Clifford分析中双Hypergenic函数的等价条件

The Equivalent Conditions of Bihypergenic Functions in Clifford Analysis

摘要: 本文首先以双hypergenic函数的定义为基础，借助Cl_n+1,0(R)空间中的一种分解，讨论了Cl_n+1,0(R)中双hypergenic函数的一个等价条件，其与复分析中的Cauchy-Riemann方程比较类似，其次通过对结果中方程的某些量进行变换得到了双hypergenic函数的又一个等价刻画，这些等价条件建立了双hypergenic函数与偏微分方程之间的联系，使Clifford分析的函数理论有了进一步发展，对于研究高维空间中的方程和算子提供了理论基础。

Abstract: In this paper, based on the definition of bihypergeneric function, we first discuss an equivalent condition of the bihypergenic function in Cl_n+1,0(R) by a decomposition in Cl_n+1,0(R) space, which is similar to the Cauchy-Riemann equation in complex analysis. Second, by changing some quantities of the equations in the results, we obtain another equivalent characterization of the bihypergenic function, which establishes the relation between the bihypergenic function and the partial differential equation functions. They further develop the function theory of Clifford analysis and provide a theoretical basis for the study of equations and operators in high-dimensional space.

1. 引言

Clifford [1] 代数是一种可结合但不可交换的代数结构，是Clifford在1878年建立的。Hypergenic函数是在修正的Dirac算子的基础上提出来的，是单复分析中全纯函数在高维空间欧式度量下的推广。2009年以来，S. L. Eriksson和H. Orelma [2] [3] 研究了实Clifford代数中的hypergenic函数并给出了它的Cauchy型积分公式。2014年，谢永红 [4] [5] [6] 研究了对偶的k-hypergenic函数的Cauchy型积分公式和hypergenic函数的相关性质，同时研究了实Clifford分析中hypergenic函数拟Cauchy型积分的边界性质，并且给出了Plemelj公式和Privalov定理。2019年，陈雪 [7] 等研究了双hypergenic函数的Cauchy型积分公式及其相关理论。2022年，边小丽 [8] 等研究了双曲调和函数的积分表示。

本文基于以上的研究结果，从 $C{l}_{n+1,0}\left(R\right)$ 空间中的函数 $f\left(x,y\right)$ 的P₀部和Q₀部分解的角度，讨论了双hypergenic函数的充分必要条件，为进一步讨论双hypergenic函数的性质及应用奠定了基础。

2.预备知识

2.1. Clifford代数 $C{l}_{n+1,0}(\; R\; )$

$C{l}_{n+1,0}\left(R\right)$ 为实Clifford代数，单位元为 $e_{\varphi }=1$， $C{l}_{n+1,0}\left(R\right)$ 的基元素是 $e_0,e_1,\cdots ,e_n;e_{\text{0}}{e}_{\text{1}},e_{\text{0}}{e}_{\text{2}},\cdots ,e_{n-1}{e}_n;\cdots ;e_{\text{0}}{e}_1\cdots e_n$，且满足 $e_i{e}_j+e_j{e}_i=2{\delta }_{ij}\left(i,j=0,1,\cdots ,n\right)$。对 $\forall a\in C{l}_{n+1,0}\left(R\right)$，有 $a={\displaystyle \underset{A}{\sum }{a}_A{e}_A}$， $a_A\in R$，其中 $A=\left\{{\alpha }_1,\cdots ,{\alpha }_h\right\}\subseteq \left\{\text{0},1,\cdots ,n\right\}$， $\text{0}\le {\alpha }_1<{\alpha }_2<\cdots <{\alpha }_h\le n$。当 $A=\varphi$， $e_A=1$。

2.2. $C{l}_{n+1,0}\left(R\right)$ 中的基本运算

设 $a,b\in C{l}_{n+1,0}\left(R\right)$，定义“ $\text{'}$ ”运算： $C{l}_{n+1,0}\left(R\right)\to C{l}_{n+1,0}\left(R\right)$， $e{\text{'}}_j=-e_j$ $\left(j=0,1,\cdots ,n\right)$， $\left(ab\right)\text{'}=a\text{'}b\text{'}$。

