})$ by the norm
$\|\,\cdot\,\|^{\prime}_{(s-1)_{1},p^{\prime},\mathbb{R}^{n}}$, then
$\|\,\cdot\,\|^{\prime}_{(-s+1)_{1},p,\mathbb{R}^{n}}$ and
$\|\,\cdot\,\|_{(-s+1)_{1},p,\mathbb{R}^{n}}$ are equivalent norms on this
space.

Let us now fix our attention upon $W^{(s-1)_{1}}_{p^{\prime}}(\mathbb{R}^{n})$
and for $1\leq j\leq k_{r-1}$ let $\tilde{\nu}_{j}$ denote the number of
distinct multi-indices $\alpha$ for which $|\alpha|\leq s_{j}-1$. Then to each
$v=(v_{1}\ldots,v_{k_{r-1}})^{T}\in
W^{(s-1)_{1}}_{p^{\prime}}(\mathbb{R}^{n})$ we can associate the vector

$\mathcal{P}^{(1)}_{p^{\prime}}v=\left(\mathcal{P}^{(1)}_{1,p^{\prime}}v_{1},\ldots,\mathcal{P}^{(1)}_{k_{r-1},p^{\prime}}v_{k_{r-1}}\right)\in
L_{p^{\prime}}(\Omega)^{\sum_{j=1}^{k_{r-1}}\tilde{\nu}_{j}},$

where for $1\leq j\leq k_{r-1},\mathcal{P}^{(1)}_{j,p^{\prime}}v$ denotes the
$\tilde{\nu}_{j}$-th vector with components
$\\{\,D^{\alpha}v_{j}\,\\}_{|\alpha|\leq s_{j}-1}$, and where the components
of $\mathcal{P}^{(1)}_{j,p^{\prime}}$ are arranged in the same way as the
entries of $\mathcal{P}_{j,p^{\prime}}$ were arranged in the proof of
Proposition 3.2. Hence if we let $\mathcal{H}^{(1)}_{p^{\prime}}$ denote the
closed subspace of
$L_{p^{\prime}}^{\sum_{j=1}^{k_{r-1}}\tilde{\nu}_{j}}(\mathbb{R}^{n})$ spanned
by the vectors $\mathcal{P}^{(1)}_{p^{\prime}}v$ for $v\in
W^{(s-1)_{1}}_{p^{\prime}}(\mathbb{R}^{n})$ and if we equip
$W^{(s-1)_{1}}_{p^{\prime}}(\mathbb{R}^{n})$ with the norm
$\|\,\cdot\,\|_{(s-1)_{1},p^{\prime},\mathbb{R}^{n}}^{\prime}$, then
$\mathcal{H}^{(1)}_{p^{\prime}}$ is isometrically isomorphic to
$W^{(s-1)_{1}}_{p^{\prime}}(\mathbb{R}^{n})$. Hence it follows, as in the
proof of Proposition 3.2, that for each
$u\in\mathcal{H}^{(-s+1)_{1}}_{p}(\mathbb{R}^{n})$ there is an
$f^{(1)}=\left(f^{(1)}_{1},\ldots,f^{(1)}_{k_{r-1}}\right)\in
L_{p}(\mathbb{R}^{n})^{\sum_{j=1}^{k_{r-1}}\tilde{\nu}_{j}}$ where
$f^{(1)}_{j}=\left(f^{(1)}_{j,1},\ldots,f^{(1)}_{j,\tilde{\nu_{j}}}\right)$
such that

$\|\,u\,\|^{\prime}_{(-s+1)_{1},p,\mathbb{R}^{n}}=\|\,f^{(1)}\,\|_{\mathbb{R}^{n}},$
(3.11)

where $\|\,\cdot\,\|_{\mathbb{R}^{n}}$ denotes the norm in
$L_{p}(\mathbb{R}^{n})^{\sum_{j=1}^{k_{r-1}}\tilde{\nu}_{j}}$. Consequently if
we fix our attention again upon (3.10) and let $P^{(1)}_{p}$ denote the
operator projecting $H^{(-s+1)}_{p}(\mathbb{R}^{n})$ onto
$H^{(-s+1)_{1}}_{p}(\mathbb{R}^{n})$ along
$W^{1}_{p}(\mathbb{R}^{n})^{N-k_{r-1}}$, then there is an $f^{(1)}\in
L_{p}(\mathbb{R}^{n})^{\sum_{j=1}^{k_{r-1}}\tilde{\nu}_{j}}$ satisfying (3.11)
with $u$ there replaced by $P^{(1)}_{p}\bigl{(}A(x,D)-\lambda\,I_{N}\bigr{)}u$
such that for $\ell\geq 1$ we have

