Integral representation formulas are very powerful tools for solving boundary value problems in Clifford analysis. In [1-15, 18-27] etc., a great deal of work about integral representation formulas and boundary value problems in Clifford analysis was well presented. In[16-17], classical theories of boundary value problems and singular integral equations were systematically built.

However, most of the work about integral representation formulas was built over bounded domains. Naturally, developing integral representation formulas over unbounded domains is important and interesting, it will serve to study the Riemann-Hilbert boundary value problems for k-regular functions over unbounded domains in Clifford analysis. Similar to Cauchy type integrals over the real axis in classical complex analysis, Cauchy type integrals over the plane in Clifford analysis framework are also valuable. In [8], Cauchy transform and Hilbert transform over $\mathcal{R}$^m were introduced; In [12-13]etc., by constructing the new Cauchy kernel function, some integral representation formulas over unbounded domains and its applications were shown. In [27], Cauchy type integral and singular integral over hyper-complex plane Ⅱ in the hyper-complex space RQ₃ were studied by using a special Möbius transform, integral representation formulas over hyper-complex plane Ⅱ for regular functions were built.

In this paper, combining the idea in [9] with the technique in [12-13], we construct the kernel functions, and then give the integral representations over hyper-complex plane Ⅱ for bi-regular functions and harmonic functions with values in a Clifford algebra.

Let $V_n,0$ be an n-dimensional ($(n\geq 1)$) real linear space with basis $\{e_1, e_2,\cdots,e_n\}$, $C(V_{n,0})$ be the $2^n$--dimensional real linear space with basis

$ \left\{ e_A, A=\{h_1,\cdots, h_r\}\in \mathcal{P}N, 1\leq h_1<\cdots<h_r \leq n \right\}, $

where N stands for the set $\{1,\cdots, n \}$ and $\mathcal{P}N$ denotes the family of all order-preserving subsets of N in the above way. we denote $e_{\emptyset}$ as $e_0$ and $e_A$ as $e_{h_1\cdots h_r}$ for $A=\{h_1,\cdots,h_r\}\in \mathcal{P}N$. The product on $C(V_{n,0})$ is defined by

$\left\{ \begin{array}{l} {e_A}{e_B} = {( - 1)^{\# (A \cap B)}}{( - 1)^{P(A,B)}}{e_{A\Delta B}},{\rm{if}},B \in {\cal P}N,\\ \lambda \mu = \sum\limits_{A \in {\cal P}N} {\sum\limits_{B \in {\cal P}N} {{\lambda _A}} } {\mu _B}{e_A}{e_B},{\rm{if}}\;\lambda = \sum\limits_{A \in {\cal P}N} {{\lambda _A}} {e_A},\mu = \sum\limits_{B \in {\cal P}N} {{\mu _B}{e_B},} \end{array} \right.$

(1.1)

where $\#(A)$ is the cardinal number of the set A, the number $P(A,B)=\sum\limits_{j\in B}P(A,j)$, $P(A,j)=\#\{i, i\in A, i>j\}$, the symmetric difference set $A\triangle B$ is also order-preserving in the above way, and $ \lambda_{A} \in \mathcal{R}$ is the coefficient of the $e_A$-component of the Clifford number $\lambda$. We also denote $\lambda_0$ as ${\rm Re}(\lambda)$. Thus $C(V_{n,0})$ is called the Clifford algebra over $V_{n,0}$.

An involution is defined by

$\left\{ {_{\bar \lambda = \sum\limits_{A \in \mathcal{P}N} {_{{\lambda _A}}\overline {{e_A}} } {\rm{,if}}{\mkern 1mu} {\mkern 1mu} \lambda = \sum\limits_{A \in \mathcal{P}N} {{\lambda _A}{e_A}} ,}^{\overline {{e_A}} = {{( - 1)}^{\sigma (A)}}{e_A},{\rm{if}}{\mkern 1mu} {\mkern 1mu} A \in \mathcal{P}N,}} \right.$

(1.2)

where $\sigma (A)=\#(A)(\#(A)+1)/2$. The $C\left(V_{n.0}\right)$-valued n-differential form

$ {\rm d}\sigma= \sum\limits_{k=0}^{n}(-1)^{k}e_k{\rm d} \widehat{x}^{N+1}_{k},\,\,\,\overline{\rm d\sigma}= \sum\limits_{k=0}^{n}(-1)^{k}\overline{e_k}{\rm d} \widehat{x}^{N+1}_{k} $

are exact, where

$ {\rm d}\widehat{x}^{N+1}_{k}= {\rm d}x_0\wedge\cdots\wedge{\rm d}x_{k-1} \wedge{\rm d}x_{k+1}\wedge\cdots\wedge{\rm d}x_n. $

In this paper, we confine n=2. The real linear space with basis $\{e_0, e_1, e_2\}$ is a subspace of $C(V_{2,0}))$, which is called the reduced quaternions and denoted by $RQ_3$. The operator D which is written as

$D=\sum\limits_{k=0}^2e_k\dfrac{\partial}{\partial\!x_k}:\, C^{(r)}(\Omega,C(V_{2,0}))\rightarrow C^{(r-1)}(\Omega,C(V_{2,0})). $

