#  $\LL_{p}$-maximal regularity for parabolic and elliptic boundary value
problems with boundary conditions of mixed differentiability orders

Björn Augner  Technische Universität Darmstadt, Fachbereich Mathematik,
Schlossgartenstraße 7, 64289 Darmstadt<EMAIL_ADDRESS>
###### Abstract.

We consider vector-valued elliptic boundary value problems and parabolic
initial-boundary value problems on abstract Banach spaces of class
$\mathcal{HT}$. For a very general class of boundary conditions, which
includes, but is not limited to the classical cases of Dirichlet, Neumann and
Robin boundary conditions, we show $\LL_{p}$-maximal regularity of the system.
We do not restrict ourselves to the homogeneous case, but also identify the
temporal and boundary trace spaces, which are both sufficient and necessary to
obtain a unique solution in the $\LL_{p}$-maximal regularity space. Our
results are based on a (partial) Fourier–Laplace transform of the problem in
the half-space case, leading to elliptic boundary value problems which have to
satisfy variants of the Lopatinskii–Shapiro condition to be solvable. The
parabolic theory is then developed with use of tools from modern harmonic
analysis such as the bounded $\mathcal{H}^{\infty}$-calculus and the
Kalton–Weis theorem.

###### Key words and phrases:

Elliptic boundary value problems, parabolic boundary value problems,
$\LL_{p}$-maximal regularity, kernel estimates, non-standard boundary
conditions.

###### 2010 Mathematics Subject Classification:

35K52, 35B65, 35D35, 35J58, 42B15

Version of August 27, 2024.

##  1\. Introduction

We will consider a quite general class of elliptic and parabolic systems with
values in a general Banach space of class $\mathcal{HT}$. To give a first
motivation for the relevance of the investigation of such systems, let us
start with a rather simple example which illustrates the main point: From the
theory of parabolic equations it is well-known that, for a bounded domain
$\Omega\subseteq\operatorname{\mathbb{R}}^{d}$ with smooth boundary, both the
_Dirichlet heat equation_

(D) $\begin{cases}\partial_{t}u_{1}-d_{1}\Delta u_{1}=f_{1}&\text{in
}(0,T)\times\Omega,\\\ u_{1}=g_{1}&\text{on }(0,T)\times\partial\Omega,\\\
u_{1}(0,\cdot)=u_{1}^{0}&\text{in }\Omega\end{cases}$

(for some $d_{1}>0$) and the _Neumann heat equation_

(N) $\begin{cases}\partial_{t}u_{2}-d_{2}\Delta u_{2}=f_{1}&\text{in
}(0,T)\times\Omega,\\\ -d_{2}\partial_{\boldsymbol{n}}u_{2}=g_{2}&\text{on
}(0,T)\times\partial\Omega,\\\ u_{2}(0,\cdot)=u_{2}^{0}&\text{in
}\Omega\end{cases}$

(for some $d_{2}>0$) have the property of $\LL_{p}$-maximal regularity on the
base space $\LL_{q}(\Omega)$, where we consider parameters $p,q\in(1,\infty)$.
This means that (D) has a unique solution $u_{1}$ in the class
$\mathrm{W}_{p}^{1}((0,T);\LL_{q}(\Omega))\cap\LL_{p}((0,T);\mathrm{W}_{q}^{2}(\Omega))$
if and only if the data $f_{1}$, $g_{1}$ and $u_{1}^{0}$ belong to the classes
$f_{1}\in\LL_{p}((0,T);\LL_{q}(\Omega))$,
$g_{1}\in\mathrm{W}_{p}^{1-1/2q}((0,T);\LL_{q}(\partial\Omega))\cap\LL_{p}((0,T);\mathrm{W}_{p}^{2-1/q}(\partial\Omega))$
and $u_{1}^{0}\in\mathrm{W}_{q}^{2-2/p}(\Omega)$ and $u_{1}^{0}$ and $g_{1}$
fullfil a certain compatibility condition. Similarly, the problem (N) has a
unique solution
$u_{2}\in\mathrm{W}_{p}^{1}((0,T);\LL_{q}(\Omega))\cap\LL_{p}((0,T);\mathrm{W}_{q}^{2}(\Omega))$
if and only if $f_{2}\in\LL_{p}((0,T);\LL_{q}(\Omega))$,
$g_{2}\in\mathrm{W}_{p}^{1/2-1/2q}((0,T);\LL_{q}(\partial\Omega))\cap\LL_{p}((0,T);\mathrm{W}_{q}^{1-1/q}(\partial\Omega))$
and $u_{2}^{0}\in\mathrm{W}_{q}^{2-2/p}(\Omega)$. Combining these two scalar
results, it easily follows that the two-component system of parabolic
equations

$\begin{cases}\partial_{t}\boldsymbol{u}-\boldsymbol{D}\Delta\boldsymbol{u}=\boldsymbol{f}&\text{in
}(0,T)\times\Omega,\\\ u_{1}=g_{1}&\text{on }(0,T)\times\partial\Omega,\\\
-d_{2}\partial_{\boldsymbol{n}}u_{2}=g_{2}&\text{on
}(0,T)\times\partial\Omega,\\\ \boldsymbol{u}(0,\cdot)=u^{0}&\text{on
}\Omega,\end{cases}$

where $\boldsymbol{u}=(u_{1},u_{2})$, $\boldsymbol{f}=(f_{1},f_{2})$,
$\boldsymbol{u}^{0}=(u_{1}^{0},u_{2}^{0})$ are two-component vector-fields and
$\boldsymbol{D}=\operatorname{diag}(d_{1},d_{2})$ is a diagonal matrix, has a
unique solution in the class
$\boldsymbol{u}\in\mathrm{W}_{p}^{1}((0,T);\LL_{q}(\Omega;\operatorname{\mathbb{K}}^{2}))\cap\LL_{p}((0,T);\mathrm{W}_{q}^{2}(\Omega;\operatorname{\mathbb{K}}^{2}))$
if and only if
$\boldsymbol{f}\in\LL_{p}((0,T);\LL_{q}(\Omega;\operatorname{\mathbb{K}}^{2}))$,
$\boldsymbol{u}^{0}\in\mathrm{W}_{q}^{2-2/p}(\Omega;\operatorname{\mathbb{K}}^{2}))$
and the components $g_{1}$ and $g_{2}$ of the function $\boldsymbol{g}$ on the
boundary satisfy
$g_{1}\in\mathrm{W}_{p}^{1-1/2q}((0,T);\LL_{q}(\partial\Omega))\cap\LL_{p}((0,T);\mathrm{W}_{q}^{2-1/q}(\partial\Omega))$
and
$g_{2}\in\mathrm{W}_{p}^{1/2-1/2q}((0,T);\LL_{q}(\partial\Omega))\cap\LL_{p}((0,T);\mathrm{W}_{q}^{1-1/q}(\partial\Omega))$
and the respective compatibility conditions. However, as soon as we allow for
non-diagonal, but, say, symmetric and positive definite,
$\boldsymbol{D}\in\operatorname{\mathbb{R}}^{2\times 2}$, the situation
becomes less clear, as then the two equations in $\Omega$ do _not decouple_
anymore. Still, one can _expect_ maximal regularity also for this case.
As similar situation occurs, if not the equations in the domain are coupled,
but the boundary conditions are of mixed type, say

$\begin{cases}\alpha_{1}u_{1}+\alpha_{2}u_{2}=g_{1}&\text{on
}(0,T)\times\partial\Omega,\\\
-\beta_{1}d_{1}\partial_{\boldsymbol{n}}u_{1}-\beta_{2}d_{2}\partial_{\boldsymbol{n}}u_{2}=g_{2}\end{cases}$

for some parameters $\alpha_{1}$, $\alpha_{2}$, $\beta_{1}$ and
$\beta_{2}\in\operatorname{\mathbb{R}}$. We will consider a general class of
such parabolic systems. More precisely, we will consider systems of the
abstract form

$\begin{cases}\partial_{t}u+\mathcal{A}(t,\boldsymbol{x},\mathrm{D})u=f&\text{in
}(0,T)\times\Omega,\\\
\mathcal{B}_{j}(t,\boldsymbol{x},\mathrm{D})u=g_{j}&\text{on
}(0,T)\times\partial\Omega,\,j=1,\ldots,m,\\\ u(0,\cdot)=u^{0}&\text{in
}\Omega.\end{cases}$

Here, $\mathcal{A}(t,\boldsymbol{x},\mathrm{D})$ is a parameter-elliptic
differential operator of order $2m$, where the coefficients may, in general,
both depend on the time variable $t\in[0,T]$ and on the spatial position
$\boldsymbol{x}\in\Omega$, whereas the operators
$\mathcal{B}_{j}(t,\boldsymbol{x},\mathrm{D})$ are boundary differential
operators of order $2m-1$ at most. The mixed type boundary conditions are
reflected by the boundary symbols
$\mathcal{B}_{j}(t,\boldsymbol{x},\boldsymbol{\xi})$ which will not
necessarily have a homogeneous (in $\boldsymbol{\xi}$) principal part
$\mathcal{B}^{\\#}_{j}(t,\boldsymbol{x},\boldsymbol{\xi})$ – in contrast to
related results in [17] and [18], which only apply for systems with
homogeneous principle parts. Note that in the abstract formulation of the
above presented second-order in space two-component example, we have $m=1$, so
that the two (Dirichlet and Neumann) boundary conditions will be described by
only _one_ boundary operator, which thus includes both these scalar boundary
conditions at once.
Before presenting further instances, for which such mixed type boundary
conditions will be of importance, let us give a short overview over the
development of the theory of elliptic boundary value problems and parabolic
initial–boundary value problems and its close connection to modern harmonic
analysis: In the middle of the 20th century some for the further development
of the theory of partial differential equations, especially evolutionary
partial differential equations, fundamentally important works were written,
which were groundbreaking not only for the linear theory, but in particular
for the treatment of semi- and quasilinear, up to fully nonlinear parabolic
systems. Ladyshenskaya, Solonnikov and Ural’ceva [32] basically completely
solved the question of $\LL_{p}$-maximal regularity for the case of
sufficiently regular bounded domains and an exhaustive class of scalar or
$\operatorname{\mathbb{R}}^{n}$-valued functions. Up to today, this work
represents a milestone in the theory of parabolic equations. At a similar
time, de Simon [21] was able to successfully treat the case of abstract
parabolic initial boundary value problems with values in an abstract Hilbert
space $X$ instead of the concrete Lebesgue space $\LL_{p}(\Omega)$, but at
first the question remained open whether and to what extent an extension of
his results to more general abstract Banach spaces is possible. From then on,
at the latest, the question of characterizing $\LL_{p}$-maximal regularity
constituted an intensively studied area of research. Sobolevskii [38] was able
to show that (for a fixed base space $X$) $\LL_{p}$-maximal regularity of an
operator does not depend on the chosen parameter $p\in(1,\infty)$, i.e. if an
operator has $\LL_{p}$-maximal regularity for one $p\in(1,\infty)$, then it
already has $\LL_{q}$-maximal regularity for all $q\in(1,\infty)$. With the
sum method for real interpolation spaces of da Prato and Grisvard [16] rich
subclasses of operators with the maximal regularity could be identified. Using
a functional calculus, Dore and Venni [22], Prüss and Sohr [37] were able to
show that any operator with bounded imaginary power of power angle
$\theta_{A}<\frac{\pi}{2}$ has $\LL_{p}$-maximal regularity, provided the
Banach space $X$ under consideration belongs to the class $\mathcal{HT}$,
i.e., Banach spaces with a bounded Hilbert transform (which – as it turned out
in works by Bourgain [11] and Burkholder [13] – are exactly the UMD spaces).
Meanwhile, further sufficient conditions for $\LL_{p}$-maximal regularity have
been provided for various relevant cases, including boundedness of the
semigroup associated to an abstract Cauchy problem on all spaces
$\LL_{q}(\Omega)$ for $1\leq q\leq\infty$ by Lamberton [33] or Gaussian kernel
estimate problems [25], [14]. The question of actual characterization of
$\LL_{p}$-maximal regularity, however, proved to be much more difficult:
Kalton and Lancien [27] first showed that de Simon’s technique cannot be
applied to abstract Banach spaces of class $\mathcal{HT}$, consequently not to
more general Banach spaces. For a generalisation of de Simon’s results it is
mandatory that the basic space $X$ is at least topologically _isomorphic_ to a
Hilbert space, which is a far too restrictive constraint for many concrete
problems. The breakthrough then came with Weis [46], who was able to
characterize maximal regularity of operators on Banach spaces of type
$\mathcal{HT}$ completely via $\mathcal{R}$-boundedness of the set
$\\{\mathrm{i}\rho(\mathrm{i}\rho+A)^{-1}:\,\rho\in\operatorname{\mathbb{R}}\setminus\\{0\\}\\}$.
This property, called _Riesz-property_ or (later) _randomized boundedness_ ,
provides just the tightening of the notion of uniform boundedness as a
requirement in the Mikhlin multiplier theorem, which is necessary for its
generalisation to Banach spaces of class $\mathcal{HT}$. Simultaneously, for
families of bounded linear operators this gave a new approach to the proof of
maximal regularity; namely, Kalton and Weis [28] showed how to obtain maximal
regularity for $\mathcal{R}$-sectorial operators using the
$\mathcal{H}^{\infty}$-calculus. Based on these developments in modern
harmonic analysis and multiplier theory on Banach spaces of class
$\mathcal{HT}$, see, e.g., Berksen and Gillespie [9], Clément, de Pagter,
Sukochev and Witvliet [15], Kalton and Weis [28], Denk, Hieber and Prüss
[17],[18] extended a large class of $\operatorname{\mathbb{R}}^{n}$-valued
results by Ladyshenskaya, Solonnikov and Ural’ceva [32] on linear parabolic
systems on bounded domains to the general vector-valued case. The boundary
conditions considered there mainly includes boundary conditions of the same
type, so either on the Dirichlet data of the functions itself or on their
normal flux through the boundary, which corresponds to homogeneity (w.r.t.
$\boldsymbol{\xi}$) of the principal part of the boundary symbols.
In a recent paper [6], cf. Example 1.1 below, however, while performing an
analysis of fast-surface chemistry and fast-adsorption limit models for
heterogeneous catalysis systems, we encountered a situation where the boundary
conditions have a combined type: Instead of a Dirichtlet or Neumann condition
in all components equally, a no-flux boundary condition occurred at certain
linear combinations of the individual normal fluxes, i.e., the normal
direction of the gradient of the individual components, together with
nonlinear boundary conditions at the boundary trace of the individual
concentrations due to equilibrium conditions in extremely fast surface
chemical reactions. Though that problem can be solved by employing the results
and techniques of Ladyshenskaya, Solonnikov and Uralceva [32], which are valid
for finite-component systems of parabolic type, this problem stimulated the
search for more abstract and more general results on vector-valued parabolic
and elliptic boundary value problems in the spirit of [17] and [18]. In
contrast to the situation where, say the boundary is decomposed into a
Dirichlet and a Neumann part, cf. e.g. the works by Auscher et al. [7], Egert
[23], Tolksdorff [41], Bechtel, Egert and Haller-Dintelmann [8] and related
works, there seems to be no extension of these results for this particular
_mixed type_ boundary conditions case, and we want to close this gap in the
present manuscript.
As a further motivation for the subsequent development of an abstract theory,
and before we comment on the organisation of this manuscript, let us shortly
discuss more examples, where such mixed type boundary condition play a role,
and for which the results of this paper shall be valuable.

######  Example 1.1 (Fast surface chemistry limits).

In [5] and [6], we derived and analysed a class of reaction-diffusion-systems
inspired by heterogeneous catalysis, which includes, for example, the
following three component model problem [5]

$\begin{cases}\partial_{t}c_{i}-d_{i}\Delta c_{i}=0,&i=1,2,3,\,t\geq
0,\,\boldsymbol{x}\in\Omega\\\ \kappa_{f}c_{1}c_{2}-\kappa_{b}c_{3}=0,&t\geq
0,\,\boldsymbol{x}\in\Sigma,\\\
-d_{1}\partial_{\nu}c_{1}+d_{2}\partial_{\nu}c_{2}=0,&t\geq
0,\,\boldsymbol{x}\in\Sigma,\\\
-d_{1}\partial_{\nu}c_{1}-d_{2}\partial_{\nu}c_{3}=0,&t\geq
0,\,\boldsymbol{x}\in\Sigma,\\\
c_{i}(0,\cdot)=c_{i}^{0},&\boldsymbol{x}\in\Omega\end{cases}$

on some bounded domain $\Omega\subseteq\operatorname{\mathbb{R}}^{n}$ with
regular boundary $\Sigma=\partial\Omega$, and parameters $k_{f}$, $k_{b}$ and
$d_{i}>0$, $i=1,2,3$. This model describes a chemical process inside a reactor
with catalytic boundary $\Sigma$, at which an instantaneous reversible
chemical reaction of the form $A_{1}^{\Sigma}+A_{2}^{\Sigma}\rightleftharpoons
A_{3}^{\Sigma}$ takes place. A linearisation of this system leads to a system
of parabolic-initial boundary value problems of the form

(1) $\begin{cases}\partial_{t}v_{i}-d_{i}\Delta v_{i}=f_{i},&i=1,2,3,\,t\geq
0,\,\boldsymbol{x}\in\Omega,\\\
\alpha_{1}v_{1}+\alpha_{2}v_{2}+\alpha_{3}v_{3}=g_{1},&t\geq
0,\,\boldsymbol{x}\in\Sigma,\\\
\beta_{1}\partial_{\nu}v_{1}+\beta_{2}\partial_{\nu}v_{2}=g_{2},&t\geq
0,\,\boldsymbol{x}\in\Sigma,\\\
\gamma_{1}\partial_{\nu}v_{1}+\gamma_{3}\partial_{\nu}v_{3}=g_{3},&t\geq
0,\,\boldsymbol{x}\in\Sigma,\\\
v_{i}(0,\cdot)=v_{i}^{0},&i=1,2,3,\,\boldsymbol{x}\in\Omega,\end{cases}$

and, evidently, the orders of the boundary differential operator differs
between the second and the two subsequent lines. Similar parabolic systems
with mixed type boundary conditions naturally appear also for more general
reaction networks on the surface, see [6]. Whereas for the particular model
(1) one may exploit the special structure of the boundary conditions on the
surface to derive maximal regularity from the scalar case, cf. the strategy
used in [5, Theorem 3.2], this will no longer be true for the latter, more
general class of reaction-networks.

######  Example 1.2 (Navier–Stokes equations).

In [10], the authors consider several energy-preserving boundary conditions of
the abstract form $\mathcal{B}(u,p)=0$ for the incompressible Navier–Stokes
equation

$\begin{cases}\rho\partial_{t}\boldsymbol{u}+\operatorname{\operatorname{div}}(\rho\boldsymbol{u}\otimes\boldsymbol{u}-\boldsymbol{S})=\rho
f,&t\in(0,T),\,\boldsymbol{x}\in\Omega,\\\
\operatorname{\operatorname{div}}\boldsymbol{u}=0,&t\in(0,T),\,\boldsymbol{x}\in\Omega,\\\
\boldsymbol{u}(0,\cdot)=\boldsymbol{u}^{0}&\boldsymbol{x}\in\Omega.\end{cases}$

The classes of boundary conditions considered there include

  * •

_no-slip boundary conditions_ leading to the homogeneous Dirichlet boundary
condition

$\boldsymbol{u}=\boldsymbol{0}\quad\text{on }\Omega\times(0,T);$

  * •

_free slip (or perfect slip) boundary conditions_ which lead to mixed type
boundary conditions of the form

$\boldsymbol{u}\cdot\boldsymbol{n}=0\quad\text{and}\quad 2\mu
P_{\Gamma}\boldsymbol{D}\nu=\boldsymbol{0}\quad\text{on}\quad\partial\Omega\times(0,T).$

Here,
$P_{\Gamma}(\boldsymbol{x})=\boldsymbol{I}-\boldsymbol{n}(\boldsymbol{x})\otimes\boldsymbol{n}(\boldsymbol{x})$
is the orthogonal projection onto the tangent space $T_{x}\Gamma$ of
$\Gamma=\partial\Omega$ at $\boldsymbol{x}$ and
$\boldsymbol{D}=\frac{1}{2}\big{(}\nabla u+(\nabla u)^{\mathsf{T}}\big{)}$ is
the _rate of deformation tensor_ ;

  * •

_Navier boundary conditions_

$\boldsymbol{u}\cdot\boldsymbol{n}=0\quad\text{and}\quad
P_{\Gamma}\boldsymbol{u}+\alpha
P_{\Gamma}\boldsymbol{S}\boldsymbol{n}=\boldsymbol{0}\quad\text{on}\quad\partial\Omega\times(0,T),$

where $\alpha>0$ and $\boldsymbol{S}=2\mu\boldsymbol{D}-p\boldsymbol{I}$ is
the total stress tensor;

  * •

_Neumann boundary conditions_

$\boldsymbol{S}\boldsymbol{n}=\boldsymbol{0}\quad\text{on}\quad\partial\Omega\times(0,T).$

The boundary symbols related to the Dirichlet resp. Neumann boundary
conditions are homogeneous of order $0$ resp. order $1$. On the other hand,
free slip conditions and the Navier boundary conditions are – in general – not
homogeneous (neither of order $0$ nor $1$) due to the involvement of the
orthogonal projection $P_{\Gamma}$ on the tangent space.
In this sense, mixed type boundary conditions appear naturally in the theory
of Navier–Stokes equations.

######  Example 1.3 (Pattern formation).

In a recent research article [30], the authors motivate and investigate the
pattern which are formed for a particular two component reaction-diffusion
system of the form

$\begin{cases}\partial_{t}u-d_{u}\Delta u=f(u,v)&\text{in
}(0,T)\times\Omega,\\\ \partial_{t}v-d_{v}\Delta v=g(u,v)&\text{in
}(0,T)\times\Omega,\end{cases}$

depending on the type of boundary conditions which are imposed. One of the
several types of boundary conditions considered there is given by the mixed
type boundary condition

$u=R\quad\text{and}\quad\partial_{\boldsymbol{n}}v=0\quad\text{on
}(0,T)\times\partial\Omega,$

where $R\in\operatorname{\mathbb{R}}$ is a constant. In their derivation of
such specific boundary conditions, they make a case for their relevance. Note
that in this particular example, maximal regularity for the quasi-autonomous
linear system follows from the scalar theory, since the diffusion process is
modelled by Fickean diffusion. However, as soon as cross-diffusion is allowed
for, in the sense of a Fick–Onsager or Maxwell–Stefan diffusion model, this
approach is not available.

