∎

11institutetext: Yang Liu 22institutetext: Department of Mathematics, Zhejiang
Normal University, Jinhua 321004, China
Fax: +86-57982298897
22email<EMAIL_ADDRESS>33institutetext: Zhihua Chen 44institutetext:
Department of Mathematics, Tongji University, Shanghai 200092, China
55institutetext: Yifei Pan 66institutetext: Department of Mathematical
Sciences, Indiana University-Purdue University Fort Wayne, Fort Wayne, Indiana
46805, USA

# A variant of Hörmander’s $L^{2}$ theorem for Dirac operator in Clifford
analysis

Yang Liu  Zhihua Chen  Yifei Pan

(Received: date / Accepted: date)

###### Abstract

In this paper, we give the Hörmander’s $L^{2}$ theorem for Dirac operator over
an open subset $\Omega\in\mathbb{R}^{n+1}$ with Clifford algebra. Some
sufficient condition on the existence of the weak solutions for Dirac operator
has been found in the sense of Clifford analysis. In particular, if $\Omega$
is bounded, then we prove that for any $f$ in $L^{2}$ space with value in
Clifford algebra, there exists a weak solution of Dirac operator such that

$\overline{D}u=f$

with $u$ in the $L^{2}$ space as well. The method is based on Hörmander’s
$L^{2}$ existence theorem in complex analysis and the $L^{2}$ weighted space
is utilised.

###### Keywords:

Hörmander’s $L^{2}$ theoremClifford analysis weak solutionDirac operator

###### MSC:

32W50 15A66

##  1 Introduction

The development of function theories on Clifford algebras has proved a useful
setting for generalizing many aspects of one variable complex function theory
to higher dimensions. The study of these function theories is referred to as
Clifford analysis Brackx et al (1982); Huang et al (2006); Gong et al (2009);
Ryan (2000), which is closely related to a number of studies made in
mathematical physics, and many applications in this area have been found in
recent years. In Ryan (1995), Ryan considered solutions of the polynomial
Dirac operator, which afforded an integral representation. Furthermore, the
author gave a Pompeiu representation for $C^{1}$-functions in a Lipschitz
bounded domain. In Ryan (1990), the author presented a classification of
linear, conformally invariant, Clifford-algebra-valued differential operators
over $\mathbb{C}^{n}$, which comprised the Dirac operator and its iterates. In
Qian and Ryan (1996), Qian and Ryan used Vahlen matrices to study the
conformal covariance of various types of Hardy spaces over hypersurfaces in
$\mathbb{R}^{n}$. In De Ridder et al (2012), the discrete Fueter polynomials
was introduced, which formed a basis of the space of discrete spherical
monogenics. Moreover, the explicit construction for this discrete Fueter
basis, in arbitrary dimension $m$ and for arbitrary homogeneity degree $k$ was
presented as well.

In Hörmander (1965), the famous Hörmander’s $L^{2}$ existence and
approximation theorems was given for the $\bar{\partial}$ operator in pseudo-
convex domains in $\mathbb{C}^{n}$. When $n=1$, the existence theorem of
complex variable can be deduced. The aim of this paper is to establish a
Hörmander’s $L^{2}$ theorem in $\mathbb{R}^{n+1}$ with Clifford analysis, and
present sufficient condition on the existence of the weak solutions for Dirac
operator in the sense of Clifford algebra.

Let $\mathcal{A}$ be a real Clifford algebra over an (n+1)-dimensional real
vector space $\mathbb{R}^{n+1}$ and the corresponding norm on $\mathcal{A}$ is
given by $|\lambda|_{0}=\sqrt{(\lambda,\lambda)_{0}}$ (see subsection 2.1).
Let $\Omega$ be an open subset of $\mathbb{R}^{n+1}$,
$L^{2}(\Omega,\mathcal{A},\varphi)$ be a right Hilbert $\mathcal{A}$-module
for a given function $\varphi\in C^{2}(\Omega,\mathbb{R})$ with the norm given
by Definition 2.9. (see subsection 2.3). $\overline{D}$ denotes the Dirac
differential operator and the dual operator $\overline{D}^{*}_{\varphi}$ of
$\overline{D}$ is given by (4). For
$x=(x_{0},x_{1},...,x_{n})\in\mathbb{R}^{n+1}$,
$\Delta=\sum_{i=0}^{n}\frac{\partial^{2}}{\partial x_{i}^{2}}$. Then we can
obtain our main results as follows.

###### Theorem 1.1

Given $f\in L^{2}(\Omega,\mathcal{A},\varphi)$, there exists $u\in
L^{2}(\Omega,\mathcal{A},\varphi)$ such that

$\begin{split}\overline{D}u=f\end{split}$ (1)

with

$\begin{split}\|u\|^{2}=\int_{\Omega}|u|^{2}_{0}e^{-\varphi}dx\leq
2^{2n}c\end{split}$ (2)

if

$\begin{split}|(f,\alpha)_{\varphi}|^{2}_{0}\leq
c\|\overline{D}^{*}_{\varphi}\alpha\|^{2}=c\int_{\Omega}|\overline{D}^{*}_{\varphi}\alpha|^{2}_{0}e^{-\varphi}dx,~{}\forall\alpha\in
C^{\infty}_{0}(\Omega,\mathcal{A}).\end{split}$ (3)

Conversely, if there exists $u\in L^{2}(\Omega,\mathcal{A},\varphi)$ such that
(1) is satisfied with

$\begin{split}\|u\|^{2}\leq c\end{split}$

Then we can get the inequality (3) for norm estimation.

The factor $2^{2n}$ in (2) comes from the definition of the norm in Clifford
analysis. If $n=1$, then the factor would disappear which gives a necessary
and sufficient condition in the theorem. From the above theorem, we give the
following sufficient condition on the existence of weak solutions for Dirac
operator.

###### Theorem 1.2

Given $\varphi\in C^{2}(\Omega,\mathbb{R})$ and $n>1$; $\Delta\varphi\geq 0$,
and $\frac{\partial^{2}\varphi}{\partial x_{j}\partial x_{i}}=0,~{}i\neq
j,~{}1\leq i,j\leq n$ and $\frac{\partial^{2}\varphi}{\partial x^{2}_{i}}\leq
0,~{}1\leq i\leq n$. Then for all $f\in L^{2}(\Omega,\mathcal{A},\varphi)$
with $\int_{\Omega}\frac{|f|^{2}_{0}}{\Delta\varphi}e^{-\varphi}dx=c<\infty$,
there exists a $u\in L^{2}(\Omega,\mathcal{A},\varphi)$ such that

$\overline{D}u=f$

with

$\|u\|^{2}=\int_{\Omega}|u|^{2}_{0}e^{-\varphi}dx\leq
2^{2n}\int_{\Omega}\frac{|f|^{2}_{0}}{\Delta\varphi}e^{-\varphi}dx.$

###### Remark 1.3

Assuming $x=(x_{0},x_{1},...,x_{n})\in\mathbb{R}^{n+1}$, it is easy to see
that $\varphi(x)=x_{0}^{2}$ satisfies the conditions in Theorem 1.2. Another
simple example would be

$\varphi(x)=(n+1)x_{0}^{2}-\sum_{i=1}^{n}x_{i}^{2}.$

It is obvious that $\Delta\varphi(x)=2$, $\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}=-2$, and $\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{i}}=0,~{}i\neq j,~{}1\leq i,j\leq n$.

###### Corollary 1.4

Given $\varphi\in C^{2}(\Omega,\mathbb{R}),$ and $\varphi(x)=\varphi(x_{0})$
with $\varphi^{\prime\prime}(x_{0})\geq 0$. Then for all $f\in
L^{2}(\Omega,\mathcal{A},\varphi)$ with
$\int_{\Omega}\frac{|f|^{2}_{0}}{\varphi^{\prime\prime}}e^{-\varphi}dx=c<\infty$,
there exists a $u\in L^{2}(\Omega,\mathcal{A},\varphi)$ such that

$\overline{D}u=f$

with

$\|u\|^{2}=\int_{\Omega}|u|^{2}_{0}e^{-\varphi}dx\leq
2^{2n}\int_{\Omega}\frac{|f|^{2}_{0}}{\varphi^{\prime\prime}}e^{-\varphi}dx.$

It is noticed that there is nothing to do with the boundary conditions of
$\Omega$ in the above results. This phenomenon is totally different with the
famous Hörmander’s $L^{2}$ existence theorems of several complex variables in
Hörmander (1965). Then we can also have the following theorem on global
solutions.

###### Theorem 1.5

Given $\varphi\in C^{2}(\mathbb{R}^{n+1},\mathbb{R})$ with all derivative
conditions in Theorem 1.1 satisfied. Then for all $f\in
L^{2}(\mathbb{R}^{n+1},\mathcal{A},\varphi)$ with
$\int_{\mathbb{R}^{n+1}}\frac{|f|^{2}_{0}}{\Delta\varphi}e^{-\varphi}dx=c<\infty$,
there exists a $u\in L^{2}(\mathbb{R}^{n+1},\mathcal{A},\varphi)$ satisfying

$\overline{D}u=f$

with

$\|u\|^{2}=\int_{\mathbb{R}^{n+1}}|u|^{2}_{0}e^{-\varphi}dx\leq
2^{2n}\int_{\mathbb{R}^{n+1}}\frac{|f|^{2}_{0}}{\Delta\varphi}e^{-\varphi}dx.$

On the other hand, if the boundary of $\Omega$ is concerned, we consider a
special kind of domain ${\Omega}_{0}=\\{x\in\mathbb{R}^{n+1}:a\leq x_{0}\leq
b\\}$ for any $a,~{}b\in\mathbb{R}$ with $a<b$, then we can get the following
theorem within $L^{2}$ space instead of $L^{2}$ weighted space.

###### Theorem 1.6

Let $f\in L^{2}({\Omega_{0}},\mathcal{A})$. Then there exists a $u\in
L^{2}({\Omega_{0}},\mathcal{A})$ such that

$\overline{D}u=f$

with

$\int_{\Omega_{0}}|u|^{2}_{0}dx\leq
2^{2n}c(a,b)\int_{\Omega_{0}}{|f|^{2}_{0}}dx$

and $c(a,b)$ is a factor depending on $a,~{}b$.

###### Proof

Let $\varphi(x)=x_{0}^{2}$. It can be obtained that
$L^{2}({\Omega_{0}},\mathcal{A})=L^{2}({\Omega_{0}},\mathcal{A},\varphi)$ for
the boundary of $x_{0}$. Then the theorem is proved with Theorem 1.2.

###### Remark 1.7

In particular, any bounded domain $\Omega$ in $\mathbb{R}^{n+1}$ can be
regarded as one type of $\Omega_{0}$. Therefore, it comes from Theorem 1.6
that for any $f\in L^{2}(\Omega,\mathcal{A})$, we can find a weak solution of
Dirac operator $\overline{D}u=f$ with $u\in L^{2}(\Omega,\mathcal{A})$.

##  2 Preliminaries

To make the paper self-contained, some basic notations and results used in
this paper are included.