2.3. $C{l}_{n+1,0}\left(R\right)$ 中的一种分解

$\forall a\in C{l}_{n+1,0}\left(R\right)$ 可唯一地分解为 $a=b+e_{0}c$，其中 $b,c\in C{l}_{n,0}\left(R\right)$。定义两个映射 $P_{\text{0}}:C{l}_{n+1,0}\left(R\right)\to C{l}_{n,0}\left(R\right)$ 和 $Q_{\text{0}}:C{l}_{n+1,0}\left(R\right)\to C{l}_{n,0}\left(R\right)$， $P_{\text{0}}a=b$， $Q_{\text{0}}a=c$，其中b和c分别称为a的 $P_{\text{0}}$ 部和 $Q_{\text{0}}$ 部。

对 $\forall a\in C{l}_{n,0}$，有

$e_{0}a\text{'}=a{e}_0$, $a\text{'}{e}_0=e_{0}a$. (2.1)

且对 $\forall a\in C{l}_{n\text{+1},0}$，有

$e_{0}a\text{'}=\hat{a}{e}_0$, $a\text{'}{e}_0=e_0\hat{a}$. (2.2)

2.4. 微分算子

设 ${\Omega }_0\subset R^{n+1}$ 为一非空连通开子集， $F_{{\Omega }_0}^{\left(\text{1}\right)}$ 表示 ${\Omega }_0$ 中 $C^r$ 函数的全体。

$F_{{\Omega }_0}^{\left(\text{1}\right)}=\left\{f|f:{\Omega }_0\to C{l}_{n+1,0}\left(R\right),f\left(x\right)={\displaystyle \underset{A}{\sum }{f}_A\left(x\right)e_A,f_A\left(x\right)\in C^r\left({\Omega }_0\right),x\in {\Omega }_0}\right\}$

Dirac算子

$D_{l}f\left(x\right)={\displaystyle \underset{i=0}{\overset{n}{\sum }}{e}_i}\frac{\partial f\left(x\right)}{\partial x_i}$,

$D_{r}f\left(x\right)={\displaystyle \underset{i=0}{\overset{n}{\sum }}\frac{\partial f\left(x\right)}{\partial x_i}{e}_i}$.

其中 $f\in F_{{\Omega }_0}^{\left(1\right)}\left({\Omega }_0\subset R^{n+1}\backslash \left\{x_0=0\right\}\right)$。可以定义修正的Dirac算子

$H_k^{l}f=D_{l}f-\frac{k}{x_0}{Q}_{0}f$,

$H_k^{r}f=D_{r}f-\frac{k}{x_0}Q{\text{'}}_{0}f$.

定义2.1 [2] 若函数 $f\in C^1\left(\Omega ,C{l}_{n+1,0}\right)$，且对 $\forall x\in \Omega \left(x_0\ne 0\right)$ 有 $H_{n-1}^{l}f\left(x\right)=\text{0}$，则称 $f\left(x\right)$ 为 $\Omega$ 上的左hypergenic函数，类似地，若 $H_{n-1}^{r}f\left(x\right)=\text{0}$，则称 $f\left(x\right)$ 为 $\Omega$ 上的右hypergenic函数。

设 ${\Omega }_1\subset R^{m+1}$， ${\Omega }_{\text{2}}\subset R^{k+1}$ 为非空联通开子集， $F_{\Omega }^{\left(r\right)}$ 表示 $\Omega ={\Omega }_1\times {\Omega }_2$ 中 $C^r$ 函数的全体。

$F_{\Omega }^{\left(r\right)}=\left\{f|\begin{array}{l}f:\Omega \to C{l}_{m+k+2}\left(R\right),f\left(x,y\right)={\displaystyle \underset{A,B}{\sum }{f}_{A,B}\left(x,y\right)e_A{e}_B},\hfill \\ f_{A,B}\left(x,y\right)\in C^r\left(\Omega \right),x\in {\Omega }_1,y\in {\Omega }_2\hfill \end{array}\right\}$.

