\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 96, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/96\hfil Singular integrals]
{Singular integrals of the time-harmonic relativistic Dirac
equation on a piecewise Liapunov surface}
\author[B. Schneider\hfil EJDE-2005/96\hfilneg]
{Baruch Schneider}  

\address{Baruch Schneider \hfill\break
Department of Mathematics\\
Izmir University of Economics\\
35330 Izmir, Turkey}
\email{baruch.schneider@ieu.edu.tr}

\date{}
\thanks{Submitted May 26, 2005. Published September 4, 2005.}
\subjclass[2000]{81Q05, 30E20, 35Q40}
\keywords{Relativistic Dirac equation; Cauchy-type integral;
\hfill\break\indent quaternionic analysis}

\begin{abstract}
 We give a short proof of a formula of Poincar\'e-Bertrand in the
 setting of time-harmonic solutions of the relativistic Dirac
 equation on a piecewise Liapunov surface, as well as for some
 versions of quaternionic analysis.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]

\section{Introduction}

Let $\Gamma$ be a closed Liapunov curve in the complex plane and
let $f$ be a H\"older function on $\Gamma\times\Gamma$. Then,
everywhere on $\Gamma$,
\begin{equation}\label{pb}
\begin{aligned}
&\frac{1}{\pi i}\int_{\Gamma_\tau}\frac{d\tau}{\tau - t}\cdot
\frac{1}{\pi
i}\int_{\Gamma_{\tau_1}}\frac{f(\tau,\tau_1)d\tau_1}{\tau_1-\tau}\\
&= f(t,t) + \frac{1}{\pi i}\int_{\Gamma_{\tau_1}} d\tau_1\cdot
\frac{1}{\pi i}\int_{\Gamma_\tau}\frac{f(\tau,\tau_1)d\tau}{(\tau
- t)(\tau_1-\tau)},
\end{aligned}
\end{equation}
which is usually called the Poincar\'e-Bertrand formula, the
integrals being understood in the sense of the Cauchy principal
value. The Poincar\'e-Bertrand formula plays a significant role in
the theory of one-dimensional singular integral equations with the
Cauhy kernel and its numerous applications. Indeed, all the
integrals in \eqref{pb} contain the (singular) Cauchy kernel, and
its importance for one-dimensional complex analysis is obvious.

It is known that the theory of solutions of the Dirac equation
reduces, in some degenerate cases, to that of complex holomorphic
functions. Hence, one may consider the former to be a
generalization of the latter. At the same time, not many facts
from the holomorphic function theory have their extensions onto
the Dirac equation theory. In the present paper we study a number
of generalization of (\ref{pb}). In realizing this study we follow
the approach first presented in \cite{krsh1:gnus} and developed in
\cite{ksh4:gnus}, \cite{rosh1:gnus}, \cite{schneider:gnus} which
are based on the intimate relation between time-harmonic bispinor
fields and quaternion-valued $\alpha$-hyperholomorphic functions,
see the book \cite{ksh4:gnus}. This approach proved to be quite
efficient and heuristic since it allows the exploitation of
profound similarity between holomorphic functions in one variable
and $\alpha$-hyperholomorphic functions.

The paper is organized as follows. In Section 2 the reader can
find  the Poincar\'e-Bertrand formula for time-harmonic Dirac
bispinors, i.e, time-harmonic solutions of the relativistic Dirac
equation. The proof can be found in the Section 6, and
is based on the contents of Sections 3-5. In section 4 we present
the Poincar\'e-Bertrand formula for $\alpha$-hyperholomorphic
quaternionic function theory on a piece-wise Liapunov surface.

Note that the Poincar\'e-Bertrand formula on closed piece-wise
smooth  manifold in $\mathbb{C}^n$ for Bochner-Martinelli type
singular integrals was studied, for example, by Liangyu Lin and
Chunhui Qiu \cite{lin:gnus}.