$\displaystyle\|\,P^{(1)}_{p}\eta_{\ell}\left(A(x,D)-\lambda\,I_{N}\right)u$
$\displaystyle\|_{(-s+1)_{1},p,Q_{\ell}}\leq\|\,P^{(1)}_{p}\eta_{\ell}\left(A(x,D)-\lambda\,I_{N}\right)u\,\|_{(-s+1)_{1},p,\mathbb{R}^{n}}$
$\displaystyle\leq
C\|\,P^{(1)}_{p}\eta_{\ell}\left(A(x,D)-\lambda\,I_{N}\right)u\,\|^{\prime}_{(-s+1)_{1},p,\mathbb{R}^{n}}$
$\displaystyle=C\,\textrm{sup}\,\bigl{|}\langle\,P^{(1)}_{p}\eta_{\ell}\left(A(x,D)-\lambda\,I_{N}\right)u,\zeta\,\rangle_{\mathbb{R}^{n}}\bigr{|}$
$\displaystyle=C\,\textrm{sup}\,\bigl{|}\sum_{j=1}^{k_{r-1}}\sum_{k=1}^{\tilde{\nu}_{j}}\left(f^{(1)}_{j,k},D^{\alpha}\eta_{\ell}\zeta_{j}\right)_{\mathbb{R}^{n}}\bigr{|}\leq
C_{1}\|\,f^{(1)}\,\|_{Q_{\ell}},$

where in each case the supremum is over the set $\bigl{\\{}\,\zeta\in
C^{\infty}_{0}(\mathbb{R}^{n})^{k_{r-1}}\,\bigl{|}\,\|\,\zeta\,\|_{(s-1)_{1},p^{\prime},\mathbb{R}^{n}}=1\,\bigr{\\}}$,
the constant $C_{1}$ has the same properties as the constant $C$, $f^{(1)}\in
L_{p}(\mathbb{R}^{n})^{\sum_{j=1}^{k_{r-1}}\tilde{\nu}_{j}}$ denotes the norm
preserving extension of $P^{(1)}_{p}\left(A(x,D)-\lambda\,I_{N}\right)u$ to a
linear functional over
$L_{p^{\prime}}(\mathbb{R}^{n})^{\sum_{j=1}^{k_{r-1}}\tilde{\nu}_{j}}$, and
the terminology used here is analogous to that used in the proof of
Proposition 3.2. If we now let
$f^{(2)}=\left(f^{(2)}_{k_{r-1}+1},\ldots,f^{(2)}_{N}\right)\in
L_{p}(\mathbb{R}^{n})^{(N-k_{r-1})(n+1)}$, where
$f^{(2)}_{j}=\left(f^{(2)}_{j,1},\ldots,f^{(2)}_{j,n+1}\right)$ and
$f^{(2)}_{j,k}=D_{k-1}u_{j}$, with $D_{0}u_{j}=u_{j}$, then we conclude from
the foregoing results that

$\displaystyle\sum_{\ell\geq 1}\|\,\eta_{\ell}\bigl{(}A(x,D)-$
$\displaystyle\lambda\,I_{N}\,\bigr{)}u\,\|^{p}_{(-s+1),p,Q_{\ell}}$
$\displaystyle\leq C\sum_{\ell\geq
1}\left(\sum_{j=1}^{k_{r-1}}\sum_{k=1}^{\tilde{\nu}_{j}}\|\,f^{(1)}_{j,k}\,\|^{p}_{0,p,Q_{\ell}}+\sum_{j=k_{r-1}+1}^{N}\sum_{k=1}^{n+1}\|\,f^{(2)}_{j,k}\,\|^{p}_{0,p,Q_{\ell}}\right)$
$\displaystyle\leq
C^{\prime}\|\,\left(A(x,D)-\lambda\,I_{N}\right)u\,\|^{p}_{(-s+1),p,\mathbb{R}^{n}},$

where the constant $C^{\prime}$ has the same properties as the constant $C$.
In light of these last inequalities, (3.9), and (3.10), we conclude that $u\in
W^{(t+1)}_{p}(\mathbb{R}^{n})$ and that the inequality (3.1) holds with $t$
and $-s$ replaced by $t+1$ and $-s+1$, respectively.