###  2.1 The Clifford algebra $\mathcal{A}$

Let $\mathcal{A}$ be a real Clifford algebra over an (n+1)-dimensional real
vector space $\mathbb{R}^{n+1}$ with orthogonal basis
$e:=\\{e_{0},e_{1},...,e_{n}\\}$, where $e_{0}=1$ is a unit element in
$\mathbb{R}^{n+1}$. Furthermore,

$\left\\{\begin{aligned} e_{i}e_{j}+e_{j}e_{i}&=0,~{}i\neq j\\\
e_{i}^{2}&=-1,~{}i=1,...,n.\end{aligned}\right.$

Then $\mathcal{A}$ has its basis

$\\{e_{A}=e_{h_{1}\cdots h_{r}}=e_{h_{1}}\cdots e_{h_{r}}:1\leq
h_{1}<...<h_{r}\leq n,1\leq r\leq n\\}.$

If $i\in\\{h_{1},...,h_{r}\\}$, we denote $i\in A$ and if
$i\not\in\\{h_{1},...,h_{r}\\}$, we denote $i\not\in A$. $A-{i}$ means
$\\{h_{1},...,h_{r}\\}\setminus\\{i\\}$ and $A+{i}$ means
$\\{h_{1},...,h_{r}\\}\cup\\{i\\}$. So the real Clifford algebra is composed
of elements having the type $a=\sum\limits_{A}x_{A}e_{A}$, in which
$x_{A}\in\mathbb{R}$ are real numbers. For $a\in\mathcal{A}$, we give the
inversion in the Clifford algebra as follows:
$a^{*}=\sum\limits_{A}x_{A}e_{A}^{*}$ where $e_{A}^{*}=(-1)^{|A|}e_{A}$ and
$|A|=n(A)$ is the $r\in\mathbb{Z}^{+}$ as $e_{A}=e_{h_{1}\cdots h_{r}}$. When
$A=\emptyset$, $e_{A}=e_{0}$, $|A|=0$. Next, we define the reversion in the
Clifford algebra, which is given by
$a^{\dagger}=\sum\limits_{A}x_{A}e_{A}^{\dagger}$ where
$e_{A}^{\dagger}=(-1)^{(|A|-1)|A|/2}e_{A}.$ Now we present the involution
which is a combination of the inversion and the reversion introduced above.

$\bar{a}=\sum\limits_{A}x_{A}\bar{e}_{A}$

where $\bar{e}_{A}=e_{A}^{*{\dagger}}=(-1)^{(|A|+1)|A|/2}e_{A}.$ From the
definition, one can easily deduce that $e_{A}\bar{e}_{A}=\bar{e}_{A}e_{A}=1.$
Furthermore, we have

$\overline{\lambda\mu}=\bar{\mu}\bar{\lambda},~{}~{}\forall\lambda,\mu\in\mathcal{A}.$

Let $a=\sum\limits_{A}x_{A}e_{A}$ be a Clifford number. The coefficient
$x_{A}$ of the $e_{A}$-component will also be denoted by $[a]_{A}$. In
particular the coefficient $x_{0}$ of the $e_{0}$-component will be denoted by
$[a]_{0}$, which is called the scalar part of the Clifford number $a$. An
inner product on $\mathcal{A}$ is defined by putting for any
$\lambda,\mu\in\mathcal{A}$,
$(\lambda,\mu)_{0}:=2^{n}[\lambda\bar{\mu}]_{0}=2^{n}\sum\limits_{A}\lambda_{A}\mu_{A}$.
The corresponding norm on $\mathcal{A}$ reads
$|\lambda|_{0}=\sqrt{(\lambda,\lambda)_{0}}$.

We define a real functional on $\mathcal{A}$ that
$\tau_{e_{A}}:\mathcal{A}\rightarrow\mathbb{R}$

$\langle\tau_{e_{A}},\mu\rangle=2^{n}(-1)^{(|A|+1)|A|/2}\mu_{A}.$

In the special case where $A=\emptyset$ we have

$\langle\tau_{e_{0}},\mu\rangle=2^{n}\mu_{0}.$

Let $\Omega$ be an open subset of $\mathbb{R}^{n+1}$. Then functions $f$
defined in $\Omega$ and with values in $\mathcal{A}$ are considered. They are
of the form

$f(x)=\sum_{A}f_{A}(x)e_{A}$

where $f_{A}(x)$ are functions with real value. Let $\overline{D}$ denotes the
Dirac differential operator

$\overline{D}=\sum_{i=0}^{n}e_{i}\partial_{x_{i}},$

its action on functions from the left and from the right being governed by the
rules

$\overline{D}f=\sum_{i,A}e_{i}e_{A}\partial_{x_{i}}f_{A}~{}\mbox{and}~{}f\overline{D}=\sum_{i,A}e_{A}e_{i}\partial_{x_{i}}f_{A}.$

$f$ is called left-monogenic if $\overline{D}f=0$ and it is called right-
monogenic if $f\overline{D}=0$. The conjugate operator is given by

$D=\sum_{i=0}^{n}\bar{e}_{i}\partial_{x_{i}}.$

It can be found that

$\overline{D}D=D\overline{D}=\Delta$

where $\Delta$ denotes the classical Laplacian in $\mathbb{R}^{n+1}$. When
$n=1$, one can think of $x_{0}$ as the real part and of $x_{1}$ as the
imaginary part of the variable and to identify $e_{1}$ with $i$. the operator
$\overline{D}$ then take the form
$\overline{D}=\partial_{x_{0}}+i\partial_{x_{1}}$, which is similar with the
operator $\bar{\partial}$ in complex analysis.

###  2.2 Modules over Clifford algebras

This subsection is to give some general information concerning a class of
topological modules over Clifford algebras. In the sequel definitions and
properties will be stated for left $\mathcal{A}$-module and their duals, the
passage to the case of right $\mathcal{A}$-module being straight-forward.

###### Definition 2.1

(unitary left $\mathcal{A}$-module) Let $X$ be a unitary left
$\mathcal{A}$-module, i.e. $X$ is abelian group and a law
$(\lambda,f)\rightarrow\lambda f:\mathcal{A}\times X\rightarrow X$ is defined
such that $\forall\lambda,\mu\in\mathcal{A}$, and $f,~{}g\in X$

  1. (1)

$(\lambda+\mu)f=\lambda f+\mu f$,

  2. (2)

$\lambda\mu f=\lambda(\mu f)$,

  3. (3)

$\lambda(f+g)=\lambda f+\lambda g$,

  4. (4)

$e_{0}f=f$.

Moreover, when speaking of a submodule $E$ of the unitary left
$\mathcal{A}$-module $X$, we mean that $E$ is a non empty subset of $X$ which
becomes a unitary left $\mathcal{A}$-module too when restricting the module
operations of $X$ to $E$.

###### Definition 2.2

(left $\mathcal{A}$-linear operator) If $X,Y$ are unitary left
$\mathcal{A}$-modules, then $T:X\rightarrow Y$ is said to be a left
$\mathcal{A}$-linear operator, if $\forall~{}f,~{}g\in X$ and
$\lambda\in\mathcal{A}$

$T(\lambda f+g)=\lambda T(f)+T(g).$

The set of all $``T"$ is denoted by $L(X,Y)$. If
$Y=\mathcal{A},~{}L(X,\mathcal{A})$ is called the algebraic dual of $X$ and
denoted by $X^{*alg}$. Its elements are called left $\mathcal{A}$-linear
functionals on $X$ and for any $T\in X^{*alg}$ and $f\in X$, we denote by
$\langle T,f\rangle$ the value of $T$ at $f$.

###### Definition 2.3

(bounded functional) An element $T\in X^{*alg}$ is called bounded, if there
exist a semi-norm $p$ on $X$ and $c>0$ such that for all $f\in X$

$|\langle T,f\rangle|_{0}\leq c\cdot p(f).$

###### Theorem 2.4

(Hahn-Banach type theorem)Brackx et al (1982) Let $X$ be a unitary left
$\mathcal{A}$-module with semi-norm $p$, $Y$ be a submodule of $X$, and $T$ be
a left $\mathcal{A}$-linear functional on $Y$ such that for some $c>0,$

$|\langle T,g\rangle|_{0}\leq c\cdot p(g),~{}~{}\forall g\in Y$

Then there exists a left $\mathcal{A}$-linear functional $\widetilde{T}$ on
$X$ such that

  1. (1)

$\widetilde{T}\mid_{Y}=T$,

  2. (2)

for some $c^{*}>0$, $|\langle\widetilde{T},f\rangle|_{0}\leq c^{*}\cdot p(f)$,
$\forall f\in X$.

###### Definition 2.5

(inner product on a unitary right $\mathcal{A}$-module) Let $H$ be a unitary
right $\mathcal{A}$-module, then a function $(~{},~{}):~{}H\times
H\rightarrow\mathcal{A}$ is said to be a inner product on $H$ if for all
$f,g,h\in H$ and $\lambda\in\mathcal{A}$,

  1. (1)

$(f,g+h)=(f,g)+(f,h)$,

  2. (2)

$(f,g\lambda)=(f,g)\lambda$,

  3. (3)

$(f,g)=\overline{(g,f)}$,

  4. (4)

$\langle\tau_{e_{0}},(f,f)\rangle\geq 0$ and
$\langle\tau_{e_{0}},(f,f)\rangle=0~{}\mbox{if and only if}~{}f=0$,

  5. (5)

$\langle\tau_{e_{0}},(f\lambda,f\lambda)\rangle\leq|\lambda|^{2}_{0}\langle\tau_{e_{0}},(f,f)\rangle$.

From the definition on inner product, putting for each $f\in H$

$\|f\|^{2}=\langle\tau_{e_{0}},(f,f)\rangle,$

then it can be obtained that for any $f,g\in H,$

$\begin{split}|\langle\tau_{e_{0}},~{}(f,g)\rangle|\leq\|f\|\|g\|,\|f+g\|\leq\|f\|+\|g\|.\end{split}$

Hence, $\|\cdot\|$ is a proper norm on $H$ turning it into a normed right
$A$-module. Moreover, we have the following Cauchy-Schwarz inequality.

###### Proposition 2.6

Brackx et al (1982) For all $f,g\in H,$ $|(f,g)|_{0}\leq\|f\|\|g\|.$

###### Definition 2.7

(right Hilbert $\mathcal{A}$-module) Let $H$ be a unitary right
$\mathcal{A}$-module provided with an inner product $(~{},~{})$. Then is it
called a right Hilbert $\mathcal{A}$-module if it is complete for the norm
topology derived from the inner product.

###### Theorem 2.8

(Riesz representation theorem)Brackx et al (1982) Let $H$ be a right Hilbert
$\mathcal{A}$-modules and $T\in H^{*alg}$. Then $T$ is bounded if and only if
there exists a (unique) element $g\in H$ such that for all $f\in H$,

$T(f):=\langle T,f\rangle=(g,f).$

###  2.3 Hilbert space of square integrable functions

Now we extend the standard Hilbert space of square integrable functions to
Clifford algebra. First, we denote $L^{1}(\Omega,\mu)$ and $L^{2}(\Omega,\mu)$
be the sets of all integrable or square integrable functions defined on the
domain $\Omega\in\mathbb{R}^{n+1}$ with respect to the measure $\mu$. Then
$L^{1}(\Omega,\mathcal{A},\mu)$ and $L^{2}(\Omega,\mathcal{A},\mu)$ are
defined as the sets of functions $f:\Omega\rightarrow\mathcal{A}$ which are
integrable or square integrable with respect to $\mu$, i.e., if
$f=\sum\limits_{A}f_{A}e_{A}$, then for each $A$, $f_{A}\in L^{1}(\Omega,\mu)$
and $f^{2}_{A}\in L^{1}(\Omega,\mu)$, respectively. Then one may easily check
that $L^{1}(\Omega,\mathcal{A},\mu)$ and $L^{2}(\Omega,\mathcal{A},\mu)$ are
unitary bi-$\mathcal{A}$-module, i.e., unitary left-$\mathcal{A}$-module and
unitary right-$\mathcal{A}$-module. Furthermore, for any $f,g\in
L^{2}(\Omega,\mathcal{A},\mu)$, $\bar{f}\in L^{2}(\Omega,\mathcal{A},\mu)$
while $\bar{f}g\in L^{1}(\Omega,\mathcal{A},\mu)$, where
$\bar{f}(x)=\overline{f(x)}$ and $(\bar{f}g)(x)=\bar{f}(x)g(x),~{}x\in\Omega$.
Consider as a right $\mathcal{A}$-module, define for $f,g\in
L^{2}(\Omega,\mathcal{A},\mu)$ that

$(f,g)=\int_{\Omega}\bar{f}(x)g(x)d\mu.$

Furthermore for any real linear functional $T$ on $\mathcal{A}$

$\langle T,(f,g)\rangle=\langle
T,\int_{\Omega}\bar{f}(x)g(x)d\mu\rangle=\int_{\Omega}\langle
T,\bar{f}(x)g(x)\rangle d\mu.$

Consequently, taking $T=\tau_{e_{0}}$ we find that

$\begin{split}\langle\tau_{e_{0}},(f,f)\rangle&=\langle\tau_{e_{0}},\int_{\Omega}\bar{f}(x)f(x)d\mu\rangle=\int_{\Omega}\langle\tau_{e_{0}},\bar{f}(x)f(x)\rangle
d\mu\\\ &=\int_{\Omega}|f(x)|^{2}_{0}d\mu.\end{split}$

Hence, for all $f\in L^{2}(\Omega,\mathcal{A},\mu)$,
$\langle\tau_{e_{0}},(f,f)\rangle\geq 0$ and
$\langle\tau_{e_{0}},(f,f)\rangle=0$ if and only if $f=0$ a.e. in $\Omega$.
Then it is easy to see that under the inner product defined, all conditions
for $L^{2}(\Omega,\mathcal{A},\mu)$ to be a unitary right inner product
$\mathcal{A}$-module are satisfied. Since
$L^{2}(\Omega,\mathcal{A},\mu)=\prod_{A}L^{2}(\Omega,\mu)$, we have that
$L^{2}(\Omega,\mathcal{A},\mu)$ is complete; in other words
$L^{2}(\Omega,\mathcal{A},\mu)$ is a right Hilbert $\mathcal{A}$-module, with
the norm

$\|f\|^{2}=\langle\tau_{e_{0}},(f,f)\rangle=\int_{\Omega}|f(x)|^{2}_{0}d\mu$

for $f\in L^{2}(\Omega,\mathcal{A},\mu)$.