\section{Time-harmonic bispinor fields theory and the Cauchy-Dirac integral}

Let $\Omega$ be a domain in $\mathbb{R}^3$,
$\Gamma:=\partial\Omega$ be its boundary. We consider the
following {\it Dirac equation} for a free massive particle of spin
$\frac{1}{2}$:
\[
\mathbb D[\Phi] := \Bigl ( \gamma_0\partial_t -
\sum^3_{k=1}\gamma_k\partial_k + im \Bigr )[\Phi] = 0,
\]
where the Dirac matrices have the standard Dirac-Pauli form
\begin{gather*}
 \gamma_0 := \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 &-1 & 0 \\
0 & 0 & 0 &-1
\end{pmatrix},\quad
\gamma_1 := \begin{pmatrix}
0 & 0 & 0 &-1 \\
0 & 0 &-1 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0
\end{pmatrix}, \\
\gamma_2 := \begin{pmatrix}
0 & 0 & 0 & i \\
0 & 0 &-i & 0 \\
0 &-i & 0 & 0 \\
i & 0 & 0 & 0
\end{pmatrix},\quad
\gamma_3 :=  \begin{pmatrix}
0 & 0 & -1 & 0 \\
0 & 0 &  0 & 1 \\
1 & 0 &  0 & 0 \\
0 &-1 &  0 & 0
\end{pmatrix},
\end{gather*}
and where $\partial_t := \frac{\partial}{\partial t}$;
$\partial_k:= \frac{\partial}{\partial x_k}$, $m\in\mathbb{R}$,
$\Phi : \mathbb{R}^4\rightarrow \mathbb{C}^4$. Suppose that the
spinor field $\Phi$ is time-harmonic (= monochromatic):
\[ %\label{Lee1}
\Phi(t,x)=q(x)e^{i\omega t},
\]
where $\omega\in\mathbb{R}$ is the frequency and $q :
\Omega\subset\mathbb{R}^3\to \mathbb{C}^4$ is the amplitude. Then
the relativistic Dirac equation is equivalent to the
\emph{time-harmonic Dirac equation}:
\[ %\label{Lee2}
\mathbb D_{\omega,m}[q] := \Bigl (i\omega\gamma_0 -
\sum^3_{k=1}\gamma_k\partial_k + im\Bigr)[q] = 0.
\]
This is the equation which we are going to consider. We shall
consider certain objects related to it in a bounded domain.
Physical phenomena which gave rise to the Dirac equation occur
usually in unbounded domains but some of them (the Casimir effect,
for instance) take place in bounded domains also. For more details
see, e.g. \cite{ksh4:gnus}.

The integral
\[
K_{\mathbb{D}_{\omega,m}}[g](x) := -\int_{\Gamma}
\check{\mathcal{K}}^x_{\mathbb{D}_{\omega,m}} [\sigma_{\mathbb{
D}_{\omega,m}}g(\tau)],\quad x\notin\Gamma,
\]
plays the role of the Cauchy-type integral in the
theory of time-harmonic bispinor fields with $g : \Gamma\rightarrow
\mathbb{C}^4$(see \cite{rosh1:gnus}) and we shall call it the Cauchy-Dirac-type integral, where
\[
\sigma_{\mathbb{D}_{\omega,m}} := \frac{1}{2}
\begin{pmatrix}
(n_2-in_1) & in_3 & in_3 & (n_2+in_1) \\
-n_3 & i(n_2+in_1) & i(n_2-in_1) & -n_3 \\
-in_3 & -(n_2+in_1) & (n_2-in_1) & in_3 \\
-i(n_2-in_1) & n_3 & -n_3 & i(n_2+in_1)
\end{pmatrix} dS,
\]
$\vec n(\tau) = (n_1(\tau),n_2(\tau),n_3(\tau))$ is an outward pointing
normal unit vector on $\Gamma$ at $\tau\in\Gamma$, and $dS$ is an element
of the surface area in $\mathbb{R}^3$. The explicit form of time-harmonic
relativistic Cauchy-Dirac kernel $\check{\mathcal{K}}^x_{\mathbb{D}_{\omega,m}}$
can be seen, e.g. in reference \cite{rosh1:gnus}.