######  Example 1.4 (Thermo-diffusion systems).

For thermo-diffusive reaction-diffusion systems like

$\begin{cases}\partial_{t}\vartheta-\kappa\Delta\vartheta=h(\vartheta,\boldsymbol{c}),&t\geq
0,\,\boldsymbol{x}\in\Omega,\\\
\partial_{t}\boldsymbol{c}-\operatorname{\operatorname{div}}(\boldsymbol{D}\nabla\boldsymbol{c})=\boldsymbol{r}(\vartheta,\boldsymbol{c}),&t\geq
0,\,\boldsymbol{x}\in\Omega\end{cases}$

it is reasonable to fix boundary conditions as follows: For the temperature
$\vartheta$, one might, for example, fix the temperature at the boundary
$\partial\Omega$, i.e. impose Dirichlet boundary conditions
$\vartheta|_{\Sigma}=\Theta$, whereas for the concentrations $c_{i}$ it might
be more natural to impose no-flux boundary conditions or controlled in-flow
conditions, i.e.
$-\boldsymbol{D}\partial_{\boldsymbol{n}}\boldsymbol{c}=\boldsymbol{g}$.
Again, this approach leads to mixed type boundary conditions for the combined
vector variable $(\vartheta,\boldsymbol{c})$.

With these examples in mind, let us comment on the structural restrictions
imposed in [17] and [18]: The $\LL_{p}$-maximal regularity results in [17,
Theorem 7.11] (half-space problem) and [17, Theorem 8.2] (domains with compact
boundary), both rely on a version of the Lopatinskii-Shapiro condition in
which all the boundary operators $\mathcal{B}_{j}$, $j=1,\ldots,N$, are
_homogeneous_ of degree $m_{j}\in\operatorname{\mathbb{N}}_{0}$ up to some
lower order perturbation. Therefore, in the definition of the principle symbol
as used there, e.g. in [17, Section 8.1], the principle part of the boundary
operators for _mixed type_ boundary conditions could not ’see’ the Dirichlet
type contribution, as their order of differentiation is $0$, i.e. strictly
smaller than the order $1$ for the Neumann type parts. However, this can _not_
be considered as a relatively compact, lower order perturbation of the first
order part of the boundary operator $\mathcal{B}_{j}$. We, therefore, extend
the results in [17] and [18] to boundary conditions of such mixed type. For
this purpose, we have to closely follow the lines of [17, Chapter 6–8] and
[18], and observe which adjustments are needed to transfer the results there
to this more general and, thus, more flexible setup. In particular, we have to
introduce range conditions for lower order terms which respect the structure
of the boundary conditions. Note that the latter obstacle does not appear in
the case of homogeneous principal parts, since in that case it is clear what
should be a lower order perturbation of the boundary condition (namely, all
linear boundary operators of strictly lesser differentiability order than the
principle part).
To conclude this introductory section, let us now give a brief overview over
this manuscript: Since we will need notation and results from modern harmonic
analysis, in Section 2 we begin by providing the basic notation as well as
introducing important notions such as $\mathcal{R}$-boundedness,
$\mathcal{R}$-sectoriality and the Kalton–Weis generalisation of Mikhlin’s
multiplier theorem. Also, we present a useful auxiliary result on holomorphic
operator families on sectors, see Proposition 2.7, where
$\mathcal{R}$-boundedness of the family of operators $T(\lambda)$ on one
sector implies $\mathcal{R}$-boundedness of $\lambda T^{\prime}(\lambda)$ on
any smaller sector. In Section 3, we review the existing theory of vector-
valued elliptic differential operators on the full space
$\operatorname{\mathbb{R}}^{n}$, where the solution can be constructed via the
Fourier transform on the full-space $\operatorname{\mathbb{R}}^{n}$, see
Theorem 3.5. Then, in Section 4 we outline the basic ideas for solving the
parabolic initial-boundary value problem on the half-space
$\operatorname{\mathbb{R}}_{+}^{n+1}$ by employing the Laplace transform with
respect to time $t>0$, and partial Fourier transform with respect to the first
$n$ variables $\boldsymbol{x}^{\prime}=(x_{1},\ldots,x_{n})$ of the vector
$\boldsymbol{x}\in\operatorname{\mathbb{R}}_{+}^{n+1}$, leading to a (vector-
valued) system of ordinary differential equations, which can be solved by
rewriting it as a first order system in $y:=x_{n+1}$ system, provided the
_Lopatinskii–Shapiro condition_ is met. The solution operator constructed in
this way formally provides the solution of the corresponding elliptic boundary
value problem, so that (in view of the multiplier theorems) we need to
establish kernel estimates and $\mathcal{R}$-bounds for these solution
operators, establishing maximal regularity for parabolic initial-boundary
value problems on the half-space. Then, in Section 5.1 by localisation
techniques, we may transfer the previous results to general domains with a
compact and sufficiently regular boundary. Whereas the previous sections have
been restricted to zero initial data $u(0,\cdot)=0$, in the last section,
Section 6, we extend the results to non-zero initial data and give the
corresponding optimal regularity results.

##  2\. Notation and preliminaries

###  2.1. Notation

In this manuscript, all Banach spaces under consideration will be Banach
spaces over the field $\operatorname{\mathbb{C}}$ of complex numbers, and will
typically be denoted by capital Roman letters. Some of the results, however,
transfer to the case of real Banach spaces $E$, possibly via complexification
of the Banach space, cf. e.g. [3]. A very important subclass of these Banach
spaces consists of the _Banach spaces $E$ of class $\mathcal{HT}$_ which are
characterised by the condition that the _Hilbert transform_

$(Hf)(x):=\frac{1}{\pi}\big{(}\operatorname{PV}\frac{1}{t}\ast
f\big{)}(x)=\frac{1}{\pi}\lim_{R\rightarrow\infty}\lim_{\varepsilon\rightarrow
0+}\int_{(-R,-\varepsilon)\cup(\varepsilon,R)}\frac{1}{t}f(x-t)\,\mathrm{d}t,\quad
f\in\mathcal{S}(\operatorname{\mathbb{R}};E),\,\boldsymbol{x}\in\operatorname{\mathbb{R}}$

extends to a bounded linear operator on the $E$-valued Lebesgue-Bochner space
$\LL_{p}(\operatorname{\mathbb{R}};E)$ for some (and then, for all)
$p\in(1,\infty)$. Important examples of such Banach spaces of class
$\mathcal{HT}$ are given by Hilbert spaces (or Banach spaces which are
isomorphic to a Hilbert space, which includes any finite dimensional Banach
space) and the standard Lebesgue spaces $\LL_{q}(\Omega)$, $q\in(1,\infty)$,
where the set $\Omega\subseteq\operatorname{\mathbb{R}}^{n}$ is open and non-
empty, see e.g. [26, Propositions 4.2.14 and 4.2.15]. Also, if $E$ is a Banach
space of class $\mathcal{HT}$, so is the Lebesgue–Bochner space
$\LL_{p}(\Omega;E)$ for any open set
$\Omega\subseteq\operatorname{\mathbb{R}}^{n}$ and $p\in(1,\infty)$.

######  Lemma 2.1.

Let $E$ be a Banach space of class $\mathcal{HT}$ and $p\in(1,\infty)$. Then
the Lebesgue–Bochner space $\LL_{p}(\Omega;E)$ is a Banach space of class
$\mathcal{HT}$.

For a proof of this result, see [26, Proposition 4.2.15(b)].

Given two Banach spaces $X$ and $Y$, we denote by
$\operatorname{\mathcal{B}}(X,Y)$ the Banach space of bounded linear operators
$T:X\rightarrow Y$. Moreover,
$\rho(A)=\\{\lambda\in\operatorname{\mathbb{C}}:(\lambda
I-A):\operatorname{\operatorname{dom}}(A)\rightarrow X\text{ is bijective and
}(\lambda I-A)^{-1}\in\operatorname{\mathcal{B}}(X)\\}$ and
$\sigma(A)=\operatorname{\mathbb{C}}\setminus\rho(A)$ denote the resolvent set
and the spectrum of a linear operator
$A:\operatorname{\operatorname{dom}}(A)\subseteq X\rightarrow X$, and
$\operatorname{\operatorname{dom}}(A)$, $\operatorname{\operatorname{ran}}(A)$
and $\operatorname{ker}(A)$ the domain, range and kernel, resp., of a linear
operator $A:\operatorname{\operatorname{dom}}(A)\subseteq X\rightarrow Y$.
For the ($E$-valued) spaces of continuous functions
$\mathrm{C}(\overline{\Omega};E)$, of $k$-times continuously differentiable
functions $\mathrm{C}^{k}(\overline{\Omega};E)$, Lebesgue-Bochner spaces
$\LL_{p}(\Omega;E)$, Sobolev spaces $\mathrm{W}_{p}^{k}(\Omega;E)$, Sobolev-
Slobodetskii spaces $\mathrm{W}_{p}^{s}(\Omega;E)$ and Bessel-potential spaces
$\mathrm{H}_{p}^{s}(\Omega;E)$, we use standard notation. Moreover, by

$\mathrm{C}_{\mathrm{l}}(\overline{\Omega}):=\\{f\in\mathrm{C}(\overline{\Omega}):\,\lim_{\left|\boldsymbol{x}\right|\rightarrow\infty}f(\boldsymbol{x})=:f(\infty)\text{
exists, if }\Omega\text{ is unbounded}\\}$

we mean the functions which are continuous on $\Omega$ up to its boundary, and
which additionally converge as $\left|\boldsymbol{z}\right|\rightarrow\infty$,
provided $\Omega$ is unbounded.
Later on, inhomogeneous Lebesgue spaces
$\LL_{p,q}(J\times\Omega):=\LL_{p}(J;\LL_{q}(\Omega))$ for $p,q\in[1,\infty)$
and any proper interval $J\subseteq\operatorname{\mathbb{R}}$ and Sobolev
spaces or Sobolev–Slobodetskii spaces
$\mathrm{W}_{p,q}^{(\tau,\sigma)}(J\times\Omega):=\mathrm{W}_{p}^{\tau}(J;\LL_{q}(\Omega))\cap\LL_{p}(J;\mathrm{W}_{q}^{\sigma}(\Omega))$
will appear, as well as Besov spaces $\mathrm{B}_{p,q}^{s}(\Omega)$ and
Triebel–Lizorkin spaces $\mathrm{F}_{p,q}^{s}(\Omega)$. As before, we denote
by $\LL_{p,q}(J\times\Omega;E)$ etc. their vector-valued versions. Moreover,
sometimes we write $\mathrm{W}_{p,q}^{(\tau,\sigma)\cdot s}$ for
$\mathrm{W}_{p,q}^{(\tau s,\sigma s)}$. For more information on these spaces,
we refer to the rich existing literature, e.g. [1], [12], [40], [42], [43],
[44], [45] and [48].

###  2.2. Preliminaries

Since the results will heavily rely on modern results on harmonic analysis,
especially the theory of $\mathcal{R}$-bounded operator families and its
relation to maximal regularity of abstract Cauchy problems, let us in a
preliminary section recall some of the (for our purpose) most relevant results
in this perspective. A fundamental role will be played by
$\mathcal{R}$-bounded subsets in a space of bounded linear operators, cf. [26,
Definition 5.3.13].

######  Definition 2.2.

Let $X$ and $Y$ be Banach spaces and
$\mathcal{T}\subseteq\operatorname{\mathcal{B}}(X,Y)$ be a family of bounded
linear operators. $\mathcal{T}$ is called _$\mathcal{R}$ -bounded_ if for some
(then: for all) $p\in[1,\infty)$ there is a constant $C_{p}>0$ such that for
every finite subfamily $\\{T_{\nu}\\}_{\nu=1}^{N}\subseteq\mathcal{T}$,
vectors $\\{x_{\nu}\\}_{\nu=1}^{N}\subseteq X$, and independent random
variables $\\{\varepsilon_{\nu}\\}_{\nu=1}^{N}$ with
$\boldsymbol{P}(\varepsilon_{\nu}=\pm 1)=\frac{1}{2}$ on some probability
space $(\Omega,\mathcal{M},\mu)$, for arbitrary
$N\in\operatorname{\mathbb{N}}$, the estimate

(2)
$\left\|\sum_{\nu=1}^{N}\varepsilon_{\nu}T_{\nu}x_{\nu}\right\|_{\LL_{p}(\Omega;Y)}\leq
C_{p}\left\|\sum_{\nu=1}^{N}\varepsilon_{\nu}x_{\nu}\right\|_{\LL_{p}(\Omega;X)}$

is valid. In this case, we write

$\mathcal{R}(\mathcal{T})=\mathcal{R}_{p}(\mathcal{T})=\inf\\{C_{p}>0:\,\eqref{eqn:R-bound}\text{
is valid }\\}$

for its $\mathcal{R}$-bound.

######  Remark 2.3.

Note the following:

  1. (1)

If $X$ is (topologically isomorphic to) a Hilbert space,
$\mathcal{T}\subseteq\operatorname{\mathcal{B}}(X,Y)$ is $\mathcal{R}$-bounded
if and only if $\mathcal{T}\subseteq\operatorname{\mathcal{B}}(X,Y)$ is
(uniformly) bounded, cf. the remarks right after [26, Definition 5.3.13]. For
general Banach spaces, this equivalence is no longer true, but in any case an
$\mathcal{R}$-bounded operator family is always (uniformly) bounded.

  2. (2)

If $X=Y=\LL_{p}(\Omega)$ for some open
$\Omega\subseteq\operatorname{\mathbb{R}}^{n}$, then via the _Khintchine
inequality_ , see [26, Theorem 3.2.23], $\mathcal{R}$-boundedness can be
characterised by the _square-function estimate_

$\left\|\big{(}\sum_{\nu=1}^{N}\left|T_{\nu}x_{\nu}\right|^{2}\big{)}^{1/2}\right\|_{\LL_{p}(\Omega)}\leq
M_{p}\left\|\big{(}\sum_{\nu=1}^{N}\left|x_{\nu}\right|^{2}\big{)}^{1/2}\right\|_{\LL_{p}(\Omega)}$

for any finite $\\{T_{\nu}\\}_{\nu=1}^{N}\subseteq\mathcal{T}$ and
$\\{x_{\nu}\\}_{\nu=1}^{N}\subseteq\LL_{p}(\Omega)$, cf. [34, Theorem 1.d.6]
for more general ($q$-concave for some $q<\infty$) Banach spaces.

  3. (3)

For any fixed $p\in[1,\infty)$, the $\mathcal{R}$-bounds are subadditive and
submultiplicative [26, Remark 5.3.14(1)]:

$\displaystyle\mathcal{R}(\mathcal{S}+\mathcal{T})$
$\displaystyle\leq\mathcal{R}(\mathcal{S})+\mathcal{S}(\mathcal{T})$
$\displaystyle\text{for
}\mathcal{S},\mathcal{T}\subseteq\operatorname{\mathcal{B}}(X,Y),$
$\displaystyle\mathcal{R}(\mathcal{S}\mathcal{T})$
$\displaystyle\leq\mathcal{R}(\mathcal{S})\mathcal{R}(\mathcal{T})$
$\displaystyle\text{for
}\mathcal{T}\subseteq\operatorname{\mathcal{B}}(X,Y),\,\mathcal{S}\subseteq\operatorname{\mathcal{B}}(Y,Z).$

  4. (4)

If $\mathcal{T}$, $\mathcal{T}_{n}\subseteq\operatorname{\mathcal{B}}(X,Y)$
are such that

$\sup_{n\geq
1}\mathcal{R}(\mathcal{T}_{n})<\infty\quad\text{and}\quad\lim_{n\rightarrow\infty}\sup_{T\in\mathcal{T}}\inf_{T_{n}\in\mathcal{T}_{n}}\left\|T-T_{n}\right\|_{\operatorname{\mathcal{B}}(X;Y)}=0,$

then

$\mathcal{R}(\mathcal{T})\leq\limsup_{n\rightarrow\infty}\mathcal{R}(\mathcal{T}_{n})<\infty.$

###### Proof.

We only prove the last statement here. Let $N\in\operatorname{\mathbb{N}}$ be
an arbitrary natural number, $\\{x_{\nu}\\}_{\nu=1,\ldots,N}\subseteq X$,
$\\{T_{\nu}\\}_{\nu=1,\ldots,N}\subseteq\mathcal{T}$ and
$\\{\varepsilon_{\nu}\\}_{\nu=1,\ldots,N}$ i.i.d. random variables with
$\boldsymbol{P}(\varepsilon_{\nu}=\pm 1)=\frac{1}{2}$ on common probability
space $(\Omega,\mathcal{M},\mu)$. Then, setting
$C_{n}:=\mathcal{R}(\mathcal{T}_{n})+\frac{1}{n}$, and choosing
$T_{n,\nu}\in\mathcal{T}_{n}$ such that
$\left\|T_{\nu}-T_{n,\nu}\right\|_{\operatorname{\mathcal{B}}(X;Y)}\leq\inf_{S_{n}\in\mathcal{T}_{n}}\left\|T_{\nu}-S_{n}\right\|+\frac{1}{n}$,
we may conclude that

$\displaystyle\left\|\sum_{\nu=1}^{N}\varepsilon_{\nu}T_{\nu}x_{\nu}\right\|_{\LL_{p}(\Omega;Y)}$
$\displaystyle\leq\left\|\sum_{\nu=1}^{N}\varepsilon_{\nu}T_{n,\nu}x_{\nu}\right\|_{\LL_{p}(\Omega;Y)}+\left\|\sum_{\nu=1}^{N}\varepsilon_{\nu}(T_{\nu}-T_{n,\nu})x_{\nu}\right\|_{\LL_{p}(\Omega;Y)}$
$\displaystyle\leq
C_{n}\left\|\sum_{\nu=1}^{N}\varepsilon_{\nu}x_{\nu}\right\|_{\LL_{p}(\Omega;X)}+\sum_{\nu=1}^{N}\left\|T_{\nu}-T_{n,\nu}\right\|_{\operatorname{\mathcal{B}}(X)}\left\|\varepsilon_{\nu}x_{\nu}\right\|_{\LL_{p}(\Omega;X)}.$

By assumption, the second sum tends to zero as $n\rightarrow\infty$, whereas
the first term can be estimated by

$\limsup_{n\rightarrow\infty}C_{n}\cdot\left\|\sum_{\nu=1}^{N}\varepsilon_{\nu}x_{\nu}\right\|_{\LL_{p}(\Omega;X)}=\limsup_{n\rightarrow\infty}\mathcal{R}(\mathcal{T}_{n})\left\|\sum_{\nu=1}^{N}\varepsilon_{\nu}x_{\nu}\right\|_{\LL_{p}(\Omega;X)}.$

Hence,
$\mathcal{R}(\mathcal{T})\leq\limsup_{n\rightarrow\infty}\mathcal{R}(\mathcal{T}_{n})$
and, in particular, $\mathcal{T}$ is $\mathcal{R}$-bounded. ∎

######  Corollary 2.4.

Let $\Lambda$ be an arbitrary index set and
$\\{T_{\lambda}:\lambda\in\Lambda\\}\subseteq\operatorname{\mathcal{B}}(X)$ be
a family of bounded linear operators on a Banach space $X$. Assume that

$\mathcal{R}(\\{T_{\lambda}:\lambda\in\Lambda\\})=:\rho<1.$

Then, the resolvent operator
$R_{\lambda}:=(I-T_{\lambda})^{-1}\in\operatorname{\mathcal{B}}(X)$ exists for
all $\lambda\in\Lambda$ and

$\mathcal{R}(\\{R_{\lambda}:\lambda\in\Lambda\\})\leq\frac{1}{1-\rho}.$

###### Proof.

Since any $\mathcal{R}$-bounded family is uniformly bounded with uniform bound
less or equal the $\mathcal{R}$-bound,
$\sup_{\lambda\in\Lambda}\left\|T_{\lambda}\right\|_{\operatorname{\mathcal{B}}(X)}\leq\rho<1$,
so that the Neumann series
$(I-T_{\lambda})^{-1}=\sum_{\nu=0}^{\infty}T_{\lambda}^{\nu}$ converges in
$\operatorname{\mathcal{B}}(X)$ for all $\lambda\in\Lambda$. Since the
$\mathcal{R}$-bounds are subadditive, for every finite
$K\in\operatorname{\mathbb{N}}$, and submultiplicative, we obtain that

$\displaystyle\mathcal{R}(\\{\sum_{\nu=0}^{K}T_{\lambda}^{\nu}:\,\lambda\in\Lambda\\})$
$\displaystyle\leq\sum_{\nu=0}^{K}\mathcal{R}(\\{T_{\lambda}^{\nu}:\,\lambda\in\Lambda\\})\leq\sum_{\nu=0}^{K}\mathcal{R}(\\{T_{\lambda}:\,\lambda\in\Lambda\\})^{\nu}$
$\displaystyle\leq\sum_{\nu=0}^{K}\rho^{\nu}=\frac{1-\rho^{K+1}}{1-\rho}.$

Therefore, by Remark 2.3

$\displaystyle\mathcal{R}(\\{R_{\lambda}:\,\lambda\in\Lambda\\})$
$\displaystyle=\mathcal{R}(\\{\sum_{\nu=0}^{\infty}T_{\lambda}^{\nu}:\,\lambda\in\Lambda\\})\leq\limsup_{K\rightarrow\infty}\mathcal{R}(\\{\sum_{\nu=0}^{K}T_{\lambda}^{\nu}:\,\lambda\in\Lambda\\})$
$\displaystyle\leq\limsup_{K\rightarrow\infty}\frac{1-\rho^{K+1}}{1-\rho}=\frac{1}{1-\rho}.$

∎

######  Remark 2.5.

In Corollary 2.4, it suffices to assume that

$\limsup_{\nu\rightarrow\infty}\big{(}\mathcal{R}\\{T_{\lambda}^{\nu}:\,\lambda\in\Lambda\\}\big{)}^{1/\nu}<1.$

(The left-hand side may be seen as a $\mathcal{R}$-boundedness based version
of the spectral radius.) In that case there are
$\nu_{0}\in\operatorname{\mathbb{N}}$, $C_{0}\geq 0$ and $\rho\in(0,1)$ such
that
$\big{(}\mathcal{R}\\{T_{\lambda}^{\nu}:\,\lambda\in\Lambda\\}\big{)}^{1/\nu}\leq\rho$
for all $\nu\geq\nu_{0}$ and
$\sum_{\nu=0}^{\nu_{0}-1}\mathcal{R}\\{T_{\lambda}^{\nu}:\,\lambda\in\Lambda\\}\leq
C_{0}$, and in this case,

$\mathcal{R}\\{R_{\lambda}:\,\lambda\in\Lambda\\}\leq C_{0}+\frac{1}{1-\rho}.$

Lutz Weis showed that, for Banach spaces of class $\mathcal{HT}$, Mikhlin’s
Multiplier Theorem can be generalised from the scalar case by demanding
$\mathcal{R}$-boundedness of the (now operator-valued) functions
$\left|\boldsymbol{\xi}\right|^{\left|\boldsymbol{\alpha}\right|}\mathrm{D}^{\boldsymbol{\alpha}}M(\boldsymbol{\xi})$
on $\operatorname{\mathbb{R}}^{n}\setminus\\{0\\}$ instead of merely uniform
boundedness as in the classical Mikhlin Multiplier Theorem. For a proof of
this result, see [39, Theorem 4.4] (or [17, Theorem 3.25] for the case
$M\in\mathrm{C}^{n}(\operatorname{\mathbb{R}}^{n}\setminus\\{0\\};\operatorname{\mathcal{B}}(X,Y))$).

