ABSTRACT

Given two left $O^c$ -analytic functions $f\mathrm{,}g$ in some open set $\Omega$ of $R^8$ , we obtain some sufficient conditions for $fg$ is also left $O^c$ -analytic in $\Omega$ . Moreover, we prove that $f\lambda$ is a left $O^c$ -analytic function for any constants $\lambda \in O^c$ if and only if $\bar{f}$ is a complex Stein-Weiss conjugate harmonic system. Some applications and connections with Cauchy- Kowalewski product are also considered.

Keywords:

Octonions, $O^c$ -Analytic Functions, Stein-Weiss Conjugate Harmonic System, Cauchy-Kowalewski Product

1. Introduction

Let $\Omega$ be an open set of $R^8$ . A function $f$ in $C^1\left(\Omega \mathrm{,}O\right)$ is said to be left (right) $O$ -analytic in $\Omega$ when

$Df={\displaystyle \underset{i=0}{\overset{7}{\sum }}}{e}_i\frac{\partial f}{\partial x_i}=0\left(fD={\displaystyle \underset{i=0}{\overset{7}{\sum }}}\frac{\partial f}{\partial x_i}{e}_i=0\right),$

where the Dirac D-operator and its adjoint $\bar{D}$ are the first-order systems of

differential operators in $C^1\left(\Omega \mathrm{,}O\right)$ defined by $D={\displaystyle {\sum }_0^7}{e}_i\frac{\partial }{\partial x_i}$ and $\bar{D}=e_0\frac{\partial }{\partial x_0}-{\displaystyle {\sum }_1^7}{e}_i\frac{\partial }{\partial x_i}$ .

If $f$ is a simultaneously left and right $O$ -analytic function, then $f$ is called an $O$ -analytic function. If $f$ is a (left) $O$ -analytic function in $R^8$ , then $f$ is called a (left) $O$ -entire function.

Since octonions is non-commutative and non-associative, the product $f\left(x\right)g\left(x\right)$ of two left $O$ -analytic functions $f\left(x\right)$ and $g\left(x\right)$ is generally no longer a left $O$ -analytic function. Furthermore, if $g\left(x\right)\equiv \lambda$ becomes an octonionic constant function, the product $f\left(x\right)\lambda$ is also probably not a left $O$ -analytic function; that is, the collection of left $O$ -analytic functions is not a right module (see [1] ).

The purpose of this paper is to study the analyticity for the product of two left $O^c$ -analytic functions in the framework of complexification of $O$ , $O^c$ . Especially, the analyticity for the product of left $O^c$ -analytic functions and $O^c$ constants will be consider more by us.

The rest of this paper is organized as follows. Section 2 is an overview of some basic facts concerning octonions and octonionic analysis. Section 3 we give some sufficient conditions for the product $f\left(x\right)g\left(x\right)$ of two left $O^c$ -analytic functions $f\left(x\right)$ and $g\left(x\right)$ is also a left $O^c$ -analytic function. In Section 3, we prove that, $f\left(x\right)\lambda$ is a left $O^c$ -analytic function for any constants $\lambda \in O^c$ if and only if $\overline{f\left(x\right)}$ is a complex Stein-Weiss conjugate harmonic system. This gives the solution of the problem in [2] . In the last section we give some applications for our results.

2. Preliminaries: Octonions and Octonionic Analysis

It is well known that there are only four normed division algebras [3] [4] [5] : the real numbers $R$ , complex numbers $C$ , quaternions $H$ and octonions $O$ , with the relations $R\subseteq C\subseteq H\subseteq O$ . In other words, for any $x=\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_n\right)$ , $y=\left(y_1\mathrm{,}\cdots \mathrm{,}{y}_n\right)\in R^n$ , if we define a product “ $xy$ ” such that $xy\in R^n$ and

$\left|x\cdot y\right|=\left|x\right|\left|y\right|$ , where $\left|x\right|=\sqrt{{\displaystyle {\sum }_1^n}{x}_i^2}$ , then the only four values of $n$ are 1,2,4,8.

Quaternions $H$ is not commutative and octonions $O$ is neither commutative nor associative. Unlike $R$ , $C$ and $H$ , the non-associative octonions can not be embedded into the associative Clifford algebras [6] .

Octonions stand at the crossroads of many interesting fields of mathematics, they have close relations with Clifford algebras, spinors, Bott periodicity, Projection and Lorentzian geometry, Jordan algebras, and exceptional Lie groups, and also, they have many applications in quantum logic, special relativity and supersymmetry [3] [4] .

Denote the set $W$ by

$W=\left\{\left(\mathrm{1,2,3}\right)\mathrm{,}\left(\mathrm{1,4,5}\right)\mathrm{,}\left(\mathrm{1,7,6}\right)\mathrm{,}\left(\mathrm{2,4,6}\right)\mathrm{,}\left(\mathrm{2,5,7}\right)\mathrm{,}\left(\mathrm{3,4,7}\right)\mathrm{,}\left(\mathrm{3,6,5}\right)\right\}\mathrm{.}$

Then the multiplication rules between the basis $e_0\mathrm{,}{e}_1\mathrm{,}\cdots \mathrm{,}{e}_7$ on octonions are given by [3] [7] :

$e_0^2=e_0,e_i{e}_0=e_0{e}_i=e_i,e_i^2=-1,i=1,2,\cdots ,7,$

and for any triple $\left(\alpha \mathrm{,}\beta \mathrm{,}\gamma \right)\in W$ ,