Let $R{Q_3} = \left\{ {x = {x_0} + {x_1}{e_1} + {x_2}{e_2}:{x_0},{x_1},{x_2} \in \mathcal{R}} \right\}$, then $RQ_3$ is identical with the usual Euclidean space$\mathcal{R}$ ³. Denote $\Pi=\left\{{\rm{x}}\in RQ_3| {x}_0=0\right\}$, $RQ_3^ + = \left\{ {{\rm{x}} \in R{Q_3}|{\rm{Re}}({\rm{x}})<0} \right\}$, $RQ_3^ - = \left\{ {{\rm{x}} \in R{Q_3}|{\rm{Re}}({\rm{x}})<0} \right\}$, $\partial B({\rm{x}},r) = \left\{ {{\rm{y}} \in R{Q_3}||y - {\rm{x}}| = r} \right\}$, then Ⅱ and $\partial B(0, 1)$ are the plane and unit sphere in hyper-complex space $RQ_3$ respectively. Denote $D(0,R) = \left\{ {{\rm{x}} \in \Pi ||\ {\rm{x}}|<R} \right\}$.

Definition 2.1 Denote $f\in \widehat{H}^{\mu}(\Pi, C(V_{2,0}))$ if $f$ is $\widehat{H}$ in $\Pi$. $f$ is called $\widehat{H}$ in $\Pi$ if $f$ satisfies the following conditions: (i) $\left| {f({\rm{x}} - f({\rm{x}}^*)} \right|{\rm{ }} \le {M_1}{\left| {{\rm{x}} - {\rm{x}}^*} \right|^\mu }$$\forall{ {\rm{x}}},{ {\rm{x}}}^*\in \overline{D(0, R_1)}$, where $R_1$ is any given sufficiently great constant, $M_1$ is independent of $\ {\rm{x}},{\rm{x}}^*$, $M_1$ depends on $R_1$, $0<\mu\leq 1$. (ii) $ | f({\rm{x}}-f({\rm{x}}^*)|\leq M_2 \left |\dfrac{1}{\ {\rm{x}}}-\dfrac{1}{\ {\rm{x}}^*}\right |^{\mu}$$\forall{ {\rm{x}}},{ {\rm{x}}}^*\in \Pi\setminus D(0, R_2)$, where $R_2$ is any given sufficiently great constant, $M_2$ is independent of $\ {\rm{x}},{\rm{x}}^*$, $M_2$ depends on $R_2$, $0<\mu\leq 1$.

Remark 2.1 $\forall{ {\rm{x}}},{ {\rm{x}}}^*\in \overline{D(0, R_2)}\setminus D(0, R_1)$, $|f({\rm{x}}-f({{{\rm{x}}}^{*}})|\le {{M}_{1}}|{\rm{x}}-{{{\rm{x}}}^{*}}{{|}^{\mu }}$ is equivalent to $| f({\rm{x}}-f({\rm{x}}^*) | \le {M_2} |{{\rm{x}}}-{{\rm{x}}^*}|{|^\mu }$, where $M_1$ and $M_2$ are given constants.

Remark 2.2 By Definition 2.1, if $f\in \widehat{H}^{\mu}(\Pi, C(V_{2,0}))$, then $\mathop {\lim }\limits_{\left| X \right| \to \infty } {\mkern 1mu} f({\rm{x}})$ exists, denote $\mathop {\lim }\limits_{\left| X \right| \to \infty } {\mkern 1mu} f({\rm{x}}) = f(\infty )$ and

$\left| {f({\rm{x}}) - f(\infty )} \right|{\rm{ }} \le \frac{{{M_2}}}{{{{\left| {\rm{x}} \right|}^\mu }}},{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \forall {\rm{x}} \in \Pi \setminus D(0,{R_2}).$

(2.1)

Deflnition 2.2 Denote $f\in \widehat{H}_0^{\mu}(\Pi, C(V_{2,0}))$ if $f$ is $\widehat{H}_0$ in $\Pi$. $f$ is called $\widehat{H}_0$ in $\Pi$ if $f$ satisfies the following conditions: (i) $f\in \widehat{H}^{\mu}(\Pi, C(V_{2,0}))$; (ii) $f(\infty)=0$.

Deflnition 2.3 A function $f\in C^(r)(\Omega, C(V_{2, 0})) (r\geq 2)$ is called bi-regular in $\Omega$ if $D^2[f]=0$ in $\Omega$, which is also called 2-regular in $\Omega$; A function $f\in C^{(r)}(\Omega, C(V_{2, 0})) (r\geq 2)$ is called harmonic in $\Omega$ if $\triangle [f]=0$ in $\Omega$, where $\triangle$ is the Laplace operator.