Let $H_{\mu}(\Gamma, \mathbb{C}^4):=\{ f\in\mathbb{C}^4 :
|f(t_1) - f(t_2)|\le L_f\cdot {|t_1-t_2|}^{\mu};\, \forall
\{t_1,t_2 \}\subset\Gamma, L_f = \text{const} \}$ denote the class of
functions satisfying the H\"older condition  with the exponent
$0<\mu\le 1$. Here $|f|$ means the Euclidean norm in $\mathbb{
C}^4=\mathbb{R}^8$ while $|t|$ is the Euclidean norm in
$\mathbb{R}^3$. Let $\Gamma$ be a surface in $\mathbb R^3$ which contains a
finite number of conical points and a finite number of
non-intersecting edges such that none of the edges contains any of
conical points. If the complement (in $\Gamma$) of the union of
conical points and edges, is a Liapunov surface, then we shall
refer to $\Gamma$ as a piece-wise Liapunov surface in $\mathbb
R^3$.

\begin{theorem}[Poincar\'e-Bertrand formula for time-harmonic bispinor field theory on a
piece-wise Liapunov surface] \label{thm2.3}
 Let $\Omega$ be a bounded domain in $\mathbb{R}^3$ with the piece-wise
 Liapunov boundary. Let $q\in H_{\mu}(\Gamma\times\Gamma,\mathbb{C}^4)$,
$0<\mu< 1$. Then the following equality holds, everywhere on $\Gamma$:
\begin{equation}\label{Jim}
\begin{aligned}
&\int_{\Gamma_{\tau_1}}\int_{\Gamma_\tau} \check{\mathcal{
K}}^{t}_{\mathbb{D}_{\omega,m}}\Bigl [ \sigma_{\mathbb{
D}_{\omega,m,\tau}} \check{\mathcal{K}}^{\tau}_{\mathbb{D}_{\omega,m}}[
\sigma_{\mathbb{D}_{\omega,m,\tau_1}} q(\tau_1,\tau)]\Bigr ] +
\frac{1-\gamma(t)}{2} q(t,t) \\
&=\int_{\Gamma_\tau} \check{\mathcal{K}}^{t}_{\mathbb{
D}_{\omega,m}}\Bigl [ \sigma_{\mathbb{D}_{\omega,m,\tau}}
\int_{\Gamma_{\tau_1}} \check{\mathcal{
K}}^{\tau}_{\mathbb{D}_{\omega,m}}[ \sigma_{\mathbb{
D}_{\omega,m,\tau_1}} q(\tau_1,\tau)]\Bigr ],
\end{aligned}
\end{equation}
where the integrals being understood in the sense of the Cauchy principal value,
 $\gamma(t):=\frac{\eta(t)}{4\pi}$; $\eta(t)$ is the measure of a solid
angle of the tangential conical surface at the point $t$
or is the solid measure of the tangential dihedral angle at the point $t$.
\end{theorem}

The proof will be presented in Section 6. Note that if $\Gamma$ is a Liapunov
surface, then formula (\ref{Jim}) coincides with the result in paper \cite{rosh1:gnus}.

\section{Basic facts of hyperholomorphic function theory}
\setcounter{equation}{0}

In this section, we provide some background on quaternionic analysis needed
in this paper. For more information, we refer the
reader to \cite{gs1:gnus}, \cite{ksh4:gnus}.

 Let $\mathbb H(\mathbb{C})$ be the set of
complex quaternions, it means that each quaternion $a$ is represented in
the form $a=\sum^3_{k=0} a_ki_k$, with the standard basis
$\{i_0:=1, i_1, i_2, i_3\}$, where
$\{a_k : k\in\mathbb N^0_3 := \mathbb N_3\cup\{0\};\ \mathbb N_3 :=
\{1,2,3\}\} \subset\mathbb{C}$. We use the Euclidean norm $|a|$ in $\mathbb H(\mathbb{C})$, defined by
$|a| := \sqrt{\sum^3_{k=0} |a_k|^2}$.