$e_{\alpha }{e}_{\beta }=e_{\gamma }=-e_{\beta }{e}_{\alpha },e_{\beta }{e}_{\gamma }=e_{\alpha }=-e_{\gamma }{e}_{\beta },e_{\gamma }{e}_{\alpha }=e_{\beta }=-e_{\alpha }{e}_{\gamma }.$

For each $x={\displaystyle {\sum }_0^7}{x}_i{e}_i\in O\left(x_i\in R\mathrm{,}i=\mathrm{0,1,}\cdots \mathrm{,7}\right)$ , $x_0$ is called the scalar part of x and $\underset{\_}{x}={\displaystyle {\sum }_1^7}{x}_i{e}_i$ is termed its vector part. Then the norm of x is $\left|x\right|={\left({\displaystyle {\sum }_0^7}{x}_i^2\right)}^{\frac{1}{2}}$ and its conjugate is defined by $\bar{x}={\displaystyle {\sum }_0^7}{x}_i{\bar{e}}_i=x_0-\underset{\_}{x}$ . We have $x\bar{x}=\bar{x}x={\displaystyle {\sum }_0^7}{x}_i^2$ , $\overline{xy}=\bar{y}\bar{x}\left(x\mathrm{,}y\in O\right)$ Hence, $x^{-1}=\frac{\bar{x}}{{\left|x\right|}^2}$ is the inverse of $x\left(\ne 0\right)$ .

Let $x={\displaystyle {\sum }_0^7}{x}_i{e}_i\mathrm{,}y={\displaystyle {\sum }_0^7}{y}_i{e}_i\in O\left(x_i\mathrm{,}{y}_i\in R\mathrm{,}i=\mathrm{0,1,}\cdots \mathrm{,7}\right)$ , then

$xy=x_0{y}_0-\underset{\_}{x}\cdot \underset{\_}{y}+x_0\underset{\_}{y}+y_0\underset{\_}{x}+\underset{\_}{x}\times \underset{\_}{y},$ (2.1)

where $\underset{\_}{x}\cdot \underset{\_}{y}:={\displaystyle {\sum }_1^7}{x}_i{y}_i$ is the inner product of vectors $\underset{\_}{x}\mathrm{,}\underset{\_}{y}$ and

$\begin{array}{c}\underset{\_}{x}\times \underset{\_}{y}:=e_1\left(A_{23}+A_{45}-A_{67}\right)+e_2\left(-A_{13}+A_{46}+A_{57}\right)+e_3\left(A_{12}+A_{47}-A_{56}\right)\\ +e_4\left(-A_{15}-A_{26}-A_{37}\right)+e_5\left(A_{14}-A_{27}+A_{36}\right)\\ +e_6\left(A_{17}+A_{24}-A_{35}\right)+e_7\left(-A_{16}+A_{25}+A_{34}\right)\end{array}$

is the cross product of vectors $\underset{\_}{x}\mathrm{,}\underset{\_}{y}$ , with

$A_{ij}=\det \left(\begin{array}{cc}{x}_i& x_j\\ y_i& y_j\end{array}\right),i,j=1,2,\cdots ,7.$

For any $x\mathrm{,}y\in O$ , the inner product and cross product of their vector parts satisfy the following rules [8] :

$\left(\underset{\_}{x}\times \underset{\_}{y}\right)\cdot \underset{\_}{x}=0,\left(\underset{\_}{x}\times \underset{\_}{y}\right)\cdot \underset{\_}{y}=0,\underset{\_}{x}\left|\right|\underset{\_}{y}\iff \underset{\_}{x}\times \underset{\_}{y}=0,\underset{\_}{x}\times \underset{\_}{y}=-\underset{\_}{y}\times \underset{\_}{x}.$

We usually utilize associator as an useful tool on ontonions since its non- associativity. Define the associator $\left[x\mathrm{,}y\mathrm{,}z\right]$ of any $x\mathrm{,}y\mathrm{,}z\in O$ by $\left[x\mathrm{,}y\mathrm{,}z\right]=\left(xy\right)z-x\left(yz\right)$ .

The octonions obey the following some weakened associative laws.

For any $x\mathrm{,}y\mathrm{,}z\mathrm{,}u\mathrm{,}v\in O$ , we have (see [7] )

$\left[x,y,z\right]=\left[y,z,x\right],\left[x,z,y\right]=-\left[x,y,z\right],\left[x,x,y\right]=0=\left[\bar{x},x,y\right]$ (2.2)

and the so-called Moufang identities [5]

$\left(uvu\right)x=u\left(v\left(ux\right)\right),x\left(uvu\right)=\left(\left(xu\right)v\right)u,u\left(xy\right)u=\left(ux\right)\left(uy\right).$

Proposition 2.1 ( [7] ). For any $i\mathrm{,}j\mathrm{,}k\in \left\{\mathrm{0,1,}\cdots \mathrm{,7}\right\}$ , $\left[e_i,e_j,e_k\right]=0\iff ijk=0$ or $\left(i-j\right)\left(j-k\right)\left(k-i\right)=0$ or $\left(e_i{e}_j\right)e_k=\pm 1$ .

Proposition 2.2 ( [7] ). Let $e_i\mathrm{,}{e}_j\mathrm{,}{e}_k$ be three different elements of $\left\{e_1\mathrm{,}{e}_2\mathrm{,}\cdots \mathrm{,}{e}_7\right\}$ and $\left(e_i{e}_j\right)e_k\ne \pm 1$ . Then $\left(e_i{e}_j\right)e_k=-e_i\left(e_j{e}_k\right)$ .

Since octonions is an alternative algebra (see [3] [9] [10] ), we have the following power-associativity of octonions.