Denote ${H_1}({\rm{y}} - {\rm{x}}) = \frac{1}{{4\pi }}\frac{{\bar {\rm{y}} - \bar {\rm{x}}}}{{{{\left| {{\rm{y}} - {\rm{x}}} \right|}^3}}}$, $H_1^*({\rm{y}} - {\rm{x}}) = \frac{1}{{4\pi }}\frac{{\bar {\rm{y}} - \overline {{S_\Pi }({\rm{x}})} }}{{{\rm{y}} - {S_\Pi }({\rm{x}}){|^3}}} = \frac{1}{{4\pi }}\frac{{\bar {\rm{y}} + {\rm{x}}}}{{|{\rm{y}} + \bar {\rm{x}}{|^3}}}$, where $S_{\Pi}({\rm{x}})=-\overline{{\rm{x}}}$, $S_{\Pi}({\rm{x}})$ is just the symmetric point of ${\rm{x}}$ with respect to $\Pi$, $y \ne {\rm{x}}$. Denote $E_1({\rm{y}}-{\rm{x}})=H_1({\rm{y}}-{\rm{x}})-H_1^*({\rm{y}}-{\rm{x}})$, $H_2({\rm{y}}-{\rm{x}})=H_1({\rm{y}}-{\rm{x}})\cdot (y_0-x_0)$, $H_2^*({\rm{y}}-{\rm{x}})=H_1^*({\rm{y}}-{\rm{x}})\cdot (y_0-x_0)$, $E_2({\rm{y}}-{\rm{x}})=E_1({\rm{y}}-{\rm{x}})\cdot (y_0-x_0)$.

Lemma 2.1 Let $H_1({\rm{y}}- {\rm{x}})$, $H_2({\rm{y}}- {\rm{x}})$, $H_1^*({\rm{y}}- {\rm{x}})$ and $H_2^*({\rm{y}}- {\rm{x}})$ be as above, then

$\left\{ {_{_{[H_2^*({\rm{y}} - {\rm{x}})]{D^2} = [H_1^*({\rm{y}} - {\rm{x}})]D = 0,}^{[{H_2}({\rm{y}} - {\rm{x}})]{D^2} = [{H_1}({\rm{y}} - {\rm{x}})]D = 0,}}^{_{{D^2}[H_2^*({\rm{y}} - {\rm{x}})] = D[H_1^*({\rm{y}} - {\rm{x}})] = 0,}^{{D^2}[{H_2}({\rm{y}} - {\rm{x}})] = D[{H_1}({\rm{y}} - {\rm{x}})] = 0,}}} \right. $

where $D=\sum\limits_{k=0}^2e_k\dfrac{\partial}{\partial\!y_k}$.

Lemma 2.2 Let $E_1({\rm{y}}-{\rm{x}})$ and $E_2({\rm{y}}- {\rm{x}})$ be as above, then

$ \left\{ {_{[{E_2}({\rm{y}} - {\rm{x}})]{D^2} = [{E_1}({\rm{y}} - {\rm{x}})]D = 0,{\rm{ }}}^{{D^2}[{E_2}({\rm{y}} - {\rm{x}})] = D[{E_1}({\rm{y}} - {\rm{x}})] = 0,}} \right.$

where $D=\sum\limits_{k=0}^2e_k\dfrac{\partial}{\partial\!y_k}$.

Denote $K({\rm{y}}- {\rm{x}})=-\dfrac{1}{4\pi}\dfrac{1}{\rho ({\rm{y}}- {\rm{x}})}$, $G({\rm{y}}- {\rm{x}}) =-\dfrac{1}{4\pi}\left (\dfrac{1}{\rho ({\rm{y}}- {\rm{x}})}-\dfrac{1}{\rho ({\rm{y}}+\overline{{\rm{x}}})}\right )$, where $\rho ({\rm{y}}- {\rm{x}})=\left (\sum\limits^2_{k=0}(y_k-x_k)^2\right )^{\frac{1}{2}}$.

Lemma 2.3 Let $K({\rm{y}}-{\rm{x}})$ be as above, then

$\left\{ {\begin{array}{*{20}{l}} {\bar D[K({\rm{y}} - {\rm{x}})] = [K({\rm{y}} - {\rm{x}})]\bar D = {H_1}({\rm{y}} - {\rm{x}}),}\\ {D[K({\rm{y}} - {\rm{x}})] = [K({\rm{y}} - {\rm{x}})]D = \overline {{H_1}} ({\rm{y}} - {\rm{x}}),} \end{array}} \right.$

where $D=\sum\limits_{k=0}^2e_k\dfrac{\partial}{\partial\!y_k}$.

Lemma 2.4 Let $G({\rm{y}}-{\rm{x}})$ be as above, then

$\left\{ \begin{array}{l} \bar D[G({\rm{y}} - {\rm{x}})] = [G({\rm{y}} - {\rm{x}})]\bar D = {E_1}({\rm{y}} - {\rm{x}}),\\ D[G({\rm{y}} - {\rm{x}})] = [G({\rm{y}} - {\rm{x}})]D = \overline {{E_1}} ({\rm{y}} - {\rm{x}}), \end{array} \right.$

where $D=\sum\limits_{k=0}^2e_k\dfrac{\partial}{\partial\!y_k}$.

Lemma 2.5 Let $E_1({\rm{y}}-{\rm{x}})$ be as above, then

$|{E_1}({\rm{y}} - {\rm{x}})| \le \frac{{|{x_0}|}}{{2\pi }}\left( {\frac{2}{{|{\rm{y}} - {\rm{x}}{|^3}}} + \frac{1}{{|{\rm{y}} - {\rm{x}}{|^2}|{\rm{y}} + \bar {\rm{x}}|}} + \frac{1}{{|{\rm{y}} - {\rm{x}}||{\rm{y}} + \bar {\rm{x}}{|^2}}}} \right).$

(2.2)

Lemma 2.6 Let $f\in C^{(2)}(\Omega, C(V_{2, 0}))C^{(1)}(\overline{\Omega}, C(V_{2,0})) $and $ D^2[f]=0$in $\Omega$, where $ \Omega $is a bounded domain with smooth boundary inRQ₃, then for any$ {\rm{x}}\notin \overline{\Omega} $,

$\int\limits_{\partial \Omega } {{H_1}} ({\rm{y - {\rm{x}}}}){\rm{d}}{\sigma _{\rm{y}}}f({\rm{y}}){\rm{ - }}\int\limits_{\partial \Omega } {{H_{\rm{2}}}} ({\rm{y - {\rm{x}}}}){\rm{d}}{\sigma _{\rm{y}}}D[f]({\rm{y}}){\rm{ = 0}}.$

(2.3)

Proof By Lemma 2.1 and Stokes' formula (see [5]), the result follows.