记 ${\Omega }_1$ 中的基元素为： $e_0^{\left(1\right)},e_1^{\left(1\right)},\cdots ,e_m^{\left(1\right)}$ ；记 ${\Omega }_{\text{2}}$ 中的基元素为： $e_0^{\left(\text{2}\right)},e_1^{\left(\text{2}\right)},\cdots ,e_k^{\left(\text{2}\right)}$。

定义2.2 [7] 设 ${\Omega }_{\text{1}}\subset R^{m+1}\backslash \left\{x_{\text{0}}\ne 0\right\}$， ${\Omega }_{\text{2}}\subset R^{k+1}\backslash \left\{y_{\text{0}}\ne 0\right\}$ 为非空连通开集， $\Omega ={\Omega }_1\times {\Omega }_2$ 如前所述， $f\left(x,y\right)\in F_{\Omega }^{\left(1\right)}$，若

$\{\begin{array}{c}{H}_x^{l}f\left(x,y\right)=D_x^{l}f\left(x,y\right)-\frac{m-1}{x_0}{Q}_{0x}^{\left(1\right)}f\left(x,y\right)=0,\\ H_y^{r}f\left(x,y\right)=D_y^{r}f\left(x,y\right)-\frac{k-1}{y_0}{Q}_{0y}^{\left(2\right)\text{'}}f\left(x,y\right)=0.\end{array}$

则称 $f\left(x,y\right)$ 为 $\Omega$ 上的双hypergenic函数。

注： $Q_{\text{0}x}^{\left(1\right)}f\left(x,y\right)$ 表示 $f\left(x,y\right)$ 只对第一个变量取 $Q_0$ 部， $P_{\text{0}x}^{\left(1\right)}f\left(x,y\right)$， $P_{\text{0}y}^{\left(2\right)}f\left(x,y\right)$， $Q_{\text{0}y}^{\left(2\right)}f\left(x,y\right)$ 表示类似的含义。

3. 主要结果

设 $f\left(x,y\right)\in F_{\Omega }^{\left(1\right)}$， $f\left(x,y\right)$ 可以分解成以下两种形式：

$f\left(x,y\right)=P_{\text{0}x}^{\left(1\right)}f\left(x,y\right)+e_{\text{0}}^{\left(1\right)}{Q}_{\text{0}x}^{\left(1\right)}f\left(x,y\right)$ (3.1)

$f\left(x,y\right)=P_{\text{0}y}^{\left(2\right)}f\left(x,y\right)+e_{\text{0}}^{\left(2\right)}{Q}_{\text{0}y}^{\left(2\right)}f\left(x,y\right)$ (3.2)

下面我们利用 $f\left(x,y\right)$ 的 $P_{\text{0}}$ 部和 $Q_{\text{0}}$ 部的分解，给出双hypergenic函数的等价条件。

引理3.1 [2] 设 $\Omega \subset {ℝ}_{+}^{n+1}$ 为非空连通开集， $f\in C^1\left(\Omega ,C{l}_{n+1,0}\right)$，则

$D_r\left(\frac{f}{x_0^{n-1}}\right)=\frac{1}{x_0^{n-1}}{H}^{r}f-\frac{n-1}{x_0^n}\left(P_{\text{0}}f\right)e_0$ (3.3)

$P_{\text{0}}\left(H^{l}f\right)=x_0^{n-1}{P}_0\left(D_l\left(\frac{f}{x_0^{n-1}}\right)\right)$ (3.4)

$Q_{\text{0}}\left(H^{l}f\right)=Q_{\text{0}}\left(D^{l}f\right)$ (3.5)

引理3.2 [2] 设 $\Omega \subset {ℝ}_{+}^{n+1}$ 为非空连通开集， $f\in C^1\left(\Omega ,C{l}_{n+1,0}\right)$，则

$D_l\left(\frac{f}{x_0^{n-1}}\right)=\frac{1}{x_0^{n-1}}{H}^{l}f-\frac{n-1}{x_0^n}{P}_{\text{0}}{}^{\prime }f{e}_0$ (3.6)

$P_{\text{0}}\left(H^{r}f\right)=x_0^{n-1}{P}_0\left(D_r\left(\frac{f}{x_0^{n-1}}\right)\right)$ (3.7)

$Q_{\text{0}}\left(H^{r}f\right)=Q_{\text{0}}\left(D^{r}f\right)$ (3.8)

定理3.1 若 $f\in C^1\left(\Omega ,C{l}_{n+1,0}\right)$， $f\left(x,y\right)$ 是双Hypergenic函数的充分必要条件是

