Riemann-Type Problem in Hermitian Clifford Analysis

Abstract

We study some properties of Hermitian-2-monogenic circulant matrix function ($\mathbf {H}$-2-monogenic function for short) and Hermitian singular integral matrix operator, i.e., Hermitian Hilbert transform. Using these properties and fixed-point theorem, we investigate a kind of generalized Riemann-type problem for $\mathbf {H}$-2-monogenic function, where the constant circulant matrix in the transmission condition is changed into a circulant matrix function. Under some assumptions, we prove that the boundary value problem has a unique solution.

1 Introduction

Euclidean Clifford analysis becomes a powerful mathematical tool for generalized the classical boundary value problems in the complex analysis. Under the framework of classical Clifford analysis [1,2,3,4,5,6,7,8,9,10], many interesting results with respect to boundary value problems for monogenic functions were presented. In [8, 11], Gürlebeck and Zhang studied Riemann boundary value problems for harmonic functions and bi-harmonic functions, on the basis of higher-order integral representations and Plemelj–Sokhotski formulae, the solutions for the problems were given in an explicit way. The idea of translating boundary value problems to the corresponding singular integral equations and using the properties of integral operators to discuss the solvability of this problem was applied by Xu [3], Shapiro and Vasoevski in [12].

In recent years, Hermitian Clifford analysis has emerged as a new and successful branch of Clifford analysis, offering yet a refinement of the Euclidean case, it focuses on the concept of so-called h-monogenic functions defined in Euclidean space of even dimension with values complex Clifford algebra $\mathbb {C}_{2n}$. The theory of Hermitian monogenicity, Hermitian Cauchy integral formulae and the matrix Hermitian Hilbert transform were constructed in the framework of circulant matrix $(2\times 2)$ functions. For more details, we refer to [11, 13,14,15,16,17]. It is natural to consider the boundary value problems in Hermitian Clifford analysis, particularly Riemann-type problems and Dirichlet- type problems. In [18], $\mathbf {R}_{-1}$ Riemann-type problems for $\mathbf {H}$-monogenic functions and (left) Helmholtz $\mathbf {H}$-monogenic functions were studied by R. Abreu Blaya, J. Bory Reyes, F. Brackx, H. De Schepper and F. Sommen. In [19], using the Hermitian Cauchy transformation, Ku and Wang solved the half Dirichlet problems for matrix functions on the unite ball in Hermitian Clifford analysis. Combining integral representation formula with orthogonal decomposition of Hermitian Sobolev spaces, solvability conditions for two Dirichlet-type boundary value problems were given by Abreu Blaya et al. [20]. In [21], applying the second-order Hermitian Borel–Pompeiu formula, Hermitian Plemelj–Sokhotski formula and Liouville-type theorem, we solved the following $R_{0}$ Riemann-type problems for Hermitian-2-monogenic function

$$\begin{aligned} \left\{ \begin{array}{ll} (\mathcal {D}_{(\underline{V},\underline{V}^{\dag })})^{2}\mathbf {G}_{2}^{1}=0, &{} \text {in }\,\mathbf {R}^{2n}{\setminus }\partial B(\underline{a}, R)\\ {[}\mathbf {G}_{2}^{1}]^{+}(\underline{Y})=[\mathbf {G}_{2}^{1}]^{-}(\underline{Y})\mathbf {A}_{2}^{1} +\mathbf {F}_{2}^{1}(\underline{Y}), &{} {\underline{Y}\in \partial B(\underline{a}, R)}\\ {[}\mathcal {D}_{(\underline{V},\underline{V}^{\dag })}\mathbf {G}_{2}^{1}]^{+}(\underline{Y}) =[\mathcal {D}_{(\underline{V},\underline{V}^{\dag })}\mathbf {G}_{2}^{1}]^{-}(\underline{Y}) \mathbf {B}_{2}^{1}+\mathbf {U}_{2}^{1}(\underline{Y}), &{} {\underline{Y}\in \partial B(\underline{a}, R)}\\ \Vert \mathbf {G}_{2}^{1}(\infty )\Vert \le M \end{array}\right. \end{aligned}$$

(1.1)

where $\mathbf {A}_{2}^{1},$$\mathbf {B}_{2}^{1}$ are invertible constant circulant matrixes and $\mathbf {F}_{2}^{1}(\underline{Y})$, $\mathbf {U}_{2}^{1}(\underline{Y})$ are given circulant matrix functions in $\mathbf {C}^{0,\alpha }(\partial B(\underline{a}, R)),$$0<\alpha \le 1.$ The explicit expression of solutions for (1.1) was constructed. However, to our knowledge, the Riemann-type problem, $\mathbf {A}_{2}^{1}$, $\mathbf {B}_{2}^{1}$ are circulant matrix functions in transmission conditions in (1.1), has not been studied in Hermitian Clifford analysis. In this paper, motivated by [3, 18, 21], we shall further consider the following linear Riemann-type problem in Hermitian Clifford analysis:

$$\begin{aligned} \left\{ \begin{array}{ll} (\mathcal {D}_{(\underline{V},\underline{V}^{\dag })})^{2}\mathbf {G}_{2}^{1}=0, &{} \text {in}\, \mathbf {R}^{2n}{\setminus }\partial \Omega ,\\ {[}\mathbf {G}_{2}^{1}]^{+}(\underline{Y})=[\mathbf {G}_{2}^{1}]^{-}(\underline{Y})\mathbf {A}_{2}^{1}(\underline{Y}) +\mathbf {F}_{2}^{1}(\underline{Y}), &{} \underline{Y}\in \partial \Omega ,\\ {[}\mathcal {D}_{(\underline{V},\underline{V}^{\dag })}\mathbf {G}_{2}^{1}]^{+}(\underline{Y}) =[\mathcal {D}_{(\underline{V},\underline{V}^{\dag })}\mathbf {G}_{2}^{1}]^{-}(\underline{Y}) \mathbf {B}_{2}^{1}+\mathbf {U}_{2}^{1}(\underline{Y}), &{} \underline{Y}\in \partial \Omega ,\\ \Vert \mathbf {G}_{2}^{1}(\infty )\Vert =0, \end{array}\right. \end{aligned}$$

(1.2)

where $\mathbf {A}_{2}^{1}(\underline{Y})$, $\mathbf {F}_{2}^{1}(\underline{Y})$, $\mathbf {U}_{2}^{1}(\underline{Y})$ are circulant matrix functions in $\mathbf {C}^{0,\alpha }(\partial \Omega ),$$0<\alpha <1$, $\mathbf {B}_{2}^{1}$ is an invertible constant circulant matrix. Under some assumptions, we will show that problem (1.2) is solvable.

This article is organized as follows: the basis of complex Clifford analysis and Hermitian Clifford analysis is recalled in Sect. 2. Based on the Hermitian integral formulae in [21], we study some properties of $\mathbf {H}$-2-monogenic functions, such as maximum modulus theorem in Sect. 3. With the help of Hermitian Plemelj–Sokhotski formulae, we consider some properties of Hermitian integral operators and their application in Sect. 4. Finally, in Sect. 5, by the aid of the above results, the Riemann-type problems (1.2) are investigated.

2 Preliminaries

We first recall some basic facts about the Clifford algebra and the Hermitian Clifford analysis needed in the sequel. More details can be found in references, e.g., [11, 13,14,15,16,17, 22,23,24,25].