Suppose next that $s_{j}=0$ for $j=1,\ldots,N$. Then
$H^{(-s+1)}_{p}(\mathbb{R}^{n})=W^{1}_{p}(\mathbb{R}^{n})^{N}$, and for this
case the proposition can be proved by arguing with
$W^{1}_{p}(\mathbb{R}^{n})^{N}$ as we argued with
$W^{1}_{p}(\mathbb{R}^{n})^{N-k_{r-1}}$ in the previous case.

If $\ell_{0}=1$, then the proof of the proposition is complete. Otherwise we
complete the proof by proceeding by induction. ∎

##  4\. Fredholm theory

In this section we are going to use the results of Section 3 to derive
information pertaining to the Fredholm theory for the Banach space operators
induced by the spectral problem (1.1). Furthermore, when in the sequel we
refer to $W^{(\tau)}_{p}(G),\tau=t\;\textrm{or}\;s$ and $H^{(-s)}_{p}(G)$ as
Banach spaces (see Section 3 for terminology), then it is to be understood
that we are equipping these spaces with their ordinary norms. If $X$ and $Y$
are Banach spaces, then we shall also use the notation $L\left(X,Y\right)$ to
denote the space of bounded linear operators mapping $X$ into $Y$ and equipped
with its usual norm.

Next let $A_{p}$ denote the operator on $H^{(-s)}_{p}(\mathbb{R}^{n})$ that
acts like $A(x,D)$ and has domain $D(A_{p})=W^{(t)}_{p}(\mathbb{R}^{n})$.

We note for later use that that if we suppose that the hypotheses of
Proposition 3.2 hold, then

$\|\,u\,\|_{(t),p,\mathbb{R}^{n}}\leq
C\left(\|\,A_{p}\,u\,\|_{(-s),p,\mathbb{R}^{n}}+\|\,u\,\|_{(-s),p,\mathbb{R}^{n}}\right)\quad\textrm{for}\quad
u\in D(A_{p}),$

where the constant $C$ does not depend upon $u$. Thus we conclude that the
operator $A_{p}:D(A_{p})\to H^{(-s)}_{p}(\mathbb{R}^{n})$ is closed. Note also
from Proposition 3.1 that $A_{p}:D(A_{p})\to H^{(-s)}_{p}(\mathbb{R}^{n})$ is
bounded.

Referring to [12, pp. 242–243] for terminology, we now have the following
result.

######  Theorem 4.1.

Suppose that the spectral problem (1.1) is weakly smooth and parameter-
elliptic in $\mathcal{L}$. Then there exists the number
$\lambda^{\dagger}\in\mathbb{R}_{+}$, where $\lambda^{\dagger}$ does not
depend upon $p$, such that $A_{p}-\lambda\,I_{N}\in
L\left(D(A_{p}),H^{(-s)}_{p}(\mathbb{R}^{n})\right)$ and is Fredholm for
$\lambda\in\mathcal{L}^{\dagger}=\bigl{\\{}\,\lambda\in\mathcal{L}\bigl{|}\,|\lambda|\geq\lambda^{\dagger}\,\bigr{\\}}$.
Furthermore, there exists the number $\lambda^{0}(p)\geq\lambda^{\dagger}$
such that $\lambda$ belongs to the resolvent set of $A_{p}$ for
$\lambda\in\mathcal{L}^{\dagger}_{p}=\bigl{\\{}\,\lambda\in\mathcal{L}^{\dagger}\,\bigl{|}|\lambda|\geq\lambda^{0}(p)\,\bigr{\\}}$,
and hence index $(A_{p}-\lambda\,I_{N})=0$ for
$\lambda\in\mathcal{L}^{\dagger}$. Finally, that part of the spectrum of
$A_{p}$ which is contained in $\mathcal{L}^{\dagger}$ consists solely of most
a finite number of distinct eigenvalues, with each being of finite algebraic
multiplicity.

###### Proof.

We know from the proof of Proposition 3.4 that for $x,\xi\in\mathbb{R}^{n}$
and $\lambda\in\mathcal{L}$ with $|\lambda|\geq\lambda_{0}$,

$|\textrm{det}\,\left(\textrm{\r{A}}(x,\xi)-\lambda\,I_{N}\right)|\geq
C_{0}\prod_{j=1}^{N}\langle\,\xi,\lambda\,\rangle_{j}^{m_{j}},$

where the constant $C_{0}$ does not depend upon $x,\xi,$ and $\lambda$.
Furthermore, it is not difficult to verify that for these values of $x,\xi,$
and $\lambda$,

