# Spectral properties of the gradient operator with nonconstant coefficients

F. Colombo  (FC) Politecnico di Milano, Dipartimento di Matematica, Via E.
Bonardi 9, 20133 Milano, Italy<EMAIL_ADDRESS>,  F. Mantovani
(FM) Politecnico di Milano, Dipartimento di Matematica, Via E. Bonardi 9,
20133 Milano, Italy<EMAIL_ADDRESS>and  P. Schlosser  (PS)
Politecnico di Milano, Dipartimento di Matematica, Via E. Bonardi 9, 20133
Milano, Italy<EMAIL_ADDRESS>
###### Abstract.

In mathematical physics, the gradient operator with nonconstant coefficients
encompasses various models, including Fourier’s law for heat propagation and
Fick’s first law, that relates the diffusive flux to the gradient of the
concentration. Specifically, consider $n\geq 3$ orthogonal unit vectors
$e_{1},\dots,e_{n}\in\mathbb{R}^{n}$, and let $\Omega\subseteq\mathbb{R}^{n}$
be some (in general unbounded) Lipschitz domain. This paper investigates the
spectral properties of the gradient operator
$T=\sum_{i=1}^{n}e_{i}a_{i}(x)\frac{\partial}{\partial x_{i}}$ with
nonconstant positive coefficients $a_{i}:\overline{\Omega}\to(0,\infty)$.
Under certain regularity and growth conditions on the $a_{i}$, we identify
bisectorial or strip-type regions that belong to the $S$-resolvent set of $T$.
Moreover, we obtain suitable estimates of the associated resolvent operator.
Our focus lies in the spectral theory on the $S$-spectrum, designed to study
the operators acting in Clifford modules $V$ over the Clifford algebra
$\mathbb{R}_{n}$, with vector operators being a specific crucial subclass. The
spectral properties related to the $S$-spectrum of $T$ are linked to the
inversion of the operator $Q_{s}(T):=T^{2}-2s_{0}T+|s|^{2}$, where
$s\in\mathbb{R}^{n+1}$ is a paravector, i.e., it is of the form
$s=s_{0}+s_{1}e_{1}+\dots+s_{n}e_{n}$. This spectral problem is substantially
different from the complex one, since it allows to associate general boundary
conditions to $Q_{s}(T)$, i.e., to the squared operator $T^{2}$.

AMS Classification 47A10, 47A60.

Keywords: Vector operators, $S$-spectrum, $S$-resolvent operator, boundary
value problem.

Acknowledgements: Fabrizio Colombo is supported by MUR grant Dipartimento di
Eccellenza 2023-2027. Peter Schlosser was funded by the Austrian Science Fund
(FWF) under Grant No. J 4685-N and by the European Union–NextGenerationEU.

##  1\. Introduction

Quaternionic and Clifford operators encompass vector operators, which arise in
numerous scientific and technological contexts. Contrary to complex
holomorphic function theory, there are different possible notions of
hyperholomorphicity in the noncommutative setting of $\mathbb{R}_{n}$.

Noncommutative spectral theories then rely on various concepts of spectra,
based on distinct Cauchy kernels depending on the type of hyperholomorphicity
being considered.

The spectral theory, grounded in the notion of $S$-spectrum and associated
with slice hyperholomorphicity, initiated its development in 2006, showing
that this noncommutative framework is not merely an extension deducible from
complex operator theory, especially when dealing with unbounded operators, see
[30]. The identification of the $S$-spectrum emerged through methodologies in
hypercomplex analysis, although its roots were initially hinted by
quaternionic quantum mechanics, see [9]. An exhaustive description on the
discovery of the $S$-spectrum can be found in the introduction of the book
[18]. The spectral theory on the $S$-spectrum was developed in several papers
and it is systematically organized in the books [2, 3, 17, 18, 24], where one
can find also the development of the theory of slice hyperholomorphic
functions. It is also important to point out that the quaternionic and the
Clifford spectral theorems are based on the notion of $S$-spectrum as well,
see [1, 20]. However, this spectral theory goes beyond the quaternionic and
the Clifford setting, as shown in [19].

Let us give a short explanation of the differences between complex and
$S$-spectral theory. The resolvent operator of a complex linear operator $A$
is directly linked to the definition of its spectrum via the invertibility of
the operator $\lambda-A$. In particular, if $A$ is bounded, the inverse of
this operator admits the series representation

$\sum_{n=0}^{\infty}A^{n}\lambda^{-1-n}=(\lambda-A)^{-1},\qquad|\lambda|>\|A\|.$

However, this identity is not true for Clifford operators $T$. In this case
there rather holds

$\displaystyle\sum_{n=0}^{\infty}T^{n}s^{-1-n}=(T^{2}-2s_{0}T+|s|^{2})^{-1}(\overline{s}-T),\qquad|s|>\|T\|,$
(1.1a)
$\displaystyle\sum_{n=0}^{\infty}s^{-1-n}T^{n}=(\overline{s}-T)(T^{2}-2s_{0}T+|s|^{2})^{-1},\qquad|s|>\|T\|,$
(1.1b)

for $s\in\mathbb{R}_{n}$ of the form $s=s_{0}+s_{1}e_{1}+\dots+s_{n}e_{n}$.
This relation suggests, that in contrast to the complex spectral theory, the
definition of the $S$-spectrum is connected to the invertibility of the second
order operator

$Q_{s}(T):=T^{2}-2s_{0}T+|s|^{2}.$ (1.2)

That is, for every $T\in\mathcal{B}(V)$, the $S$-spectrum and the
$S$-resolvent set of $T$ are defined as

$\rho_{S}(T):=\Set{s\in\mathbb{R}^{n+1}}{Q_{s}(T)^{-1}\in\mathcal{B}(V)}\qquad\text{and}\qquad\sigma_{S}(T):=\mathbb{R}^{n+1}\setminus\rho_{S}(T).$

Here $V$ is a Clifford module and $\mathcal{B}(V)$ the set of all bounded
right-linear operators $T:V\to V$. For every $s\in\rho_{S}(T)$, the inverse
$Q_{s}(T)^{-1}$ is called the pseudo $S$-resolvent operator.

A second main difference between complex and $S$-spectral theory, lies in the
fact that

$\rho_{S}(T)\ni s\mapsto Q_{s}(T)^{-1},$

is not a (operator valued) slice hyperholomorphic function. The operator
$Q_{s}(T)$ is only used for the definition of the $S$-spectrum, while the left
and the right $S$-resolvent operators are motivated by the series
representations (1.1a) and (1.1b)

$S_{L}^{-1}(s,T):=Q_{s}(T)^{-1}(\overline{s}-T)\qquad\text{and}\qquad
S_{R}^{-1}(s,T):=(\overline{s}-T)Q_{s}(T)^{-1},\qquad s\in\rho_{S}(T).$

These resolvent operators now preserve the slice hyperholomorphicity and are
used in the definition of the $S$-functional calculus for Clifford operators.

Clifford operators that are sectorial, bi-sectorial or strip-operators, can be
defined in analogy to complex operators, but in this case the estimates are
associated with the $S$-resolvents

$\|S_{L}^{-1}(s,T)\|\leq\frac{C}{|s|}\qquad\text{and}\qquad\|S_{R}^{-1}(s,T)\|\leq\frac{C}{|s|},$

where the parameter $s$ belongs to suitable sectorial, bi-sectorial or strip-
type subsets of $\rho_{S}(T)$.

In this paper we will study the spectral properties of the first order
differential operator

$T=\sum_{i=1}^{n}e_{i}a_{i}(x)\frac{\partial}{\partial x_{i}},\qquad
x\in\Omega,$ (1.3)

with Dirichlet, Neumann and Robin boundary conditions associated to
$Q_{s}(T)$. Here $e_{1},\dots,e_{n}$ are the imaginary units of the Clifford
algebra $\mathbb{R}_{n}$ and the coefficients $a_{1},\dots,a_{n}$ are defined
on some minimally smooth domain $\Omega\subseteq\mathbb{R}^{n}$ in the sense
of [45, Section VI.3.3], in $n\geq 3$ dimensions. The restriction to $n\geq 3$
is necessary since the crucial Sobolev inequality in Lemma 2.3 only holds true
in those dimensions. For the coefficients of $T$ in (1.3) we require that
$a_{1},\dots,a_{n}:\overline{\Omega}\mapsto(0,\infty)$ belong to
$C^{1}(\overline{\Omega})$ and satisfy the bounds

$\displaystyle m_{a}:=$
$\displaystyle\min\limits_{i\in\\{1,\dots,n\\}}\inf\limits_{x\in\Omega}a_{i}(x)>0,$
(1.4a) $\displaystyle M_{a}:=$
$\displaystyle\Big{(}\sum\nolimits_{i=1}^{n}\|a_{i}\|_{\infty}^{2}\Big{)}^{\frac{1}{2}}<\infty,$
(1.4b) $\displaystyle M_{a}^{\prime}:=$
$\displaystyle\Big{(}\sum\nolimits_{i,j=1}^{n}\Big{\|}a_{j}\frac{\partial
a_{i}}{\partial x_{j}}\Big{\|}^{2}_{L^{n}}\Big{)}^{\frac{1}{2}}<\infty,$
(1.4c)

where $\|\cdot\|_{L^{n}}$ is the usual Lebesgue norm of real valued functions
and $\|\cdot\|_{\infty}$ is the supremum norm of continuous functions. The
Clifford module in which we will be working, is

$L^{2}(\Omega)=\Set{f:\Omega\to\mathbb{R}_{n}}{\int_{\Omega}|u(x)|^{2}dx<\infty},$

that is the space of square integrable functions with values in the Clifford
algebra $\mathbb{R}_{n}$.

In this paper we will interpret the invertibility of the operator $Q_{s}(T)$
in (1.2) in the weak sense. That is, we ask whether the partial differential
equation

$Q_{s}(T)u(x)=(T^{2}-2s_{0}T+|s|^{2})u(x)=f(x),\qquad x\in\Omega,$ (1.5)

equipped with boundary conditions of Dirichlet, Neumann or Robin type, admits
a unique weak solution for every $f\in L^{2}(\Omega)$. In order to derive this
weak formulation, we first note that the square of the operator $T$ in (1.3)
is given by

$\displaystyle T^{2}$
$\displaystyle=\sum_{i,j=1}^{n}e_{j}a_{j}\frac{\partial}{\partial
x_{j}}\Big{(}e_{i}a_{i}\frac{\partial}{\partial x_{i}}\Big{)}$
$\displaystyle=-\sum_{i=1}^{n}a_{i}\frac{\partial}{\partial
x_{i}}\Big{(}a_{i}\frac{\partial}{\partial x_{i}}\Big{)}+\sum_{i,j=1,j\neq
i}^{n}e_{j}e_{i}a_{j}\frac{\partial a_{i}}{\partial
x_{j}}\frac{\partial}{\partial x_{i}}+\underbrace{\sum_{i,j=1,j\neq
i}^{n}e_{j}e_{i}a_{j}a_{i}\frac{\partial^{2}}{\partial x_{j}\partial
x_{i}}}_{\makebox[0.0pt]{=0}}$
$\displaystyle=-\sum_{i=1}^{n}a_{i}\frac{\partial}{\partial
x_{i}}\Big{(}a_{i}\frac{\partial}{\partial
x_{i}}\Big{)}-\sum_{i=1}^{n}e_{i}\Big{(}B_{i}-e_{i}a_{i}\frac{\partial
a_{i}}{\partial x_{i}}\Big{)}\frac{\partial}{\partial x_{i}},$ (1.6)

where the third sum in the second line vanishes due to anti-commutation
property of the units of the Clifford algebra $e_{j}e_{i}=-e_{i}e_{j}$ for
every $j\neq i$, and in the last line we used the abbreviation

$B_{i}:=\sum_{j=1}^{n}e_{j}a_{j}\frac{\partial a_{i}}{\partial x_{j}},\qquad
i\in\\{1,\dots,n\\}.$ (1.7)

Applying this operator to some $u\in H^{2}(\Omega)$, testing with $v\in
H^{1}(\Omega)$ and using integration by parts with the outer unit normal
vector $\vec{\nu}=(\nu_{1},\dots,\nu_{n})$ of the boundary $\partial\Omega$,
gives

$\displaystyle\langle T^{2}u,v\rangle_{L^{2}}$
$\displaystyle=-\sum_{i=1}^{n}\Big{\langle}a_{i}\frac{\partial}{\partial
x_{i}}\Big{(}a_{i}\frac{\partial u}{\partial
x_{i}}\Big{)},v\Big{\rangle}_{L^{2}}-\sum_{i=1}^{n}\Big{\langle}e_{i}\Big{(}B_{i}-e_{i}a_{i}\frac{\partial
a_{i}}{\partial x_{i}}\Big{)}\frac{\partial u}{\partial
x_{i}},v\Big{\rangle}_{L^{2}}$
$\displaystyle=\sum_{i=1}^{n}\Big{\langle}a_{i}\frac{\partial u}{\partial
x_{i}},\frac{\partial}{\partial
x_{i}}(a_{i}v)\Big{\rangle}_{L^{2}}-\sum_{i=1}^{n}\Big{\langle}\nu_{i}a_{i}\frac{\partial
u}{\partial x_{i}},a_{i}v\Big{\rangle}_{L^{2}(\partial\Omega)}$
$\displaystyle\quad-\sum_{i=1}^{n}\Big{\langle}\frac{\partial u}{\partial
x_{i}},\Big{(}B_{i}-e_{i}a_{i}\frac{\partial a_{i}}{\partial
x_{i}}\Big{)}e_{i}v\Big{\rangle}_{L^{2}}$
$\displaystyle=\sum_{i=1}^{n}\Big{\langle}\frac{\partial u}{\partial
x_{i}},a_{i}^{2}\frac{\partial v}{\partial
x_{i}}-B_{i}e_{i}v\Big{\rangle}_{L^{2}}-\sum_{i=1}^{n}\Big{\langle}\frac{\partial
u}{\partial x_{i}},a_{i}^{2}v\Big{\rangle}_{L^{2}(\partial\Omega)}.$

Consequently, the weak formulation of the problem (1.5) is connected to the
form

$q_{s}(u,v):=\sum_{i=1}^{n}\Big{\langle}\frac{\partial u}{\partial
x_{i}},a_{i}^{2}\frac{\partial v}{\partial
x_{i}}+(2s_{0}a_{i}-B_{i})e_{i}v\Big{\rangle}_{L^{2}}+|s|^{2}\langle
u,v\rangle_{L^{2}}-\sum_{i=1}^{n}\Big{\langle}\nu_{i}\frac{\partial
u}{\partial x_{i}},a_{i}^{2}v\Big{\rangle}_{L^{2}(\partial\Omega)}.$ (1.8)

The primary objective of this paper is to delve into this spectral problem
with Dirichlet, Neumann and Robin-type boundary conditions, establishing bi-
sectorial estimates for the solution and hence the $S$-resolvent operator.
These findings are an extension of the quaternionic framework explored in
prior publications such as [8, 14, 15, 21], and in some cases contains results
never investigated in previous works.