######  Theorem 2.6 (Mikhlin’s Theorem in $n$ variables).

Let $X,Y$ be Banach spaces of class $\mathcal{HT}$. Assume that a function
$M:\operatorname{\mathbb{R}}^{n}\setminus\\{0\\}\rightarrow\operatorname{\mathcal{B}}(X,Y)$
has distributional derivates $\mathrm{D}^{\boldsymbol{\alpha}}M$ which for
each $\boldsymbol{\alpha}\in\\{0,1\\}^{n}$ can be represented by a function
and satisfy

$\mathcal{R}(\\{\left|\boldsymbol{\xi}\right|^{\left|\boldsymbol{\alpha}\right|}\mathrm{D}^{\boldsymbol{\alpha}}M(\boldsymbol{\xi}):\quad\boldsymbol{\xi}\in\operatorname{\mathbb{R}}^{n}\setminus\\{0\\}\\})=:\kappa_{\boldsymbol{\alpha}}<\infty$

for all $\boldsymbol{\alpha}\in\\{0,1\\}^{n}$. Then the operator $T_{M}$
defined by
$T_{M}f:=\mathcal{F}^{-1}M\mathcal{F}f\in\mathcal{S}^{\prime}(\operatorname{\mathbb{R}}^{n};Y)$
for all $\mathcal{F}f\in\mathcal{D}(\operatorname{\mathbb{R}}^{n};X)$ (here,
$\mathcal{F}$ denotes the Fourier transform on
$\operatorname{\mathbb{R}}^{n}$) extends to a bounded linear operator
$\LL_{p}(\operatorname{\mathbb{R}}^{n};X)\rightarrow\LL_{p}(\operatorname{\mathbb{R}}^{n};Y)$
with operator norm
$\left\|T_{M}\right\|_{\operatorname{\mathcal{B}}(\LL_{p}(\operatorname{\mathbb{R}}^{n};X);\LL_{p}(\operatorname{\mathbb{R}}^{n};Y))}\leq
C\sum_{\boldsymbol{\alpha}\in\\{0,1\\}^{n}}\kappa_{\boldsymbol{\alpha}}$.

As a rule of thumb, the $\mathcal{R}$-boundedness of the function itself
typically is the most critical step in checking the assumptions of Mikhlin’s
theorem. In fact, we can provide the following result for operator families
which are holomorphic and $\mathcal{R}$-bounded on sectors. We did not find
this precise statement in the existing literature, although Robert Denk draw
our attention to a paper of Kunstmann and Weis, who used a similar technique
in the proof of [31, Example 2.16].
For every $\vartheta\in(0,\pi]$ we define the _sector_
$\Sigma_{\vartheta}\subseteq\operatorname{\mathbb{C}}$ as

$\Sigma_{\vartheta}=\\{z\in\operatorname{\mathbb{C}}\setminus\\{0\\}:\,\left|\arg(z)\right|<\vartheta\\},$

see Figure 1.

Figure 1. A sector in $\operatorname{\mathbb{C}}$.

######  Proposition 2.7.

Let $X$ be a complex Banach space, $\phi\in(0,\pi)$ and let
$T\in\mathrm{C}(\overline{\Sigma_{\pi-\phi}};\operatorname{\mathcal{B}}(X))\cap\mathcal{H}(\Sigma_{\pi-\phi};\operatorname{\mathcal{B}}(X))$
be holomorphic on a sector and continuous up to its boundary. If

$\mathcal{R}(\\{T(\lambda):\,\lambda\in\Sigma_{\pi-\phi}\\})=:C<\infty,$

then for all $\phi^{\prime}\in(\phi,\pi)$ it holds that

$\mathcal{R}(\\{\lambda
T^{\prime}(\lambda):\,\lambda\in\Sigma_{\pi-\phi^{\prime}}\\})\leq\frac{C}{\sin(\phi^{\prime}-\phi)^{2}}.$

Consequently, for every $k\in\operatorname{\mathbb{N}}$ and
$\phi^{\prime}\in(\phi,\pi)$,
$\mathcal{R}(\\{\lambda^{k}T^{(k)}(\lambda):\,\lambda\in\Sigma_{\pi-\phi^{\prime}}\\})<\infty$.

######  Remark 2.8.

Note that it suffices to demand weak holomorphy since every weakly holomorphic
function $T:\Sigma_{\pi-\phi}\rightarrow\operatorname{\mathcal{B}}(X)$ is
already holomorphic.

###### Proof of Proposition 2.7.

Let us fix some $\phi^{\prime}\in(\phi,\pi)$ and first consider any
$\lambda\in\Sigma_{\pi-\phi^{\prime}}$ and estimate its distance to the
boundary $\partial\Sigma_{\pi-\phi}$. In fact, we can distinguish between two
cases, namely $\lambda\in\overline{\Sigma_{\frac{\pi}{2}-\phi}}$ (only
possible if $\phi^{\prime}<\frac{\pi}{2}$) and
$\lambda\in\Sigma_{\pi-\phi^{\prime}}\setminus\overline{\Sigma_{\frac{\pi}{2}-\phi}}$.
In the first case, the distance from $\lambda$ to the boundary
$\partial\Sigma_{\pi-\phi}$ can easily be seen to be $\left|\lambda\right|$ as
$0$ is the closest point to $\lambda$ on $\partial\Sigma_{\pi-\phi}$. In
particular, in that case
$B_{\left|\lambda\right|}(\lambda)\subseteq\Sigma_{\pi-\phi}$. On the other
hand, if $\lambda\not\in\overline{\Sigma_{\frac{\pi}{2}-\phi}}$ this will not
be the case, but the closest point to $\lambda$ on $\partial\Sigma_{\pi-\phi}$
will be the orthogonal projection onto the line
$\\{r\mathrm{e}^{\pm\mathrm{i}(\pi-\phi)}:\,r>0\\}$, where the sign is
positive or negative depending on whether $\operatorname{Im}\,\lambda_{0}$ is
positive or negative, resp. W.l.o.g. let us assume that
$\operatorname{Im}\,\lambda>0$ (the case $\operatorname{Im}\,\lambda<0$
follows similarly). Then,

$\displaystyle\operatorname{dist}(\lambda,\partial\Sigma_{\pi-\phi})=\operatorname{Im}\,(\mathrm{e}^{\mathrm{i}\phi}\lambda)\geq\sin(\phi^{\prime}-\phi)\left|\lambda\right|,$

since
$\mathrm{e}^{\mathrm{i}\phi}\lambda\in\Sigma_{\phi^{\prime}-\phi}\cap\\{z\in\operatorname{\mathbb{C}}:\,\operatorname{Re}\,z<0,\,\operatorname{Im}\,z>0\\}$,
and, hence,
$B_{\sin(\phi^{\prime}-\phi)\left|\lambda\right|}(\lambda)\subseteq\Sigma_{\pi-\phi}$.
Using the Cauchy integral formula and the contraction principle for
$\mathcal{R}$-bounds of parameter-dependent integrals, we infer that, fixing
an arbitrary $\varepsilon\in(0,1)$,

$\displaystyle\mathcal{R}(\\{\lambda
T^{\prime}(\lambda):\,\lambda\in\Sigma_{\pi-\phi^{\prime}}\\})$
$\displaystyle=\mathcal{R}\big{(}\big{\\{}\frac{\lambda}{2\pi\mathrm{i}}\int_{\partial
B_{\varepsilon\sin(\phi^{\prime}-\phi)\left|\lambda\right|}(\lambda)}\frac{T(\omega)}{(\lambda-\omega)^{2}}\,\mathrm{d}\omega:\,\lambda\in\Sigma_{\pi-\phi^{\prime}}\big{\\}}\big{)}$
$\displaystyle\leq\mathcal{R}\big{(}\big{\\{}\frac{\left|\lambda\right|^{2}}{\left|\lambda-\omega\right|^{2}}:\,\lambda\in\Sigma_{\pi-\phi^{\prime}},\,\omega\in\partial
B_{\varepsilon\sin(\phi^{\prime}-\phi)\left|\lambda\right|}(\lambda)\\}\cdot\mathcal{R}\big{(}\big{\\{}T(\omega):\,\lambda\in\Sigma_{\pi-\phi^{\prime}},\,\omega\in\partial
B_{\varepsilon\sin(\phi^{\prime}-\phi)\left|\lambda\right|}(\lambda)\big{\\}}\big{)}$
$\displaystyle=\frac{1}{(\varepsilon\sin(\phi^{\prime}-\phi))^{2}}\mathcal{R}(\\{T(\lambda):\,\lambda\in\Sigma_{\pi-\phi}\\})=\frac{C}{\varepsilon^{2}\sin(\phi^{\prime}-\phi)^{2}}.$

Letting $\varepsilon\rightarrow 1-$, we may, thus conclude that

$\mathcal{R}(\\{\lambda
T^{\prime}(\lambda):\,\lambda\in\Sigma_{\pi-\phi^{\prime}}\\})\leq\frac{C}{\sin(\phi^{\prime}-\phi)^{2}}.$

The $\mathcal{R}$-boundedness of the sets
$\\{\lambda^{k}T^{(k)}(\lambda):\,\lambda\in\Sigma_{\pi-\phi^{\prime}}\\}$
follows by induction over $k\in\operatorname{\mathbb{N}}$. ∎

This property comes in handy for elliptic boundary value problems and
parabolic initial-boundary value problems, since for these usually
$\mathcal{R}$-boundedness can be guaranteed on some complex sector rather than
merely on $(0,\infty)$, cf. the characterisation of $\LL_{p}$-maximal
regularity in Theorem 2.12 below. In fact, in the theory of parabolic initial-
boundary value problems, sectorial operators appear naturally as generators of
analytic semigroups, see e.g. [24, Theorem II.4.6]. Combining the property of
sectoriality with the notion of $\mathcal{R}$-bounded operators then leads to
the (in general much smaller) class of $\mathcal{R}$-sectorial operators,
which – as we will immediately see in Theorem 2.12 below – appears inevitably
when looking for $\LL_{p}$-maximal regularity of abstract Cauchy problems.

######  Definition 2.9 ($\mathcal{R}$-sectorial operators).

A sectorial operator $A$ on a Banach space $X$ (write $A\in\mathcal{S}(X)$) is
called $\mathcal{R}$-sectorial (write $A\in\mathcal{R}\mathcal{S}(X)$), if

$\mathcal{R}_{A}(0):=\mathcal{R}(\\{\lambda(\lambda+A)^{-1}:\,\lambda>0\\})<\infty.$

In this case, the _$\mathcal{R}$ -angle_ $\phi_{A}^{\mathcal{R}}$ of $A$ is
defined as

$\phi_{A}^{\mathcal{R}}:=\inf\\{\theta\in(0,\pi):\mathcal{R}_{A}(\pi-\theta)<\infty\\},$

where

$\mathcal{R}_{A}(\theta):=\mathcal{R}(\\{\lambda(\lambda+A)^{-1}:\,\left|\arg\lambda\right|\leq\theta\\}).$

######  Remark 2.10.

By expanding the resolvent operator $(\lambda-A)^{-1}$ as a Taylor series
around some $s>0$, using the Neumann series and the algebraic properties
(subadditive, submultiplicative) of the $\mathcal{R}$-bounds, one finds that
$\phi_{A}^{\mathcal{R}}>0$ for every $\mathcal{R}$-sectorial operator, i.e.
the definition of the $\mathcal{R}$-angle is meaningful.

###### Proof.

Let $\eta\in(0,\frac{1}{\mathcal{R}_{A}(0)})$. Then, for every $s>0$ and
$\lambda\in\operatorname{\mathbb{C}}$ such that $\operatorname{Re}\,\lambda=s$
and $\left|\operatorname{Im}\,\lambda\right|<\eta s$ (i.e.,
$\lambda\in\Sigma_{\vartheta}$ for $\vartheta=\arctan(\eta)>0$) it holds that
$\lambda+A=(s+A)+(\lambda-s)$, hence,

$(s+A)^{-1}(\lambda+A)=(\lambda+A)(s+A)^{-1}=I+(\lambda-s)(s+A)^{-1},$

where

$\mathcal{R}(\\{(\lambda-s)(s+A)^{-1}:\,s>0,\,\operatorname{Re}\,\lambda=s,\,\left|\operatorname{Im}\,\lambda\right|<\eta
s\\})\leq\eta\mathcal{R}(\\{s(s+A)^{-1}:s>0\\})=\eta\mathcal{R}_{A}(0)<1,$

so that

$\mathcal{R}(\\{(I+(\lambda-s)(s+A)^{-1})^{-1}=\sum_{k=0}^{\infty}(\lambda-s)^{k}(s+A)^{-k}:\,s>0,\,\operatorname{Re}\,\lambda=s,\,\left|\operatorname{Im}\,\lambda\right|<\eta
s\\})<\frac{1}{1-\eta\mathcal{R}_{A}(0)},$

and, thus, we may deduce that

$\displaystyle\mathcal{R}(\\{\lambda(\lambda+A)^{-1}:\operatorname{Re}\,\lambda=s,\,\left|\operatorname{Im}\,\lambda\right|<\eta
s,\,s>0\\})$
$\displaystyle\leq\mathcal{R}(\\{\frac{\lambda}{\operatorname{Re}\,\lambda}:\,\left|\operatorname{Im}\,\lambda\right|<\eta\operatorname{Re}\,\lambda,\,\operatorname{Re}\,\lambda>0\\})\cdot\mathcal{R}(\\{s(s+A)^{-1}:\,s>0\\})$
$\displaystyle\quad\cdot\mathcal{R}(\\{(I+(\lambda-s)(s+A)^{-1})^{-1}:\,\operatorname{Re}\,\lambda=s>0,\,\left|\operatorname{Im}\,\lambda\right|<\eta
s\\})$
$\displaystyle<\sqrt{1+\eta^{2}}\cdot\mathcal{R}_{A}(0)\cdot\frac{1}{1-\eta\mathcal{R}_{A}(0)}<\infty.$

∎

Similar to the perturbation theorems for contractive or analytic semigroup
generators, $\mathcal{R}$-sectorial operators allow for a relative bounded
perturbation theorem, see [17, Propositions 4.2 and 4.3].

######  Proposition 2.11 (Relatively bounded perturbation of
$\mathcal{R}$-sectorial operators).

Suppose that $A$ is an $\mathcal{R}$-sectorial operator on a Banach space $X$
and set

$a:=\mathcal{R}(\\{A(\lambda+A)^{-1}:\,\lambda\in\Sigma_{\theta}\\})<\infty$

for some fixed $\theta\in(0,\pi-\phi_{A}^{\mathcal{R}})$.

  1. (1)

Assume that $B$ is a linear operator with
$\operatorname{\operatorname{dom}}(B)\supseteq\operatorname{\operatorname{dom}}(A)$
and such that, for some $\alpha\in(0,\nicefrac{{1}}{{a}})$,

$\left\|Bu\right\|\leq\alpha\left\|Au\right\|\quad\text{for all
}u\in\operatorname{\operatorname{dom}}(A).$

Then the operator $A+B$ is $\mathcal{R}$-sectorial with $\mathcal{R}$-angle
$\phi_{A+B}^{\mathcal{R}}\geq\pi-\theta$ and

$\mathcal{R}(\\{\lambda(\lambda+A+B)^{-1}:\,\lambda\in\Sigma_{\theta}\\})\leq\frac{1}{1-\gamma}\mathcal{R}(\\{\lambda(\lambda+A)^{-1}:\,\lambda\in\Sigma_{\theta}\\})$

where
$\gamma:=\left\|BA^{-1}\right\|_{\operatorname{\mathcal{B}}(X)}\mathcal{R}(\\{A(\lambda+A)^{-1}:\,\lambda\in\Sigma_{\theta}\\})$.

  2. (2)

Let

$C_{A}=\sup_{\lambda>0}\left\|A(\lambda+A)^{-1}\right\|_{\operatorname{\mathcal{B}}(X)},\quad
M_{A}=\sup_{\lambda>0}\left\|\lambda(\lambda+A)^{-1}\right\|_{\operatorname{\mathcal{B}}(X)}<\infty$

and assume that $B$ is a linear operator with
$\operatorname{\operatorname{dom}}(B)\supseteq\operatorname{\operatorname{dom}}(A)$
and such that, for some $\alpha,\beta>0$,

$\left\|Bu\right\|\leq\alpha\left\|Au\right\|+\beta\left\|u\right\|\quad\text{for
all }u\in\operatorname{\operatorname{dom}}(A).$

Whenever $\alpha<\frac{1}{1+a}C_{A}$ and $\mu>\beta M_{A}\frac{1+a}{1-\alpha
C_{A}(1+a)}$, then $A+B$ is $\mathcal{R}$-sectorial with $\mathcal{R}$-angle
$\phi_{A+B}^{\mathcal{R}}\geq\pi-\theta$ and

$\mathcal{R}(\\{\lambda(\lambda+\mu+A+B)^{-1}:\,\lambda\in\Sigma_{\theta}\\})<\infty.$

As announced above, $\mathcal{R}$-sectorial operators and $\LL_{p}$-maximal
regularity of abstract Cauchy problems are closely related. In fact, the
following result is known for Banach spaces of class $\mathcal{HT}$, see [17,
Theorem 4.4].

######  Theorem 2.12 (Characterisation of $\LL^{p}$-maximal regularity).

Let $X$ be a Banach space of class $\mathcal{HT}$, $p\in(1,\infty)$ and let
$A$ be a sectorial operator on $X$ with spectral angle
$\phi_{A}^{\sigma}=\inf\\{\phi>0:\,\sigma(A)\subseteq\Sigma_{\pi-\phi}\\}<\frac{\pi}{2}$.
In this case, the _abstract Cauchy problem_ with homogeneous initial data

($\mathrm{ACP}_{0}$)
$\begin{cases}\frac{\mathrm{d}}{\mathrm{d}t}u(t)+Au(t)&=f(t)\quad\text{for
}t\geq 0,\\\ u(0)&=0\end{cases}$

admits $\LL^{p}$-maximal regularity on the ray
$\operatorname{\mathbb{R}}_{+}$, i.e. ($\mathrm{ACP}_{0}$) has a unique
solution in class $u:\operatorname{\mathbb{R}}_{+}\rightarrow X$ with
$Au\in\LL_{p}(\operatorname{\mathbb{R}}_{+};X)$ if and only if
$f\in\LL_{p}(\operatorname{\mathbb{R}}_{+};X)$, if and only if $A$ is
$\mathcal{R}$-sectorial with angle $\phi_{A}^{\mathcal{R}}<\frac{\pi}{2}$.
More precisely, the following statements are equivalent:

  1. (1)

The abstract Cauchy problem ($\mathrm{ACP}_{0}$) has $\LL^{p}$-maximal
regularity on $\operatorname{\mathbb{R}}_{+}$.

  2. (2)

The set
$\\{A(\mathrm{i}\rho+A)^{-1}:\,\rho\in\operatorname{\mathbb{R}}\setminus\\{0\\}\\}$
is $\mathcal{R}$-bounded.

  3. (3)

The set $\\{A(\lambda+A)^{-1}:\,\lambda\in\Sigma_{\theta}\\}$ is
$\mathcal{R}$-bounded for some $\theta>\frac{\pi}{2}$.

  4. (4)

The set $\\{\mathrm{e}^{-A\lambda}:\,\lambda\in\Sigma_{\vartheta}\\}$ is
$\mathcal{R}$-bounded for some $\vartheta>0$.

  5. (5)

The sets $\\{\mathrm{e}^{-At}:\,t>0\\}$ and $\\{tA\mathrm{e}^{-At}:\,t>0\\}$
are $\mathcal{R}$-bounded.

######  Remark 2.13.

It may be interesting to compare the conditions on $\LL_{p}$-maximal
regularity with those of the (weaker) conditions on an operator $A$ to
generate a bounded analytic semigroup:

  1. (1)

First note that, for the case that $A$ is invertible, to have maximal
regularity on the (unbounded) ray $\operatorname{\mathbb{R}}_{+}$ one has to
expect exponential stability of the $C_{0}$-semigroup
$(\mathrm{e}^{-tA})_{t\geq 0}$ generated by $-A$, since then from
$Au\in\LL_{p}(\operatorname{\mathbb{R}}_{+};X)$ it follows
$u\in\LL_{p}(\operatorname{\mathbb{R}}_{+};X)$, and for $f=0$ and by the Datko
lemma this implies exponential stability of the semigroup.

  2. (2)

In the context of bounded analytic semigroups, the following assertions are
equivalent, see e.g. [24, Theorem II.4.6]:

     * •

$-A$ generates a bounded (on some sector $\Sigma_{\theta}$, $\theta>0$)
analytic $C_{0}$-semigroup $(\mathrm{e}^{-\lambda
A})_{\lambda\in\Sigma_{\theta}}$.