$|\textrm{det}\left(\textrm{\r{A}}(x,\xi)-\lambda\,I_{N}\right)-\textrm{det}\left(A(x,\xi)-\lambda\,I_{N}\right)|\leq
C|\lambda|^{-1/m_{1}}\prod_{j=1}^{N}\langle\,\xi,\lambda\,\rangle_{j}^{m_{j}},$

where the constant $C$ does not depend upon $x,\xi,$ and $\lambda$. Hence it
follows that we can choose the number $\lambda^{\dagger}\in\mathbb{R}_{+}$
such that

$|\textrm{det}\left(A(x,\xi)-\lambda\,I_{N}\right)|\geq
C_{0}/2\prod_{j=1}^{N}\langle\,\xi,\lambda\,\rangle_{j}^{m_{j}}$

for $x,\xi\in\mathbb{R}^{n}$ and $\lambda\in\mathcal{L}$ with
$|\lambda|\geq\lambda^{\dagger}$. Thus all but the final assertions of the
proposition now follow from this last result, [19], Proposition 3.3, from what
was said in the text preceding the statement of this theorem, and from the
fact that $\mathcal{L}^{\dagger}$ is contained in a component of the Fredholm
domain of $A_{p}$ (see [12, pp. 242–243]).

Turning now to the final assertions of the proposition, let
$\lambda_{1},\lambda_{2}\in\mathcal{L}^{\dagger}$ with
$|\lambda_{1}|\geq\lambda^{0}(p)$ and $|\lambda_{2}|<\lambda^{0}(p)$. Then
there is a polygonal arc $\gamma=\bigl{\\{}\,\gamma(t),0\leq t\leq
1\,\bigr{\\}}$ joining $\lambda_{1}$ to $\lambda_{2}$ and lying entirely in
$\mathcal{L}^{\dagger}$ such that $\gamma(0)=\lambda_{1}$ and
$\gamma(1)=\lambda_{2}$. But this implies that either
$N(t)=\textrm{dim\,ker}\,\left(A_{p}-\gamma(t)\,I_{N}\right)=0$ for $0\leq
t<1$ or there is a $\tau_{1},0<\tau_{1}<1$ such that $N(t)=0$ for $0\leq
t<\tau_{1},N(\tau_{1})>0$. However for either case we know from [12, Theorem
5.3.1, p. 241] that for some $\epsilon>0$, dim ker$(A_{p}-\lambda\,I_{N})=0$
for $0<|\lambda-\lambda_{2}|<\epsilon$ if the first case occurs and for
$0<|\lambda-\gamma(\tau_{1})|<\epsilon$ if the second case occurs.
Furthermore, if the second case occurs, then either $N(t)=0$ for $0\leq t<1$
except for $t=\tau_{1}$, or there is a $\tau_{2},0<\tau_{1}<\tau_{2}<1$, such
that $N(t)=0$ for $0\leq t\leq\tau_{2}$ except for $t=\tau_{j},j=1,2$. But as
before, we know that in either case there is an $\epsilon>0$ such that dim
ker$\left(A_{p}-\lambda\,I_{N}\right))=0$ for $0<|\lambda-\lambda_{2}|<0$ if
the first case holds and for $0<|\lambda-\gamma(\tau_{2})|<\epsilon$ if the
second case holds. Carrying on in this manner we finally arrive at the
situation where either there is a finite sequence
$\\{\,\tau_{j}\,\\}_{1}^{k},0<\tau_{1}<\cdots<\tau_{k}<1$ such that $N(t)=0$
for $0\leq t<1$ except for $t=\tau_{j},j=1,\ldots,k$, or there is an infinite
sequence $\\{\,\tau_{j}\,\\}_{1}^{\infty},0<\tau_{1}<\tau_{2}<\cdots<1$, such
that $N(t)=0$ for $0\leq t<\tau$ except for $t=\tau_{j},j\geq 1$, where
$\tau=\lim_{j\to\infty}\tau_{j}$. Then we see from [12] that the second case
is not possible, while for the first case there is an $\epsilon>0$ such that
dim ker$\left(A_{p}-\lambda\,I_{N}\right)=0$ for
$0<|\lambda-\lambda_{2}|<\epsilon$. Thus we have shown that if
$\mu\in\mathcal{L}^{\dagger}$ and dim ker$\left(A_{p}-\mu\,I_{N}\right)>0$,
then there is an $\epsilon>0$ such that dim ker$(A_{p}-\lambda\,I_{N})=0$ for
$0<|\lambda-\mu|<\epsilon$. The final assertions of the theorem follows from
this last result, and [12], which concludes the proof. ∎

We are now going to investigate how the eigenvalues, if any, of $A_{p}$ which
lie in $\mathcal{L}^{\dagger}$ vary with $p$.