The structure of the paper is outlined as follows: In Section 2, we present
the main results concerning modules over the Clifford algebra
$\mathbb{R}_{n}$. Among those is the Lax-Milgram lemma, which will be crucial
for the solvability of the form (1.8) in Section 3 and Section 4. Section 3
focuses on the $S$-spectral problem (1.5) with Dirichlet boundary conditions
$u=0$ on $\partial\Omega$, and similarly, Section 4 investigates Robin-type
boundary conditions $\sum_{i=1}^{n}\nu_{i}a_{i}\frac{\partial u}{\partial
x_{i}}+bu=0$ on $\partial\Omega$. Additional to the unique weak solvability,
for both kinds of boundary conditions, $L^{2}$\- and $H^{1}$-estimates of the
solution are established. Consequently sectoriality estimates of the resolvent
operators can be deduced. The paper concludes with some concluding remarks on
the current state of this theory in Section 5, as well as an auxiliary
maximization problem in the Appendix A.

##  2\. Functional setting in Clifford modules over $\mathbb{R}_{n}$

In this section we consider for $n\in\mathbb{N}$, $n\geq 3$ the Clifford
algebra $\mathbb{R}_{n}$, generated by $n$ imaginary units $e_{1},\dots,e_{n}$
which satisfy the relations

$e_{i}^{2}=-1\qquad\text{and}\qquad e_{i}e_{j}=-e_{j}e_{i},\qquad i\neq
j\in\\{1,\dots,n\\}.$

More precisely, $\mathbb{R}_{n}$ is given by

$\mathbb{R}_{n}:=\Set{\sum\nolimits_{A\in\mathcal{A}}x_{A}e_{A}}{(x_{A})_{A}\in\mathbb{R}},$
(2.1)

using the index set

$\mathcal{A}:=\Set{(i_{1},\dots,i_{r})}{r\in\\{0,\dots,n\\},\,1\leq
i_{1}<\dots<i_{r}\leq n},$

and the basis vectors $e_{A}:=e_{i_{1}}\dots e_{i_{r}}$. Note, for
$A=\emptyset$ the empty product of imaginary units is the real number
$e_{\emptyset}:=1$. Moreover, we will consider the set of all paravectors

$\mathbb{R}^{n+1}:=\Set{x_{0}+\sum\nolimits_{i=1}^{n}x_{i}e_{i}}{x_{0},x_{1},\dots,x_{n}\in\mathbb{R}}.$

For any Clifford number $x\in\mathbb{R}_{n}$, we define

$\displaystyle\operatorname{Sc}(x)$ $\displaystyle:=x_{\emptyset},$ (scalar
part) $\displaystyle\overline{x}$
$\displaystyle:=\sum\nolimits_{A\in\mathcal{A}}x_{A}\overline{e_{A}},$
(conjugate) $\displaystyle|x|$
$\displaystyle:=\Big{(}\sum\nolimits_{A\in\mathcal{A}}|x_{A}|^{2}\Big{)}^{\frac{1}{2}}=\operatorname{Sc}(x\overline{x})=\operatorname{Sc}(\overline{x}x),$
(norm)

where $\overline{e_{A}}=\overline{e_{i_{r}}}\dots\overline{e_{i_{1}}}$ and
$\overline{e_{i}}=-e_{i}$. It is now obvious that for any Clifford number
$x\in\mathbb{R}_{n}$ one can calculate its coefficients $x_{A}$ by

$\operatorname{Sc}(x\overline{e_{A}})=\operatorname{Sc}\Big{(}\sum\nolimits_{B\in\mathcal{A}}x_{B}e_{B}\overline{e_{A}}\Big{)}=\sum\nolimits_{B\in\mathcal{A}}x_{B}\operatorname{Sc}(e_{B}\overline{e_{A}})=x_{A},$
(2.2)

where in the last equation we used that

$\operatorname{Sc}(e_{B}\overline{e_{A}})=\begin{cases}1,&\text{if }B=A,\\\
0,&\text{if }B\neq A.\end{cases}$ (2.3)

For any real Hilbert space $V_{\mathbb{R}}$ with inner product
$\langle\cdot,\cdot\rangle_{\mathbb{R}}$ and norm
$\|\cdot\|_{\mathbb{R}}^{2}=\langle\cdot,\cdot\rangle_{\mathbb{R}}$, we define
the generated Clifford module

$V:=\Set{\sum\nolimits_{A\in\mathcal{A}}v_{A}\otimes e_{A}}{v_{A}\in
V_{\mathbb{R}}}.$

For any vector $v=\sum_{A\in\mathcal{A}}v_{A}\otimes e_{A}\in V$ and any
Clifford number $x=\sum_{A\in\mathcal{A}}x_{A}e_{A}\in\mathbb{R}_{n}$, we
equip this space with a left and a right scalar multiplication

$\displaystyle xv:=$
$\displaystyle\sum\nolimits_{A,B\in\mathcal{A}}(x_{B}v_{A})\otimes(e_{B}e_{A}),$
(left-multiplication) (2.4a) $\displaystyle vx:=$
$\displaystyle\sum\nolimits_{A,B\in\mathcal{A}}(v_{A}x_{B})\otimes(e_{A}e_{B}).$
(right-multiplication) (2.4b)

Moreover, we define the inner product

$\langle v,w\rangle:=\sum\nolimits_{A,B\in\mathcal{A}}\langle
v_{A},w_{B}\rangle_{\mathbb{R}}\,\overline{e_{A}}e_{B},\qquad v,w\in V,$ (2.5)

and the norm

$\|v\|^{2}:=\sum\nolimits_{A\in\mathcal{A}}\|v_{A}\|_{\mathbb{R}}^{2}=\operatorname{Sc}\langle
v,v\rangle,\qquad v\in V.$ (2.6)

The sesquilinear form (2.5) is clearly right-linear in the second, and right-
antilinear in the first argument, i.e. for every $u,v,w\in V$,
$x\in\mathbb{R}_{n}$, there holds

$\displaystyle\langle u,v+w\rangle$ $\displaystyle=\langle u,v\rangle+\langle
u,w\rangle,$ $\displaystyle\langle w,vx\rangle=\langle w,v\rangle x,$
$\displaystyle\langle v+w,u\rangle$ $\displaystyle=\langle v,u\rangle+\langle
w,u\rangle,$ $\displaystyle\langle vx,w\rangle=\overline{x}\langle
v,w\rangle.$

Next we provide some basic properties of the inner product (2.5) and the norm
(2.6).

######  Lemma 2.1.

For every $v,w\in V$, $x\in\mathbb{R}_{n}$, there holds

  1. i)

$\|vx\|\leq 2^{\frac{n}{2}}|x|\|v\|$,

  2. ii)

$\|vx\|=|x|\|v\|$, if $x\in\mathbb{R}^{n+1}$ is a paravector,

  3. iii)

$|\langle v,w\rangle|\leq 2^{\frac{n}{2}}\|v\|\,\|w\|$,

  4. iv)

$|\operatorname{Sc}\langle v,w\rangle|\leq\|v\|\,\|w\|$.

###### Proof.

i) Analogously to (2.2), one can write for every $A\in\mathcal{A}$ the
component $(vx)_{A}$ as

$(vx)_{A}=\operatorname{Sc}(vx\overline{e_{A}})=\operatorname{Sc}\bigg{(}\sum\nolimits_{B,C\in\mathcal{A}}(v_{B}(x\overline{e_{A}})_{C})\otimes(e_{B}e_{C})\bigg{)}=\sum\nolimits_{B\in\mathcal{A}}(-1)^{\frac{|B|(|B|+1)}{2}}v_{B}(x\overline{e_{A}})_{B},$

where in the last equation we used that $e_{B}e_{C}\in\mathbb{R}$ if and only
if $B=C$. In this form, we can estimate every component of the product as

$\displaystyle\|(vx)_{A}\|_{\mathbb{R}}$
$\displaystyle\leq\sum\nolimits_{B\in\mathcal{A}}\|v_{B}(x\overline{e_{A}})_{B}\|_{\mathbb{R}}=\sum\nolimits_{B\in\mathcal{A}}\|v_{B}\|_{\mathbb{R}}|(x\overline{e_{A}})_{B}|$
$\displaystyle\leq\Big{(}\sum\nolimits_{B\in\mathcal{A}}\|v_{B}\|^{2}_{\mathbb{R}}\Big{)}^{\frac{1}{2}}\Big{(}\sum\nolimits_{B\in\mathcal{A}}|(x\overline{e_{A}})_{B}|^{2}\Big{)}^{\frac{1}{2}}=\|v\|\,|x\overline{e_{A}}|=\|v\|\,|x|.$

Consequently the norm of the whole product admits the upper bound

$\|vx\|^{2}=\sum\nolimits_{A\in\mathcal{A}}\|(vx)_{A}\|_{\mathbb{R}}^{2}\leq\sum\nolimits_{A\in\mathcal{A}}\|v\|^{2}|x|^{2}=2^{n}\|v\|^{2}|x|^{2}.$

ii) If we assume that $x$ is a paravector, it clearly satisfies
$x\overline{x}=|x|^{2}$, and we have

$\|vx\|^{2}=\operatorname{Sc}\langle
vx,vx\rangle=\operatorname{Sc}\big{(}\overline{x}\langle v,v\rangle
x\big{)}=\operatorname{Sc}\big{(}x\overline{x}\langle
v,v\rangle\big{)}=|x|^{2}\operatorname{Sc}\langle
v,v\rangle=|x|^{2}\|v\|^{2}.$

iii) Using the representation (2.2) for the quaternionic number $\langle
v,w\rangle$ as well as the already proven results i) and ii), gives

$|\langle v,w\rangle|^{2}=\sum\nolimits_{A\in\mathcal{A}}\langle
v,w\rangle_{A}^{2}=\sum\nolimits_{A\in\mathcal{A}}\big{(}\operatorname{Sc}\langle
v,w\,\overline{e_{A}}\rangle\big{)}^{2}\leq\sum\nolimits_{A\in\mathcal{A}}\|v\|^{2}\|w\,\overline{e_{A}}\|^{2}=2^{n}\|v\|^{2}\|w\|^{2}.$

iv) It follows from (2.3) and the Cauchy-Schwartz inequality of
$\langle\cdot,\cdot\rangle_{\mathbb{R}}$, that

$\displaystyle|\operatorname{Sc}\langle u,v\rangle|$
$\displaystyle=\Big{|}\operatorname{Sc}\sum\nolimits_{A,B\in\mathcal{A}}\langle
v_{A},w_{B}\rangle_{\mathbb{R}}\overline{e_{A}}e_{B}\Big{|}=\Big{|}\sum\nolimits_{A\in\mathcal{A}}\langle
v_{A},w_{A}\rangle_{\mathbb{R}}\Big{|}\leq\sum\nolimits_{A\in\mathcal{A}}\|v_{A}\|_{\mathbb{R}}\|w_{A}\|_{\mathbb{R}}$
$\displaystyle\leq\Big{(}\sum\nolimits_{A\in\mathcal{A}}\|v_{A}\|_{\mathbb{R}}^{2}\Big{)}^{\frac{1}{2}}\Big{(}\sum\nolimits_{A\in\mathcal{A}}\|w_{A}\|_{\mathbb{R}}^{2}\Big{)}^{\frac{1}{2}}=\|v\|\,\|w\|.\qed$

Next, we come to the main result of this first section, the Lax-Milgram lemma
for Clifford modules. We will need this result in the later sections to proof
unique solvability of the spectral problem for the $S$-spectrum, in particular
to detect regions of $S$-resolvent set and related estimates of the
$S$-resolvent operator. Although the Lax-Milgram lemma is a well known result
in complex functional analysis, for its Clifford algebra version, we cannot
provide a precise reference, hence we give a short proof.