Lemma 2.7 (see [9]) Let $f\in C^{(2)}(\Omega, C(V_{2, 0})) C^{(1)}(\overline{\Omega}, C(V_{2,0}))$ and $D^2[f]=0 in \Omega $, where $\Omega $is a bounded domain with smooth boundary in RQ₃, then for any ${\rm{x}}\in \Omega$,

$f({\rm{x}}) = \int\limits_{\partial \Omega } {{H_1}} ({\rm{y}} - {\rm{x}}){\rm{d}}{\sigma _{\rm{y}}}f({\rm{y}}){\rm{ - }}\int\limits_{\partial \Omega } {{H_{\rm{2}}}} ({\rm{{\rm{y}} - {\rm{x}}}}){\rm{d}}{\sigma _{\rm{y}}}D[f]({\rm{y}}).$

(2.4)

Lemma 2.8 Let $f\in C^{(2)}(\Omega, C(V_{2, 0})) \bigcap C^{(1)}(\overline{\Omega}, C(V_{2,0})) $ and $ \triangle [f]=0 $in $\Omega$, where $\Omega$is a bounded domain with smooth boundary in RQ₃, then for any${\rm{x}}\notin \overline{\Omega}$

$\int\limits_{\partial \Omega } {{H_1}} ({\rm{y}} - {\rm{x}}){\rm{d}}{\sigma _{\rm{y}}}f({\rm{y}}){\rm{ - }}\int\limits_{\partial \Omega } K ({\rm{y - {\rm{x}}}}){\overline {{\rm{d}}\sigma } _{\rm{y}}}D[f]({\rm{y}}){\rm{ = 0}}.$

(2.5)

Proof By Lemma 2.1, Lemma 2.3 and Stokes' formula, the result follows.

Lemma 2.9 Let $f\in C^{(2)}(\Omega, C(V_{2, 0}))\bigcap C^{(1)}(\overline{\Omega}, C(V_{2,0})) $and $\triangle [f]=0 $in $ \Omega $, where $\Omega $is a bounded domain with smooth boundary in RQ₃, then for any${\rm{x}}\notin \overline{\Omega}$

$\int\limits_{\partial \Omega } {\overline {{H_1}} } ({\rm{y}} - {\rm{x}}){\overline {d\sigma } _y}f({\rm{y}}) - \int\limits_{\partial \Omega } K ({\rm{y}} - {\rm{x}}){\rm{d}}{\sigma _{\rm{y}}}\bar D[f]({\rm{y}}){\rm{ = 0}}.$

(2.6)

Proof By Lemma 2.1, Lemma 2.3 and Stokes' formula, the result follows.

Lemma 2.10 Let $f\in C^{(2)}(\Omega, C(V_{2, 0})) \bigcap C^{(1)}(\overline{\Omega}, C(V_{2,0}))$and $\triangle [f]=0$in$ \Omega $, where$ \Omega $is a bounded domain with smooth boundary in RQ₃, then for any$ {\rm{x}}\in \Omega $,

$f({\rm{x}})) = \int\limits_{\partial \Omega } {{H_1}} ({\rm{y}} - {\rm{x}}){\rm{d}}{\sigma _{\rm{y}}}f({\rm{y}}){\rm{ - }}\int\limits_{\partial \Omega } K ({\rm{y - {\rm{x}}}}){\overline {d\sigma } _{\rm{y}}}D[f]({\rm{y}}).$

(2.7)

Proof By Lemma 2.8, it can be similarly proved as in Lemma 2.7.

Lemma 2.11 Let $f\in C^{(2)}(\Omega, C(V_{2, 0})) \bigcap C^{(1)}(\overline{\Omega}, C(V_{2,0}))$and $ \triangle [f]=0 $in$ \Omega $, where $ \Omega $is a bounded domain with smooth boundary inRQ₃, then for any${\rm{x}}\in \Omega $,

$f({\rm{x}}) = \int\limits_{\partial \Omega } {\overline {{H_1}} } ({\rm{y}} - {\rm{x}}){\overline {d\sigma } _y}f({\rm{y}}) - \int\limits_{\partial \Omega } K ({\rm{y}} - {\rm{x}}){\rm{d}}{\sigma _{\rm{y}}}{\rm{\bar D}}[f]({\rm{y}}).$

(2.8)

Proof By Lemma 2.9, it can be similarly proved as in Lemma 2.7.