Let $Cl(V_{2n,0})$ be the $2^{2n}$-dimensional Clifford algebra constructed from the orthogonal basis $\{e_{1}, e_{2}, \ldots , e_{2n}\}$ which satisfies the relationship $e_{j}e_{k}+e_{k}e_{j}=-2\delta _{jk}$, $j, k=1, \ldots , 2n$. Any element $a\in Cl(V_{2n,0})$ has form $a=\sum _{A\in \mathcal {P}N}a_{A}e_{A}$, where N stands for the set $\{1, \ldots , 2n\}$ and $\mathcal {P}N$ denotes the family of all order-preserving subsets of N, $A=\{l_{1}, \ldots , l_{r}\}\in \mathcal {P}N$, $e_{A}=e_{l_{1}\ldots l_{r}}=e_{l_{1}}\ldots {e_{l_{r}}}$ with $1\le l_{1}<\cdots <l_{r}\le 2n$. The Euclidean space $\mathbf {R}^{2n}$ is embedded in $Cl(V_{2n,0})$ by identifying $(X_{1},\ldots ,X_{2n})$ with the Clifford vector $\underline{X}$ is defined by $\underline{X}=\sum ^{2n}_{j=1}e_{j}X_{j}.$ Note that the square of $\underline{X}$ is scalar valued and equals the norm squared up to a minus sign: $\underline{X}^{2}=-<\underline{X},\underline{X}>=-|\underline{X}|^{2}.$ The dual of $\underline{X}$ is the vector value first-order differential operator, $\partial _{\underline{X}}=\sum _{j=1}^{2n}e_{j}\partial _{X_{j}}$, called Dirac operator. A differentiable function f defined in an open region $\Omega$ of $\mathbf {R}^{2n}$ and taking values in $Cl(V_{2n,0})$ is called (left) monogenic in $\Omega$ if $\partial _{\underline{X}}[f]=0.$ When allowing for complex constants, the set of generators $\{e_{1},\ldots ,e_{2n}\},$ produces the complex Clifford algebra $\mathbb {C}_{2n},$ being the complexification of the real Clifford algebra $Cl(V_{2n,0}),$ i.e., $\mathbb {C}_{2n}=Cl(V_{2n,0})\bigoplus i Cl(V_{2n,0}).$ Any complex Clifford number $\lambda \in \mathbb {C}_{2n}$ may be written as $\lambda =a+ib,$$a, b\in Cl(V_{2n,0})$. An observation leading to the definition of the Hermitian conjugation $\lambda ^{\dag }=(a+ib)^{\dag }=\overline{a}-i\overline{b},$ where the bar notation stands for the usual Clifford conjugation in $Cl(V_{2n,0}),$ i.e., the main anti-involution for which $\overline{e}_{j}=-e_{j},$$j=1,\ldots ,2n.$ This Hermitian conjugation also leads to a Hermitian inner product and its associated norm on $\mathbb {C}_{2n}$ given by $(\lambda ,\mu )=[\lambda ^{\dag }\mu ]_{0}$ and $|\lambda |=\sqrt{[\lambda ^{\dag }\lambda ]_{0}}=(\sum _{A}|\lambda _{A}|^{2})^{\frac{1}{2}}.$

The following the framework is so-called Hermitian Clifford analysis, yet a refinement of orthogonal Clifford analysis. An elegant way for introducing this setting consists in considering a so-called complex structure, i.e., a specific $SO(2n;\mathbf {R})$-element J for which $J^{2}=-\mathbf {1}$ (see [11, 13,14,15,16,17]). Here, J is chosen to act upon the generators $e_{1},\ldots ,e_{2n}$ of the Clifford algebra as $J[e_{j}]=-e_{n+j}$ and $J[e_{n+j}]=e_{j}$, $j=1,\ldots ,n$. We identify a vector $\underline{X}=(\underline{X}_{1},\ldots ,\underline{X}_{2n})$ in $\mathbf {R}^{2n}$ with the Clifford vector $\underline{X}=\sum \nolimits _{j=1}^{n}(e_{j}x_{j}+e_{n+j}y_{j})$, and we denote by $\underline{X}|$ the action of the complex structure J on $\underline{X}$, i.e., $\underline{X}|=J[\underline{X}]=\sum _{j=1}^{n}(e_{j}y_{j}-e_{n+j}x_{j}).$ The actions of the projection operators on the Clifford vector $\underline{X}$ then produce the Hermitian Clifford variables $\underline{Z}$ and its Hermitian conjugate $\underline{Z}^{\dag }:$

$$\begin{aligned} \underline{Z}= & {} \frac{1}{2}(\mathbf {1}+i J)[\underline{X}]=\frac{1}{2}(\underline{X}+i\underline{X}|),\\ \underline{Z}^{\dag }= & {} -\frac{1}{2}(\mathbf {1}-i J)[\underline{X}]=-\frac{1}{2}(\underline{X}-i\underline{X}|). \end{aligned}$$

Finally, the Hermitian Dirac operators $\partial _{\underline{Z}}$ and $\partial _{\underline{Z}^{\dag }}$ are derived from the orthogonal Dirac operator $\partial _{\underline{X}}:$

$$\begin{aligned} \partial _{\underline{Z}^{\dag }}= & {} \frac{1}{4}(\mathbf {1}+i J)[\partial _{\underline{X}}]=\frac{1}{4} (\partial _{\underline{X}}+i\partial _{\underline{X}|}),\\ \partial _{\underline{Z}}= & {} -\frac{1}{4}(\mathbf {1}-i J)[\partial _{\underline{X}}]=-\frac{1}{4} (\partial _{\underline{X}}-i\partial _{\underline{X}|}),\\ \end{aligned}$$

where we have introduced

$$\begin{aligned} \partial _{\underline{X}|}=J[\partial _{\underline{X}}]=\sum \limits _{j=1}^{n}(e_{j}\partial _{y_{j}}-e_{n+j}\partial _{x_{j}}). \end{aligned}$$

The respective fundamental solutions of $\partial _{\underline{X}}$ and $\partial _{\underline{X}|}$ are given by

$$\begin{aligned} E(\underline{X})=\frac{1}{\omega _{2n}}\frac{\overline{\underline{X}}}{|\underline{X}|^{2n}},~~~ E|(\underline{X})=\frac{1}{\omega _{2n}}\frac{\overline{\underline{X}}|}{|\underline{X}|^{2n}} ,&\underline{X}\in \mathbf {R}^{2n}{\setminus }\{0\} \end{aligned}$$

where $\omega _{2n}$ denotes the area of the unit sphere $S^{2n-1}$ in $\mathbf {R}^{2n}.$ The transition from Hermitian Clifford analysis to a circulant matrix approach is essentially based on the following observation. Introducing the particular circulant $(2\times 2)$ matrixes

$$\begin{aligned} \mathcal {D}_{(\underline{Z},\underline{Z}^{\dag })}=\left( \begin{array}{cc} \partial _{\underline{Z}} &{}\quad \partial _{\underline{Z}^{\dag }} \\ \partial _{\underline{Z}^{\dag }} &{} \quad \partial _{\underline{Z}} \\ \end{array} \right) ~~~\text {and}~~~ {{\varvec{E}}}=\left( \begin{array}{cc} \mathcal {E} &{} \quad \mathcal {E}^{\dag }\\ \mathcal {E}^{\dag } &{}\quad \mathcal {E}\\ \end{array} \right) , \end{aligned}$$

where $\mathcal {E}=-(E+iE|)$ and $\mathcal {E}^{\dag }=(E-iE|)$. Then $\mathcal {D}_{(\underline{Z},\underline{Z}^{\dag })}\mathbf {E}=\delta _{2}^{1}$ , where $\delta _{2}^{1}$ is the diagonal matrix with the Dirac delta distribution $\delta$ on the diagonal, whence may be considered as a fundamental solution of the matrix operator $\mathcal {D}_{(\underline{Z},\underline{Z}^{\dag })}$.

For further use, we introduce the Hermitian-oriented surface elements $\mathrm{d}\sigma _{\underline{Z}}$ and $\mathrm{d}\sigma _{\underline{Z}^{\dag }}$ as follows:

$$\begin{aligned} \mathrm{d}\sigma _{\underline{Z}}= & {} -\frac{1}{4}(-1)^{\frac{n(n+1)}{2}}(2i)^{n} (\widetilde{\mathrm{d}\sigma }_{\underline{X}}-i\widetilde{\mathrm{d}\sigma }_{\underline{X}|}),\\ \mathrm{d}\sigma _{\underline{Z}^{\dag }}= & {} -\frac{1}{4}(-1)^{\frac{n(n+1)}{2}}(2i)^{n} (\widetilde{\mathrm{d}\sigma }_{\underline{X}}+i\widetilde{\mathrm{d}\sigma }_{\underline{X}|}), \end{aligned}$$

where $\widetilde{\mathrm{d}\sigma }_{\underline{X}}$ denotes the vector-valued oriented surface element and $\widetilde{\mathrm{d}\sigma }_{\underline{X}|}=J[\widetilde{\mathrm{d}\sigma }_{\underline{X}}].$ They are explicitly given by means of the following differential forms of order $2n-1$

here

the corresponding oriented volume elements then read

$$\begin{aligned} \widetilde{d V}(\underline{X})= & {} dx_{1}\wedge \cdots \wedge dx_{n}\wedge dy_{1}\wedge \cdots \wedge dy_{n}\\ \widetilde{d V}(\underline{X}|)= & {} dy_{1}\wedge \cdots \wedge dy_{n}\wedge (-dx_{1})\wedge \cdots \wedge (-dx_{n}). \end{aligned}$$

We still introduce the matrix

$$\begin{aligned} \mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })}=\left( \begin{array}{cc} \mathrm{d}\sigma _{\underline{Z}} &{}\quad -\mathrm{d}\sigma _{\underline{Z}^{\dag }} \\ -\mathrm{d}\sigma _{\underline{Z}^{\dag }} &{} \quad \mathrm{d}\sigma _{\underline{Z}} \\ \end{array} \right) \end{aligned}$$

which will play the role of the differential form.