######  Lemma 2.2 (Lemma of Lax-Milgram).

Let $q:V\times V\to\mathbb{R}_{n}$ be right-linear in the second and right-
antilinear in the first argument. Moreover, we assume that there exists
constants $C\geq 0$ and $\kappa>0$, such that

$q(v,v)\geq\kappa\|v\|^{2}\qquad\text{and}\qquad|q(v,w)|\leq
C\|v\|\|w\|,\qquad v,w\in V.$

Then for every right-linear bounded functional $\varphi:V\to\mathbb{R}_{n}$,
there exists a unique vector $u_{\varphi}\in V$, satisfying

$\|u_{\varphi}\|\leq\frac{1}{\kappa}\|\operatorname{Sc}\varphi\|\qquad\text{and}\qquad
q(u_{\varphi},v)=\varphi(v),\qquad\text{for all }v\in V.$

###### Proof.

Note that it is possible to consider $V$ together with the inner product
$\operatorname{Sc}\langle\cdot,\cdot\rangle$ as a real Hilbert space, by
simply restricting the scalar multiplication to real numbers. In this sense,
the scalar part $\operatorname{Sc}q(\cdot,\cdot):V\times V\to\mathbb{R}$ turns
out to be a real bilinear mapping, which satisfies for every $v,w\in V$ the
estimates

$\operatorname{Sc}q(v,v)=q(v,v)\geq\kappa\|v\|^{2}\qquad\text{and}\qquad|\operatorname{Sc}q(v,w)|\leq|q(v,w)|\leq
C\|v\|\,\|w\|.$

Since also $\operatorname{Sc}\varphi:V\to\mathbb{R}$ is a linear and bounded
functional, the classical Lax-Milgram lemma ensures the existence of a unique
$u_{\varphi}\in V$, satisfying

$\|u_{\varphi}\|\leq\frac{1}{\kappa}\|\operatorname{Sc}\varphi\|\qquad\text{and}\qquad\operatorname{Sc}q(u_{\varphi},v)=\operatorname{Sc}\varphi(v),\qquad\text{for
all }v\in V.$

Using now the components
$\varphi(v)_{A}=\operatorname{Sc}\varphi(v\overline{e_{A}})$ and
$q(u_{\varphi},v)_{A}=\operatorname{Sc}q(u_{\varphi},v\overline{e_{A}})$, see
(2.2), we can extend this representation result to

$q(u_{\varphi},v)=\sum\nolimits_{A\in\mathcal{A}}\operatorname{Sc}q(u_{\varphi},v\overline{e_{A}})e_{A}=\sum\nolimits_{A\in\mathcal{A}}\operatorname{Sc}\varphi(v\overline{e_{A}})e_{A}=\varphi(v),\qquad\text{for
all }v\in V.\qed$

We will end this section with two basic lemmas, which will be crucial
throughout the paper, the Sobolev and the Poincaré inequality. In order to
state these inequalities we will need for every function $u\in H^{1}(\Omega)$
the Sobolev-seminorm

$\|u\|_{D}:=\bigg{(}\sum_{i=1}^{n}\Big{\|}\frac{\partial u}{\partial
x_{i}}\Big{\|}_{L^{2}}^{2}\bigg{)}^{1/2}.$

######  Lemma 2.3 (Sobolev inequality).

Let $n\geq 3$ and $\Omega\subseteq\mathbb{R}^{n}$ be an minimally smooth
domain in the sense of [45, Section VI.3.3]. Then there exists a constant
$C_{S}\geq 0$, such that

$\displaystyle\|u\|_{L^{\frac{2n}{n-2}}}$ $\displaystyle\leq
C_{S}\|u\|_{D},\hskip 27.03003ptu\in H_{0}^{1}(\Omega),$ (2.7a)
$\displaystyle\|u\|_{L^{\frac{2n}{n-2}}}$ $\displaystyle\leq
C_{S}\|u\|_{H^{1}},\qquad u\in H^{1}(\Omega).$ (2.7b)

In (2.7a) the constant is explicitly given by $C_{S}=\frac{1}{\sqrt{\pi
n(n-2)}}\big{(}\frac{\Gamma(n)}{\Gamma(\frac{n}{2})}\big{)}^{\frac{1}{n}}$.

###### Proof.

The basic Gagliardo-Nirenberg-Sobolev inequality on $\Omega=\mathbb{R}^{n}$ is

$\|u\|_{L^{\frac{2n}{n-2}}}\leq C_{S}\|u\|_{D},\qquad u\in
H^{1}(\mathbb{R}^{n}),$ (2.8)

with the optimal constant $C_{S}=\frac{1}{\sqrt{\pi
n(n-2)}}\big{(}\frac{\Gamma(n)}{\Gamma(\frac{n}{2})}\big{)}^{1/n}$, see for
example [46]. If we now consider $u\in H_{0}^{1}(\Omega)$, the zero-extension
of $u$ is an element in $H^{1}(\mathbb{R}^{n})$ and the inequality (2.7a)
follows immediately from (2.8) with the same constant. If on the other hand
$u\in H^{1}(\Omega)$, we have to use the extension operator
$E:H^{1}(\Omega)\to H^{1}(\mathbb{R}^{n})$ from [45, Section VI.3.1,Theorem
5], to get

$\|u\|_{L^{\frac{2n}{n-2}}(\Omega)}\leq\|Eu\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^{n})}\leq
C_{S}\|Eu\|_{H^{1}(\mathbb{R}^{n})}\leq C_{S}\|E\|\|u\|_{H^{1}(\Omega)}.\qed$

######  Lemma 2.4 (Poincaré inequality).

[10, Corollary 9.19 & Remark 21] Let $\Omega\subseteq\mathbb{R}^{n}$ be open
and either bounded in one direction or of finite measure. Then there exists a
constant $C_{P}>0$, such that

$\|u\|_{L^{2}}\leq C_{P}\|u\|_{D},\qquad u\in H_{0}^{1}(\Omega).$ (2.9)

##  3\. Dirichlet boundary conditions

In this section we study the weak formulation of the spectral problem (1.5)
with Dirichlet boundary conditions, on some minimally smooth domain
$\Omega\subseteq\mathbb{R}^{n}$, $n\geq 3$, in the sense of Stein [45, Section
VI.3.3]. Precisely, we equip the form (1.8) with Dirichlet boundary
conditions, that is, for every $s\in\mathbb{R}^{n+1}$ we consider the form

$q_{s}^{D}(u,v):=\sum_{i=1}^{n}\Big{\langle}\frac{\partial u}{\partial
x_{i}},a_{i}^{2}\frac{\partial v}{\partial
x_{i}}+(2s_{0}a_{i}-B_{i})e_{i}v\Big{\rangle}_{L^{2}}+|s|^{2}\langle
u,v\rangle_{L^{2}},$ (3.1)

with $\operatorname{dom}q_{s}^{D}:=H_{0}^{1}(\Omega)\times H_{0}^{1}(\Omega)$.
We study the connection between regularity and growth conditions of the
coefficients of the operator $T$ in (1.3) and the most extended portion of the
$S$-resolvent set of $T$.

######  Problem 3.1.

Determine assumptions on the coefficients $a_{1},\dots,a_{n}$ in (1.3) and on
the spectral parameter $s\in\mathbb{R}^{n+1}$, such that for every $f\in
L^{2}(\Omega)$ there exists a unique solution $u_{f}\in H_{0}^{1}(\Omega)$ of

$q_{s}^{D}(u_{f},v)=\langle f,v\rangle_{L^{2}},\qquad\text{for all }v\in
H_{0}^{1}(\Omega).$

Furthermore, determine $L^{2}$\- and $H^{1}$-estimates of $u_{f}$, depending
on the parameter $s$.

######  Theorem 3.2.

Let the bounds (1.4) of the coefficients $a_{1},\dots,a_{n}$ in (1.4) satisfy

$m_{a}^{2}>C_{S}M_{a}^{\prime},$

with $C_{S}\geq 0$ from Lemma 2.3. Let us set
$K_{a}:=\frac{M_{a}}{\sqrt{m_{a}^{2}-C_{S}M_{a}^{\prime}}}$ and consider
$s\in\mathbb{R}^{n+1}$ with

$|s|>K_{a}|s_{0}|.$ (3.2)

Then for every $f\in L^{2}(\Omega)$ there exists a unique $u_{f}\in
H_{0}^{1}(\Omega)$ such that

$q_{s}^{D}(u_{f},v)=\langle f,v\rangle_{L^{2}},\qquad\text{for all }v\in
H_{0}^{1}(\Omega).$ (3.3)

Moreover, this solution satisfies the bounds

$\|u_{f}\|_{L^{2}}\leq\frac{\|f\|_{L^{2}}}{|s|^{2}-K_{a}^{2}s_{0}^{2}}\qquad\text{and}\qquad\|u_{f}\|_{D}\leq\frac{K_{a}|s|\|f\|_{L^{2}}}{M_{a}(|s|^{2}-K_{a}^{2}s_{0}^{2})}.$
(3.4)

$|s|=K_{a}|s_{0}|$$s_{0}$$\operatorname{Im}(s)$

$s_{0}$$\operatorname{Im}(s)$

Figure 1. Both pictures show the double cone which arises from the condition
(3.2). The two graphics also visualize the values of the $\|\cdot\|_{L^{2}}$\-
and the $\|\cdot\|_{D}$-norm bounds (3.4), ($\|\cdot\|_{L^{2}}$ on the left,
$\|\cdot\|_{D}$ on the right). Plotted are the level lines of these upper
bounds, where dark colors indicate large values and light colors small values.
One in particular sees the singularities of both estimates on the boundary of
the double cone and also the $\mathcal{O}(|s|^{-2})$-behaviour of the
$\|\cdot\|_{L^{2}}$-estimate compared to the slower
$\mathcal{O}(|s|^{-1})$-behaviour of the $\|\cdot\|_{D}$-estimate for
$|s|\to\infty$ but also for $|s|\to 0$.

###### Proof.

For the proof of the weak solvability (3.3), we will verify that the form
$q_{s}^{D}$ satisfies the assumptions of Lemma 2.2. For the boundedness, we
use the Cauchy-Schwartz type inequality in Lemma 2.1 iii), as well as the
bound $M_{a}$ in (1.4b), to obtain

$\displaystyle|q_{s}^{D}(u,v)|$ $\displaystyle\leq
2^{\frac{n}{2}}\sum_{i=1}^{n}\Big{\|}\frac{\partial u}{\partial
x_{i}}\Big{\|}_{L^{2}}\Big{(}M_{a}^{2}\Big{\|}\frac{\partial v}{\partial
x_{i}}\Big{\|}_{L^{2}}+2M_{a}|s_{0}|\|v\|_{L^{2}}+\|B_{i}e_{i}v\|_{L^{2}}\Big{)}+2^{\frac{n}{2}}|s|^{2}\|u\|_{L^{2}}\|v\|_{L^{2}}$
$\displaystyle\leq
2^{\frac{n}{2}}\|u\|_{D}\bigg{(}M_{a}^{2}\|v\|_{D}+2M_{a}|s_{0}|\|v\|_{L^{2}}+\bigg{(}\sum_{i=1}^{n}\|B_{i}e_{i}v\|_{L^{2}}^{2}\bigg{)}^{1/2}\bigg{)}+2^{\frac{n}{2}}|s|^{2}\|u\|_{L^{2}}\|v\|_{L^{2}}.$

Since $B_{i}$ and $e_{i}$ are both paravectors, we conclude from Lemma 2.1
ii), that

$|B_{i}e_{i}v|=|B_{i}||e_{i}v|=|B_{i}||e_{i}|\|v\|=|B_{i}|\|v\|.$

Hölder’s inequality with $p=\frac{n}{2}$ and $q=\frac{2}{2-n}$, the Sobolev
inequality (2.7a), as well as the bound $M_{a}^{\prime}$ in (1.4c), then give
the estimate of the term

$\displaystyle\sum_{i=1}^{n}\|B_{i}e_{i}v\|^{2}_{L^{2}}$
$\displaystyle=\sum_{i=1}^{n}\int_{\Omega}|B_{i}e_{i}v|^{2}dx=\sum_{i=1}^{n}\int_{\Omega}|B_{i}|^{2}|v|^{2}dx=\sum_{i,j=1}^{n}\int_{\Omega}\Big{|}a_{j}\frac{\partial
a_{i}}{\partial x_{j}}\Big{|}^{2}|v|^{2}dx$
$\displaystyle\leq\sum_{i,j=1}^{n}\Big{\|}a_{j}\frac{\partial a_{i}}{\partial
x_{j}}\Big{\|}_{L^{n}}^{2}\|v\|_{L^{\frac{2n}{n-2}}}^{2}\leq
C_{S}^{2}\sum_{i,j=1}^{n}\Big{\|}a_{j}\frac{\partial a_{i}}{\partial
x_{j}}\Big{\|}^{2}_{L^{n}}\|v\|^{2}_{D}\leq C_{S}^{2}M_{a}^{\prime
2}\|v\|^{2}_{D}.$ (3.5)

Hence, we conclude the boundedness

$\displaystyle|q_{s}^{D}(u,v)|$ $\displaystyle\leq
2^{\frac{n}{2}}\|u\|_{D}\big{(}M_{a}^{2}\|v\|_{D}+2M_{a}|s_{0}|\|v\|_{L^{2}}+C_{S}M_{a}^{\prime}\|v\|_{D}\big{)}+2^{\frac{n}{2}}|s|^{2}\|u\|_{L^{2}}\|v\|_{L^{2}}$
$\displaystyle\leq
2^{\frac{n}{2}}\big{(}M_{a}^{2}+2M_{a}|s_{0}|+C_{S}M_{a}^{\prime}+|s|^{2}\big{)}\|u\|_{H^{1}}\|v\|_{H^{1}}.$
(3.6)

Next, for the coercivity of the scalar part $\operatorname{Sc}q_{s}^{D}$, we
use the lower bound $m_{a}$ in (1.4a) and the Cauchy-Schwartz type inequality
of Lemma 2.1 iv), to get