Proposition 2.3. Let $x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_k\in O$ , $\left(l_1\mathrm{,}\cdots \mathrm{,}{l}_n\right)$ be $n$ elements out of

$\left\{\mathrm{1,}\cdots \mathrm{,}k\right\}$ repetitions being allowed and let ${\left(x_{l_1}{x}_{l_2}\cdots x_{l_n}\right)}_{{\otimes }_n}$ be the product of $n$ octonions in a fixed associative order ${\otimes }_n$ . Then ${\displaystyle \underset{\pi \left(l_1,\cdots ,l_n\right)}{\sum }}{\left(x_{l_1}{x}_{l_2}\cdots x_{l_n}\right)}_{{\otimes }_n}$ is

independent of the associative order ${\otimes }_n$ , where the sum runs over all distinguishable permutations of $\left(l_1\mathrm{,}\cdots \mathrm{,}{l}_n\right)$

Proof. Let $x={\lambda }_1{x}_1+{\lambda }_2{x}_2+\cdots +{\lambda }_k{x}_k$ , then ${\displaystyle \underset{\pi \left(l_1,\cdots ,l_n\right)}{\sum }}{\left(x_{l_1}{x}_{l_2}\cdots x_{l_n}\right)}_{{\otimes }_n}$ is just the coefficient of ${\lambda }_{l_1}{\lambda }_{l_2}\cdots {\lambda }_{l_n}$ in the product of $x^n=\underset{nx\text{s}}{\underset{︸}{\left(xx\cdots x\right)}}{}_{{\otimes }_n}$ . By induction and (2.2), one can easily prove that $x^n=\underset{nx\text{s}}{\underset{︸}{\left(xx\cdots x\right)}}{}_{{\otimes }_n}$ is independent of the associative order ${\otimes }_n$ for any $x\in O$ . Hence ${\displaystyle \underset{\pi \left(l_1,\cdots ,l_n\right)}{\sum }}{\left(x_{l_1}{x}_{l_2}\cdots x_{l_n}\right)}_{{\otimes }_n}$ is also independent of the associative order ${\otimes }_n$ . ,

$\mu =\left({\mu }_0\mathrm{,}{\mu }_1\mathrm{,}\cdots \mathrm{,}{\mu }_n\right)$ is called a Stein-Weiss conjugate harmonic system if they satisfy the following equations (see [11] ):

${\displaystyle \underset{i=0}{\overset{n}{\sum }}}\frac{\partial {\mu }_i}{\partial x_i}=0,\frac{\partial {\mu }_i}{\partial x_j}=\frac{\partial {\mu }_j}{\partial x_i}\left(0\le i<j\le n\right).$

It is easy to see that if $F\left(x_0\mathrm{,}{x}_1\mathrm{,}\cdots \mathrm{,}{x}_7\right)=\left(f_0\mathrm{,}{f}_1\mathrm{,}\cdots \mathrm{,}{f}_7\right)$ is a Stein-Weiss conjugate harmonic system in an open set $\Omega$ of $R^8$ , then there exists a real- valued harmonic function $\Phi$ in $\Omega$ such that F is the gradient of $\Phi$ . Thus $\bar{F}=f_0{e}_0-f_1{e}_1-\cdots -f_7{e}_7=\bar{D}\Phi$ is an $O$ -analytic function. But inversely, this is not true [12] .

Example. Observe the $O$ -analytic function $g\left(x\right)=\left(x_6^2-x_7^2\right)e_2-2x_6{x}_7{e}_3$ . Since

$\frac{\partial g_2}{\partial x_6}=2x_6\ne 0=\frac{\partial g_6}{\partial x_2},$

$\bar{g}$ is not a Stein-Weiss conjugate harmonic system.

In [13] Li and Peng proved the octonionic analogue of the classical Taylor theorem. Taking account of Proposition 2.3, we obtain an improving of Taylor type theorem for $O$ -analytic functions (see [14] [15] ).

Theorem A (Taylor). If $f\left(x\right)$ is a left $O$ -analytic function in $\Omega$ which containing the origin, then it can be developed into Taylor series

$f\left(x\right)={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}}{\displaystyle \underset{\left(l_1,\cdots ,l_k\right)}{\sum }}{V}_{l_1\cdots l_k}\left(x\right){\partial }_{x_{l_1}}\cdots {\partial }_{x_{l_k}}f\left(0\right),$

and if $f\left(x\right)$ is a right $O$ -analytic function, then the Taylor series of $f$ at the origin is given by

$f\left(x\right)={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}}{\displaystyle \underset{\left(l_1,\cdots ,l_k\right)}{\sum }}{\partial }_{x_{l_1}}\cdots {\partial }_{x_{l_k}}f\left(0\right)V_{l_1\cdots l_k}\left(x\right),$

where $\left(l_1,\cdots ,l_k\right)$ runs over all possible combinations of $k$ elements out of $\left\{\mathrm{1,}\cdots \mathrm{,7}\right\}$ repetitions being allowed.

The polynomials $V_{l_1\cdots l_k}$ of order $k$ in Theorem A is defined by

$V_{l_1\cdots l_k}\left(x\right)=\frac{1}{k!}{\displaystyle \underset{\pi \left(l_1,\cdots ,l_k\right)}{\sum }}\left(\cdots \left(\left(z_{l_1}{z}_{l_2}\right)z_{l_3}\right)\cdots \right)z_{l_k},$

where the sum runs over all distinguishable permutations of $\left(l_1,\cdots ,l_k\right)$ and $z_{l_j}=x_{l_j}{e}_0-x_0{e}_{l_j},j=1,\cdots ,k$ .

We have the following uniqueness theorem for $O$ -analytic functions [7] .

