An Integral Representation Formula for Multi Meta- \(\varphi \) -Monogenic Functions of Second Class

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• 作者：Eusebio Ariza García ; Antonio Di Teodoro
• 关键词：Monogenic function ; Metamonogenic function ; Multi ; meta ; monogenic function ; Multi meta ; \(\varphi \) ; monogenic function ; Clifford algebras ; Clifford type algebras depending on parameters
• 刊名：Complex Analysis and Operator Theory
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：11
• 期：2
• 页码：375-392
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general; Operator Theory; Analysis;
• 出版者：Springer International Publishing
• ISSN：1661-8262
• 卷排序：11

文摘

Consider the following operator $$\begin{aligned} \displaystyle \mathcal {D}_{\varphi ^{(j)},\lambda ^{(j)}}=\varphi _{0}^{(j)}\partial _{x_{0}^{(j)}}+\displaystyle \sum _{i=1}^{m_{j}} \varphi _{i}^{(j)}(x) e_{i+a_j}\partial _{i+a_j}-\lambda ^{(j)} \end{aligned}$$where \(\varphi ^{(j)}\) is a Clifford-valued function and \(\lambda ^{(j)}\) is a Clifford-constant defined by $$\begin{aligned} \varphi ^{(j)} = \varphi _0^{(j)} + \displaystyle \sum _{i=1}^{m_{j}}\varphi _{i}^{(j)}e_{i+a_j},\quad \lambda ^{(j)} = \lambda _0^{(j)} + \displaystyle \sum _{i=1}^{m_j}\lambda _{i}^{(j)}e_{i+a_j} \end{aligned}$$with \(m=m_1+\cdots +m_n\), \(a_1=m_0=0\) and \(a_j=m_{1}+\cdots +m_{j-1}\) for \(j=2,\ldots ,n\); and \(\varphi _{i}^{(j)}\) can be real-valued functions defined in \({\mathbb {R}}^{m_1+1}\times {\mathbb {R}}^{m_{2}+1}\times \cdots \times {\mathbb {R}}^{m_{n}+1}\). \(\lambda _i^{(j)}\) are real numbers for \(i=0,1,\ldots ,m_j\) and \(j=1,\ldots ,n\). A function u is multi meta\(-\varphi \)-monogenic of second class, in several variables \(x^{(j)}\), for \(j=1,\ldots ,n\), if $$\begin{aligned} \displaystyle \mathcal {D}_{{\varphi ^{(j)},\lambda ^{(j)}}u=0.} \end{aligned}$$In this paper we give a Cauchy-type integral formula for multi meta-\(\varphi \)-monogenic of second class operator in one way by iteration and in the second way by the use of the construction of the Levi function. Also, in this work, we define a multi meta-\(\varphi \)-monogenic function of first class with the help of the Clifford type algebras depending on parameters.KeywordsMonogenic functionMetamonogenic functionMulti-meta-monogenic functionMulti meta-\(\varphi \)-monogenic functionClifford algebrasClifford type algebras depending on parametersCommunicated by Frank Sommen.