# On monogenic functions and the Dirac complex of two vector variables

Yun Shi  Department of Mathematics
Zhejiang University of Science and Technology
Hangzhou 310023, China<EMAIL_ADDRESS>,  Wei Wang  Department of
Mathematics
Zhejiang University
Hangzhou 310027, China<EMAIL_ADDRESS>and  Qingyan Wu  Department of
Mathematics
Linyi University
Linyi 276005, China<EMAIL_ADDRESS>
###### Abstract.

A monogenic function of two vector variables is a function annihilated by the
operator consisting of two Dirac operators, which are associated to two
variables, respectively. We give the explicit form of differential operators
in the Dirac complex resolving this operator and prove its ellipticity
directly. This open the door to apply the method of several complex variables
to investigate this kind of monogenic functions. We prove the Poincaré lemma
for this complex, i.e. the non-homogeneous equations are solvable under the
compatibility condition by solving the associated Hodge Laplacian equations of
fourth order. As corollaries, we establish the Bochner–Martinelli integral
representation formula for this differential operator and the Hartogs’
extension phenomenon for monogenic functions. We also apply abstract duality
theorem to the Dirac complex to obtain the generalization of Malgrange’s
vanishing theorem and establish the Hartogs–Bochner extension phenomenon for
monogenic functions under the moment condition.

###### Key words and phrases:

Dirac complex; Non-homogeneous equations; Bochner–Martinelli formula; Hartogs’
phenomenon; The Hartogs–Bochner extension for monogenic functions

###### 1991 Mathematics Subject Classification:

58J10; 15A66; 34L40

##  1\. Introduction

Since Pertici [21] proved Hartogs’ phenomenon for regular functions of several
quaternionic variables, analysis of this kind of regular functions developed
rapidly. The method for this analysis is to solve non-homogeneous Cauchy-
Fueter equation, which is overdetermined. Therefore it is necessary to find
its compatibility condition, more generally, the resolution of the
Cauchy–Fueter operator. The search of this complex began in 1990s by using
computer method (cf. e.g. [1, 2, 8] and references therein). Later, it was
realized that the Penrose transformation can be applied to construct
resolutions after complexification (cf. [3, 6, 7, 9, 30, 32] and reference
therein). Both method can be applied to monogenic functions of several vector
variables, which are annihilated by several Dirac operators, and the
construction of their resolutions [10, 16, 17, 18, 25, 26]. On the other hand,
Ren and H.-Y. Wang [23] have already solved the non-homogeneous several Dirac
equation with a compatibility condition in the integral form, proved the
Bochner–Martinelli formula and Hartogs’ phenomenon for monogenic functions of
several vector variables by generalizing method used by Pertici.

Let $\mathbb{R}_{n}$ be the real Clifford algebra. The _Dirac operator_ acts
on functions $f:\mathbb{R}^{n}\rightarrow\mathbb{R}_{n}$ as the linear
operator

(1)
$\displaystyle\partial_{\mathbf{x}}=\sum_{i=1}^{n}e_{i}{\partial_{x_{i}}},$

where $\mathbf{x}=\left(x_{1},\cdots,x_{n}\right)\in\mathbb{R}^{n},$ and
$e_{i},i=1,\cdots,n,$ is Clifford number. The equation
$\partial_{\mathbf{x}}f=g$ has a smooth solution $f$ for smooth $g$ on
suitable open sets (cf. e.g. [4, 11]). Now consider the Cartesian product
$\mathbb{R}^{kn}.$ If we write
$\mathbf{x}_{A}=\left(x_{A1},\cdots,x_{An}\right)$ as the vector variable in
the $A$-th copy of $\mathbb{R}^{n},$
$\partial_{Ai}:=\frac{\partial}{\partial{x_{Ai}}},A=0,\cdots,k-1,i=1,\cdots,n,$
for simplicity. The _several Dirac operator_ is

(2) $\displaystyle\partial_{\mathbf{x}_{A}}=\sum_{i=1}^{n}e_{i}\partial_{Ai},$

acting on functions $f:\Omega\subset\mathbb{R}^{kn}\rightarrow\mathbb{R}_{n}.$
$f$ is called _monogenic_ on $\Omega$ if it satisfies

(3) $\displaystyle\partial_{\mathbf{x}_{A}}f=0,$

on $\Omega$ for $A=0,\cdots,k-1.$

As in the complex and quaternionic case, to investigate this kind of monogenic
functions, a fundamental method is to solve the associated inhomogeneous
system

(4)
$\displaystyle\left\\{\begin{array}[]{rcl}\partial_{\mathbf{x}_{0}}f=&g_{0},\\\
\partial_{\mathbf{x}_{1}}f=&g_{1},\\\ \vdots&\\\
\partial_{\mathbf{x}_{k-1}}f=&g_{k-1},\end{array}\right.$

where $f$ and $g_{i}$ are in a suitable space of generalized functions. (4)
can only be solved under a compatibility condition since it is overdetermined.
Thus the solution of this problem can be obtained if one can provide a
description of the so-called Dirac complex.

It is more simple to consider function $f$ in (3) valued in an irreducible
module of ${\rm Spin}(n),$ in particular, the spinor modules
$\mathbb{S}^{\pm}.$ In this paper $\mathbb{S}^{\pm}$ denote the two spinor
modules for $n$ even, and the same symbols are used for $n$ odd with the
convention that $\mathbb{S}^{+}$ and $\mathbb{S}^{-}$ are isomorphic.

For two vector variables, this complex is explicitly known by Damiano-
Sabadini-Souček [10] as

(5) $\displaystyle
0\rightarrow\Gamma\left(\Omega,\mathscr{V}_{0}\right)\xrightarrow{\mathscr{D}_{0}}\Gamma\left(\Omega,\mathscr{V}_{1}\right)\xrightarrow{\mathscr{D}_{1}}\Gamma\left(\Omega,\mathscr{V}_{2}\right)\xrightarrow{\mathscr{D}_{2}}\Gamma\left(\Omega,\mathscr{V}_{3}\right)\rightarrow
0,$

where $\Omega$ is a domain in $\mathbb{R}^{2n},$ and

$\displaystyle\mathscr{V}_{0}=\mathbb{V}_{00}^{+},\quad\mathscr{V}_{1}=\mathbb{V}_{10}^{-},\quad\mathscr{V}_{2}=\mathbb{V}_{21}^{-},\quad\mathscr{V}_{3}=\mathbb{V}_{22}^{+}.$

Here $\mathbb{V}_{\lambda}^{\pm}=V_{\lambda}\otimes\mathbb{S}^{\pm},$ and
$V_{\lambda}$ is an irreducible ${\rm GL}(2)$-module with the highest
$\lambda.$ It is known that

$V_{00}\cong\mathbb{C},\quad V_{10}\cong\mathbb{C}^{2},\quad
V_{21}\cong\mathbb{C}^{2},\quad V_{22}\cong\mathbb{C},$

are representations of ${\rm GL}(2)$ [12]. As complex vector spaces, we have,

(6) $\displaystyle\mathscr{V}_{0}=\mathbb{S}^{+},\
\mathscr{V}_{1}=\mathbb{C}^{2}\otimes\mathbb{S}^{-},\
\mathscr{V}_{2}=\mathbb{C}^{2}\otimes\mathbb{S}^{+},\
\mathscr{V}_{3}=\mathbb{S}^{-}.$

Although operators $\mathscr{D}_{j}$ in the complex (5) are known to be
invariant operators, but their explicit form were not given in [10]. Denote
the linear operator

(7) $\displaystyle\nabla_{A}:=\sum_{j=1}^{n}\gamma_{j}\partial_{Aj},$

for $\mathbf{x}_{A}=\left(x_{A1},\cdots,x_{An}\right),$ where
$\gamma_{j}:\mathbb{S}^{\pm}\rightarrow\mathbb{S}^{\mp}$ are Dirac matrices. A
section in $\Gamma\left(\Omega,\mathscr{V}_{0}\right)$ is an
$\mathbb{S}^{+}$-valued function on $\Omega$, while a section in
$\Gamma\left(\Omega,\mathscr{V}_{1}\right)$ is written as
$g=\left(\begin{matrix}g_{0}\\\ g_{1}\end{matrix}\right)$ for some
$\mathbb{S}^{-}$-valued functions $g_{0}$ and $g_{1}$ on $\Omega.$ Similarly,
a section in $\Gamma\left(\Omega,\mathscr{V}_{2}\right)$ is also written as
$h=\left(\begin{matrix}h_{0}\\\ h_{1}\end{matrix}\right)$ for some
$\mathbb{S}^{+}$-valued function $h_{0}$ and $h_{1}$ on $\Omega.$ The explicit
form of operators $\mathscr{D}_{j}$’s are as follows

(8) $\displaystyle\left(\mathscr{D}_{0}f\right)_{A}:=$
$\displaystyle\nabla_{A}f,\quad$ $\displaystyle{\rm for}\
f\in\Gamma\left(\Omega,\mathscr{V}_{0}\right),$
$\displaystyle\left(\mathscr{D}_{1}g\right)_{A}:=$
$\displaystyle\nabla_{0}\nabla_{A}g_{1}-\nabla_{1}\nabla_{A}g_{0},\quad$
$\displaystyle{\rm for}\ g\in\Gamma\left(\Omega,\mathscr{V}_{1}\right),$
$\displaystyle\mathscr{D}_{2}h:=$ $\displaystyle
2\nabla_{[0}h_{1]}:=\nabla_{0}h_{1}-\nabla_{1}h_{0},\quad$ $\displaystyle{\rm
for}\ h\in\Gamma\left(\Omega,\mathscr{V}_{2}\right),$

where $A=0,1.$

The Dirac complex on $\mathbb{R}^{kn}$ for $k=3,n\geq 6$ is given in [10], but
their operators are more complicated than the case of $k=2$. The case of $k=4$
is discussed by Krump in [17]. For the stable range $n\geq 2k,$ the Dirac
complex is known [14]. But for the unstable range, it is an open problem to
construct the Dirac complex. However some results can be found in [18].

We prove the ellipticity of this differential complex (5) directly. As in the
case of several quaternionic variables (cf. e.g. [20, 24, 29, 31, 33, 34]),
once we know differential complex explicitly, which is also elliptic, the
method of several complex variables can be applied to obtain many profound
results about monogenic functions. We define the associated Hodge Laplacian
operators of fourth order:

(9) $\displaystyle\Box_{0}:=$
$\displaystyle\left(\mathscr{D}_{0}^{*}\mathscr{D}_{0}\right)^{2},$
$\displaystyle\Box_{1}:=$
$\displaystyle\left(\mathscr{D}_{0}\mathscr{D}_{0}^{*}\right)^{2}+\mathscr{D}_{1}^{*}\mathscr{D}_{1},$
$\displaystyle\Box_{2}:=$
$\displaystyle\mathscr{D}_{1}\mathscr{D}_{1}^{*}+\left(\mathscr{D}_{2}^{*}\mathscr{D}_{2}\right)^{2},$

which are elliptic operators on $\Gamma\left(\Omega,\mathscr{V}_{j}\right),$
where $\mathscr{D}_{j}^{*}$ is the formal adjoint of $\mathscr{D}_{j}.$ These
operators have very simply form

(12)
$\displaystyle\Box_{0}=\Delta^{2},\quad\Box_{1}=\Box_{2}=\left(\begin{array}[]{cc}\Delta^{2}&\\\
&\Delta^{2}\end{array}\right),$

where $\Delta=\sum_{j=1}^{n}\left(\partial_{0j}^{2}+\partial_{1j}^{2}\right)$
is the Laplacian operator on $\mathbb{R}^{2n}.$ So their fundamental solution
are given by $\frac{1}{|\mathbf{x}|^{2n-4}}$ up to a constant. This allows us
to derive Bochner–Martinelli formula directly in a very simple way and solve
the non-homogeneous Dirac equation

(13) $\displaystyle\mathscr{D}_{j}u=f,$

under the compatibility condition

(14) $\displaystyle\mathscr{D}_{j+1}f=0.$

######  Theorem 1.1.

Suppose that $f\in L^{2}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right)$
satisfies the compatibility condition (14) in sense of distributions. Then
there exists a function $u\in
W^{1,2}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right)$ satisfying the non-
homogeneous equation (13). Furthermore, if $f\in
C_{0}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right)$ with $\mathscr{D}_{1}f=0$
in the sense of distributions, then there exists a function $u\in
C_{0}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right)\cap
W^{1,2}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right)$ satisfying (13) and
vansihing on the unbounded connected component of
$\mathbb{R}^{2n}\setminus{\rm supp}f.$

As a corollary, we can prove the following Hartogs’ phenomenon. A function
$u:\Omega\rightarrow\mathscr{V}_{0}$ is called _monogenic_ on $\Omega$ if

$\displaystyle\mathscr{D}_{0}u(\mathbf{x})=0,$

for any $\mathbf{x}\in\Omega.$ The space of all monogenic functions on
$\Omega$ is denoted by $\mathcal{O}(\Omega).$

######  Theorem 1.2.

Let $\Omega$ be an open set in $\mathbb{R}^{2n}$ and let $K$ be a compact
subset of $\Omega$ such that $\Omega\setminus K$ is connected. Then for each
$u\in\mathcal{O}(\Omega\setminus K),$ we can find $U\in\mathcal{O}(\Omega)$
such that $U=u$ in $\Omega\setminus K.$

For the Dirac complex ${\mathscr{V}}_{\bullet},$ let
$\mathcal{E}\left(\mathbb{R}^{2n},{\mathscr{V}}_{\bullet}\right)$ be the space
of smooth ${\mathscr{V}}_{\bullet}$-valued functions with the topology of
uniform convergence on compact sets of the functions and all their
derivatives. Let
$\mathcal{D}\left(\mathbb{R}^{2n},{\mathscr{V}}_{\bullet}\right)$ be the space
of compactly supported elements of
$\mathcal{E}\left(\mathbb{R}^{2n},{\mathscr{V}}_{\bullet}\right).$ Denote
$\mathcal{E}^{\prime}\left(\mathbb{R}^{2n},{\mathscr{V}}_{\bullet}\right),$
the dual of $\mathcal{E}\left(\mathbb{R}^{2n},{\mathscr{V}}_{\bullet}\right).$
By using abstract duality theorem, we have the following generalization of
Malgrange’s vanishing theorem.