###### Definition 2.9

(weighted $L^{2}$ space) Similar with $L^{2}(\Omega,\mathcal{A},\mu)$, we can
define the weighted $L^{2}(H,\mathcal{A},\varphi)$ for a given function
$\varphi\in C^{2}(\Omega,\mathbb{R})$. First, let

$L^{2}(\Omega,\varphi)=\big{\\{}f|f:\Omega\rightarrow\mathbb{R},~{}\int_{\Omega}|f(x)|^{2}e^{-\varphi}~{}dx<+\infty\big{\\}}.$

Then we denote

$L^{2}(H,\mathcal{A},\varphi)=\\{f|f:\Omega\rightarrow\mathcal{A},~{}f=\sum\limits_{A}f_{A}e_{A},~{}f_{A}\in
L^{2}(\Omega,\varphi)\\}.$

Moreover, for all $f,g\in L^{2}(H,\mathcal{A},\varphi)$, we define

$(f,g)_{\varphi}=\int_{\Omega}\bar{f}(x)g(x)e^{-\varphi}dx.$

Then it is also easy to see $L^{2}(\Omega,\mathcal{A},\varphi)$ is a right
Hilbert $\mathcal{A}$-module, with the norm

$\begin{split}\|f\|^{2}=\langle\tau_{e_{0}},(f,f)_{\varphi}\rangle=\int_{\Omega}|f(x)|^{2}_{0}e^{-\varphi}dx\end{split}$

for $f\in L^{2}(\Omega,\mathcal{A},\varphi)$.

###  2.4 Cauchy’s integral formula

Let $M$ be an (n+1)-dimensional differentiable and oriented manifold contained
in some open subset $\Sigma$ of $\mathbb{R}^{n+1}$. By means of the n-forms

$d\hat{x}_{i}=dx_{0}\wedge\cdots\wedge dx_{i-1}\wedge
dx_{x_{i+1}}\wedge\cdots\wedge dx_{n},~{}i=0,1,...,n,$

an $\mathcal{A}$-valued n-form is introduced by putting

$d\sigma=\sum_{i=0}^{n}(-1)^{i}e_{i}d\hat{x}_{i},$

similarly, denote

$d\bar{\sigma}=\sum_{i=0}^{n}(-1)^{i}\bar{e}_{i}d\hat{x}_{i}.$

Furthermore the volume-element

$dx=dx_{0}\wedge\cdots\wedge dx_{n}$

is used.

###### Proposition 2.10

(Stokes-Green Theorem)Brackx et al (1982) If $f,g\in
C^{1}(\Sigma,\mathcal{A})$ then for any (n+1)-chain $\Omega$ on
$M\subset\Sigma$,

$\int_{\partial\Omega}fd\sigma
g=\int_{\Omega}(f\overline{D})gdx+\int_{\Omega}f(\overline{D}g)dx,$
$\int_{\partial\Omega}fd\bar{\sigma}g=\int_{\Omega}(fD)gdx+\int_{\Omega}f(Dg)dx.$

###### Remark 2.11

Denote $C^{\infty}_{0}(\Omega,\mathbb{R})$ as the set of all smooth real-
valued functions with compact support in $\Omega$ and
$C^{\infty}_{0}(\Omega,\mathcal{A}):=\\{f|f:\Omega\rightarrow\mathcal{A},~{}f=\sum\limits_{A}f_{A}e_{A},~{}f_{A}\in
C^{\infty}_{0}(\Omega,\mathbb{R})\\}.$ If $f$ or $g\in
C^{\infty}_{0}(\Omega,\mathcal{A})$, then we have from the Stokes-Green
theorem that

$\int_{\Omega}(f\overline{D})gdx=-\int_{\Omega}f(\overline{D}g)dx,$
$\int_{\Omega}(fD)gdx=-\int_{\Omega}f(Dg)dx.$

###### Lemma 2.12

If $u(x)\in C^{1}(\Omega,\mathcal{A})$, then
$\overline{\overline{D}u}=\bar{u}D$.

###### Proof

Let $u(x)=\sum_{A}e_{A}u_{A}$. Then

$\begin{split}\overline{\overline{D}u}=\sum_{i,A}\overline{e_{i}e_{A}}\partial_{x_{i}}u_{A}=\sum_{i,A}\bar{e}_{A}\bar{e}_{i}\partial_{x_{i}}u_{A}=\bar{u}D.\end{split}$

###### Lemma 2.13

Huang et al (2006) If $u(x)=\sum_{A}e_{A}u_{A}$,
$v(x)=\sum_{i=0}^{n}e_{i}v_{i}$, then

$\overline{D}(uv)=(\overline{D}u)v+u(\overline{D}v)+\sum\limits^{n}_{j=1}(e_{j}u-ue_{j})\partial_{x_{j}}v.$

###  2.5 Weak solutions

Let
$L_{loc}^{1}(\Omega,\mathcal{A}):=\\{f|f:\Omega\rightarrow\mathcal{A},~{}f=\sum\limits_{A}f_{A}e_{A},~{}f_{A}\in
L_{loc}^{1}(\Omega,\mathbb{R})\\}$. Then we define the weak solution in the
sense of Clifford algebra as follows.

###### Definition 2.14

($\overline{D}$ solution in weak sense) If $f\in
L_{loc}^{1}(\Omega,\mathcal{A})$, $u:\Omega\rightarrow\mathcal{A}$ is a weak
solution of

$\overline{D}u=f~{}(\mbox{or}~{}{D}u=f)$

if for any $\alpha\in C^{\infty}_{0}(\Omega,\mathcal{A})$,

$\int_{\Omega}\alpha
fdx=-\int_{\Omega}(\alpha\overline{D})udx~{}(\mbox{or}~{}\int_{\Omega}\alpha
fdx=-\int_{\Omega}(\alpha{D})udx).$

It should be noticed that if $u$ is a weak solution of Dirac equation
$\overline{D}u=0$, in addition, if $u$ is smooth in $\Omega$, then it is left-
monogenic. Now it is natural to give the definition of $\Delta$ solution in
the weak sense.

###### Definition 2.15

($\Delta$ solution in weak sense) If $f\in L_{loc}^{1}(\Omega,\mathcal{A})$,
$u:\Omega\rightarrow\mathcal{A}$ is a weak solution of

$\Delta u=f$

if for any $\alpha\in C^{\infty}_{0}(\Omega,\mathcal{A})$,

$\int_{\Omega}\alpha fdx=\int_{\Omega}({\Delta}\alpha)udx.$

###### Theorem 2.16

If $f\in L_{loc}^{1}(\Omega,\mathcal{A})$, and $\overline{D}f=0$ in weak
sense, then $f$ is left-monogenic at any point of $\Omega$.

###### Proof

: Since $\overline{D}f=0$ in weak sense, then $\Delta f=0$ in weak sense. By
Weyl’s lemma, $f$ is smooth in $\Omega$ and has $\Delta f=0$ in classical
sense, then of course $f$ is left-monogenic at any point of $\Omega$.

###### Remark 2.17

This is useful to deal with uniqueness of weak solutions. for example, if
$u,~{}v\in L_{loc}^{1}(\Omega,\mathcal{A})$ are two weak solutions of
$\overline{D}u=f$, then $u=v+w$ with any $w$ left-monogenic.

###### Remark 2.18

An important example of a left monogenic function is the generalized Cauchy
kernel

$G(x)=\frac{1}{\omega_{n+1}}\frac{\overline{x}}{|x|^{n+1}},$

where $\omega_{n+1}$ denotes the surface area of the unit ball in
$\mathbb{R}^{n+1}$. This function obviously belongs to
$L_{loc}^{1}(\Omega,\mathcal{A})$ and is a fundamental solution of the Dirac
equation in the classical sense at any point of $\mathbb{R}^{n+1}$ except 0.
However, it is not a weak solution of the Dirac operator. In fact, if it
satisfies $\overline{D}f=0$ in the weak sense, then from Theorem 2.16, it must
be left-monogenic in the any point of $\Omega$ which could include $0$.
Therefore, we get a contradiction.

For $f\in L^{2}(\Omega,\mathcal{A},\varphi)$,
$u:\Omega\rightarrow\mathcal{A}$. If $\overline{D}u=f$, based on the Stokes-
Green theorem, we can define the dual operator $\overline{D}^{*}_{\varphi}$ of
$\overline{D}$ under the inner product of $L^{2}(\Omega,\mathcal{A},\varphi)$.
For any $\alpha\in C^{\infty}_{0}(\Omega,\mathcal{A})$,

$\begin{split}(\alpha,f)_{\varphi}=&~{}\int_{\Omega}\bar{\alpha}fe^{-\varphi}dx=\int_{\Omega}\bar{\alpha}e^{-\varphi}fdx\\\
=&~{}\int_{\Omega}(\bar{\alpha}e^{-\varphi})(\overline{D}u)dx\\\
=&~{}-\int_{\Omega}\big{(}(\bar{\alpha}e^{-\varphi})\overline{D}\big{)}udx\\\
=&~{}-\int_{\Omega}\big{(}(\bar{\alpha}e^{-\varphi})\overline{D}\big{)}e^{\varphi}ue^{-\varphi}dx\\\
=&~{}\int_{\Omega}\overline{-e^{\varphi}D(\alpha
e^{-\varphi})}ue^{-\varphi}dx\\\ =&~{}(-e^{-\varphi}D(\alpha
e^{-\varphi}),u)_{\varphi}\triangleq(\overline{D}^{*}_{\varphi}\alpha,u)_{\varphi},\end{split}$
(4)

where $\overline{D}^{*}_{\varphi}\alpha=-e^{\varphi}D(\alpha
e^{-\varphi})=\alpha(D\varphi)-D\alpha$, i.e.

$(\alpha,\overline{D}u)_{\varphi}=(\overline{D}^{*}_{\varphi}\alpha,u)_{\varphi}.$

In the same way, we also have

$(\overline{D}u,\alpha)_{\varphi}=(u,\overline{D}^{*}_{\varphi}\alpha)_{\varphi}.$

##  3 The proof of Theorem 1.1

Now we are in the position of proving Theorem 1.1.