 Let $\lambda\in\mathbb H(\mathbb{C})\setminus\{0\}$, and let $\alpha$ be its
complex-quaternionic square root:  $\alpha\in\mathbb H(\mathbb{C})$,
$\alpha^2 = \lambda$. The function
$f: \Omega\subset\mathbb R^3\to\mathbb H(\mathbb{C})$ is called
{\it left-$\alpha$-hyperholomorphic} if
\[
D_{\alpha}f := f\alpha + i_1\frac{\partial}{\partial x_1} f + i_2\frac{\partial}{\partial x_2}f + i_3\frac{\partial}{\partial x_3}f = 0.
\]
Let $\alpha\in\mathbb H(\mathbb{C})$ and let $\theta_\alpha$ be the fundamental
solution of the Helmholtz operator $\Delta_\lambda := \Delta + I\lambda$,
where $\Delta := \sum_{k=1}^3 \frac{\partial^2}{\partial x_k^2}$ and $I$ is the identity operator. Then the fundamental
solution of the operator
$D_{\alpha}$, $\mathcal{K}_{\alpha}$, is given by the
formula (see \cite{ksh4:gnus}):
\[
\mathcal{K}_{\alpha}(x) := -D_\alpha\theta_\alpha(x),
\]
and its explicit form can be seen, e.g., in \cite{schsh1:gnus}.
We shall use the notation
$C^p(\Omega,\mathbb H(\mathbb{C}))$, $p\in\mathbb N\cup\{0\}$ which has the
usual component-wise meaning.
Denote by ${\mathfrak G}$ the set of zero
divisors from $\mathbb H (\mathbb{C})$, i.e.,
${\mathfrak G} := \{ a\in\mathbb H(\mathbb{C})\ | \ a\not= 0; \
\exists b\not= 0 : \ ab = 0 \}$.
Let $\sigma_{\tau} = \sum_{k=1}^3 (-1)^{k-1}i_k {dx}_{[k]}$,
where ${dx}_{[k]}$ denotes as usual the differential form
${dx}_1\wedge {dx}_2\wedge {dx}_3$ with the factor ${dx}_k$
omitted.
Let $\Omega=\Omega^+$ be a domain in $\mathbb R^3$ with the boundary $\Gamma$
which is assumed to be a piece-wise Liapunov surface; denote
$\Omega^- := \mathbb R^3\setminus ({\Omega}^+\cup\Gamma)$. If $f$ is a
H\"older function then its $\alpha$-hyperholomorphic left
Cauchy-type integral is defined (see \cite[Subsection 4.16]{ksh4:gnus}):
\[
K_{\alpha}[f](x):=  -\int_{\Gamma} \check{\mathcal{K}}^x_{\alpha}
[\sigma_{\tau}f(\tau)],\ x\in\Omega^{\pm},
\]
where
\begin{enumerate}
\item If $\alpha=\alpha_0\in\mathbb{C}$, then
\[
\check{\mathcal{K}}^x_{\alpha}[f](\tau):= \mathcal{K}_{\alpha_0}(x-\tau)f(\tau).
\]
\item If $\alpha\notin{\mathfrak G}$, \ ${\vec {\alpha}}^2\neq 0$, then
\begin{equation}\label{1}
\check{\mathcal{K}}^x_{\alpha}[f](\tau):= \frac{1}{2\sqrt{{\vec\alpha}^2}}\mathcal{K}_{{\xi}_+}(x)f(\tau)(\sqrt{{\vec\alpha}^2} + \vec\alpha) +
\frac{1}{2\sqrt{{\vec\alpha}^2}}\mathcal{K}_{{\xi}_-}(x)f(\tau)(\sqrt{{\vec\alpha}^2} - \vec\alpha).
\end{equation}
\item If $\alpha\notin{\mathfrak G}, \ {\vec {\alpha}}^2 = 0$, then
\begin{equation}\label{2}
\check{\mathcal{K}}^x_{\alpha}[f](\tau) := \mathcal{K}_{{\alpha}_0}(x)f(\tau) + \frac{\partial}{\partial {\alpha_0}}[\mathcal{K}_{\alpha_0}](x)f(\tau)\vec {\alpha}.
\end{equation}
\item If $\alpha\in{\mathfrak G}, \ \alpha_0\neq 0$, then
\begin{equation}\label{3}
\check{\mathcal{K}}^x_{\alpha}[f](\tau):= \frac{1}{2\alpha_0}\mathcal{K}_{2\alpha_0}(x)f(\tau)\alpha + \frac{1}{2\alpha_0}\mathcal{K}_{0}(x)f(\tau)\overline{\alpha}.
\end{equation}
\item If $\alpha\in{\mathfrak G},\ \alpha_0 = 0$, then
\begin{equation}\label{4}
\check{\mathcal{K}}^x_{\alpha}[f](\tau):= \mathcal{K}_{0}(x)f(\tau) + \theta_0(x)f(\tau)\alpha.
\end{equation}
\end{enumerate}
For more information about $\alpha$-hyperholomorphic functions, we refer
the reader to \cite{gs1:gnus}, \cite{ksh4:gnus}, \cite{roshsom:gnus}.