$\{\begin{array}{l}\frac{\partial \left(P_{0y}^{\left(2\right)}{}^{\prime }f\left(x,y\right)\right)}{\partial y_0}+{\displaystyle \underset{j=1}{\overset{k}{\sum }}\frac{\partial \left(Q_{0y}^{\left(2\right)}f\left(x,y\right)\right)}{\partial y_j}{e}_j^{\left(2\right)}}=0\hfill \\ {\displaystyle \underset{j=1}{\overset{k}{\sum }}\frac{\partial P_{0y}^{\left(2\right)}f\left(x,y\right)}{\partial y_j}{e}_j^{\left(2\right)}}+\frac{\partial Q_{0y}^{\left(2\right)}{}^{\prime }f\left(x,y\right)}{\partial y_0}+\left(1-k\right)\frac{Q_{0y}^{\left(2\right)}{}^{\prime }f\left(x,y\right)}{y_0}=0\hfill \\ \frac{\partial P_{0x}^{\left(1\right)}f\left(x,y\right)}{\partial x_0}-{\displaystyle \underset{i=1}{\overset{m}{\sum }}{e}_i^{\left(1\right)}\frac{\partial Q_{0x}^{\left(1\right)}f\left(x,y\right)}{\partial x_i}}=0\hfill \\ {\displaystyle \underset{i=1}{\overset{m}{\sum }}{e}_i^{\left(1\right)}\frac{\partial P_{0x}^{\left(1\right)}f\left(x,y\right)}{\partial x_i}+\frac{\partial Q_{0x}^{\left(1\right)}f\left(x,y\right)}{\partial x_0}+\left(1-m\right)\frac{Q_{0x}^{\left(1\right)}f\left(x,y\right)}{x_0}}=0\hfill \end{array}$ (3.9)

证明 由

$\begin{array}{c}{H}_y^{r}f\left(x,y\right)=P_{0y}^{\left(2\right)}\left(H_y^{r}f\left(x,y\right)\right)+e_0^{\left(2\right)}{Q}_{0y}^{\left(2\right)}\left(H_y^{r}f\left(x,y\right)\right)\\ =y_0^{k-1}{P}_{0y}^{\left(2\right)}\left(D_y^r\left(\frac{f\left(x,y\right)}{y_0^{k-1}}\right)\right)+e_0^{\left(2\right)}{Q}_{0y}^{\left(2\right)}\left(D_y^{r}f\left(x,y\right)\right)\end{array}$ (3.10)

又

$\begin{array}{c}{y}_0^{k-1}{P}_{0y}^{\left(2\right)}\left(D_y^r\left(\frac{f\left(x,y\right)}{y_0^{k-1}}\right)\right)=y_0^{k-1}{P}_{0y}^{\left(2\right)}{\displaystyle \underset{j=0}{\overset{k}{\sum }}\frac{\partial \left(\frac{f\left(x,y\right)}{y_0^{k-1}}\right)}{\partial y_j}{e}_j^{\left(2\right)}}\\ =y_0^{k-1}{P}_{0y}^{\left(2\right)}{\displaystyle \underset{j=0}{\overset{k}{\sum }}\frac{\partial \left(\frac{P_{0y}^{\left(2\right)}f\left(x,y\right)+e_0^{\left(2\right)}{Q}_{0y}^{\left(2\right)}f\left(x,y\right)}{y_0^{k-1}}\right)}{\partial y_j}{e}_j^{\left(2\right)}}\\ =y_0^{k-1}{P}_{0y}^{\left(2\right)}{\displaystyle \underset{j=0}{\overset{k}{\sum }}\frac{\partial \left(\frac{P_{0y}^{\left(2\right)}f\left(x,y\right)}{y_0^{k-1}}\right)}{\partial y_j}}{e}_j^{\left(2\right)}+y_0^{k-1}{P}_{0y}^{\left(2\right)}{\displaystyle \underset{j=0}{\overset{k}{\sum }}{e}_0^{\left(2\right)}\frac{\partial \left(\frac{Q_{0y}^{\left(2\right)}f\left(x,y\right)}{y_0^{k-1}}\right)}{\partial y_j}{e}_j^{(\; 2\; )}}\end{array}$