Proposition 2.4. If $f$ is left (right) $O$ -analytic in an open connect set $\Omega \subset R^8$ and vanishes in the open set $E\subset \Omega \cap \left\{x_0=a_0\right\}\ne \varnothing$ , then $f$ is identically zero in $\Omega$ .

Proof. Without loss of generality, we let $E$ which containing the origin and let $x_0=0$ . Then $f$ can be developed into Taylor series

$f\left(x\right)={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}}{\displaystyle \underset{\left(l_1,\cdots ,l_k\right)}{\sum }}{V}_{l_1\cdots l_k}\left(x\right){\partial }_{x_{l_1}}\cdots {\partial }_{x_{l_k}}f\left(0\right).$

Thus we have

$f\left(\underset{\_}{x}\right)={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}}{\displaystyle \underset{\left(l_1,\cdots ,l_k\right)}{\sum }}{x}_{l_1}{x}_{l_2}\cdots x_{l_k}{\partial }_{x_{l_1}}\cdots {\partial }_{x_{l_k}}f\left(0\right)\equiv 0.$

By the uniqueness of the Taylor series for the real analytic function, we have ${\partial }_{x_{l_1}}\cdots {\partial }_{x_{l_k}}f\left(0\right)=0$ for any $\left(l_1,\cdots ,l_k\right)\in {\left\{\mathrm{1,2,}\cdots \mathrm{,7}\right\}}^7$ and $k\in N$ . This shows that $f$ is identically zero in $E$ and also in $\Omega$ . ,

For more references about octonions and octonionic analysis, we refer the reader to [7] [13] - [20] .

3. Sufficient Conditions

In what follows we consider the complexification of $O$ , it is denoted by $O^c$ .

Thus, $\mathbb{Z}\in O^c$ is of the form $\mathbb{Z}={\displaystyle {\sum }_0^7}{\mathbb{Z}}_i{e}_i\mathrm{,}{\mathbb{Z}}_i\in C$ . ${\mathbb{Z}}_0$ and $\underset{\_}{\mathbb{Z}}={\displaystyle {\sum }_0^7}{\mathbb{Z}}_i{e}_i$ are still

called the scalar part and vector part, respectively. The norm of $\mathbb{Z}\in O^c$ is

$\left|\mathbb{Z}\right|={\left({\displaystyle {\sum }_0^7}{\left|{\mathbb{Z}}_i\right|}^2\right)}^{\frac{1}{2}}$ and its conjugate is defined by $\bar{\mathbb{Z}}={\displaystyle {\sum }_0^7}{\bar{\mathbb{Z}}}_i{\bar{e}}_i$ , where ${\bar{\mathbb{Z}}}_i$ is of the

conjugate in the complex numbers. We can easily show that for any $\mathbb{Z}\mathrm{,}{\mathbb{Z}}^{\prime }\in O^c$ , $\left|\mathbb{Z}{\mathbb{Z}}^{\prime }\right|\le \sqrt{2}\left|\mathbb{Z}\right|\left|{\mathbb{Z}}^{\prime }\right|$ . For any $\mathbb{Z}\in O^c$ , we may rewrite $\mathbb{Z}$ as $\mathbb{Z}=x+iy$ , where $x\mathrm{,}y\in O$ . The multiplication rules in $O^c$ is the same as in (2.1). Note that $O^c$ is no longer a division algebra. Finally, the properties of associator in (2.2) except that $\left[\mathbb{Z},\bar{\mathbb{Z}},\mathbb{U}\right]=0$ are also true for any $\mathbb{Z}\mathrm{,}\mathbb{U}\mathrm{,}\mathbb{V}\in O^c$ :

$\left[\mathbb{Z}\mathrm{,}\mathbb{U}\mathrm{,}\mathbb{V}\right]=\left[\mathbb{U}\mathrm{,}\mathbb{V}\mathrm{,}\mathbb{Z}\right],\left[\mathbb{Z}\mathrm{,}\mathbb{V}\mathrm{,}\mathbb{U}\right]=-\left[\mathbb{Z}\mathrm{,}\mathbb{U}\mathrm{,}\mathbb{V}\right],\left[\mathbb{Z},\mathbb{Z},\mathbb{U}\right]=0.$ (3.1)

Example. Let $\mathbb{Z}=e_1+i{e}_2,\mathbb{U}=e_4$ , then

$\left[\mathbb{Z},\bar{\mathbb{Z}},\mathbb{U}\right]=\left[e_1+i{e}_2,-e_1+i{e}_2,e_4\right]=i\left[e_1,e_2,e_4\right]-i\left[e_2,e_1,e_4\right]=4i{e}_7\ne 0.$

By (3.1) we can get the following lemma, which is useful to deduce our results.

Lemma 3.1. Let $\mathbb{Z}\mathrm{,}\mathbb{U}\mathrm{,}\mathbb{V}\in O^c$ and there exists complex numbers $\lambda$ and $\mu \left(\left|\lambda \right|+\left|\mu \right|\ne 0\right)$ such that $\lambda \underset{\_}{\mathbb{Z}}+\mu \underset{\_}{\mathbb{U}}=0$ or $\lambda \underset{\_}{\mathbb{U}}+\mu \underset{\_}{\mathbb{V}}=0$ or $\lambda \underset{\_}{\mathbb{V}}+\mu \underset{\_}{\mathbb{Z}}=0$ , then $\left[\mathbb{Z},\mathbb{U},\mathbb{V}\right]=0$ .

For functions, f, under study will be defined in an open set $\Omega$ of $R^8$ and

take values in $O^c$ , with the form $f\left(x\right)={\displaystyle {\sum }_0^7}{f}_i\left(x\right)e_i$ , where $f_i\left(x\right)\left(i=0,1,\cdots ,7\right)$

are the complex-valued functions.