###### Proof

($Sufficiency$) From the definition of dual operator and Cauchy-Schwarz
inequality in Proposition 2.6, we have

$\begin{split}|(f,\alpha)_{\varphi}|^{2}_{0}=&|(\overline{D}u,\alpha)_{\varphi}|^{2}_{0}=|(u,\overline{D}^{*}_{\varphi}\alpha)_{\varphi}|^{2}_{0}\\\
\leq&~{}\|u\|^{2}\cdot\|\overline{D}^{*}_{\varphi}\alpha\|^{2}\\\
\leq&~{}c\cdot\|\overline{D}^{*}_{\varphi}\alpha\|^{2}.\end{split}$

($necessity$) We aim to prove the necessity with Riesz representation theorem.
First, we denote the submodule

$E=\\{\overline{D}^{*}_{\varphi}\alpha,~{}\alpha\in
C^{\infty}_{0}(\Omega,\mathcal{A}),~{}\varphi\in
C^{2}(\Omega,\mathbb{R})\\}\subset L^{2}(\Omega,\mathcal{A},\varphi).$

Then we define a linear functional $L_{f}$ on $E$, i.e., $L_{f}\in E^{*alg}$
for a fixed $f\in L^{2}(\Omega,\mathcal{A},\varphi)$ as follows,

$\langle
L_{f},\overline{D}^{*}_{\varphi}\alpha\rangle=(f,\alpha)_{\varphi}=\int_{\Omega}\bar{f}\cdot\alpha\cdot
e^{-\varphi}dx\in\mathcal{A}.$

From (3), we have

$|\langle
L_{f},\overline{D}^{*}_{\varphi}\alpha\rangle|_{0}=|(f,\alpha)_{\varphi}|_{0}\leq\sqrt{c}\cdot\|\overline{D}^{*}_{\varphi}\alpha\|,$

which meas that $L_{f}$ is a bounded functional from Definition 2.3. By the
Hahn-Banach type theorem in Theorem 2.4, $L_{f}$ can be extended to a linear
functional $\widetilde{L}_{f}$ on $L^{2}(\Omega,\mathcal{A},\varphi)$, and
with

$\begin{split}|\langle\widetilde{L}_{f},g\rangle|_{0}\leq\sqrt{c^{*}}\|g\|,~{}\forall
g\in L^{2}(\Omega,\mathcal{A},\varphi),\end{split}$ (5)

where $\sqrt{c^{*}}=\sqrt{c}\cdot|e_{0}|_{0}$, since $|e_{A}|_{0}=2^{n/2}$,
then $c^{*}=2^{n}c$ from Brackx et al (1982). Now we are in the position to
use the Riesz representation theorem for the operator $\widetilde{L}_{f}$.
From Theorem 2.8, there exists a $u\in L^{2}(\Omega,\mathcal{A},\varphi)$ such
that

$\begin{split}\langle\widetilde{L}_{f},g\rangle=(u,g)_{\varphi},~{}\forall
g\in L^{2}(\Omega,\mathcal{A},\varphi).\end{split}$ (6)

For $\forall\alpha\in C^{\infty}_{0}(\Omega,\mathcal{A})$, let
$g=\overline{D}^{*}_{\varphi}\alpha$. Then

$\begin{split}(f,\alpha)_{\varphi}=&\langle\widetilde{L}_{f},\overline{D}^{*}_{\varphi}\alpha\rangle=(u,\overline{D}^{*}_{\varphi}\alpha)_{\varphi}=(\overline{D}u,\alpha)_{\varphi},\end{split}$

which deduces that

$\int_{\Omega}\bar{f}\alpha
e^{-\varphi}dx=\int_{\Omega}\overline{(\overline{D}u)}{\alpha}e^{-\varphi}dx.$

Conjugating both sides of above equation leads to

$\int_{\Omega}\bar{\alpha}f\cdot
e^{-\varphi}dx=\int_{\Omega}\bar{\alpha}(\overline{D})ue^{-\varphi}dx.$

Let $\alpha=\bar{\alpha}e^{\varphi}$, it can be obtained that

$\int_{\Omega}\alpha
fdx=\int_{\Omega}\alpha(\overline{D}u)dx,~{}\forall\alpha\in
C^{\infty}_{0}(\Omega,\mathcal{A}).$

Therefore,

$\overline{D}u=f$

is proved from the definition of weak solutions.

Next, we give the bound for the norm of $u$. Let $g=u=\sum_{A}e_{A}u_{A}\in
L^{2}(\Omega,\mathcal{A},\varphi)$, from (5) and (6), we get that

$\begin{split}|(u,u)_{\varphi}|_{0}\leq\sqrt{c^{*}}\|u\|.\end{split}$ (7)

On the other hand,

$\begin{split}|(u,u)_{\varphi}|_{0}^{2}=&\big{|}\int_{\Omega}\bar{u}ue^{-\varphi}dx\big{|}^{2}_{0}\\\
=&~{}2^{n}\cdot\big{[}\int_{\Omega}\bar{u}ue^{-\varphi}dx\cdot\overline{\int_{\Omega}\bar{u}ue^{-\varphi}dx}\big{]}_{0}\\\
=&~{}2^{n}\big{[}\int_{\Omega}(\sum\limits_{A}u^{2}_{A}+\sum\limits_{A\neq
B}\bar{e}_{A}e_{B}u_{A}u_{B})e^{-\varphi}dx\cdot\overline{\int_{\Omega}(\sum\limits_{A}u^{2}_{A}+\sum\limits_{A\neq
B}\bar{e}_{A}e_{B}u_{A}u_{B})e^{-\varphi}dx}\big{]}_{0}\\\
=&~{}2^{n}\big{[}(\int_{\Omega}\sum\limits_{A}u^{2}_{A}e^{-\varphi}dx)^{2}+(\int_{\Omega}\sum\limits_{A\neq
B}u_{A}u_{B}e^{-\varphi}dx)^{2}\big{]},\end{split}$

and

$\begin{split}\|u\|^{2}=&~{}\int_{\Omega}|u|^{2}_{0}e^{-\varphi}dx=2^{n}\int_{\Omega}[\bar{u}u]_{0}e^{-\varphi}dx=2^{n}\int_{\Omega}\sum\limits_{A}u^{2}_{A}\cdot
e^{-\varphi}dx\end{split}$

So we have $\|u\|^{4}=2^{2n}\cdot(\int_{\Omega}\sum\limits_{A}u^{2}_{A}\cdot
e^{-\varphi}dx)^{2}$. Hence,

$|(u,u)_{\varphi}|_{0}^{2}=2^{n}[(\int_{\Omega}\sum\limits_{A}u^{2}_{A}\cdot
e^{-\varphi}dx)^{2}+(\int_{\Omega}\sum\limits_{A\neq
B}u_{A}u_{B}e^{-\varphi}dx)^{2}]\geq 2^{-n}\|u\|^{4}.$

Combining with (7), it is obtained that

$\|u\|^{2}\leq 2^{n/2}|(u,u)_{\varphi}|_{0}\leq 2^{n/2}\sqrt{c^{*}}\|u\|,$

and

$\|u\|^{2}\leq 2^{2n}{c}.$

The proof is completed.

##  4 The proof of Theorem 1.2

It should be noticed that inequality (3) in Theorem 1.1 is related with
$\alpha\in C^{\infty}_{0}(\Omega,\mathcal{A})$. In the following, we will give
another sufficient condition that has nothing to do with the space
$C^{\infty}_{0}(\Omega,\mathcal{A})$. First, we need to compute the norm of
$\|\overline{D}^{*}_{\varphi}\alpha\|$ for any $\alpha\in
C^{\infty}_{0}(\Omega,\mathcal{A}).$

$\begin{split}\|\overline{D}^{*}_{\varphi}\alpha\|^{2}=&\int_{\Omega}|\overline{D}^{*}_{\varphi}\alpha|^{2}_{0}e^{-\varphi}dx\\\
=&\int_{\Omega}\langle\tau_{e_{0}},\overline{\overline{D}^{*}_{\varphi}\alpha}\cdot\overline{D}^{*}_{\varphi}\alpha\rangle
e^{-\varphi}dx\\\
=&\langle\tau_{e_{0}},\int_{\Omega}\overline{\overline{D}^{*}_{\varphi}\alpha}\cdot\overline{D}^{*}_{\varphi}\alpha
e^{-\varphi}dx\rangle\\\
=&\langle\tau_{e_{0}},(\overline{D}^{*}_{\varphi}\alpha,\overline{D}^{*}_{\varphi}\alpha)_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\overline{D}\overline{D}^{*}_{\varphi}\alpha)_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\overline{D}(\alpha(D\varphi)-D\alpha))_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\overline{D}\alpha(D\varphi)+\alpha\Delta\varphi-\Delta\alpha+\sum\limits^{n}_{j=1}(e_{j}\alpha-\alpha
e_{j})\frac{\partial}{\partial x_{j}}(D\varphi))_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\overline{D}^{*}_{\varphi}(\overline{D}\alpha)+\alpha\Delta\varphi+\sum\limits^{n}_{j=1}(e_{j}\alpha-\alpha
e_{j})\frac{\partial}{\partial x_{j}}(D\varphi))_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\overline{D}^{*}_{\varphi}(\overline{D}\alpha))_{\varphi}+(\alpha,\alpha\Delta\varphi)_{\varphi}+(\alpha,\sum\limits^{n}_{j=1}(e_{j}\alpha-\alpha
e_{j})\frac{\partial}{\partial x_{j}}(D\varphi))_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\overline{D}^{*}_{\varphi}(\overline{D}\alpha))_{\varphi}\rangle+\langle\tau_{e_{0}},(\alpha,\alpha\Delta\varphi)_{\varphi}\rangle+\langle\tau_{e_{0}},(\alpha,\sum\limits^{n}_{j=1}(e_{j}\alpha-\alpha
e_{j})\frac{\partial}{\partial x_{j}}(D\varphi))_{\varphi}\rangle\\\
=&I_{1}+I_{2}+I_{3},\end{split}$

where

$\begin{split}I_{1}=&\langle\tau_{e_{0}},(\alpha,\overline{D}^{*}_{\varphi}(\overline{D}\alpha))_{\varphi}\rangle=\langle\tau_{e_{0}},(\overline{D}\alpha,\overline{D}\alpha)_{\varphi}\rangle=\|\overline{D}\alpha\|^{2},\\\
I_{2}=&\langle\tau_{e_{0}},(\alpha,\alpha\Delta\varphi)_{\varphi}\rangle=\int_{\Omega}|\alpha|^{2}_{0}\Delta\varphi
e^{-\varphi}dx,\end{split}$

and

$\begin{split}I_{3}=&\langle\tau_{e_{0}},(\alpha,\sum\limits^{n}_{j=1}(e_{j}\alpha-\alpha
e_{j})\frac{\partial}{\partial x_{j}}(D\varphi))_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\sum\limits^{n}_{j=1}(e_{j}\alpha-\alpha
e_{j})\frac{\partial}{\partial
x_{j}}(\sum_{i=0}^{n}\bar{e}_{i}\frac{\partial\varphi}{\partial
x_{i}}))_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},(\alpha,\sum\limits^{n}_{j=1}\sum_{i=0}^{n}(e_{j}\alpha\bar{e}_{i}-\alpha
e_{j}\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{i}})_{\varphi}\rangle\\\
=&\langle\tau_{e_{0}},\int_{\Omega}\bar{\alpha}\sum\limits^{n}_{j=1}\sum_{i=0}^{n}(e_{j}\alpha\bar{e}_{i}-\alpha
e_{j}\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{i}}e^{-\varphi}dx\rangle\\\
=&\int_{\Omega}\langle\tau_{e_{0}},\bar{\alpha}\sum\limits^{n}_{j=1}\sum_{i=0}^{n}(e_{j}\alpha\bar{e}_{i}-\alpha
e_{j}\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{i}}\rangle e^{-\varphi}dx.\end{split}$