$\displaystyle\operatorname{Sc}q_{s}^{D}(u,u)$
$\displaystyle\geq\sum\limits_{i=1}^{n}\bigg{(}m_{a}^{2}\Big{\|}\frac{\partial
u}{\partial x_{i}}\Big{\|}_{L^{2}}^{2}-2M_{a}|s_{0}|\Big{\|}\frac{\partial
u}{\partial x_{i}}\Big{\|}_{L^{2}}\|u\|_{L^{2}}-\Big{\|}\frac{\partial
u}{\partial
x_{i}}\Big{\|}_{L^{2}}\|B_{i}e_{i}u\|_{L^{2}}\bigg{)}+|s|^{2}\|u\|^{2}_{L^{2}}$
$\displaystyle\geq(m_{a}^{2}-C_{S}M_{a}^{\prime})\|u\|_{D}^{2}-2M_{a}|s_{0}|\|u\|_{D}\|u\|_{L^{2}}+|s|^{2}\|u\|^{2}_{L^{2}},$
(3.7)

where in the second inequality we again used (3.5). By the assumption (3.2) on
$s$, there is

$\delta_{0}:=\frac{K_{a}^{2}|s_{0}|}{M_{a}}<\frac{|s|^{2}}{M_{a}|s_{0}|}=:\delta_{1}.$
(3.8)

Note, that we interpret $\delta_{1}=\infty$ if $s_{0}=0$. Hence, for
$\delta\in(\delta_{0},\delta_{1})$, we use Young’s inequality

$\|u\|_{D}\|u\|_{L^{2}}\leq\frac{1}{2\delta}\|u\|_{D}^{2}+\frac{\delta}{2}\|u\|^{2}_{L^{2}},$

to further estimate the lower bound (3.7) by

$\operatorname{Sc}q_{s}^{D}(u,u)\geq\Big{(}\frac{M_{a}^{2}}{K_{a}^{2}}-\frac{M_{a}|s_{0}|}{\delta}\Big{)}\|u\|^{2}_{D}+\big{(}|s|^{2}-M_{a}|s_{0}|\delta\big{)}\|u\|^{2}_{L^{2}}$
(3.9)

Note, that $\delta_{0}$ and $\delta_{1}$ in (3.8) are chosen exactly in the
way that both brackets on the right hand side of (3.9) are positive. Hence we
have proven that $\operatorname{Sc}q_{s}^{D}$ is coercive in
$H_{0}^{1}(\Omega)$. Fixing now any function $f\in L^{2}(\Omega)$, we can
consider the corresponding functional

$\varphi_{f}(v):=\langle f,v\rangle_{L^{2}},\qquad v\in H_{0}^{1}(\Omega).$

Then by Lemma 2.1 iii), this functional is bounded in $H_{0}^{1}(\Omega)$,
i.e.

$|\varphi_{f}(v)|=|\langle f,v\rangle_{L^{2}}|\leq
2^{\frac{n}{2}}\|f\|_{L^{2}}\|v\|_{L^{2}}\leq
2^{\frac{n}{2}}\|f\|_{L^{2}}\|v\|_{H^{1}},\qquad v\in H_{0}^{1}(\Omega).$
(3.10)

Hence, the assumptions of Lemma 2.2 are satisfied and there exists a unique
weak solution $u_{f}\in H_{0}^{1}(\Omega)$, which satisfies

$q_{s}^{D}(u_{f},v)=\varphi_{f}(v)=\langle f,v\rangle_{L^{2}},\qquad\text{for
all }v\in H_{0}^{1}(\Omega).$ (3.11)

For the $L^{2}$-estimate in (3.4), we test (3.11) with $v=u_{f}$. Using also
Lemma 2.1 iv), this gives

$\operatorname{Sc}q_{s}^{D}(u_{f},u_{f})=\operatorname{Sc}\langle
f,u_{f}\rangle_{L^{2}}\leq\|f\|_{L^{2}}\|u_{f}\|_{L^{2}}.$ (3.12)

Combining this inequality with the coercivity estimate (3.9), furthermore
gives

$\big{(}|s|^{2}-M_{a}|s_{0}|\delta\big{)}\|u_{f}\|^{2}_{L^{2}}\leq\operatorname{Sc}q_{s}^{D}(u_{f},u_{f})\leq\|f\|_{L^{2}}\|u_{f}\|_{L^{2}}.$

However, since $\delta\in(\delta_{0},\delta_{1})$ was chosen arbitrary, we can
maximize the left hand side by taking the limit
$\delta\to\delta_{0}=\frac{K_{a}^{2}|s_{0}|}{M_{a}}$, and get

$\|u_{f}\|_{L^{2}}\leq\frac{\|f\|_{L^{2}}}{|s|^{2}-K_{a}^{2}s_{0}^{2}}.$
(3.13)

For the $\|\cdot\|_{D}$-estimate, we get from (3.9), (3.12) and (3.13) for
every $\delta\in(\delta_{0},\delta_{1})$ the estimate

$\Big{(}\frac{M_{a}^{2}}{K_{a}^{2}}-\frac{M_{a}|s_{0}|}{\delta}\Big{)}\|u_{f}\|^{2}_{D}\leq\operatorname{Sc}q_{s}^{D}(u_{f},u_{f})\leq\|f\|_{L^{2}}\|u_{f}\|_{L^{2}}\leq\frac{\|f\|_{L^{2}}^{2}}{|s|^{2}-K_{a}^{2}s_{0}^{2}}.$

Maximizing the left hand side with respect to
$\delta\in(\delta_{0},\delta_{1})$, i.e. taking the limit
$\delta\to\delta_{1}=\frac{|s|^{2}}{M_{a}|s_{0}|}$, gives

$\Big{(}\frac{M_{a}^{2}}{K_{a}^{2}}-\frac{M_{a}^{2}s_{0}^{2}}{|s|^{2}}\Big{)}\|u_{f}\|^{2}_{D}\leq\|f\|_{L^{2}}\|u_{f}\|_{L^{2}}\leq\frac{\|f\|_{L^{2}}^{2}}{|s|^{2}-K_{a}^{2}s_{0}^{2}}$

Rearranging this inequality gives exactly the second estimate in (3.4). ∎

In the special case of minimally smooth domains
$\Omega\subseteq\mathbb{R}^{n}$, which are either bounded in one direction or
of finite measure, we even get stronger results. In particular, the double
sector of Theorem 3.2 opens at the origin $s=0$, which means that the
Dirichlet form $q_{s}^{D}$ is uniquely solvable also in a vertical strip
around the imaginary axis.

######  Theorem 3.3.

Let $\Omega\subseteq\mathbb{R}^{n}$ be a bounded minimally smooth domain in
$n\geq 3$ dimensions and let the bounds (1.4) of the coefficients
$a_{1},\dots,a_{n}$ in (1.4) satisfy

$m_{a}^{2}>C_{S}M_{a}^{\prime},$

with $C_{S}\geq 0$ from Lemma 2.3. Let us set
$K_{a}:=\frac{M_{a}}{\sqrt{m_{a}^{2}-C_{S}M_{a}^{\prime}}}$, use the Poincaré
constant $C_{P}\in(0,1]$ from Lemma 2.4, and consider $s\in\mathbb{R}^{n+1}$
with

$|s|^{2}>\begin{cases}\frac{2M_{a}|s_{0}|}{C_{P}}-\frac{M_{a}^{2}}{C_{P}^{2}K_{a}^{2}},&\text{if
}|s_{0}|\leq\frac{M_{a}}{C_{P}K_{a}^{2}},\\\ K_{a}^{2}s_{0}^{2},&\text{if
}|s_{0}|\geq\frac{M_{a}}{C_{P}K_{a}^{2}}.\end{cases}$ (3.14)

Then for every $f\in L^{2}(\Omega)$ there exists a unique $u_{f}\in
H_{0}^{1}(\Omega)$, such that

$q_{s}^{D}(u_{f},v)=\langle f,v\rangle_{L^{2}},\qquad{\rm for\ all}\ \ v\in
H_{0}^{1}(\Omega).$ (3.15)

Moreover, this solution admits the norm estimates

$\|u_{f}\|_{L^{2}}\leq\frac{\|f\|_{L^{2}}}{\kappa_{0}(s)}\qquad\text{and}\qquad\|u_{f}\|_{D}\leq\frac{\|f\|_{L^{2}}}{\sqrt{\kappa_{0}(s)\kappa_{1}(s)}},$
(3.16)

using the bounds

$\displaystyle\kappa_{0}(s):=$
$\displaystyle\begin{cases}|s|^{2}-\frac{2M_{a}|s_{0}|}{C_{P}}+\frac{M_{a}^{2}}{C_{P}^{2}K_{a}^{2}},&\text{if
}|s_{0}|\leq\frac{M_{a}}{C_{P}K_{a}^{2}},\\\
|s|^{2}-K_{a}^{2}s_{0}^{2},&\text{if
}|s_{0}|\geq\frac{M_{a}}{C_{P}K_{a}^{2}},\end{cases}$ (3.17)
$\displaystyle\kappa_{1}(s):=$
$\displaystyle\begin{cases}C_{P}^{2}\big{(}|s|^{2}-\frac{2M_{a}|s_{0}|}{C_{P}}+\frac{M_{a}^{2}}{C_{P}^{2}K_{a}^{2}}\big{)},&\text{if
}|s|^{2}\leq\frac{M_{a}|s_{0}|}{C_{P}},\\\
\frac{M_{a}^{2}(|s|^{2}-K_{a}^{2}s_{0}^{2})}{K_{a}^{2}|s|^{2}},&\text{if
}|s|^{2}\geq\frac{M_{a}|s_{0}|}{C_{P}}.\end{cases}$ (3.18)

$s_{0}$$\frac{M_{a}}{C_{P}}$$\frac{M_{a}\sqrt{1-K_{a}^{-2}}}{C_{P}}$$|s|=K_{a}|s_{0}|$$\operatorname{Im}(s)$

$s_{0}$$\frac{M_{a}}{C_{P}K_{a}^{2}}$$\frac{M_{a}}{2C_{P}}$$\frac{M_{a}}{2C_{P}}$$\operatorname{Im}(s)$

Figure 2. Both pictures show the double cone with an opening at the origin,
which is characteristic for the Poincaré inequality. The two graphics also
visualize (3.16), the $\|\cdot\|_{L^{2}}$-norm bound on the left and the
$\|\cdot\|_{D}$-norm bound on the right. Plotted are also the level lines of
these upper bounds, where dark colors indicate large and light colors small
values of the upper bound. One in particular sees the singularities of both
estimates on the boundary, and also the faster decay $\mathcal{O}(|s|^{-2})$
of the $\|\cdot\|_{L^{2}}$-estimate compared to the slower
$\mathcal{O}(|s|^{-1})$-decay of the $\|\cdot\|_{D}$-estimate whenever
$|\operatorname{Im}(s)|\rightarrow\infty$.

###### Proof.

In the same way as in (3.6), one proves the boundedness

$|q_{s}^{D}(u,v)|\leq
2^{\frac{n}{2}}\big{(}M_{a}^{2}+2M_{a}|s_{0}|+C_{S}M_{a}^{\prime}+|s|^{2}\big{)}\|u\|_{H^{1}}\|v\|_{H^{1}},\qquad
u,v\in H_{0}^{1}(\Omega).$

Also for the coercivity of $\operatorname{Sc}q_{s}^{D}$, we start as in (3.7)
with the inequality

$\operatorname{Sc}q_{s}(u,u)\geq\frac{M_{a}^{2}}{K_{a}^{2}}\|u\|^{2}_{D}-2M_{a}|s_{0}|\|u\|_{D}\|u\|_{L^{2}}+|s|^{2}\|u\|^{2}_{L^{2}},$
(3.19)

where $K_{a}^{2}=\frac{M_{a}^{2}}{m_{a}^{2}-C_{S}M_{a}^{\prime}}$. For any
$\alpha\in[0,1]$ and $\delta>0$, we now use Young’s inequality

$\|u\|_{D}\|v\|_{L^{2}}\leq\frac{C_{P}}{2\delta}\|u\|_{D}^{2}+\frac{\delta}{2C_{P}}\|u\|_{L^{2}}^{2}$

as well as the Poincaré inequality (2.9) in the coercivity estimate (3.19), to
get

$\displaystyle\operatorname{Sc}q_{s}^{D}(u,u)$
$\displaystyle\geq\frac{M_{a}^{2}}{K_{a}^{2}}\|u\|_{D}^{2}-2\alpha
M_{a}|s_{0}|\|u\|_{D}\|u\|_{L^{2}}-2(1-\alpha)M_{a}|s_{0}|\|u\|_{D}\|u\|_{L^{2}}+|s|^{2}\|u\|^{2}_{L^{2}}$
$\displaystyle\geq M_{a}\Big{(}\frac{M_{a}}{K_{a}^{2}}-\frac{\alpha
C_{P}|s_{0}|}{\delta}-2(1-\alpha)C_{P}|s_{0}|\Big{)}\|u\|^{2}_{D}+\Big{(}|s|^{2}-\frac{\alpha\delta
M_{a}|s_{0}|}{C_{P}}\Big{)}\|u\|^{2}_{L^{2}}.$ (3.20)

In order to get an $H^{1}$-coercivity lower bound, both brackets on the right
hand side have to be positive, i.e. we have to choose $\alpha\in[0,1]$ and
$\delta>0$ such that

$\displaystyle\text{I}:\;\;$
$\displaystyle\frac{M_{a}}{K_{a}^{2}}-\frac{\alpha
C_{P}|s_{0}|}{\delta}-2(1-\alpha)C_{P}|s_{0}|>0$
$\displaystyle\Rightarrow\quad\frac{1}{\delta}<\frac{M_{a}-2(1-\alpha)C_{P}K_{a}^{2}|s_{0}|}{\alpha
C_{P}K_{a}^{2}|s_{0}|},$ $\displaystyle\text{II}:\;\;$
$\displaystyle|s|^{2}-\frac{\alpha\delta M_{a}|s_{0}|}{C_{P}}>0$
$\displaystyle\Rightarrow\quad\delta<\frac{C_{P}|s|^{2}}{\alpha
M_{a}|s_{0}|}.$