     * •

$-A$ generates a bounded (on $(0,\infty)$) $C_{0}$-semigroup
$(\mathrm{e}^{-tA})_{t\geq 0}$ such that
$\operatorname{\operatorname{ran}}(\mathrm{e}^{-tA})\subseteq\operatorname{\operatorname{dom}}(A)$
for every $t>0$ and

$M:=\sup_{t>0}\left\|tA\mathrm{e}^{-tA}\right\|<\infty.$

     * •

There is $\vartheta\in(0,\tfrac{\pi}{2})$ such that
$\pm\mathrm{e}^{\mathrm{i}\vartheta}(-A)$ generate bounded $C_{0}$-semigroups.

     * •

$-A$ generates a bounded $C_{0}$-semigroup and

$\sup_{\lambda\in\operatorname{\mathbb{C}}_{0}^{+}}\left\|(\operatorname{Im}\,\lambda)(A+\lambda)^{-1}\right\|<\infty,$

where
$\operatorname{\mathbb{C}}_{\omega}^{+}:=\\{z\in\operatorname{\mathbb{C}}:\,\operatorname{Re}\,z>\omega\\}$
for $\omega\in\operatorname{\mathbb{R}}$.

For families of integral operators, $\mathcal{R}$-bounds on their kernel
functions may help to derive $\mathcal{R}$-boundedness of the familiy, see
[17, Proposition 4.12].

######  Proposition 2.14.

Let $X$ and $Y$ be Banach spaces,
$\Omega\subseteq\operatorname{\mathbb{R}}^{n}$ be open and $p\in(1,\infty)$.
Further, let
$\mathcal{K}\subseteq\operatorname{\mathcal{B}}(\LL_{p}(\Omega;X);\LL_{p}(\Omega;Y))$
be a family of kernel operators, i.e. for each $K\in\mathcal{K}$ there is a
measurable kernel
$k:\Omega\times\Omega\rightarrow\operatorname{\mathcal{B}}(X;Y)$ such that

$(Kf)(\boldsymbol{x})=\int_{\Omega}k(\boldsymbol{x},\boldsymbol{x}^{\prime})f(\boldsymbol{x}^{\prime})\,\mathrm{d}\boldsymbol{x}^{\prime}\quad\text{for
all }f\in\LL_{p}(\Omega;X)\text{ and a.e.\ }\boldsymbol{x}\in\Omega.$

Further, assume that the family of kernels can be $\mathcal{R}$-bounded
pointwise

$\mathcal{R}(\\{k(\boldsymbol{x},\boldsymbol{x}^{\prime}):\,k\in\mathcal{K}\\})\leq\kappa_{0}(\boldsymbol{x},\boldsymbol{x}^{\prime})\quad\text{for
a.e.\ }\boldsymbol{x},\boldsymbol{x}^{\prime}\in\Omega$

by some scalar, measurable function
$\kappa_{0}:\Omega\times\Omega\rightarrow\operatorname{\mathbb{R}}_{+}$ which
itself is the kernel of a scalar integral operator
$K_{0}\in\operatorname{\mathcal{B}}(\LL_{p}(\Omega))$. Then the family
$\mathcal{K}\subseteq\operatorname{\mathcal{B}}(\LL_{p}(\Omega;X);\LL_{p}(\Omega;Y))$
is $\mathcal{R}$-bounded with $\mathcal{R}$-bound
$\mathcal{R}(\mathcal{K})\leq\left\|K_{0}\right\|_{\operatorname{\mathcal{B}}(\LL_{p}(\Omega))}$.

######  Remark 2.15.

For comparison, note that a similar result is true if all $\mathcal{R}$-bounds
are replaced by (uniform) bounds.

######  Definition 2.16 (The classes $\mathcal{H}^{\infty}(X)$,
$\mathcal{R}\mathcal{H}^{\infty}(X)$ and $\mathcal{BIP}(X)$).

A sectorial operator $A\in\mathcal{S}(X)$ is said to have _bounded imaginary
powers_ (write $A\in\mathcal{BIP}(X)$), if $A^{\mathrm{i}s}$ (defined via the
Dunford calculus for sectorial operators) is in
$\operatorname{\mathcal{B}}(X)$ for every $s\in\operatorname{\mathbb{R}}$ and
there is a constant $C>0$ such that
$\left\|A^{\mathrm{i}s}\right\|_{\operatorname{\mathcal{B}}(X)}\leq C$ for all
$s\in[-1,1]$, cf. [17, Definition 2.4].
A sectorial operator $A\in\mathcal{S}(X)$ is said to have _bounded holomorphic
functional calculus_ (write $A\in\mathcal{H}^{\infty}(X)$), if there are
$\phi>0$ and $C_{\phi}>0$ such that for all
$h\in\mathcal{H}_{0}(\Sigma_{\phi}):=\\{h:\Sigma_{\phi}\rightarrow\operatorname{\mathbb{C}}:\,h\text{
holomorphic and bounded}\\}$ which tend polynomially to zero as
$\left|\boldsymbol{z}\right|\rightarrow\infty$, i.e. there are $C,s>0$ such
that

$\left|h(z)\right|\leq C(1+\left|z\right|)^{-s},\quad z\in\Sigma_{\phi}$

the operator $h(A)\in\operatorname{\mathcal{B}}(X)$, which in this case can be
defined via the Dunford calculus, satisfies the norm estimate

$\left\|h(A)\right\|_{\operatorname{\mathcal{B}}(X)}\leq
C_{\phi}\left\|h\right\|_{\infty}.$

We set $\phi_{A}^{\infty}$ to be the infimum of those $\phi$ for which such a
constant $C_{\phi}>0$ exists, cf. [17, Definition 2.9].
An operator $A\in\mathcal{H}^{\infty}(X)$ is said to have an
$\mathcal{R}$-bounded functional calculus (write
$A\in\mathcal{R}\mathcal{H}^{\infty}(X)$), if

$\mathcal{R}(\\{h(A):\,h\in\mathcal{H}_{0}(\Sigma_{\phi}),\,\left\|h\right\|_{\infty}\leq
1\\})<\infty$

for some $\phi>0$. The infimum of those $\phi$ which are admissible for
$\mathcal{R}$-boundedness of the set on the left-hand side is denoted by
$\phi_{A}^{\mathcal{R},\infty}$, cf. [17, Definition 4.9].

######  Remark 2.17 (Hilbert space and Banach space case).

If $X$ is a Hilbert space, then

$\mathcal{R}\mathcal{H}^{\infty}(X)=\mathcal{H}^{\infty}(X)=\mathcal{BIP}(X)\subseteq\mathcal{R}\mathcal{S}(X)=\mathcal{S}(X),$

where the inclusion of the (set of) operators of bounded imaginary powers in
the (set of) $\mathcal{R}$-sectorial operators is strict, in general. For
general Banach spaces the equalities may become strict inclusions, too. In any
case, for every $A\in\mathcal{R}\mathcal{H}^{\infty}(X)$

$\phi_{A}^{\mathcal{R},\infty}\geq\phi_{A}^{\infty}\geq\theta_{A}\geq\phi_{A}^{\mathcal{R}}\geq\phi_{A}^{\sigma},$

but in general these inequalities may be strict.

A nice property of operators with $\mathcal{R}$-bounded
$\mathcal{H}^{\infty}$-calculus is the following, see [17, Proposition 4.10].

######  Proposition 2.18.

Let $X$ be a Banach space, $A\in\mathcal{R}\mathcal{H}^{\infty}(X)$ and let
$\\{h_{\lambda}\\}_{\lambda\in\Lambda}$ be a family of bounded, holomorphic,
scalar functions which is uniformly bounded in
$\mathcal{H}^{\infty}(\Sigma_{\theta})$ for some
$\theta>\phi_{A}^{\mathcal{R},\infty}$ and $\Lambda$ be an arbitrary index
set. Then the family of operators $\\{h_{\lambda}(A)\\}_{\lambda\in\Lambda}$
given by the $\mathcal{H}^{\infty}$-calculus for $A$ is $\mathcal{R}$-bounded.

##  3\. Elliptic differential operators on $\operatorname{\mathbb{R}}^{n}$

Our primary goal is to consider abstract initial-boundary value problems
(IBVP) of parabolic type. However, it is well-known that the Laplace transform
of parabolic equation with respect to the time variable leads to elliptic
boundary value problems (BVP), which in many cases are easier to handle than
directly starting with the parabolic equations themselves. Therefore, we begin
our investigation with elliptic problems arising from parabolic inital-
boundary value problems by formally taking the Laplace transform of these.

###  3.1. The homogeneous, constant coefficient case:

Let $k\in\operatorname{\mathbb{N}}$, $E$ be an arbitrary Banach space and
$\mathcal{A}(\boldsymbol{\xi})$ be a $\operatorname{\mathcal{B}}(E)$-valued
polynomial in $\operatorname{\mathbb{R}}^{n}$ which is homogeneous of order
$k$, i.e.

$\mathcal{A}(\boldsymbol{\xi})=\sum_{\left|\boldsymbol{\alpha}\right|=k}a_{\boldsymbol{\alpha}}\boldsymbol{\xi}^{\boldsymbol{\alpha}}$

for some operator-valued constant coefficients
$a_{\boldsymbol{\alpha}}\in\operatorname{\mathcal{B}}(E)$. To this we
associate the differential operator $\mathcal{A}(\mathrm{D})$ by formally
plugging in
$\mathrm{D}=-\mathrm{i}\nabla=(\mathrm{D}_{i})_{i=1,\ldots,n}=(-\mathrm{i}\partial_{x_{i}})_{i=1,\ldots,n}$
as $\boldsymbol{\xi}$:

$\mathcal{A}(\mathrm{D})=\sum_{\left|\boldsymbol{\alpha}\right|=k}a_{\boldsymbol{\alpha}}\mathrm{D}^{\boldsymbol{\alpha}}=\sum_{\left|\boldsymbol{\alpha}\right|=k}a_{\boldsymbol{\alpha}}(-\mathrm{i})^{k}\prod_{i=1}^{n}\frac{\partial^{\alpha_{i}}}{\partial
x_{i}^{\alpha_{i}}}.$

If, for each given function $f$ which is sufficiently regular and integrable
and suitable $\lambda\in\operatorname{\mathbb{C}}$, the vector-valued partial
differential equation

(3) $\lambda
u(\boldsymbol{x})+\mathcal{A}(\mathrm{D})u(\boldsymbol{x})=f(\boldsymbol{x})\quad\text{for
}\boldsymbol{x}\in\operatorname{\mathbb{R}}^{n}$

has a unique solution which can be represented by an integral formula

$u(\boldsymbol{x})=\int_{\operatorname{\mathbb{R}}^{n}}\gamma_{\lambda}(\boldsymbol{x}-\boldsymbol{z})f(\boldsymbol{z})\,\mathrm{d}\boldsymbol{z}$

for some kernel function
$\gamma_{\lambda}:\operatorname{\mathbb{R}}^{n}\rightarrow\operatorname{\mathbb{C}}$,
then, by homogeneity of degree $k$ for the polynomial $\mathcal{A}$, the
kernel function $\gamma_{\lambda}$ must scale as

$\gamma_{\lambda}(\boldsymbol{x})=\left|\lambda\right|^{\tfrac{n}{k}-1}\tilde{\gamma}_{\theta}(\left|\lambda\right|^{1/k}\boldsymbol{x})\quad\text{for
}\boldsymbol{x}\in\operatorname{\mathbb{R}}^{n},\,\theta=\arg(\lambda),\,\lambda\neq
0$

and $\tilde{\gamma}_{\theta}=\gamma_{\mathrm{e}^{\mathrm{i}\theta}}$ is the
fundamental solution of the vector-valued partial differential equation (3),
i.e.

$\mathrm{e}^{\mathrm{i}\theta}\tilde{\gamma}_{\theta}+\mathcal{A}(\mathrm{D})\tilde{\gamma}_{\theta}=\delta_{0}$

in the sense of distributions. To see this, assume that
$\lambda=\rho\mathrm{e}^{\mathrm{i}\theta}$ for some radius $\rho>0$ and some
angle $\theta\in(-\pi,\pi)$ and let $u$ be the solution to the elliptic PDE

$\mathrm{e}^{\mathrm{i}\theta}u+\mathcal{A}(\mathrm{D})u=f$

for some given function $f:\operatorname{\mathbb{R}}^{n}\rightarrow E$ and
which has the integral representation

$u(\boldsymbol{x})=\int_{\operatorname{\mathbb{R}}^{n}}\tilde{\gamma}_{\theta}(\boldsymbol{x}-\boldsymbol{x}^{\prime})f(\boldsymbol{x}^{\prime})\,\mathrm{d}\boldsymbol{x}^{\prime},\quad\boldsymbol{x}\in\operatorname{\mathbb{R}}^{n}.$

Then, since $\mathcal{A}(\boldsymbol{\xi})$ is homogeneous of degree $k$ in
$\boldsymbol{\xi}$, the function
$u_{\lambda}(\boldsymbol{x}):=u(\rho^{1/k}\boldsymbol{x})$ solves the elliptic
PDE

$\displaystyle\lambda
u_{\lambda}(\boldsymbol{x})+\mathcal{A}(\mathrm{D})u_{\lambda}(\boldsymbol{x})$
$\displaystyle=\lambda
u(\rho^{1/k}\boldsymbol{x})+\rho(\mathcal{A}(\mathrm{D})u)(\rho^{1/k}\boldsymbol{x})$
$\displaystyle=\rho\big{[}\mathrm{e}^{\mathrm{i}\theta}u(\rho^{1/k}\boldsymbol{x})+(\mathcal{A}(\mathrm{D})u)(\rho^{1/k}\boldsymbol{x})\big{]}=\rho
f(\rho^{1/k}\boldsymbol{x})$

and has the integral representation

$\displaystyle u_{\lambda}(\boldsymbol{x})$
$\displaystyle=u(\rho^{1/k}\boldsymbol{x})=\int_{\operatorname{\mathbb{R}}^{n}}\tilde{\gamma}_{\theta}(\rho^{1/k}\boldsymbol{x}-\boldsymbol{z})f(\boldsymbol{z})\,\mathrm{d}\boldsymbol{z}$
$\displaystyle=\int_{\operatorname{\mathbb{R}}^{n}}\rho^{\frac{n}{k}}\tilde{\gamma}_{\theta}(\rho^{1/k}(\boldsymbol{x}-\boldsymbol{y}))f(\rho^{1/k}\boldsymbol{y})\,\mathrm{d}\boldsymbol{y}$
$\displaystyle=\int_{\operatorname{\mathbb{R}}^{n}}\rho^{\frac{n}{k}-1}\tilde{\gamma}_{\theta}(\rho^{\frac{1}{k}}(\boldsymbol{x}-\boldsymbol{y}))\rho
f(\rho^{\frac{1}{k}}\boldsymbol{y})\,\mathrm{d}\boldsymbol{y}.$

Thus, we may conclude that
$\gamma_{\lambda}(\boldsymbol{x})=\rho^{\frac{n}{k}-1}\tilde{\gamma}_{\theta}(\rho^{1/k}\boldsymbol{x})$
has the scaling property which we claimed.

######  Definition 3.1 (Parameter-ellipticity).

A homogeneous, $\operatorname{\mathcal{B}}(E)$-valued polynomial
$\mathcal{A}(\boldsymbol{\xi})$ is called parameter-elliptic, if there is an
angle $\phi\in[0,\pi)$ such that the spectrum
$\sigma(\mathcal{A}(\boldsymbol{\xi}))$ of
$\mathcal{A}(\boldsymbol{\xi})\in\operatorname{\mathcal{B}}(E)$ is contained
in the sector $\Sigma_{\phi}$ for every
$\boldsymbol{\xi}\in\mathbb{S}^{n-1}\subseteq\operatorname{\mathbb{R}}^{n}$.
The infimum $\phi_{\mathcal{A}}^{\mathrm{ellipt}}$ of these $\phi>0$ is called
_angle of ellipticity_.

######  Remark 3.2.

Let us make two short remarks on why later on we will choose $k=2m$ to be
even, and how parameter-ellipticity and the Lopatinskii–Shapiro condition
introduced below in Assumption 4.3 will carry over from real vectors
$\boldsymbol{\xi}\in\operatorname{\mathbb{R}}^{n}$ to complex vectors
$\boldsymbol{\xi}+\mathrm{i}\boldsymbol{\eta}\in\operatorname{\mathbb{C}}^{n}$
with sufficiently small imaginary parts of the components.

  1. (1)

To have an angle of ellipticity
$\phi_{\mathcal{A}}^{\mathrm{ellipt}}<\frac{\pi}{2}$, the degree $k$ of the
polynomial $\mathcal{A}(\boldsymbol{\xi})$ must necessarily be even, see,
e.g., the remarks after [17, Definition 5.1]. Therefore, soon we will restrict
ourselves to the most relevant case that $k=2m$ for some
$m\in\operatorname{\mathbb{N}}$.

  2. (2)

Although parameter-ellipticity is first formulated for real unit vectors
$\boldsymbol{\xi}\in\mathbb{S}^{n-1}$, by homogeneity of $\mathcal{A}$ of
order $k$, the spectrum
$\sigma(\mathcal{A}(\boldsymbol{\xi}))\subseteq\Sigma_{\phi}$ is contained in
some sector $\Sigma_{\phi}$, uniformly for all
$\boldsymbol{\xi}\in\operatorname{\mathbb{R}}^{n}\setminus\\{\boldsymbol{0}\\}$.
Moreover, and this will become relevant when considering the
Lopatinskii–Shapiro condition for elliptic boundary value problems, for any
$\boldsymbol{\xi}\in\operatorname{\mathbb{R}}^{n}$ and
$\boldsymbol{\eta}\in\operatorname{\mathbb{R}}^{n}$ sufficiently small
compared to $\boldsymbol{\xi}$, say,
$\left|\boldsymbol{\eta}\right|\leq\varepsilon\left|\boldsymbol{\xi}\right|$
for some $\varepsilon>0$, we may deduce that

$\sigma(\mathcal{A}(\boldsymbol{\xi}+\mathrm{i}\boldsymbol{\eta}))=\sigma\big{(}\sum_{\left|\boldsymbol{\alpha}\right|=k}a_{\boldsymbol{\alpha}}\boldsymbol{\xi}^{\boldsymbol{\alpha}}+\sum_{\left|\boldsymbol{\alpha}\right|=k}a_{\boldsymbol{\alpha}}\sum_{\boldsymbol{0}\lneq\boldsymbol{\beta}\leq\boldsymbol{\alpha}}\boldsymbol{\xi}^{\boldsymbol{\alpha}-\boldsymbol{\beta}}(\mathrm{i}\boldsymbol{\eta})^{\boldsymbol{\beta}}\big{)}\subseteq\Sigma_{\phi^{\prime}}$

for some, in general smaller, angle $\phi^{\prime}\in(0,\phi)$ (which depends
on the chosen $\varepsilon>0$), and where we write
$\boldsymbol{\beta}\leq\boldsymbol{\alpha}$ if component-wise
$\beta_{i}\leq\alpha_{i}$ for $i=1,\ldots,n$.

For some kernel estimates which we will derive right below, we employ the
functions

$p_{k,\nu}^{n}:\operatorname{\mathbb{R}}_{+}\rightarrow\operatorname{\mathbb{R}},\quad
p_{k,\nu}^{n}(r):=\int_{0}^{\infty}\frac{s^{n-2}}{(1+s)^{k-1-\nu}}\mathrm{e}^{-(s+1)r}\,\mathrm{d}s,\quad
k,n,\nu\in\operatorname{\mathbb{N}},\,r>0.$

These are _completely monotone_ , i.e.
$(-1)^{p}\frac{\mathrm{d}^{p}}{\mathrm{d}r^{p}}p_{k,\nu}^{l}\geq 0$ for all
$p\in\operatorname{\mathbb{N}}_{0}$, hence, by the Bernstein–Widder theorem,
see e.g. [47, Theorem 17], is the Laplace transform of a non-negative measure,
i.e.
$p_{k,\nu}^{l}(x)=\int_{0}^{\infty}\mathrm{e}^{-sx}\,\mathrm{d}\mu_{j,\nu}^{l}(s)$,
and

$\int_{0}^{\infty}r^{n+\rho-1}p_{k,n}^{k}(r)\,\mathrm{d}r<\infty\quad\text{if
and only if}\quad\rho>\max\\{-n,\nu-k\\}.$

Moreover, the following integral relation between the functions
$p_{k,\nu}^{n}$ is valid.

######  Lemma 3.3.

For all $c,y>0$ and $k,n,\nu\in\operatorname{\mathbb{N}}_{0}$, the following
identity is valid:

$\int_{0}^{\infty}p_{k,\nu}^{n}(c(y+r))r^{n-1}\,\mathrm{d}r=\frac{(n-1)!}{c^{n}}p_{k+n,\nu}^{n}(cy).$

This identity can be validated straightforwardly via integration by parts, see
[17, Corollary 5.3].