Hence, we say that, a function $f\left(x\right)=g\left(x\right)+ih\left(x\right)$ is left $O^c$ -analytic in an open set $\Omega$ of $R^8$ , if $g\left(x\right)$ and $h\left(x\right)$ are the left $O$ -analytic functions, since

$Df=0\iff Dg=Dh=0,$

where $D={\displaystyle {\sum }_{i=0}^7}\frac{\partial }{\partial x_i}{e}_i$ is the Dirac operator as in Section 1.

In the case of $O^c$ , we call $f\left(x\right)=g\left(x\right)+ih\left(x\right)$ a complex Stein-Weiss conjugate harmonic system, if $g\left(x\right)\mathrm{,}h\left(x\right)$ are the Stein-Weiss conjugate harmonic systems. A left (right) $O^c$ -analytic functions $g\left(x\right)$ also have the Taylor expansion as in Theorem A.

Now we consider the product $f\left(x\right)g\left(x\right)$ of two left $O^c$ -analytic functions $f\left(x\right),g\left(x\right)$ in $\Omega$ . In general, $f\left(x\right)g\left(x\right)$ is no longer left $O^c$ -analytic in $\Omega$ . But, in some particular cases, the product $f\left(x\right)g\left(x\right)$ can maintain the analyticity for two left $O^c$ -analytic functions $f\left(x\right)$ and $g\left(x\right)$ .

Theorem 3.2. Let $f\left(x\right),g\left(x\right)$ be two left $O^c$ -analytic functions in $\Omega$ . Then $f\left(x\right)g\left(x\right)$ is also left $O$ -analytic in $\Omega$ if $f\left(x\right),g\left(x\right)$ satisfy one of the following conditions:

1) $f\left(x\right)$ or $g\left(x\right)$ is a complex constant function.

2) $\overline{f\left(x\right)}$ is a complex Stein-Weiss conjugate harmonic system in $\Omega$ and $g\left(x\right)$ is an $O^c$ -constant function.

3) $f\left(x\right)$ is of the form $f\left(x\right)=f_0{e}_0+f_i{e}_i\left(i\in \left\{1,2,\cdots ,7\right\}\right)$ and $f\left(x\right),g\left(x\right)$ depend only on $x_0$ and $x_i$ , where $f_0\mathrm{,}{f}_i$ are the complex-valued functions.

4) $f\left(x\right)$ and $g\left(x\right)$ belong to the following class

$\mathfrak{S}=\left\{h\left(x\right)\mathrm{<mo>\; |\; </mo>}Dh\left(x\right)=\mathrm{0,}\underset{\_}{h\left(x\right)}={\displaystyle \underset{i=1}{\overset{7}{\sum }}}{h}_1\left(x\right)e_i\mathrm{,}{h}_1\left(x\right)\in C^1\left(\Omega \mathrm{,}C\right)\right\}\mathrm{.}$ (3.2)

5) $f\left(x\right)$ is of the form $f\left(x\right)=f_0{e}_0+f_{\alpha }{e}_{\alpha }+f_{\beta }{e}_{\beta }+f_{\gamma }{e}_{\gamma }$ , $g=c_0{e}_0+c_{\alpha }{e}_{\alpha }+c_{\beta }{e}_{\beta }+c_{\gamma }{e}_{\gamma }$ is a constant function, where $\left(\alpha \mathrm{,}\beta \mathrm{,}\gamma \right)\in W$ , $c_0\mathrm{,}{c}_{\alpha }\mathrm{,}{c}_{\beta }\mathrm{,}{c}_{\gamma }\in C$ and $f\left(x\right)$ depends only on $x_0\mathrm{,}{x}_{\alpha }\mathrm{,}{x}_{\beta }\mathrm{,}{x}_{\gamma }$ .

Proof. 1) The proof is trivial.

2) In view of Proposition 2.1 we have $\left[e_i,e_j,\lambda \right]=0$ when $i=0$ or $j=0$ or $i=j$ for any $\lambda \in O^c$ . Then we have

$\begin{array}{c}D\left(f\lambda \right)={\displaystyle \underset{i,j=0}{\overset{7}{\sum }}}\frac{\partial f_j}{\partial x_i}{e}_i\left(e_j\lambda \right)\\ ={\displaystyle \underset{i,j=0}{\overset{7}{\sum }}}\frac{\partial f_j}{\partial x_i}\left(e_i{e}_j\right)\lambda -{\displaystyle \underset{i,j=0}{\overset{7}{\sum }}}\frac{\partial f_j}{\partial x_i}\left[e_i,e_j,\lambda \right]\\ =\left(Df\right)\lambda -{\displaystyle \underset{i,j=0}{\overset{7}{\sum }}}\frac{\partial f_j}{\partial x_i}\left[e_i,e_j,\lambda \right]\\ =\left(Df\right)\lambda -{\displaystyle \underset{1\le i\ne j\le 7}{\sum }}\frac{\partial f_j}{\partial x_i}\left[e_i,e_j,\lambda \right].\end{array}$

Since $\bar{f}$ is a complex Stein-Weiss conjugate harmonic system, thus $Df=0$

and $\frac{\partial f_j}{\partial x_i}=\frac{\partial f_i}{\partial x_j}$ for $i\mathrm{,}j\ge \mathrm{1,}i\ne j$ . But $\left[e_j,e_i,\lambda \right]=-\left[e_i,e_j,\lambda \right]$ , therefore