It should be noticed that if $n=1$, i.e., the space $\mathbb{R}^{2}$ is
considered, then $I_{3}=0.$

Since for $1\leq i,j\leq n$ and $i\neq j$,
$e_{j}\bar{e}_{i}=-e_{j}{e}_{i}=e_{i}{e}_{j}=-{e}_{i}\bar{e}_{j}$. For
simplicity, let

$\begin{split}I_{4}=&\langle\tau_{e_{0}},\bar{\alpha}\sum\limits^{n}_{j=1}\sum\limits_{i=0}^{n}(e_{j}\alpha\bar{e}_{i}-\alpha
e_{j}\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{i}}\rangle\\\
=&\langle\tau_{e_{0}},\sum\limits^{n}_{j=1}\sum\limits_{i=1}^{n}(\bar{\alpha}e_{j}\alpha\bar{e}_{i}-\bar{\alpha}\alpha
e_{j}\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{i}}\rangle+\langle\tau_{e_{0}},\sum\limits^{n}_{j=1}(\bar{\alpha}e_{j}\alpha\bar{e}_{0}-\bar{\alpha}\alpha
e_{j}\bar{e}_{0})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{0}}\rangle\\\
=&\langle\tau_{e_{0}},\sum\limits_{i=1}^{n}(\bar{\alpha}e_{i}\alpha\bar{e}_{i}-\bar{\alpha}\alpha
e_{i}\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\rangle+\langle\tau_{e_{0}},\sum\limits^{n}_{j\neq
i}(\bar{\alpha}e_{j}\alpha\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial
x_{j}\partial x_{i}}\rangle\\\
&+\langle\tau_{e_{0}},\sum\limits^{n}_{j=1}(\bar{\alpha}e_{j}\alpha\bar{e}_{0}-\bar{\alpha}\alpha
e_{j}\bar{e}_{0})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{0}}\rangle\\\
=&\langle\tau_{e_{0}},\sum\limits_{i=1}^{n}(\bar{\alpha}e_{i}\alpha\bar{e}_{i}-\bar{\alpha}\alpha)\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\rangle+\langle\tau_{e_{0}},\sum\limits^{n}_{j\neq
i}(\bar{\alpha}e_{j}\alpha\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial
x_{j}\partial x_{i}}\rangle\\\
&+\langle\tau_{e_{0}},\sum\limits^{n}_{j=1}(\bar{\alpha}e_{j}\alpha\bar{e}_{0}-\bar{\alpha}\alpha
e_{j}\bar{e}_{0})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{0}}\rangle\\\ =&I_{5}+I_{6}+I_{7}.\end{split}$

Assume
$\alpha=\sum\limits_{A}\alpha_{A}e_{A}\in\mathcal{A},~{}\bar{\alpha}=\sum\limits_{A}\alpha_{A}\bar{e}_{A}$,
then for any $1\leq i\leq n,$

$\begin{split}\bar{\alpha}e_{i}\alpha\bar{e}_{i}=&~{}\sum\limits_{A}\alpha_{A}\bar{e}_{A}e_{i}\cdot\sum\limits_{A}\alpha_{A}e_{A}\bar{e}_{i}\\\
=&~{}\sum\limits_{A}(-1)^{\frac{|A|(|A|+1)}{2}}\alpha_{A}e_{A}e_{i}\cdot\sum\limits_{A}(-1)\alpha_{A}e_{A}e_{i}\end{split}$

Therefore

$\begin{split}I_{5}=&\langle\tau_{e_{0}},\sum\limits_{i=1}^{n}(\bar{\alpha}e_{i}\alpha\bar{e}_{i}-\bar{\alpha}\alpha)\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\rangle\\\
=&\langle\tau_{e_{0}},\sum\limits_{i=1}^{n}(\bar{\alpha}e_{i}\alpha\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\rangle-\langle\tau_{e_{0}},\sum\limits_{i=1}^{n}(\bar{\alpha}\alpha)\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\rangle\\\
=&\langle\tau_{e_{0}},\sum\limits_{i=1}^{n}(\sum\limits_{A}(-1)^{\frac{|A|(|A|+1)}{2}}\alpha_{A}e_{A}e_{i}\cdot\sum\limits_{A}(-1)\alpha_{A}e_{A}e_{i})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\rangle-\langle\tau_{e_{0}},\sum\limits_{i=1}^{n}(\bar{\alpha}\alpha)\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\rangle\\\
=&2^{n}\sum\limits_{i=1}^{n}(\sum\limits_{A}(-1)^{\frac{|A|(|A|+1)}{2}+1}\alpha_{A}^{2}e_{A}e_{i}e_{A}e_{i})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}-\sum\limits_{i=1}^{n}|\alpha|^{2}_{0}\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\\\ =&2^{n}\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in
A}(-1)^{\frac{|A|(|A|+1)}{2}+1}\alpha^{2}_{A}\cdot\overline{e_{A}e_{i}}\cdot
e_{A}e_{i}\cdot(-1)^{\frac{(|A|+1)(|A|+2)}{2}}\\\ &+\sum\limits_{i\in
A}(-1)^{\frac{|A|(|A|+1)}{2}+1}\cdot\alpha^{2}_{A}\cdot\overline{e_{A-{i}}}\cdot
e_{A-{i}}\cdot(-1)^{\frac{(|A|-1)(|A|)}{2}})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}-\sum\limits_{i=1}^{n}|\alpha|^{2}_{0}\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\\\ =&2^{n}\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in
A}(-1)^{\frac{|A|(|A|+1)}{2}+1+\frac{(|A|+1)(|A|+2)}{2}}\cdot\alpha^{2}_{A}\\\
&+\sum\limits_{i\in
A}(-1)^{\frac{|A|(|A|+1)}{2}+1+\frac{(|A|-1)(|A|)}{2}}\cdot\alpha^{2}_{A})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}-\sum\limits_{i=1}^{n}|\alpha|^{2}_{0}\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\\\ =&2^{n}\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in
A}(-1)^{|A|^{2}}\cdot\alpha^{2}_{A}+\sum\limits_{i\in
A}(-1)^{|A|^{2}+1}\cdot\alpha^{2}_{A})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}-\sum\limits_{i=1}^{n}|\alpha|^{2}_{0}\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\\\ =&2^{n}\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in
A,|A|^{2}~{}\mbox{is odd}}(-2)\alpha^{2}_{A}+\sum\limits_{i\in
A,|A|^{2}~{}\mbox{is
even}}(-2)\alpha^{2}_{A})\frac{\partial^{2}\varphi}{\partial x^{2}_{i}}\\\
=&-2^{n+1}\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in A,|A|^{2}~{}\mbox{is
odd}}\alpha^{2}_{A}+\sum\limits_{i\in A,|A|^{2}~{}\mbox{is
even}}\alpha^{2}_{A})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}.\end{split}$ (8)

To consider $I_{7}$, we first study $\bar{\alpha}e_{j}\alpha$ for any $1\leq
j\leq n$. Without loss of generality, let
$e_{j}=e_{1},~{}\bar{\alpha}=\sum\limits_{A}\alpha_{A}\bar{e}_{A},~{}\alpha=\sum\limits_{A}\alpha_{A}e_{A}$.
Then
$\bar{\alpha}e_{1}\alpha=(\sum\limits_{A}\alpha_{A}\bar{e}_{A})e_{1}(\sum\limits_{A}\alpha_{A}e_{A})$.

When $e_{A}=e_{1}e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$, where
$1<h_{2}<h_{3}<\cdots<h_{r}$ and $1<r\leq n.$

$\begin{split}\alpha_{A}\bar{e}_{A}e_{1}=&\alpha_{1h_{2}\cdots
h_{r}}(-1)^{\frac{r(r+1)}{2}}\cdot e_{1}e_{h_{2}}e_{h_{3}}\cdots
e_{h_{r}}\cdot e_{1}\\\ =&\alpha_{1h_{2}\cdots
h_{r}}(-1)^{\frac{r(r+1)}{2}+r}e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}\\\
\alpha_{A}e_{A}e_{1}=&\alpha_{1h_{2}\cdots h_{r}}e_{1}e_{h_{2}}\cdots
e_{h_{r}}\cdot e_{1}=\alpha_{1h_{2}\cdots h_{r}}(-1)^{r}e_{h_{2}}\cdots
e_{h_{r}}.\end{split}$ (9)

When $e_{A}=e_{1}$,

$\begin{split}\alpha_{A}\bar{e}_{A}e_{1}=&\alpha_{1}\\\
\alpha_{A}e_{A}e_{1}=&-\alpha_{1}.\end{split}$ (10)

When $e_{A}=e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$, where
$1<h_{2}<h_{3}<\cdots<h_{r}$ and $1<r\leq n.$

$\begin{split}\alpha_{A}\bar{e}_{A}e_{1}=&\alpha_{h_{2}\cdots
h_{r}}(-1)^{\frac{(r-1)(r)}{2}}\cdot e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}\cdot
e_{1}\\\ =&\alpha_{h_{2}\cdots
h_{r}}(-1)^{\frac{(r-1)(r)}{2}+r-1}e_{1}e_{h_{2}}\cdots e_{h_{r}}\\\
\alpha_{A}e_{A}e_{1}=&\alpha_{h_{2}\cdots h_{r}}e_{h_{2}}\cdots e_{h_{r}}\cdot
e_{1}=\alpha_{h_{2}\cdots h_{r}}(-1)^{r-1}e_{1}e_{h_{2}}\cdots
e_{h_{r}}.\end{split}$ (11)

When $e_{A}=e_{0}$,

$\begin{split}\alpha_{A}\bar{e}_{A}e_{1}=&\alpha_{0}e_{1}\\\
\alpha_{A}e_{A}e_{1}=&\alpha_{0}e_{1}.\end{split}$ (12)

To compute $I_{7}$, one needs to know the coefficient for $e_{0}$ of
$\bar{\alpha}e_{1}\alpha-\bar{\alpha}\alpha e_{1}$. It means that we should
find out the corresponding terms of $e_{1}e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$
and $e_{h_{2}}\cdots e_{h_{r}}$ in $\bar{\alpha}e_{1}$ and $\alpha$, in
$\bar{\alpha}$ and $\alpha e_{1}$.

Case a1. For $\bar{\alpha}e_{1}\alpha$, from (11), the corresponding terms of
$e_{1}e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$ with $1<h_{2}<h_{3}<\cdots<h_{r}$
and $1<r\leq n$ in
$\bar{\alpha}e_{1}=(\sum\limits_{A}\alpha_{A}\bar{e}_{A})e_{1}$ and
$\alpha=\sum\limits_{A}\alpha_{A}e_{A}$ are $\alpha_{h_{2}\cdots
h_{r}}(-1)^{\frac{(r-1)(r)}{2}+r-1}e_{1}e_{h_{2}}\cdots e_{h_{r}}$ and
$\alpha_{1h_{2}\cdots h_{r}}e_{1}e_{h_{2}}\cdots e_{h_{r}}$, respectively.
Multiplying these terms leads to

$\begin{split}(-1)&{}^{\frac{(r-1)(r)}{2}+r-1}e_{1}e_{h_{2}}\cdots
e_{h_{r}}\cdot e_{1}e_{h_{2}}\cdots e_{h_{r}}\cdot\alpha_{1h_{2}\cdots
h_{r}}\cdot\alpha_{h_{2}\cdots h_{r}}\\\
=&~{}(-1)^{\frac{(r-1)(r)}{2}+r-1}(-1)^{\frac{(r)(r+1)}{2}}\cdot\overline{e_{1}\cdots
e_{h_{r}}}\cdot e_{1}e_{h_{2}}\cdots e_{h_{r}}\cdot\alpha_{1h_{2}\cdots
h_{r}}\alpha_{h_{2}\cdots h_{r}}\\\
=&~{}(-1)^{\frac{(r)(r+1)}{2}+r-1+\frac{(r-1)(r)}{2}}\cdot\alpha_{1h_{2}\cdots
h_{r}}\alpha_{h_{2}\cdots h_{r}}.\end{split}$ (13)

On the other hand, for $\bar{\alpha}e_{1}\alpha$, from (9), the corresponding
terms of $e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$ with
$1<h_{2}<h_{3}<\cdots<h_{r}$ and $1<r\leq n$ in $\bar{\alpha}e_{1}$ and
$\alpha$ are $\alpha_{1h_{2}\cdots
h_{r}}(-1)^{\frac{r(r+1)}{2}+r}e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$ and
$\alpha_{h_{2}\cdots h_{r}}e_{h_{2}}\cdots e_{h_{r}}$, respectively.
Multiplying these terms leads to

$\begin{split}(-1)&{}^{\frac{(r)(r+1)}{2}+r}e_{h_{2}\cdots h_{r}}\cdot
e_{h_{2}\cdots h_{r}}\cdot\alpha_{1h_{2}\cdots h_{r}}\cdot\alpha_{h_{2}\cdots
h_{r}}\\\
=&~{}(-1)^{\frac{(r)(r+1)}{2}+r}(-1)^{\frac{(r-1)(r)}{2}}\cdot\overline{e_{h_{2}\cdots
h_{r}}}\cdot e_{h_{2}\cdots h_{r}}\cdot\alpha_{1h_{2}\cdots
h_{r}}\alpha_{h_{2}\cdots h_{r}}\\\
=&~{}(-1)^{\frac{(r)(r+1)}{2}+r+\frac{(r-1)(r)}{2}}\cdot\alpha_{1h_{2}\cdots
h_{r}}\alpha_{h_{2}\cdots h_{r}}.\end{split}$ (14)

From (13) and (14), these two terms vanish.