Definition 2.1

A continuously differentiable function f on an open region $\Omega$ of $\mathbf {R}^{2n}$ with values in $\mathbb {C}_{2n}$ is called (left) h-monogenic function in $\Omega$ if and only if it satisfies in $\Omega$ the system

$$\begin{aligned} \partial _{\underline{X}}f=0=\partial _{\underline{X}|}f \end{aligned}$$

or equivalent, the system

$$\begin{aligned} \partial _{\underline{Z}}f=0=\partial _{\underline{Z}^{\dag }}f. \end{aligned}$$

Let $g_{1}$, $g_{2}$ be continuously differentiable functions defined in $\Omega$ and taking values in $\mathbb {C}_{2n},$ and consider the corresponding $(2\times 2)$ circulant matrix function

$$\begin{aligned} \mathbf {G}^{1}_{2}(\underline{X})=\left( \begin{array}{cc} g_{1}(\underline{X}) &{}\quad g_{2}(\underline{X}) \\ g_{2}(\underline{X}) &{} \quad g_{1}(\underline{X}) \\ \end{array} \right) . \end{aligned}$$

The ring of such matrix functions over $\mathbb {C}_{2n}$ is denoted by $\mathbf {C^{1}M}^{2\times 2}.$ In what follows, $\mathbf {O}$ will denoted the matrix in $\mathbf {C^{1}M}^{2\times 2}$ with zero entries.

Definition 2.2

The matrix function $\mathbf {G}_{2}^{1}\in \mathbf {C^{1}M}^{2\times 2}$ is called (left) $\mathbf {H}$-monogenic in $\Omega$ if and only if it satisfies in $\Omega$ the system $\mathcal {D}_{(\underline{Z},\underline{Z}^{\dag })}[\mathbf {G}_{2}^{1}]=\mathbf {O}.$

Definition 2.3

The function g: $\Omega (\subset \mathbf {R}^{2n})\rightarrow \mathbb {C}_{2n}$ is said to be Hölder continuous if and only if there are constants $C>0$ and $0<\alpha \le 1$ satisfying for arbitrary $\underline{X}$, $\underline{Y}\in \Omega$,

$$\begin{aligned} |g(\underline{X})-g(\underline{Y})|\le C|\underline{X}-\underline{Y}|^{\alpha }. \end{aligned}$$

Let $H^{\alpha }(\Omega , \mathbb {C}_{2n})$ be the set of all Hölder continuous functions on $\Omega$. Notions of continuity, differentiability and integrability of $\mathbf {G}^{1}_{2}\in \mathbf {C^{1}M}^{2\times 2}$ have the usual component-wise meaning. In particular, we will need to define in this way the classes $\mathbf {C}^\mathbf {r}(\Omega )$, $\mathbf {r}\in \mathbf {N}{\setminus }\{0\},$ of $\mathbf {r}$ times continuously differentiable functions over some suitable subset $\Omega$ of $\mathbf {R}^{2n}$. Introducing the non-negative function

$$\begin{aligned} \Vert \mathbf {G}_{2}^{1}(\underline{X})\Vert =\left( {\sum _{i=1}^{2}|g_{i}(x)|^{2}}\right) ^{\frac{1}{2}}. \end{aligned}$$

Furthermore, in this way the set $\mathbf {C}^{0,\alpha }(\Omega )$$(0<\alpha <1)$ stands for Hölder continuous circulant matrix functions over $\Omega$. We define the norm in $\mathbf {C}^{0,\alpha }(\Omega )$ as

$$\begin{aligned} \Vert \mathbf {G}_{2}^{1}\Vert _{(\alpha , \Omega )}\triangleq \sup \limits _{\underline{X}\in \Omega }\Vert \mathbf {G}_{2}^{1}(\underline{X})\Vert +\sup \limits _{\mathop {\underline{X}_{1}, \underline{X}_{2}\in \Omega }\limits _ {\underline{X}_{1}\ne \underline{X}_{2}}}\frac{\Vert \mathbf {G}_{2}^{1}(\underline{X}_{1})-\mathbf {G}_{2}^{1}(\underline{X}_{2})\Vert }{|\underline{X}_{1}-\underline{X}_{2}|^{\alpha }}. \end{aligned}$$

The space of circulant matrix functions $\mathbf {C}^{0,\alpha }(\Omega )$ is a Banach space.

Definition 2.4

The matrix function $\mathbf {G}_{2}^{1}\in \mathbf {C^{r}M}^{2\times 2}$$(\mathbf {r}\ge 2)$ is called (left) $\mathbf {H}$-2-monogenic in $\Omega$ if and only if it satisfies in $\Omega$ the system $(\mathcal {D}_{(\underline{Z},\underline{Z}^{\dag })})^{2}[\mathbf {G}_{2}^{1}]=\mathbf {O}.$

3 Some Properties for $\mathbf {H}$-2-Monogenic Function

In what follows, we denote

$$\begin{aligned} B(\underline{Y}, R)= & {} \{\underline{X}\in \mathbf {R}^{2n}| |\underline{X}-\underline{Y}|<R\},\nonumber \\ \mathbf {I}= & {} \left( \begin{array}{cc} 1 &{} \quad 0\\ 0 &{} \quad 1\\ \end{array}\right) , \end{aligned}$$

(3.1)

$$\begin{aligned} {{\varvec{E}}}_{1}(\underline{Z}-\underline{V})= & {} \frac{1}{\omega _{2n}(2-2n)} \left( \begin{array}{cc} 0 &{} \quad \frac{4}{|\underline{X}-\underline{Y}|^{2n-2}} \\ \frac{4}{|\underline{X}-\underline{Y}|^{2n-2}} &{} \quad 0 \\ \end{array} \right) ,\underline{X}\in \mathbf {R}^{2n}{\setminus }\{\underline{Y}\}, \end{aligned}$$

(3.2)

where $\omega _{2n}$ denotes the area of the unit sphere in $\mathbf {R}^{2n}.$

Lemma 3.1

[21] Suppose $\Gamma \subset \Omega$ is a 2n-dimensional compact differentiable and oriented manifold with $C^{\infty }$ smooth boundary $\partial \Gamma$, $g_{1}$ and $g_{2}$ are functions in $C^{2}(\Omega ,\mathbb {C}_{2n})$ and $\mathbf {G}_{2}^{1}(\underline{X})= \left( \begin{array}{cc} g_{1}(\underline{X}) &{}\quad g_{2}(\underline{X})\\ g_{2}(\underline{X}) &{} \quad g_{1}(\underline{X})\\ \end{array}\right)$ is the matrix function. It then holds that

$$\begin{aligned}&\int \limits _{\partial \Gamma }{{\varvec{E}}}(\underline{Z}-\underline{V})\mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })} \mathbf {G}_{2}^{1}(\underline{X})-\int \limits _{\partial \Gamma }{{\varvec{E}}}_{1}(\underline{Z}-\underline{V}) \mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })}\mathcal {D}_{(\underline{Z},\underline{Z}^{\dag })}\mathbf {G}_{2}^{1}(\underline{X}) \nonumber \\&\qquad +\int \limits _{\Gamma }{{\varvec{E}}}_{1}(\underline{Z}-\underline{V}) [(\mathcal {D}_{(\underline{Z},\underline{Z}^{\dag })})^{2}\mathbf {G}_{2}^{1}{(\underline{X})}] \mathrm{d}W_{(\underline{Z},\underline{Z}^{\dag })}\nonumber \\&\quad = \left\{ \begin{array}{ll} \mathbf {O}, &{}\quad \mathrm{if }\,\underline{Y}\in \Gamma ^{-},\\ (-1)^\frac{n(n+1)}{2}(2i)^{n}\mathbf {G}_{2}^{1}(\underline{Y}), &{}\quad \mathrm{if}\, \underline{Y}\in \Gamma ^{+}. \end{array} \right. \end{aligned}$$

(3.3)

Lemma 3.2

[21] If the matrix function $\mathbf {G}^{1}_{2}(\underline{X})=\left( \begin{array}{cc} g_{1}(\underline{X}) &{} \quad g_{2}(\underline{X}) \\ g_{2}(\underline{X}) &{} \quad g_{1}(\underline{X}) \\ \end{array} \right)$ is $\mathbf {H}$-2-monogenic in $\Omega$, then

$$\begin{aligned} \frac{2n}{\omega _{2n} R^{2n}}\int _{B(\underline{Y},R)}\mathbf {G}_{2}^{1}(\underline{X})\widetilde{dV}(\underline{X})=\mathbf {G}^{1}_{2}(\underline{Y}) \end{aligned}$$