######  Theorem 1.3.

The cohomology groups
$H^{3}\left(\mathcal{E}\left(\mathbb{R}^{2n},{\mathscr{V}}_{\bullet}\right)\right)$
and
$H^{3}\left(\mathcal{D}^{\prime}\left(\mathbb{R}^{2n},{\mathscr{V}}_{\bullet}\right)\right)$
both vanish.

Moreover, we give the Hartogs–Bochner extension for monogenic functions under
the moment condition.

######  Theorem 1.4.

Let $\Omega$ be a domain of $\mathbb{R}^{2n}$ with smooth boundary such that
$\mathbb{R}^{2n}\setminus\overline{\Omega}$ connected. Suppose that $f$ is the
restriction to $\partial\Omega$ of a
$C^{2}\left(\Omega,\mathscr{V}_{0}\right)$ function, with $\mathscr{D}_{0}f$
vanishing to the second order on $\partial\Omega,$ and satisfies the moment
condition

(15) $\displaystyle\int_{\partial\Omega}\left\langle
f,n_{0}G_{0}+n_{1}G_{1}\right\rangle_{\mathbb{S}^{+}}{{\rm d}S}=0,$

for any $G=\left(\begin{matrix}G_{0}\\\
G_{1}\end{matrix}\right)\in\ker{\mathscr{D}}^{*}_{0}\subset\mathcal{E}\left(\mathbb{R}^{2n},{\mathscr{V}}_{1}\right),$
${\mathscr{D}}^{*}_{0}G=0$ on $\overline{\Omega},$ where
$n_{A}=\sum_{j=1}^{n}\gamma_{j}n_{Aj}$ for $A=0,1,$ and
$n=\left(n_{01},\cdots,n_{0n},n_{11},\cdots,n_{1n}\right)$ is the unit outer
normal vector to $\partial\Omega.$ Then there exists a monogenic function
$\tilde{f}\in\mathcal{O}(\Omega)$ such that $\tilde{f}=f$ on $\partial\Omega.$

Here the vanishing of $\mathscr{D}_{0}f$ to the second order on
$\partial\Omega$ and the moment condition (15) are also the necessary
condition for the Hartogs–Bochner extension (see Remark 4.2).

The paper is organized as follows. In Sect. 2, we give the preliminaries on
Clifford algebras, the Dirac operator, spin modules and the differential
complex of two vector variables. Then we write down operators of the Dirac
complex of two vector variables and prove its ellipticity. In Sect. 3, we give
the Bochner–Martinelli integral representation formula for $\mathscr{D}_{0}$
by using the fundamental solution to the Hodge Laplacian $\Box_{j}.$ As
applications, we prove Theorem 1.1 and Theorem 1.2. In Sect. 4, we apply
abstract duality theorem to the Dirac complex on $\mathbb{R}^{2n}$ to obtain
the generalization of Malgrange’s vanishing theorem and the Hartogs–Bochner
extension for monogenic functions under the moment condition.

##  2\. The Dirac complex of two vector variables

###  2.1. Clifford algebra and spin modules $\mathbb{S}^{\pm}$

The real Clifford algebra $\mathbb{R}_{n}$ is the associative algebra
generated by the $n$ basis elements of $\mathbb{R}^{n}$ satisfying

$\displaystyle e_{i}e_{j}+e_{j}e_{i}=-2\delta_{ij},$

for $i,j=1,\cdots,n.$ The basis of $\mathbb{R}_{n}$ is

$\displaystyle e_{0}=1,\quad e_{\alpha}=e_{a_{1}}\cdots e_{a_{i}},$

for $1\leq i\leq n,1\leq a_{1}<\cdots<a_{i}\leq n.$ For $n=2m,$

$W={\rm span}_{\mathbb{C}}\left\\{f_{j},j=1,\cdots,m,\right\\},\quad
W^{\prime}={\rm span}_{\mathbb{C}}\left\\{\bar{f}_{j},j=1,\cdots,m,\right\\},$

are maximally isotropic subspaces of _complex Clifford algebra_
$\mathbb{C}_{n}$ and $\mathbb{C}^{n}=W\oplus W^{\prime},$ where

$\displaystyle f_{j}=\frac{1}{2}\left(e_{2j-1}+\mathbf{i}e_{2j}\right),\
\bar{f}_{j}=-\frac{1}{2}\left(e_{2j-1}-\mathbf{i}e_{2j}\right),$

(cf. [11, (4.11)]). We have two spin modules

$\displaystyle\mathbb{S}^{+}=\wedge^{\rm
even}W,\quad\mathbb{S}^{-}=\wedge^{\rm odd}W,$

which are the irreducible representations of $\mathfrak{so}(2m,\mathbb{C})$
[11, p. 118, Theorem 1]. For $n=2m+1,$ $W={\rm
span}_{\mathbb{C}}\left\\{f_{j},j=1,\cdots,m,\right\\},$ $W^{\prime}={\rm
span}_{\mathbb{C}}$ $\left\\{\bar{f}_{j},j=1,\cdots,m,\right\\},$
$U=\left\\{f_{2m+1}\right\\},$ are maximally isotropic subspaces of
$\mathbb{C}_{n}$ and $\mathbb{C}^{n}=W\oplus W^{\prime}\oplus U,$ where $U$ is
a $1$-dimensional space perpendicular to them and

$\displaystyle f_{m+1}=\mathbf{i}e_{2m+1}.$

We have one spin module

$\displaystyle\mathbb{S}=\wedge^{\bullet}W,$

which is the irreducible representation of $\mathfrak{so}(2m+1,\mathbb{C}).$
$f_{j}$’s satisfy relations

(16) $\displaystyle f_{j}f_{k}+f_{k}f_{j}=0,\
\bar{f}_{j}\bar{f}_{k}+\bar{f}_{k}\bar{f}_{j}=0,\
\bar{f}_{j}f_{k}+f_{k}\bar{f}_{j}=0,$ $\displaystyle{\rm for}\ j\neq k,$
$\displaystyle f_{m+1}f_{k}+f_{k}f_{m+1}=0,\
f_{m+1}\bar{f}_{k}+\bar{f}_{k}\bar{f}_{m+1}=0,$ $\displaystyle{\rm for\ any}\
k.$

The action of Clifford algebra on the spin module $\mathbb{S}^{\pm}$ is given
by $\gamma\left(f_{s}\right)\in{\rm End}\ \mathbb{S}^{\pm}.$ For any
$a\in\mathbb{C}_{n},$ $\gamma(a):(\wedge W)I\rightarrow(\wedge W)I$ is given
by

$\displaystyle\gamma(a)(bI)=abI,$

where $I=I_{1}I_{2}\cdots I_{m},$ with $I_{j}=\bar{f}_{j}f_{j},j=1,\cdots,m,$
is a primitive idempotent in $\mathbb{C}_{n}$ (cf. [11, p. 115]). A primitive
idempotent of $\mathbb{C}_{n}$ is a nonzero idempotent $I$ such that
$I\mathbb{C}_{n}$ is indecomposable as a right $\mathbb{C}_{n}$-module; that
is, such that $I\mathbb{C}_{n}$ is not a direct sum of two nonzero submodules.
Equivalently, $I$ is a primitive idempotent if it cannot be written as
$I=J+K,$ where $J$ and $K$ are nonzero orthogonal idempotents in
$\mathbb{C}_{n}.$ Then $\mathbb{S}^{+},\mathbb{S}^{-}$ and $\mathbb{S}$ can be
realized as $\wedge^{\rm even}WI,\wedge^{\rm odd}WI$ and $\wedge^{\bullet}WI,$
respectively. For any
$\alpha=\left\\{\alpha_{1},\cdots,\alpha_{h}\right\\}\subset\\{1,\cdots,m\\}$
with $\alpha_{1}<\alpha_{2}<\cdots<\alpha_{h},$ set

(17) $\displaystyle f_{\alpha}:=f_{\alpha_{1}}\wedge\cdots\wedge
f_{\alpha_{h}},$

which constitute a basis of $\mathbb{C}_{n}.$

######  Theorem 2.1.

(cf. [11, p.116, Theorem 1]) For $j=1,\cdots,m,$
(1) $\gamma\left(f_{j}\right)f_{\alpha}I=\left(f_{j}\wedge
f_{\alpha}\right)I;$
(2)
$\gamma\left(\bar{f}_{j}\right)f_{\alpha}I=\bar{f}_{j}\left(f_{\alpha}\right)I;$
(3) $\gamma\left(f_{m+1}\right)f_{\alpha}I=(-1)^{\sharp\alpha}f_{\alpha}I,$
where

$\displaystyle\bar{f}_{j}\left(f_{\alpha}\right)=\left\\{\begin{array}[]{lcl}&0,&{\rm
if}\ j\notin\alpha,\\\ &(-1)^{t+1}f_{\alpha^{\prime}},&{\rm if}\
j=\alpha_{t},\alpha^{\prime}=\alpha\setminus\\{\alpha_{t}\\}\end{array}\right.,$

and $\sharp\alpha=h$ if $f_{\alpha}=f_{\alpha_{1}}\wedge\cdots\wedge
f_{\alpha_{h}}.$

Denote $\gamma_{j}:=\gamma\left(e_{j}\right).$ Since

$\displaystyle e_{2j-1}=f_{j}-\bar{f}_{j},\quad
e_{2j}=-\mathbf{i}\left(f_{j}-\bar{f}_{j}\right),\quad
e_{2m+1}=-\mathbf{i}f_{m+1},$

it follows form (16) that
$\gamma_{j}:\mathbb{S}^{\pm}\rightarrow\mathbb{S}^{\mp},$ and

(18) $\displaystyle\gamma_{j}\gamma_{k}+\gamma_{k}\gamma_{j}=-2\delta_{jk}\bf
1.$

Define an Hermitian inner product on $\mathbb{S}^{\pm}$ by

$\displaystyle\left\langle
f_{\alpha},f_{\beta}\right\rangle_{\mathbb{S}^{\pm}}:=\delta_{\alpha\beta}.$

######  Lemma 2.1.

The adjoint $\gamma_{k}^{*}$ of $\gamma_{k}$ with respect to this inner
product is given by

(19) $\displaystyle\gamma_{k}^{*}=-\gamma_{k}.$

###### Proof.

We only need to prove

$\displaystyle\left\langle\gamma_{k}f_{\alpha},f_{\beta}\right\rangle_{\mathbb{S}^{\pm}}=-\left\langle
f_{\alpha},\gamma_{k}f_{\beta}\right\rangle_{\mathbb{S}^{\mp}},$

where $f_{\alpha}$ is the basis of $\mathbb{C}_{n}$ given in (17). It is easy
to see that
$\left\langle\gamma_{2j-1}f_{\alpha},f_{\beta}\right\rangle_{\mathbb{S}^{\mp}}$
$\neq 0$ only if $\alpha\cup\\{j\\}=\beta$ or $\beta\cup\\{j\\}=\alpha.$ If
$\alpha\cup\\{j\\}=\beta$ and $j$ is the $k$-th element of $\beta,$ we have

$\left\langle\gamma_{2j-1}f_{\alpha},f_{\beta}\right\rangle_{\mathbb{S}^{\pm}}=(-1)^{k-1}=-\left\langle
f_{\alpha},\gamma_{2j-1}f_{\beta}\right\rangle_{\mathbb{S}^{\mp}},$

by Theorem 2.1. So $\gamma_{2j-1}^{*}=-\gamma_{2j-1},$ for $j=1,\cdots,m.$
Similarly, we can prove the $\beta\cup\\{j\\}=\alpha$ case.

Also
$\left\langle\gamma_{2j}f_{\alpha},f_{\beta}\right\rangle_{\mathbb{S}^{\mp}}\neq
0$ only if $\alpha\cup\\{j\\}=\beta$ or $\beta\cup\\{j\\}=\alpha.$ If
$\alpha\cup\\{j\\}=\beta$ and $j$ is the $k$-th element of $\beta,$ we have

$\left\langle\gamma_{2j}f_{\alpha},f_{\beta}\right\rangle_{\mathbb{S}^{\pm}}=-\mathbf{i}(-1)^{k-1}=-\left\langle
f_{\alpha},\gamma_{2j}f_{\beta}\right\rangle_{\mathbb{S}^{\mp}},$

by Theorem 2.1. So $\gamma_{2j}^{*}=-\gamma_{2j},$ for $j=1,\cdots,m.$
Similarly, we can prove the $\beta\cup\\{j\\}=\alpha$ case.

$\left\langle\gamma_{2m+1}f_{\alpha},f_{\beta}\right\rangle_{\mathbb{S}^{\mp}}\neq
0$ only if $\alpha=\beta.$ In this case

$\left\langle\gamma_{2m+1}f_{\alpha},f_{\alpha}\right\rangle_{\mathbb{S}^{\pm}}=-\mathbf{i}(-1)^{\sharp\alpha}=-\left\langle
f_{\alpha},\gamma_{2m+1}f_{\alpha}\right\rangle_{\mathbb{S}^{\mp}},$

by Theorem 2.1. So $\gamma_{2m+1}^{*}=-\gamma_{2m+1}.$ The lemma is proved. ∎

###  2.2. Operators in the Dirac complex of two vector variables

Note that

(20) $\displaystyle\nabla_{A}\nabla_{A}=$
$\displaystyle\sum_{i,j=1}^{n}\gamma_{i}{\partial_{Ai}}\gamma_{j}\partial_{Aj}=-\sum_{j=1}^{n}{\partial^{2}_{Aj}}:=\Delta_{A},$

by (18), for $A=0,1.$

Define the Hermitian inner product on
$L^{2}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right),$ for $j=1,3,$ by

$(\phi,\psi)_{j}:=\sum_{A=0,1}\int_{\mathbb{R}^{2n}}\left\langle\phi_{A}(\mathbf{x}),{\psi_{A}(\mathbf{x})}\right\rangle_{\mathbb{S}^{-}}\hbox{d}V(\mathbf{x}),$

for $\phi,\psi\in L^{2}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right),$ and
define the Hermitian inner product on
$L^{2}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right),$ for $j=0,2,$ by

$(\Phi,\Psi)_{j}=\int_{\mathbb{R}^{2n}}\left\langle\Phi(\mathbf{x}),{\Psi(\mathbf{x})}\right\rangle_{\mathbb{S}^{+}}\hbox{d}V(\mathbf{x}),$

for $\Phi,\Psi\in L^{2}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right),$ where

$\displaystyle\hbox{d}V(\mathbf{x})=\hbox{d}x_{01}\wedge\cdots\wedge\hbox{d}x_{0n}\wedge\hbox{d}x_{11}\wedge\cdots\wedge\hbox{d}x_{1n},$

is the standard volume form on $\mathbb{R}^{2n}.$ We can rewrite the operator
in (8) into the matrix form as follows.
$\mathscr{D}_{0}:\Gamma\left(\mathbb{S}^{+}\right)\rightarrow\Gamma\left(\mathbb{C}^{2}\otimes\mathbb{S}^{-}\right)$
is given by

(21) $\displaystyle\left(\begin{matrix}\nabla_{0}\\\
\nabla_{1}\end{matrix}\right)=\left(\begin{matrix}\sum_{j}\gamma_{j}{\partial_{0j}}\\\
\sum_{k}\gamma_{k}{\partial_{1k}}\end{matrix}\right),$

$\mathscr{D}_{1}:\Gamma\left(\mathbb{C}^{2}\otimes\mathbb{S}^{-}\right)\rightarrow\Gamma\left(\mathbb{C}^{2}\otimes\mathbb{S}^{+}\right)$
is given by

$\displaystyle\left(\begin{array}[]{cc}-\nabla_{1}\nabla_{0}&\nabla_{0}\nabla_{0}\\\
-\nabla_{1}\nabla_{1}&\nabla_{0}\nabla_{1}\end{array}\right)=\left(\begin{array}[]{cc}-\sum_{j,k}\gamma_{k}\gamma_{j}{\partial_{0j}}{\partial_{1k}}&\Delta_{0}\\\
-\Delta_{1}&\sum_{j,k}\gamma_{j}\gamma_{k}{\partial_{0j}}{\partial_{1k}}\end{array}\right),$

while
$\mathscr{D}_{2}:\Gamma\left(\mathbb{C}^{2}\otimes\mathbb{S}^{+}\right)\rightarrow\Gamma\left(\mathbb{S}^{-}\right)$
is given by

(22)
$\displaystyle\left(-\nabla_{1},\nabla_{0}\right)=\left(-\sum_{k}\gamma_{k}{\partial_{1k}},\sum_{j}\gamma_{j}{\partial_{0j}}\right).$

Let $\mathscr{D}_{j}^{*}$ be the formal adjoint of the operator
$\mathscr{D}_{j}$ in (8), i.e.