Lemma 2.12 (see [27]) Let $f\in \widehat{H}^{\mu}(\Pi, C(V_{2,0}))$, then for all ${\rm{y}}_*,{{\rm{y}}}_{**}\in \Pi$,

$\mathop {\lim }\limits_{R \to + \infty } \left( {\frac{1}{{4\pi }}\int \int\limits_{D({{\bf{y}}_*},R)} {\frac{{\overline {\bf{y}} - \overline {\bf{{\rm{x}}}} }}{{|{\bf{y}} - {\bf{{\rm{x}}}}{|^3}}}} f({\bf{y}}){\rm{d}}S - \frac{1}{{4\pi }}\int \int\limits_{D({{\bf{y}}_{**}},R)} {\frac{{\overline {\bf{y}} - \overline {\bf{{\rm{x}}}} }}{{|{\bf{y}} - {\bf{{\rm{x}}}}{|^3}}}} f({\bf{y}}){\rm{d}}S} \right) = 0,$

(2.9)

where ${\rm{x}}\in RQ_3.$.

In this section, we shall give the integral representations over $\Pi$ for bi-regular functions. For $f({\rm{x}})\in \widehat{H}^{\mu}(\Pi, C(V_{2,0}))$, the Cauchy type integral Cf over $\Pi$ is defined by

$Cf({\rm{x}}) = - \frac{1}{{4\pi }}\int {} \int\limits_\Pi {\frac{{\bar {\rm{y}} - \bar {\rm{x}}}}{{|{\rm{y}} - {\rm{x}}{|^3}}}f({\rm{y}})dS,{\rm{x}} \notin \Pi ,} $

(3.1)

where

$- \frac{1}{{4\pi }}\int {} \int\limits_\Pi {} \frac{{\bar {\rm{y}} - \bar {\rm{x}}}}{{{{\left| {{\rm{y}} - {\rm{x}}} \right|}^3}}}f({\rm{y}})dS = \mathop {\lim }\limits_{R \to + \infty } - \frac{1}{{4\pi }}\int {} \int\limits_{D(0,R)} {} \frac{{\bar {\rm{y}} - \bar {\rm{x}}}}{{{{\left| {{\rm{y}} - {\rm{x}}} \right|}^3}}}f({\rm{y}})dS.$

(3.2)

Lemma 3.13 (see [27]) Let $f({\rm{y}})\in \widehat{H}^{\mu}(\Pi, C(V_{2,0}))$, $Cf({\rm{x}})$ be defined as in (3.1), then $Cf({\rm{x}})$ exists and

$Cf({\rm{x}}) = \left\{ {_{ - \frac{{f(\infty )}}{2} - \frac{1}{{4\pi }}\int {} \int\limits_\Pi {} \frac{{\bar {\rm{y}} - \bar {\rm{x}}}}{{|{\rm{y}} - {\rm{x}}{|^3}}}(f({\rm{y}}) - f(\infty ))dS,{\rm{x}} \in RQ_3^ - .{\rm{ }}}^{\frac{{f(\infty )}}{2} - \frac{1}{{4\pi }}\int {} \int\limits_\Pi {} \bar - \frac{{\bar {\rm{y}} - \bar {\rm{x}}}}{{|{\rm{y}} - {\rm{x}}{|^3}}}(f({\rm{y}}) - f(\infty ))dS,{\rm{x}} \in RQ_3^ + ,}} \right.$

(3.3)

Theorem 3.1 Let $f\in C^{(2)}(\Omega, C(V_{2, 0})) \bigcap C^{(1)}(\overline{\Omega}, C(V_{2,0}))$ and$ D^2[f]=0 $in$ \Omega $, where$ \Omega $is a bounded domain with smooth boundary in RQ₃, then for any$ {\rm{x}}\in \Omega $,

$f({\rm{x}})=\displaystyle\int\limits_{\partial\Omega}E_1({\rm{y}}- {\rm{x}})d\sigma_yf({\rm{y}})- \displaystyle\int\limits_{\partial\Omega}E_2({\rm{y}}- {\rm{x}}) d\sigma_yD[f]({\rm{y}}).$

(3.4)

Proof By Lemma 2.2, Lemma 2.7 and Stokes' formula, the result follows.

Denote ${\partial ^ + }B({\rm{x}},R) = {\rm{ }}\{ {\rm{y}}|{\rm{y}} \in \partial B({\rm{x}},R),{\mkern 1mu} {\rm{Re}}({\rm{y}}) > 0{\rm{\} }}$.

Lemma 3.14 For any ${\rm{x}}\in RQ^+_3$,

$\mathop {\lim }\limits_{R \to + \infty } \int\limits_{{\partial ^ + }({\rm{x}},R)} {} {H_1}({\rm{y}} - {\rm{x}})d{\sigma _y} = \mathop {\lim }\limits_{R \to + \infty } \int\limits_{{\partial ^ + }B({\rm{x}},R)} {} H_1^*({\rm{y}} - {\rm{x}})d{\sigma _y} = \frac{1}{2}.$

(3.5)

Proof It can be proved by Lemma 2.5.

Lemma 3.15 Let $f\in \widehat{H}^{\mu}(\overline{RQ^+_3}, C(V_{2,0}))$, then for any ${\rm{x}}\in RQ^+_3$,

$\mathop {\lim }\limits_{R \to + \infty } \int\limits_{{\partial ^ + }B({\rm{x}},R)} {} {H_1}({\rm{y}} - {\rm{x}})d{\sigma _y}f({\rm{y}}) = \mathop {\lim }\limits_{R \to + \infty } \smallint \int\limits_{{\partial ^ + }({\rm{x}},R)} {} H^*_1({\rm{y}} - {\rm{x}}){\sigma _y}f({\rm{y}}) = \frac{1}{2}f(\infty ).$

(3.6)

Proof It can be proved by Lemma 3.14.