\section{The Poincar\'e-Bertrand formula for $\alpha$-hyperholomorphic function theory on a piece-wise Liapunov surface}

\begin{theorem}[Poincar\'e-Bertrand formula for $\alpha$-hyperholomorphic
function theory on a piece-wise Liapunov surface] \label{thm4.1}
Let $\Omega$ be a bounded domain in $\mathbb R^3$ with piece-wise Liapunov boundary and let $f\in H_\mu(\Gamma\times\Gamma,\mathbb H(\mathbb{C}))$.
Then the following equality holds everywhere on $\Gamma$:
\begin{equation}\label{chernij}
\begin{aligned}
&\int_{\Gamma_{\tau_1}}\int_{\Gamma_\tau} \check{\mathcal{
K}}^t_{\alpha}[\sigma_{\tau}\check{\mathcal{K}}^\tau_{\alpha}
[\sigma_{\tau_1}f(\tau_1,\tau)]] + \frac{1-\gamma(t)}{2}f(t,t) \\
&=
\int_{\Gamma_\tau} \check{\mathcal{K}}^t_{\alpha} \Bigl [
\sigma_{\tau} \int_{\Gamma_{\tau_1}} \check{\mathcal{
K}}^\tau_{\alpha}[\sigma_{\tau_1}f(\tau_1,\tau)] \Bigr ].
\end{aligned}
\end{equation}
\end{theorem}

\begin{proof}
Let $\Gamma$ be as above and $f\in H_{\mu}(\Gamma\times\Gamma,
\mathbb H(\mathbb{C}))$.
We begin with $\alpha=\alpha_0\in\mathbb{C}$. Then the formulas
(\ref{chernij}) takes the form
\begin{align*} %\label{chernij1}
&\int_{\Gamma_{\tau_1}}\int_{\Gamma_\tau} \mathcal{K}_{\alpha_0}(t-\tau)\sigma_{\tau}
\mathcal{K}_{\alpha_0}(\tau-\tau_1)\sigma_{\tau_1}f(\tau_1,\tau)
+ \frac{1-\gamma(t)}{2}f(t,t) \\
&=\int_{\Gamma_\tau}\int_{\Gamma_{\tau_1}} \mathcal{K}_{\alpha_0}(t-\tau)\sigma_{\tau}
\mathcal{K}_{\alpha_0}(\tau-\tau_1)\sigma_{\tau_1}f(\tau_1,\tau),
\end{align*}
and it was proved in \cite{baruch:gnus}. Thus the case $\alpha=\alpha_0\in\mathbb{C}$ is covered. For other possible
situations, the argument is similar to the proof of
 \cite[Theorem 3.1]{rosh1:gnus}
\end{proof}