######  Theorem 4.2.

Suppose that the spectral problem (1.1) is weakly smooth and parameter-
elliptic in $\mathcal{L}$. Suppose in addition that $a^{jk}_{\alpha}\in
C^{s_{j}+n}(\overline{\mathbb{R}^{n}})$ for $|\alpha|\leq s_{j}+t_{k},1\leq
j,k\leq N$. Lastly suppose that $\lambda_{1}\in\mathcal{L}^{\dagger}$ is an
eigenvalue of $A_{p}$ and $u^{(1)}$ a corresponding eigenvector. Then
$\lambda_{1}$ is an eigenvalue and $u^{(1)}\in W^{(t)}_{q}(\mathbb{R}^{n})$ a
corresponding eigenvector of $A_{q}$ for every $q$ satisfying $p<q<\infty$.
Consequently ker$\,\left(A_{p}-\lambda_{1}\,I_{N}\right)\subset$
ker$\,\left(A_{q}-\lambda_{1}\,I_{N}\right)$.

###### Proof.

As a consequence of Proposition 3.6 we see that $u^{(1)}\in
W^{(t+n)}_{p}(\mathbb{R}^{n})$. Hence it follows from the Sobolev embedding
theorem (see [1, Theorem 5.4, p. 97]) that $u^{(1)}\in
W^{(t)}_{q}(\mathbb{R}^{n})$ for every $q$ satisfying $p<q<\infty$, and all
the assertions of the theorem are immediate consequences of this fact. ∎

Under some further restrictions Theorem 4.2 can be improved.

######  Theorem 4.3.

Suppose that the hypotheses of Theorem 4.2 hold with $s_{j}=0$ and $t_{j}$
even for $j=1,\ldots,N$. Suppose in addition that for $1\leq j,k\leq N$,
$a^{jk}_{\alpha}\in C^{|\alpha|+n}(\overline{\mathbb{R}^{n}})\cap
C^{|\alpha|+1,0}(\overline{\mathbb{R}^{n}})$ for $|\alpha|\leq t_{k}$ with
$a^{jk}_{\alpha}(x)=0$ for $x\in\mathbb{R}^{n}$ if $j>k,t_{j}<t_{k}$, and
$|\alpha|\geq t_{j}$ and also if $j<k,t_{j}>t_{k}$, and $|\alpha|=t_{k}$. Then
the eigenvalues of $A_{p}$ lying in $\mathcal{L}^{\dagger}$, as well as their
geometric and algebraic multiplicities, are the same for all these values of
p.

In order to prove the theorem, a preliminary result is required.

######  Proposition 4.4.

Suppose that the hypotheses of Theorem 4.3 hold and let $1<q<p$. Also let
$\lambda_{1}\in\mathcal{L}^{\dagger}$ and put
$n_{p}=\textrm{dim\,ker}\,(A_{p}-\lambda_{1}\,I_{N}),d_{p}=\textrm{dim\,coker}\,(A_{p}-\lambda_{1}\,I_{N})$.
Then
(1) ker $(A_{p}-\lambda_{1}\,I_{N})\subset$ ker $(A_{q}-\lambda_{1}\,I_{N})$
if $0\leq d_{p}\leq n_{p}<\infty$;
(2) ker $(A_{p}-\lambda_{1}\,I_{N})\subset$ ker $(A_{q}-\lambda_{1}\,I_{N})$
if $0\leq n_{p}<d_{p}<\infty$.

###### Proof.

To begin with let us introduce the multiplication operators $T_{p}(\tau)$ and
$S_{p}(\tau)$, $\tau\geq 0$, acting on $D(A_{p})$ and
$L_{p}(\mathbb{R}^{n})^{N}$ ($=H^{(-s)}_{p}(\mathbb{R}^{n})$), respectively,
where $T_{p}(\tau)u(x)=e^{-\tau\,\langle\,x\,\rangle}u(x)$ for $u\in
D(A_{p})$, $S_{p}(\tau)v(x)=e^{-\tau\,\langle\,x\,\rangle}v(x)$ for $v\in
L_{p}(\mathbb{R}^{n})^{N}$, where $\langle\,x\,\rangle=(1+|x|^{2})^{1/2}$.