Lemma 3.16 Let $D[f]\in \widehat{H}_0^{\mu}(\overline{RQ^+_3}, C(V_{2,0}))$, then for any ${\rm{x}}\in RQ^+_3$,

$\mathop {\lim }\limits_{R \to + \infty } \int\limits_{{\partial ^ + }B({\rm{x}},R)} {} {E_2}({\rm{y}} - {\rm{x}})d{\sigma _y}D[f]({\rm{y}}) = 0.$

(3.7)

Proof It can be proved by Lemma 2.5.

Lemma 3.17 Let $f\in \widehat{H}^{\mu}(\Pi, C(V_{2,0}))$, then for all $y_*,{ y}_{**}\in \Pi$,

$\mathop {\lim }\limits_{R \to + \infty } \left( {{\mkern 1mu} \int {} \int\limits_{D({{\rm{y}}_*},R)} {} {E_2}({\rm{y}} - {\rm{x}})f({\rm{y}}){\rm{dS}} - \int\limits_{D({{\rm{y}}_{**}},R)} {} {E_2}({\rm{y}} - {\rm{x}})f({\rm{y}}){\rm{dS}}} \right){\rm{ = 0}},$

(3.8)

where ${\rm{x}}\in RQ_3.$

Proof By Lemma 2.5, it can be similarly proved as in Lemma 2.12.

Theorem 3.2 Let $f\in C^{(1)}(\overline{RQ^+_3}, C(V_{2,0}))\bigcap C^{(2)}(RQ^+_3, C(V_{2,0}))$, $f\in \widehat{H}^{\mu}(\overline{RQ^+_3}, C(V_{2,0}))$, $D[f]\in \widehat{H}_0^{\mu}(\overline{RQ^+_3}, C(V_{2,0}))$ and $D^2[f]=0$ in $RQ^+_3$, then for any ${\rm{x}}\in RQ^+_3$,

$f({\rm{x}})=-\int{{}}\int\limits_{\Pi }{{{E}_{1}}}({\rm{y}}-{\rm{x}})f({\rm{y}})dS+\int{{}}\int\limits_{\Pi }{{{E}_{2}}}({\rm{y}}-{\rm{x}})D[f]({\rm{y}})dS.$

(3.9)

Proof For any ${\rm{x}}\in RQ^+_3$, denote $\Omega=B( {\rm{x}}, R)\bigcap RQ^+_3$, by Theorem 3.1, we have

$f({\rm{x}})=\displaystyle\int\limits_{\partial\Omega}E_1({\rm{y}}- {\rm{x}}) d\sigma_yf({\rm{y}})- \displaystyle\int\limits_{\partial\Omega}E_2({\rm{y}}- {\rm{x}})d\sigma_yD[f]({\rm{y}}).$

(3.10)

By Lemma 2.12 and Lemma 3.15, it can be proved that

$\lim\limits_{R\rightarrow +\infty}\displaystyle\int\limits_{\partial\Omega}E_1({\rm{y}}- {\rm{x}})d\sigma_yf({\rm{y}})= -\displaystyle\int\displaystyle\int\limits_{\Pi}E_1({\rm{y}}- {\rm{x}}) f({\rm{y}})dS.$

(3.11)

In view of

$\int\limits_{\partial \Omega }{{{E}_{2}}}({\rm{y}}-{\rm{x}})d{{\sigma }_{y}}D[f]({\rm{y}}) \\ =\int\limits_{{{\partial }^{+}}B({\rm{x}},R)}{{{E}_{2}}({\rm{y}}-{\rm{x}})d{{\sigma }_{y}}D[f]({\rm{y}})-\int{{}}\int\limits_{D({\rm{Imx}},\sqrt{{{R}^{2}}-x_{0}^{2}})}{{}}}{{E}_{2}}({\rm{y}}-{\rm{x}}D[f]({\rm{y}})dS,$

(3.12)

where ${ {\rm{Imx}}}=x_1e_1+x_2e_2$. By Lemma 3.17, in view of $D[f]\in \widehat{H}_0^{\mu}(\overline{RQ^+_3}, C(V_{2,0}))$, we have

$\underset{R\to +\infty }{\mathop{\lim \limits}}\,\int\int\limits_{D({\rm{Imx}},\sqrt{{{R}^{2}}-x_{0}^{2}})}{{}}{{E}_{2}}({\rm{y}}-{\rm{x}})D[f]({\rm{y}}){\rm{d}}S=\int{{}}\int\limits_{\Pi }{{{E}_{2}}}({\rm{y}}-{\rm{x}})D[f]({\rm{y}}){\rm{d}}S.$

(3.13)

By Lemma 3.16, Combining (3.12) with (3.13), we have

$\lim\limits_{R\rightarrow +\infty}\displaystyle\int\limits_{\partial\Omega}E_2({\rm{y}}- {\rm{x}})d\sigma_yD[f]({\rm{y}})= -\displaystyle\int\displaystyle\int\limits_{\Pi}E_2({\rm{y}}- {\rm{x}}) D[f]({\rm{y}})dS.$

(3.14)

Combining (3.10), (3.11) with (3.14), taking $R\rightarrow +\infty$ in (3.10), the result follows.