Case a2. For $\bar{\alpha}e_{1}\alpha$, from (12), the corresponding terms of
$e_{1}$ in $\bar{\alpha}e_{1}$ and $\alpha$ are $\alpha_{0}e_{1}$ and
$\alpha_{1}e_{1}$, respectively. Multiplying these terms leads to

$\begin{split}\alpha_{0}e_{1}\alpha_{1}e_{1}=-\alpha_{0}\alpha_{1}.\end{split}$
(15)

On the other hand, for $\bar{\alpha}e_{1}\alpha$, from (10), the corresponding
terms of $e_{0}$ in $\bar{\alpha}e_{1}$ and $\alpha$ are $\alpha_{1}$ and
$\alpha_{0}$, respectively. Multiplying these terms leads to
$\alpha_{0}\alpha_{1}$. Combining with (15), these two terms also vanish.

From Cases a1 and a2, one can obtain that the coefficient for $e_{0}$ of
$\bar{\alpha}e_{1}\alpha$ equals zero, i.e.,

$\begin{split}\langle\tau_{e_{0}},\sum\limits^{n}_{j=1}(\bar{\alpha}e_{j}\alpha\bar{e}_{0})\frac{\partial^{2}\varphi}{\partial
x_{j}\partial x_{0}}\rangle=0.\end{split}$ (16)

Case b1. For $\bar{\alpha}\alpha e_{1}$, from (11), the corresponding terms of
$e_{1}e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$ with $1<h_{2}<h_{3}<\cdots<h_{r}$
and $1<r\leq n$ in ${\alpha}e_{1}=(\sum\limits_{A}\alpha_{A}{e}_{A})e_{1}$ and
$\bar{\alpha}=\sum\limits_{A}\alpha_{A}\bar{e}_{A}$ are $\alpha_{h_{2}\cdots
h_{r}}(-1)^{r-1}e_{1}e_{h_{2}}\cdots e_{h_{r}}$ and $\alpha_{1h_{2}\cdots
h_{r}}\overline{e_{1}e_{h_{2}}\cdots e_{h_{r}}}$, respectively. Multiplying
these terms leads to

$\begin{split}(\alpha_{1h_{2}\cdots h_{r}}&\overline{e_{1}e_{h_{2}}\cdots
e_{h_{r}}})\cdot(\alpha_{h_{2}\cdots h_{r}}e_{h_{2}}\cdots e_{h_{r}}\cdot
e_{1})\\\ =&~{}(\alpha_{1h_{2}\cdots h_{r}}\overline{e_{1}e_{h_{2}}\cdots
e_{h_{r}}})\cdot((-1)^{r-1}e_{1}e_{h_{2}}\cdots
e_{h_{r}}\cdot\alpha_{h_{2}\cdots h_{r}})\\\
=&~{}(-1)^{r-1}\alpha_{1h_{2}\cdots h_{r}}\cdot\alpha_{h_{2}\cdots
h_{r}}.\end{split}$ (17)

On the other hand, for $\bar{\alpha}\alpha e_{1}$, from (9), the corresponding
terms of $e_{h_{2}}e_{h_{3}}\cdots e_{h_{r}}$ with
$1<h_{2}<h_{3}<\cdots<h_{r}$ and $1<r\leq n$ in ${\alpha}e_{1}$ and
$\bar{\alpha}$ are $\alpha_{1h_{2}\cdots h_{r}}(-1)^{r}e_{h_{2}}\cdots
e_{h_{r}}$ and $\alpha_{h_{2}\cdots h_{r}}\overline{e_{h_{2}}\cdots
e_{h_{r}}}$, respectively. Multiplying these terms leads to

$\begin{split}(\alpha_{h_{2}\cdots h_{r}}&\overline{e_{h_{2}}\cdots
e_{h_{r}}})\cdot(\alpha_{1h_{2}\cdots h_{r}}e_{1}\cdots e_{h_{r}}\cdot
e_{1})\\\ =&~{}(\alpha_{h_{2}\cdots h_{r}}\overline{e_{h_{2}}\cdots
e_{h_{r}}})\cdot((-1)^{r}e_{h_{2}}\cdots e_{h_{r}}\cdot\alpha_{1h_{2}\cdots
h_{r}})\\\ =&~{}(-1)^{r}\alpha_{h_{2}\cdots h_{r}}\cdot\alpha_{1h_{2}\cdots
h_{r}}.\end{split}$ (18)

From (17) and (18), these two terms vanish.

Case b2. For $\bar{\alpha}\alpha e_{1}$, from (12), the corresponding terms of
$e_{1}$ in ${\alpha}e_{1}$ and $\bar{\alpha}$ are $\alpha_{0}e_{1}$ and
$\alpha_{1}\bar{e}_{1}$, respectively. Multiplying these terms leads to

$\begin{split}\alpha_{0}e_{1}\alpha_{1}\bar{e}_{1}=\alpha_{0}\alpha_{1}.\end{split}$
(19)

On the other hand, for $\bar{\alpha}\alpha e_{1}$, from (10), the
corresponding terms of $e_{0}$ in ${\alpha}e_{1}$ and $\bar{\alpha}$ are
$-\alpha_{1}$ and $\alpha_{0}$, respectively. Multiplying these terms leads to
$-\alpha_{0}\alpha_{1}$. Combining with (19), these two terms also cancel.

From Cases b1 and b2, one can obtain that the coefficient for $e_{0}$ of
$\bar{\alpha}e_{1}\alpha$ equals zero, i.e.,

$\begin{split}\langle\tau_{e_{0}},\sum\limits^{n}_{j=1}(\bar{\alpha}\alpha
e_{j}\bar{e}_{0})\frac{\partial^{2}\varphi}{\partial x_{j}\partial
x_{0}}\rangle=0.\end{split}$ (20)

Thus, $I_{7}=0$ from (16) and (20).

To compute $I_{6}$, i.e., to get $[\bar{\alpha}e_{i}\alpha\bar{e}_{j}]_{0}$
for $i\neq j$, similar with the analysis of $I_{7}$, we should divide the
vectors in $\bar{\alpha}e_{i}$ and $\alpha\bar{e}_{j}$ into four cases.

Case c1. $i\in A,~{}j\not\in A$ for $e_{A}$ in $\bar{\alpha}$ and $i\not\in
B,~{}j\in B$ for $e_{B}$ in ${\alpha}$ with $A-{i}=B-{j}$.

For this case, firstly, we assume $e_{A}=e_{h_{1}\cdots h_{p(i)}\cdots h_{r}}$
and $h_{p(i)}=i$, $e_{B}=e_{h_{1}\cdots h_{p(j)}\cdots h_{r}}$ and
$h_{p(j)}=j$. We have

$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}}\cdot
e_{h_{1}}\cdots e_{i}\cdots e_{h_{r}}\cdot e_{i}\\\
=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}+r-p(i)}e_{h_{1}}\cdots e_{i}^{2}\cdots
e_{h_{r}},\\\ =&\alpha_{A}(-1)^{\frac{r(r+1)}{2}+r-p(i)+1}e_{A-{i}},\\\
\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{B}e_{h_{1}}\cdots e_{j}\cdots
e_{h_{r}}\cdot\bar{e}_{j}\\\ =&\alpha_{B}(-1)^{r-p(j)}e_{h_{1}}\cdots
e_{j}\bar{e}_{j}\cdots e_{h_{r}},\\\
=&\alpha_{B}(-1)^{r-p(j)}e_{B-{j}}.\end{split}$

Then

$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}+r-p(i)+1}e_{A-{i}}\alpha_{B}(-1)^{r-p(j)}e_{B-{j}}\\\
=&\alpha_{A}\alpha_{B}(-1)^{\frac{r(r+1)}{2}+r-p(i)+1+r-p(j)+\frac{r(r-1)}{2}}\overline{e_{A-{i}}}e_{B-{j}}\\\
=&\alpha_{A}\alpha_{B}(-1)^{r^{2}+1-p(i)-p(j)}.\end{split}$ (21)

Case c2. $i\not\in A,~{}j\in A$ for $e_{A}$ in $\bar{\alpha}$ and $i\in
B,~{}j\not\in B$ for $e_{B}$ in ${\alpha}$ with $A+{i}=B+{j}$.

We assume $e_{A}=e_{h_{1}\cdots h_{p(j)}\cdots h_{r}}$ and $h_{p(j)}=j$,
$e_{B}=e_{h_{1}\cdots h_{p(i)}\cdots h_{r}}$ and $h_{p(i)}=i$. We have

$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}}\cdot
e_{h_{1}}\cdots e_{j}\cdots e_{h_{r}}\cdot e_{i},\\\
\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{B}e_{h_{1}}\cdots e_{i}\cdots
e_{h_{r}}\cdot\bar{e}_{j}\\\ =&-\alpha_{B}e_{h_{1}}\cdots e_{i}\cdots
e_{h_{r}}\cdot{e}_{j}\\\ =&\alpha_{B}e_{h_{1}}\cdots e_{j}\cdots
e_{h_{r}}\cdot e_{i}.\end{split}$

Then

$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}}\cdot
e_{h_{1}}\cdots e_{j}\cdots e_{h_{r}}\cdot e_{i}\alpha_{B}e_{h_{1}}\cdots
e_{j}\cdots e_{h_{r}}\cdot e_{i}\\\
=&\alpha_{A}\alpha_{B}(-1)^{\frac{r(r+1)}{2}+\frac{(r+1)(r+2)}{2}}\overline{e_{h_{1}}\cdots
e_{j}\cdots e_{h_{r}}\cdot e_{i}}e_{h_{1}}\cdots e_{j}\cdots e_{h_{r}}\cdot
e_{i}\\\ =&\alpha_{A}\alpha_{B}(-1)^{r^{2}+1}.\end{split}$

Case c3. $i\in A,~{}j\in A$ for $e_{A}$ in $\bar{\alpha}$ and $i\not\in
B,~{}j\not\in B$ for $e_{B}$ in ${\alpha}$ with $A-{i}=B+{j}$.