$\begin{array}{l}=y_0^{k-1}{P}_{0y}^{\left(2\right)}{\displaystyle \underset{j=1}{\overset{k}{\sum }}\frac{\partial \left(\frac{P_{0y}^{\left(2\right)}f\left(x,y\right)}{y_0^{k-1}}\right)}{\partial y_j}{e}_j^{\left(2\right)}}+y_0^{k-1}{P}_{0y}^{\left(2\right)}\left(\frac{\partial \left(\frac{P_{0y}^{\left(2\right)}f\left(x,y\right)}{y_0^{k-1}}\right)}{\partial y_0}{e}_0^{\left(2\right)}\right)\\ +y_0^{k-1}{P}_{0y}^{\left(2\right)}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{e}_0^{\left(2\right)}\frac{\partial \left(\frac{Q_{0y}^{\left(2\right)}f\left(x,y\right)}{y_0^{k-1}}\right)}{\partial y_j}{e}_j^{\left(2\right)}}+y_0^{k-1}{P}_{0y}^{\left(2\right)}\left(e_0^{\left(2\right)}\frac{\partial \left(\frac{Q_{0y}^{\left(2\right)}f\left(x,y\right)}{y_0^{k-1}}\right)}{\partial y_0}{e}_0^{\left(2\right)}\right)\\ ={\displaystyle \underset{j=1}{\overset{k}{\sum }}\frac{\partial P_{0y}^{\left(2\right)}f\left(x,y\right)}{\partial y_j}{e}_j^{\left(2\right)}}+y_0^{k-1}\frac{\partial \frac{Q_{0y}^{\left(2\right)}{}^{\prime }f\left(x,y\right)}{y_0^{k-1}}}{\partial y_0}\\ ={\displaystyle \underset{j=1}{\overset{k}{\sum }}\frac{\partial P_{0y}^{\left(2\right)}f\left(x,y\right)}{\partial y_j}{e}_j^{\left(2\right)}}+\frac{\partial Q_{0y}^{\left(2\right)}{}^{\prime }f\left(x,y\right)}{\partial y_0}+\left(1-k\right)\frac{Q_{0y}^{\left(2\right)}{}^{\prime }f\left(x,y\right)}{y_0}\end{array}$

及

$\begin{array}{c}{Q}_{0y}^{\left(2\right)}\left(D_y^{r}f\left(x,y\right)\right)=Q_{0y}^{\left(2\right)}{D}_y^r\left(P_{0y}^{\left(2\right)}f\left(x,y\right)+e_0^{\left(2\right)}{Q}_{0y}^{\left(2\right)}f\left(x,y\right)\right)\\ =Q_{0y}^{\left(2\right)}{\displaystyle \underset{j=0}{\overset{k}{\sum }}\frac{\partial \left(P_{0y}^{\left(2\right)}f\left(x,y\right)+e_0^{\left(2\right)}{Q}_{0y}^{\left(2\right)}f\left(x,y\right)\right)}{\partial y_j}{e}_j^{\left(2\right)}}\\ =Q_{0y}^{\left(2\right)}{\displaystyle \underset{j=0}{\overset{k}{\sum }}\frac{\partial \left(P_{0y}^{\left(2\right)}f\left(x,y\right)\right)}{\partial y_j}{e}_j^{\left(2\right)}}+Q_{0y}^{\left(2\right)}{\displaystyle \underset{j=0}{\overset{k}{\sum }}{e}_0^{\left(2\right)}\frac{\partial \left(Q_{0y}^{\left(2\right)}f\left(x,y\right)\right)}{\partial y_j}{e}_j^{(\; 2\; )}}\end{array}$

$\begin{array}{l}=Q_{0y}^{\left(2\right)}{\displaystyle \underset{j=1}{\overset{k}{\sum }}\frac{\partial \left(P_{0y}^{\left(2\right)}f\left(x,y\right)\right)}{\partial y_j}{e}_j^{\left(2\right)}}+Q_{0y}^{\left(2\right)}\left(\frac{\partial \left(P_{0y}^{\left(2\right)}f\left(x,y\right)\right)}{\partial y_0}{e}_0^{\left(2\right)}\right)\\ +Q_{0y}^{\left(2\right)}{\displaystyle \underset{j=1}{\overset{k}{\sum }}{e}_0^{\left(2\right)}\frac{\partial \left(Q_{0y}^{\left(2\right)}f\left(x,y\right)\right)}{\partial y_j}{e}_j^{\left(2\right)}}+Q_{0y}^{\left(2\right)}\left(e_0^{\left(2\right)}\frac{\partial \left(Q_{0y}^{\left(2\right)}f\left(x,y\right)\right)}{\partial y_0}{e}_0^{\left(2\right)}\right)\\ =Q_{0y}^{\left(2\right)}\left(e_0^{\left(2\right)}\frac{\partial \left(P_{0y}^{\left(2\right)}{}^{\prime }f\left(x,y\right)\right)}{\partial y_0}\right)+{\displaystyle \underset{j=1}{\overset{k}{\sum }}\frac{\partial \left(Q_{0y}^{\left(2\right)}f\left(x,y\right)\right)}{\partial y_j}{e}_j^{\left(2\right)}}\\ =\frac{\partial \left(P_{0y}^{\left(2\right)}{}^{\prime }f\left(x,y\right)\right)}{\partial y_0}+{\displaystyle \underset{j=1}{\overset{k}{\sum }}\frac{\partial \left(Q_{0y}^{\left(2\right)}f\left(x,y\right)\right)}{\partial y_j}{e}_j^{(\; 2\; )}}\end{array}$