However, such a choice is only possible if

$M_{a}-2(1-\alpha)C_{P}K_{a}^{2}|s_{0}|>0\qquad\text{and}\qquad\frac{\alpha
C_{P}K_{a}^{2}|s_{0}|}{M_{a}-2(1-\alpha)C_{P}K_{a}^{2}|s_{0}|}<\frac{C_{P}|s|^{2}}{\alpha
M_{a}|s_{0}|}.$

If we now set

$x:=\frac{M_{a}}{C_{P}K_{a}^{2}|s_{0}|}\qquad\text{and}\qquad
y:=\frac{C_{P}|s|^{2}}{M_{a}|s_{0}|},$ (3.21)

these conditions turn into

$x-2+2\alpha>0\qquad\text{and}\qquad(x-2)y+2\alpha y-\alpha^{2}>0.$ (3.22)

Taking now Lemma A.1, the second condition is satisfied if and only if

$y>\begin{cases}\frac{1}{x},&\text{if }x\leq 1,\\\ 2-x,&\text{if }x\geq
1,\end{cases}$

which, plugging back in (3.21), turns into the assumed condition (3.14).
Moreover, if we choose the values $\alpha_{0}<\alpha_{1}\in[0,1]$ from (A.3),
then for every $\alpha\in(\alpha_{0},\alpha_{1})$ also the first condition in
(3.22) is satisfied. Altogether, for every $\alpha\in(\alpha_{0},\alpha_{1})$
and $\delta$ in between

$\delta_{0}(\alpha):=\frac{\alpha
C_{P}K_{a}^{2}|s_{0}|}{M_{a}-2(1-\alpha)C_{P}K_{a}^{2}|s_{0}|}<\delta<\frac{C_{P}|s|^{2}}{\alpha
M_{a}|s_{0}|}=:\delta_{1}(\alpha),$

both brackets in (3.20) are positive and we have proven the coercivity of the
scalar part $\operatorname{Sc}q_{s}^{D}$. Let now $f\in L^{2}(\Omega)$, and
consider the functional $\varphi_{f}(v):=\langle f,v\rangle_{L^{2}}$, $v\in
H_{0}^{1}(\Omega)$, which is bounded in $H_{0}^{1}(\Omega)$ by the same
estimate as in (3.10). Lemma 2.2 then states the existence of a unique
$u_{f}\in H_{0}^{1}(\Omega)$, which satisfies

$q_{s}^{D}(u_{f},v)=\varphi_{f}(v)=\langle f,v\rangle_{L^{2}},\qquad\text{for
all }v\in H_{0}^{1}(\Omega).$ (3.23)

Next, we will prove the bounds (3.16) of this solution. Using the Poincaré
inequality (2.9) in the form
$\|u_{f}\|_{D}\geq\frac{1}{C_{P}}\|u_{f}\|_{L^{2}}$ on the coercivity estimate
(3.20), gives

$\operatorname{Sc}q_{s}^{D}(u_{f},u_{f})\geq\Big{(}|s|^{2}-\frac{2M_{a}|s_{0}|}{C_{P}}+\frac{M_{a}^{2}}{C_{P}^{2}K_{a}^{2}}+\frac{\alpha
M_{a}|s_{0}|}{C_{P}}\Big{(}2-\frac{1}{\delta}-\delta\Big{)}\Big{)}\|u_{f}\|^{2}_{L^{2}}.$
(3.24)

This lower bound can now be maximized with respect to the two free parameters
$\alpha\in(\alpha_{0},\alpha_{1})$ and
$\delta\in(\delta_{0}(\alpha),\delta_{1}(\alpha))$. However, the maximization
problem (A.5b) with the values (3.21), already states that

$\sup\limits_{\stackrel{{\scriptstyle\alpha\in(\alpha_{0},\alpha_{1})}}{{{\delta\in(\delta_{0}(\alpha),\delta_{1}(\alpha))}}}}\alpha\Big{(}2-\frac{1}{\delta}-\delta\Big{)}=\begin{cases}0,&\text{if
}|s_{0}|\leq\frac{M_{a}}{C_{P}K_{a}^{2}},\\\
2-\frac{M_{a}}{C_{P}K_{a}^{2}|s_{0}|}-\frac{C_{P}K_{a}^{2}|s_{0}|}{M_{a}},&\text{if
}|s_{0}|\geq\frac{M_{a}}{C_{P}K_{a}^{2}}.\end{cases}$

This maximization problem now transforms (3.24) into

$\operatorname{Sc}q_{s}^{D}(u_{f},u_{f})\geq\|u_{f}\|_{L^{2}}^{2}\begin{cases}|s|^{2}-\frac{2M_{a}|s_{0}|}{C_{P}}+\frac{M_{a}^{2}}{C_{P}^{2}K_{a}^{2}},&\text{if
}|s_{0}|\leq\frac{M_{a}}{C_{P}K_{a}^{2}},\\\
|s|^{2}-K_{a}^{2}s_{0}^{2},&\text{if
}|s_{0}|\geq\frac{M_{a}}{C_{P}K_{a}^{2}},\end{cases}=\kappa_{0}(s)\|u_{f}\|_{L^{2}}^{2},$

where in the second equality we used the constant $\kappa_{0}(s)$ from (3.17).
Combining this inequality with (3.12), then gives the final $L^{2}$-estimate
of the solution

$\|u_{f}\|_{L^{2}}^{2}\leq\frac{\operatorname{Sc}q_{s}^{D}(u_{f},u_{f})}{\kappa_{0}(s)}\leq\frac{\|f\|_{L^{2}}\|u_{f}\|_{L^{2}}}{\kappa_{0}(s)},$
(3.25)

which is exactly the $\|\cdot\|_{L^{2}}$-norm estimate in (3.16). For the
$\|\cdot\|_{D}$-estimate of the solution $u_{f}$, we once more use the
coercivity (3.20), use the nonnegativity of the second term, and get

$\operatorname{Sc}q_{s}^{D}(u_{f},u_{f})\geq\Big{(}\frac{M_{a}^{2}}{K_{a}^{2}}-\frac{\alpha
C_{P}M_{a}|s_{0}|}{\delta}-2(1-\alpha)C_{P}M_{a}|s_{0}|\Big{)}\|u_{f}\|_{D}^{2}.$

Also this lower bound will now be maximized with respect to the free
parameters $\alpha$ and $\delta$. Maximizing with respect to $\delta$, means
choosing the maximal possible value, i.e. the limit
$\delta\to\delta_{1}(\alpha)=\frac{C_{P}|s|^{2}}{\alpha M_{a}|s_{0}|}$. This
gives

$\operatorname{Sc}q_{s}^{D}(u_{f},u_{f})\geq\Big{(}\frac{M_{a}^{2}}{K_{a}^{2}}-2C_{P}M_{a}|s_{0}|+\alpha
C_{P}M_{a}|s_{0}|\Big{(}2-\frac{\alpha
M_{a}|s_{0}|}{C_{P}|s|^{2}}\Big{)}\Big{)}\|u_{f}\|_{D}^{2}.$ (3.26)

The maximization problem (A.5a) with the values (3.21) gives

$\sup\limits_{\alpha\in(\alpha_{0},\alpha_{1})}\alpha\Big{(}2-\frac{\alpha
M_{a}|s_{0}|}{C_{P}|s|^{2}}\Big{)}=\begin{cases}\frac{C_{P}|s|^{2}}{M_{a}|s_{0}|},&\text{if
}|s|^{2}\leq\frac{M_{a}|s_{0}|}{C_{P}},\\\
2-\frac{M_{a}|s_{0}|}{C_{P}|s|^{2}},&\text{if
}|s|^{2}\geq\frac{M_{a}|s_{0}|}{C_{P}}.\end{cases}$

This maximization problem allows us to also maximize (3.10) with respect to
$\alpha$, and get

$\operatorname{Sc}q_{s}^{D}(u_{f},u_{f})\geq\|u_{f}\|_{D}^{2}\begin{cases}\frac{M_{a}^{2}}{K_{a}^{2}}-2C_{P}M_{a}|s_{0}|+C_{P}^{2}|s|^{2},&\text{if
}|s|^{2}\leq\frac{M_{a}|s_{0}|}{C_{P}},\\\
\frac{M_{a}^{2}}{K_{a}^{2}}-\frac{M_{a}^{2}s_{0}^{2}}{|s|^{2}},&\text{if
}|s|^{2}\geq\frac{M_{a}|s_{0}|}{C_{P}},\end{cases}=\kappa_{1}(s)\|u_{f}\|_{D}^{2},$

where in the second equality we used the constant $\kappa_{1}(s)$ from (3.18).
Rearranging now this inequality and using again (3.12) and the already
calculated $L^{2}$-estimate (3.25), finally gives the $\|\cdot\|_{D}$-estimate
of the solution

$\|u_{f}\|_{D}^{2}\leq\frac{\operatorname{Sc}q_{s}^{D}(u_{f},u_{f})}{\kappa_{1}(s)}\leq\frac{\|f\|_{L^{2}}\|u_{f}\|_{L^{2}}}{\kappa_{1}(s)}\leq\frac{\|f\|_{L^{2}}^{2}}{\kappa_{0}(s)\kappa_{1}(s)}.\qed$

##  4\. Robin-like boundary conditions

In this section we study the weak formulation of the spectral problem (1.5)
with Robin-like boundary conditions

$\sum_{i=1}^{n}\nu_{i}a_{i}^{2}\frac{\partial u}{\partial
x_{i}}+bu=0,\qquad\text{on }\partial\Omega.$ (4.1)

The domain $\Omega\subseteq\mathbb{R}^{n}$, in $n\geq 3$ dimensions is assumed
to be minimally smooth in the sense of Stein [45, Section VI.3.3]. Precisely,
if we plug in the boundary conditions (4.1) into (1.8), means that for every
$s\in\mathbb{R}^{n+1}$ we are dealing with form

$q_{s}^{R}(u,v):=\sum\limits_{i=1}^{n}\Big{\langle}\frac{\partial u}{\partial
x_{i}},a_{i}^{2}\frac{\partial v}{\partial
x_{i}}+(2s_{0}a_{i}-B_{i})e_{i}v\Big{\rangle}_{L^{2}}+|s|^{2}\langle
u,v\rangle_{L^{2}}+\langle
b\,\tau_{D}u,\tau_{D}v\rangle_{L^{2}(\partial\Omega)},$ (4.2)

defined on $\operatorname{dom}q_{s}^{R}:=H^{1}(\Omega)\times H^{1}(\Omega)$.
Here $\tau_{D}:H^{1}(\Omega)\rightarrow L^{2}(\partial\Omega)$ is the bounded
Dirichlet trace operator, see [41, Equation (4)].

######  Problem 4.1.

Determine assumptions on the coefficients $a_{1},\dots,a_{n}$ in (1.3), the
Robin parameter $b$ in (4.1) and on the spectral parameter
$s\in\mathbb{R}^{n+1}$, such that for every $f\in L^{2}(\Omega)$ there exists
a unique solution $u_{f}\in H^{1}(\Omega)$ of the problem

$q_{s}^{R}(u_{f},v)=\langle f,v\rangle_{L^{2}},\qquad\text{for all }v\in
H^{1}(\Omega).$

Furthermore, determine the $L^{2}$\- and $H^{1}$-estimates of $u_{f}$,
depending on the parameter $s$.

######  Theorem 4.2.

Let $b\in L^{\infty}(\partial\Omega)$ be real valued and $a_{1},\dots,a_{n}\in
C^{1}(\overline{\Omega})$ be positive coefficients. With the Sobolev constant
$C_{S}\geq 0$ from (2.7b) and the upper bound $M_{a}^{\prime}$ from (1.4c), we
set
$D_{a,b}:=C_{S}M_{a}^{\prime}+\|\min\\{b,0\\}\|_{L^{\infty}(\partial\Omega)}\|\tau_{D}\|^{2}$,
and assume that there holds

$m_{a}^{2}>D_{a,b}.$

We now consider $s\in\mathbb{R}^{n+1}$ with

$|s|^{2}>\frac{M_{a}^{2}s_{0}^{2}}{m_{a}^{2}-D_{a,b}}+D_{a,b}.$ (4.3)

Then for every $f\in L^{2}(\Omega)$ there exists a unique $u_{f}\in
H^{1}(\Omega)$ such that

$q_{s}^{R}(u_{f},v)=\langle f,v\rangle_{L^{2}},\qquad\text{for all }v\in
H^{1}(\Omega).$ (4.4)

Moreover, this solution satisfies the bound

$\|u_{f}\|_{H^{1}}\leq\frac{2\|f\|_{L^{2}}}{|s|^{2}+m_{a}^{2}-2D_{a,b}-\sqrt{(|s|^{2}-m_{a}^{2})^{2}+4M_{a}^{2}s_{0}^{2}}}.$
(4.5)

$s_{0}$$\operatorname{Im}(s)$$\sqrt{D_{a,b}}$$m_{a}$ Figure 3. The complete
shaded hyperbolic region indicates the condition (4.3), for which (4.4) is
uniquely solvable. The gap of size $2\sqrt{D_{a,b}}$ between the two
hyperbolas comes on the one hand from the negative part $\min\\{b,0\\}$ of the
Robin parameter and also from the upper bound $M_{a}^{\prime}$ of the
derivative of the coefficients. Moreover, the color visualizes the level lines
of the $H^{1}$-estimate (4.5). Dark color indicates large values and light
color small values of the upper bound.