$D\left(f\lambda \right)=-{\displaystyle \underset{1\le i\ne j\le 7}{\sum }}\frac{\partial f_j}{\partial x_i}\left[e_i,e_j,\lambda \right]={\displaystyle \underset{1\le i<j\le 7}{\sum }}\left(\frac{\partial f_i}{\partial x_j}-\frac{\partial f_j}{\partial x_i}\right)\left[e_i,e_j,\lambda \right]=0.$

3) Since $f\left(x\right)\mathrm{,}g\left(x\right)$ are only related to variables $x_0$ and $x_i$ , we have

$\begin{array}{c}D\left(fg\right)=\left(\frac{\partial }{\partial x_0}+\frac{\partial }{\partial x_i}{e}_i\right)\left(\left(f_0+f_i{e}_i\right)g\right)\\ =\frac{\partial f}{\partial x_0}g+e_i\left(\left(\frac{\partial f_0}{\partial x_i}+\frac{\partial f_i}{\partial x_i}{e}_i\right)g\right)+f\frac{\partial g}{\partial x_0}+e_i\left(\left(f_0+f_i{e}_i\right)\frac{\partial g}{\partial x_i}\right).\end{array}$

By Lemma 3.1 it follows that

$e_i\left(\left(\frac{\partial f_0}{\partial x_i}+\frac{\partial f_i}{\partial x_i}{e}_i\right)g\right)=\left(e_i\left(\frac{\partial f_0}{\partial x_i}+\frac{\partial f_i}{\partial x_i}{e}_i\right)\right)g=\left(e_i\frac{\partial f}{\partial x_i}\right)g$

and

$e_i\left(\left(f_0+f_i{e}_i\right)\frac{\partial g}{\partial x_i}\right)=\left(e_i\left(f_0+f_i{e}_i\right)\right)\frac{\partial g}{\partial x_i}=\left(\left(f_0+f_i{e}_i\right)e_i\right)\frac{\partial g}{\partial x_i}=\left(f_0+f_i{e}_i\right)\left(e_i\frac{\partial g}{\partial x_i}\right).$

Thus we get

$D\left(fg\right)=\frac{\partial f}{\partial x_0}g+\left(e_i\frac{\partial f}{\partial x_i}\right)g+f\frac{\partial g}{\partial x_0}+f\left(e_i\frac{\partial g}{\partial x_i}\right)=\left(Df\right)g+f\left(Dg\right)=0.$

4) Let $f\left(x\right)=f_0{e}_0+{\displaystyle {\sum }_{i=1}^7}{f}_1{e}_i$ and $g\left(x\right)=g_0{e}_0+{\displaystyle {\sum }_{i=1}^7}{g}_1{e}_i$ , then we have

$\begin{array}{c}D\left(f\left(x\right)g\left(x\right)\right)={\displaystyle \underset{j=0}{\overset{7}{\sum }}}{e}_j\frac{\partial }{\partial x_j}\left(\left(f_0{e}_0+f_1{\displaystyle \underset{i=1}{\overset{7}{\sum }}}{e}_i\right)\left(g_0{e}_0+g_1{\displaystyle \underset{i=1}{\overset{7}{\sum }}}{e}_i\right)\right)\\ ={\displaystyle \underset{j=0}{\overset{7}{\sum }}}{e}_j\left(\left(\frac{\partial f_0}{\partial x_j}{e}_0+\frac{\partial f_1}{\partial x_j}{\displaystyle \underset{i=1}{\overset{7}{\sum }}}{e}_i\right)\left(g_0{e}_0+g_1{\displaystyle \underset{i=1}{\overset{7}{\sum }}}{e}_i\right)\right)\\ +{\displaystyle \underset{j=0}{\overset{7}{\sum }}}{e}_j\left(\left(f_0{e}_0+f_1{\displaystyle \underset{i=1}{\overset{7}{\sum }}}{e}_i\right)\left(\frac{\partial g_0}{\partial x_j}{e}_0+\frac{\partial g_1}{\partial x_j}{\displaystyle \underset{i=1}{\overset{7}{\sum }}}{e}_i\right)\right).\end{array}$

By Lemma 3.1 we get

$e_j\left(\left(\frac{\partial f_0}{\partial x_j}{e}_0+\frac{\partial f_1}{\partial x_j}{\displaystyle \underset{i=1}{\overset{7}{\sum }}}{e}_i\right)\left(g_0{e}_0+g_1{\displaystyle \underset{i=1}{\overset{7}{\sum }}}{e}_i\right)\right)=\left(e_j\frac{\partial f}{\partial x_j}\right)g$

and

$\begin{array}{l}{e}_j\left(\left(f_0{e}_0+f_1{\displaystyle \underset{i=1}{\overset{7}{\sum }}}{e}_i\right)\left(\frac{\partial g_0}{\partial x_j}{e}_0+\frac{\partial g_1}{\partial x_j}{\displaystyle \underset{i=1}{\overset{7}{\sum }}}{e}_i\right)\right)\\ =e_j\left(\left(\frac{\partial g_0}{\partial x_j}{e}_0+\frac{\partial g_1}{\partial x_j}{\displaystyle \underset{i=1}{\overset{7}{\sum }}}{e}_i\right)\left(f_0{e}_0+f_1{\displaystyle \underset{i=1}{\overset{7}{\sum }}}{e}_i\right)\right)\\ =\left(e_j\frac{\partial g}{\partial x_j}\right)f.\end{array}$

Hence we obtain

$D\left(f\left(x\right)g\left(x\right)\right)={\displaystyle \underset{j=0}{\overset{7}{\sum }}}\left(\left(e_j\frac{\partial f}{\partial x_j}\right)g+\left(e_j\frac{\partial g}{\partial x_j}\right)f\right)=\left(Df\right)g+\left(Dg\right)f=0.$

5) This case is equivalent to a left quaternionic analytic function right- multiplying by a quaternionic constant, the analyticity is obvious since the multiplication of the quaternion is associative.