(3.4)

for each $R>0$ such that $\overline{B}(\underline{Y},R)\subset \Omega .$

Theorem 3.3

Let $\mathbf {G}_{2}^{1}\in \mathbf {C}^{2}(\mathbf {R}^{2n}{\setminus } B(\underline{a}, R))\bigcap \mathbf {C}^{1}(\overline{\mathbf {R}^{2n}{\setminus } B(\underline{a}, R)})$ and the matrix function $\mathbf {G}_{2}^{1}$ be $\mathbf {H}$-2-monogenic in $\mathbf {R}^{2n}{\setminus } B(\underline{a}, R)$ with $\mathbf {G}_{2}^{1}(\underline{X})\rightarrow \mathbf {O}$, $|\underline{X}|\rightarrow \infty$. Then for all $\underline{Y}\in \mathbf {R}^{2n}{\setminus }\overline{B(\underline{a}, R)}$

$$\begin{aligned} (-1)^\frac{n(n+1)}{2}(2i)^{n}\mathbf {G}_{2}^{1}(\underline{Y})= & {} -\int \limits _{\partial B(\underline{a}, R)}{{\varvec{E}}}(\underline{Z}-\underline{V})\mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })} \mathbf {G}_{2}^{1}(\underline{X})\nonumber \\&+\int \limits _{\partial B(\underline{a}, R)}{{\varvec{E}}}_{1}(\underline{Z}-\underline{V}) \mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })}\mathcal {D}_{(\underline{Z},\underline{Z}^{\dag })}\mathbf {G}_{2}^{1}(\underline{X}). \end{aligned}$$

(3.5)

Proof

Applying Lemma 3.1 and the spherical coordinates transform, the result follows.$\square$

Theorem 3.4

Let the matrix function $\mathbf {G}_{2}^{1}(\underline{X})$ be a $\mathbf {H}$-2-monogenic in the open and connected set $\Omega .$ If there exists a point $\underline{A}\in \Omega$ such that

$$\begin{aligned} \Vert \mathbf {G}_{2}^{1}(\underline{X})\Vert \le \Vert \mathbf {G}_{2}^{1}(\underline{A})\Vert \end{aligned}$$

for all $\underline{X}\in \Omega$, then $\mathbf {G}_{2}^{1}$ must be constant circulant matrix in $\Omega .$

Proof

Using Lemma 3.2, the proof is similar to method in Theorem 3.3 in [21]. $\square$

Theorem 3.5

Let $\Omega$ be a bounded open and connected set of $\mathbf {R}^{2n}$ with a Lyapunov boundary $\partial \Omega$, $g_{1}(\underline{X})$, $g_{2}(\underline{X})\in C^{2}(\Omega , \mathbb {C}_{2n})\bigcap C^{1}(\overline{\Omega },\mathbb {C}_{2n})$ and the matrix function

$$\begin{aligned} \mathbf {G}_{2}^{1}(\underline{X})=\left( \begin{array}{cc} g_{1}(\underline{X}) &{}\quad g_{2}(\underline{X}) \\ g_{2}(\underline{X}) &{}\quad g_{1}(\underline{X}) \\ \end{array} \right) \end{aligned}$$

be $\mathbf {H}$-2-monogenic in $\Omega$. Then

$$\begin{aligned} \max \limits _{\underline{X}\in \overline{\Omega }}\{\Vert \mathbf {G}_{2}^{1}(\underline{X})\Vert \} =\max \limits _{\underline{X}\in \partial \Omega }\{\Vert \mathbf {G}_{2}^{1}(\underline{X})\Vert \}. \end{aligned}$$

Proof

In view of $\overline{\Omega }$ is compact and $\Vert \mathbf {G}_{2}^{1}(\cdot )\Vert$ is a continuous function on $\overline{\Omega }$, there is a point $\underline{A}\in \overline{\Omega }$ such that

$$\begin{aligned} \max \limits _{\underline{X}\in \overline{\Omega }}\{\Vert \mathbf {G}_{2}^{1}(\underline{X})\Vert \} =\Vert \mathbf {G}_{2}^{1}(\underline{A})\Vert . \end{aligned}$$

If $\underline{A}\in \partial \Omega$, then the assertion is true. So assume that $\underline{A}\in \Omega$. Decompose $\Omega$ into its components $\Omega =\bigcup \nolimits _{i=1}^{\infty }\Omega _{i}$ where all of them being bounded, open connected. Then $\overline{A}$ belongs to one of components, denote $\Omega _{i}$. As for any $\underline{X}\in \Omega _{i}$, we have

$$\begin{aligned} \Vert \mathbf {G}_{2}^{1}(\underline{X})\Vert \le \Vert \mathbf {G}_{2}^{1}(\underline{A})\Vert . \end{aligned}$$

Using Theorem 3.4, we then have $\mathbf {G}_{2}^{1}(\underline{X})$ is a constant circulant matrix in $\Omega _{i}$. Hence

$$\begin{aligned} \max \limits _{\underline{X}\in \overline{\Omega }}\{\Vert \mathbf {G}_{2}^{1}(\underline{X})\Vert \}= & {} \Vert \mathbf {G}_{2}^{1}(\underline{A})\Vert \nonumber \\= & {} \max \limits _{\underline{X}\in \partial \Omega _{i}} \{\Vert \mathbf {G}_{2}^{1}(\underline{X})\Vert \}\nonumber \\\le & {} \max \limits _{\underline{X}\in \partial \Omega }\{\Vert \mathbf {G}_{2}^{1}(\underline{X})\Vert \}\nonumber \\\le & {} \max \limits _{\underline{X}\in \overline{\Omega }}\{\Vert \mathbf {G}_{2}^{1}(\underline{X})\Vert \}. \end{aligned}$$

The proof is done. $\square$

In what follows we denote

$$\begin{aligned} M[r, \mathbf {G}_{2}^{1}]\triangleq \max \limits _{|\underline{Y}|=r}\{\Vert \mathbf {G}_{2}^{1}(\underline{Y})\Vert \} \end{aligned}$$

Theorem 3.6

If the matrix function $\mathbf {G}_{2}^{1}(\underline{X})=\left( \begin{array}{cc} g_{1}(\underline{X}) &{}\quad g_{2}(\underline{X})\\ g_{2}(\underline{X})&{}\quad g_{1}(\underline{X})\\ \end{array} \right)$ is $\mathbf {H}$-2-monogenic in $\mathbf {R}^{2n}$ and $\liminf \limits _{r\rightarrow \infty }M[r, \mathbf {G}_{2}^{1}]<\infty$, then $\liminf \limits _{r\rightarrow \infty }M[r, \mathcal {D}_{(\underline{Z},\underline{Z}^{\dag })}[\mathbf {G}_{2}^{1}]]=0$.

Proof

Applying Lemma 3.2 and the Stokes formula in [14], we obtain

$$\begin{aligned} \mathcal {D}_{(\underline{Z},\underline{Z}^{\dag })}[\mathbf {G}_{2}^{1}](\underline{Y})= & {} \frac{2n}{\omega _{2n}R^{2n}}\int _{B(\underline{Y}, R)} \mathcal {D}_{(\underline{Z},\underline{Z}^{\dag })}[\mathbf {G}_{2}^{1}](\underline{X}) \widetilde{\mathrm{d}V}(\underline{X})\nonumber \\= & {} \frac{1}{V_{2n}R^{2n}}\int _{\partial \Omega (\underline{Y}, R)} \mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })}\mathbf {G}_{2}^{1}(\underline{X}) \end{aligned}$$

(3.6)

Taking $R=|\underline{X}|$ in (3.6), in view of Theorem 3.4, we get

$$\begin{aligned} \Vert D_{(\underline{Z},\underline{Z}^{\dag })}[\mathbf {G}_{2}^{1}](\underline{Y})\Vert \le \frac{2n\cdot 2^{2n}}{|\underline{Y}|}M[2|\underline{Y}|, \mathbf {G}_{2}^{1}]. \end{aligned}$$

(3.7)

Denoting $|\underline{Y}|=r$, we have

$$\begin{aligned} r\Vert D_{(\underline{Z},\underline{Z}^{\dag })}[\mathbf {G}_{2}^{1}](\underline{Y})\Vert \le 2n\cdot 2^{2n}M[2r, \mathbf {G}_{2}^{1}]. \end{aligned}$$

(3.8)

Using $\liminf \limits _{r\rightarrow \infty }M[r, \mathbf {G}_{2}^{1}]<\infty$, the result follows immediately from (3.8). $\square$

4 Some Properties for Hermitian Integral Operators

Suppose $\Omega$ is an open bounded non-empty subset of $\mathbf {R}^{2n}$ with a Lyapunov boundary $\partial \Omega ,$ we usually write $\Omega ^{+}=\Omega$ and $\Omega ^{-}=\mathbf {R}^{2n} {\setminus }\overline{\Omega }.$ The notations $\underline{Y}$ and $\underline{Y}|$ will be reserved for Clifford vectors associated with points $\Omega ^{+}$, while their Hermitian counterparts are denoted $\underline{V}=\frac{1}{2}(\underline{Y}+i\underline{Y}|)$ and $\underline{V}^{\dag }=-\frac{1}{2}(\underline{Y}-i\underline{Y}|).$