(23)
$\displaystyle\left(\mathscr{D}_{j}\phi,\psi\right)_{j+1}=\left(\phi,\mathscr{D}_{j}^{*}\psi\right)_{j}$

for any $\phi\in
C_{0}^{\infty}\left(\mathbb{R}^{2n},{\mathscr{V}_{j}}\right),\psi\in
C_{0}^{\infty}\left(\mathbb{R}^{2n},{\mathscr{V}_{j+1}}\right).$

######  Proposition 2.1.

For any $\phi\in C_{0}^{\infty}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right),$
we have

$\displaystyle\mathscr{D}_{0}^{*}\mathscr{D}_{0}\phi=\Delta\phi,$

where $\Delta=\Delta_{0}+\Delta_{1}$ is the Laplacian operator on
$\mathbb{R}^{2n}.$

###### Proof.

For $\phi\in C_{0}^{\infty}\left(\mathbb{R}^{2n},\mathbb{S}^{\pm}\right),$ and
$\psi\in C_{0}^{\infty}\left(\mathbb{R}^{2n},\mathbb{S}^{\mp}\right),$ we have

$\displaystyle\int_{\mathbb{R}^{2n}}\left\langle\nabla_{A}\phi,{\psi_{A}}\right\rangle_{\mathbb{S}^{\mp}}\hbox{d}V(\mathbf{x})=$
$\displaystyle\int_{\mathbb{R}^{2n}}\sum_{i}\left\langle\gamma_{i}\partial_{Ai}\phi,{\psi_{A}}\right\rangle_{\mathbb{S}^{\mp}}\hbox{d}V(\mathbf{x})$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{2n}}\sum_{i}\left\langle\phi,\gamma_{i}\partial_{Ai}{\psi_{A}}\right\rangle_{\mathbb{S}^{\pm}}\hbox{d}V(\mathbf{x})$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{2n}}\left\langle\phi,{\nabla_{A}\psi_{A}}\right\rangle_{\mathbb{S}^{\pm}}\hbox{d}V(\mathbf{x})$

by using (19) and Stokes’s formula, i.e. the formal adjoint $\nabla_{A}^{*}$
of $\nabla_{A}$ satisfies

(24) $\displaystyle\nabla_{A}^{*}=\nabla_{A}.$

Thus, we have

$\displaystyle\left(\mathscr{D}_{0}\phi,\psi\right)=\sum_{A=0,1}\int_{\mathbb{R}^{2n}}\left\langle\nabla_{A}\phi,{\psi_{A}}\right\rangle_{\mathbb{S}^{-}}\hbox{d}V(\mathbf{x})=\left(\phi,\mathscr{D}_{0}^{*}\psi\right),$

i.e.

(25) $\displaystyle\mathscr{D}_{0}^{*}\psi=\sum_{A=0,1}\nabla_{A}\psi_{A}.$

Thus

(26)
$\displaystyle\mathscr{D}_{0}^{*}\mathscr{D}_{0}\phi=\sum_{A=0,1}\nabla_{A}\nabla_{A}\phi=\sum_{A=0,1}\sum_{j,k=0}^{n-1}\gamma_{j}{\partial_{Aj}}\gamma_{k}{\partial_{Ak}}\phi=\Delta\phi.$

The proposition is proved. ∎

Recall that the symbol of a matrix differential operator

$\mathscr{D}=\sum_{|\alpha|\leq
n}B_{\alpha_{A_{1}k_{1}}\cdots\alpha_{A_{N}k_{N}}}(\mathbf{x})\partial^{\alpha_{A_{1}k_{1}}}_{A_{1}k_{1}}\cdots\partial^{\alpha_{A_{N}k_{N}}}_{A_{N}k_{N}}:\Gamma(\Omega,V)\rightarrow\Gamma(\Omega,V^{\prime})$

at $(\mathbf{x},\nu)$ for
$\mathbf{x}\in\Omega\subset\mathbb{R}^{2n},\mathbf{0}\neq\nu\in\mathbb{R}^{2n}$
is defined as

(27)
$\displaystyle\sigma(\mathscr{D})(\mathbf{x};\nu):=\sum_{|\alpha|=n}B_{\alpha_{A_{1}k_{1}}\cdots\alpha_{A_{N}k_{N}}}(\mathbf{x})\left(\frac{\nu_{A_{1}k_{1}}}{\mathbf{i}}\right)^{\alpha_{A_{1}k_{1}}}\cdots\left(\frac{\nu_{A_{N}k_{N}}}{\mathbf{i}}\right)^{\alpha_{A_{N}k_{N}}}:V\rightarrow
V^{\prime},$

where $A_{l}\in\\{0,1\\},k_{l}\in\\{1,\cdots,n\\},$ and
$B_{\alpha_{A_{1}k_{1}}\cdots\alpha_{A_{N}k_{N}}}$ is a linear transformation
from vector space $V$ to $V^{\prime}.$ A differential complex

$0\rightarrow
C^{\infty}\left(\Omega,V_{0}\right)\xrightarrow{\mathscr{D}_{0}}\cdots\xrightarrow{\mathscr{D}_{n-1}}C^{\infty}\left(\Omega,V_{n}\right)\rightarrow
0,$

is called _elliptic_ if its symbol sequence

$0\rightarrow
V_{0}\xrightarrow{\sigma\left(\mathscr{D}_{0}\right){(\mathbf{x};\nu)}}\cdots\xrightarrow{\sigma\left(\mathscr{D}_{n-1}\right){(\mathbf{x};\nu)}}V_{n}\rightarrow
0,$

is exact for any
$\mathbf{x}\in\Omega,\nu\in\mathbb{R}^{2n}\setminus\\{\mathbf{0}\\},$ i.e.

$\ker\sigma\left(\mathscr{D}_{l}\right){(\mathbf{x};\nu)}={\rm Im}\
\sigma\left(\mathscr{D}_{l-1}\right){(\mathbf{x};\nu)}.$

The symbol of differential operator $\nabla_{A}$ is

(28)
$\displaystyle\sigma\left(\nabla_{A}\right)(\mathbf{x};\nu)=-{\mathbf{i}}\sum_{j=1}^{n}{\gamma_{j}\nu_{Aj}}:=\nu_{A}$

######  Theorem 2.2.

The complex (5) with operators given by (8) is an elliptic differential
complex.

###### Proof.

Firstly, we prove

(29) $\displaystyle\mathscr{D}_{l+1}\circ\mathscr{D}_{l}=0$

for each $l=0,1.$

When $l=0,$ noting that $\mathscr{D}_{0}f=\left(\begin{matrix}\nabla_{0}f\\\
\nabla_{1}f\end{matrix}\right)$ for $f\in\Gamma\left(\mathscr{V}_{0}\right),$
we have

$\displaystyle\left(\mathscr{D}_{1}\circ\mathscr{D}_{0}f\right)_{0}=$
$\displaystyle\left(\nabla_{0}\nabla_{0}\nabla_{1}-\nabla_{1}\nabla_{0}\nabla_{0}\right)f=\left(\Delta_{0}\nabla_{1}-\nabla_{1}\Delta_{0}\right)f=0,$
$\displaystyle\left(\mathscr{D}_{1}\circ\mathscr{D}_{0}f\right)_{1}=$
$\displaystyle\left(\nabla_{0}\nabla_{1}\nabla_{1}-\nabla_{1}\nabla_{1}\nabla_{0}\right)f=\left(\nabla_{0}\Delta_{1}-\Delta_{1}\nabla_{0}\right)f=0,$

by the definition of $\mathscr{D}_{1}$ in (8) and using (20) and

(30) $\displaystyle\nabla_{B}\Delta_{A}=\Delta_{A}\nabla_{B},\quad A,B=0,1,$

since $\Delta_{A}$ is a scalar differential operator of constant coefficients.
So $\mathscr{D}_{1}\circ\mathscr{D}_{0}=0.$

When $l=1,$ for $g=\left(\begin{matrix}g_{0}\\\
g_{1}\end{matrix}\right)\in\Gamma\left(\mathscr{V}_{1}\right),$ we have

$\displaystyle\mathscr{D}_{2}\circ\mathscr{D}_{1}g=$
$\displaystyle\nabla_{0}\left(\mathscr{D}_{1}g\right)_{1}-\nabla_{1}\left(\mathscr{D}_{1}g\right)_{0}$
$\displaystyle=$
$\displaystyle\nabla_{0}\left(\nabla_{0}\nabla_{1}g_{1}-\nabla_{1}\nabla_{1}g_{0}\right)-\nabla_{1}\left(\nabla_{0}\nabla_{0}g_{1}-\nabla_{1}\nabla_{0}g_{0}\right)$
$\displaystyle=$
$\displaystyle\Delta_{0}\nabla_{1}g_{1}-\nabla_{0}\Delta_{1}g_{0}-\nabla_{1}\Delta_{0}g_{1}+\Delta_{1}\nabla_{0}g_{0}=0,$

by the definition of $\mathscr{D}_{2}$ in (8) and (30). So
$\mathscr{D}_{2}\circ\mathscr{D}_{1}=0.$

Secondly, let us prove that (5) is elliptic, i.e. the symbol sequence

(31) $\displaystyle
0\rightarrow\mathbb{S}^{+}\xrightarrow{\sigma_{0}(\nu)}\mathbb{C}^{2}\otimes\mathbb{S}^{+}\xrightarrow{\sigma_{1}(\nu)}\mathbb{C}^{2}\otimes\mathbb{S}^{-}\xrightarrow{\sigma_{2}(\nu)}\mathbb{S}^{+}\rightarrow
0,$

is exact, $\nu\in\mathbb{R}^{2n}\setminus\\{\mathbf{0}\\},$ where
$\sigma_{l}(\nu)=\sigma\left(\mathscr{D}_{l}\right)(\mathbf{x};\nu)$ is
independent of $\mathbf{x}\in\mathbb{R}^{2n}.$

It follows from $\mathscr{D}_{l+1}\circ\mathscr{D}_{l}=0$ that
$\sigma_{l+1}\circ\sigma_{l}=0,$ i.e. ${\rm Im}\
\sigma_{l}(\nu)\in\ker\sigma_{l+1}(\nu).$ So we only need to prove that
$\sigma_{0}$ is injective, $\ker\sigma_{l}\subset{\rm Im}\ \sigma_{l-1},$ for
$l=1,2,$ and $\sigma_{2}$ is surjective.
(1) Denote $\left|\nu_{A}\right|^{2}:=\nu_{A1}^{2}+\cdots+\nu_{An}^{2}.$ Then

(32)
$\displaystyle\nu_{A}\nu_{A}=-{\mathbf{i}}\sum_{k}{\gamma_{k}\nu_{Ak}}\left(-{\mathbf{i}}\sum_{j}{\gamma_{j}\nu_{Aj}}\right)=\left|\nu_{A}\right|^{2}{\rm
id}_{\mathbb{S}^{\pm}}.$

By definition, for $\eta\in\mathscr{V}_{0}$

$\sigma_{0}(\nu)\eta=\left(\begin{matrix}\nu_{0}\eta\\\
\nu_{1}\eta\end{matrix}\right)\in\mathbb{C}^{2}\otimes\mathbb{S}^{+}.$

For any $\eta\in\ker\sigma_{0}(\nu),$ we have $\nu_{A}\eta=0,$ for $A=0,1.$
Then

(33) $\displaystyle 0=\nu_{A}\nu_{A}\eta=|\nu_{A}|^{2}\eta,$

by (32). So $\eta=0.$ Hence $\sigma_{0}$ is injective.