In this section, we shall give the integral representations over Ⅱ for harmonic functions. Denote ${{K}^{*}}({\rm{y}}-{\rm{x}})=-\frac{1}{4\pi }\frac{1}{\rho ({\rm{y}}-{{S}_{\Pi }}({\rm{x}}))}=-\frac{1}{4\pi }\frac{1}{\rho ({\rm{y}}+\bar{{\rm{x}}})}$.

Theorem 4.3 Let$f\in C^{(2)}(\Omega, C(V_{2, 0})) \bigcap C^{(1)}(\overline{\Omega}, C(V_{2,0}))$ and $ \triangle [f]=0 $in$ \Omega $, where$ \Omega $is a bounded domain with smooth boundary in$ RQ^+_3 $, then for any${\rm{x}}\in \Omega $,

$ f({\rm{x}})=\int\limits_{\partial \Omega }{{{E}_{1}}}({\rm{y}}-{\rm{x}})\text{d}{{\sigma }_{\text{y}}}f(\text{y})\text{-}\int\limits_{\partial \Omega }G({\rm{y}}-{\rm{x}}){{\overline{d\sigma }}_{\text{y}}}D[f](\text{y}).$

(4.1)

Proof By Stokes' formula and Lemma 2.3, for any $ {\rm{x}}\in \Omega$, we have

$\int\limits_{\partial \Omega } {H_1^*} ({\rm{y}} - {\rm{x}}){\rm{d}}{\sigma _{\rm{y}}}f({\rm{y}}){\rm{0}}\int\limits_{\partial \Omega } {{K^{\rm{*}}}} ({\rm{y}} - {\rm{x}}){\overline {d\sigma } _{\rm{y}}}D[f]({\rm{y}}){\rm{ = 0}}.$

(4.2)

Combining Lemma 2.10 with (4.2), the result follows.

Theorem 4.4 Let $f\in C^{(2)}(\Omega, C(V_{2, 0})) \bigcap C^{(1)}(\overline{\Omega}, C(V_{2,0})) $ and$ \triangle [f]=0 $in$ \Omega $, where$ \Omega $is a bounded domain with smooth boundary in$ RQ^+_3 $, then for any${\rm{x}}\in \Omega$,

$f({\rm{x}})=\int\limits_{\partial \Omega }{\overline{{{E}_{1}}}}({\rm{y}}-{\rm{x}}){{\overline{d\sigma }}_{y}}f({\rm{y}})-\int\limits_{\partial \Omega }{G}({\rm{y}}-{\rm{x}})\text{d}{{\sigma }_{\text{y}}}D[f](\text{y}).$

(4.3)

Proof By Stokes' formula and Lemma 2.3, for any ${\rm{x}}\in \Omega$, we have

$\int\limits_{\partial \Omega }{\overline{H_{1}^{*}}}({\rm{y}}-{\rm{x}}){{\overline{d\sigma }}_{y}}f({\rm{y}})-\int\limits_{\partial \Omega }{{{K}^{*}}}({\rm{y}}-{\rm{x}})\text{d}{{\sigma }_{\text{y}}}D[f](\text{y})\text{=0}$

(4.4)

Combining Lemma 2.11 with (4.4), the result follows.

Lemma 4.18 Let $f\in \widehat{H}^{\mu}(\Pi, C(V_{2,0}))$, then for all $y_*,{ y}_{**}\in \Pi$,

$\mathop {\lim }\limits_{R \to + \infty } {\mkern 1mu} {\mkern 1mu} \left( {{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \int {\int\limits_{D({{\rm{y}}_{\rm{*}}},{\rm{R}})} {} } G({\rm{y}} - {\rm{x}})f({\rm{y}}){\rm{dS - }}\int {} \int\limits_{D({{\rm{y}}_{{\rm{**}}}},{\rm{R}})} {} {\rm{G}}({\rm{y - {\rm{x}}}})f({\rm{y}}){\rm{dS}}} \right){\rm{ = 0}},$

(4.5)

where ${\rm{x}}\in RQ_3.$

Proof It can be similarly proved as in Lemma 2.12.

Theorem 4.5 Let $f\in C^{(1)}(\overline{RQ^+_3}, C(V_{2,0}))\bigcap C^{(2)}(RQ^+_3, C(V_{2,0}))$, $f\in \widehat{H}^{\mu}(\overline{RQ^+_3}, C(V_{2,0}))$, $D[f]\in \widehat{H}_0^{\mu}(\overline{RQ^+_3}, C(V_{2,0}))$ and $\triangle [f]=0$ in $RQ^+_3$, then for any ${\rm{x}}\in RQ^+_3$,

$f({\rm{x}}) = - \int {} \int\limits_\Pi {{E_1}} ({\rm{y}} - {\rm{x}})f({\rm{y}}){\rm{dS + }}\int {} \int\limits_\Pi {\rm{G}} ({\rm{y - {\rm{x}}}})D[f]({\rm{y}}){\rm{dS}}.$

(4.6)

Proof For any ${\rm{x}}\in RQ^+_3$, denote $\Omega=B({\rm{x}}, R)\bigcap RQ^+_3$, by Theorem 4.3, we have