We shall introduce the following matrix operators:

$$\begin{aligned} \mathcal {C}[\mathbf {G}_{2}^{1}](\underline{Y})= & {} \int \limits _{\partial \Omega }{{\varvec{E}}} (\underline{Z}-\underline{V})\mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })} \mathbf {G}_{2}^{1}(\underline{X}),~~~~~~\underline{Y}\in \Omega ^{\pm }, \end{aligned}$$

(4.1)

$$\begin{aligned} \mathcal {H}_{\partial \Omega }[\mathbf {G}_{2}^{1}](\underline{Y})= & {} \frac{1}{2}\left( \begin{array}{cc} H_{\partial \Omega }+H|_{\partial \Omega } &{} \quad -H_{\partial \Omega }+H|_{\partial \Omega } \\ -H_{\partial \Omega }+H|_{\partial \Omega } &{} \quad H_{\partial \Omega }+H|_{\partial \Omega } \end{array}\right) \left( \begin{array}{cc} g_{1}(\underline{Y}) &{}\quad g_{2}(\underline{Y})\\ g_{2}(\underline{Y}) &{}\quad g_{1}(\underline{Y}) \end{array}\right) \nonumber \\= & {} (-1)^{\frac{n(n+1)}{2}}\left( -\frac{i}{2}\right) ^{n}\int \limits _{\partial \Omega }2{{\varvec{E}}} (\underline{Z}-\underline{V})\mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })} \mathbf {G}_{2}^{1}(\underline{X}),~~~~~~\underline{Y}\in \partial \Omega ,\nonumber \\ \end{aligned}$$

(4.2)

where $\mathbf {G}_{2}^{1}(\underline{X})\in \mathbf {C}^{0,\alpha }(\partial \Omega ).$

Lemma 4.1

[15, 18] Let $\mathbf {G}_{2}^{1}(\underline{X})\in \mathbf {C}^{0,\alpha }(\partial \Omega ).$ Then the boundary values of the Hermitian Cauchy integral $\mathcal {C}[\mathbf {G}_{2}^{1}]$ are given by

$$\begin{aligned} \mathcal {C}[\mathbf {G}_{2}^{1}]^{\pm }(\underline{U})\triangleq & {} \lim \limits _{\mathop {\underline{Y}\rightarrow \underline{U}\in \partial \Omega }\limits _ {\underline{Y}\in \Omega ^{\pm }}}\mathcal {C}[\mathbf {G}_{2}^{1}](\underline{Y})\nonumber \\= & {} (-1)^{\frac{n(n+1)}{2}}(2i)^{n}\left( \pm \frac{1}{2}\mathbf {G}_{2}^{1}(\underline{U})+ \frac{1}{2}\mathcal {H}_{\partial \Omega }[\mathbf {G}_{2}^{1}](\underline{U})\right) . \end{aligned}$$

(4.3)

Furthermore, we have

$$\begin{aligned} \Vert \mathcal {C}[\mathbf {G}_{2}^{1}]^{\pm }\Vert _{(\alpha , \partial \Omega )} \le C_{(\alpha , \partial \Omega )}\Vert \mathbf {G}_{2}^{1}\Vert _{(\alpha , \partial \Omega )}, \end{aligned}$$

(4.4)

where $C_{(\alpha , \partial \Omega )}$ depending on $\alpha$ and $\partial \Omega$.

Lemma 4.2

Let $\mathbf {G}_{2}^{1}(\underline{Y})\in \mathbf {C}^{0,\alpha }(\partial \Omega )$, $0<\alpha <1.$ Then for any $\underline{Y}_{1}$, $\underline{Y}_{2}\in \partial \Omega$, there exists a positive constant $C(\alpha , n)$ which does not depend on $\underline{Y}_{1}$ and $\underline{Y}_{2}$ such that

$$\begin{aligned} \Vert \mathcal {H}_{\partial \Omega }[\mathbf {G}_{2}^{1}](\underline{Y}_{1})-\mathcal {H}_{\partial \Omega }[\mathbf {G}_{2}^{1}](\underline{Y}_{2})\Vert \le C(\alpha , n)|\underline{Y}_{1}-\underline{Y}_{2}|^{\alpha }. \end{aligned}$$

(4.5)

Proof

For $\underline{Y}_{1}$, $\underline{Y}_{2}\in \partial \Omega$, it is enough to consider the case of $|\underline{Y}_{1}-\underline{Y}_{2}|$ being sufficiently small. In view of some properties of the Hilbert transform that can be found in [3, 15, 18, 26], we get

$$\begin{aligned} |H_{\partial \Omega }[u](\underline{Y}_{1})-H_{\partial \Omega }[u](\underline{Y}_{2})|\le C_{1}(\alpha , n)|\underline{Y}_{1}-\underline{Y}_{2}|^{\alpha } \end{aligned}$$

(4.6)

and

$$\begin{aligned} |H|_{\partial \Omega }[u](\underline{Y}_{1})-H|_{\partial \Omega }[u](\underline{Y}_{2})|\le C_{1}(\alpha , n)|\underline{Y}_{1}-\underline{Y}_{2}|^{\alpha }, \end{aligned}$$

(4.7)

where $C_{1}(n)$ is a positive constant independent of $\underline{Y}_{1}$, $\underline{Y}_{2}$. Since $\mathbf {G}_{2}^{1}(\underline{Y})\in \mathbf {C}^{0,\alpha }(\partial \Omega )$, we have $g_{1}\pm g_{2}\in H^{\alpha }(\partial \Omega , \mathbb {C}_{2n})$. Denote

$$\begin{aligned} A_{1}(\underline{Y}_{1},\underline{Y}_{2}):= & {} H_{\partial \Omega }[g_{1}-{g}_{2}] (\underline{Y}_{1})-H_{\partial \Omega }[g_{1}-{g}_{2}](\underline{Y}_{2})\nonumber \\&+H|_{\partial \Omega }[g_{1}+{g}_{2}](\underline{Y}_{1})-H|_{\partial \Omega }[g_{1}+{g}_{2}](\underline{Y}_{2}), \end{aligned}$$

(4.8)

$$\begin{aligned} A_{2}(\underline{Y}_{1},\underline{Y}_{2}):= & {} H_{\partial \Omega }[g_{2}-{g}_{1}] (\underline{Y}_{1})-H_{\partial \Omega }[g_{2}-{g}_{1}](\underline{Y}_{2})\nonumber \\&+H|_{\partial \Omega }[g_{1}+{g}_{2}](\underline{Y}_{1})-H|_{\partial \Omega }[g_{1}+{g}_{2}](\underline{Y}_{2}), \end{aligned}$$

(4.9)

combining (4.6) with (4.7), we have

$$\begin{aligned} \Vert \mathcal {H}_{\partial \Omega }[\mathbf {G}_{2}^{1}](\underline{Y}_{1})-\mathcal {H}_{\partial \Omega }[\mathbf {G}_{2}^{1}](\underline{Y}_{2})\Vert= & {} \left\| \frac{1}{2}\left( \begin{array}{cc} A_{1}(\underline{Y}_{1},\underline{Y}_{2}) &{}\quad A_{2}(\underline{Y}_{1},\underline{Y}_{2}) \\ A_{2}(\underline{Y}_{1},\underline{Y}_{2}) &{} \quad A_{1}(\underline{Y}_{1},\underline{Y}_{2}) \end{array}\right) \right\| \nonumber \\= & {} \left( \sum \limits _{i=1}^{2}|A_{i}(\underline{Y}_{1},\underline{Y}_{2})|^{2}\right) ^{\frac{1}{2}}\nonumber \\\le & {} C(\alpha , n)|\underline{Y}_{1}-\underline{Y}_{2}|^{\alpha }. \end{aligned}$$

(4.10)

The proof is done. $\square$

In particular, using Lemmas 4.1 and 4.2, we have the following property.

Theorem 4.3

The matrix operator $\mathcal {H}_{\partial \Omega }$: $\mathbf {C}^{0,\alpha }(\partial \Omega )\rightarrow \mathbf {C}^{0,\alpha }(\partial \Omega )$, $0<\alpha <1$, defined by (4.2) is bounded, i.e., for any $\mathbf {G}_{2}^{1}(\underline{Y})\in \mathbf {C}^{0,\alpha }(\partial \Omega )$, there exists a positive constant $C_{(\alpha , n, \partial \Omega )}$ which does not depend on $\mathbf {G}_{2}^{1}(\underline{Y})$ such that

$$\begin{aligned} \Vert \mathcal {H}_{\partial \Omega }[\mathbf {G}_{2}^{1}]\Vert _{(\alpha ,\partial \Omega )}\le C_{(\alpha , n, \partial \Omega )}\Vert \mathbf {G}_{2}^{1}\Vert _{(\alpha ,\partial \Omega )}. \end{aligned}$$

(4.11)

Firstly, as some applications of Lemmas 4.1, 4.2 and Theorem 4.3 we solve the problem of finding a sectionally $\mathbf {H}$-monogenic circulant $(2\times 2)$ matrix function (H-monogenic function for short) with a given discontinuity on $\partial \Omega$. Secondly, we state necessary and sufficient conditions for $\mathbf {H}$-monogenic function in $\Omega ^{+}$ or $\mathbf {R}^{2n}{\setminus }\overline{\Omega }$ with given boundary values. Finally, we can prove a property of the matrix operator $\mathcal {H}_{\partial \Omega }$ in Hermitian Clifford analysis.