(2) By definition, for $\xi=\left(\begin{matrix}\xi_{0}\\\
\xi_{1}\end{matrix}\right)\in\mathscr{V}_{1},$

$\sigma_{1}(\nu)\xi=\left(\begin{matrix}\nu_{0}\nu_{0}\xi_{1}-\nu_{1}\nu_{0}\xi_{0}\\\
\nu_{0}\nu_{1}\xi_{1}-\nu_{1}\nu_{1}\xi_{0}\end{matrix}\right)\in\mathbb{C}^{2}\otimes\mathbb{S}^{-}.$

Note that

(34) $\displaystyle\left(\sigma_{1}(\nu)\xi\right)_{0}=$
$\displaystyle|\nu_{0}|^{2}\xi_{1}-\nu_{1}\nu_{0}\xi_{0},$
$\displaystyle\left(\sigma_{1}(\nu)\xi\right)_{1}=$
$\displaystyle\nu_{0}\nu_{1}\xi_{1}-|\nu_{1}|^{2}\xi_{0},$

by (32). If $\xi=\left(\begin{matrix}\xi_{0}\\\
\xi_{1}\end{matrix}\right)\in\ker\sigma_{1}(\nu)$ with
$\left|\nu_{0}\right|\neq 0,$ let

$\displaystyle\eta=\frac{\nu_{0}\xi_{0}}{|\nu_{0}|^{2}}\in\mathscr{V}_{0}.$

We claim that $\sigma_{0}\eta=\xi.$ Since $\xi=\left(\begin{matrix}\xi_{0}\\\
\xi_{1}\end{matrix}\right)\in\ker\sigma_{1}(\nu),$ we have
$|\nu_{0}|^{2}\xi_{1}-\nu_{1}\nu_{0}\xi_{0}=0$ by (34). Then

$\displaystyle\nu_{1}|\nu_{0}|^{2}\xi_{1}=\nu_{1}\nu_{1}\nu_{0}\xi_{0}=\nu_{0}|\nu_{1}|^{2}\xi_{0},$

by (32), i.e.

(35)
$\displaystyle\frac{\nu_{0}\xi_{0}}{|\nu_{0}|^{2}}=\frac{\nu_{1}\xi_{1}}{|\nu_{1}|^{2}}=\eta,$

if $\left|\nu_{1}\right|\neq 0.$ Then

(36)
$\displaystyle\left(\sigma_{0}(\nu)\eta\right)_{0}=\frac{\nu_{0}\nu_{0}}{|\nu_{0}|^{2}}\xi_{0}=\xi_{0},$
$\displaystyle\left(\sigma_{0}(\nu)\eta\right)_{1}=\frac{\nu_{1}\nu_{1}}{|\nu_{0}|^{2}}\xi_{1}=\xi_{1},$

by (32) and (35). If $\left|\nu_{1}\right|=0,$ we must have $\xi_{1}=0,$ and
so $\left(\sigma_{0}(\nu)\eta\right)_{1}=\nu_{1}\eta=0=\xi_{1},$ i.e. (36)
also holds. Thus $\sigma_{0}\eta=\xi$ and so $\ker\sigma_{1}\subset{\rm Im}\
\sigma_{0}.$

If $\left|\nu_{0}\right|=0,$ we get $\left|\nu_{1}\right|^{2}\xi_{0}=0$ by
(34). Since $\nu\neq\mathbf{0},$ we have $\xi_{0}=0.$ Then we have

$\sigma_{0}(\nu)\left(\nu_{1}^{-1}\xi_{1}\right)=\left(\begin{matrix}0\\\
\xi_{1}\end{matrix}\right),$

since $\nu_{1}$ is reversible.

(3) By definition, for $\zeta=\left(\begin{matrix}\zeta_{0}\\\
\zeta_{1}\end{matrix}\right)\in\mathscr{V}_{1},$ we have

$\sigma_{2}(\nu)\zeta=\nu_{0}\zeta_{1}-\nu_{1}\zeta_{0}\in\mathbb{C}\otimes\mathbb{S}^{+}.$

For any $\zeta=\left(\begin{matrix}\zeta_{0}\\\
\zeta_{1}\end{matrix}\right)\in\ker\sigma_{2}(\nu),$ let
$\xi=\left(\begin{matrix}\xi_{0}\\\
\xi_{1}\end{matrix}\right)\in\mathscr{V}_{1}$ given by

(37)
$\displaystyle\xi_{0}=-\frac{\zeta_{1}}{|\nu_{0}|^{2}+|\nu_{1}|^{2}},\quad\xi_{1}=\frac{\zeta_{0}}{|\nu_{0}|^{2}+|\nu_{1}|^{2}}.$

Since $\zeta=\left(\begin{matrix}\zeta_{0}\\\
\zeta_{1}\end{matrix}\right)\in\ker\sigma_{2}(\nu),$ we have

(38) $\displaystyle\sigma_{2}(\nu)\zeta=\nu_{0}\zeta_{1}-\nu_{1}\zeta_{0}=0.$

Then we have

$\displaystyle\left(\sigma_{1}(\nu)\xi\right)_{0}=$
$\displaystyle\nu_{0}\nu_{0}\xi_{1}-\nu_{1}\nu_{0}\xi_{0}=\frac{|\nu_{0}|^{2}\zeta_{0}}{|\nu_{0}|^{2}+|\nu_{1}|^{2}}+\frac{\nu_{1}\nu_{0}\zeta_{1}}{|\nu_{0}|^{2}+|\nu_{1}|^{2}}$
$\displaystyle=$
$\displaystyle\frac{|\nu_{0}|^{2}\zeta_{0}}{|\nu_{0}|^{2}+|\nu_{1}|^{2}}+\frac{\nu_{1}\nu_{1}\zeta_{0}}{|\nu_{0}|^{2}+|\nu_{1}|^{2}}=\zeta_{0},$

by using (37)-(38). Similarly,

$\displaystyle\left(\sigma_{1}(\nu)\xi\right)_{1}=$
$\displaystyle\nu_{0}\nu_{1}\xi_{1}-\nu_{1}\nu_{1}\xi_{0}=\frac{\nu_{0}\nu_{1}\zeta_{0}}{|\nu_{0}|^{2}+|\nu_{1}|^{2}}+\frac{|\nu_{1}|^{2}\zeta_{1}}{|\nu_{0}|^{2}+|\nu_{1}|^{2}}$
$\displaystyle=$
$\displaystyle\frac{\nu_{0}\nu_{0}\zeta_{1}}{|\nu_{0}|^{2}+|\nu_{1}|^{2}}+\frac{|\nu_{1}|^{2}\zeta_{1}}{|\nu_{1}|^{2}+|\nu_{1}|^{2}}=\zeta_{1}.$

Thus $\sigma_{1}(\nu)\xi=\zeta$ and so $\ker\sigma_{2}(\nu)\subset{\rm
Im}\sigma_{1}.$

(4) For any $\rho\in\mathscr{V}_{3},$ let
$\zeta=\left(\begin{matrix}\zeta_{0}\\\
\zeta_{1}\end{matrix}\right)\in\mathscr{V}_{2}$ be given by

$\displaystyle\zeta_{0}=-\frac{\nu_{1}\rho}{|\nu_{0}|^{2}+|\nu_{1}|^{2}},\quad\zeta_{1}=\frac{\nu_{0}\rho}{|\nu_{0}|^{2}+|\nu_{1}|^{2}}.$

Then we have

$\displaystyle\sigma_{2}(\nu)\zeta=$
$\displaystyle\nu_{0}\zeta_{1}-\nu_{1}\zeta_{0}=\frac{\nu_{0}\nu_{0}\rho}{|\nu_{0}|^{2}+|\nu_{1}|^{2}}+\frac{\nu_{1}\nu_{1}\rho}{|\nu_{0}|^{2}+|\nu_{1}|^{2}}$
$\displaystyle=$
$\displaystyle\frac{|\nu_{0}|^{2}\rho}{|\nu_{0}|^{2}+|\nu_{1}|^{2}}+\frac{|\nu_{1}|^{2}\rho}{|\nu_{0}|^{2}+|\nu_{1}|^{2}}=\rho.$

So $\sigma_{2}$ is surjective. ∎

##  3\. Solutions to the non-homogeneous Dirac equations and the Hartogs’
phenomenon

###  3.1. Solutions to the non-homogeneous equations

The natural Hodge Laplacian associated to the differential complex (5) should
be

$\displaystyle\widetilde{\Box}_{j}=\mathscr{D}_{j-1}\mathscr{D}_{j-1}^{*}+\mathscr{D}_{j}^{*}\mathscr{D}_{j},$

for $j=1,2.$ But $\mathscr{D}_{1}$ is a differential operator of second order
while $\mathscr{D}_{0},\mathscr{D}_{2}$ are of first order, the principal
symbols of $\widetilde{\Box}_{1}$ and $\widetilde{\Box}_{2}$ are degenerate
and so $\widetilde{\Box}_{1}$ and $\widetilde{\Box}_{2}$ are not uniformly
elliptic. So it is better to consider the Laplacians (9) of forth order
associated to the differential complex (5).

######  Proposition 3.1.

(39)
$\displaystyle\Box_{1}=\left(\mathscr{D}_{0}\mathscr{D}_{0}^{*}\right)^{2}+\mathscr{D}_{1}^{*}\mathscr{D}_{1}=\left(\begin{array}[]{cc}\Delta^{2}&\\\
&\Delta^{2}\end{array}\right),$
$\displaystyle\Box_{2}=\mathscr{D}_{1}\mathscr{D}_{1}^{*}+\left(\mathscr{D}_{2}^{*}\mathscr{D}_{2}\right)^{2}=\left(\begin{array}[]{cc}\Delta^{2}&\\\
&\Delta^{2}\end{array}\right).$

###### Proof.

By using (24), it is direct to check by definition (23) that the formal
adjoints of $\mathscr{D}_{0},\mathscr{D}_{1},\mathscr{D}_{2}$ are given by

(40) $\displaystyle\mathscr{D}_{0}^{*}=$
$\displaystyle\left(\nabla_{0},\nabla_{1}\right),$
$\displaystyle\mathscr{D}_{1}^{*}=$
$\displaystyle\sum_{j,k}\left(\begin{array}[]{cc}-\nabla_{0}\nabla_{1}&-\Delta_{1}\\\
\Delta_{0}&\nabla_{1}\nabla_{0}\end{array}\right),$
$\displaystyle\mathscr{D}_{2}^{*}=$
$\displaystyle\left(\begin{matrix}-\nabla_{1}\\\
\nabla_{0}\end{matrix}\right),$

respectively. Recall $\mathscr{D}_{0}^{*}\mathscr{D}_{0}=\Delta$ by (26).
Compositions of matrix-valued differential operators give us that

(41) $\displaystyle\mathscr{D}_{0}\mathscr{D}_{0}^{*}=$
$\displaystyle\sum_{j,k}\left(\begin{array}[]{cc}\Delta_{0}&\nabla_{0}\nabla_{1}\\\
\nabla_{1}\nabla_{0}&\Delta_{1}\end{array}\right),$
$\displaystyle\mathscr{D}_{1}^{*}\mathscr{D}_{1}=$
$\displaystyle\sum_{j,k}\left(\begin{array}[]{cc}\Delta_{0}\Delta_{1}+\Delta_{1}^{2}&-\nabla_{0}\nabla_{1}\left(\Delta_{0}+\Delta_{1}\right)\\\
-\nabla_{1}\nabla_{0}\left(\Delta_{0}+\Delta_{1}\right)&\Delta_{0}\Delta_{1}+\Delta_{0}^{2}\end{array}\right),$
$\displaystyle\mathscr{D}_{1}\mathscr{D}_{1}^{*}=$
$\displaystyle\sum_{j,k}\left(\begin{array}[]{cc}\Delta_{0}\Delta_{1}+\Delta_{0}^{2}&\nabla_{1}\nabla_{0}\left(\Delta_{0}+\Delta_{1}\right)\\\
\nabla_{0}\nabla_{1}\left(\Delta_{0}+\Delta_{1}\right)&\Delta_{0}\Delta_{1}+\Delta_{1}^{2}\end{array}\right),$
$\displaystyle\mathscr{D}_{2}^{*}\mathscr{D}_{2}=$
$\displaystyle\left(\begin{array}[]{cc}\Delta_{1}&-\nabla_{1}\nabla_{0}\\\
-\nabla_{0}\nabla_{1}&\Delta_{0}\end{array}\right),$

by expressions in (40) and (21)-(22), and then we have

$\displaystyle\left(\mathscr{D}_{0}^{*}\mathscr{D}_{0}\right)^{2}=$
$\displaystyle\Delta^{2},$
$\displaystyle\left(\mathscr{D}_{0}\mathscr{D}_{0}^{*}\right)^{2}=$
$\displaystyle\sum_{j,k}\left(\begin{array}[]{cc}\Delta_{0}^{2}+\Delta_{0}\Delta_{1}&\nabla_{0}\nabla_{1}\left(\Delta_{0}+\Delta_{1}\right)\\\
\nabla_{1}\nabla_{0}\left(\Delta_{0}+\Delta_{1}\right)&\Delta_{0}\Delta_{1}+\Delta_{1}^{2}\end{array}\right),$
$\displaystyle\left(\mathscr{D}_{2}^{*}\mathscr{D}_{2}\right)^{2}=$
$\displaystyle\sum_{j,k}\left(\begin{array}[]{cc}\Delta_{1}^{2}+\Delta_{0}\Delta_{1}&-\nabla_{1}\nabla_{0}\left(\Delta_{0}+\Delta_{1}\right)\\\
-\nabla_{0}\nabla_{1}\left(\Delta_{0}+\Delta_{1}\right)&\Delta_{0}\Delta_{1}+\Delta_{0}^{2}\end{array}\right).$

Thus (39) follows. ∎

So $\Box_{j}$ are uniformly elliptic differential operator of fourth order. It
is known that

(42) $\displaystyle G_{0}=-\frac{C_{2n}}{|\mathbf{x}|^{2n-4}},$

is the fundamental solution of the operator $\Delta^{2}$ on $\mathbb{R}^{2n}$
for some positive constant $C_{2n}.$ Let

(45) $\displaystyle G_{1}=G_{2}=\left(\begin{array}[]{cc}G_{0}&\\\
&G_{0}\end{array}\right).$

The following proposition for homogeneous distributions and singular integral
operators is well known.

######  Proposition 3.2.

([13, Proposition 2.4.7]) Let $K\in
C^{\infty}\left(\mathbb{R}^{2n}\setminus\\{\mathbf{0}\\}\right)$ be a
homogeneous function of degree $k-2n.$ Let $\mathbf{K}$ be the operator
defined by $\mathbf{K}\phi=\phi*K.$ Then, for $\phi\in
C_{0}^{\infty}\left(\mathbb{R}^{2n}\right)$,
$A_{1},\cdots,A_{k}\in\\{0,1\\},j_{1},\cdots,j_{k}\in\\{1,\cdots,2n\\},$

(46)
$\displaystyle\partial_{A_{1}j_{1}}\cdots\partial_{A_{k}j_{k}}(\mathbf{K}\phi)=P.V.\left(\phi*\partial_{A_{1}j_{1}}\cdots\partial_{A_{k}j_{k}}K\right)+a_{A_{1}j_{1}\cdots
A_{k}j_{k}}\phi,$

and each term in (46) is $C^{\infty}$ and the identity holds as $C^{\infty}$
functions, where $a_{A_{1}j_{1}\cdots A_{k}j_{k}}$ is a constant.