从而

$\begin{array}{c}{H}_y^{r}f\left(x,y\right)={\displaystyle \underset{j=1}{\overset{k}{\sum }}\frac{\partial P_{0y}^{\left(2\right)}f\left(x,y\right)}{\partial y_j}{e}_j^{\left(2\right)}}+\frac{\partial Q_{0y}^{\left(2\right)}{}^{\prime }f\left(x,y\right)}{\partial y_0}+\left(1-k\right)\frac{Q_{0y}^{\left(2\right)}{}^{\prime }f\left(x,y\right)}{y_0}\\ +e_0^{\left(2\right)}\left(\frac{\partial P_{0y}^{\left(2\right)}{}^{\prime }f\left(x,y\right)}{\partial y_0}+{\displaystyle \underset{j=1}{\overset{k}{\sum }}\frac{\partial \left(Q_{0y}^{\left(2\right)}f\left(x,y\right)\right)}{\partial y_j}{e}_j^{(\; 2\; )}}\right)\end{array}$

故 $H_y^{r}f\left(x,y\right)=0$ 成立的充要条件是(3.9)的前两式成立。

同理可得

$\begin{array}{c}{H}_x^{l}f\left(x,y\right)=P_{0x}^{\left(1\right)}\left(H_x^{l}f\left(x,y\right)\right)+e_0^{\left(1\right)}{Q}_{0x}^{\left(l\right)}\left(H_x^{l}f\left(x,y\right)\right)\\ =x_0^{m-1}{P}_{0x}^{\left(1\right)}\left(D_x^l\left(\frac{f\left(x,y\right)}{x_0^{m-1}}\right)\right)+e_0^{\left(1\right)}{Q}_{0x}^{\left(1\right)}\left(D_x^{l}f\left(x,y\right)\right)\\ =\left({\displaystyle \underset{i=1}{\overset{m}{\sum }}{e}_i^{\left(1\right)}\frac{\partial P_{0x}^{\left(1\right)}f\left(x,y\right)}{\partial x_i}+\frac{\partial Q_{0x}^{\left(1\right)}f\left(x,y\right)}{\partial x_0}+\left(1-m\right)\frac{Q_{0x}^{\left(1\right)}f\left(x,y\right)}{x_0}}\right)\\ \begin{array}{cc}& +\end{array}{e}_0^{\left(1\right)}\left(\frac{\partial P_{0x}^{\left(1\right)}f\left(x,y\right)}{\partial x_0}-{\displaystyle \underset{i=1}{\overset{m}{\sum }}{e}_i^{\left(1\right)}\frac{\partial Q_{0x}^{\left(1\right)}f\left(x,y\right)}{\partial x_i}}\right)\end{array}$

因而 $H_x^{l}f\left(x,y\right)=0$ 的充要条件为(3.9)的后两式成立，综上 $f\left(x,y\right)$ 是双hypergenic函数的充分必要条件是(3.9)成立，得证。

定理2.1 类似于复分析中的Cauchy-Riemann方程，建立了双Hypergenic函数与偏微分方程之间的联系。下面对方程(3.9)的某些量进行变换得到了双Hypergenic函数的又一个等价条件。