Theorem 4.4

Assume $\mathbf {G}_{2}^{1}(\underline{X})\in \mathbf {C}^{0, \alpha }(\partial \Omega )$. Then there exists a unique circulant $(2\times 2)$ matrix function $\mathbf {F}_{2}^{1}$ which is $\mathbf {H}$-monogenic in $\Omega ^{+}$ and $\Omega ^{-}$, which can be extended continuously from $\Omega ^{+}$ into $\overline{\Omega ^{+}}$ and from $\Omega ^{-}$ into $\overline{\Omega ^{-}}$ satisfying the boundary condition

$$\begin{aligned}{}[\mathbf {F}_{2}^{1}]^{+}(\underline{X})-[\mathbf {F}_{2}^{1}]^{-}(\underline{X}) =\mathbf {G}_{2}^{1}(\underline{X})&\text {on }\,\partial \Omega , \end{aligned}$$

(4.12)

and for which $\mathbf {F}_{2}^{1}(\underline{X})\rightarrow \mathbf {O}$, $|\underline{X}|\rightarrow \infty$, uniformly for all directions. This matrix function is given by

$$\begin{aligned} \mathbf {F}_{2}^{1}(\underline{Y})=(-1)^{\frac{n(n+1)}{2}}\left( -\frac{i}{2}\right) ^{n}\int \limits _{\partial \Omega }{\mathbf {E}} (\underline{Z}-\underline{V})\mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })} \mathbf {G}_{2}^{1}(\underline{X}), \underline{Y}\in \mathbf {R}^{2n}{\setminus }\partial \Omega .\qquad \quad \end{aligned}$$

(4.13)

Proof

The matrix functions $\mathbf {F}_{2}^{1}(\underline{Y})$ is given by

$$\begin{aligned} \mathbf {F}_{2}^{1}(\underline{Y})=(-1)^{\frac{n(n+1)}{2}}\left( -\frac{i}{2}\right) ^{n}\int \limits _{\partial \Omega }{{\varvec{E}}} (\underline{Z}-\underline{V})\mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })} \mathbf {G}_{2}^{1}(\underline{X}),~~~~~~\underline{Y}\in \mathbf {R}^{2n}{\setminus }\partial \Omega . \end{aligned}$$

Using Lemma 4.1, we have the required properties. To establish uniqueness, we suppose there exists $\widetilde{\mathbf {F}_{2}^{1}}(\underline{Y})$ satisfying the required properties. Denote by

$$\begin{aligned} \mathbf {H}_{2}^{1}(\underline{Y})=\mathbf {F}_{2}^{1}(\underline{Y})-\widetilde{\mathbf {F}_{2}^{1}}(\underline{Y})&\text {on }\,\partial \Omega . \end{aligned}$$

(4.14)

Using the Hermitian Borel–Pompeiu formula and Lemma 4.1, we conclude that $\mathbf {H}_{2}^{1}(\underline{Y})$ is $\mathbf {H}$-monogenic in $\mathbf {R}^{2n}$. It follows from $\Vert \mathbf {H}_{2}^{1}(\infty )\Vert =0$ by Liouville theorem (Theorem 3.2 in [21]) that $\mathbf {H}_{2}^{1}(\underline{Y})=\mathbf {O}$ in $\mathbf {R}^{2n}$. $\square$

Remark 4.5

From the above proof, we see that the general sectionally $\mathbf {H}$-monogenic function $\mathbf {F}_{2}^{1}(\underline{Y})$ satisfying condition (4.12) is obtained from (4.13) by adding an arbitrary $\mathbf {H}$-monogenic function in $\mathbf {R}^{2n}$.

Theorem 4.6

For a given matrix function $\mathbf {G}_{2}^{1}(\underline{Y})\in \mathbf {C}^{0,\alpha }(\partial \Omega )$, there exists a matrix function $\mathbf {F}_{2}^{1}(\underline{Y})$$\mathbf {H}$-monogenic in $\Omega ^{+}$ and continuous in $\overline{\Omega ^{+}}$ with boundary values $\mathbf {F}_{2}^{1}=\mathbf {G}_{2}^{1}$ on $\partial \Omega$ if and only if $\mathbf {G}_{2}(\underline{Y})$ is a solution of the integral equation

$$\begin{aligned} \mathbf {G}_{2}^{1}(\underline{Y})-\mathcal {H}_{\partial \Omega }[\mathbf {G}_{2}^{1}](\underline{Y})=\mathbf {O}. \end{aligned}$$

(4.15)

Proof

Let $\mathbf {F}_{2}^{1}$ be $\mathbf {H}$-monogenic in $\Omega$ with $\mathbf {F}_{2}^{1}(\underline{X})=\mathbf {G}_{2}^{1}(\underline{X})$ on $\partial \Omega$. Then by Hermitian Borel–Pompeiu formula, we have

$$\begin{aligned} \mathbf {F}_{2}^{1}(\underline{Y})= & {} (-1)^{\frac{n(n+1)}{2}}\left( -\frac{i}{2}\right) \int \limits _{\partial \Omega } {{\varvec{E}}} (\underline{Z}-\underline{V})\mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })} \mathbf {F}_{2}^{1}(\underline{X})\nonumber \\= & {} (-1)^{\frac{n(n+1)}{2}}\left( -\frac{i}{2}\right) \int \limits _{\partial \Omega }{{\varvec{E}}} (\underline{Z}-\underline{V})\mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })} \mathbf {G}_{2}^{1}(\underline{X}),~~~~\underline{Y}\in \Omega . \end{aligned}$$

Applying Lemma 4.1, we have

$$\begin{aligned} 2\mathbf {G}_{2}^{1}(\underline{Y})=2[\mathbf {F}_{2}^{1}]^{+}(\underline{Y}) =\mathbf {G}_{2}^{1}(\underline{Y})+\mathcal {H}_{\partial \Omega }[\mathbf {G}_{2}^{1}](\underline{Y}), ~~~~\underline{Y}\in \partial \Omega , \end{aligned}$$

therefore $\mathbf {G}_{2}^{1}(\underline{Y})-\mathcal {H}_{\partial \Omega }[\mathbf {G}_{2}^{1}](\underline{Y})=\mathbf {O}$.

Conversely, if $\mathbf {G}_{2}^{1}(\underline{X})$ is a solution of $\mathbf {G}_{2}^{1}(\underline{X})-\mathcal {H}_{\partial \Omega }[\mathbf {G}_{2}^{1}](\underline{X})=\mathbf {O}$, we denote

$$\begin{aligned} \mathbf {F}_{2}^{1}(\underline{Y})= & {} (-1)^{\frac{n(n+1)}{2}}\left( -\frac{i}{2}\right) \int \limits _{\partial \Omega } {{\varvec{E}}} (\underline{Z}-\underline{V})\mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })} \mathbf {G}_{2}^{1}(\underline{X}),~~\underline{Y}\in \Omega ^{+}, \end{aligned}$$

again by Lemma 4.1 the matrix function $\mathbf {F}_{2}^{1}(\underline{Y})$ has boundary values

$$\begin{aligned} 2[\mathbf {F}_{2}^{1}]^{+}=\mathcal {H}_{\partial \Omega }[\mathbf {G}_{2}^{1}](\underline{Y})+\mathbf {G}_{2}^{1}(\underline{Y}) =2\mathbf {G}_{2}^{1}(\underline{Y}),&\text {on}\, \partial \Omega . \end{aligned}$$

The proof is done. $\square$

Remark 4.7

For the corresponding exterior problem in $\Omega ^{-}$ with $\mathbf {F}_{2}^{1}(\underline{X})\rightarrow \mathbf {O}$, $|\underline{X}|\rightarrow \infty$, we have the integral equation

$$\begin{aligned} \mathbf {G}_{2}^{1}(\underline{Y})+\mathcal {H}_{\partial \Omega }[\mathbf {G}_{2}^{1}](\underline{Y})=\mathbf {O}. \end{aligned}$$

(4.16)

Theorem 4.8

The matrix operator $\mathcal {H}_{\partial \Omega }$ satisfies $\mathcal {H}_{\partial \Omega }^{2}=\mathbf {I}$, where $\mathbf {I}$ denotes the $(2\times 2)$ identity matrix operator.