Let $u\in D(A_{p})$, denote by $u_{k}$ its $k$-th component, and let $\alpha$
denote a multi-index such that $|\alpha|\leq t_{k}$. Then it follows from the
Leibnitz formula that

$D^{\alpha}e^{\pm\tau\,\langle\,x\,\rangle}u_{k}(x)=e^{\pm\tau\,\langle\,x\,\rangle}D^{\alpha}u_{k}(x)+\sum_{\begin{subarray}{c}\beta\leq\alpha\\\
|\beta|>0\end{subarray}}\binom{\alpha}{\beta}\left(D^{\beta}e^{\pm\tau\,\langle\,x\,\rangle)}\right)D^{\alpha-\beta}u_{k}(x),$

while direct calculations show that

$D^{\beta}e^{\pm\tau\,\langle\,x\,\rangle}=e^{\pm\tau\langle\,x\,\rangle}\sum\,^{\prime}\prod\,^{\prime}\left(\pm\tau\,D^{\gamma_{j}}\langle\,x\,\rangle\right)$

and that

$|D^{\gamma_{j}}\langle\,x\,\rangle|\leq
C(\gamma_{j})\langle\,x\,\rangle^{1-|\gamma_{j}|},$

where $\prod^{\prime}$ indicates that the product is taken over a set of
distinct multi-indices $\gamma_{j}$ for which $|\gamma_{j}|>0$ and
$\sum_{j}|\gamma_{j}|=|\beta|$, $\sum^{\prime}$ indicates that the sum is
taken over all such sets, and $C(\gamma_{j})$ denotes a constant depending
upon $\gamma_{j}$. In light of these facts it is not difficult to deduce that
for $\tau\in[0,\infty)$, $T_{p}(\tau)\in L\left(D(A_{p}),D(A_{p})\right)$ and
$S(\tau)\in L\left(L_{p}(\mathbb{R}^{n})^{N},L_{p}(\mathbb{R}^{n})^{N}\right)$
are $C_{0}$ semigroups such that for each $\tau,T(\tau)$ and $S(\tau)$ are
injective,while $S(\tau)^{-1}A_{p}T(\tau)\in
L\left(D(A_{p}),L_{p}(\mathbb{R}^{n})^{N}\right)$ and that
$S(\tau)^{-1}A_{p}T(\tau)$ converges in norm to $A_{p}$ as $\tau\to 0$.

Let us next fix our attention upon the assertions of the proposition, and to
begin with let us consider assertion (1) with $n_{p}>0$ and $d_{p}=0$. Then it
follows from [12, Theorem 5.22, p. 236] that for some
$\tau_{0}>0,V=S(\tau_{0})^{-1}(A_{p}-\lambda_{1}\,I_{N})T(\tau_{0})$ is semi-
Fredholm with dim ker $V=n_{p}$, and dim coker $V=0$. Hence by arguing as in
[19] and applying Holder’s inequality, we readly deduce that
ker$(A_{p}-\lambda_{1}\,I_{N})\subset$ ker$(A_{q}-\lambda_{1}\,I_{N})$.
Obviously the same result is true if we assume instead that $n_{p}=0$.

Finally assertions (1), with $d_{p}>0$, as well as assertion (2), can be
proved by modifying the above arguments as in [19]. ∎

Proof of Theorem 4.3. Referring to Proposition 4.4 for notation, let us
firstly fix our attention upon the case $1<q<p$ and prove all but the final
assertions of the theorem for this case. Accordingly, to begin with, let us
observe from Theorem 4.2 and Proposition 4.4 that
ker$(A_{q}-\lambda_{1}\,I_{N})=$ ker$(A_{p}-\lambda_{1}\,I_{N})$.

We now turn our attention to cokernels. Then it is not difficult to show that
under our assumptions the spectral problem formally adjoint to the spectral
problem (1.1),

$A^{\prime}(x,D)u(x)-\overline{\lambda_{1}}u(x)=f(x)\quad\textrm{for}\quad
x\in\mathbb{R}^{n},$ (4.1)