Moreover, $\partial_{A_{1}j_{1}}\cdots\partial_{A_{k}j_{k}}K$ is a
Calderon–Zygmund kernel on $\mathbb{R}^{2n}.$ The singular integral operator
$f\rightarrow P.V.$
$\left(f*\partial_{A_{1}j_{1}}\cdots\partial_{A_{k}j_{k}}K\right)$ is bounded
on $L^{p}$ for $1<p<\infty.$ (46) holds as $L^{p}$ functions.

######  Proposition 3.3.

The Laplacian $\Box_{j}$ has the inverse $\mathbf{G}_{j}$ in
$L^{2}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right),j=0,1,2,$ which is a
convolution operator with kernel $G_{j}.$ The kernel $G_{j}$ is a smooth
homogeneous functions of degree $4-2n.$ Moreover, $\mathbf{G}_{j}$ can be
extended to a bounded linear operator from
$L^{p}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right)$ to
$W^{4,p}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right),$ for $1<p<\infty.$ If a
$\mathscr{V}_{j}$-valued function $f$ is in
$C_{0}^{k}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right),$ for any non-negative
integer $k,$ we have $\mathbf{G}_{j}f\in
C^{k+3}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right).$

###### Proof.

For $f\in C_{0}^{\infty}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right).$ Let

(47)
$\displaystyle\mathbf{G}_{j}f=\int_{\mathbb{R}^{2n}}G_{j}(\mathbf{x}-\mathbf{y})f(\mathbf{y})\hbox{d}V(\mathbf{y}),\
j=0,1,2.$

Note that for $u\in C_{0}^{\infty}\left(\mathbb{R}^{2n}\right)$

$\displaystyle\int_{\mathbb{R}^{2n}}G_{0}(\mathbf{x}-\mathbf{y})\Delta^{2}u(\mathbf{y}){\rm
d}V(\mathbf{y})=u(\mathbf{x}),$

since $G_{0}$ is the fundamental solution to $\Delta^{2}.$ Then
$\mathbf{G}_{j}$ is the inverse operator of Laplacian $\Box_{j},$ i.e.

$\displaystyle\mathbf{G}_{j}\Box_{j}f=\Box_{j}\mathbf{G}_{j}f=f.$

$G_{0}$ satisfies the following decay estimates

(48)
$\displaystyle\left|\partial_{{A_{1}j_{1}}}\cdots\partial_{{A_{m}j_{m}}}G_{0}\right|\leq\frac{C_{A_{1}j_{1}\cdots
A_{m}j_{m}}}{|\mathbf{x}|^{2n-4+m}},$

for some constant $C_{A_{1}j_{1}\cdots A_{m}j_{m}}>0$ depending on
$A_{1},\cdots,A_{m}\in\\{0,1\\},$ $j_{1},\cdots,$ $j_{m}\in\\{1,\cdots,n\\}.$
By Proposition 3.2, the convolution with a homogeneous function of degree
$-2n+k$ can be extended to a bounded linear operator from
$L^{p}\left(\mathbb{R}^{2n}\right)$ to $W^{k,p}\left(\mathbb{R}^{2n}\right).$

When $f\in C_{0}^{k}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right),$ it follows
from (46) that $\mathbf{G}_{j}f\in
C^{k+3}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right)$ by differentiation. The
proposition is proved. ∎

######  Proposition 3.4.

A monogenic function $f\in\mathcal{O}(\Omega)$ on a domain
$\Omega\in\mathbb{R}^{2n}$ is real analytic.

###### Proof.

Note that

(49)
$\displaystyle\Box_{0}f=\left(\mathscr{D}_{0}^{*}\mathscr{D}_{0}\right)^{2}f=\Delta^{2}f=0,$

in the sense of distributions. Thus $f$ is biharmonic and so it is real
analytic at each point $\mathbf{x}\in\Omega.$ ∎

Now we can prove the Theorem 1.1.

Proof of Theorem 1.1. We prove the case $j=1,$ the case $j=2$ is similar.
Recall that $\mathbf{G}_{1}$ is the inverse operator of $\Box_{1}$ on
$C_{0}^{\infty}\left(\mathbb{R}^{2n},\mathscr{V}_{1}\right).$ Set

$\displaystyle
u:=\mathscr{D}_{0}^{*}\mathscr{D}_{0}\mathscr{D}_{0}^{*}\mathbf{G}_{1}f\in
C^{\infty}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right).$

Note that

(50) $\displaystyle\mathscr{D}_{1}\Box_{1}=$
$\displaystyle\mathscr{D}_{1}\left(\mathscr{D}_{1}^{*}\mathscr{D}_{1}+\left(\mathscr{D}_{0}\mathscr{D}_{0}^{*}\right)^{2}\right)=\mathscr{D}_{1}\mathscr{D}_{1}^{*}\mathscr{D}_{1}$
$\displaystyle=$
$\displaystyle\left(\mathscr{D}_{1}\mathscr{D}_{1}^{*}+\left(\mathscr{D}_{2}^{*}\mathscr{D}_{2}\right)^{2}\right)\mathscr{D}_{1}=\Box_{2}\mathscr{D}_{1},$

on $C_{0}^{\infty}\left(\mathbb{R}^{2n},\mathscr{V}_{2}\right)$ by
$\mathscr{D}_{1}\mathscr{D}_{0}=0,\mathscr{D}_{2}\mathscr{D}_{1}=0.$ Then we
have

$\displaystyle\Box_{2}\left(\mathbf{G}_{2}\mathscr{D}_{1}f-\mathscr{D}_{1}\mathbf{G}_{1}f\right)=\mathscr{D}_{1}f-\mathscr{D}_{1}\Box_{1}\mathbf{G}_{1}f=0,$

by using (50). By Proposition 3.1, each entry of the $2\times 2$ matrix
$\mathbf{G}_{2}\mathscr{D}_{1}f-\mathscr{D}_{1}\mathbf{G}_{1}f$ is biharmonic.
On the other hand, we have

$\left\|\mathbf{G}_{2}\mathscr{D}_{1}f(\mathbf{x})-\mathscr{D}_{1}\mathbf{G}_{1}f(\mathbf{x})\right\|\leq\frac{C}{\left(1+|\mathbf{x}|\right)^{2n-2}},$

for some constant $C>0$ by (48). Thus
$\mathbf{G}_{2}\mathscr{D}_{1}f-\mathscr{D}_{1}\mathbf{G}_{1}f$ is constant by
Liouville-type theorem [15], i.e.

(51)
$\displaystyle\mathbf{G}_{2}\mathscr{D}_{1}=\mathscr{D}_{1}\mathbf{G}_{1},$

on $C_{0}^{\infty}\left(\mathbb{R}^{2n},\mathscr{V}_{2}\right).$ Thus

(52)
$\displaystyle\mathscr{D}_{0}u=\mathscr{D}_{0}\mathscr{D}_{0}^{*}\mathscr{D}_{0}\mathscr{D}_{0}^{*}\mathbf{G}_{1}f=\left(\left(\mathscr{D}_{0}\mathscr{D}_{0}^{*}\right)^{2}+\mathscr{D}_{1}^{*}\mathscr{D}_{1}\right)\mathbf{G}_{1}f=f,$

by

(53)
$\displaystyle\mathscr{D}_{1}\mathbf{G}_{1}f=\mathbf{G}_{2}\mathscr{D}_{1}f=0.$

Thus $u=\mathscr{D}_{0}^{*}\mathscr{D}_{0}\mathscr{D}_{0}^{*}\mathbf{G}_{1}f$
satisfies (13) for $f\in
C_{0}^{\infty}\left(\mathbb{R}^{2n},\mathscr{V}_{1}\right).$

Note that $\mathbf{G}_{j},j=0,1,2,$ is formally self-adjoint in the following
sense:

$\displaystyle\left(\mathbf{G}_{j}\phi,\psi\right)_{\mathscr{V}_{j}}=\left(\phi,\mathbf{G}_{j}\psi\right)_{\mathscr{V}_{j}},$

for any $\phi,\psi\in
C_{0}^{\infty}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right),$ by the explicit
expression of $G_{j},j=0,1,2,$ (42)-(45). Also $\mathbf{G}_{j}$ is bounded
from $L^{2}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right)$ to
$W^{4,2}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right)$ by Proposition 3.3. So
$\mathbf{G}_{j}$ can be extended to a bounded operator from
$W^{-4,2}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right)$ to
$L^{2}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right)$ by the duality argument.
In particular, it is bounded from
$W^{-2,2}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right)$ to
$L^{2}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right).$ Therefore, the identity
(51) holds as boun-ded linear operators on
$L^{2}\left(\mathbb{R}^{2n},\mathscr{V}_{1}\right)$ and so the identity (53)
holds for $f\in L^{2}\left(\mathbb{R}^{2n},\mathscr{V}_{1}\right).$ Thus (52)
holds as $L^{2}$ functions for $f\in
L^{2}\left(\mathbb{R}^{2n},\mathscr{V}_{1}\right)$ satisfying
$\mathscr{D}_{1}f=0$ in the sense of distributions. Thus
$u=\mathscr{D}_{0}^{*}\mathscr{D}_{0}\mathscr{D}_{0}^{*}\mathbf{G}_{1}f$
satisfies the equation (13).

Suppose that $f$ is supported in $\Omega\Subset\mathbb{R}^{2n}.$ Since the
integral kernel $K(\mathbf{x})$ of
$\mathscr{D}_{0}^{*}\mathscr{D}_{0}\mathscr{D}_{0}^{*}\mathbf{G}_{1}$ decays
as $|\mathbf{x}|^{-2n+1}$ for large $|\mathbf{x}|$ by decay estimate (48) and
$f$ is compactly supported, we see that

(54)
$\displaystyle\left|u(\mathbf{x})\right|=\left|\int_{\mathbb{R}^{2n}}K\left(\mathbf{x}-{\mathbf{y}}\right)f\left({\mathbf{y}}\right)\hbox{d}V\left({\mathbf{y}}\right)\right|\leq\frac{C}{\left(1+|\mathbf{x}|\right)^{2n-1}},$

for some constant $C>0.$ So
$\lim_{|\mathbf{x}|\rightarrow\infty}u(\mathbf{x})=0.$ For
$\mathbf{x}=\left(\mathbf{x}_{0},\mathbf{x}_{1}\right),$ if $|\mathbf{x}_{0}|$
is so large that

$\left(\left\\{\mathbf{x}_{0}\right\\}\times\mathbb{R}^{n}\right)\cap\bar{\Omega}=\emptyset,$

then $u\left(\mathbf{x}_{0},\mathbf{x}_{1}\right)$ is a monogenic function in
$\mathbf{x}_{1},$ which vanishes at infinity. Since

(55) $\displaystyle\Delta_{1}u\left(\mathbf{x}_{0},\mathbf{x}_{1}\right)=0,$

by (20) each component of $u\left(\mathbf{x}_{0},\cdot\right)$ is a biharmonic
function on $\mathbb{R}^{n}$ vanishing at infinity and so is bounded. Hence
$u\left(\mathbf{x}_{0},\cdot\right)\equiv 0$ for $|\mathbf{x}_{0}|$ large by
Liouville-type theorem again. By (49), we see that $u(\cdot)$ is monogenic on
$\mathbb{R}^{2n}\setminus\Omega.$ So $u\equiv 0$ on the unbounded connected
component of $\mathbb{R}^{2n}\setminus\Omega$ by the identity theorem for real
analytic function, since $u$ is real analytic on
$\mathbb{R}^{2n}\setminus\Omega$ by Proposition 3.4. The continuity of $u$
follows from the formula (54). The theorem is proved. ∎

###  3.2. The Bochner–Martinelli formula on $\mathbb{R}^{2n}$

Denote

(56) $\displaystyle
H(\mathbf{x}):=\mathscr{D}_{0}^{*}\mathscr{D}_{0}\mathscr{D}_{0}^{*}G_{1}(\mathbf{x}),$

whose entries are
$C^{\infty}\left(\mathbb{R}^{2n}\setminus\\{\mathbf{0}\\}\right)$ homogeneous
functions of degree $1-2n,$ where $G_{1}(\cdot)$ is given by (45). By (41), we
have

$\displaystyle H(\mathbf{x})=$
$\displaystyle\mathscr{D}_{0}^{*}\mathscr{D}_{0}\mathscr{D}_{0}^{*}\left(\begin{array}[]{cc}G_{0}&\\\
&G_{0}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\nabla_{0},\nabla_{1}\right)\left(\begin{array}[]{cc}\Delta_{0}&\nabla_{0}\nabla_{1}\\\
\nabla_{1}\nabla_{0}&\Delta_{1}\end{array}\right)\left(\begin{array}[]{cc}G_{0}&\\\
&G_{0}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\nabla_{0}\Delta,\nabla_{1}\Delta\right)\left(\begin{array}[]{cc}G_{0}&\\\
&G_{0}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\sum_{j}\gamma_{j}\partial_{0j}\Delta
G_{0},\sum_{k}\gamma_{k}\partial_{1k}\Delta G_{0}\right)$ $\displaystyle=$
$\displaystyle
4(n-2)C_{2n}\left(\sum_{j}\gamma_{j}\partial_{0j}|\mathbf{x}|^{2-2n},\sum_{k}\gamma_{k}\partial_{1k}|\mathbf{x}|^{2-2n}\right)$
$\displaystyle=$
$\displaystyle-8(n-1)(n-2)C_{2n}\left(\sum_{j}\frac{\gamma_{j}x_{0j}}{|\mathbf{x}|^{2n}},\sum_{k}\frac{\gamma_{k}x_{1k}}{|\mathbf{x}|^{2n}}\right),$

for $\mathbf{x}\in\mathbb{R}^{2n}\setminus\\{\mathbf{0}\\},$ where $C_{2n}$ is
the positive constant in (42).