Proof

For any $\mathbf {G}_{2}^{1}(\underline{Y})\in \mathbf {C}^{0, \alpha }(\partial \Omega )$, define $\mathbf {F}_{2}^{1}$ by

$$\begin{aligned} \mathbf {F}_{2}^{1}(\underline{Y})= & {} (-1)^{\frac{n(n+1)}{2}}\left( -\frac{i}{2}\right) \int \limits _{\partial \Omega } {{\varvec{E}}} (\underline{Z}-\underline{V})\mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })} \mathbf {G}_{2}^{1}(\underline{X}),~~\underline{Y}\in \mathbf {R}^{2n}{\setminus }\partial \Omega . \end{aligned}$$

Then, by Theorem 4.6 and Remark 4.7, we have

$$\begin{aligned}{}[\mathbf {F}_{2}^{1}]^{+}(\underline{Y})-\mathcal {H}_{\partial \Omega }[\mathbf {F}_{2}^{1}]^{+}(\underline{Y})=\mathbf {O} \end{aligned}$$

and

$$\begin{aligned}{}[\mathbf {F}_{2}^{1}]^{-}(\underline{Y})+\mathcal {H}_{\partial \Omega }[\mathbf {F}_{2}^{1}]^{-}(\underline{Y})=\mathbf {O}, \end{aligned}$$

applying Lemma 4.1 we derive

$$\begin{aligned} \mathcal {H}_{\partial \Omega }^{2}[\mathbf {G}_{2}^{1}](\underline{Y})= & {} \mathcal {H}_{\partial \Omega }[[\mathbf {F}_{2}^{1}]^{+}(\underline{Y})+[\mathbf {F}_{2}^{1}]^{-}(\underline{Y})]\nonumber \\= & {} [\mathbf {F}_{2}^{1}]^{+}(\underline{Y})-[\mathbf {F}_{2}^{1}]^{-}(\underline{Y})\nonumber \\= & {} \mathbf {G}_{2}^{1}(\underline{Y}), \end{aligned}$$

(4.17)

and this is the desired result. $\square$

5 Riemann-Type Problem for $\mathbf {H}$-2-Monogenic Function

In the following, we need to make slight generalization of Theorem 3.6. This step is necessary to study generalized Riemann-type problem for $\mathbf {H}$-2-monogenic function in Hermitian Clifford analysis. In what follows, suppose $\Omega$ is an open bounded non-empty subset of $\mathbf {R}^{2n}$ with a Lyapunov boundary $\partial \Omega$.

Theorem 5.1

Let $g_{1}$, $g_{2}\in C^{2}(\Omega ^{-}, \mathbb {C}_{2n})\bigcap C^{1}(\overline{\Omega ^{-}}, \mathbb {C}_{2n})$, $(\mathcal {D}_{(\underline{Z},\underline{Z}^{\dag })})^{2}[\mathbf {G}_{2}^{1}]=\mathbf {O}$ in $\Omega ^{-}$, and $\mathbf {G}_{2}^{1}(\underline{X})$ satisfies the following conditions:

$$\begin{aligned} \begin{array}{ll} \mathbf {G}_{2}^{1}(\underline{X})\in \mathbf {C}^{0, \alpha }(\partial \Omega ),\\ \mathcal {D}_{(\underline{Z},\underline{Z}^{\dag })}[\mathbf {G}_{2}^{1}](\underline{X})\in \mathbf {C}^{0, \beta }(\partial \Omega ),\\ \liminf \limits _{r\rightarrow \infty }M[r,\mathbf {G}_{2}^{1}]\le \infty , \end{array} \end{aligned}$$

where $0<\alpha , \beta \le 1$. Then $\liminf \limits _{r\rightarrow \infty }M[r,\mathcal {D}_{(\underline{Z},\underline{Z}^{\dag })}[\mathbf {G}_{2}^{1}]]=0$.

Proof

Let $\mathbf {G}_{2}^{1}(\underline{X})=-\Psi _{2}^{1}(\underline{X})$, $\mathcal {D}_{(\underline{Z},\underline{Z}^{\dag })}[\mathbf {G}_{2}^{1}](\underline{X}) =-\widetilde{\Psi }_{2}^{1}(\underline{X})$, $\underline{X}\in \partial \Omega$. For $\underline{Y}\in \mathbf {R}^{2n}$, denote

$$\begin{aligned} \Phi _{2}^{1}(\underline{Y})= & {} (-1)^{\frac{n(n+1)}{2}}\left( -\frac{i}{2}\right) ^{n} \left[ \int _{\partial \Omega }{{\varvec{E}}} (\underline{Z}-\underline{V})\mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })}\Psi _{2}^{1}(\underline{X})\right. \nonumber \\&\left. -\int _{\partial \Omega }{{\varvec{E}}}_{1}(\underline{Z}-\underline{V})\mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })} \widetilde{\Psi }_{2}^{1}(\underline{X})\right] \end{aligned}$$

(5.1)

and

$$\begin{aligned} \mathbf {G}_{2}^{1*}(\underline{Y})=\left\{ \begin{array}{ll} -\Phi _{2}^{1}(\underline{Y}), &{} \underline{Y}\in \Omega ^{+},\\ \mathbf {G}_{2}^{1}(\underline{Y})-\Phi _{2}^{1}(\underline{Y}), &{} \underline{Y}\in \Omega ^{-}. \end{array}\right. \end{aligned}$$

(5.2)

By Lemma 4.1, we have

$$\begin{aligned} \begin{array}{ll} {[}\mathbf {G}_{2}^{1*}]^{+}(\underline{Y})=[\mathbf {G}_{2}^{1*}]^{+}(\underline{Y})\in \mathbf {C}^{0,\widetilde{\alpha }}(\partial \Omega ),\\ {[}\mathcal {D}_{(\underline{V},\underline{V}^{\dag })}[\mathbf {G}_{2}^{1*}]]^{+}(\underline{Y})= {[}\mathcal {D}_{(\underline{V},\underline{V}^{\dag })}[\mathbf {G}_{2}^{1*}]]^{-}(\underline{Y})\in \mathbf {C}^{0,\widetilde{\beta }}(\partial \Omega ), \end{array} \end{aligned}$$

(5.3)

where $0<\widetilde{\alpha }, \widetilde{\beta }\le 1$, then $\mathcal {D}_{(\underline{Z},\underline{Z}^{\dag })}[\mathbf {G}_{2}^{1*}]=\mathbf {O}$ in $\mathbf {R}^{2n}{\setminus }\partial \Omega$. It is clear that

$$\begin{aligned} \liminf \limits _{r\rightarrow \infty } M[r, \mathbf {G}_{2}^{1*}]<\infty . \end{aligned}$$

Furthermore, using Theorem 3.6, we get

$$\begin{aligned} \liminf \limits _{r\rightarrow \infty } M[r, \mathcal {D}_{(\underline{Z},\underline{Z}^{\dag })}[\mathbf {G}_{2}^{1*}]]=0. \end{aligned}$$

(5.4)

For $\underline{Y}\in \Omega ^{-}$,

$$\begin{aligned}&\mathcal {D}_{(\underline{V},\underline{V}^{\dag })}[\mathbf {G}_{2}^{1}](\underline{Y}) =\mathcal {D}_{(\underline{V},\underline{V}^{\dag })}[\mathbf {G}_{2}^{1*}](\underline{Y})\nonumber \\&\quad +\,(-1)^{\frac{n(n+1)}{2}}\left( -\frac{i}{2}\right) ^{n} \int _{\partial \Omega }{{\varvec{E}}} (\underline{Z}-\underline{V})\mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })}\widetilde{\Psi }_{2}^{1}(\underline{X}). \end{aligned}$$

(5.5)

Combining (5.4) with (5.5), the result follows. $\square$

In this section, we will investigate the Riemann-type problem (1.2).

Theorem 5.2

Suppose that circulant matrix functions $\mathbf {A}_{2}^{1}(\underline{Y})$, $\mathbf {F}_{2}^{1}(\underline{Y})$, $\mathbf {U}_{2}^{1}(\underline{Y})\in \mathbf {C}^{0, \alpha }(\partial \Omega )$, $0<\alpha <1$, and $\mathbf {A}_{2}^{1}(\underline{Y})$ satisfies the following condition

$$\begin{aligned} 2^{2n-2}\Vert \mathbf {I}-\mathbf {A}_{2}^{1}\Vert _{(\alpha , \partial \Omega )}\cdot C_{(\alpha , n, \partial \Omega )}<1, \end{aligned}$$

(5.6)

where $C_{(\alpha , n, \partial \Omega )}$ is a positive constant appearing in Theorem 4.3. Then there exists a solution to Riemann boundary type problem (1.2).