######  Theorem 3.4 (Kernel estimate for parameter-elliptic operators).

Let $k,n\in\operatorname{\mathbb{N}}$ be natural numbers, $E$ be a Banach
space and operator-valued coefficients
$a_{\boldsymbol{\alpha}}\in\operatorname{\mathcal{B}}(E)$ be given such that

$\mathcal{A}(\boldsymbol{\xi})=\sum_{\left|\boldsymbol{\alpha}\right|=k}a_{\boldsymbol{\alpha}}\boldsymbol{\xi}^{\boldsymbol{\alpha}}\quad\text{for
}\boldsymbol{\xi}\in\operatorname{\mathbb{R}}^{n}$

is parameter-elliptic with angle of ellipticity
$\phi_{\mathcal{A}}^{\mathrm{ellipt}}<\pi$. Then, for each
$\phi>\phi_{\mathcal{A}}^{\mathrm{ellipt}}$ and
$\nu\in\operatorname{\mathbb{N}}_{0}$ there exist constants
$c_{\phi},C_{\phi,\nu}>0$ such that the fundamental solution
$\gamma_{\lambda}$ of

$\lambda u+\mathcal{A}(\mathrm{D})u=\delta_{0}$

satisfies the kernel estimates

$\displaystyle\left|\mathrm{D}^{\boldsymbol{\beta}}\gamma_{\lambda}(\boldsymbol{x})\right|\leq
C_{\phi}\left|\lambda\right|^{(\nu+1)\tfrac{n}{k}-1}p_{k,\nu}^{n}(c_{\phi}\left|\lambda\right|^{1/k}\left|\boldsymbol{x}\right|)\quad\text{for
all
}\boldsymbol{x}\in\operatorname{\mathbb{R}}^{n}\setminus\\{0\\},\,\lambda\in\Sigma_{\pi-\phi}\text{
and }\left|\boldsymbol{\beta}\right|=\nu.$

###### Proof.

By [17, Corollary 5.3],

$\left|\mathrm{D}^{\boldsymbol{\beta}}\tilde{\gamma}_{\theta}(\boldsymbol{x})\right|\leq
C_{\phi,\nu}p_{k,\nu}^{n}(c_{\phi}\left|\boldsymbol{x}\right|),$

so from
$\gamma_{\lambda}(\boldsymbol{x})=\left|\lambda\right|^{\frac{n}{k}-1}\tilde{\gamma}_{\theta}(\left|\lambda\right|^{1/k}\boldsymbol{x})$
we find that

$\displaystyle\left|\mathrm{D}^{\boldsymbol{\beta}}\gamma_{\lambda}(\boldsymbol{x})\right|$
$\displaystyle=\left|\lambda\right|^{(\nu+1)\frac{n}{k}-1}\left|(\mathrm{D}^{\boldsymbol{\beta}}\tilde{\gamma}_{\theta})(\left|\lambda\right|^{1/k}\boldsymbol{x})\right|$
$\displaystyle\leq\left|\lambda\right|^{(\nu+1)\frac{n}{k}-1}C_{\phi,\nu}p_{k,\nu}^{n}(c_{\phi}\left|\lambda\right|^{1/k}\left|\boldsymbol{x}\right|).$

∎

The most important results on the full space problem with constant
coefficients can be summarised as follows, cf. [17, Theorems 5.4 and 5.5 and
Corollary 5.6].

######  Theorem 3.5 (Full-space problem).

Let $n,m\in\operatorname{\mathbb{N}}$ and $p\in(1,\infty)$. Assume that the
interior symbol

$\mathcal{A}(\boldsymbol{\xi})=\sum_{\left|\boldsymbol{\alpha}\right|=2m}a_{\boldsymbol{\alpha}}\boldsymbol{\xi}^{\boldsymbol{\alpha}},\quad\boldsymbol{\xi}\in\operatorname{\mathbb{R}}^{n+1}$

is parameter-elliptic with angle of ellipticity
$\phi_{\mathcal{A}}^{\mathrm{ellipt}}<\pi$.

  1. (1)

The $\LL_{p}(\operatorname{\mathbb{R}}^{n+1})$-realisation
$A_{\operatorname{\mathbb{R}}^{n+1}}$ of $\mathcal{A}$, defined as the closure
$A_{\operatorname{\mathbb{R}}^{n+1}}=\overline{A_{0}}$ of the minimal
realisation of the interior symbol on
$\LL_{p}(\operatorname{\mathbb{R}}^{n+1})$

$\displaystyle A_{0}:\quad\operatorname{\operatorname{dom}}(A_{0})$
$\displaystyle=\mathrm{W}_{p}^{2m}(\operatorname{\mathbb{R}}^{n};E)\subseteq\LL_{p}(\operatorname{\mathbb{R}}^{n};E)\rightarrow\LL_{p}(\operatorname{\mathbb{R}}^{n};E),$
$\displaystyle[A_{0}u](\boldsymbol{x})$
$\displaystyle=\mathcal{A}(\mathrm{D})u(\boldsymbol{x}),\quad\text{a.e.\
}\boldsymbol{x}\in\operatorname{\mathbb{R}}^{n+1}$

is sectorial with _spectral angle_
$\phi_{A_{\operatorname{\mathbb{R}}^{n+1}}}^{\sigma}\leq\phi_{\mathcal{A}}^{\mathrm{ellipt}}$.
For its domain
$\operatorname{\operatorname{dom}}(A_{\operatorname{\mathbb{R}}^{n+1}})$ the
inclusions

$\mathrm{W}_{p}^{2m}(\operatorname{\mathbb{R}}^{n};E)\subseteq\operatorname{\operatorname{dom}}(A_{\operatorname{\mathbb{R}}^{n+1}})\subseteq\mathrm{W}_{p}^{2m-1}(\operatorname{\mathbb{R}}^{n};E)$

are valid.

  2. (2)

If $E$ is a Banach space of class $\mathcal{HT}$, then
$A_{\operatorname{\mathbb{R}}^{n+1}}=A_{0}$, i.e.
$\operatorname{\operatorname{dom}}(A_{\operatorname{\mathbb{R}}^{n+1}})=\mathrm{W}_{p}^{2m}(\operatorname{\mathbb{R}}^{n+1})$,
and
$A_{\operatorname{\mathbb{R}}^{n+1}}\in\mathcal{H}^{\infty}(\LL_{p}(\operatorname{\mathbb{R}}^{n+1};E))$
with $\mathcal{H}^{\infty}$-angle
$\phi_{A_{\operatorname{\mathbb{R}}^{n+1}}}^{\infty}\leq\phi_{\mathcal{A}}^{\mathrm{ellipt}}$.
In particular,
$A_{\operatorname{\mathbb{R}}^{n+1}}\in\mathcal{R}\mathcal{S}(\LL_{p}(\operatorname{\mathbb{R}}^{n};E))$
with angle
$\phi_{A_{\operatorname{\mathbb{R}}^{n+1}}}^{\mathcal{R}}\leq\phi_{\mathcal{A}}^{\mathrm{ellipt}}$
and $-A_{\operatorname{\mathbb{R}}^{n+1}}$ generates a bounded analytic
semigroup $(\mathrm{e}^{-tA_{\operatorname{\mathbb{R}}^{n+1}}})_{t\geq 0}$ on
$\LL_{p}(\operatorname{\mathbb{R}}^{n+1};E)$. Moreover, for every angle
$\phi>\phi_{\mathcal{A}}^{\mathrm{ellipt}}$,

$\mathcal{R}(\\{\lambda^{1-\tfrac{\nu}{k}}\mathrm{D}^{\boldsymbol{\beta}}(\lambda+A_{\operatorname{\mathbb{R}}^{n+1}})^{-1}:\,\lambda\in\Sigma_{\pi-\phi},\,0\leq\left|\boldsymbol{\beta}\right|=\nu\leq
2m\\})<\infty.$

For the proof of this theorem, we refer to [17, Theorem 5.5 and Corollary
5.6]. It crucially uses that the $n$-dimensional sphere
$\mathbb{S}^{n}\subseteq\operatorname{\mathbb{R}}^{n+1}$ is compact and
$\mathcal{A}(\boldsymbol{\xi})$ is holomorphic and homogeneous of degree
$k=2m$, so that by Proposition 2.18 the family
$\\{\boldsymbol{\xi}^{\boldsymbol{\alpha}}\mathcal{A}(\boldsymbol{\xi})^{-1}:\,\boldsymbol{\xi}\in\operatorname{\mathbb{R}}^{n+1}\setminus\\{0\\}\\}$
is $\mathcal{R}$-bounded for each fixed
$\boldsymbol{\alpha}\in\operatorname{\mathbb{N}}_{0}^{n+1}$ with
$\left|\boldsymbol{\alpha}\right|=k$, and similarly
$\\{\boldsymbol{\xi}^{\boldsymbol{\beta}}\mathrm{D}^{\boldsymbol{\beta}}[\boldsymbol{\xi}^{\boldsymbol{\alpha}}\mathcal{A}(\boldsymbol{\xi})^{-1}]:\boldsymbol{\xi}\in\operatorname{\mathbb{R}}^{n+1}\setminus\\{0\\}\\}$
is $\mathcal{R}$-bounded as well. The bounded $\mathcal{H}^{\infty}$-calculus
is then established via Cauchy’s theorem and again Proposition 2.18.

###  3.2. Spatially dependent coefficients and lower order perturbations.

Next, we consider the variable coefficient case with lower order coefficients,
i.e.

$\mathcal{A}(\boldsymbol{x},\boldsymbol{\xi})=\sum_{\left|\boldsymbol{\alpha}\right|\leq
k}a_{\boldsymbol{\alpha}}(\boldsymbol{x})\mathrm{D}^{\boldsymbol{\alpha}}$

for some spatial-dependent operator-valued coefficients
$a_{\boldsymbol{\alpha}}(\boldsymbol{x})\in\operatorname{\mathcal{B}}(E)$,
$k\in\operatorname{\mathbb{N}}$, and its minimal
$\LL_{p}(\operatorname{\mathbb{R}}^{n};X)$-realisation

$[A_{0}u](\boldsymbol{x})=\mathcal{A}(\boldsymbol{x},\mathrm{D})u(\boldsymbol{x})\quad\text{for
}u\in\operatorname{\operatorname{dom}}(A_{0})=\mathrm{W}_{p}^{k}(\operatorname{\mathbb{R}}^{n};E)\text{
and a.e.\ }\boldsymbol{x}\in\operatorname{\mathbb{R}}^{n}.$

Using perturbation arguments, the following result can be derived, provided
the Banach space $E$ is of class $\mathcal{HT}$, see [17, Theorem 5.7].

######  Theorem 3.6.

Let $E$ be a Banach space of class $\mathcal{HT}$,
$n,k\in\operatorname{\mathbb{N}}$, $p\in(1,\infty)$ and $\phi_{0}\in(0,\pi)$.
Assume that
$\mathcal{A}(\boldsymbol{x},\mathrm{D})=\sum_{\left|\boldsymbol{\alpha}\right|\leq
k}a_{\boldsymbol{\alpha}}(\boldsymbol{x},\mathrm{D})$ is a differential
operator of order $k$ with variable coefficients which are subject to the
following assumptions:

  1. (1)

$a_{\boldsymbol{\alpha}}\in\mathrm{C}_{\mathrm{l}}(\operatorname{\mathbb{R}}^{n};\operatorname{\mathcal{B}}(E))$
for the top order coefficients where
$\boldsymbol{\alpha}\in\operatorname{\mathbb{N}}_{0}^{n}$ has length
$\left|\boldsymbol{\alpha}\right|=k$.

  2. (2)

The principal part
$\mathcal{A}_{\\#}(\boldsymbol{x},\boldsymbol{\xi})=\sum_{\left|\boldsymbol{\alpha}\right|=k}a_{\boldsymbol{\alpha}}(\boldsymbol{x})\boldsymbol{\xi}^{\boldsymbol{\alpha}}$
is parameter elliptic with angle of ellipticity less or equal $\phi_{0}$, for
every $\boldsymbol{x}\in\operatorname{\mathbb{R}}^{n}\cup\\{\infty\\}$, where
we formally write
$a_{\boldsymbol{\alpha}}(\infty):=\lim_{\left|\boldsymbol{x}\right|\rightarrow\infty}a_{\boldsymbol{\alpha}}(\boldsymbol{x})$.

  3. (3)

$a_{\boldsymbol{\alpha}}\in[\LL_{\infty}+\LL_{r_{\nu}}](\operatorname{\mathbb{R}}^{n};\operatorname{\mathcal{B}}(E))$
for each lower order multi-index
$\boldsymbol{\alpha}\in\operatorname{\mathbb{N}}_{0}^{n}$ of length
$\left|\boldsymbol{\alpha}\right|=\nu<k$ for some $r_{\nu}\geq p$ such that
$k-\nu>\nicefrac{{n}}{{r_{\nu}}}$.

Then, for each angle $\phi>\phi_{0}$ there is a constant $\mu_{\phi}\geq 0$
such that the shifted operator $\mu_{\phi}+A_{\operatorname{\mathbb{R}}^{n}}$
is $\mathcal{R}$-sectorial with $\mathcal{R}$-angle
$\phi^{\mathcal{R}}_{\mu_{\phi}+A_{\operatorname{\mathbb{R}}^{n}}}\leq\phi$.
In particular, if $\phi_{0}<\nicefrac{{\pi}}{{2}}$, then the abstract Cauchy
problem

$\frac{\,\mathrm{d}}{\,\mathrm{d}t}u(t)+A_{\operatorname{\mathbb{R}}^{n}}u(t)=f(t),\quad
t\in(0,T),\quad u(0)=0$

has maximal regularity in the class
$\LL_{q}((0,T);\LL_{p}(\operatorname{\mathbb{R}}^{n};E))$ for each
$q\in(1,\infty)$, i.e. has a unqiue solution in the class
$u\in\mathrm{W}_{q,p}^{(1,2)}((0,T)\times\operatorname{\mathbb{R}}^{n};E)$ if
and only if $f\in\LL_{q,p}((0,T)\times\operatorname{\mathbb{R}}^{n};E)$.

##  4\. Elliptic vector-valued PDE: The half-space problem

In the preceding section, we have mainly recalled results from [17] on the
full-space problem, and no adjustments have been necessary. Next, we consider
the half-space problem as a prototypical domain and an intermediate step
towards general domains. In this context, we need to adjust the notation and
revise the proofs in [17] to cover the more general class of systems with
mixed type boundary conditions.

###  4.1. Partial Fourier transforms

Starting with the prototypical case of a half-space
$\operatorname{\mathbb{R}}_{+}^{n+1}:=\operatorname{\mathbb{R}}^{n}\times(0,\infty)$
instead of a general domain $\Omega\subseteq\operatorname{\mathbb{R}}^{n}$, we
are interested in well-posedness and maximal regularity properties of linear
systems of parabolic equations of the form

(PIBVP)
$\begin{cases}\partial_{t}u(t,\boldsymbol{x})+\mathcal{A}(\mathrm{D})u(t,\boldsymbol{x})=f(t,\boldsymbol{x})&\text{for
}(t,\boldsymbol{x})\in(0,\infty)\times\operatorname{\mathbb{R}}_{+}^{n+1},\\\
\mathcal{B}_{j}(\mathrm{D})u(t,\boldsymbol{x})=g_{j}(t,\boldsymbol{x})&\text{for
}(t,\boldsymbol{x})\in(0,\infty)\times\partial\operatorname{\mathbb{R}}_{+}^{n+1},\end{cases}$

accompanied with some suitable initial conditions at $t=0$. Revisiting the
strategy in [17], for the moment, let us assume that the differential
operators $\mathcal{A}(\mathrm{D})$ in the interior and
$\mathcal{B}_{j}(\mathrm{D})$ on the boundary are homogeneous of degrees $2m$
and $m_{j}<2m$, respectively, i.e.

$\mathcal{A}(\mathrm{D})=\sum_{\left|\boldsymbol{\alpha}\right|=2m}a_{\boldsymbol{\alpha}}\mathrm{D}^{\boldsymbol{\alpha}},\quad\mathcal{B}_{j}(\mathrm{D})=\sum_{\left|\boldsymbol{\beta}\right|=m_{j}}b_{j,\boldsymbol{\beta}}\mathrm{D}^{\boldsymbol{\beta}},$

where $m\in\operatorname{\mathbb{N}}$ is a fixed natural number and
$m_{j}\in\\{0,1,\ldots,2m-1\\}$ and corresponds to the differentiability order
of the operator $\mathcal{B}_{j}(\mathrm{D})$. The coefficients
$a_{\boldsymbol{\alpha}},b_{j,\boldsymbol{\beta}}\in\operatorname{\mathcal{B}}(E)$,
where $E$ may be some arbitrary Banach space, are assumed to be spatially and
temporally constant. We employ the multi-index notation
$\mathrm{D}^{\boldsymbol{\alpha}}=\prod_{i=1}^{n+1}\mathrm{D}_{x_{i}}^{\alpha_{i}}$
with $\mathrm{D}_{x_{i}}=-\mathrm{i}\frac{\partial}{\partial x_{i}}$ and write
$\left|\boldsymbol{\alpha}\right|=\sum_{i=1}^{n+1}\alpha_{i}$ for the length
of a vector of natural numbers
$\boldsymbol{\alpha}=(\alpha_{1},\ldots,\alpha_{n+1})^{\mathsf{T}}\in\operatorname{\mathbb{N}}_{0}^{n+1}$.
Then, the interior symbol $\mathcal{A}(\boldsymbol{\xi})$ and each of the
boundary symbols $\mathcal{B}_{j}(\boldsymbol{\xi})$

$\mathcal{A}(\boldsymbol{\xi})=\sum_{\left|\boldsymbol{\alpha}\right|=2m}a_{\boldsymbol{\alpha}}\boldsymbol{\xi}^{\boldsymbol{\alpha}},\quad\mathcal{B}_{j}(\boldsymbol{\xi})=\sum_{\left|\boldsymbol{\beta}\right|=m_{j}}b_{j,\boldsymbol{\beta}}\boldsymbol{\xi}^{\boldsymbol{\beta}}\quad(j=1,\ldots,m)$

are homogeneous in $\boldsymbol{\xi}\in\operatorname{\mathbb{R}}^{n+1}$ of
orders $2m$ for $\mathcal{A}$ and $m_{j}$ for $\mathcal{B}_{j}$,
$j=1,\ldots,m$, respectively. (Throughout, we use the short-hand notation
$\boldsymbol{\xi}^{\boldsymbol{\alpha}}:=\prod_{i=1}^{n+1}\xi_{i}^{\alpha_{i}}$
etc.) To establish an existence and uniqueness theory for solutions of the
parabolic equation PIBVP, in [17, Subsection 6.1], one may, first only
formally, apply the Laplace transform w.r. to the time variable
$t\in(0,\infty)$, i.e. we consider

$(\mathcal{L}u)(\lambda,\boldsymbol{x})=\int_{0}^{\infty}\mathrm{e}^{-\lambda
t}u(t,\boldsymbol{x})\,\mathrm{d}t\quad\text{for some set of parameters
}\lambda\in U\subseteq\operatorname{\mathbb{C}}\text{ and
}\boldsymbol{x}\in\operatorname{\mathbb{R}}_{+}^{n+1}$

so that the parabolic system transforms into a family of elliptic boundary
value problems on the half-space $\operatorname{\mathbb{R}}_{+}^{n+1}$ which
are given by

(4)
$\displaystyle\lambda(\mathcal{L}u)(\lambda,\boldsymbol{x})+\mathcal{A}(\mathrm{D})(\mathcal{L}u)(\lambda,\boldsymbol{x})$
$\displaystyle=(\mathcal{L}f)(\lambda,\boldsymbol{x})+u(0,\boldsymbol{x})$
$\displaystyle\text{for }(\lambda,\boldsymbol{x})\in
U\times\operatorname{\mathbb{R}}_{+}^{n+1},$ (5)
$\displaystyle\mathcal{B}_{j}(\mathrm{D})(\mathcal{L}u)(\lambda,\boldsymbol{x})$
$\displaystyle=(\mathcal{L}g_{j})(\lambda,\boldsymbol{x})$
$\displaystyle\text{for }(\lambda,\boldsymbol{x})\in
U\times\partial\operatorname{\mathbb{R}}_{+}^{n+1},\,j=1,\ldots,m.$

In [17], this provides the first step towards more general interior and
exterior symbols. More general results are then derived using operator-
theoretic results from modern harmonic analysis and employing perturbation
methods. In this manuscript, however, we want to cover situations similar to
those in [6], where demanding homogeneity of the (adjusted notion of the)
principle part of the boundary symbols cannot be guaranteed. Therefore, in
comparison with [17], we relax the conditions on the _principle parts_ of the
boundary operators in the following way:

######  Assumption 4.1.

The interior symbol
$\mathcal{A}(\boldsymbol{\xi})=\sum_{\left|\boldsymbol{\alpha}\right|=2m}a_{\boldsymbol{\alpha}}\boldsymbol{\xi}^{\boldsymbol{\alpha}}$
is homogeneous of degree $2m$ for some $m\in\operatorname{\mathbb{N}}$ and
parameter-elliptic with angle of ellipticity
$\phi^{\mathrm{ellipt}}_{\mathcal{A}}\in[0,\pi)$, i.e. the spectrum is
uniformly contained in some sector,

(6)
$\sigma(\mathcal{A}(\boldsymbol{\xi}))\subseteq\Sigma_{\pi-\phi}\quad\text{for
all }\boldsymbol{\xi}\in\operatorname{\mathbb{R}}^{n+1}\text{ with
}\left|\boldsymbol{\xi}\right|=1,$

for some $\phi\in[0,\pi)$ and where we put
$\phi^{\mathrm{ellipt}}_{\mathcal{A}}\in[0,\pi)$ to be the infimum of all
$\phi$ which are admissible for this inclusion.
Moreover, we assume that for $j=1,\ldots,m$ there exist continuous, linear
projections $\mathcal{P}_{j,k}\in\operatorname{\mathcal{B}}(E)$,
$k=0,1,\ldots,m_{j}<2m$ such that
$\mathcal{P}_{j,k}\mathcal{P}_{j,k^{\prime}}=0$ for $k\neq k^{\prime}$. The
boundary symbols $\mathcal{B}_{j}(\boldsymbol{\xi})$, $j=1,\ldots,m$ are
defined by

$\mathcal{B}_{j}(\boldsymbol{\xi})=\sum_{k=0}^{m_{j}}\mathcal{B}_{j,k}(\boldsymbol{\xi})\quad\text{with}\quad\mathcal{B}_{j,k}(\boldsymbol{\xi})=\sum_{\left|\boldsymbol{\beta}\right|=k}b_{j,k,\boldsymbol{\beta}}\boldsymbol{\xi}^{\boldsymbol{\beta}}\mathcal{P}_{j,k}\quad\text{for
}j=1,\ldots,m\text{ and }k=0,1,\ldots,m_{j}.$

We assume that $a_{\boldsymbol{\alpha}}\in\operatorname{\mathcal{B}}(E)$ and
$b_{j,k,\boldsymbol{\beta}}\in\operatorname{\mathcal{B}}(E)$ denote constant
coefficients and such that
$b_{j,k,\boldsymbol{\beta}}(\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k}))\subseteq\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k})$
for all $\boldsymbol{\beta}\in\operatorname{\mathbb{N}}_{0}^{n+1}$ with
$\left|\boldsymbol{\beta}\right|=k$.