###### Proof.

For the proof of the weak solvability (4.4), we will verify that the form
$q_{s}^{R}$ satisfies the assumptions of the Lemma 2.2. We start by estimating
the third term in (4.2) by

$\displaystyle|\langle b\,\tau_{D}u,\tau_{D}v\rangle_{L^{2}(\partial\Omega)}|$
$\displaystyle\leq
2^{\frac{n}{2}}\|b\|_{L^{\infty}(\partial\Omega)}\|\tau_{D}u\|_{L^{2}(\partial\Omega)}\|\tau_{D}v\|_{L^{2}(\partial\Omega)}$
$\displaystyle\leq
2^{\frac{n}{2}}\|b\|_{L^{\infty}(\partial\Omega)}\|\tau_{D}\|^{2}\|u\|_{H^{1}}\|v\|_{H^{1}}.$

Moreover, similar to (3.5), we can use the Sobolev inequality (2.7b), to
estimate

$\sum_{i=1}^{n}\|B_{i}e_{i}v\|_{L^{2}}^{2}\leq\sum_{i,j=1}^{n}\Big{\|}a_{j}\frac{\partial
a_{i}}{\partial x_{j}}\Big{\|}_{L^{n}}^{2}\|v\|_{L^{\frac{2n}{n-2}}}^{2}\leq
C_{S}^{2}\sum_{i,j=1}^{n}\Big{\|}a_{j}\frac{\partial a_{i}}{\partial
x_{j}}\Big{\|}_{L^{n}}^{2}\|v\|_{H^{1}}^{2}\leq C_{S}^{2}M_{a}^{\prime
2}\|v\|^{2}_{H^{1}}.$ (4.6)

Equipped with these two estimates, we arrive in the same way as in (3.6) with
the boundedness

$|q_{s}^{R}(u,v)|\leq
2^{\frac{n}{2}}\big{(}M_{a}^{2}+2M_{a}|s_{0}|+C_{S}M_{a}^{\prime}+|s|^{2}+\|b\|_{L^{\infty}(\partial\Omega)}\|\tau_{D}\|^{2}\big{)}\|u\|_{H^{1}}\|v\|_{H^{1}}.$
(4.7)

For the coercivity of the scalar part $\operatorname{Sc}q_{s}^{R}$, we start
with the estimate

$\displaystyle\operatorname{Sc}\langle
b\,\tau_{D}u,\tau_{D}u\rangle_{L^{2}(\partial\Omega)}$
$\displaystyle=\int_{\partial\Omega}b\,|\tau_{D}u|^{2}d\sigma\geq\int_{\partial\Omega}\min\\{b,0\\}|\tau_{D}u|^{2}d\sigma$
$\displaystyle\geq-\|\min\\{b,0\\}\|_{L^{\infty}(\partial\Omega)}\|\tau_{D}u\|_{L^{2}(\partial\Omega)}^{2}\geq-\|\min\\{b,0\\}\|_{L^{\infty}(\partial\Omega)}\|\tau_{D}\|^{2}\|u\|_{H^{1}}^{2}.$

Using also the estimate (4.6), we end up with the coercivity

$\displaystyle\operatorname{Sc}q_{s}^{R}(u,u)$
$\displaystyle\geq\sum_{i=1}^{n}\bigg{(}m_{a}^{2}\Big{\|}\frac{\partial
u}{\partial x_{i}}\Big{\|}_{L^{2}}^{2}-2M_{a}|s_{0}|\Big{\|}\frac{\partial
u}{\partial x_{i}}\Big{\|}_{L^{2}}\|u\|_{L^{2}}-\Big{\|}\frac{\partial
u}{\partial x_{i}}\Big{\|}_{L^{2}}\|B_{i}e_{i}u\|_{L^{2}}\bigg{)}$
$\displaystyle\quad+|s|^{2}\|u\|_{L^{2}}^{2}+\operatorname{Sc}\langle
b\,\tau_{D}u,\tau_{D}u\rangle_{L^{2}(\partial\Omega)}$ $\displaystyle\geq
m_{a}^{2}\|u\|_{D}^{2}-2M_{a}|s_{0}|\|u\|_{D}\|u\|_{L^{2}}-C_{S}M_{a}^{\prime}\|u\|_{D}\|u\|_{H^{1}}$
$\displaystyle\quad+|s|^{2}\|u\|_{L^{2}}^{2}-\|\min\\{b,0\\}\|_{L^{\infty}(\partial\Omega)}\|\tau_{D}\|^{2}\|u\|_{H^{1}}^{2}$
$\displaystyle\geq
m_{a}^{2}\|u\|_{D}^{2}-2M_{a}|s_{0}|\|u\|_{D}\|u\|_{L^{2}}+|s|^{2}\|u\|_{L^{2}}^{2}-D_{a,b}\|u\|_{H^{1}}^{2}.$

For any $\delta>0$ we now use Young’s inequality

$\|u\|_{D}\|u\|_{L^{2}}\leq\frac{1}{2\delta}\|u\|_{D}^{2}+\frac{\delta}{2}\|u\|_{L^{2}}^{2},$

to further estimate this lower bound by

$\operatorname{Sc}q_{s}^{R}(u,u)\geq\Big{(}m_{a}^{2}-\frac{M_{a}|s_{0}|}{\delta}\Big{)}\|u\|_{D}^{2}+(|s|^{2}-\delta
M_{a}|s_{0}|)\|u\|_{L^{2}}^{2}-D_{a,b}\|u\|_{H^{1}}^{2}.$ (4.8)

The optimal choice on $\delta$ to get an $H^{1}$-lower bound, is when the two
brackets in (4.8), i.e. for

$\delta=\frac{|s|^{2}-m_{a}^{2}+\sqrt{(|s|^{2}-m_{a}^{2})^{2}+4M_{a}^{2}s_{0}^{2}}}{2M_{a}|s_{0}|}.$

Clearly, there is $\delta>0$ and if we plug it into (4.8), we get

$\operatorname{Sc}q_{s}^{R}(u,u)\geq\frac{1}{2}\Big{(}|s|^{2}+m_{a}^{2}-2D_{a,b}-\sqrt{(|s|^{2}-m_{a}^{2})^{2}+4M_{a}^{2}s_{0}^{2}}\Big{)}\|u\|_{H^{1}}^{2}.$
(4.9)

This bracket term is positive if and only if

$\displaystyle|s|^{2}+m_{a}^{2}-2D_{a,b}-\sqrt{(|s|^{2}-m_{a}^{2})^{2}+4M_{a}^{2}s_{0}^{2}}>0$
$\displaystyle\hskip
56.9055pt\Leftrightarrow\quad\big{(}|s|^{2}+m_{a}^{2}-2D_{a,b}\big{)}^{2}>(|s|^{2}-m_{a}^{2})^{2}+4M_{a}^{2}s_{0}^{2}\quad\&\quad|s|^{2}+m_{a}^{2}>2D_{a,b}$
$\displaystyle\hskip
56.9055pt\Leftrightarrow\quad|s|^{2}>\frac{M_{a}^{2}s_{0}^{2}}{m_{a}^{2}-D_{a,b}}+D_{a,b}\quad\&\quad|s|^{2}+m_{a}^{2}>2D_{a,b}$
$\displaystyle\hskip
56.9055pt\Leftrightarrow\quad|s|^{2}>\frac{M_{a}^{2}s_{0}^{2}}{m_{a}^{2}-D_{a,b}}+D_{a,b},$
(4.10)

where the last equivalence follows from
$\frac{M_{a}^{2}s_{0}^{2}}{m_{a}^{2}-D_{a,b}}+D_{a,b}\geq 2D_{a,b}-m_{a}^{2}$,
which is true by (4.3). However, the condition (4.10) is exactly the stated
assumption (4.3), which means that we have proven the coercivity of the Robin
form $q_{s}^{R}$. Let now $f\in L^{2}(\Omega)$, and consider the functional
$\varphi_{f}(v):=\langle f,v\rangle_{L^{2}}$, $v\in H^{1}(\Omega)$, which is
bounded in $H^{1}(\Omega)$ by the same estimate as in (3.10). Lemma 2.2 then
states the existence of a unique $u_{f}\in H^{1}(\Omega)$, which satisfies

$q_{s}^{R}(u_{f},v)=\varphi_{f}(v)=\langle f,v\rangle_{L^{2}},\qquad\text{for
all }v\in H^{1}(\Omega).$

Testing this equation with $v=u_{f}$, using the coercivity estimate (4.9) as
well as the analogous estimate (3.12) for $q_{s}^{R}$, gives the $H^{1}$-norm
estimate of the solution

$\|u_{f}\|_{H^{1}}^{2}\leq\frac{2\|f\|_{L^{2}}\|u_{f}\|_{L^{2}}}{|s|^{2}+m_{a}^{2}-2D_{a,b}-\sqrt{(|s|^{2}-m_{a}^{2})^{2}+4M_{a}^{2}s_{0}^{2}}},$

which is exactly (4.5) if we estimate $\|u_{f}\|_{L^{2}}\leq\|u_{f}\|_{H^{1}}$
on the right hand side. ∎

If the coefficients $a_{1},\dots,a_{n}$ of the operator $T$ in (1.3) are
constant, and if the Robin parameter $b\geq 0$ is nonnegative, which in
particular includes Neumann boundary conditions $b\equiv 0$, the results of
Theorem 4.2 can be refined. In particular the gap at $s=0$ closes, see Figure
3 and Figure 4, and also the norm estimate (4.5) of the solution can be
improved.

######  Corollary 4.3.

Consider $b\in L^{\infty}(\partial\Omega)$ with $b\geq 0$ and constant
coefficients $a_{1},\dots,a_{n}\in(0,\infty)$ in (1.3). In this case we have
$m_{a}=\min\\{a_{1},\dots,a_{n}\\}$, $M_{a}=\max\\{a_{1},\dots,a_{n}\\}$ and
$M_{a}^{\prime}=0$, for the values of (1.4). Let us consider
$s\in\mathbb{R}^{n+1}$ with

$|s|>\frac{M_{a}}{m_{a}}|s_{0}|.$ (4.11)

Then, for every $f\in L^{2}(\Omega)$, there exists a unique $u_{f}\in
H^{1}(\Omega)$ such that

$q_{s}^{R}(u_{f},v)=\langle f,v\rangle_{L^{2}},\qquad\text{for all }v\in
H^{1}(\Omega).$ (4.12)

Moreover, this solution satisfies the bounds

$\|u_{f}\|_{L^{2}}\leq\frac{\|f\|_{L^{2}}}{|s|^{2}-\frac{M_{a}^{2}s_{0}^{2}}{m_{a}^{2}}}\qquad\text{and}\qquad\|u_{f}\|_{D}\leq\frac{|s|\,\|f\|_{L^{2}}}{m_{a}\big{(}|s|^{2}-\frac{M_{a}^{2}s_{0}^{2}}{m_{a}^{2}}\big{)}}.$
(4.13)

$|s|=K_{a}|s_{0}|$$s_{0}$$\operatorname{Im}(s)$

$s_{0}$$\operatorname{Im}(s)$

Figure 4. Both pictures show the double cone which arises from the condition
(4.3). The two graphics also visualize the values of the $\|\cdot\|_{L^{2}}$\-
and the $\|\cdot\|_{D}$-norm bounds (4.5), ($\|\cdot\|_{L^{2}}$ on the left,
$\|\cdot\|_{D}$ on the right). Plotted are the level lines of these upper
bounds, where dark colors indicate large values and light colors small values.
One in particular sees the singularities of both estimates on the boundary of
the double cone and also the $\mathcal{O}(|s|^{-2})$-behaviour of the
$\|\cdot\|_{L^{2}}$-estimate compared to the slower
$\mathcal{O}(|s|^{-1})$-behaviour of the $\|\cdot\|_{D}$-estimate for
$|s|\to\infty$ but also for $|s|\to 0$.

###### Proof.

Since we assumed $b\geq 0$ and $a_{1},\dots,a_{n}\in(0,\infty)$ being positive
and constant, it is clear that
$\|\min\\{b,0\\}\|_{L^{\infty}(\partial\Omega)}=0$ and $M_{a}^{\prime}=0$.
Hence also the constant $D_{a,b}=0$ in Theorem 4.2 vanishes and the condition
(4.3) turns into (4.11). This immediately yields the unique solvability (4.12)
from Theorem 4.2. Moreover, to verify the estimates (4.13), we note that the
coercivity estimate (4.8) looks like

$\operatorname{Sc}q_{s}^{R}(u_{f},u_{f})\geq\Big{(}m_{a}^{2}-\frac{M_{a}|s_{0}|}{\delta}\Big{)}\|u_{f}\|_{D}^{2}+(|s|^{2}-\delta
M_{a}|s_{0}|)\|u_{f}\|_{L^{2}}^{2},$ (4.14)

for every $\delta>0$. For the $L^{2}$-estimate in (4.13), we choose
$\delta=\frac{M_{a}|s_{0}|}{m_{a}^{2}}$, which turns (4.14) into

$\operatorname{Sc}q_{s}^{R}(u_{f},u_{f})\geq\Big{(}|s|^{2}-\frac{M_{a}^{2}s_{0}^{2}}{m_{a}^{2}}\Big{)}\|u_{f}\|_{L^{2}}^{2}.$

Together with the analog estimate (3.12) for $q_{s}^{R}$, this leads to the
stated bound

$\|u_{f}\|_{L^{2}}^{2}\leq\frac{\operatorname{Sc}q_{s}^{R}(u_{f},u_{f})}{|s|^{2}-\frac{M_{a}^{2}s_{0}^{2}}{m_{a}^{2}}}\leq\frac{\|f\|_{L^{2}}\|u_{f}\|_{L^{2}}}{|s|^{2}-\frac{M_{a}^{2}s_{0}^{2}}{m_{a}^{2}}}.$

For the $\|\cdot\|_{D}$-estimate in (4.13), we choose
$\delta=\frac{|s|^{2}}{M_{a}|s_{0}|}$, which turns (4.14) into

$\operatorname{Sc}q_{s}^{R}(u_{f},u_{f})\geq\frac{m_{a}^{2}}{|s|^{2}}\Big{(}|s|^{2}-\frac{M_{a}^{2}s_{0}^{2}}{m_{a}^{2}}\Big{)}\|u_{f}\|_{D}^{2}.$

Together with the analog estimate (3.12) for $q_{s}^{R}$, this leads to the
second bound in (4.13), namely

$\|u_{f}\|_{D}^{2}\leq\frac{|s|^{2}\operatorname{Sc}q_{s}^{R}(u_{f},u_{f})}{m_{a}^{2}\big{(}|s|^{2}-\frac{M_{a}^{2}s_{0}^{2}}{m_{a}^{2}}\big{)}}\leq\frac{|s|^{2}\|f\|_{L^{2}}\|u_{f}\|_{L^{2}}}{m_{a}^{2}\big{(}|s|^{2}-\frac{M_{a}^{2}s_{0}^{2}}{m_{a}^{2}}\big{)}}\leq\frac{|s|^{2}\|f\|_{L^{2}}^{2}}{m_{a}^{2}\big{(}|s|^{2}-\frac{M_{a}^{2}s_{0}^{2}}{m_{a}^{2}}\big{)}^{2}},$

where in the last inequality we used the already calculated $L^{2}$-estimate
in (4.13). ∎

##  5\. Concluding remarks

We conclude this paper with some considerations that contextualize the
estimates we have derived within the framework of the $H^{\infty}$-functional
calculus, which manifests in various settings. In the realm of hypercomplex
settings, depending on the spectral theory under consideration, this calculus
takes on several distinct formulations and the estimates for sectorial and bi-
sectorial operators are crucial for establishing $H^{\infty}$-functional
calculi.