\section{Function theory for the quaternionic Dirac operator}

 We start this Section with a brief description of
the relations between the time-harmonic spinor fields theory and
the theory of $\alpha$-hyperholomorphic functions. One can find
more about this in \cite{ksh4:gnus}, \cite{rosh1:gnus}.
The standard Dirac matrices have the well-known properties:
\begin{gather*}
\gamma^2_0=E_4,\quad \gamma^2_k=-E_4,\quad  k\in\mathbb{N}_3 :=\{1,2,3\}, \\
\gamma_j\gamma_k+\gamma_k\gamma_j=0,\quad
 j,k\in\mathbb{N}^0_3:=\mathbb{N}_3\cup\{0\},\quad j\neq k,
\end{gather*}
where $E_4$ is the $4\times 4$ identity matrix.
The products of the Dirac matrices
\[
\hat{i}_0:=E_4,\quad \hat{i}_1:=\gamma_3\gamma_2,\quad
 \hat{i}_2:=\gamma_1\gamma_3,\quad \hat{i}_3:=\gamma_1\gamma_2,
\quad \hat{i}:=\gamma_0\gamma_1\gamma_2\gamma_3,
\]
have the following properties:
\begin{gather*}
\hat{i}_0^2=\hat{i}_0=-\hat{i}_k^2,\quad
\hat{i}_0\hat{i}_k=\hat{i}_k\hat{i}_0=\hat{i}_k,\quad k\in\mathbb{N}_3,
\\
\hat{i}_1\hat{i}_2=-\hat{i}_2\hat{i}_1=\hat{i}_3,\quad
\hat{i}_2\hat{i}_3=-\hat{i}_3\hat{i}_2=\hat{i}_1,\quad
\hat{i}_3\hat{i}_1=-\hat{i}_1\hat{i}_3=\hat{i}_2,
\\
\hat{i}\cdot\hat{i}_k=\hat{i}_k\cdot\hat{i},\quad k\in\mathbb{N}^0_3.
\end{gather*}
For $b\in\mathbb H(\mathbb{C})$, set
\[
B_l(b) := \begin{pmatrix}
b_0 & -b_1 & -b_2 & -b_3 \\
b_1 & b_0 & -b_3 & b_2 \\
b_2 & b_3 & b_0 & -b_1 \\
b_3 & -b_2 & b_1 & b_0
\end{pmatrix}.
\]
Matrix subalgebra $\mathcal{B}_l(\mathbb{C}) := \{ B_l(b) : b\in\mathbb H(\mathbb{C})\}$
and $\mathbb H(\mathbb{C})$ are isomorphic as complex algebras. Abusing a little
we shall not distinguish, sometimes, between $B_l(b)$, the column
$\begin{pmatrix}
b_0 \\
b_1 \\
b_2 \\
b_3
\end{pmatrix}$ and the quaternion $b$.
Set
\[
\mathcal{D}:=i\omega\gamma_0-E_4\partial_1-\gamma_1\partial_2-\gamma_3\partial_3
+im.
\]
We shall consider $\mathcal{D}$ on the set $C^1(\Omega,\mathcal{B}_l(\mathbb{C}))$ of
corresponding matrices. Hence for us
\[
\mathcal{D} : C^1(\Omega,\mathcal{B}_l(\mathbb{C}))\to
C^0(\Omega,\mathcal{B}_l(\mathbb{C})).
\]
In \cite[Section 12]{ksh4:gnus} (see also \cite[page 7563]{kr:gnus})
there was introduced the map $\mathcal{UA}$ which
transforms a function $q: \tilde\Omega\subset\mathbb R^3\to\mathbb{C}^4$
into the function $\rho: \Omega\subset\mathbb R^3\to\mathbb H(\mathbb{C})$
by the rule:
\[
\rho = \mathcal{UA}[q]:= \frac{1}{2}[-(\tilde{q}_1-\tilde{q}_2)i_0
+i(\tilde{q}_0-\tilde{q}_3)i_1 - (\tilde{q}_0+\tilde{q}_3)i_2
+i(\tilde{q}_1+\tilde{q}_2)i_3],
\]
where $\tilde q(x):=q(x_1,x_2,-x_3)$, the domain $\tilde\Omega$ is obtained
from $\Omega\subset\mathbb R^3$ by the reflection $x_3\to -x_3$.
The corresponding  inverse transform is given as follows:
\[
(\mathcal{UA})^{-1}[\rho] = \mathcal{A}^{-1}\mathcal{U}^{-1}[\rho]
:= (-i{\tilde\rho}_1-{\tilde\rho}_2,-{\tilde\rho}_0-i{\tilde\rho}_3,
{\tilde\rho}_0-i{\rho}_3,i{\tilde\rho}_1-{\tilde\rho}_2).
\]
The maps $\mathcal{UA}$ and $(\mathcal{UA})^{-1}$ may be represented in
a matrix form (see \cite[Subsection 12.13]{ksh4:gnus}):
\begin{gather*}
\rho = \mathcal{UA}[q] :=\frac{1}{2}  \begin{pmatrix}
 0 & -1 & 1 &  0 \\
 i &  0 & 0 & -i \\
-1 &  0 & 0 & -1 \\
 0 &  i & i &  0
\end{pmatrix} \begin{pmatrix}
\tilde{q}_0 \\
\tilde{q}_1 \\
\tilde{q}_2 \\
\tilde{q}_3
\end{pmatrix}, \\
q = (\mathcal{UA})^{-1}[\rho] :=\begin{pmatrix}
 0 & -i & -1 &  0 \\
-1 &  0 &  0 & -i \\
 1 &  0 &  0 & -i \\
 0 &  i & -1 &  0
\end{pmatrix}
\begin{pmatrix}
{\tilde\rho}_0 \\
{\tilde\rho}_1 \\
{\tilde\rho}_2 \\
{\tilde\rho}_3
\end{pmatrix}.
\end{gather*}
Direct computation leads to the equality
\begin{equation}\label{D1}
\mathbb D_{\omega,m}=-\gamma_0\hat{i}(\mathcal{UA})^{-1}\mathcal{D}\hat{i}_2(\mathcal{UA}),
\end{equation}
on $C^1(\Omega,\mathbb{C}^4)$. Also we get
$\mathcal{D}\hat{i}_2 = D_{\alpha}$,
on $\mathcal{B}_l(\mathbb{C})$,
where $\alpha := -(i\omega i_1+mi_2)$. By these reasons $\mathcal{D}$
is termed ``the quaternionic relativistic Dirac operator". Thus,
\[
\ker \mathcal{D} =\begin{pmatrix}
 0 & 1 & 0 & 0 \\
-1 & 0 & 0 & 0 \\
 0 & 0 & 0 & 1 \\
 0 & 0 &-1 & 0
\end{pmatrix} \ker D_{\alpha}.
\]
There exists a one-to-one correspondence between elements of
$\ker \mathcal{D}$ (which are matrices) and matrices of the form
$B_l(q)$, with $q=q_0i_0+q_1i_1+q_2i_2+q_3i_3$ being
$\alpha$-hyperholomorphic function.