定理3.2 $f\left(x,y\right)$ 是双Hypergenic函数的充分必要条件是

$\{\begin{array}{l}\frac{\partial f\text{'}\left(x,y\right)}{\partial y_0}+D_y^r\left(Q_{0y}^{\left(2\right)}f\left(x,y\right)\right)=0\hfill \\ D_y^r\left(P_{0y}^{\left(2\right)}f\left(x,y\right)\right)-e_0^{\left(2\right)}\frac{\partial f\text{'}\left(x,y\right)}{\partial y_0}+\frac{1-k}{y_0}{Q}_{0y}^{\left(2\right)}{}^{\prime }f\left(x,y\right)=0\hfill \\ \frac{\partial f\left(x,y\right)}{\partial x_0}-D_x^l\left(Q_{0x}^{\left(1\right)}f\left(x,y\right)\right)=0\hfill \\ D_x^l\left(P_{0x}^{\left(1\right)}f\left(x,y\right)\right)+e_0^{\left(1\right)}\frac{\partial f\left(x,y\right)}{\partial x_0}-2e_0^{\left(1\right)}\frac{\partial P_{0x}^{\left(1\right)}f\left(x,y\right)}{\partial x_0}+\frac{\left(1-m\right)}{x_0}{Q}_{0x}^{\left(1\right)}f\left(x,y\right)=0\hfill \end{array}$ (3.11)

证明 由定理3.1知，我们只需证明方程组(3.9)等价于方程组(2.11)即可。

由

$\begin{array}{c}\frac{\partial P_{0y}^{\left(2\right)}\text{'}f\left(x,y\right)}{\partial y_0}=\frac{\partial \left(f\left(x,y\right)-e_0^{\left(2\right)}{Q}_{0y}^{\left(2\right)}f\left(x,y\right)\right)\text{'}}{\partial y_0}\\ =\frac{\partial f\text{'}\left(x,y\right)}{\partial y_0}+e_0^{\left(2\right)}\frac{\partial Q_{0y}^{\left(2\right)}\text{'}f\left(x,y\right)}{\partial y_0}\\ =\frac{\partial f\text{'}\left(x,y\right)}{\partial y_0}+\frac{\partial Q_{0y}^{\left(2\right)}f\left(x,y\right)}{\partial y_0}{e}_0^{\left(2\right)}\end{array}$ (3.12)

$\begin{array}{c}\frac{\partial Q_{0y}^{\left(2\right)}\text{'}f\left(x,y\right)}{\partial y_0}=\frac{\partial \left(e_0^{\left(2\right)}\left(f\left(x,y\right)-P_{0y}^{\left(2\right)}f\left(x,y\right)\right)\right)\text{'}}{\partial y_0}\\ =-e_0^{\left(2\right)}\left(\frac{\partial f\text{'}\left(x,y\right)}{\partial y_0}-\frac{\partial P_{0y}^{\left(2\right)}\text{'}f\left(x,y\right)}{\partial y_0}\right)\\ =-e_0^{\left(2\right)}\frac{\partial f\text{'}\left(x,y\right)}{\partial y_0}+\frac{\partial P_{0y}^{\left(2\right)}f\left(x,y\right)}{\partial y_0}{e}_0^{\left(2\right)}\end{array}$ (3.13)

同理可得

$\frac{\partial P_{0x}^{\left(\text{1}\right)}f\left(x,y\right)}{\partial x_0}=\frac{\partial f\left(x,y\right)}{\partial x_0}-e_0^{\left(1\right)}\frac{\partial Q_{0x}^{\left(1\right)}f\left(x,y\right)}{\partial x_0}$ (3.14)

$\frac{\partial Q_{0x}^{\left(\text{1}\right)}f\left(x,y\right)}{\partial x_0}=e_0^{\left(1\right)}\frac{\partial f\left(x,y\right)}{\partial x_0}-e_0^{\left(1\right)}\frac{\partial P_{0x}^{\left(1\right)}f\left(x,y\right)}{\partial x_0}$ (3.15)

将(3.12)，(3.13)，(3.14)，(3.15)分别代入方程组(3.9)，即得(3.11)，反之也成立。

故定理得证。

4. 结论

双hypergenic函数是Clifford分析中hypergenic函数的进一步推广，双hypergenic函数的等价条件的研究对于研究高维空间中的方程和算子提供了重要的理论基础。在本文中，我们研究了双hypergenic函数的充分必要条件，这些判别条件类似于复分析中的Cauchy-Riemann方程，建立了双Hypergenic函数与偏微分方程之间的联系。这些结果丰富了Clifford分析的理论基础，但关于双hypergenic函数还有很多问题有待进一步探讨，后续我将继续研究双hypergenic函数方面的内容，争取有进一步的研究成果。

基金项目

国家自然科学基金11802208。