The $H^{\infty}$-functional calculus for complex operators on Banach spaces,
introduced by A. McIntosh [42] and further investigated in [5, 6, 7, 31],
holds significant relevance across multiple disciplines. For more information
and the developments one can consult the books [36, 37, 38].

Regarding the spectral theory on the $S$-spectrum [2, 3, 17, 18, 24, 25, 33,
34], which has its roots in the theory of slice hyperholomorphic functions,
the $H^{\infty}$-functional calculus has been considered in the papers [4,
16].

The spectral theory concerning the monogenic spectrum, introduced in [39],
relies on monogenic functions [29]. Further exploration of this topic,
including the $H^{\infty}$-functional calculus within this framework can be
found in the books [35, 40].

In recent times, a novel branch of spectral theory concerning the $S$-spectrum
has emerged in connection with the factorizations of the second mapping in the
Fueter-Sce extension theorem, see [23, 32, 43, 44]. This branch, referred to
as the fine structures on the $S$-spectrum, revolves around sets of functions
characterized by integral representations, from which the corresponding
functional calculi are defined. The development of this new theory and its
associated functional calculi is elaborated in the papers [11, 12, 13, 26,
27], while recent investigations into the $H^{\infty}$-functional calculus
linked with the functional calculi of the fine structures are studied in [22,
28].

##  Appendix A An auxiliary maximization problem

In this section we collect an auxiliary maximization problem that we have used
in the proofs of the main theorem.

######  Lemma A.1.

For every $x,y>0$, there holds the equivalence

$\exists\alpha\in(0,1):(x-2)y+2\alpha y-\alpha^{2}>0\Leftrightarrow
y>\begin{cases}\frac{1}{x},&\text{if }x\leq 1,\\\ 2-x,&\text{if }x\geq
1.\end{cases}$ (A.1)

More precisely, if the two equivalent conditions in (A.1) are satisfied, then
$x+y>2$, and

$\Set{\alpha\in(0,1)}{(x-2)y+2\alpha y-\alpha^{2}>0}=(\alpha_{0},\alpha_{1}),$
(A.2)

where $\alpha_{0}<\alpha_{1}\in[0,1]$ are given by

$\alpha_{0}=\begin{cases}y-\sqrt{y(x+y-2)},&\text{if }x\leq 2,\\\ 0,&\text{if
}x\geq
2,\end{cases}\quad\text{and}\quad\alpha_{1}:=\begin{cases}y+\sqrt{y(x+y-2)},&\text{if
}y\leq\frac{1}{x},\\\ 1,&\text{if }y\geq\frac{1}{x}.\end{cases}$ (A.3)

For every $\alpha\in(\alpha_{0},\alpha_{1})$ we now have $x-2+2\alpha>0$ and
there exists a gap between

$\delta_{0}(\alpha):=\frac{\alpha}{x-2+2\alpha}<\frac{y}{\alpha}=:\delta_{1}(\alpha).$
(A.4)

Moreover, for every $0<C\leq 1$ there holds the two maximization problems

$x$$y$$\frac{1}{x}$$1$$2$$1$

$\displaystyle\sup\limits_{\alpha\in(\alpha_{0},\alpha_{1})}\alpha\Big{(}2-\frac{\alpha}{y}\Big{)}$
$\displaystyle=\begin{cases}y,&\text{if }y\leq 1,\\\ 2-\frac{1}{y},&\text{if
}y\geq 1,\end{cases}$ (A.5a)
$\displaystyle\sup\limits_{\stackrel{{\scriptstyle\alpha\in(\alpha_{0},\alpha_{1})}}{{{\delta\in(\delta_{0}(\alpha),\delta_{1}(\alpha))}}}}\alpha\Big{(}2-\frac{1}{\delta}-\delta\Big{)}$
$\displaystyle=\begin{cases}0,&\text{if }x\geq 1,\\\ 2-x-\frac{1}{x},&\text{if
}x\leq 1.\end{cases}$ (A.5b)

###### Proof.

Let us define the polynomial

$p(\alpha):=(x-2)y+2\alpha y-\alpha^{2},\qquad\alpha\in(0,1).$ (A.6)

In the first step we calculate the maximal value
$\sup_{\alpha\in(0,1)}p(\alpha)$. To do so, we first note, that the (possible)
zero of the derivative

$p^{\prime}(\alpha)=2(y-\alpha),$ (A.7)

is given by $\alpha=y$. However, since we only consider the polynomial on the
interval $\alpha\in(0,1)$, the maximum value is attained at $\alpha=y$ if
$y<1$ and at the right endpoint $\alpha\to 1$ if $y\geq 1$. This then gives
the maximal value

$\sup\limits_{\alpha\in(0,1)}p(\alpha)=\begin{cases}p(y),&\text{if }y<1,\\\
p(1),&\text{if }y\geq 1,\end{cases}=\begin{cases}y(x+y-2),&\text{if }y\leq
1,\\\ xy-1,&\text{if }y\geq 1.\end{cases}$ (A.8)

In the second step we will prove the equivalence (A.1). Using the already
calculated maximal value (A.8) allows us to rewrite (A.1) as

$0<\begin{cases}y(x+y-2),&\text{if }y\leq 1,\\\ xy-1,&\text{if }y\geq
1,\end{cases}\Leftrightarrow y>\begin{cases}\frac{1}{x},&\text{if }x\leq 1,\\\
2-x,&\text{if }x\geq 1.\end{cases}$ (A.9)

The proof of this equivalence will be done for the two cases of the left hand
side separately.

$\circ$ If $y\leq 1$, the equivalence looks like

$0<x+y-2\Leftrightarrow y>\begin{cases}\frac{1}{x},&\text{if }x\leq 1,\\\
2-x,&\text{if }x\geq 1.\end{cases}$

It is true, because for $x\geq 1$ both inequalities are the same, and for
$x\leq 1$ both inequalities are never satisfied.

$\circ$ If $y>1$, the equivalence (A.9) looks like

$0<xy-1\Leftrightarrow y>\begin{cases}\frac{1}{x},&\text{if }x\leq 1,\\\
2-x,&\text{if }x\geq 1.\end{cases}$

It is true, because for $x\leq 1$ both inequalities are the same, and for
$x\geq 1$ both inequalities are always satisfied.

In the third step we prove (A.2). The zeros of $p$, if we forget about the
constraint $\alpha\in(0,1)$ for the moment, are given by

$\alpha_{\pm}=y\pm\sqrt{y(x+y-2)}.$ (A.10)

Since we assumed that (A.1) is satisfied, there exists at least one point
$\alpha\in(0,1)$, for which $p(\alpha)>0$. Since moreover
$\lim_{\alpha\to\pm\infty}p(\alpha)=-\infty$, the zeros (A.10) have to be real
valued, i.e. $x+y>2$ is satisfied and the interval where $p$ is positive is
exactly between those zeros,

$\Set{\alpha\in\mathbb{R}}{p(\alpha)>0}=(\alpha_{-},\alpha_{+}).$

Restricting now again to the interval $\alpha\in(0,1)$, we get

$\Set{\alpha\in(0,1)}{p(\alpha)>0}=\big{(}\max\\{0,\alpha_{-}\\},\min\\{1,\alpha_{+}\\}\big{)}.$

It is left to check when there holds $\alpha_{-}\geq 0$ and $\alpha_{+}\leq 1$
respectively. The condition $\alpha_{-}\geq 0$ is equivalent to

$\alpha_{-}\geq 0\quad\Leftrightarrow\quad
y\geq\sqrt{y(x+y-2)}\quad\Leftrightarrow\quad\sqrt{y}\geq\sqrt{x+y-2}\quad\Leftrightarrow\quad
x\leq 2,$ (A.11)

and the condition $\alpha_{+}\leq 1$ is equivalent to

$\displaystyle\alpha_{+}\leq 1\quad$ $\displaystyle\Leftrightarrow\quad
y+\sqrt{y(x+y-2)}\leq 1$ $\displaystyle\Leftrightarrow\quad 1-y\geq
0\quad\text{and}\quad y(x+y-2)\leq(1-y)^{2}$
$\displaystyle\Leftrightarrow\quad y\leq 1\quad\text{and}\quad xy\leq 1$
$\displaystyle\Leftrightarrow\quad y\leq\frac{1}{x},$ (A.12)

where in the last equivalence we used that the combination $y>1$ and
$y\leq\frac{1}{x}$ is not allowed, since in this case we have
$1<y\leq\frac{1}{x}$, consequently $x<1$, which is a contradiction to the
assumed (A.9), see also the shaded region in (A.5). The two equivalences
(A.11) and (A.12) now show that $\max\\{0,\alpha_{-}\\}=\alpha_{0}$ and
$\min\\{1,\alpha_{+}\\}=\alpha_{1}$ coincide with the values in (A.3). Hence
(A.2) is proven.

In the fourth step, we will solve the first maximization problem (A.5a), i.e.
we will maximize

$f(\alpha):=\alpha\Big{(}2-\frac{\alpha}{y}\Big{)},\qquad\alpha\in(\alpha_{0},\alpha_{1}).$

The zero of the derivative

$f^{\prime}(\alpha)=2\Big{(}1-\frac{\alpha}{y}\Big{)},$ (A.13)

is given by $\alpha=y$. We now have to check if this value $y$ is contained in
the interval $(\alpha_{0},\alpha_{1})$ of allowed values. First,

$y>\begin{cases}y-\sqrt{y(x+y-2)},&\text{if }x\leq 2,\\\ 0,&\text{if }x\geq
2,\end{cases}=\alpha_{0},\qquad\text{is always satisfied.}$ (A.14)

The second condition $y<\alpha_{1}$ can be characterized by

$\begin{split}y<\alpha_{1}\quad&\Leftrightarrow\quad
y<\begin{cases}y+\sqrt{y(x+y-2)},&\text{if }y\leq\frac{1}{x}\\\ 1,&\text{if
}y\geq\frac{1}{x}\end{cases}\\\
&\Leftrightarrow\quad\Big{(}y\leq\frac{1}{x}\Big{)}\quad\text{or}\quad\Big{(}y\geq\frac{1}{x}\quad\text{and}\quad
y<1\Big{)}\\\ &\Leftrightarrow\quad y<1,\end{split}$ (A.15)

where in the last equivalence we used that the combination $y>1$ and
$y\leq\frac{1}{x}$ is not allowed, similar to the argument below equation
(A.12). Hence, the maximal value of $f$ is attained at $\alpha=y$ if $y<1$ and
at the right endpoint $\alpha\to\alpha_{1}=1$ if $y\geq 1$. This means, we get

$\sup\limits_{\alpha\in(\alpha_{0},\alpha_{1})}f(\alpha)=\begin{cases}f(y),&\text{if
}y<1,\\\ f(1),&\text{if }y\geq 1,\end{cases}=\begin{cases}y,&\text{if }y\leq
1,\\\ 2-\frac{1}{y},&\text{if }y\geq 1.\end{cases}$

In the fifth step we first note that for every
$\alpha\in(\alpha_{0},\alpha_{1})$, there holds

$x-2+2\alpha>x-2+2\alpha_{0}=\begin{cases}x-2+2y-2\sqrt{y(x+y-2)},&\text{if
}x\leq 2,\\\ x-2,&\text{if }x\geq 2,\end{cases}\geq 0,$

where the right hand side is nonnegative because $a+b-2\sqrt{ab}\geq 0$, for
every $a,b\geq 0$. Moreover, there holds

$\delta_{0}(\alpha):=\frac{\alpha}{x-2+2\alpha}<\frac{y}{\alpha}=:\delta_{1}(\alpha),\qquad\alpha\in(\alpha_{0},\alpha_{1}),$

which is a simple rearrangement of the inequality $(x-2)y+2\alpha
y-\alpha^{2}>0$ in (A.2). Now, we will solve the second maximization problem
(A.5b), i.e. we want to maximize the function