The ``quaternionic relativistic Cauchy-Dirac kernel", i.e., the fundamental
solution of $\mathcal{D}$, is given by
\[ %\label{qrdck}
\mathcal{K}_{\mathcal{D},\alpha} := \hat{i}_2\mathcal{K}_{\alpha}.
\]
The integral
\[
K_{\mathcal{D},\alpha}[f](x) := -\int_\Gamma
\check{\mathcal{K}}^x_{\mathcal{D},\alpha}[\sigma_{\mathcal{D},
\tau}f(\tau)],\ x\in\Omega^{\pm},
\]
plays the role of the Cauchy-type integral, the one with the
quaternionic relativistic Cauchy-Dirac kernel (see
\cite{rosh1:gnus}, \cite{ksh4:gnus}); with $f: \Gamma\to \mathcal{
B}_l(\mathbb{C})$ and
\[
\sigma_{\mathcal{D},\tau} :=
 \begin{pmatrix}
-n_1(\tau) & 0 & n_3(\tau) & n_2(\tau) \\
0 & -n_1(\tau) & n_2(\tau) & -n_3(\tau) \\
-n_3(\tau) & -n_2(\tau) & -n_1(\tau) & 0 \\
-n_2(\tau) & n_3(\tau) & 0 & -n_1(\tau)
\end{pmatrix}dS.
\]
We shall call also $K_{\mathcal{D},\alpha}[f]$ the quaternionic
relativistic Cauchy-Dirac-type integral.