With a slight abuse of notation, for the remainder of this section we focus on
the elliptic boundary value problem and simply write $u$, $f$ and $g_{j}$ for
what should actually be the Laplace-transformed versions $\mathcal{L}u$,
$\mathcal{L}f-u(0,\cdot)$ and $\mathcal{L}g_{j}$, respectively. We fix an
integrability exponent from the reflexive range $p\in(1,\infty)$ and an angle
$\phi>\phi^{\mathrm{ellipt}}_{\mathcal{A}}$, and consider the boundary value
problem

$\begin{cases}\lambda u+\mathcal{A}(\mathrm{D})u=f&\text{in
}\operatorname{\mathbb{R}}_{+}^{n+1}\\\
\mathcal{B}_{j}(\mathrm{D})u=g_{j}&\text{on
}\partial\operatorname{\mathbb{R}}_{+}^{n+1},\quad j=1,\ldots,m\end{cases}$

for a given parameter $\lambda\in\Sigma_{\pi-\phi}$, and given data
$f\in\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$ and
$g_{j}=\sum_{k=0}^{m_{j}}g_{j,k}$ such that
$g_{j,k}\in\mathrm{W}_{p}^{2m-k-1/p}(\operatorname{\mathbb{R}}^{n};\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k}))$
for $j=1,\ldots,m$ and $k=0,1,\ldots,m_{j}<2m$. The function $f$ can be
extended trivially by zero to
$\mathcal{E}_{0}f\in\LL_{p}(\operatorname{\mathbb{R}}^{n+1};E)$ on the full
space $\operatorname{\mathbb{R}}^{n+1}$; below $\mathcal{E}_{0}$ will always
denote this trivial extension operator. Moreover, by assumption we may write
$g_{j}=\sum_{k=0}^{m_{j}}g_{j,k}=\sum_{k=0}^{m_{j}}\mathcal{P}_{j,k}g_{j}$
(which is a consistent choice since all operators $\mathcal{P}_{j,k}$ are
projections such that $\mathcal{P}_{j,k}\mathcal{P}_{j,k^{\prime}}=0$ for
$k\neq k^{\prime}$). To solve the elliptic boundary value problem, we employ
the following strategy (cf. [17]): First, we solve the inhomogeneous problem
on the full space

$\lambda
w^{\mathrm{fs}}+\mathcal{A}(\mathrm{D})w^{\mathrm{fs}}=\mathcal{E}_{0}f\quad\text{on
}\operatorname{\mathbb{R}}^{n+1}.$

This is somehow the easiest part, since we may employ the full space theory
developed in [17, Section 5], to write
$\boldsymbol{\xi}=(\boldsymbol{\xi}^{\prime},y)$ and

$w^{\mathrm{fs}}(\boldsymbol{\xi}^{\prime},y,\lambda)=(P_{\operatorname{\mathbb{R}}_{+}^{n+1}}(\lambda+A_{\operatorname{\mathbb{R}}^{n+1}})^{-1}\mathcal{E}_{0}f)(\boldsymbol{\xi}^{\prime},y).$

Here, we write $P_{\operatorname{\mathbb{R}}_{+}^{n+1}}$ for the restriction
from the full space $\operatorname{\mathbb{R}}^{n+1}$ to the half-space
$\operatorname{\mathbb{R}}_{+}^{n+1}$, and
$A_{\operatorname{\mathbb{R}}^{n+1}}$ for the
$\LL_{p}(\operatorname{\mathbb{R}}^{n+1})$-realisation of the differential
operator $\mathcal{A}(\mathrm{D})$, i.e.
$A_{\operatorname{\mathbb{R}}^{n+1}}u=\mathcal{A}(\mathrm{D})u$ on
$\operatorname{\operatorname{dom}}(A_{\operatorname{\mathbb{R}}^{n+1}})=\\{u\in\LL_{p}(\operatorname{\mathbb{R}}^{n+1}):\,\mathcal{A}(\mathrm{D})u\in\LL_{p}(\operatorname{\mathbb{R}}^{n+1})\\}$.
We refer to [17] for details, but in view of the inhomogeneous boundary
problem on the half-space considered next, let us sketch the basic idea first:
Using the Fourier transformation on $\operatorname{\mathbb{R}}^{n+1}$,
initially defined for functions
$u\in\LL^{1}(\operatorname{\mathbb{R}}^{n+1};E)$ by

$(\mathcal{F}_{\operatorname{\mathbb{R}}^{n+1}}u)(\boldsymbol{\xi})=\int_{\operatorname{\mathbb{R}}^{n+1}}u(\boldsymbol{x})\mathrm{e}^{-\mathrm{i}\boldsymbol{x}\cdot\boldsymbol{\xi}}\,\mathrm{d}\boldsymbol{x},$

the (elliptic) full space problem
$(\lambda+\mathcal{A}(\mathrm{D}))w^{\mathrm{fs}}=\mathcal{E}_{0}f$ in Fourier
space becomes the parameter
$\boldsymbol{\xi}\in\operatorname{\mathbb{R}}^{n+1}$ dependent algebraic
system of linear equations on $E$

$(\lambda+\mathcal{A}(\boldsymbol{\xi}))(\mathcal{F}_{\operatorname{\mathbb{R}}^{n+1}}w^{\mathrm{fs}})(\lambda,\boldsymbol{\xi})=(\mathcal{F}_{\operatorname{\mathbb{R}}^{n+1}}\mathcal{E}_{0}f)(\lambda,\boldsymbol{\xi})\quad\text{for
}\lambda\in\Sigma_{\pi-\phi}\text{ and
}\boldsymbol{\xi}\in\operatorname{\mathbb{R}}^{n+1}.$

(Note that the solution to this problem is not subject to any boundary
conditions prescribed on $\partial\operatorname{\mathbb{R}}_{+}^{n+1}$, so
that in a further step, we have to subtract a corrective solution to the
problem $(\lambda+\mathcal{A}(\mathrm{D}))w^{\mathrm{corr}}=0$ in
$\operatorname{\mathbb{R}}_{+}^{n+1}$ and
$\mathcal{B}_{j}(\mathrm{D})w^{\mathrm{corr}}=-\mathcal{B}_{j}(\mathrm{D})w^{\mathrm{fs}}$
on $\partial\operatorname{\mathbb{R}}_{+}^{n+1}$ from the constructed
solution.) Subsequently we consider the half-space problem for the difference
$w:=u-P_{\operatorname{\mathbb{R}}_{+}^{n+1}}w^{\mathrm{fs}}$ between the
solution $u$ to the original problem which we seek for, and the full-space
problem with right-hand side $\mathcal{E}_{0}f$. This new problem is
homogeneous in $\operatorname{\mathbb{R}}_{+}^{n+1}$, but inhomogeneous at the
boundary $\partial\operatorname{\mathbb{R}}_{+}^{n+1}$: For $u$ to be a
solution to (4)–(5), this function $w$ has to solve the elliptic boundary
value problem

(EBVP) $\begin{cases}\lambda w+\mathcal{A}(\mathrm{D})w=0&\text{in
}\operatorname{\mathbb{R}}_{+}^{n+1},\\\
\mathcal{B}_{j}(\mathrm{D})w=g_{j}-\mathcal{B}_{j}(\mathrm{D})w^{\mathrm{fs}}&\text{on
}\partial\operatorname{\mathbb{R}}_{+}^{n+1},\,j=1,\ldots,m.\end{cases}$

Due to the special structure of the half-space
$\operatorname{\mathbb{R}}_{+}^{n+1}=\operatorname{\mathbb{R}}^{n}\times(0,\infty)$,
this inhomogeneous boundary value problem can be approached by performing the
_partial Fourier transform_ in $\boldsymbol{x}^{\prime}$, i.e. in the first
$n$ variables of the vector
$\boldsymbol{x}\in\operatorname{\mathbb{R}}_{+}^{n+1}$: Starting with
functions
$u=u(\boldsymbol{x}^{\prime},y)\in\LL_{1}(\operatorname{\mathbb{R}}_{+}^{n+1})$
we introduce the partial Fourier transform in $\boldsymbol{x}^{\prime}$ as

$\mathcal{F}u(\boldsymbol{\xi}^{\prime},y)=\int_{\operatorname{\mathbb{R}}^{n}}\mathrm{e}^{-\mathrm{i}\boldsymbol{x}^{\prime}\cdot\boldsymbol{\xi}^{\prime}}u(\boldsymbol{x}^{\prime},y)\,\mathrm{d}\boldsymbol{x}^{\prime}.$

Formally applying the partial Fourier transformation to the boundary value
problem (EBVP), leads to the elliptic boundary value problem in the
Laplace–Fourier space

(7)
$\displaystyle\lambda\mathcal{F}u(\boldsymbol{\xi}^{\prime},y)+\sum_{l=0}^{2m}\tilde{a}_{l}(\boldsymbol{\xi}^{\prime})\mathrm{D}_{y}^{2m-l}\mathcal{F}u(\boldsymbol{\xi}^{\prime},y)$
$\displaystyle=\mathcal{F}f(\boldsymbol{\xi}^{\prime},y)$
$\displaystyle\text{in }\operatorname{\mathbb{R}}_{+}^{n+1},$ (8)
$\displaystyle\mathcal{F}(\mathcal{B}_{j}u)(\boldsymbol{\xi}^{\prime},0)=\sum_{k=0}^{m_{j}}\sum_{l=0}^{k}\tilde{b}_{j,k,l}(\boldsymbol{\xi}^{\prime})\mathrm{D}_{y}^{k-l}\mathcal{P}_{j,k}\mathcal{F}u(\boldsymbol{\xi}^{\prime},0)$
$\displaystyle=\mathcal{F}g_{j}(\boldsymbol{\xi}^{\prime},0)$
$\displaystyle\text{on
}\partial\operatorname{\mathbb{R}}_{+}^{n+1},\,j=1,\ldots,m,$

where we merge coefficients and factors of the same $(n+1)$-st index
$\alpha_{n+1}$ and $\beta_{n+1}$, resp., and write

$\displaystyle\tilde{a}_{l}(\boldsymbol{\xi}^{\prime})$
$\displaystyle=\sum_{\left|\boldsymbol{\alpha}^{\prime}\right|=l}a_{(\boldsymbol{\alpha}^{\prime},2m-l)}(\boldsymbol{\xi}^{\prime})^{\boldsymbol{\alpha}^{\prime}},$
$\displaystyle l=0,1,\ldots,2m,$
$\displaystyle\tilde{b}_{j,k,l}(\boldsymbol{\xi}^{\prime})$
$\displaystyle=\sum_{\left|\boldsymbol{\beta}^{\prime}\right|=l}b_{j,k,(\boldsymbol{\beta}^{\prime},k-l)}(\boldsymbol{\xi}^{\prime})^{\boldsymbol{\beta}^{\prime}},$
$\displaystyle j=1,\ldots,m,\,k=1,\ldots,m_{j},\,l=0,1,\ldots,k.$

Note that by construction the terms $\tilde{a}_{l}(\boldsymbol{\xi}^{\prime})$
and $\tilde{b}_{j,k,l}(\boldsymbol{\xi}^{\prime})$ are homogeneous of order
$l$ in $\boldsymbol{\xi}^{\prime}\in\operatorname{\mathbb{R}}^{n}$. The
vector-valued ordinary differential equation (7) is equivalent to a first
order system (in $y$) on the space $E^{2m}$, as can be seen by introducing the
$E^{2m}$–valued functions

$\mathcal{F}\boldsymbol{v}(\lambda,\boldsymbol{\xi}^{\prime},y):=(\mathcal{F}u(\lambda,\boldsymbol{\xi}^{\prime},y),\frac{1}{\rho}\mathrm{D}_{y}\mathcal{F}u(\lambda,\boldsymbol{\xi}^{\prime},y),\ldots,\frac{1}{\rho^{2m-1}}\mathrm{D}_{y}^{2m-1}\mathcal{F}u(\lambda,\boldsymbol{\xi}^{\prime},y))^{\mathsf{T}},$

where $\rho>0$ is some additional parameter. To exploit the homogeneity of
$\mathcal{A}(\boldsymbol{\xi})$, $\mathcal{B}_{j,k}(\boldsymbol{\xi})$ in
$\boldsymbol{\xi}$ later on, we closely follow the lines of [17] and introduce

$\sigma:=\frac{\lambda}{\rho^{2m}}\in\Sigma_{\pi-\phi}\quad\text{and}\quad\boldsymbol{b}:=\frac{\boldsymbol{\xi}^{\prime}}{\rho}\in\operatorname{\mathbb{R}}^{n}$

as new variables in the Laplace–Fourier space. From now on, whenever $\sigma$
and $\boldsymbol{b}$ will appear, they are related to the variables
$\lambda\in\Sigma_{\pi-\phi}$ and
$\boldsymbol{\xi}^{\prime}\in\operatorname{\mathbb{R}}^{n}$ and the parameter
$\rho>0$ by these defining relations: In particular, we have

$(\left|\lambda\right|^{1/m}+\left|\boldsymbol{\xi}^{\prime}\right|^{2})^{1/2}=\rho(\left|\sigma\right|^{1/m}+\left|\boldsymbol{b}\right|^{2})^{1/2}\quad\text{and}\quad(\left|\lambda\right|+\left|\boldsymbol{\xi}^{\prime}\right|^{2m})^{1/2m}=\rho(\left|\sigma\right|+\left|\boldsymbol{b}\right|^{2m})^{1/2}.$

######  Remark 4.2.

Later on we will choose the parameter
$\rho=\rho(\lambda,\boldsymbol{\xi}^{\prime})$ depending on the variables
$\lambda\in\Sigma_{\pi-\phi}$ and
$\boldsymbol{\xi}^{\prime}\in\operatorname{\mathbb{R}}^{n}$. The particular
choice of $\rho=\rho(\lambda,\boldsymbol{\xi}^{\prime})$ will be
$\rho=(\lambda+\left|\boldsymbol{\xi}^{\prime}\right|^{2m})^{1/2m}$, in line
with the choice in [18], but in contrast to the earlier choice
$\rho=(\left|\lambda\right|^{1/m}+\left|\boldsymbol{\xi}^{\prime}\right|^{2})^{1/2}$
in [17]. The reason behind this are the respective interpretations of
$\rho(\lambda,\boldsymbol{\xi}^{\prime})$ as a multiplication operator in
Fourier space: For the choices just mentioned,
$\rho(\lambda,\boldsymbol{\xi}^{\prime})$ will be the Fourier symbol of the
pseudo-differential operators $((-\Delta)^{m}+\lambda)^{1/2m}$ and
$((-\Delta)+\left|\lambda\right|^{1/m})^{1/2}$, respectively. As it turned
out, the former choice is more practical, see [18].

Let us continue with the formulation of the elliptic BVP (EBVP) as a first
order system and introduce a matrix-valued function
$\boldsymbol{A}_{0}:\operatorname{\mathbb{R}}^{n}\times\Sigma_{\pi-\phi}\ni(\boldsymbol{\xi}^{\prime},\lambda)\mapsto\boldsymbol{A}_{0}(\boldsymbol{\xi}^{\prime},\lambda)\in\operatorname{\mathcal{B}}(E^{2m})$
by setting

$\boldsymbol{A}_{0}(\boldsymbol{\xi}^{\prime},\lambda)=\left[\begin{array}[]{ccccc}0&I_{E}&0&\cdots&0\\\
0&0&I_{E}&&0\\\ \vdots&\vdots&\,\ddots&\,\ddots&\\\ 0&\cdots&&0&I_{E}\\\
c_{2m}(\boldsymbol{\xi}^{\prime},\lambda)&c_{2m-1}(\boldsymbol{\xi}^{\prime})&\cdots&&c_{1}(\boldsymbol{\xi}^{\prime})\end{array}\right],$

where we choose the (operator-valued) entries of the operator matrix
$\boldsymbol{A}_{0}(\boldsymbol{\xi}^{\prime},\lambda)$ as

$c_{j}(\boldsymbol{\xi}^{\prime})=-a_{0}^{-1}\tilde{a}_{j}(\boldsymbol{\xi}^{\prime})\quad\text{for
}j=1,\ldots,2m-1\text{ and
}c_{2m}(\boldsymbol{\xi}^{\prime},\lambda)=-a_{0}^{-1}(\tilde{a}_{2m}(\boldsymbol{\xi}^{\prime})+\lambda
I_{E}),$

and $I_{E}$ denotes the identity map on $E$. By parameter-ellipticity of the
interior symbol $\mathcal{A}(\boldsymbol{\xi})$, the
($\boldsymbol{\xi}^{\prime}$-independent) coefficient
$a_{0}(\boldsymbol{\xi}^{\prime})=a_{0}\in\operatorname{\mathcal{B}}(E)$ is
invertible because

$a_{0}(\boldsymbol{\xi}^{\prime})=a_{(\boldsymbol{0},2m)}=\mathcal{A}(\boldsymbol{0},2m)\quad\text{for
all }\boldsymbol{\xi}^{\prime}\in\operatorname{\mathbb{R}}^{n},$

where $(\boldsymbol{0},2m)=(0,\ldots,0,2m)\in\operatorname{\mathbb{R}}^{n+1}$.
Since each of the symbols $\tilde{a}_{l}(\boldsymbol{\xi}^{\prime})$ is
homogeneous in $\boldsymbol{\xi}^{\prime}\in\operatorname{\mathbb{R}}^{n}$ of
order $l$, we may, reformulate the elliptic BVP (7) (with the homogeneous
choice $f=0$) equivalently as a first order in $y\in(0,\infty)$ system of
ordinary differential equations, viz.

$\frac{\partial}{\partial
y}\mathcal{F}\boldsymbol{v}^{\mathrm{hs}}(\lambda,\boldsymbol{\xi}^{\prime},y)=\mathrm{i}\rho\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)\mathcal{F}\boldsymbol{v}^{\mathrm{hs}}(\lambda,\boldsymbol{\xi}^{\prime},y),\quad\lambda\in\Sigma_{\pi-\phi},\,\boldsymbol{\xi}^{\prime}\in\operatorname{\mathbb{R}}^{n},\,y>0.$

In contrast to the situation in [17], this time, the boundary symbols
$\mathcal{B}_{j}(\boldsymbol{\xi})=\sum_{k=0}^{m_{j}}\mathcal{B}_{j,k}(\boldsymbol{\xi})$
are _not_ homogeneous in $\boldsymbol{\xi}\in\operatorname{\mathbb{R}}^{n+1}$
of order $m_{j}$, in general, but only each of the symbols
$\mathcal{B}_{j,k}(\boldsymbol{\xi}):=\mathcal{B}_{j}(\boldsymbol{\xi})\mathcal{P}_{j,k}=\mathcal{P}_{j,k}\mathcal{B}_{j}(\boldsymbol{\xi})\mathcal{P}_{j,k}$
is homogeneous in $\boldsymbol{\xi}\in\operatorname{\mathbb{R}}^{n+1}$ of
order $k$, for each $j=1,\ldots,m$ and $k=0,1,\ldots,m_{j}$. Hence, each
(operator-valued) coefficient
$\tilde{b}_{j,k,l}(\boldsymbol{\xi}^{\prime})\in\operatorname{\mathcal{B}}(E)$
is homogeneous in $\boldsymbol{\xi}^{\prime}\in\operatorname{\mathbb{R}}^{n}$
of degree $l$, and we may write

$\displaystyle\mathcal{F}g_{j}(\rho\boldsymbol{b},0)$
$\displaystyle=\mathcal{F}\mathcal{B}_{j}u(\rho\boldsymbol{b},0)=\sum_{k=0}^{m_{j}}\sum_{l=0}^{k}\tilde{b}_{j,k,l}(\rho\boldsymbol{b})\mathrm{D}_{y}^{k-l}\mathcal{P}_{j,k}\mathcal{F}u(\rho\boldsymbol{b},0)$
$\displaystyle=\sum_{k=0}^{m_{j}}\sum_{l=0}^{k}\rho^{l}\tilde{b}_{j,k,l}(\boldsymbol{b})\mathrm{D}_{y}^{k-l}\mathcal{P}_{j,k}\mathcal{F}u(\rho\boldsymbol{b},0)$
$\displaystyle=\sum_{k=0}^{m_{j}}\sum_{l=0}^{k}\rho^{l}\mathcal{P}_{j,k}\tilde{b}_{j,k,l}(\boldsymbol{b})\mathrm{D}_{y}^{k-l}\mathcal{P}_{j,k}\mathcal{F}u(\rho\boldsymbol{b},0)\quad\text{for
each }\boldsymbol{b}\in\SS^{n-1}\text{ and }\rho>0.$

Therefore, the boundary condition (8) is equivalently expressed as

$\boldsymbol{\mathcal{B}}_{j,k}^{0}(\boldsymbol{b})\mathcal{F}\boldsymbol{v}(\rho\boldsymbol{b},0)=\frac{\mathcal{P}_{j,k}\mathcal{F}g_{j}(\rho\boldsymbol{b},0)}{\rho^{k}}\quad\text{for
}j=1,\ldots,m\text{ and }k=0,1,\ldots,m_{j},$

where we define the row vector
$\boldsymbol{\mathcal{B}}_{j,k}^{0}(\boldsymbol{b})$ as follows:

$\boldsymbol{\mathcal{B}}_{j,k}^{0}(\boldsymbol{b})=(\tilde{b}_{j,k,k}(\boldsymbol{b})\mathcal{P}_{j,k},\ldots,\tilde{b}_{j,k,0}\mathcal{P}_{j,k},0,\ldots,0)\quad\text{for
}j=1,\ldots,m\text{ and }k=0,1,\ldots,m_{j}.$

(At this point, note that the inclusion
$\tilde{b}_{j,k,l}(\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k}))\subseteq\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k})$
for the range of the projections $\mathcal{P}_{j,k}$ is valid, since we
assumed that
$b_{j,k,\boldsymbol{\beta}}(\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k}))\subseteq\operatorname{\operatorname{ran}}(\mathcal{P}_{j,k})$
for each $\boldsymbol{\beta}\in\operatorname{\mathbb{N}}_{0}^{n+1}$ of length
$\left|\boldsymbol{\beta}\right|=k$.) Then [17, Proposition 6.1] applies and,
therefore, the spectrum of
$\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)\in\operatorname{\mathcal{B}}(E^{2m})$
is strictly separated by the real axis, i.e.
$\sigma(\boldsymbol{A}_{0}(\boldsymbol{b},\sigma))\cap\operatorname{\mathbb{R}}=\emptyset$
and, hence,
$\mathrm{i}\sigma(\boldsymbol{A}_{0}(\boldsymbol{b},\sigma))=\sigma(\mathrm{i}\boldsymbol{A}_{0}(\boldsymbol{b},\sigma))=S_{+}(\boldsymbol{b},\sigma)\cup
S_{-}(\boldsymbol{b},\sigma)$ splits into two parts
$S_{\pm}(\boldsymbol{b},\sigma)$ which are _uniformly_ (for
$(\boldsymbol{b},\sigma)\in\operatorname{\mathbb{R}}^{n}\times\Sigma_{\pi-\phi}$
with $\left|\boldsymbol{b}\right|,\left|\sigma\right|\leq 1$) separated by a
strip around the imaginary axis. This means that, there are constants
$c_{1},c_{2}>0$ such that

(9) $\displaystyle\inf\\{\operatorname{Re}\,\mu:\,\mu\in
S_{+}(\boldsymbol{b},\sigma)\\}$ $\displaystyle\geq c_{1},$
$\displaystyle\left|\boldsymbol{b}\right|\leq 1,\,\left|\sigma\right|\leq 1,$
(10) $\displaystyle\sup\\{\operatorname{Re}\,\mu:\,\mu\in
S_{-}(\boldsymbol{b},\sigma)\\}$ $\displaystyle\leq-c_{2},$
$\displaystyle\left|\boldsymbol{b}\right|\leq 1,\,\left|\sigma\right|\leq 1.$

Figure 2. Spectral gap between parts $S_{\pm}(\boldsymbol{b},\sigma)$ of the
spectrum $\mathrm{i}\sigma(\boldsymbol{A}_{0}(\boldsymbol{b},\Sigma))$.