######  Theorem 3.1.

Suppose that $\Omega$ is a bounded domain in $\mathbb{R}^{2n}$ with $C^{2}$
boundary, and suppose that $f:\Omega\rightarrow\mathscr{V}_{1}$ is a
continuous function of class $W^{1,2}(U)$ for a domain $U$ with
$\overline{\Omega}\subset U\subset\mathbb{R}^{2n}.$ Let $n(\mathbf{x})$ be the
unit outer normal vector to the surface $\partial\Omega,$ we have

$\displaystyle
f(\mathbf{x})=-\int_{\partial\Omega}H(\mathbf{x}-\mathbf{y})\left(\begin{matrix}n_{0}\\\
n_{1}\end{matrix}\right)f(\mathbf{y}){\rm
d}S(\mathbf{y})+\int_{\Omega}H(\mathbf{x}-\mathbf{y})\mathscr{D}_{0}f(\mathbf{y}){\rm
d}V(\mathbf{y}),$

where ${\rm d}S$ is the surface measure and $\left(\begin{matrix}n_{0}\\\
n_{1}\end{matrix}\right)=\left(\begin{matrix}\sum_{j}\gamma_{j}n_{0j}\\\
\sum_{j}\gamma_{j}n_{1j}\end{matrix}\right).$

###### Proof.

Since

$\displaystyle\mathscr{D}_{0}\Box_{0}=\mathscr{D}_{0}\mathscr{D}_{0}^{*}\mathscr{D}_{0}\mathscr{D}_{0}^{*}\mathscr{D}_{0}=\left(\left(\mathscr{D}_{0}\mathscr{D}_{0}^{*}\right)^{2}+\mathscr{D}_{1}^{*}\mathscr{D}_{1}\right)\mathscr{D}_{0}=\Box_{1}\mathscr{D}_{0},$

by $\mathscr{D}_{1}\mathscr{D}_{0}=0.$ As in the proof of Theorem 1.1, we have
$\mathbf{G}_{1}\mathscr{D}_{0}=\mathscr{D}_{0}\mathbf{G}_{0},$ which holds as
operator from $L^{2}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right)$ to
$W^{3,2}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right)$ by Proposition 3.3. So

(57) $\displaystyle{\rm
id}=\left(\mathscr{D}_{0}^{*}\mathscr{D}_{0}\right)^{2}\mathbf{G}_{0}=\mathscr{D}_{0}^{*}\mathscr{D}_{0}\mathscr{D}_{0}^{*}\mathbf{G}_{1}\mathscr{D}_{0}$

on $L^{2}.$ It is sufficient to show the theorem for $f\in C^{\infty}(U).$
Suppose that $\epsilon$ is sufficiently small, let
$\phi_{\epsilon}(\mathbf{x})=\epsilon^{-2n}\phi\left(\epsilon^{-1}\mathbf{x}\right).$
Then $\left(\chi_{\Omega}*\phi_{\epsilon}\right)\cdot f\in
C_{0}^{\infty}\left(\mathbb{R}^{2n},\mathscr{V}_{1}\right).$ Apply (57) to
$\left(\chi_{\Omega}*\phi_{\epsilon}\right)\cdot f$ to get

(58) $\displaystyle\left(\chi_{\Omega}*\phi_{\epsilon}\cdot
f\right)(\mathbf{x})=$
$\displaystyle\mathscr{D}_{0}^{*}\mathscr{D}_{0}\mathscr{D}_{0}^{*}\mathbf{G}_{1}\mathscr{D}_{0}\left(\chi_{\Omega}*\phi_{\epsilon}\cdot
f\right)(\mathbf{x})$ $\displaystyle=$
$\displaystyle\mathscr{D}_{0}^{*}\mathscr{D}_{0}\mathscr{D}_{0}^{*}\int_{\mathbb{R}^{2n}}G_{1}(\mathbf{x}-\mathbf{y})\chi_{\Omega}*\phi_{\epsilon}(\mathbf{y})\mathscr{D}_{0}f(\mathbf{y}){\rm
d}V(\mathbf{y})$
$\displaystyle+\mathscr{D}_{0}^{*}\mathscr{D}_{0}\mathscr{D}_{0}^{*}\int_{\mathbb{R}^{2n}}G_{1}(\mathbf{x}-\mathbf{y})\chi_{\Omega}*\left(\mathscr{D}_{0}\phi_{\epsilon}(\mathbf{y})\right)f(\mathbf{y}){\rm
d}V(\mathbf{y}).$

It follows from the definition of $H(\cdot)$ in (56) that its entries are
$C^{\infty}\left(\mathbb{R}^{2n}\right.$
$\left.\setminus\\{\mathbf{0}\\}\right)$ homogeneous functions of degree
$1-2n,$ since $G_{1}(\cdot)$ is a matrix with each entry a
$C^{\infty}\left(\mathbb{R}^{2n}\setminus\\{\mathbf{0}\\}\right)$ homogeneous
function of degree $4-2n$ by Proposition 3.3. Note that for a fixed
$\mathbf{x}\in\Omega,$ $\chi_{\Omega}*\phi_{\epsilon}(\mathbf{x})=1$ if
$\epsilon$ is sufficiently small. (58) can be rewritten as

(59) $\displaystyle f(\mathbf{x})=$
$\displaystyle\int_{\mathbb{R}^{2n}}H(\mathbf{x}-\mathbf{y})\chi_{\Omega}*\phi_{\epsilon}(\mathbf{y})\mathscr{D}_{0}f(\mathbf{y}){\rm
d}V(\mathbf{y})$
$\displaystyle+\int_{\mathbb{R}^{2n}}H(\mathbf{x}-\mathbf{y})\chi_{\Omega}*\mathscr{D}_{0}\phi_{\epsilon}(\mathbf{y})f(\mathbf{y}){\rm
d}V(\mathbf{y}).$

The first term in (59) converges obviously for smooth $f$ as
$\epsilon\rightarrow 0.$ For the second term, to see
$\mathscr{D}_{0}\phi_{\epsilon}*\chi_{\Omega}$ converging to a matrix-valued
measure on $\partial\Omega,$ note that

(60) $\displaystyle\frac{\partial}{\partial
x_{Aj}}\left(\phi_{\epsilon}*\chi_{\Omega}\right)(\mathbf{x})=$
$\displaystyle\int_{\Omega}\frac{\partial\phi_{\epsilon}}{\partial
x_{Aj}}(\mathbf{x}-\mathbf{y}){\rm d}V(\mathbf{y})$ $\displaystyle=$
$\displaystyle-\int_{\Omega}\frac{\partial}{\partial
y_{Aj}}\left(\phi_{\epsilon}(\mathbf{x}-\mathbf{y})\right){\rm
d}V(\mathbf{y})$ $\displaystyle=$
$\displaystyle-\int_{\partial\Omega}\phi_{\epsilon}(\mathbf{x}-\mathbf{y})n_{Aj}(\mathbf{y}){\rm
d}S(\mathbf{y}),$

where $n_{Aj}(\mathbf{y})$ is the $(Aj)$-th component of the unit outer normal
vector to $\partial\Omega,A=0,1,j=1,\cdots,n.$ By (60), we see that the
support of $\mathscr{D}_{0}\phi_{\epsilon}*\chi_{\Omega}$ is a small
neighborhood of $\partial\Omega$ if $\epsilon$ is sufficiently small. So for a
function $g$ continuous on a neighborhood of $\partial\Omega,$ we have

$\displaystyle\int_{\mathbb{R}^{2n}}g(\mathbf{x})\mathscr{D}_{0}\phi_{\epsilon}*\chi_{\Omega}(\mathbf{x}){\rm
d}V(\mathbf{x})$ $\displaystyle=$
$\displaystyle-\int_{\mathbb{R}^{2n}}g(\mathbf{x}){\rm
d}V(\mathbf{x})\int_{\partial\Omega}\phi_{\epsilon}(\mathbf{x}-\mathbf{y})\left(\begin{matrix}n_{0}\\\
n_{1}\end{matrix}\right)(\mathbf{y}){\rm d}S(\mathbf{y})$ $\displaystyle=$
$\displaystyle-\int_{\partial\Omega}\left(\int_{\mathbb{R}^{2n}}g(\mathbf{x})\phi_{\epsilon}(\mathbf{x}-\mathbf{y}){\rm
d}V(\mathbf{x})\right)\left(\begin{matrix}n_{0}\\\
n_{1}\end{matrix}\right)(\mathbf{y}){\rm d}S(\mathbf{y})$
$\displaystyle\rightarrow$
$\displaystyle-\int_{\partial\Omega}g(\mathbf{y})\left(\begin{matrix}n_{0}\\\
n_{1}\end{matrix}\right)(\mathbf{y}){\rm d}S(\mathbf{y}),$

as $\epsilon\rightarrow 0,$ by
$\int_{\mathbb{R}^{2n}}g(\mathbf{x})\phi_{\epsilon}(\mathbf{x}-\mathbf{y}){\rm
d}V(\mathbf{x})=g*\phi_{\epsilon}(\mathbf{y})\rightarrow g(\mathbf{y}).$ The
result follows.∎

######  Remark 3.1.

By Theorem 3.1, for a monogenic function $f\in
C\left(\overline{\Omega},\mathbb{S}^{+}\right)\cap\
C^{1}\left({\Omega},\mathbb{S}^{+}\right),$ where $\Omega$ is an open bounded
set of $\mathbb{R}^{2n}$ with $C^{1}$ boundary, the Cauchy-type formula holds,
i.e.

$\displaystyle
f(\mathbf{x})=-\int_{\partial\Omega}H(\mathbf{x}-\mathbf{y})\left(\begin{matrix}n_{0}\\\
n_{1}\end{matrix}\right)f(\mathbf{y}){\rm d}S(\mathbf{y}).$

###  3.3. Hartogs’ extension phenomenon for monogenic functions

Now we can prove the Hartogs’ extension phenomenon for monogenic functions.

Proof of Theorem 1.2. By Proposition 2.1, a monogenic function is biharmonic
and so it is $C^{\infty}.$ Let $\chi\in C_{0}^{\infty}(\Omega)$ be equal to
$1$ in a neighborhood of $K.$ Set $\tilde{u}:=(1-\chi)u,$ which vanishes on
$K.$ Then $\tilde{u}\in C^{\infty}(\Omega,\mathbb{S}^{+}).$ Let $\tilde{U}$ be
the solution to the non-homogeneous

$\displaystyle\mathscr{D}_{0}\tilde{U}=\mathscr{D}_{0}\tilde{u}=-\mathscr{D}_{0}\chi
u=f,$

where $f,$ defined as $0$ in $K$ and outside $\Omega,$ has components in
$C_{0}^{\infty}\left(\mathbb{R}^{2n},\mathbb{S}^{+}\right)$ and satisfies the
compatible condition
$\mathscr{D}_{1}f=\mathscr{D}_{1}\mathscr{D}_{0}\tilde{u}=0.$ It follows from
Theorem 1.1 that there exists such a solution $\tilde{U}$ which vanishes in
the unbounded component of the complement of the support of $\chi.$ Then the
function

$\displaystyle U=\tilde{u}-\tilde{U}$

is monogenic in $\Omega$ since
$\mathscr{D}_{0}\left(\tilde{u}-\tilde{U}\right)=0$ on $\Omega.$ Note that
$\tilde{U}$ on $\tilde{\Omega}=\mathbb{R}^{2n}\setminus{\rm supp}\chi$ and
supp$\chi\subset K^{\prime}$ for some compact set $K\subset
K^{\prime}\Subset\Omega.$ So $U=u$ on $\Omega\setminus K^{\prime}.$ Then $U=u$
on $\Omega\setminus K$ by the identity theorem. The theorem is proved. ∎

##  4\. The generalization of Malgrange’s vanishing theorem and the
Hartogs–Bochner extension for monogenic functions

A _cohomological complex_ of topological vector spaces is a pair
$(E^{\bullet},{\rm d}),$ where
$E^{\bullet}=\left(E^{q}\right)_{q\in\mathbb{Z}}$ is a sequence of topological
vector spaces and ${\rm d}=\left(\hbox{d}^{q}\right)_{q\in\mathbb{Z}}$ is a
sequence of continuous linear maps $\hbox{d}^{q}:E^{q}\rightarrow E^{q+1}$
satisfying $\hbox{d}^{q+1}\circ\hbox{d}^{q}=0.$ Its cohomology groups
$H^{q}\left(E^{\bullet}\right)$ are the quotient spaces $\ker\hbox{d}^{q}/{\rm
Im}\ \hbox{d}^{q-1},$ endowed with the quotient topology. A _homological
complex_ of topological vector spaces is a pair
$\left(E_{\bullet},\hbox{d}\right)$ where
$E_{\bullet}=\left(E_{q}\right)_{q\in\mathbb{Z}}$ is a sequence of topological
vector spaces and ${\rm d}=\left(\hbox{d}_{q}\right)_{q\in\mathbb{Z}}$ is a
sequence of continuous linear maps $\hbox{d}_{q}:E_{q}\rightarrow E_{q-1}$
satisfying $\hbox{d}_{q-1}\circ\hbox{d}_{q}=0.$ Its homology groups
$H_{q}\left(E_{\bullet}\right)$ are the quotient spaces
$\ker\hbox{d}_{q-1}/{\rm Im}\ \hbox{d}_{q},$ endowed with the quotient
topology. The dual complex of a cohomological complex $(E^{\bullet},{\rm d})$
of topological vector spaces is the homological complex
$(E^{\prime}_{\bullet},{\rm d}^{\prime}),$ where
$E^{\prime}_{\bullet}=\left(E^{\prime}_{q}\right)_{q\in\mathbb{Z}}$ with
$E^{\prime}_{q}$ the dual of $E_{q}$ and
$\hbox{d}^{\prime}=\left(\hbox{d}_{q}^{\prime}\right)_{q\in\mathbb{Z}}$ with
$\hbox{d}_{q}^{\prime}$ the transpose map of $\hbox{d}_{q}.$

Recall that a _Fréchet–Schwartz space_ is a topological vector space whose
topology is defined by an increasing sequence of seminorms such that the unit
ball with respect to the seminorm is relatively compact for the topology
associated to the previous seminorm. We need the following abstract duality
theorem.