The proof of Theorem 3.2 is complete. ,

From Theorem 3.2(d), if $f\left(x\right)\mathrm{,}g\left(x\right)\in \mathfrak{S}$ , then $f\left(x\right)g\left(x\right)\in \mathfrak{S}$ ; that is, the multiply operation in $\mathfrak{S}$ is closed. Also, the division operation is closed in $\mathfrak{S}$ .

Actually, let $f\left(x\right)=f_0\left(x\right)+{\displaystyle {\sum }_{i=1}^7}{f}_1\left(x\right)e_i\in \mathfrak{S}$ , assume $f_0^2+7f_1^2\ne 0$ , then

${\left(f\left(x\right)\right)}^{-1}=\frac{f_0-f_1\left(e_1+e_2+\cdots +e_7\right)}{f_0^2+7f_1^2}\mathrm{.}$

Thus we have

$\begin{array}{l}D{\left(f\left(x\right)\right)}^{-1}={\displaystyle \underset{i=0}{\overset{7}{\sum }}}{e}_i\frac{\partial {\left(f\left(x\right)\right)}^{-1}}{\partial x_i}\\ ={\displaystyle \underset{i=0}{\overset{7}{\sum }}}{e}_i({\left(f_0^2+7f_1^2\right)}^{-1}\left(\frac{\partial f_0}{\partial x_i}-\frac{\partial f_1}{\partial x_i}\left(e_1+e_2+\cdots +e_7\right)\right)\\ -\left(f_0-f_1\left(e_1+e_2+\cdots +e_7\right)\right)\left(2f_0\frac{\partial f_0}{\partial x_i}+14f_1\frac{\partial f_1}{\partial x_i}\right){\left(f_0^2+7f_1^2\right)}^{-2})\\ ={\displaystyle \underset{i=0}{\overset{7}{\sum }}}{e}_i\left(\left(\frac{\partial f_0}{\partial x_i}+\frac{\partial f_1}{\partial x_i}\left(e_1+\cdots +e_7\right)\right)\left(7f_1^2-f_0^2+2f_0{f}_1\left(e_1+\cdots +e_7\right)\right){\left(f_0^2+7f_1^2\right)}^{-2}\right)\end{array}$

$\begin{array}{l}={\displaystyle \underset{i=0}{\overset{7}{\sum }}}\left(e_i\left(\frac{\partial f_0}{\partial x_i}+\frac{\partial f_1}{\partial x_i}\left(e_1+\cdots +e_7\right)\right)\right)\left(7f_1^2-f_0^2+2f_0{f}_1\left(e_1+\cdots +e_7\right)\right){\left(f_0^2+7f_1^2\right)}^{-2}\\ =\left(Df\left(x\right)\right)\left(7f_1^2-f_0^2+2f_0{f}_1\left(e_1+\cdots +e_7\right)\right){\left(f_0^2+7f_1^2\right)}^{-2}\\ =0.\end{array}$

An element belongs to $\mathfrak{S}$ is the exponential function:

$\exp \left(x\right)={\text{e}}^{x_1+\cdots +x_7}\left(\cos \left(x_0\sqrt{7}\right)e_0+\left(-\frac{1}{\sqrt{7}}\left(e_1+\cdots +e_7\right)\right)\sin \left(x_0\sqrt{7}\right)\right).$ (3.3)

The results in Theorem 3.2 also hold on octonions(no complexification), since $O^c$ contains $O$ . If one switch the locations of $f\left(x\right)\mathrm{,}g\left(x\right)$ , and the “left” change into “right” in Theorem 3.2, then this theorem is also true, since left and right is symmetric. These principles also hold in the rest of this paper.

4. Necessary and Sufficient Conditions

If we consider the product of a left $O^c$ -analytic function and an $O^c$ -constant, we can get the necessary and sufficient conditions for the analyticity(these results obtained in this section for $O$ -analytic functions are also described in [19] ).

Applying Theorem 3.2(a) and (b), if $f\left(x\right)$ is a left $O^c$ -analytic function and $\lambda$ is a complex constant, or $\overline{f\left(x\right)}$ is a complex Stein-Weiss conjugate harmonic system and $\lambda$ is an $O^c$ -constant, then $f\left(x\right)\lambda$ is a left $O^c$ - analytic function. In what follows we will see that these conditions are also necessary in some sense.

Theorem 4.1. Let $\lambda \in O^c$ , then $f\lambda$ is a left $O^c$ -analytic function for any left $O^c$ -analytic functions $f$ if and only if $\lambda \in C$ .

Proof. We only prove the necessity. Taking a left $O^c$ -analytic function $f=x_1{e}_2-x_0{e}_3$ , then

$\begin{array}{c}D\left(f\lambda \right)=-{\displaystyle \underset{i,j,k=0}{\overset{7}{\sum }}}\frac{\partial f_j}{\partial x_i}{\lambda }_k\left[e_i,e_j,e_k\right]={\displaystyle \underset{k=1}{\overset{7}{\sum }}}\frac{\partial f_2}{\partial x_1}{\lambda }_k\left[e_2,e_1,e_k\right]={\displaystyle \underset{k=4}{\overset{7}{\sum }}}{\lambda }_k\left[e_2,e_1,e_k\right]\\ ={\lambda }_4\left[e_2,e_1,e_4\right]+{\lambda }_5\left[e_2,e_1,e_5\right]+{\lambda }_6\left[e_2,e_1,e_6\right]+{\lambda }_7\left[e_2,e_1,e_7\right]\\ =-2{\lambda }_4{e}_7+2{\lambda }_5{e}_6-2{\lambda }_6{e}_5+2{\lambda }_7{e}_4.\end{array}$

Thus ${\lambda }_4={\lambda }_5={\lambda }_6={\lambda }_7=0$ . A similar technique yields ${\lambda }_1={\lambda }_2={\lambda }_3=0$ . Hence $\lambda \in C$ . ,

Theorem 4.2. Let $f\in C^1\left(\Omega \mathrm{,}{O}^c\right)$ . Then $D\left(f\lambda \right)=0$ for any $\lambda \in O^c$ if and only if $f$ is a complex Stein-Weiss conjugate harmonic system in $\Omega$ .