In particular, for each $(\boldsymbol{b},\sigma)$ the spectral projections
$P_{\pm}(\boldsymbol{b},\sigma)\in\operatorname{\mathcal{B}}(E^{2m})$ onto
$S_{\pm}(\boldsymbol{b},\sigma)$ exist.

###  4.2. The Lopatinskii-Shapiro condition on the half-space

The _Lopatinskii–Shapiro condition_ provides a solvability and uniqueness
property for the Fourier space representation of the elliptic boundary value
problem (4)–(5). It can be stated as follows:

######  Assumption 4.3 (Lopatinskii–Shapiro condition, half-space version).

For each $\lambda\in\overline{\Sigma}_{\pi-\phi}$ and
$\boldsymbol{\xi}^{\prime}\in\operatorname{\mathbb{R}}^{n}$ such that
$(\lambda,\boldsymbol{\xi}^{\prime})\neq(0,\boldsymbol{0})$, the initial value
problem

$\begin{cases}\lambda
v(y)+\mathcal{A}(\boldsymbol{\xi}^{\prime},\mathrm{D}_{y})v(y)=0&\text{for
}y>0,\\\
\mathcal{B}_{j}(\boldsymbol{\xi}^{\prime},\mathrm{D}_{y})v(0)=g_{j}&\text{for
}j=1,\ldots,m\end{cases}$

admits a unique solution in the class
$v\in\mathrm{C}_{0}(\operatorname{\mathbb{R}}_{+};E)=\\{v\in\mathrm{C}(\operatorname{\mathbb{R}}_{+};E):\,v(y)\rightarrow
0\text{ as }y\rightarrow\infty\\}$, for each given
$\boldsymbol{g}=(g_{1},\ldots,g_{m})^{\mathsf{T}}\in E^{m}$.

######  Remark 4.4.

Below, we introduce a map
$\tilde{D}\subseteq\operatorname{\mathbb{C}}^{n+1}\rightarrow\operatorname{\mathcal{B}}(E^{m};E^{2m})$
which (roughly) maps, for each set of parameters
$(\boldsymbol{\xi}^{\prime},\lambda)\in\tilde{D}$, any given data $g_{j}\in E$
to the time trace at zero $\boldsymbol{v}(0)\in E^{2m}$ of the unique solution
to the elliptic initial value problem from the Lopatinskii–Shapiro condition.
For technical reasons, we want $\tilde{D}$ to be an open _complex_
neighbourhood of
$(\operatorname{\mathbb{R}}^{n}\times\Sigma_{\pi-\phi})\setminus\\{\boldsymbol{0}\\}\subseteq\operatorname{\mathbb{C}}^{n+1}$.
We then can prove complex differentiability of this map, provided we know that
the Lopatinskii–Shapiro condition is actually also valid for values
$(\lambda,\boldsymbol{\xi}^{\prime})$ from this _complex_ neighbourhood
$\tilde{D}$. This technical detail had not been addressed in the original work
[17], however, see [29, Lemma 5.6] in a master thesis supervised by Robert
Denk. By this, the Lopatinskii–Shapiro condition is automatically valid in a
complex neighbourhood $\tilde{D}$ of
$(\operatorname{\mathbb{R}}^{n}\times\overline{\Sigma_{\pi-\phi}})\setminus\\{\boldsymbol{0}\\}\subseteq\operatorname{\mathbb{C}}^{n+1}$,
provided it is valid on
$(\operatorname{\mathbb{R}}^{n}\times\overline{\Sigma_{\pi-\phi}})\setminus\\{\boldsymbol{0}\\}$.

To proceed, we modify the proofs of [17, Propositions 6.2–6.4] and transfer
them to the situation considered here. We, thereby, obtain the following
preliminary result on existence and uniqueness of solutions and a solution
formula for the Fourier space representation of the elliptic boundary value
problems in half-space which satisfy the Lopatinskii–Shapiro condition.

######  Proposition 4.5.

Let $\mathcal{A}(\mathrm{D})$ be a homogeneous, parameter-elliptic operator of
order $2m$ with angle of ellipticity
$\phi_{\mathcal{A}}^{\mathrm{ellipt}}\in[0,\pi)$. Let
$\phi>\phi_{\mathcal{A}}^{\mathrm{ellipt}}$ and
$\lambda\in\overline{\Sigma}_{\pi-\phi}$ be given and assume that the
Lopatinskii-Shapiro condition is valid. Then there is a unique solution
$\mathcal{F}u$ of the system (7)–(8), which is given by the sum

$\mathcal{F}u=\mathcal{F}w^{\mathrm{fs}}+\mathcal{F}w^{\mathrm{hs}}+\mathcal{F}w^{\mathrm{corr}}=\mathcal{F}w^{\mathrm{fs}}+(\mathcal{F}\boldsymbol{v}^{\mathrm{hs}})_{1}+(\mathcal{F}\boldsymbol{v}^{\mathrm{corr}})_{1}.$

Here, $\mathcal{F}w^{\mathrm{fs}}$ denotes the solution to the full-space
problem with right-hand side $\mathcal{F}\mathcal{E}_{0}f$,
$\mathcal{F}w^{\mathrm{hs}}$ is the solution of (7)–(8) for homogeneous right-
hand side $f=0$ in $\operatorname{\mathbb{R}}_{+}^{n+1}$ and boundary
conditions $\mathcal{B}_{j}w^{\mathrm{hs}}=g_{j}$ on
$\partial\operatorname{\mathbb{R}}_{+}^{n+1}$, $j=1,\ldots,m$, whereas
$\mathcal{F}w^{\mathrm{corr}}$ is a correction term with boundary conditions
$\mathcal{B}_{j}\mathcal{F}w^{\mathrm{corr}}=-\mathcal{B}_{j}\mathcal{F}w^{\mathrm{fs}}$
matching those from the solution to the full-space problem on
$\partial\operatorname{\mathbb{R}}_{+}^{n+1}$ (and $f=0$ in
$\operatorname{\mathbb{R}}_{+}^{n+1}$). Moreover, $(\cdot)_{1}$ denotes the
first component of any vector in $E^{2m}$. The function
$\mathcal{F}\boldsymbol{v}^{\mathrm{hs}}$ can be computed using the uniformly
continuous semigroup generated by
$\mathrm{i}\rho\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)$,

(11)
$\mathcal{F}\boldsymbol{v}^{\mathrm{hs}}(\boldsymbol{\xi}^{\prime},y,\lambda)=\mathrm{e}^{\mathrm{i}\rho\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)y}\mathcal{F}\boldsymbol{v}^{\mathrm{hs}}(\boldsymbol{\xi}^{\prime},0,\lambda)\quad\text{for
}\lambda=\rho^{2m}\sigma\in\Sigma_{\pi-\phi},\,\boldsymbol{\xi}^{\prime}=\rho\boldsymbol{b}\in\operatorname{\mathbb{R}}^{n},\,y>0,$

where the initial value
$\mathcal{F}\boldsymbol{v}^{\mathrm{hs}}(\boldsymbol{\xi}^{\prime},0,\lambda)$
has to be chosen according to

$\mathcal{F}\boldsymbol{v}^{\mathrm{hs}}(\boldsymbol{\xi}^{\prime},0,\lambda)=\boldsymbol{M}(\boldsymbol{b},\sigma)\mathcal{F}\boldsymbol{g}_{\rho}^{\mathrm{hs}}(\boldsymbol{\xi}^{\prime},0).$

The (operator-valued) function
$\boldsymbol{M}:\tilde{D}\mapsto\operatorname{\mathcal{B}}(E^{m};E^{2m})$ is
jointly holomorphic in $(\boldsymbol{b},\sigma)\in\tilde{D}$ in a complex
neighbourhood $\tilde{D}$ of
$(\operatorname{\mathbb{R}}^{n}\times\overline{\Sigma_{\pi-\phi}})\setminus\\{\boldsymbol{0}\\}\subseteq\operatorname{\mathbb{C}}^{n+1}$,
and $\boldsymbol{M}(\boldsymbol{b},\sigma)$ maps any given data

$\mathcal{F}\boldsymbol{g}_{\rho}^{\mathrm{hs}}(\boldsymbol{\xi}^{\prime},0)=\left(\sum_{k=0}^{m_{1}}\tfrac{\mathcal{P}_{1,k}\mathcal{F}g_{1}(\boldsymbol{\xi}^{\prime},0)}{\rho^{k}},\ldots,\sum_{k=0}^{m_{m}}\tfrac{\mathcal{P}_{m,k}\mathcal{F}g_{m}(\boldsymbol{\xi}^{\prime},0)}{\rho^{k}}\right),\quad\boldsymbol{\xi}^{\prime}=\rho\boldsymbol{b}\in\operatorname{\mathbb{R}}^{n},\,\rho>0.$

to the corresponding unique solution of the algebraic problem

$\begin{cases}P_{+}(\boldsymbol{b},\sigma)\mathcal{F}\boldsymbol{v}(\boldsymbol{\xi}^{\prime},0,\lambda)=0,&\boldsymbol{\xi}^{\prime}\in\operatorname{\mathbb{R}}^{n},\\\
B_{0}^{j,k}(\boldsymbol{b})\mathcal{F}\boldsymbol{v}(\boldsymbol{\xi}^{\prime},0,\lambda)=\frac{\mathcal{P}_{j,k}\mathcal{F}g_{j}(\boldsymbol{\xi}^{\prime},0)}{\rho^{k}},&j=1,\ldots,2m,\,k=0,1,\ldots,m_{j}.\end{cases}$

On the other hand, the correction term
$\mathcal{F}\boldsymbol{v}^{\mathrm{corr}}$ is given by

$\mathcal{F}\boldsymbol{v}^{\mathrm{corr}}(\boldsymbol{\xi}^{\prime},y,\lambda)=-\mathrm{e}^{\mathrm{i}\rho\boldsymbol{A}_{0}(\boldsymbol{b},y)}\boldsymbol{M}(\boldsymbol{b},\sigma)(\mathcal{F}\boldsymbol{g}_{\rho}^{\mathrm{corr}})(\boldsymbol{\xi}^{\prime},0)\quad\text{for
}\lambda=\rho^{2m}\sigma\in\Sigma_{\pi-\phi},\,\boldsymbol{\xi}^{\prime}=\rho\boldsymbol{b}\in\operatorname{\mathbb{R}}^{n},\,y>0,$

where

$\mathcal{F}\boldsymbol{g}_{\rho}^{\mathrm{corr}}(\boldsymbol{\xi}^{\prime},0)=\left(\sum_{k=0}^{m_{1}}\tfrac{\mathcal{F}g^{\mathrm{corr}}_{1,k}(\boldsymbol{\xi}^{\prime},0)}{\rho^{k}},\ldots,\sum_{k=0}^{m_{m}}\tfrac{\mathcal{F}g^{\mathrm{corr}}_{m,k}(\boldsymbol{\xi}^{\prime},0)}{\rho^{k}}\right),\quad\boldsymbol{\xi}^{\prime}=\rho\boldsymbol{b}\in\operatorname{\mathbb{R}}^{n},\,\rho>0.$

Here, the functions

$\mathcal{F}g^{\mathrm{corr}}_{j,k}(\boldsymbol{\xi}^{\prime},0)=\int_{0}^{\infty}h^{\mathrm{corr}}_{j,k}(\boldsymbol{\xi}^{\prime},s)(\mathcal{F}f)(\boldsymbol{\xi}^{\prime},s)\,\mathrm{d}s,\quad
j=1,\ldots,m,\,k=0,1,\ldots,m_{j},\,\boldsymbol{\xi}^{\prime}\in\operatorname{\mathbb{R}}^{n}$

are defined using the integral operator kernels

$h^{\mathrm{corr}}_{j,k}(\boldsymbol{\xi}^{\prime},s)=\int_{\operatorname{\mathbb{R}}}\mathcal{B}_{j}(\boldsymbol{\xi}^{\prime},\eta)\mathcal{P}_{j,k}(\lambda+\mathcal{A}(\boldsymbol{\xi}^{\prime},\eta))^{-1}\mathrm{e}^{-\mathrm{i}\eta
s}\,\mathrm{d}\eta\quad\text{for
}\boldsymbol{\xi}^{\prime}\in\operatorname{\mathbb{R}}^{n},\,s>0.$

###### Proof.

To establish Proposition 4.5, we proceed as follows and analogously to [17]:
The linear problem is decomposed into three sub-problems, each of which for
itself is easier to handle than the original problem:

$\displaystyle\lambda\mathcal{F}w^{\mathrm{fs}}+\mathcal{A}(\mathrm{D})\mathcal{F}w^{\mathrm{fs}}$
$\displaystyle=\mathcal{E}_{0}\mathcal{F}f$ $\displaystyle\text{in
}\operatorname{\mathbb{R}}^{n+1}$ (obviously, without any boundary
conditions),
$\displaystyle\lambda\mathcal{F}w^{\mathrm{hs}}+\mathcal{A}(\mathrm{D})\mathcal{F}w^{\mathrm{hs}}$
$\displaystyle=0$ $\displaystyle\text{in
}\operatorname{\mathbb{R}}_{+}^{n+1},$
$\displaystyle\mathcal{B}_{j}(\mathrm{D})\mathcal{F}w^{\mathrm{hs}}$
$\displaystyle=g_{j},$ $\displaystyle\text{on
}\partial\operatorname{\mathbb{R}}_{+}^{n+1},\,j=1,\ldots,m,$ and
$\displaystyle\lambda\mathcal{F}w^{\mathrm{corr}}+\mathcal{A}(\mathrm{D})\mathcal{F}w^{\mathrm{corr}}$
$\displaystyle=0$ $\displaystyle\text{in
}\operatorname{\mathbb{R}}_{+}^{n+1},$
$\displaystyle\mathcal{B}_{j}(\mathrm{D})\mathcal{F}w^{\mathrm{corr}}$
$\displaystyle=-\mathcal{B}_{j}(\mathrm{D})\mathcal{F}w^{\mathrm{fs}}$
$\displaystyle\text{on
}\partial\operatorname{\mathbb{R}}_{+}^{n+1},\,j=1,\ldots,m.$

The second problem, i.e. the case where $f=0$, but inhomogeneous boundary
conditions are imposed, may be handled similar to [17, Proposition 5.3] which
provides us with a unique solution
$(\mathcal{F}\boldsymbol{v}^{\mathrm{hs}})_{1}$ to the problem (4)–(5) for
$f=0$, and which has the form (11), where
$\boldsymbol{M}(\boldsymbol{b},\sigma)$ is jointly holomorphic in the
variables
$(\boldsymbol{b},\sigma)\in\tilde{D}\supset(\operatorname{\mathbb{R}}^{n}\times\overline{\Sigma_{\pi-\phi}})\setminus\\{\boldsymbol{0}\\}$.
Since the projections $\mathcal{P}_{j,k}$ project the problem to subspaces on
which the corresponding system for the boundary operators is homogeneous in
$\boldsymbol{\xi}^{\prime}$ of degree $k$, uniqueness and existence of
solutions, as well as the representation formula for the solution can be
extracted from the equivalent reformulation as a first order (in
$y\in(0,\infty)$) system. It, therefore, remains to ensure holomorphy of the
map $\boldsymbol{M}$. This can be done almost literally as in the proof of
[17, Proposition 6.2]: First, we employ the closed graph theorem to conclude
that for each $\boldsymbol{z}\in D$ the operator
$\boldsymbol{M}(\boldsymbol{z})$ is a linear and closed map $E^{m}\rightarrow
E^{2m}$, and it is uniformly bounded, first on the set
$D=\overline{\mathrm{B}_{1}(0)}\times\overline{\Sigma}_{\pi-\phi}\subseteq\operatorname{\mathbb{R}}^{n}\times\operatorname{\mathbb{C}}$,
and, thereafter, in any complex neighbourhood $\tilde{D}$ of $D$, which is
sufficiently close to $D$. With use of the spectral projections
$P_{-}(\boldsymbol{b},\sigma)$ and the Lopatinskii-Shapiro condition we may
then prove continuity of the map
$\boldsymbol{z}\mapsto\boldsymbol{v}(\boldsymbol{z})=\boldsymbol{M}(\boldsymbol{z})\boldsymbol{g}$,
for any fixed $\boldsymbol{g}\in E^{m}$, on complex lines. This uses, in
particular, that the Lopatinskii–Shapiro condition is valid on a complex
neighbourhood $\tilde{D}$ of
$(\operatorname{\mathbb{R}}^{n}\times\overline{\Sigma_{\pi-\phi}})\setminus\\{\boldsymbol{0}\\}$.
Moreover, it uses that the spectral projections $P_{-}$ and the operators
$\boldsymbol{\mathcal{B}}_{j,k}^{0}$ are continuous. Complex differentiability
follows.

To derive an integral representation of the solution
$\mathcal{F}w^{\mathrm{corr}}$ to the correction problem, we use that,
analogous to the half-space problem with given boundary data,
$\mathcal{F}w^{\mathrm{corr}}$ is the first component of the $E^{2m}$-valued
function $\mathcal{F}\boldsymbol{v}^{\mathrm{corr}}$, which can be formally
written down as

$(\mathcal{F}\boldsymbol{v}^{\mathrm{corr}})(\lambda,\boldsymbol{\xi}^{\prime},y)=\mathrm{e}^{\mathrm{i}\rho\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)y}\boldsymbol{M}(\boldsymbol{b},\sigma)\mathcal{F}\boldsymbol{g}_{\rho}^{\mathrm{corr}}(\lambda,\boldsymbol{\xi}^{\prime}).$

The $\rho$-weighted version of the boundary data is

$\mathcal{F}\boldsymbol{g}_{\rho}^{\mathrm{corr}}(\lambda,\boldsymbol{\xi}^{\prime})=\big{(}\sum_{k=0}^{m_{1}}\frac{\mathcal{F}g_{1,k}^{\mathrm{corr}}(\lambda,\boldsymbol{\xi}^{\prime})}{\rho^{k}},\ldots,\sum_{k=0}^{m_{m}}\frac{\mathcal{F}g_{m,k}^{\mathrm{corr}}(\lambda,\boldsymbol{\xi}^{\prime})}{\rho^{k}}\big{)},$

where the boundary data $\mathcal{F}g_{j,k}^{\mathrm{corr}}$ result from
taking the trace of
$\mathcal{B}_{j,k}(\boldsymbol{\xi}^{\prime},\mathrm{D}_{y})\mathcal{F}w^{\mathrm{fs}}(\boldsymbol{\xi}^{\prime},y)$
at $y=0$. We note that

$(\mathcal{F}_{\boldsymbol{x}^{\prime},y}w^{\mathrm{fs}})(\boldsymbol{\xi}^{\prime},\eta)=(\lambda+\mathcal{A}(\boldsymbol{\xi}^{\prime},\eta))^{-1}\mathcal{F}_{\boldsymbol{x}^{\prime},y}(\mathcal{E}_{0}f)(\boldsymbol{\xi}^{\prime},y)=(\lambda+\mathcal{A}(\boldsymbol{\xi},\eta))^{-1}(\mathcal{L}_{y}\mathcal{F}_{\boldsymbol{x}^{\prime}}f)(\boldsymbol{\xi}^{\prime},\mathrm{i}y)$

since by definition $(\mathcal{E}_{0}f)(\boldsymbol{x}^{\prime},y)=0$ for
$y<0$. (By $\mathcal{F}_{\boldsymbol{x}^{\prime}}$ and
$\mathcal{F}_{\boldsymbol{x}^{\prime},y}$ we mean the Fourier transform w.r.t.
$\boldsymbol{x}^{\prime}$ and $\boldsymbol{x}=(\boldsymbol{x}^{\prime},y)$,
respectively.) Thus,

$\displaystyle\mathcal{B}_{j,k}(\boldsymbol{\xi}^{\prime},\eta)\mathcal{F}_{\boldsymbol{x}^{\prime},y}w^{\mathrm{fs}}(\boldsymbol{\xi}^{\prime},\eta)$
$\displaystyle=\mathcal{F}_{\boldsymbol{x}^{\prime},y}\big{(}\mathcal{B}_{j,k}(\boldsymbol{D})w^{\mathrm{fs}}\big{)}(\boldsymbol{\xi}^{\prime},\eta)$
$\displaystyle=\mathcal{B}_{j,k}(\boldsymbol{\xi}^{\prime},\eta)(\lambda+\mathcal{A}(\boldsymbol{\xi}^{\prime},\eta))^{-1}\mathcal{F}_{\boldsymbol{x}^{\prime},y}(\mathcal{E}_{0}f)(\boldsymbol{\xi}^{\prime},\eta)$
$\displaystyle=\int_{\operatorname{\mathbb{R}}}\mathcal{B}_{j,k}(\boldsymbol{\xi}^{\prime},\eta)(\lambda+\mathcal{A}(\boldsymbol{\xi}^{\prime},\eta))^{-1}\mathcal{F}_{\boldsymbol{x}^{\prime}}(\mathcal{E}_{0}f)(\boldsymbol{\xi}^{\prime},y)\mathrm{e}^{-\mathrm{i}\eta
y}\,\mathrm{d}y.$