######  Theorem 4.1.

([19, Theorem 1.6]) Let $(E^{\bullet},{\rm d})$ be a cohomological complex of
Fréchet–Schwartz spaces or of dual of Fréchet–Schwartz spaces and let
$(E_{\bullet},{\rm d})$ be its dual complex. For each $q\in\mathbb{Z}$, the
following assertions are equivalent:
(1) ${\rm Im}\ {\rm d}^{q}=\left\\{g\in E^{q+1}|\langle g,f\rangle=0{\rm\ for\
any}\ f\in\ker{\rm d}_{q}^{\prime}\right\\};$
(2) $H^{q+1}\left(E^{\bullet}\right)$ is separated;
(3) ${\rm d}^{q}$ is a topological homomorphism;
(4) ${\rm d}^{\prime}_{q}$ is a topological homomorphism;
(5) $H_{q}\left(E^{\bullet}\right)$ is separated;
(6) ${\rm Im}\ {\rm d}^{\prime}_{q}=\left\\{f\in E^{\prime}_{q}|\langle
f,g\rangle=0{\rm\ for\ any}\ g\in\ker{\rm d}^{q}\right\\}.$

A continuous linear map $\Psi$ between topological vector spaces $L_{1}$ and
$L_{2}$ is called a topological homomorphism if for each open subset $U\subset
L_{1},$ the image $\Psi(U)$ is an open subset of $\Psi\left(L_{1}\right)$. It
is known that if $L_{1}$ is a Fréchet space, $\Psi$ is a topological
homomorphism if and only if $\Psi\left(L_{1}\right)$ is closed [27]. See e.g.
[5, 19, 34] for applications of abstract duality theorem to $\bar{\partial}$
or $\bar{\partial}_{b}$-complex or tangential $k$-Cauchy–Fueter complex,
respectively. We adapt their methods to the Dirac complex.

For a complex vector space $V,$ let
$\mathcal{E}\left(\mathbb{R}^{2n},V\right)$ be the space of smooth $V$-valued
functions with the topology of uniform convergence on compact sets of the
functions and all their derivatives. Endowed with this topology
$\mathcal{E}\left(\mathbb{R}^{2n},V\right)$ is a Fréchet–Schwartz space. Let
$\mathcal{D}\left(\mathbb{R}^{2n},V\right)$ be the space of compactly
supported elements of $\mathcal{E}\left(\mathbb{R}^{2n},V\right).$ For a
compact subset $K$ of $\mathbb{R}^{2n},$ let
$\mathcal{D}_{K}\left(\mathbb{R}^{2n},V\right)$ be the closed subspace of
$\mathcal{E}\left(\mathbb{R}^{2n},V\right)$ with support in $K$ endowed with
the induced topology. Choose $\left\\{K_{n}\right\\}_{n\in\mathbb{N}}$ an
exhausting sequence of compact subsets of $\mathbb{R}^{2n}.$ Then
$\mathcal{D}\left(\mathbb{R}^{2n},V\right)=\cup_{n=1}^{\infty}\mathcal{D}_{K_{n}}\left(\mathbb{R}^{2n},V\right)$
[19]. We put on $\mathcal{D}\left(\mathbb{R}^{2n},V\right)$ the strict
inductive limit topology defined by the Fréchet–Schwartz spaces
$\mathcal{D}_{K_{n}}\left(\mathbb{R}^{2n},V\right).$ Denote by
$\mathcal{E}^{\prime}\left(\mathbb{R}^{2n},V\right)$ the dual of
$\mathcal{E}\left(\mathbb{R}^{2n},V\right)$ and
$\mathcal{D}^{\prime}\left(\mathbb{R}^{2n},V\right)$ the dual of
$\mathcal{D}\left(\mathbb{R}^{2n},V\right).$

Noting that the dual of $V$ is $V$ itself as a finite dimensional complex
vector space. The dual of the complex

$0\rightarrow\mathcal{D}\left(\mathbb{R}^{2n},{\mathscr{V}}_{0}\right)\rightarrow\mathcal{D}\left(\mathbb{R}^{2n},{\mathscr{V}}_{1}\right)\rightarrow\mathcal{D}\left(\mathbb{R}^{2n},{\mathscr{V}}_{2}\right)\rightarrow\mathcal{D}\left(\mathbb{R}^{2n},{\mathscr{V}}_{3}\right)\rightarrow
0,$

is

(61) $\displaystyle
0\leftarrow\mathcal{D}^{\prime}\left(\mathbb{R}^{2n},{\mathscr{V}}_{0}\right)\xleftarrow{\widehat{\mathscr{D}}_{0}}\mathcal{D}^{\prime}\left(\mathbb{R}^{2n},{\mathscr{V}}_{1}\right)\xleftarrow{\widehat{\mathscr{D}}_{1}}\mathcal{D}^{\prime}\left(\mathbb{R}^{2n},{\mathscr{V}}_{2}\right)\xleftarrow{\widehat{\mathscr{D}}_{2}}\mathcal{D}^{\prime}\left(\mathbb{R}^{2n},{\mathscr{V}}_{3}\right)\leftarrow
0.$

The dual can be realization as follows. For
$F\in\mathcal{E}\left(\mathbb{R}^{2n},{\mathscr{V}}_{j}\right),$ we can define
a functional on $\mathcal{D}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right)$ by

$\displaystyle\langle F,\phi\rangle:=\int_{\mathbb{R}^{2n}}\langle
F,\phi\rangle_{\mathscr{V}_{j}}\hbox{d}V,$

for $\phi\in\mathcal{D}\left(\mathbb{R}^{2n},\mathscr{V}_{j}\right).$ Then for
$\mathbb{F}=\left(\begin{matrix}\mathbb{F}_{1}\\\
\mathbb{F}_{2}\end{matrix}\right)\in\mathcal{E}\left(\mathbb{R}^{2n},{\mathscr{V}}_{1}\right)$
and $f\in\mathcal{D}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right),$ we have

$\displaystyle\left\langle\widehat{\mathscr{D}}_{0}\mathbb{F},f\right\rangle=$
$\displaystyle\left\langle\mathbb{F},\mathscr{D}_{0}f\right\rangle=\int_{\mathbb{R}^{2n}}\left\langle\mathbb{F},\mathscr{D}_{0}f\right\rangle_{\mathscr{V}_{1}}\hbox{d}V$
$\displaystyle=$
$\displaystyle\sum_{A=0,1}\int_{\mathbb{S}^{+}}\left\langle\mathbb{F}_{A},\nabla_{A}f\right\rangle_{\mathbb{S}^{+}}\hbox{d}V=\int_{\mathbb{R}^{2n}}\left\langle\sum_{A=0,1}\nabla_{A}\mathbb{F}_{A},{f}\right\rangle_{\mathscr{V}_{0}}\hbox{d}V.$

So when acting on smooth elements $\widehat{\mathscr{D}}_{0}$ in (61) is a
differential operator:

(62)
$\displaystyle\widehat{\mathscr{D}}_{0}\mathbb{F}=\sum_{A=0,1}\nabla_{A}\mathbb{F}_{A}.$

######  Proposition 4.1.

Suppose $K\Subset\mathbb{R}^{2n}.$ Then

$\displaystyle\|\mathscr{D}_{0}f\|_{L^{2}}^{2}=\|f\|_{W^{1,2}}^{2}-\|f\|_{L^{2}}^{2},$

for $f\in C_{0}^{\infty}\left(K,\mathscr{V}_{0}\right).$

###### Proof.

By (26),

$\displaystyle\|\mathscr{D}_{0}f\|_{L^{2}}^{2}=$
$\displaystyle\int_{\mathbb{R}^{2n}}\left\langle\mathscr{D}_{0}f,\mathscr{D}_{0}f\right\rangle_{\mathscr{V}_{1}}{\rm
d}V=\int_{\mathbb{R}^{2n}}\left\langle\mathscr{D}_{0}^{*}\mathscr{D}_{0}f,f\right\rangle_{\mathbb{S}^{+}}{\rm
d}V$ $\displaystyle=$ $\displaystyle\int_{\mathbb{R}^{2n}}\left\langle\Delta
f,f\right\rangle_{\mathbb{S}^{+}}{\rm d}V=\sum_{A,j}\left\|\frac{\partial
f}{\partial
x_{Aj}}\right\|_{L^{2}}^{2}=\|f\|_{W^{1,2}}^{2}-\|f\|_{L^{2}}^{2},$

by the definition of $W^{1,2}$-norm. The proposition is proved. ∎

Then we have the following estimate by the standard procedure (cf. [22]).

######  Corollary 4.1.

For any $K\Subset\mathbb{R}^{2n},$ there are constants $C_{s,K}>0,c_{s,K}\geq
0$ such that

(63) $\displaystyle
C_{s,K}\|f\|_{W^{s,2}}^{2}+\left\|\mathscr{D}_{0}f\right\|_{W^{s,2}}^{2}\geq
c_{s,K}\|f\|_{W^{s+1,2}}^{2},$

for any $f\in\mathcal{E}^{\prime}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right)$
with $\mathscr{D}_{0}f\in W^{s,2}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right)$
and supp$f\subset K.$

######  Theorem 4.2.

$\mathscr{D}_{0}:\mathcal{E}^{\prime}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right)\rightarrow\mathcal{E}^{\prime}\left(\mathbb{R}^{2n},\mathscr{V}_{1}\right)$
and
$\mathscr{D}_{0}:\mathcal{D}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right)\rightarrow\mathcal{D}\left(\mathbb{R}^{2n},\mathscr{V}_{1}\right)$
have closed ranges.

###### Proof.

Let $\\{f_{\nu}\\}$ be a sequence in
$\mathcal{E}^{\prime}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right)$ such that
$\mathscr{D}_{0}f_{\nu}$ convergence in
$\mathcal{E}^{\prime}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right),$ i.e. all
$\mathscr{D}_{0}f_{\nu}$ are supported in a fixed compact subset
$K\Subset\mathbb{R}^{2n}$ and there is a $s\in\mathbb{R}$ such that
$\mathscr{D}_{0}f_{\nu}\in
W^{s,2}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right),$
supp$\left(\mathscr{D}_{0}f_{\nu}\right)\subset K$ for all $\nu$ and
$\mathscr{D}_{0}f_{\nu}\rightarrow g$ in
$W^{s,2}\left(\mathbb{R}^{2n},\mathscr{V}_{1}\right)$ [28]. Then
$\mathscr{D}_{0}f_{\nu}=0$ outside of $K,$ i.e. $f_{\nu}$ is monogenic on
$\mathbb{R}^{2n}\setminus K.$ We can assume that $\mathbb{R}^{2n}\setminus K$
has no compact connected component. By Proposition 3.2 each component of a
monogenic function annihilated by $\Delta$ and so it is biharmonic on
$\mathbb{R}^{2n}\setminus K$ and real analytic. Consequently, as compacted
supported distributions, monogenic functions
$\left.f_{\nu}\right|_{\mathbb{R}^{2n}\setminus K}$ vanish on
$\mathbb{R}^{2n}\setminus K$ and thus $\left\\{f_{\nu}\right\\}$ are also
supported in $K.$ This argument also implies that (63) holds with $C_{s,K}=0.$
If this is not true, there exist a sequence $h_{\nu}\in
W^{s,2}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right),$ with supp$h_{\nu}\subset
K,$ such that

$\displaystyle\left\|\mathscr{D}_{0}h_{\nu}\right\|_{W^{s,2}}^{2}<\frac{1}{\nu}\left\|h_{\nu}\right\|_{W^{s+1,2}}^{2}.$

By rescaling we can assume that $\left\|h_{\nu}\right\|_{W^{s,2}}=1$ for each
$\nu.$ By (63)

$\displaystyle
C_{s,K}\geq\left(c_{s,K}-\frac{1}{\nu}\right)\left\|h_{\nu}\right\|_{W^{s+1,2}}^{2}.$

Thus $\left\\{h_{\nu}\right\\}$ is bounded in the Sobolev space
$W^{s+1,2}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right).$ By the well known
compactness of the inclusion
$W^{s+1,2}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right)\subset
W^{s,2}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right),$ there is a subsequence
that converges to a function $h_{\infty}\in
W^{s,2}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right).$ We have
$\left\|h_{\infty}\right\|_{W^{s,2}}=1,\ \mathscr{D}_{0}h_{\infty}\equiv 0.$
Then $\Delta^{2}h_{\infty}=0$ and $h_{\infty}$ is also compactly supported in
$K$. So $h_{\infty}=0$ by analytic continuation, which contradicts to
$\left\|h_{\infty}\right\|_{W^{s,2}}^{2}=1.$

By the estimate (63) with $C_{s,K}=0,$ we see that $\left\\{f_{\nu}\right\\}$
is uniformly bounded in
$W^{s+1,2}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right)$, and hence contains a
subsequence which converges to a compactly supported weak solution $f\in
W^{s,2}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right)$ of $\mathscr{D}_{0}f=g.$
Namely, the image of $\mathscr{D}_{0}$ in
$\mathcal{E}^{\prime}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right)$ is closed.
The closedness of the image of $\mathscr{D}_{0}$ in
$\mathcal{D}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right)$ follows from the
proved result for
$\mathcal{E}^{\prime}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right)$ and the
elliptic regularity. ∎

Then we can prove the generalization of Malgrange’s vanishing Theorem 1.3. See
[5, 19, 22] for Malgrange’s vanishing theorem on CR manifolds and [34, Theorem
1.3] for this theorem for tangential $k$-Cauchy–Fueter operator over right
group, respectively.