Proof

Denoting $\mathbf {W}_{2}^{1}(\underline{Y})=\mathcal {D}_{(\underline{V},\underline{V}^{\dag })}\mathbf {G}_{2}^{1}(\underline{Y})$, we then have

$$\begin{aligned}{}[\mathbf {W}_{2}^{1}]^{+}(\underline{Y})=[\mathbf {W}_{2}^{1}]^{-}(\underline{Y}) \mathbf {B}_{2}^{1}+\mathbf {U}_{2}^{1}(\underline{Y}),&\underline{Y}\in \partial \Omega . \end{aligned}$$

(5.7)

Moreover, applying Theorem 5.1 we get $\mathcal {D}_{(\underline{V},\underline{V}^{\dag })}\mathbf {G}_{2}^{1}(\infty )=\mathbf {O},$ we have

$$\begin{aligned} C(n)\mathbf {W}_{2}^{1}(\underline{Y})= \left\{ \begin{array}{ll} \int \limits _{\partial \Omega }{{\varvec{E}}}(\underline{Z}-\underline{V}) \mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })}\mathbf {U}_{2}^{1}(\underline{X}) , &{} \underline{Y}\in \Omega ^{+},\\ \int \limits _{\partial \Omega }{{\varvec{E}}}(\underline{Z}-\underline{V}) \mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })} \mathbf {U}_{2}^{1}(\underline{X})[\mathbf {B}_{2}^{1}]^{-1} , &{} \underline{Y}\in \Omega ^{-}, \end{array}\right. \end{aligned}$$

(5.8)

where $C(n)=(-1)^{\frac{n(n+1)}{2}}(2i)^{n}$.

Assume

$$\begin{aligned} C(n)\mathbf {J}_{2}^{1}(\underline{Y})= \left\{ \begin{array}{ll} -\int \limits _{\partial \Omega }{{\varvec{E}}}_{1}(\underline{Z}-\underline{V}) \mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })}\mathbf {U}_{2}^{1}(\underline{X}) , &{} \underline{Y}\in \Omega ,\\ -\int \limits _{\partial \Omega }{{\varvec{E}}}_{1}(\underline{Z}-\underline{V}) \mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })} \mathbf {U}_{2}^{1}(\underline{X})[\mathbf {B}_{2}^{1}]^{-1} , &{} \underline{Y}\in \Omega . \end{array}\right. \end{aligned}$$

(5.9)

Combining (5.8) with (5.9), we get

$$\begin{aligned} \mathcal {D}_{(\underline{V},\underline{V}^{\dag })}[\mathbf {G}_{2}^{1}-\mathbf {J}_{2}^{1}](\underline{Y})=0,&\underline{Y}\in \mathbf {R}^{2n}{\setminus }\partial \Omega . \end{aligned}$$

(5.10)

Denoting $\mathbf {G}_{2}^{1}-\mathbf {J}_{2}^{1}\triangleq {\varvec{\Phi }}_{2}^{1}(\underline{Y}),$ where $\underline{Y}\in \mathbf {R}^{2n}{\setminus }\partial \Omega$ and using

$$\begin{aligned}{}[\mathbf {G}_{2}^{1}]^{+}(\underline{Y})=[\mathbf {G}_{2}^{1}]^{-}(\underline{Y})\mathbf {A}_{2}^{1} +\mathbf {F}_{2}^{1}(\underline{Y}),~~~~~~\underline{Y}\in \partial B(\underline{a}, R), \end{aligned}$$

we obtain

$$\begin{aligned}{}[{\varvec{\Phi }}_{2}^{1}]^{+}(\underline{Y})=[{\varvec{\Phi }}_{2}^{1}]^{-}(\underline{Y})\mathbf {A}_{2}^{1} +\widetilde{\mathbf {F}}_{2}^{1}(\underline{Y}),&\underline{Y}\in \partial \Omega , \end{aligned}$$

(5.11)

where

$$\begin{aligned} \widetilde{\mathbf {F}}_{2}^{1}(\underline{Y})= & {} \mathbf {F}_{2}^{1}(\underline{Y})- (-1)^{\frac{n(n+1)}{2}}\left( -\frac{i}{2}\right) ^{n}\int \limits _{\partial \Omega } {{\varvec{E}}}_{1}(\underline{Z}-\underline{V})\mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })} \mathbf {U}_{2}^{1}(\underline{X})\nonumber \\&+(-1)^{\frac{n(n+1)}{2}}\left( -\frac{i}{2}\right) ^{n}\int \limits _{\partial \Omega } {{\varvec{E}}}_{1}(\underline{Z}-\underline{V})\mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })} \mathbf {U}_{2}^{1}(\underline{X})[\mathbf {B}_{2}^{1}]^{-1}\mathbf {A}_{2}^{1},\nonumber \\&\underline{Y}\in \partial \Omega . \end{aligned}$$

(5.12)

It is clear that $\Vert {\varvec{\Phi }}_{2}^{1}(\infty )\Vert =0$. Combining (5.10) with (5.11), we have

$$\begin{aligned} \begin{array}{ll} \mathcal {D}_{(\underline{V},\underline{V}^{\dag })}[{\varvec{\Phi }}_{2}^{1}](\underline{Y})=0, &{}\text {in }\,\mathbf {R}^{2n}{\setminus }\partial \Omega ,\\ {[}{\varvec{\Phi }}_{2}^{1}]^{+}(\underline{Y})=[{\varvec{\Phi }}_{2}^{1}]^{-}(\underline{Y})\mathbf {A}_{2}^{1}(\underline{Y}) +\widetilde{\mathbf {F}_{2}^{1}}(\underline{Y}), &{} \underline{Y}\in \partial \Omega ,\\ \Vert {\varvec{\Phi }}_{2}^{1}(\infty )\Vert =0. \end{array} \end{aligned}$$

(5.13)

We only need to consider the existence to (5.13). The solution of this problem may be written in the form:

$$\begin{aligned} C(n)\cdot {\varvec{\Phi }}_{2}^{1}(\underline{Y})=\int \limits _{\partial \Omega } {{\varvec{E}}}_{1}(\underline{Z}-\underline{V})\mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })} \widetilde{{\varvec{\Phi }}}_{2}^{1}(\underline{X}), \end{aligned}$$

(5.14)

where $\widetilde{{\varvec{\Phi }}}_{2}^{1}(\underline{X})$ is a Hölder continuous circulant matrix function to be determined on $\partial \Omega$. Using Lemma 4.1, (5.14) will be reduced to an equivalent integral equation for $\widetilde{{\varvec{\Phi }}}_{2}^{1}(\underline{X})$, i.e.,

$$\begin{aligned}&\widetilde{{\varvec{\Phi }}}_{2}^{1}(\underline{Y})= \widetilde{{\varvec{\Phi }}}_{2}^{1}(\underline{Y})\left( \frac{\mathbf {I}-\mathbf {A}_{2}^{1} (\underline{Y})}{2}\right) + \widetilde{\mathbf {F}}_{2}^{1}(\underline{Y}) \nonumber \\&- (-1)^{\frac{n(n+1)}{2}}\left( -\frac{i}{2}\right) \int \limits _{\partial \Omega } {{\varvec{E}}} (\underline{Z}-\underline{V})\mathrm{d}\Sigma _{(\underline{Z},\underline{Z}^{\dag })} \widetilde{{\varvec{\Phi }}}_{2}^{1}(\underline{X})(\mathbf {I}-\mathbf {A}_{2}^{1}(\underline{Y})). \end{aligned}$$

(5.15)

Let T be an integral operator defined by the right-hand side of (5.15), and we get

$$\begin{aligned}{}[T\widetilde{{\varvec{\Phi }}}_{2}^{1}](\underline{Y}) =[\widetilde{{\varvec{\Phi }}}_{2}^{1}(\underline{Y})-\mathcal {H}_{\partial \Omega }[\widetilde{{\varvec{\Phi }}}_{2}^{1}](\underline{Y})] \left( \frac{\mathbf {I}-\mathbf {A}_{2}^{1}(\underline{Y})}{2}\right) +\widetilde{\mathbf {F}}_{2}^{1}(\underline{Y}). \end{aligned}$$

(5.16)

For any ${\varvec{\Phi }}_{2}^{1}$, ${\varvec{\Psi }}_{2}^{1}\in \mathbf {C}^{0, \alpha }(\partial \Omega ),$

$$\begin{aligned} \Vert T{\varvec{\Phi }}_{2}^{1}-T{\varvec{\Psi }}_{2}^{1}\Vert _{(\alpha , \partial \Omega )} \le \frac{2^{2n-1}}{2}\Vert {\varvec{\Phi }}_{2}^{1}-{\varvec{\Psi }}_{2}^{1}\Vert _{(\alpha , \partial \Omega )} \Vert \mathbf {I}-\mathbf {A}_{2}^{1}\Vert _{(\alpha , \partial \Omega )}(1+C_{(\alpha , n, \partial \Omega )}) \end{aligned}$$

From (5.6), the integral operator T be as in (5.16) on the Banach space $\mathbf {C}^{0,\alpha }(\partial \Omega )$ satisfies the contractive mapping principle, and hence, there exists a unique fixed point for the operator T and thus also a unique solution of (5.13). The proof is done. $\square$