For this case, we assume $e_{A}=e_{h_{1}\cdots h_{p(i)}\cdots h_{p(j)}\cdots
h_{r+2}}$ with $h_{p(i)}=i,~{}h_{p(j)}=j$. Without loss of generality, we
assume $i<j$. Furthermore, let $e_{B}=e_{h_{1}\cdots h_{r}}$. We have

$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}=&\alpha_{A}(-1)^{\frac{(r+2)(r+3)}{2}}\cdot
e_{h_{1}}\cdots e_{i}\cdots e_{j}\cdots e_{h_{r+2}}\cdot e_{i}\\\
=&\alpha_{A}(-1)^{\frac{(r+2)(r+3)}{2}+r+2-h(i)}\cdot e_{h_{1}}\cdots
e_{j}\cdots e_{h_{r+2}}\cdot e^{2}_{i}\\\
=&\alpha_{A}(-1)^{\frac{(r+2)(r+3)}{2}+r+1-h(i)}\cdot e_{h_{1}}\cdots
e_{j}\cdots e_{h_{r+2}}\\\
=&\alpha_{A}(-1)^{\frac{(r+2)(r+3)}{2}+r+1-h(i)+r+2-h(j)}\cdot e_{h_{1}}\cdots
e_{h_{r+2}}\cdot e_{j},\\\
\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{B}e_{h_{1}}\cdots
e_{h_{r}}\cdot\bar{e}_{j}\\\ =&-\alpha_{B}e_{h_{1}}\cdots
e_{h_{r}}\cdot{e}_{j}.\end{split}$

Then

$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{A}(-1)^{\frac{(r+2)(r+3)}{2}+r+1-h(i)+r+2-h(j)}\cdot
e_{h_{1}}\cdots e_{h_{r+2}}\cdot e_{j}(-1)\alpha_{B}e_{h_{1}}\cdots
e_{h_{r}}\cdot{e}_{j}\\\
=&\alpha_{A}\alpha_{B}(-1)^{\frac{(r+2)(r+3)}{2}-h(i)-h(j)}\cdot
e_{h_{1}}\cdots e_{h_{r+2}}\cdot e_{j}e_{h_{1}}\cdots e_{h_{r}}\cdot e_{j}\\\
=&\alpha_{A}\alpha_{B}(-1)^{\frac{(r+2)(r+3)}{2}-h(i)-h(j)+\frac{(r+1)(r+2)}{2}}\cdot\overline{e_{h_{1}}\cdots
e_{h_{r+2}}\cdot e_{j}}e_{h_{1}}\cdots e_{h_{r}}\cdot e_{j}\\\
=&\alpha_{A}\alpha_{B}(-1)^{r^{2}-h(j)-h(i)}.\end{split}$

Case c4. $i\not\in A,~{}j\not\in A$ for $e_{A}$ in $\bar{\alpha}$ and $i\in
B,~{}j\in B$ for $e_{B}$ in ${\alpha}$ with $A+{i}=B-{j}$.

For this case, we assume $e_{A}=e_{h_{1}\cdots h_{r}}$, $e_{B}=e_{h_{1}\cdots
h_{p(i)}\cdots h_{p(j)}\cdots h_{r+2}}$ with $h_{p(i)}=i,~{}h_{p(j)}=j$ and
$i<j$. We have

$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}}\cdot
e_{h_{1}}\cdots e_{h_{r}}\cdot e_{i},\\\
\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{B}e_{h_{1}}\cdots e_{i}\cdots
e_{j}\cdots e_{h_{r+2}}\cdot\bar{e}_{j}\\\ =&\alpha_{B}(-1)^{r+2-h(j)}\cdot
e_{h_{1}}\cdots e_{i}\cdots e_{h_{r+2}}\cdot e_{j}\bar{e}_{j}\\\
=&\alpha_{B}(-1)^{r+2-h(j)+r+2-h(i)-1}\cdot e_{h_{1}}\cdots e_{h_{r+2}}\cdot
e_{i}\\\ =&\alpha_{B}(-1)^{1-h(j)-h(i)}\cdot e_{h_{1}}\cdots e_{h_{r+2}}\cdot
e_{i}\\\ \end{split}$

Then

$\begin{split}\alpha_{A}\bar{e}_{A}e_{i}\alpha_{B}{e}_{B}\bar{e}_{j}=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}}\cdot
e_{h_{1}}\cdots e_{h_{r}}\cdot e_{i}\alpha_{B}(-1)^{1-h(j)-h(i)}\cdot
e_{h_{1}}\cdots e_{h_{r+2}}\cdot e_{i}\\\
=&\alpha_{A}\alpha_{B}(-1)^{\frac{r(r+1)}{2}+1-h(j)-h(i)}\cdot e_{h_{1}}\cdots
e_{h_{r}}\cdot e_{i}\cdot e_{h_{1}}\cdots e_{h_{r+2}}\cdot e_{i}\\\
=&\alpha_{A}\alpha_{B}(-1)^{\frac{r(r+1)}{2}+1-h(j)-h(i)+\frac{(r+1)(r+2)}{2}}\cdot\overline{e_{h_{1}}\cdots
e_{h_{r}}\cdot e_{i}}\cdot e_{h_{1}}\cdots e_{h_{r+2}}\cdot e_{i}\\\
=&\alpha_{A}\alpha_{B}(-1)^{r^{2}-h(j)-h(i)}.\end{split}$

Combining cases c1-c4, we have

$\begin{split}I_{6}=&\langle\tau_{e_{0}},\sum\limits^{n}_{j\neq
i}(\bar{\alpha}e_{j}\alpha\bar{e}_{i})\frac{\partial^{2}\varphi}{\partial
x_{j}\partial x_{i}}\rangle\\\ =&\langle\tau_{e_{0}},\sum\limits^{n}_{j\neq
i}\big{(}(\sum_{A}\bar{e_{A}}\alpha_{A})e_{j}(\sum_{B}{e_{B}}\alpha_{B})\bar{e}_{i}\big{)}\frac{\partial^{2}\varphi}{\partial
x_{j}\partial x_{i}}\rangle\\\ =&\langle\tau_{e_{0}},\sum\limits^{n}_{j\neq
i}\big{(}(\sum_{A}\bar{e_{A}}\alpha_{A})e_{i}(\sum_{B}{e_{B}}\alpha_{B})\bar{e}_{j}\big{)}\frac{\partial^{2}\varphi}{\partial
x_{i}\partial x_{j}}\rangle\\\ =&\sum\limits^{n}_{j\neq
i}\langle\tau_{e_{0}},(\sum_{A}\bar{e_{A}}\alpha_{A})e_{i}(\sum_{B}{e_{B}}\alpha_{B})\bar{e}_{j}\rangle\frac{\partial^{2}\varphi}{\partial
x_{i}\partial x_{j}}\\\ =&\sum\limits^{n}_{j\neq
i}\langle\tau_{e_{0}},(\sum_{A}\bar{e_{A}}\alpha_{A})e_{i}(\sum_{B}{e_{B}}\alpha_{B})\bar{e}_{j}\rangle\frac{\partial^{2}\varphi}{\partial
x_{i}\partial x_{j}}\\\ =&2^{n}\sum\limits^{n}_{j\neq i}\Big{(}\sum_{i\in
A,~{}j\not\in A;A-{i}=B-{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}+1-p(i)-p(j)}\\\
&+\sum_{i\not\in A,~{}j\in A;A+{i}=B+{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}+1}\\\
&+\sum_{i\in A,~{}j\in
A;A-{i}=B+{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}-h(j)-h(i)}\\\ &+\sum_{i\not\in
A,~{}j\not\in
A;A+{i}=B-{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}-h(j)-h(i)}\Big{)}\frac{\partial^{2}\varphi}{\partial
x_{i}\partial x_{j}}.\end{split}$

In all,

$\begin{split}I_{3}=&\int_{\Omega}I_{4}e^{-\varphi}dx\\\
=&\int_{\Omega}(I_{5}+I_{6}+I_{7})e^{-\varphi}dx\\\
=&-2^{n+1}\int_{\Omega}\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in
A,|A|^{2}~{}\mbox{is odd}}\alpha^{2}_{A}+\sum\limits_{i\in
A,|A|^{2}~{}\mbox{is even}}\alpha^{2}_{A})\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}e^{-\varphi}dx\\\ &+2^{n}\int_{\Omega}\sum\limits^{n}_{j\neq
i}\Big{(}\sum_{i\in A,~{}j\not\in
A;A-{i}=B-{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}+1-p(i)-p(j)}\\\ &+\sum_{i\not\in
A,~{}j\in A;A+{i}=B+{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}+1}\\\ &+\sum_{i\in
A,~{}j\in A;A-{i}=B+{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}-h(j)-h(i)}\\\
&+\sum_{i\not\in A,~{}j\not\in
A;A+{i}=B-{j}}\alpha_{A}\alpha_{B}(-1)^{r^{2}-h(j)-h(i)}\Big{)}\frac{\partial^{2}\varphi}{\partial
x_{i}\partial x_{j}}e^{-\varphi}dx.\end{split}$

Then

$\begin{split}\|\overline{D}^{*}_{\varphi}\alpha\|^{2}=\|\overline{D}\alpha\|^{2}+\int_{\Omega}|\alpha|^{2}_{0}\Delta\varphi
e^{-\varphi}dx+I_{3}.\end{split}$ (22)

If $\frac{\partial^{2}\varphi}{\partial x_{j}\partial x_{i}}=0,~{}i\neq
j,~{}1\leq i,j\leq n$ and $\frac{\partial^{2}\varphi}{\partial x^{2}_{i}}\leq
0,~{}1\leq i\leq n$, we have $I_{3}\geq 0$, and

$\|\overline{D}^{*}_{\varphi}\alpha\|^{2}\geq\int_{\Omega}|\alpha|^{2}_{0}\Delta\varphi
e^{-\varphi}dx.$

With the above analysis, we can prove Theorem 1.2 easily.

###### Proof

It is sufficient to prove the theorem if condition (3) in Theorem 1.1 is
presented. By Cauchy-Schwarz inequality in Proposition 2.6, we have for any
$\alpha\in C^{\infty}_{0}(\Omega,\mathcal{A})$ that

$\begin{split}|({f},\alpha)_{\varphi}|^{2}_{0}=&\big{|}\int_{\Omega}\bar{f}\cdot\alpha
e^{-\varphi}dx\big{|}^{2}_{0}\\\
=&~{}\big{|}\int_{\Omega}\bar{f}\cdot\frac{1}{\sqrt{\Delta\varphi}}\cdot\alpha\cdot\sqrt{\Delta\varphi}\cdot
e^{-\varphi}dx\big{|}^{2}_{0}\\\
\leq&~{}\big{\|}\bar{f}\frac{1}{\sqrt{\Delta\varphi}}\big{\|}^{2}\cdot\big{\|}\alpha\cdot\sqrt{\Delta\varphi}\big{\|}^{2}\\\
=&~{}\int_{\Omega}\big{|}\frac{\bar{f}}{\sqrt{\Delta\varphi}}\big{|}^{2}_{0}e^{-\varphi}dx\cdot\int_{\Omega}\big{|}\alpha\cdot\sqrt{\Delta\varphi}\big{|}^{2}_{0}e^{-\varphi}dx\\\
\leq&c\|\overline{D}^{*}_{\varphi}\alpha\|^{2}.\end{split}$

The proof is completed with Theorem 1.1.

It should be noticed that when $n=1$, $I_{3}=0$. Then it comes from equation
(22) that the Hörmander’s $L^{2}$ theorem in $\mathbb{R}^{2}$ could be
described which equals the classical Hörmander’s $L^{2}$ theorem in
$\mathbb{C}$.