Proof of Theorem 1.3. By Theorem 4.2 and its proof, the sequences

(64) $\displaystyle
0\rightarrow\mathcal{D}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right)\xrightarrow{{\mathscr{D}}_{0}}\mathcal{D}\left(\mathbb{R}^{2n},\mathscr{V}_{1}\right),$
$\displaystyle
0\rightarrow\mathcal{E}^{\prime}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right)\xrightarrow{{\mathscr{D}}_{0}}\mathcal{E}^{\prime}\left(\mathbb{R}^{2n},\mathscr{V}_{1}\right),$

are both exact and have closed ranges. Thus $\mathscr{D}_{0}$’s in
(LABEL:aco1) are topological homomorphisms. We can apply abstract duality
theorem 4.1 (6) to sequences in (LABEL:aco1) to get exact sequences

(65) $\displaystyle
0\leftarrow\mathcal{D}^{\prime}\left(\mathbb{R}^{2n},{\mathscr{V}}_{0}\right)\xleftarrow{\widehat{\mathscr{D}}_{0}}\mathcal{D}^{\prime}\left(\mathbb{R}^{2n},{\mathscr{V}}_{1}\right),$
$\displaystyle
0\leftarrow\mathcal{E}\left(\mathbb{R}^{2n},{\mathscr{V}}_{0}\right)\xleftarrow{\widehat{\mathscr{D}}_{0}}\mathcal{E}\left(\mathbb{R}^{2n},{\mathscr{V}}_{1}\right),$

i.e. $\widehat{\mathscr{D}}_{0}$’s are surjective, since
$\ker\mathscr{D}_{0}=\\{0\\}$ in (LABEL:aco1). Compare the formula (62) of
$\widehat{\mathscr{D}}_{0}$ and the definition of $\mathscr{D}_{2}$ in (21),
$\widehat{\mathscr{D}}_{0}:\mathcal{E}\left(\mathbb{R}^{2n},{\mathscr{V}}_{1}\right)\rightarrow\mathcal{E}\left(\mathbb{R}^{2n},{\mathscr{V}}_{0}\right)$
can be identified with
${\mathscr{D}}_{2}:\mathcal{E}\left(\mathbb{R}^{2n},{\mathscr{V}}_{2}\right)\rightarrow\mathcal{E}\left(\mathbb{R}^{2n},{\mathscr{V}}_{3}\right)$
if we take linear isomorphism $\mathscr{V}_{2}\rightarrow\mathscr{V}_{1}$
given by $\left(\begin{matrix}h_{0}\\\
h_{1}\end{matrix}\right)\rightarrow\left(\begin{matrix}h_{1}\\\
-h_{0}\end{matrix}\right)$ and identify $\mathbb{S}^{+}$ with $\mathbb{S}^{-}$
as complex vector spaces, although they are different as representations of
${\rm Spin}(n)$. Then we have the exact sequences

$\displaystyle\mathcal{E}\left(\mathbb{R}^{2n},\mathscr{V}_{2}\right)\xrightarrow{{\mathscr{D}}_{0}}\mathcal{E}\left(\mathbb{R}^{2n},\mathscr{V}_{3}\right)\rightarrow
0.$

Similarly,
$\mathcal{D}^{\prime}\left(\mathbb{R}^{2n},\mathscr{V}_{2}\right)\xrightarrow{{\mathscr{D}}_{0}}\mathcal{D}^{\prime}\left(\mathbb{R}^{2n},\mathscr{V}_{3}\right)\rightarrow
0.$ The theorem is proved. ∎

######  Remark 4.1.

The exactness of (LABEL:aco1) is equivalent to
$H^{0}\left(\mathcal{D}\left(\mathbb{R}^{2n},{\mathscr{V}}_{\bullet}\right)\right)=$
$H^{0}\left(\mathcal{E}^{\prime}\left(\mathbb{R}^{2n},{\mathscr{V}}_{\bullet}\right)\right)=\\{0\\},$
while Theorem 1.3 is equivalent to
$H_{0}\left(\mathcal{D}^{\prime}\left(\mathbb{R}^{2n},{\mathscr{V}}_{\bullet}\right)\right)$
$=H_{0}\left(\mathcal{E}\left(\mathbb{R}^{2n},{\mathscr{V}}_{\bullet}\right)\right)=\\{0\\}.$

Now we can prove the Hartogs–Bochner extension Theorem 1.4 for monogenic
functions.

Proof of Theorem 1.4. Since $\mathscr{D}_{0}f$ vanishes to the second order on
$\partial\Omega,$ we can extend $\mathscr{D}_{0}f$ by $0$ outside of
$\overline{\Omega}$ to get a $\mathscr{D}_{1}$-closed $C^{2}$ element
$F\in\mathcal{E}^{\prime}\left(\mathbb{R}^{2n},\mathscr{V}_{1}\right)$
supported in $\overline{\Omega}.$ Since
$H^{0}\left(\mathcal{E}^{\prime}\left(\mathbb{R}^{2n},{\mathscr{V}}_{\bullet}\right)\right)$
vanish by Remark 4.1, it is separated. Thus, we can apply abstract duality
theorem 4.1 (1) to the second sequences in (LABEL:aco1) and (LABEL:aco2) to
see that

(66) $\displaystyle{\rm Im}\
\mathscr{D}_{0}=\left\\{\tilde{F}\in\mathcal{E}^{\prime}\left(\mathbb{R}^{2n},\mathscr{V}_{1}\right)|\left\langle\tilde{F},G\right\rangle=0\
{\rm for\ any}\ G\in\ker\widehat{\mathscr{D}}_{0}\right\\}.$

Consequently, we have $F\in{\rm Im}\ \mathscr{D}_{0}.$ This is because
$\widehat{\mathscr{D}}_{0}=\mathscr{D}_{0}^{*}$ on
$\mathcal{E}\left(\mathbb{R}^{2n},\mathscr{V}_{1}\right)$ and so for any
$G\in\ker{\mathscr{D}}^{*}_{0}\subset\mathcal{E}\left(\mathbb{R}^{2n},{\mathscr{V}}_{1}\right),$
we have

(67) $\displaystyle\langle F,G\rangle=$
$\displaystyle\int_{\Omega}\left\langle\mathscr{D}_{0}f,G\right\rangle_{\mathscr{V}_{1}}{\rm
d}V=\sum_{A=0,1}\int_{\Omega}\left\langle\nabla_{A}f,G_{A}\right\rangle_{\mathbb{S}^{-}}{\rm
d}V$ $\displaystyle=$
$\displaystyle\sum_{A=0,1}\sum_{j}\int_{\Omega}\left\langle\gamma_{j}\partial_{Aj}f,G_{A}\right\rangle_{\mathbb{S}^{-}}{\rm
d}V$ $\displaystyle=$
$\displaystyle-\sum_{A=0,1}\sum_{j}\int_{\Omega}\left\langle\partial_{Aj}f,\gamma_{j}G_{A}\right\rangle_{\mathbb{S}^{+}}{\rm
d}V$ $\displaystyle=$ $\displaystyle\sum_{A=0,1}\int_{\Omega}\left\langle
f,\nabla_{A}G_{A}\right\rangle_{\mathbb{S}^{+}}{\rm
d}V-\sum_{A=0,1}\sum_{j}\int_{\partial\Omega}\left\langle
f,\gamma_{j}G_{A}\right\rangle_{\mathbb{S}^{+}}n_{Aj}{{\rm d}S}$
$\displaystyle=$ $\displaystyle\left\langle
f,{\mathscr{D}}^{*}_{0}G\right\rangle-\sum_{A=0,1}\sum_{j}\int_{\partial\Omega}\left\langle
f,\gamma_{j}G_{A}\right\rangle_{\mathbb{S}^{+}}n_{Aj}{{\rm d}S}=0,$

by using (62) and Stokes-type formula, where $n$ is the unit outer normal
vector to $\partial\Omega.$ Hence, by (66) there exists a distribution
$H\in\mathcal{E}^{\prime}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right)$ such
that $F=\mathscr{D}_{0}H.$

Recall that a distribution in $\mathcal{E}^{\prime}$ always has compact
support. Now by estimate (63) with $C_{s,K}=0$ and $s=0,$ proved in the proof
of Theorem 4.2, $H\in W^{1,2}\left(\mathbb{R}^{2n},\mathscr{V}_{0}\right).$
Then $H$ is monogenic on the connected open set
$\mathbb{R}^{2n}\setminus\overline{\Omega},$ since supp
$F\in\overline{\Omega}.$ By real analyticity of monogenic functions, $H$
vanishes on $\mathbb{R}^{2n}\setminus\overline{\Omega}.$ Hence,
$\tilde{f}=f-H$ gives us the required extension. ∎

######  Remark 4.2.

The conditions $\mathscr{D}_{0}f$ vanishing to the second order on
$\partial\Omega$ and the moment condition (15) are also necessary for such
extension in Theorem 1.4. It is clear that if $f$ is monogenic, for any
$G\in\ker{\mathscr{D}}^{*}_{0}\subset\mathcal{E}\left(\mathbb{R}^{2n},{\mathscr{V}}_{1}\right),$

$\displaystyle\sum_{A=0,1}\sum_{j}\int_{\partial\Omega}\left\langle
f,\gamma_{j}G_{A}\right\rangle_{\mathbb{S}^{+}}n_{Aj}{{\rm d}S}=\left\langle
f,{\mathscr{D}}^{*}_{0}G\right\rangle-\left\langle\mathscr{D}_{0}f,G\right\rangle=0,$

as in (67).

### Acknowledgment

The first author is partially supported by Nature Science Foundation of
Zhejiang province (No. LY22A010013) and National Nature Science Foundation in
China (Nos. 11801508, 11971425); The second author is partially supported by
National Nature Science Foundation in China (No. 11971425); The third author
is partially supported by National Nature Science Foundation in China (No.
12071197), the Natural Science Foundation of Shandong Province (Nos.
ZR2019YQ04, 2020KJI002).

## References

  * [1] Adams, W., Berenstein, C., Loustaunau, P., Sabadini, I. and Struppa, D., Regular functions of several quaternionic variables and the Cauchy–Fueter complex, _J. Geom. Anal._ 9 (1999), 1-15.
  * [2] Adams, W. and Loustaunau, P., Analysis of the module determining the properties of regular functions of several quaternionic variables, _Pacific J. Math._ 196 (2001), 1-15.
  * [3] Baston, R.,  Quaternionic complexes,  J. Geom. Phys. 8 (1992), 29-52.
  * [4] Brackx, F., Delanghe, R. and Sommen, F., _Clifford Analysis,_ Pitman Research Notes in Mathematics, 76, Boston, MA, USA: Pitman Books Limited, 1982.
  * [5] Brinkschulte, J., Hill, C. and Nacinovich, M., Malgrange’s vanishing theorem for weakly pseudoconcave CR manifolds, _Manuscr. Math._ 131 (2010), 503-506.
  * [6] Bureš, J., Damiano, A. and Sabadini, I.,  Explicit resolutions for the complex of several Fueter operators,  J. Geom. Phys. 57 (2007), 765-775.
  * [7] Bureš, J. and Souček, V.,  Complexes of invariant operators in several quaternionic variables,  Complex Var. Elliptic Equ. 51 (2006), 463-485.
  * [8] Colombo, F., Sabadini, I., Sommen, F. and Struppa, D., _Analysis of Dirac systems and computational algebra,_ Progress in Mathematical Physics, 39. Birkhäuser Boston, MA, 2004.
  * [9] Colombo, F., Souček, V. and Struppa, D.,  Invariant resolutions for several Fueter operators,  J. Geom. Phys. 56 (2006), 1175-1191.
  * [10] Damiano, A., Sabadini, I. and Souček, V., Different approaches to the complex of three Dirac operators, _Ann. Global Anal. Geom._ 46 (2014), 313-334.
  * [11] Delanghe, R., Sommen, F. and Souček, V., Clifford algebra and spinor-valued functions, Mathematics and its Applications 53,  Kluwer Academic Publishers Group, Dordrecht, 1992.
  * [12] Fulton, W. and Harris, J., Representation theory, A first course, Graduate Texts in Mathematics 129, Springer-Verlag, New York, 1991.
  * [13] Grafakos, L., _Classical and Modern Fourier Analysis,_ Prentice Hall, Pearson, 2004.
  * [14] Goodmann, R. and Wallach, N.R. _Representations and invariants of the classical groups._ Cambridge University Press, Cambridge, 1998.
  * [15] Huilgol, R., On Liouville’s theorem for biharmonic functions, _Siam J. Appl. Math._ 20 (1971), 37-39.
  * [16] Krump, L. and Souček, V., The generalized Dolbeault complex in two Clifford variables, _Adv. Appl. Clifford Algebr._ 17 (2007), 537-548.
  * [17] Krump, L., The generalised Dolbeault complex for four Dirac operators in the stable rank, In: Proceedings of the ICNAAM, Kos 2008, AIP Conference Proceedings 2008, pp. 670-673.
  * [18] Krump, L., A resolution for the Dirac operator in four variables in dimension $6.$ _Adv. Appl. Clifford Algebr._ 19 (2009), 365-374.
  * [19] Laurent-Thiébaut, C. and Leiterer, J., Some applications of Serre duality in CR manifolds, _Nagoya Math. J._ 154 (1999), 141-156.
  * [20] Liu, S.-Q. and Wang, W., On pluripotential theory associated to quaternionic m-subharmonic functions, _J. Geom. Anal._ 33 (2023), Paper No. 143, 39 pp.
  * [21] Pertici, D. Funzioni regolari di piu variabili quaternioniche, _Ann. Mat. Pura Appl._ 151 (1988), 39-65.
  * [22] Nacinovich, M. and Porten, E., Weak $q$-concavity conditions for CR manifolds, _Ann. Mat. Pura Appl._ 196 (2017), 1779-1817.
  * [23] Ren, G.-B. and Wang, H.-Y., Theory of several paravector variables: Bochner–Martinelli formula and Hartogs theorem, _Sci. China Math._ 57 (2014), 2347-2360.
  * [24] Ren, G.-B. and Zhang, Y.-C., Hartogs phenomena for discrete $k$-Cauchy–Fueter operators, _J. Geom. Anal._ 33 (2023), Paper No. 248, 51 pp.
  * [25] Sabadini, I., Sommen, F., Struppa, D. and van Lancker, P., Complexes of Dirac operators in Clifford algebras, _Math. Z._ 239 (2002), 293-320.
  * [26] Salač, T., $k$-Dirac operator and parabolic geometries. _Complex Anal. Oper. Theory_ 8 (2014), 383-408.
  * [27] Schaefer, H. and Wolff, M., _Topological vector spaces,_ Graduate Texts in Mathathematics 3, Springer-Verlag, New York, Berlin, 1999.
  * [28] Schwartz, L., _Théorie des distributions._ Nouveau tirage, Paris, Hermann, 1998.
  * [29] Shi, Y. and Wang, W., The tangential $k$-Cauchy–Fueter complexes and Hartogs’ phenomenon over the right quaternionic Heisenberg group, _Ann. Mat. Pura Appl._ 199 (2020), 651-680.
  * [30] Shi, Y. and Wu, Q.-Y., The Penrose transform and the exactness of the tangential $k$-Cauchy–Fueter complex on the Heisenberg group, _Adv. Appl. Clifford Algebr._ 31 (2021), 1-22.
  * [31] Wang, W., On non-homogeneous Cauchy–Fueter equations and Hartogs’ phenomenon in several quaternionic variables, J. Geom. Phys. 58 (2008), 1203-1210.
  * [32] Wang, W., The $k$-Cauchy–Fueter complexes, Penrose transformation and Hartogs’ phenomenon for quaternionic $k$-regular functions, J. Geom. Phys. 60 (2010), 513-530.
  * [33] Wang, W., The Neumann problem for the $k$-Cauchy-Fueter complex over $k$-pseudoconvex domains in $\mathbb{R}^{4}$ and the $L^{2}$ estimate, _J. Geom. Anal._ 29 (2019), 1233-1258.
  * [34] Wang, W., On the boundary complex of the $k$-Cauchy–Fueter complex, _Ann. Mat. Pura Appl._ 202 (2023), 2255-2291.