$\mathcal{B}_{j,k}(\boldsymbol{\xi}^{\prime},\mathrm{D}_{y})\mathcal{F}_{\boldsymbol{x}^{\prime}}w^{\mathrm{fs}}(\boldsymbol{\xi}^{\prime},0)$
can, thus, be obtained by taking the inverse Laplace transform

$\displaystyle\mathcal{B}_{j,k}(\boldsymbol{\xi}^{\prime},\mathrm{D}_{y})\mathcal{F}_{\boldsymbol{\xi}^{\prime}}w^{\mathrm{fs}}(\boldsymbol{\xi}^{\prime},0)$
$\displaystyle=\int_{0}^{\infty}\int_{\operatorname{\mathbb{R}}}\mathcal{B}_{j,k}(\boldsymbol{\xi}^{\prime},\eta)(\lambda+\mathcal{A}(\boldsymbol{\xi}^{\prime},\eta))^{-1}\mathrm{e}^{-\mathrm{i}\eta
y}(\mathcal{F}_{\boldsymbol{x}^{\prime}}\mathcal{E}_{0}f)(\boldsymbol{\xi}^{\prime},y)\,\mathrm{d}y\,\mathrm{d}\eta.$

For the last of the three sub-problems, the crucial point is to derive an
integral representation of the boundary values
$\mathcal{F}g_{j}^{\mathrm{corr}}(\boldsymbol{\xi}^{\prime},0)=(\mathcal{B}_{j}(\mathrm{D})\mathcal{F}P_{\operatorname{\mathbb{R}}_{+}^{n+1}}(\lambda+A_{\operatorname{\mathbb{R}}^{n+1}})^{-1}\mathcal{E}_{0}f)(\boldsymbol{\xi}^{\prime},0)$
as
$\mathcal{F}g_{j}^{\mathrm{corr}}(\boldsymbol{\xi}^{\prime},0)=\sum_{k=0}^{m_{j}}\mathcal{F}g^{\mathrm{corr}}_{j,k}(\boldsymbol{\xi},0)$
with

$\displaystyle\mathcal{F}g^{\mathrm{corr}}_{j,k}(\boldsymbol{\xi}^{\prime},0)$
$\displaystyle=\int_{0}^{\infty}h^{\mathrm{corr}}_{j,k}(\boldsymbol{\xi}^{\prime},s)(\mathcal{F}f)(\boldsymbol{\xi}^{\prime},s)\,\mathrm{d}s$
$\displaystyle h^{\mathrm{corr}}_{j,k}(\boldsymbol{\xi}^{\prime},s)$
$\displaystyle=\int_{\operatorname{\mathbb{R}}}\mathcal{B}_{j}(\boldsymbol{\xi}^{\prime},\eta)\mathcal{P}_{j,k}(\lambda+\mathcal{A}(\boldsymbol{\xi}^{\prime},\eta))^{-1}\mathrm{e}^{-\mathrm{i}\eta
s}\,\mathrm{d}\eta,\quad 1\leq j\leq m,\,0\leq k\leq m_{j}<2m$

and to ensure that the kernels $h^{\mathrm{corr}}_{j,k}$ have good decay
properties. ∎

###  4.3. Kernel estimates

To actually solve the elliptic boundary value problem (EBVP) we may now use
the kernel representation of the solution corresponding to the boundary value
problem in Fourier space, and perform the inverse Laplace–Fourier transform.
To check that the solution operators defined by this procedure are at least
continuous on $\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$, we may employ
theory of integral operators. Roughly speaking, the following is true: If a
linear operator on $\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E)$ has a
kernel representation, estimates on the kernel imply estimates on its operator
norm. For a precise formulation of this statement, see e.g. [17, Lemma 6.7].
Its proof relies on Young’s and Hölder’s inequality.

######  Lemma 4.6.

Let $p,p^{\prime}\in(1,\infty)$ such that $\frac{1}{p}+\frac{1}{p^{\prime}}=1$
and $E$ be a Banach space and
$k\in\LL_{p,p^{\prime},1}(\operatorname{\mathbb{R}}^{n}\times(0,\infty)\times(0,\infty))$.
Then,

$(Kf)(\boldsymbol{x}):=\int_{0}^{\infty}\int_{\operatorname{\mathbb{R}}^{n}}k(\boldsymbol{x}^{\prime}-\bar{\boldsymbol{x}}^{\prime},y,\bar{y})f(\bar{\boldsymbol{x}}^{\prime},\bar{y})\,\mathrm{d}\bar{\boldsymbol{x}}^{\prime}\,\mathrm{d}\bar{y}\quad\text{for
}f\in\LL_{p}(\operatorname{\mathbb{R}}^{n+1};E),\,\text{a.e.\
}\boldsymbol{x}=(\boldsymbol{x}^{\prime},y)\in\operatorname{\mathbb{R}}_{+}^{n+1}$

defines a bounded linear operator
$K\in\operatorname{\mathcal{B}}(\LL_{p}(\operatorname{\mathbb{R}}_{+}^{n+1};E))$
with operator norm which is less or equal the norm
$\left\|k\right\|_{p,p^{\prime},1}$ of the kernel $k$.

For the kernel functions considered here, it proves helpful to define
(analogously to [17, Section 6.3]) weighted versions of the functions
$\mathcal{F}\boldsymbol{g}_{\rho}^{\mathrm{hs}}$ and the semigroup associated
to $\mathrm{i}\rho\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)$ according to

$\displaystyle\mathcal{F}\boldsymbol{g}_{\rho}^{2m}(\boldsymbol{\xi}^{\prime},0)$
$\displaystyle=\rho^{2m}\mathcal{F}\boldsymbol{g}_{\rho}^{\mathrm{hs}}(\boldsymbol{\xi}^{\prime},0)\in
E^{m}$ $\displaystyle\text{for
}\boldsymbol{\xi}^{\prime}\in\operatorname{\mathbb{R}}^{n},\,\rho>0,$
$\displaystyle\mathcal{F}\boldsymbol{V}_{\rho}^{2m}(\boldsymbol{\xi}^{\prime},y,\lambda)$
$\displaystyle=\frac{1}{\rho^{2m}}\mathrm{e}^{\mathrm{i}\rho\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)y}\boldsymbol{M}(\boldsymbol{b},\sigma)\in\operatorname{\mathcal{B}}(E^{m},E^{2m})$
$\displaystyle\text{for
}\boldsymbol{\xi}^{\prime}\in\operatorname{\mathbb{R}}^{n},\,y>0\text{ and
}\rho>0.$

By formula (11), the identity

$\mathcal{F}\boldsymbol{v}^{\mathrm{hs}}(\boldsymbol{\xi}^{\prime},y,\lambda)=\mathcal{F}\boldsymbol{V}_{\rho}^{2m}(\boldsymbol{\xi}^{\prime},y,\lambda)\mathcal{F}\boldsymbol{g}_{\rho}^{2m}(\boldsymbol{\xi}^{\prime},0)\in
E^{2m}$

is valid. Further, let us denote by

(12)
$\boldsymbol{K}^{\lambda}(\boldsymbol{x}^{\prime},y)=\mathcal{F}^{-1}(\mathcal{F}\boldsymbol{V}_{\rho}^{2m})(\boldsymbol{x}^{\prime},y)=\frac{1}{(2\pi)^{n}}\int_{\operatorname{\mathbb{R}}^{n}}\mathrm{e}^{\mathrm{i}\boldsymbol{x}^{\prime}\cdot\boldsymbol{\xi}^{\prime}}\mathcal{F}\boldsymbol{V}_{\rho}^{2m}(\boldsymbol{\xi}^{\prime},y,\lambda)\,\mathrm{d}\boldsymbol{\xi}^{\prime}\in\operatorname{\mathcal{B}}(E^{2m}),\quad\boldsymbol{x}^{\prime}\in\operatorname{\mathbb{R}}^{n},\,y>0$

the inverse (partial) Fourier transform (w.r.t.
$\boldsymbol{\xi}^{\prime}\in\operatorname{\mathbb{R}}^{n}$) of the
$\mathcal{B}(E)$-valued function $\mathcal{F}\boldsymbol{V}_{\rho}^{2m}$.
Then, the solution to the elliptic boundary value problem

$\begin{cases}(\lambda+\mathcal{A}(\mathrm{D}))w^{\mathrm{hs}}=0&\text{in
}\operatorname{\mathbb{R}}_{+}^{n+1},\\\
\mathcal{B}_{j}(\mathrm{D})w^{\mathrm{hs}}=g_{j}&\text{on
}\partial\operatorname{\mathbb{R}}_{+}^{n+1},\,j=1,\ldots,m\end{cases}$

is (formally) given by

$w^{\mathrm{hs}}(\lambda,\boldsymbol{x}^{\prime},y)=\int_{\operatorname{\mathbb{R}}^{n}}[\boldsymbol{K}^{\lambda}(\boldsymbol{x}-\boldsymbol{x}^{\prime},y)]_{1}\mathcal{F}^{-1}\boldsymbol{g}_{\rho}^{2m}(\boldsymbol{x}^{\prime})\,\mathrm{d}\boldsymbol{x}^{\prime}\quad\text{for
}\lambda\in\Sigma_{\pi-\phi},\,\boldsymbol{x}^{\prime}\in\operatorname{\mathbb{R}}^{n},\,y>0$

where we denote by
$[\boldsymbol{K}^{\lambda}(\boldsymbol{x}^{\prime},y^{\prime})]_{1}$ the first
row of $\boldsymbol{K}^{\lambda}(\boldsymbol{x}^{\prime},y^{\prime})$. Here,
$\mathcal{F}\boldsymbol{g}_{\rho}^{2m}(\boldsymbol{\xi}^{\prime},y)\in E^{2m}$
with components

$(\mathcal{F}\boldsymbol{g}_{\rho}^{2m}(\boldsymbol{\xi}^{\prime},y))_{j}=\sum_{k=0}^{m_{j}}\rho^{2m-k}\mathcal{F}g_{j,k}(\boldsymbol{\xi}^{\prime},y)=\sum_{k=0}^{m_{j}}(\left|\xi^{\prime}\right|^{2m}+\left|\lambda\right|)^{(2m-k)/2m}\mathcal{F}g_{j,k}(\boldsymbol{\xi}^{\prime},y).$

Since $(\left|\xi^{\prime}\right|^{2m}+\left|\lambda\right|)^{(2m-k)/2m}$ is
the Fourier symbol of the pseudo-differential operator
$((-\Delta_{\operatorname{\mathbb{R}}^{n}})^{m}+\left|\lambda\right|)^{(2m-k)/2m}$
for the realisation of the Laplacian $\Delta_{\operatorname{\mathbb{R}}^{n}}$
in $\LL_{p}(\operatorname{\mathbb{R}}^{n};E)$, we infer that

$\mathcal{F}^{-1}(\mathcal{F}\boldsymbol{g}_{\rho}^{2m})_{j}(\boldsymbol{x}^{\prime},y)=\sum_{k=0}^{m_{j}}((-\Delta_{\operatorname{\mathbb{R}}^{n}})^{m}+\left|\lambda\right|)^{(2m-k)/2m}g_{j,k}(\boldsymbol{x}^{\prime},y).$

Denoting by $K^{\lambda}_{j}:=(\boldsymbol{K}^{\lambda})_{1,j}$ the components
$(1,j)$ of the inverse Fourier transform $\boldsymbol{K}^{\lambda}$ of the
operator-valued function $\mathcal{F}\boldsymbol{V}_{\rho}^{2m}$ as defined by
(12) and $K^{\lambda}_{j,k}:=K^{\lambda}_{j}\mathcal{P}_{j,k}$, $j=1,\ldots,m$
and $k=0,1,\ldots,m_{j}$, we observe the following:

$\mathcal{F}\boldsymbol{v}^{\mathrm{hs}}=\mathcal{F}\boldsymbol{V}_{\rho}^{2m}\mathcal{F}\boldsymbol{g}_{\rho}^{2m},\quad\text{then}\quad\boldsymbol{v}^{\mathrm{hs}}=\boldsymbol{K}^{\lambda}\ast\mathcal{F}^{-1}(\mathcal{F}\boldsymbol{g}_{\rho}^{2m}).$

With the help of Lemma 3.3, we may now derive kernel estimates.

######  Proposition 4.7.

Let $\mathcal{A}(\mathrm{D})$ be a parameter-elliptic operator of order $2m$
with angle of ellipticity $\phi_{\mathcal{A}}^{\mathrm{ellipt}}\in[0,\pi)$.
Let $\phi>\phi_{\mathcal{A}}^{\mathrm{ellipt}}$ and assume that the
Lopatinskii-Shapiro condition is valid. Then, for every multi-index
$\boldsymbol{\alpha}\in\operatorname{\mathbb{N}}_{0}^{n+1}$, there are
constants $M,c>0$ such that

$\displaystyle\left|\mathrm{D}^{\boldsymbol{\alpha}}\boldsymbol{K}^{\lambda}(\boldsymbol{x}^{\prime},y)\right|$
$\displaystyle\leq
M\cdot\left|\lambda\right|^{\frac{n-2m+\left|\boldsymbol{\alpha}\right|}{2m}}p_{2m,\left|\boldsymbol{\alpha}\right|-1}^{n+1}\left(c\left|\lambda\right|^{1/(2m)}\left(\left|\boldsymbol{x}^{\prime}\right|+\left|y\right|\right)\right),\quad\boldsymbol{x}^{\prime}\in\operatorname{\mathbb{R}}^{n},\,y>0,\,\lambda\in\Sigma_{\pi-\phi}.$

###### Proof.

We proceed as in the proof of [17, Propositions 6.5 & 6.6]. The proof heavily
relies on theory of complex functions and, in particular, Cauchy’s Theorem. We
use that the identity

$\mathcal{F}\boldsymbol{v}^{\mathrm{hs}}(\boldsymbol{\xi}^{\prime},y,\lambda)=\mathcal{F}\boldsymbol{V}_{\rho}^{2m}(\boldsymbol{\xi}^{\prime},y,\lambda)\mathcal{F}\boldsymbol{g}_{\rho}^{2m}(\boldsymbol{\xi}^{\prime},0),\quad\boldsymbol{\xi}^{\prime}\in\operatorname{\mathbb{R}}^{n},\,y>0,\,\lambda\in\Sigma_{\pi-\phi}$

is valid independent of the particular choice of $\rho>0$, so that later on,
we may choose variables $(a,r)$ and then choose $\rho$ depending on $(a,r)$ as
$\rho=(\left|\lambda\right|+\left|\boldsymbol{\xi}^{\prime}\right|^{2m})^{1/2m}=(\left|\lambda\right|+(a^{2}+r^{2})^{m})^{\nicefrac{{1}}{{2m}}}$
for $\lambda\in\Sigma_{\pi-\phi}$ and
$\boldsymbol{\xi}^{\prime}\in\operatorname{\mathbb{R}}^{n}$ such that
$\left|\lambda\right|+\left|\boldsymbol{\xi}^{\prime}\right|>0$. Fix
$\boldsymbol{x}^{\prime}\in\operatorname{\mathbb{R}}^{n}$ and let
$\boldsymbol{Q}\in\operatorname{\mathbb{R}}^{n\times n}$ be the (uniquely
determined) rotation in $\operatorname{\mathbb{R}}^{n}$ which maps the vector
$\boldsymbol{x}^{\prime}$ to a positive multiple of the first standard basis
vector
$\boldsymbol{Q}\boldsymbol{x}^{\prime}=(\left|\boldsymbol{x}^{\prime}\right|,0,\ldots,0)^{\mathsf{T}}=:\left|\boldsymbol{x}^{\prime}\right|\boldsymbol{e}_{1}$.
Using spherical coordinates in $\operatorname{\mathbb{R}}^{n}$, we then write
$\boldsymbol{Q}\boldsymbol{\xi}^{\prime}=(a,r\boldsymbol{\varphi})^{\mathsf{T}}$
where $a\in\operatorname{\mathbb{R}}$, $r>0$ and
$\boldsymbol{\varphi}\in\mathbb{S}^{n-2}=\\{\boldsymbol{z}^{\prime}\in\operatorname{\mathbb{R}}^{n-1}:\,\left|\boldsymbol{z}^{\prime}\right|=1\\}$
(the $n-2$-dimensional sphere), hence,
$a^{2}+r^{2}=\left|\boldsymbol{\xi}^{\prime}\right|^{2}$. Since, by
convention, $\sigma=\tfrac{\lambda}{\rho^{2m}}$ and
$\boldsymbol{b}=\tfrac{\boldsymbol{\xi}^{\prime}}{\rho}$, the transformation
rule will help us to estimate the kernels and their derivatives: Using
orthogonality of the rotation matrix $\boldsymbol{Q}$ and the transformation
rule, the integral kernel operator
$\boldsymbol{K}^{\lambda}=\mathcal{F}^{-1}(\boldsymbol{V}_{\rho}^{2m})$ can be
written as

$\displaystyle\boldsymbol{K}^{\lambda}(\boldsymbol{x}^{\prime},\lambda)$
$\displaystyle=\frac{1}{(2\pi)^{n}}\int_{\operatorname{\mathbb{R}}^{n}}\mathrm{e}^{\mathrm{i}\boldsymbol{x}^{\prime}\cdot\boldsymbol{\xi}^{\prime}}\mathcal{F}\boldsymbol{V}_{\rho}^{2m}(\boldsymbol{\xi}^{\prime},y,\lambda)\,\mathrm{d}\boldsymbol{\xi}^{\prime}$
$\displaystyle=\frac{1}{(2\pi)^{n}}\int_{\operatorname{\mathbb{R}}^{n}}\mathrm{e}^{\mathrm{i}\boldsymbol{Q}\boldsymbol{x}^{\prime}\cdot\boldsymbol{Q}\boldsymbol{\xi}^{\prime}}\mathrm{e}^{\mathrm{i}\rho\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)y}\boldsymbol{M}(\boldsymbol{b},\sigma)\frac{1}{\rho^{2m}}\,\mathrm{d}\boldsymbol{\xi}^{\prime}$
$\displaystyle=\frac{1}{(2\pi)^{n}}\int_{\mathbb{S}^{n-2}}\int_{0}^{\infty}r^{n-2}\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i}\left|\boldsymbol{x}^{\prime}\right|a}\mathrm{e}^{\mathrm{i}\rho\boldsymbol{A}_{0}(\boldsymbol{Q}^{\mathsf{T}}(\tfrac{a}{\rho},\tfrac{r}{\rho}\boldsymbol{\varphi}),\tfrac{\lambda}{\rho^{2m}})y}\boldsymbol{M}(\boldsymbol{Q}^{\mathsf{T}}(\tfrac{a}{\rho},\tfrac{r}{\rho}\boldsymbol{\varphi}),\tfrac{\lambda}{\rho^{2m}})\frac{1}{\rho^{2m}}\,\mathrm{d}a\,\mathrm{d}r\,\mathrm{d}\sigma(\boldsymbol{\varphi})$
$\displaystyle=\frac{1}{(2\pi)^{n}}\int_{\mathbb{S}^{n-2}}\int_{0}^{\infty}r^{n-2}\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i}\left|\boldsymbol{x}^{\prime}\right|a}\frac{1}{(\left|\lambda\right|+(a^{2}+r^{2})^{m})^{2}}\boldsymbol{F}_{1}(\lambda,a,r,\boldsymbol{\varphi},y)\boldsymbol{F}_{2}(\lambda,a,r,\boldsymbol{\varphi})\,\mathrm{d}a\,\mathrm{d}r\,\mathrm{d}\sigma(\boldsymbol{\varphi})$

for operator-valued functions $\boldsymbol{F}_{1}$ and $\boldsymbol{F}_{2}$
which are defined by

$\displaystyle\boldsymbol{F}_{1}(\lambda,a,r,\boldsymbol{\varphi},y)$
$\displaystyle=\mathrm{e}^{\mathrm{i}(\left|\lambda\right|+(a^{2}+r^{2})^{m})^{1/2m}\boldsymbol{A}_{0}((\left|\lambda\right|+(a^{2}+r^{2})^{m})^{-\nicefrac{{1}}{{2m}}}\boldsymbol{Q}^{\mathsf{T}}(a,r\boldsymbol{\varphi}),\lambda(\left|\lambda\right|+(a^{2}+r^{2})^{m})^{-1})y}\in\operatorname{\mathcal{B}}(E^{2m}),$
$\displaystyle\boldsymbol{F}_{2}(\lambda,a,r,\boldsymbol{\varphi})$
$\displaystyle=\boldsymbol{M}((\left|\lambda\right|+(a^{2}+r^{2})^{m})^{\nicefrac{{1}}{{2m}}}\boldsymbol{Q}^{\mathsf{T}}(a,r\boldsymbol{\varphi}),\lambda(\left|\lambda\right|+(a^{2}+r^{2})^{m})^{-\nicefrac{{1}}{{2m}}})\in\operatorname{\mathcal{B}}(E^{m};E^{2m})$

for $\lambda\in\overline{\Sigma_{\pi-\phi}}$, $a\in\operatorname{\mathbb{R}}$,
$r\geq 0$, $\varphi\in\mathbb{S}^{n-1}$ and $y>0$. The inner integrand can be
interpreted as a function depending holomorphically on
$a\in\operatorname{\mathbb{C}}$ as the maps
$\boldsymbol{M}(\boldsymbol{b},\sigma)$ depend holomorphically on
$(\boldsymbol{b},\sigma)$, and $\boldsymbol{M}(\boldsymbol{b},\sigma)$ maps
into the stable part (corresponding to $S_{-}(\boldsymbol{b},\sigma)$) of
$\big{(}\mathrm{e}^{\mathrm{i}\rho\boldsymbol{A}_{0}(\boldsymbol{b},\sigma)y}\big{)}_{y\geq
0}$. Thus, we may and will replace the inner real integral