$g(\alpha,\delta):=\alpha\Big{(}2-\frac{1}{\delta}-\delta\Big{)},\qquad\alpha\in(\alpha_{0},\alpha_{1}),\,\delta\in(\delta_{0}(\alpha),\delta_{1}(\alpha)).$

First, we will fix $\alpha\in(\alpha_{0},\alpha_{1})$ and maximize
$g(\alpha,\cdot\,)$ with respect to $\delta$. The only positive zero of the
partial derivative

$\frac{\partial
g}{\partial\delta}(\alpha,\delta)=\alpha\Big{(}\frac{1}{\delta^{2}}-1\Big{)},$

is given by $\delta=1$. The maximum value of $g(\alpha,\cdot\,)$ now depends
on the location of this zero with respect to the interval
$(\delta_{0}(\alpha),\delta_{1}(\alpha))$ of allowed $\delta$-values. If
$1\leq\delta_{0}(\alpha)$, the maximum is attained the left endpoint
$\delta\to\delta_{0}(\alpha)^{+}$, if
$1\in(\delta_{0}(\alpha),\delta_{1}(\alpha))$ is an allowed value, the maximum
is attained at exactly this point $\delta=1$, and if
$1\geq\delta_{1}(\alpha)$, the maximum value is attained at the right endpoint
$\delta\to\delta_{1}(\alpha)$. I.e., we get

$\displaystyle
h(\alpha):=\sup\limits_{\delta\in(\delta_{0}(\alpha),\delta_{1}(\alpha))}g(\alpha,\delta)$
$\displaystyle=\begin{cases}g(\alpha,\delta_{0}(\alpha)^{+})&\text{if
}1\leq\delta_{0}(\alpha),\\\ g(\alpha,\frac{1}{C}),&\text{if
}\delta_{0}(\alpha)<1<\delta_{1}(\alpha),\\\
g(\alpha,\delta_{1}(\alpha)),&\text{if }1\geq\delta_{1}(\alpha),\end{cases}$
$\displaystyle=\begin{cases}2-x-\frac{\alpha^{2}}{x-2+2\alpha},&\text{if
}\alpha\leq 2-x,\\\ 0,&\text{if }2-x\leq\alpha\leq y,\\\
2\alpha-\frac{\alpha^{2}}{y}-y,&\text{if }\alpha\geq y.\end{cases}$

Note, that we were allowed to replace the $<$-inequalities by
$\leq$-inequalities in the second case since $h(\alpha)$ is continuous at
$\alpha=2-x$ and $\alpha=y$. Next, we want to maximize this function $h$ with
respect to the parameter $\alpha$. Therefore, we calculate its derivative

$h^{\prime}(\alpha)=\begin{cases}\frac{2\alpha(2-x-\alpha)}{(x-2+2\alpha)^{2}},&\text{if
}\alpha\leq 2-x,\\\ 0,&\text{if }2-x\leq\alpha\leq y,\\\
2(1-\frac{\alpha}{y}),&\text{if }\alpha\geq y.\end{cases}$ (A.16)

In the first two cases $\alpha\leq 2-x$ and $2-x\leq\alpha\leq y$ in (A.16),
we obviously have a nonnegative derivative $h^{\prime}(\alpha)\geq 0$. Only in
the third case $\alpha\geq y$, there exists the possible zero
$\alpha_{\max}=y$ of the derivative. It is already calculated in (A.14) and
(A.15), that

$\alpha_{\max}\in(\alpha_{0},\alpha_{1})\Leftrightarrow
y<1\qquad\text{and}\qquad\alpha_{\max}\geq\alpha_{1}\Leftrightarrow y\geq 1.$

Hence, we conclude that for $y\geq 1$ the function $h(\alpha)$ is monotone
increasing on the whole interval $(\alpha_{0},\alpha_{1})$, which means that
its supremum is attained at the right endpoint $\alpha\to\alpha_{1}=1$. In the
case $y<1$ on the other hand, the maximal value is attained at the interior
point $\alpha=\alpha_{\max}=y$. The solution of the maximization problem is
then given by

$\displaystyle\sup\limits_{\alpha\in(\alpha_{0},\alpha_{1})}h(\alpha)$
$\displaystyle=\begin{cases}h(1),&\text{if }y\geq 1,\\\ h(y),&\text{if
}y<1,\end{cases}=\begin{cases}2-x-\frac{1}{x},&\text{if }y\geq
1\;\;\&\;\;x\leq 1,\\\ 0,&\text{if }y\geq 1\;\;\&\;\;x\geq 1,\\\
2-\frac{1}{y}-y,&\text{if }y\geq 1\;\;\&\;\;1\geq y,\\\ 0,&\text{if
}y<1,\end{cases}$ $\displaystyle=\begin{cases}2-x-\frac{1}{x},&\text{if }x\leq
1,\\\ 0,&\text{if }x\geq 1.\end{cases}\qed$

## Declarations and statements

Data availability. The research in this paper does not imply use of data.

Conflict of interest. The authors declare that there is no conflict of
interest.

## References

  * [1] D. Alpay, F. Colombo, D.P. Kimsey: The spectral theorem for quaternionic unbounded normal operators based on the $S$-spectrum. J. Math. Phys. 57(2), 023503, 27 (2016).
  * [2] D. Alpay, F. Colombo, I. Sabadini: Slice hyperholomorphic Schur analysis. Volume 256 of Operator Theory: Advances and Applications, Basel, Birkhäuser/Springer (2016).
  * [3] D. Alpay, F. Colombo, I. Sabadini: Quaternionic de Branges spaces and characteristic operator function. SpringerBriefs in Mathematics, Springer, Cham (2020).
  * [4] D. Alpay, F. Colombo, T. Qian, I. Sabadini: The $H^{\infty}$ functional calculus based on the $S$-spectrum for quaternionic operators and for $n$-tuples of noncommuting operators. J. Funct. Anal. 271(6) (2016) 1544–1584.
  * [5] P. Auscher, A. Axelsson, A. McIntosh: On a quadratic estimate related to the Kato conjecture and boundary value problems. Harmonic analysis and partial differential equations, Contemp. Math., 505, Amer. Math. Soc., Providence, RI (2010) 105-129.
  * [6] P. Auscher, A. McIntosh, A. Nahmod. Holomorphic functional calculi of operators, quadratic estimates and interpolation. Indiana Univ. Math. J. 46 (1997) 375–403.
  * [7] A. Axelsson, S. Keith, A. McIntosh: Quadratic estimates and functional calculi of perturbed Dirac operators. Invent. Math. 163 (2006) 455–497.
  * [8] L. Baracco, F. Colombo, M.M. Peloso, S. Pinton: Fractional powers of higher-order vector operators on bounded and unbounded domains. Proc. Edinb. Math. Soc. 4(2) 65 (2022) 912-937.
  * [9] G. Birkhoff, J. von Neumann: The logic of quantum mechanics. Ann. of Math. (2) 37(4) (1936) 823–843.
  * [10] H. Brezies: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer Science+Business Media (2011).
  * [11] F. Colombo, A. De Martino, S. Pinton: Harmonic and polyanalytic functional calculi on the $S$-spectrum for unbounded operators. Banach J. Math. Anal. 17, no. 4, Paper No. 84 (2023) 41 pp.
  * [12] F. Colombo, A. De Martino, S. Pinton, I. Sabadini: Axially harmonic functions and the harmonic functional calculus on the $S$-spectrum. J. Geom. Anal. 33(2) (2023), 54 pp.
  * [13] F. Colombo, A. De Martino, S. Pinton, I. Sabadini: The fine structure of the spectral theory on the $S$-spectrum in dimension five. J. Geom. Anal. 33, no. 9, Paper No. 300 (2023) 73 pp.
  * [14] F. Colombo, D. Deniz González, S. Pinton: Fractional powers of vector operators with first order boundary conditions. J. Geom. Phys. 151, 18 (2020) 103618.
  * [15] F. Colombo, D. Deniz González, S. Pinton: The noncommutative fractional Fourier law in bounded and unbounded domains. Complex Anal. Oper. Theory 15(7), Paper No. 114, 27 (2021).
  * [16] F. Colombo. J. Gantner: An application of the $S$-functional calculus to fractional diffusion processes. Milan J. Math. 86(2) (2018) 225–303.
  * [17] F. Colombo, J. Gantner: Quaternionic closed operators, fractional powers and fractional diffusion processes. Operator Theory: Advances and Applications 274, Birkhäuser/Springer, Cham (2019) viii+322.
  * [18] F. Colombo, J. Gantner, D.P. Kimsey: Spectral theory on the $S$-spectrum for quaternionic operators. Volume 270 of Operator Theory: Advances and Applications, Birkhäuser/Springer, Cham (2018) ix+356.
  * [19] F. Colombo, J. Gantner, D.P. Kimsey, I. Sabadini: Universality property of the $S$-functional calculus, noncommuting matrix variables and Clifford operators. Adv. Math. 410, Paper No. 108719, (2022) 39 pp.
  * [20] F. Colombo, D.P. Kimsey: The spectral theorem for normal operators on a Clifford module. Anal. Math. Phys., 12(1), Paper No. 25, 92 (2022).
  * [21] F. Colombo, M.M. Peloso, S. Pinton: The structure of the fractional powers of the noncommutative Fourier law. Math. Methods Appl. Sci. 42(18) (2019) 6259–6276.
  * [22] F. Colombo, S. Pinton, P. Schlosser: The $H^{\infty}$-functional calculi for the quaternionic fine structures of Dirac type. To appear in Milan J. Math.
  * [23] F. Colombo, I. Sabadini, D.C. Struppa: Michele Sce’s works in hypercomplex analysis – A translation with commentaries. Birkhäuser/Springer, Cham (2020).
  * [24] F. Colombo, I. Sabadini, D.C. Struppa: Noncommutative functional calculus. Theory and applications of slice hyperholomorphic functions. Volume 289 of Progress in Mathematics, Birkhäuser/Springer Basel AG, Basel (2011) vi+221 pp.
  * [25] F. Colombo, I. Sabadini, D.C. Struppa: A new functional calculus for noncommuting operators. J. Funct. Anal. 254(8) (2008) 2255–2274.
  * [26] A. De Martino, S. Pinton: A polyanalytic functional calculus of order 2 on the $S$-spectrum. Proc. Amer. Math. Soc. 151 (2023) 2471–2488.
  * [27] A. De Martino, S. Pinton: Properties of a polyanalytic functional calculus on the $S$-spectrum. Math. Nachr. 296 (2023) 5190-5226.
  * [28] A. De Martino, S. Pinton, P. Schlosser: The harmonic $H^{\infty}$-functional calculus based on the $S$-spectrum. To appear in Journal of Spectral Theory.
  * [29] R. Delanghe, F. Sommen, V. Souček: Clifford algebra and spinor-valued functions. Mathematics and its Applications 53, Kluwer Academic Publishers Group, Dordrecht (1992).
  * [30] N. Dunford, J. Schwartz: Linear operators, part I: general theory. J. Wiley and Sons (1988).
  * [31] E. Franks, A. McIntosh: Discrete quadratic estimates and holomorphic functional calculi in Banach spaces. Bull. Austral. Math. Soc. 58 (1998) 271–290.
  * [32] R. Fueter: Die Funktionentheorie der Differentialgleichungen $\Delta u=0$ und $\Delta\Delta u=0$ mit vier reellen Variablen. Comment. Math. Helv. 7(1) (1934) 307–330.
  * [33] J. Gantner: Operator theory on one-sided quaternion linear spaces: intrinsic S-functional calculus and spectral operators. Mem. Amer. Math. Soc. 267, no. 1297 (2020) iii+101 pp.
  * [34] J. Gantner: A direct approach to the $S$-functional calculus for closed operators. J. Operator Theory. 77 no.2 (2017) 287-331.
  * [35] B. Jefferies: Spectral properties of noncommuting operators. Volume 1843, Lecture Notes in Mathematics. Springer-Verlag, Berlin (2004).
  * [36] M. Haase: The functional calculus for sectorial operators. Operator Theory: Advances and Applications 169, Birkhäuser Verlag, Basel (2006).
  * [37] T. Hytönen, J. van Neerven, M. Veraar, L. Weis: Analysis in Banach spaces. Vol. II. Probabilistic methods and operator theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 67, Springer, Cham (2017) xxi+616 pp.
  * [38] T. Hytönen, J. van Neerven, M. Veraar, and L. Weis: Analysis in Banach spaces. Vol. I. Martingales and Littlewood-Paley theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 63, Springer, Cham (2016) xvi+614 pp.
  * [39] B. Jefferies, A. McIntosh, J. Picton-Warlow: The monogenic functional calculus. Studia Math. 136 (1999) 99-119.
  * [40] P. Li, T. Qian: Singular integrals and Fourier theory on Lipschitz boundaries. Science Press Beijing, Beijing, Springer, Singapore (2019) xv+306 pp.
  * [41] J. Marschall: The trace of Sobolev-Slobodeckij spaces on Lipschitz domains. Manuscripta Math. 58 (1987) 47-65.
  * [42] A. McIntosh: Operators which have an $H^{\infty}$ functional calculus. Miniconference on operator theory and partial differential equations (North Ryde, 1986), 210–231, Proc. Centre Math. Anal. Austral. Nat. Univ., 14, Austral. Nat. Univ., Canberra, (1986).
  * [43] T. Qian: Generalization of Fueter’s result to $\textbf{R}^{n+1}$. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 8(2) (1997) 111–117.
  * [44] M. Sce: Osservazioni sulle serie di potenze nei moduli quadratici. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. 23(8) (1957) 220–225.
  * [45] E. Stein: Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, New Yersey (1970).
  * [46] G. Talenti: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110 (1976) 353-372.