$f({\rm{x}})=\displaystyle\int\limits_{\partial\Omega}E_1({\rm{y}}- {\rm{x}})d\sigma_yf({\rm{y}})- \displaystyle\int\limits_{\partial\Omega}G({\rm{y}}- {\rm{x}})\overline{d\sigma_y}D[f]({\rm{y}}).$

(4.7)

By Lemma 2.12 and Lemma 3.15, it can be proved that

$\mathop {\lim }\limits_{R \to + \infty } \int\limits_{\partial \Omega } {{E_1}} ({\rm{y}} - {\rm{x}})d{\sigma _y}f({\rm{y}}) = - \int {} \int\limits_\Pi {{E_1}} ({\rm{y}} - {\rm{x}})f({\rm{y}})dS.$

(4.8)

In view of

$\int\limits_{\partial \Omega } G ({\rm{y}} - {\rm{x}})\overline {d{\sigma _y}} D[f]({\rm{y}})\\ = \smallint \int\limits_{{\partial ^ + }B({\rm{x}},R)} {} G({\rm{y}} - {\rm{x}})\overline {d{\sigma _y}} D[f]({\rm{y}}) - \int {\int\limits_{D({\rm{Imx}},\sqrt {{R^2} - x_0^2} )} {} } G({\rm{y}} - {\rm{x}})D[f]({\rm{y}})dS,$

(4.9)

where ${\rm{Imx}}=x_1e_1+x_2e_2$. In view of $D[f]\in \widehat{H}_0^{\mu}(\overline{RQ^+_3}, C(V_{2,0}))$, it can be proved that

$\mathop {\lim }\limits_{R \to + \infty } \int\limits_{{\partial ^ + }B({\rm{x}},R)} {} G({\rm{y}} - {\rm{x}})\overline {d{\sigma _y}} D[f]({\rm{y}}) = 0.$

(4.10)

By Lemma 4.18, in view of $D[f]\in \widehat{H}_0^{\mu}(\overline{RQ^+_3}, C(V_{2,0}))$, we have

$\lim\limits_{R\rightarrow +\infty}\displaystyle\int\hspace{-13mm}\displaystyle\int\limits_{D({\rm{Imx}}, \sqrt{R^2-x_0^2})}G({\rm{y}}- {\rm{x}}) D[f]({\rm{y}})dS=\displaystyle\int\displaystyle\int\limits_{\Pi}G({\rm{y}}- {\rm{x}}) D[f]({\rm{y}})dS $

(4.11)

Combining (4.9), (4.10) with (4.11), we have

$\lim\limits_{R\rightarrow +\infty}\displaystyle\int\limits_{\partial\Omega}G({\rm{y}}- {\rm{x}})\overline{d\sigma_y}D[f]({\rm{y}})= -\displaystyle\int\displaystyle\int\limits_{\Pi}G({\rm{y}}- {\rm{x}}) D[f]({\rm{y}}) dS. $

(4.12)

Combining (4.7), (4.8) with (4.12), taking $R\rightarrow +\infty$ in (4.7), the result follows.

Corollary 4.1 Let $f\in C^{(1)}(\overline{RQ^+_3}, C(V_{2,0}))\bigcap C^{(2)}(RQ^+_3, C(V_{2,0}))$, $f\in \widehat{H}^{\mu}(\overline{RQ^+_3}, C(V_{2,0}))$ and $D[f]=0$ in $RQ^+_3$, then for any $ {\rm{x}}\in RQ^+_3$,

$f( {\rm{x}})=-\displaystyle\int\displaystyle\int\limits_{\Pi}E_1({\rm{y}}- {\rm{x}}) f({\rm{y}})dS.$

(4.13)

Theorem 4.6 Let $f\in C^{(1)}(\overline{RQ^+_3}, C(V_{2,0}))\bigcap C^{(2)}(RQ^+_3, C(V_{2,0}))$, $f\in \widehat{H}^{\mu}(\overline{RQ^+_3}, C(V_{2,0}))$, $\overline{D}[f]\in \widehat{H}_0^{\mu}(\overline{RQ^+_3}, C(V_{2,0}))$ and $\triangle [f]=0$ in $RQ^+_3$, then for any ${\rm{x}}\in RQ^+_3$,

$f({\rm{x}})=-\displaystyle\int\displaystyle\int\limits_{\Pi}\overline{E_1}({\rm{y}}- {\rm{x}}) f({\rm{y}})dS+\displaystyle\int\displaystyle\int\limits_{\Pi}G({\rm{y}}- {\rm{x}}) \overline{D}[f]({\rm{y}}) dS.$

(4.14)

Proof By Theorem 4.4, it can be similarly proved as in Theorem 4.5.

Corollary 4.2 Let $f\in C^{(1)}(\overline{RQ^+_3}, C(V_{2,0}))\bigcap C^{(2)}(RQ^+_3, C(V_{2,0}))$, $f\in \widehat{H}^{\mu}(\overline{RQ^+_3}, C(V_{2,0}))$ and $\overline{D}[f]=0$ in $RQ^+_3$, then for any ${\rm{x}}\in RQ^+_3$,

$f({\rm{x}})=-\displaystyle\int\displaystyle\int\limits_{\Pi}\overline{E_1}({\rm{y}}- {\rm{x}}) f({\rm{y}})dS.$

(4.15)