###### Corollary 4.1

Given $\varphi\in C^{2}(\Omega,\mathbb{R})$ with $\Omega$ being an open subset
of $\mathbb{R}^{2}$; $\Delta\varphi\geq 0$. Then for all $f\in
L^{2}(\Omega,\mathcal{A},\varphi)$ with
$\int_{\Omega}\frac{|f|^{2}_{0}}{\Delta\varphi}e^{-\varphi}dx=c<\infty$, there
exists a $u\in L^{2}(\Omega,\mathcal{A},\varphi)$ such that

$\overline{D}u=f$

with

$\|u\|^{2}\leq\int_{\Omega}\frac{|f|^{2}_{0}}{\Delta\varphi}e^{-\varphi}dx.$

##  5 Conclusion

In this paper, based on the Hörmander’s $L^{2}$ theorem in complex analysis,
the Hörmander’s $L^{2}$ theorem for Dirac operator in $\mathbb{R}^{n+1}$ has
been obtained by Clifford algebra. When $n=1$, the result is equivalent to the
classical Hörmander’s $L^{2}$ theorem in complex variable. Moreover, for any
$f$ in $L^{2}$ space over a bounded domain with value in Clifford algebra,
there is a weak solution of Dirac operator with the solution in the $L^{2}$
space as well. The potential applications of the results will be studied in
our future work.

###### Acknowledgements.

This work was supported by the National Natural Science Foundations of China
(No. 11171255, 11101373) and Doctoral Program Foundation of the Ministry of
Education of China (No. 20090072110053).

## References

  * Brackx et al (1982) Brackx F, Delanghe R, Sommen F (1982) Clifford Analysis, Research Notes in Mathematics. London, Pitman
  * De Ridder et al (2012) De Ridder H, De Schepper H, Sommen F (2012) Fueter polynomials in discrete Clifford analysis. Mathematische Zeitschrift 272 (2012) :253–268.
  * Gong et al (2009) Gong Y, Leong IT, Qian T (2009) Two integral operators in Clifford analysis. Journal of Mathematical Analysis and Applications 354(2):435–444
  * Hörmander (1965) Hörmander L (1965) $l^{2}$ estimates and existence theorems for the operator. Acta Mathematica 113(1):89–152
  * Huang et al (2006) Huang S, Qiao YY, Wen GC (2006) Real and Complex Clifford Analysis, Advances in Complex Analysis and Its Applications. New York, Springer
  * Qian and Ryan (1996) Qian T, Ryan J (1996) Conformal transformations and Hardy spaces arising in Clifford analysis. Journal of Operator Theory 35(2):349–372
  * Ryan (1990) Ryan J (1990) Iterated Dirac operators in $c^{n}$. Zeitschrift für Analysis und ihre Anwendungen 9:385–401
  * Ryan (1995) Ryan J (1995) Cauchy-Green type formulae in Clifford analysis. Transactions of the American Mathematical Society 347(4):1331–1342
  * Ryan (2000) Ryan J (2000) Basic Clifford analysis. Cubo Matemática Educacional 2:226–256


Copyright for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0).

Proceedings of the Demonstration & Resources Track, Best BPM Dissertation
Award, and Doctoral Consortium at BPM 2021 co-located with the 19th
International Conference on Business Process Management, BPM 2021, Rome,
Italy, September 6-10, 2021

[orcid=0000-0001-7161-6927<EMAIL_ADDRESS>]
[orcid=0000-0002-8223-6097<EMAIL_ADDRESS>]

[orcid=0000-0002-0955-6940<EMAIL_ADDRESS>]

# PC4PM: A Tool for Privacy/Confidentiality Preservation in Process Mining

Majid Rafiei  Alexander Schnitzler  Wil M.P. van der Aalst  Chair of Process
and Data Science, RWTH Aachen University, Aachen, Germany

(2021)

###### Abstract

Process mining enables business owners to discover and analyze their actual
processes using event data that are widely available in information systems.
Event data contain detailed information which is incredibly valuable for
providing insights. However, such detailed data often include highly
confidential and private information. Thus, concerns of privacy and
confidentiality in process mining are becoming increasingly relevant and new
techniques are being introduced. To make the techniques easily accessible, new
tools need to be developed to integrate the introduced techniques and direct
users to appropriate solutions based on their needs. In this paper, we present
a Python-based infrastructure implementing and integrating state-of-the-art
privacy/confidentiality preservation techniques in process mining. Our tool
provides an easy-to-use web-based user interface for privacy-preserving data
publishing, risk analysis, and data utility analysis. The tool also provides a
set of anonymization operations that can be utilized to support
privacy/confidentiality preservation. The tool manages both standard XES event
logs and non-standard event data. We also store and manage privacy metadata to
track the changes made by privacy/confidentiality preservation techniques.

###### keywords:

process mining privacy preservation confidentiality event data

##  1 Introduction

Process mining techniques employ event logs to provide insights into actual
processes [1]. Event logs contain detailed information about operational
processes and can be extracted from various types of information systems,
e.g., EPR systems. Events are considered as the smallest units of process
execution which are distinguished by their attributes. The main attributes of
events are as follows: activity, case id, timestamp, and resource. For
instance, a heart surgery (activity) performed by Dr. John (resource) for a
patient with id=10 (case id) at timestamp 2021.06.10-10:00:00 is an event
recorded by an information system in a hospital. The attributes that directly
or indirectly refer to individuals raise privacy concerns. For example, in the
healthcare context, the case id attribute may refer to the patients whose data
are processed, and the resource attribute may refer to the employees who
perform activities for the patients, e.g., nurses. Furthermore, other
attributes can also be considered as confidential information, e.g., the
activity attribute may contain a confidential activity name that must not be
exposed. Respect for privacy when analyzing personal data is also dictated by
regulations, e.g., the European General Data Protection Regulation
(GDPR)111http://data.europa.eu/eli/reg/2016/679/oj. Such legitimate and
ethical requirements have recently resulted in more attention to privacy and
confidentiality issues in process mining [2, 3, 4, 5]. Some tools have also
been introduced to provide specific privacy/confidentiality requirements [6,
7, 8].

Figure 1: The general overview of privacy-related activities in process
mining.

Figure 1 shows the general overview of privacy-related activities in process
mining including Privacy-Preserving Data Publishing (PPDP), Privacy-Preserving
Process Mining (PPPM), and Privacy Analysis (PrAn). PPDP tries to obscure the
identity and/or sensitive data of individuals to preserve their privacy. PPDP
techniques often apply one or more anonymization operations, e.g.,
suppression, generalization, etc., to provide the desired privacy
requirements. PPPM intends to expand existing process mining algorithms to
cope with intermediate results, so-called abstractions [9], generated by some
PPDP techniques. Note that PPPM algorithms are closely linked with the
corresponding PPDP approaches, and PPPM may refer to the entire privatization
process, starting with an event log and finishing with process mining
findings. PrAn, indicated with dashed lines in Figure 1, includes two types of
activities: risk analysis and utility analysis. Both PrAn activities could be
done for data and results. In this paper, we introduce a tool, named PC4PM,
mainly focusing on the activities indicated by the check-boxes in Figure 1.
PC4PM is the successor of the privacy tool introduced in [7], and it offers
new privacy preservation techniques, privacy analysis, a set of anonymization
operations, and user guidance that directs users to the right techniques based
on their requirements. In the rest of the paper, we demonstrate the
functionality and characteristics of PC4PM. We also describe the maturity and
availability of the tool.

##  2 Functionality and Characteristics

PC4PM is implemented in Python using Django
framework222https://www.djangoproject.com/. Figure 2 shows a high-level view
of the architecture. PC4PM includes eight main Django applications and each
application provides at least one main privacy-related activity implemented as
a Python package. The Django templates, accessible from any web browser,
provide a web interface for the applications. Implementing each technique as
an independent Django application enables users to simultaneously run
different techniques on event logs. Such architecture makes the process of
maintenance and integration simple. To integrate new techniques, one can
create a Python package and integrate it as an independent application.
Moreover, Python packages can independently be imported into other Python-
based tools.

Figure 2: The architecture of PC4PM.

Figure 3 shows the home page of PC4PM. The left menu shows the main Django
applications including event data management, privacy-aware role mining,
connector method, TLKC-privacy, TLKC-privacy extended, anonymization
operations, PRIPEL, and privacy analysis. The event data management
application manages both standard XES event logs and non-standard event data,
called event log abstraction [9]. The privacy-aware role mining application
implements the decomposition method, proposed in [10], to discover roles from
event logs while preserving privacy. This method perturbs the frequency of
activities in an event log to eliminate frequency-based attacks. The connector
method is an encryption-based method for securely discovering directly follows
graphs from event logs. This method breaks down traces into a collection of
directly follows relations to prevent linkage attacks.

Figure 3: The home page of PC4PM.

The TLKC-privacy application implements the TLKC-privacy model providing
group-based privacy guarantees for process discovery and performance analysis.
The TLKC-privacy extended application extends the TLKC-privacy model and
considers all the main perspectives of process mining [3]. The anonymization
operation application, implements all the main anonymization operations
proposed in [9] including suppression, addition, substitution, condensation,
swapping, generalization, and cryptography. The PRIPEL application presents
the PRIPEL method [2] which applies the notion of differential privacy to
provide privacy guarantees for event logs. The privacy analysis application
includes three components for analyzing disclosure risks, data utility [11],
and FCB-anonymity [12].

PC4PM supports users with a four-step user guide to help them choose the right
technique(s) based on their needs. The user guidance works based on a four-
dimension signature assigned to each technique. The signature reflects the
following aspects: process mining perspective (PMPS), process mining activity
(PMAC), privacy perspective (PRPS), and privacy activity (PRAC). PMPS
indicates the process mining perspective that a privacy technique focuses on,
e.g., control-flow. PMAC shows the process mining activity, e.g, process
discovery, for which the utility of event data is preserved. PRPS shows the
privacy perspective of a privacy technique, i.e., resource or case. PRAC
indicates the main privacy-related activity of a privacy preservation
technique, i.e., PPDP, PPPM, or PrAn. Moreover, PC4PM helps users with help
tooltips provided for the parameters used by techniques. PC4PM also inherits
all the characteristics of its predecessor [7]. Some of those are as follows:
(1) Each Django application provides the results in an independent output
section, (2) It enables a cycle of privacy/confidentiality preservation
techniques such that the results from one technique can be added to the event
data repository and used as an input for other techniques, (3) The privacy
metadata [9] which specify the order and type of the main anonymization
operations are added to anonymized event logs.

##  3 Availability and Maturity

The source code, a screencast, a user manual, and all other resources are
available in our GitHub repository: https://github.com/m4jidRafiei/PC4PM. Each
privacy/confidentiality Python package is linked to a separate GitHub project.
The main GitHub project contains links to all those projects. In the
corresponding GitHub project of each privacy/confidentiality Python package,
one can find the name of the Python package, the link to the main paper, and a
sample source code that shows the usage. In terms of performance and time
complexity, each privacy preservation technique which is linked to a Django
application behaves differently w.r.t. the size of the input event log. Based
on our experiments, the applications are able to handle real-world event logs,
e.g., BPI challenge datasets: https://data.4tu.nl/. Moreover, all the
complicated and time-consuming functions, developed in the Python packages,
have a parameter to be run using multi-processing which is enabled by default.
In this case, the input event log is divided into smaller pieces w.r.t. the
cores of the processor hosting PC4PM. PC4PM is provided as a Docker container
that can simply be hosted by users: https://hub.docker.com/r/m4jid/pc4pm. The
Docker usage is also explained in the GitHub repository.

##  4 Conclusion

In this paper, we introduced a tool for publishing event data w.r.t. privacy
concerns. Our web-based tool is mainly focused on privacy-/confidentiality-
preserving data publishing and privacy analysis considering both data utility
and disclosure risk analyses. PC4PM can be considered as a sanitizer that
provides sanitized event logs that can be used by any process mining tool. The
architecture has been designed in such a way that other privacy preservation
techniques can easily be integrated, e.g., we integrated PRIPEL as an external
library. The goal of PC4PM is to provide a comprehensive set of techniques
that can cover all the aspects of privacy-related activities for different
perspectives of process mining. We invite other researchers to integrate their
solutions as independent applications into the provided framework.

###### Acknowledgements.

Funded under the Excellence Strategy of the Federal Government and the Länder.
We also thank the Alexander von Humboldt (AvH) Stiftung for supporting our
research.

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