is well defined. Here
$A^{\prime}(x,D)=\left(A^{\prime}_{jk}(x,D\right)_{j,k=1}^{N}$ is the formal
adjoint of $A(x,D)$, the $A^{\prime}_{jk}(x,D)$ are linear differential
operators defined on $\mathbb{R}^{n}$ of order not exceeding
$s^{\prime}_{j}+t^{\prime}_{k}$ where $s^{\prime}_{j}=0,t^{\prime}_{k}=t_{k}$
for $1\leq j,k\leq N$. Note also that the top order operators
$\textrm{\r{A}}(x,D)$ and $\textrm{\r{A}}^{\prime}(x,D)$ are block diagonal.
Then it is not difficult to verify that the analogues of Propositions 3.1–3.3
for the spectral problem (4.1) also hold and we can take the constant
$\lambda^{\sharp}$ of Proposition 3.1 to remain the same. We henceforth let
$A^{\prime}_{p^{\prime}}$ denote the operator on
$L_{p^{\prime}}(\mathbb{R}^{n})^{N}$ that acts like $A^{\prime}(x,D)$ and has
domain $D(A^{\prime}_{p^{\prime}})=W^{(t)}_{p^{\prime}}(\mathbb{R}^{n})$; and
we can readily verify that the analogues of Theorems 4.1, 4.2 and Proposition
4.4 hold for $A^{\prime}_{p^{\prime}}$. Hence we can now argue as above with
$A^{\prime}_{q^{\prime}}-\overline{\lambda_{1}}\,I_{N}$ and
$A^{\prime}_{p^{\prime}}-\overline{\lambda_{1}}\,I_{N}$ in place of
$A_{p}-\lambda_{1}\,I_{N}$ and $A_{q}-\lambda_{1}\,I_{N}$, respectively, to
deduce that coker $(A_{q}-\lambda_{1}\,I_{N})$ = coker
$(A_{p}-\lambda_{1}\,I_{N})$. This proves all but the final assertions of the
theorem for the case $q<p$, and the analogous result for the case $q>p$ can be
proved in a similar fashion (i.e., by arguing with the adjoint operators).

Turning to the final assertions of the theorem, let
$\lambda_{1}\in\mathcal{L}^{\dagger}$ be an eigenvalue of $A_{p}$ and
$u^{(1)}$ a corresponding eigenvector. Then it is clear from what has already
been proved that we need only prove that the algebraic multiplicity of
$\lambda_{1}$ is the same for all $p$. Accordingly, let us suppose firstly
that $q>p$ and let $\bigl{\\{}\,u^{(1,j)}\,\bigr{\\}}_{j=0}^{m-1}$ be a chain
of length $m$ consisting of the eigenvector $u^{(1,0)}=u^{(1)}$ and the
associated vectors $\bigl{\\{}\,u^{(1j)}\,\bigr{\\}}_{j=1}^{m-1}$
corresponding to the eigenvalue $\lambda_{1}$ of $A_{p}$ (see [16, pp.
60–61]). Thus it follows from Proposition 3.6 and the Sobolev embedding
theorem (see [1, Theorem 5.4, p. 97]) that
$\bigl{\\{}\,u^{(1,j)}\,\bigr{\\}}_{j=0}^{m-1}$ is a chain of length $m$
consisting of the eigenvector $u^{(1)}=u^{(1,0)}$ and the associated vectors
$\bigl{\\{}\,u^{(1,j)}\,\bigr{\\}}_{j=1}^{m-1}$ corresponding to the
eigenvalue $\lambda_{1}$ of $A_{q}$. We conclude immediately that the
algebraic multiplicity of $\lambda_{1}$ as an eigenvalue of $A_{p}$ does not
exceed the algebraic multiplicity of $\lambda_{1}$ as an eigenvalue of
$A_{q}$. On the other hand we can appeal to the analogue of Proposition 3.6
for the spectral problem (4.1) to show that the algebraic multiplicity of
$\overline{\lambda_{1}}$ as an eigenvalue of $A^{\prime}_{q^{\prime}}$ does
not exceed the algebraic multiplicity of $\overline{\lambda_{1}}$ as an
eigenvalue of $A^{\prime}_{p^{\prime}}$. Hence we conclude that the algebraic
multiplicity of $\lambda_{1}$ as an eigenvalue of $A_{p}$ and of $A_{q}$ are
the same (see [12, p. 184]). Since similar arguments give the same result for
$q<p$, the proof of the theorem is complete. $\square$

##  5\. An example

In this section we fix our attention upon the spectral problem (1.1) with
$A(x,D)$ a $2\times 2$ matrix operator whose entries $A_{jk}(x,D)$ are linear
differential operators defined on $\mathbb{R}^{n}$ of order not exceeding
$s_{j}+t_{k}$, where $s_{j}=0$ for $j=1,2$, and $t_{1}=4,t_{2}=2$. To be more
precise we now take