\begin{theorem}[Poincar\'e-Bertrand formula for the quaternionic
relativistic Cauchy\-Dirac integral on a piece-wise Liapunov surface]
\label{thm5.3}
Let $\Omega$ be a bounded domain in $\mathbb R^3$ with the piece-wise
Liapunov boundary and let
$f\in H_{\mu}(\Gamma\times\Gamma,\mathcal{B}_l(\mathbb{C})), 0<\mu<1$.
The following equality holds everywhere on $\Gamma$:
\begin{equation}\label{may}
\begin{aligned}
& \int_{\Gamma_{\tau_1}}\int_{\Gamma_\tau}\check{\mathcal{K}}^t_{\mathcal{D},\alpha}[\sigma_{\mathcal{D},\tau}
\check{\mathcal{K}}^\tau_{\mathcal{D},\alpha}[\sigma_{\mathcal{D},\tau_1}
f(\tau_1,\tau)]] + \frac{1-\gamma(t)}{2}f(t,t)  \\
& = \int_{\Gamma_\tau}\check{\mathcal{K}}^t_{\mathcal{
D},\alpha}\Bigl[\sigma_{\mathcal{D},\tau}
\int_{\Gamma_{\tau_1}}\check{\mathcal{K}}^\tau_{\mathcal{
D},\alpha}[\sigma_{\mathcal{D},\tau_1}f(\tau_1,\tau)]\Bigr].
\end{aligned}
\end{equation}
\end{theorem}

\begin{proof}
Let $f\in H_{\mu}(\Gamma\times\Gamma,\mathcal{B}_l(\mathbb{C}))$, consider $\check{\mathcal{K}}^x_{\mathcal{D},\alpha}$.
It was proved that
\[
\check{\mathcal{K}}^x_{\mathcal{D},\alpha}[\sigma_{\mathcal{
D}}f] = \hat{i}_2\check{\mathcal{K}}^x_{\alpha}\bigl [\sigma\bigl
(-\hat{i}_2\bigr )f\bigr ].
\]
Hence using formula (\ref{chernij}) and after not complicated computation
we obtain (\ref{may}).
\end{proof}

\section{Proof of the Theorem \ref{thm2.3}}

In this Section we use results form Section 4.
For the reader's convenience, recall some information from \cite{rosh1:gnus}:
\begin{gather*}
-\gamma_0\hat{i}{(\mathcal{UA})}^{-1} = - \gamma_1{(\mathcal{UA})}^{-1}
\bigl ({\hat{i}}_2 \bigr )^{-1}, \\
 \mathcal{K}_{\mathbb{D}_{\omega,m}}
= {(\mathcal{UA})}^{-1}\bigl ({\hat{i}}_2 \bigr )^{-1}
\mathcal{K}_{\mathcal{D},\alpha} = {(\mathcal{UA})}^{-1}\mathcal{K}_{\alpha}, \\
 \sigma_{\mathbb{D}_{\omega,m}} = \sigma_\mathcal{D}{\hat{i}}_2(\mathcal{UA})=\sigma(\mathcal{UA}), \\
 \check{\mathcal{K}}^x_{\mathbb{D}_{\omega,m}} = {(\mathcal{
UA})}^{-1}\bigl ({\hat{i}}_2 \bigr )^{-1}
\check{\mathcal{K}}^x_{\mathcal{D},\alpha}.
\end{gather*}
The proof of Theorem \ref{thm2.3} follows from Theorem \ref{thm5.3}
 taking into account the above relation between the class of
the time-harmonic spinor fields and $\alpha$-hyperholomorphic functions.

\subsection*{Acknowledgments}
The author would like to thank the anonymous referee for his/her rigorous and
constructive analysis of the original manuscript, which led to essential 
improvements on this article.

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\end{document}