Now we postpone the proof of Theorem 4.2 and consider a problem under certain conditions weaker than Theorem 4.2. In [2] the authors proposed an open problem as follows:

Find the necessary and sufficient conditions for an $O^c$ -valued function $f$ , such that the equality $\left[\lambda ,f\left(x\right),D\right]=0$ holds for any constant $\lambda \in O^c$ .

Note that this problem is of no meaning for an associative system, but octonions is a non-associative algebra, therefore we usually encounter some difficulties while disposing some problems in octonionic analysis. In [2] the authors added the condition $\left[\lambda ,f\left(x\right),D\right]=0$ for $f\left(x\right)$ to study the Cauchy integrals on Lipschitz surfaces in octonions and then prove the analogue of Calderón’s conjecture in octonionic space.

Next we give the answer to the Open Problem as follows.

Theorem 4.3. Let $f\in C^1\left(\Omega \mathrm{,}{O}^c\right)$ . Then $\left[D\mathrm{,}f\mathrm{,}\lambda \right]=0\left(\left[\lambda \mathrm{,}f\mathrm{,}D\right]=0\right)$ for any $\lambda \in O^c$ if and only if

$\frac{\partial f_i}{\partial x_j}=\frac{\partial f_j}{\partial x_i},i,j=1,2,\cdots ,7.$ (4.1)

Proof. By Proposition 2.1, we have

$\left[D,f,\lambda \right]={\displaystyle \underset{i,j=0}{\overset{7}{\sum }}}\frac{\partial f_j}{\partial x_i}\left[e_i,e_j,\lambda \right]={\displaystyle \underset{1\le i<j\le 7}{\sum }}\left(\frac{\partial f_j}{\partial x_i}-\frac{\partial f_i}{\partial x_j}\right)\left[e_i,e_j,\lambda \right].$

If $f$ satisfies (4.1), then $\left[D,f,\lambda \right]=0$ .

Inversely, let $\left(\alpha \mathrm{,}\beta \mathrm{,}\gamma \right)\in W$ , $\left\{\mathrm{1,2,}\cdots \mathrm{,7}\right\}\backslash \left\{\alpha \mathrm{,}\beta \mathrm{,}\gamma \right\}=\left\{t_1\mathrm{,}{t}_2\mathrm{,}{t}_3\mathrm{,}{t}_4\right\}$ and

$e_{t_1}{e}_{t_2}=e_{\gamma }=-e_{t_2}{e}_{t_1},e_{t_3}{e}_{t_4}=e_{\gamma }=-e_{t_4}{e}_{t_3}.$

From Propositions 2.1 and 2.2 we have $\left[e_{\alpha },e_{\beta },e_t\right]=0$ and $\left[e_{\alpha },e_{\beta },e_t\right]=2\left(e_{\alpha }{e}_{\beta }\right)e_t=2e_{\gamma }{e}_t$ when $t=\alpha ,\beta ,\gamma$ and $t=t_1,t_2,t_3,t_4$ , respectively. Hence, taking $\lambda =e_{t_1}$ it follows that

$\begin{array}{l}\left[D,f,e_{t_1}\right]\\ ={\displaystyle \underset{1\le i<j\le 7}{\sum }}\left(\frac{\partial f_j}{\partial x_i}-\frac{\partial f_i}{\partial x_j}\right)\left[e_i,e_j,e_{t_1}\right]\\ =\left(\frac{\partial f_{\beta }}{\partial x_{\alpha }}-\frac{\partial f_{\alpha }}{\partial x_{\beta }}\right)\left[e_{\alpha },e_{\beta },e_{t_1}\right]+\left(\frac{\partial f_{t_4}}{\partial x_{t_3}}-\frac{\partial f_{t_3}}{\partial x_{t_4}}\right)\left[e_{t_3},e_{t_4},e_{t_1}\right]+{\displaystyle \underset{s\ne t_2}{\sum }}{g}_s{e}_s\\ =2\left(\frac{\partial f_{\beta }}{\partial x_{\alpha }}-\frac{\partial f_{\alpha }}{\partial x_{\beta }}+\frac{\partial f_{t_4}}{\partial x_{t_3}}-\frac{\partial f_{t_3}}{\partial x_{t_4}}\right)e_{t_2}+{\displaystyle \underset{s\ne t_2}{\sum }}{g}_s{e}_s.\end{array}$ (4.2)

Similarly, we take $\lambda =e_{t_3}$ , then

$\left[D,f,e_{t_3}\right]=2\left(\frac{\partial f_{\beta }}{\partial x_{\alpha }}-\frac{\partial f_{\alpha }}{\partial x_{\beta }}+\frac{\partial f_{t_2}}{\partial x_{t_1}}-\frac{\partial f_{t_1}}{\partial x_{t_2}}\right)e_{t_4}+{\displaystyle \underset{s\ne t_4}{\sum }}{h}_s{e}_s,$ (4.3)