$A(x,D)=A_{0}(x,D)-cI_{2}+\left(\tilde{A}_{jk}(x,D)\right)^{2}_{j,k=1},$ (5.1)

where $A_{0}(x,D)$ = diag $\left(\Delta^{2},-\Delta\right)$, $\Delta$ denotes
the Laplacian in $\mathbb{R}^{n},c\in\overline{\mathbb{R}_{+}}$ denotes a
constant,
$\tilde{A}_{jk}(x,D)=\sum_{|\alpha|\leq\sigma_{jk}}a^{jk}_{\alpha}(x)D^{\alpha}$
with $\sigma_{jk}=1$ if $j\neq k,\sigma_{11}=3$, $\sigma_{22}=1$, and the
$a^{jk}_{\alpha}\in C_{0}^{\infty}(\mathbb{R}^{n})$. Then with $A(x,D)$ given
by (5.1), let us now investigate the spectral theory connected with the
problem (1.1).

Accordingly, let us fix an $\epsilon$ satisfying $0<\epsilon<\pi/2$ and let
$\mathcal{L}$ denote the sector in the complex plane with vertex at the origin
determined by the inequalities $\epsilon\leq\arg\,\lambda\leq 2\pi-\epsilon$.
Then we can readily verify that the hypotheses of Theorem 4.3 concerning the
spectral problem (1.1) are satisfied, and hence if we let $A_{p}$ denote the
operator on $L_{p}(\mathbb{R}^{n})^{2}$ that acts like $A(x,D)$ and has domain
$D(A_{p})=W^{(t)}_{p}(\mathbb{R}^{n})$, the all the assertions of Theorems 4.1
and 4.3 hold.

In order to derive more information concerning the spectral properties of
$A_{p}$, we are now going to fix our attention upon the case $p=2$, and let
$A^{(1)}_{2}$ (resp. $A^{(2)}_{2}$) denote the operator on
$L_{2}(\mathbb{R}^{n})$ that acts like $\Delta^{2}$ (resp. $-\Delta$) and with
domain $D(A^{(1)}_{2})=W^{4}_{2}(\mathbb{R}^{n})$ (resp.
$D(A^{(2)}_{2})=W^{2}_{2}(\mathbb{R}^{n})$. Then direct calculations show that
$A^{(1)}_{2}$ is a symmetric operator on $L_{2}(\mathbb{R}^{n})$ whose
numerical range is contained in the interval $[0,\infty)$. Furthermore, since
it is shown in [10] that the analogue of Theorem 3.3 holds when the spectral
problem (1.1) is replaced by a scalar spectral problem, we conclude from these
results that $A^{(1)}_{2}$ is a selfadjoint operator on
$L_{2}(\mathbb{R}^{n})$ whose spectrum is contained in the interval
$[0,\infty)$. Similarly we can show that $A^{(2)}_{2}$ is a selfadjoint
operator on $L_{2}(\mathbb{R}^{n})$ whose spectrum is contained in
$[0,\infty)$, and furthermore, we know from [6, p. 416] (see also [20, p. 158]
for the case $n=2$) that its spectrum is precisely $[0,\infty)$.

Thus we have shown that if we let $A_{0,2}$ = diag
$(A^{(1)}_{2},A^{(2)}_{2}\,)$, then $A_{0,2}-c\,I_{2}$ is a selfadjoint
operator on $L_{2}(\mathbb{R}^{n})^{2}$ with domain
$D(A_{0,2})=W^{(t)}_{2}(\mathbb{R}^{n})$ and whose spectrum is precisely
$[-c,\infty)$. Furthermore, we can appeal to Proposition 3.2 to show that
$A_{2}-(A_{0,2}-c\,I_{2})$ is relatively compact with respect to
$A_{0,2}-c\,I_{2}$, and hence it follows from [12, Theorem 5.35, p. 244] that
$A_{2}$ is a closed operator on $L_{2}(\mathbb{R}^{n})^{2}$ with essential
spectrum $[-c,\infty)$ and with semi-Fredholm domain
$\mathbb{C}\setminus[-c,\infty)$. Hence referring again to Theorem 4.1, it is
clear that we must have $\lambda^{\dagger}>c$, while it also follows from the
arguments used in the proof of the final assertion of that theorem that the
Fredholm domain of $A_{2}$ is precisely $\mathcal{C}\setminus[-c,\infty)$,
that $\mathcal{C}\setminus[-c,\infty)$ consists of only one component, and
that index $(A_{2}-\lambda\,I_{2})=0$ for $\lambda$ lying in this component.

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