Integral formulas for hypermonogenic functions

Abstract

Let $C\ell_{n}$ be the (universal) Clifford algebra generated by $e_{1},...,e_{n}$ satisfying $e_{i}e_{j}+e_{j}e_{i}=-2\delta_{ij}$, $i,j=1,...,n$. The Dirac operator in $C\ell_{n}$ is defined by $D=\sum_{i=0}^{n}e_{i}\frac{\partial}{\partial x_{i}}$, where $e_{0}=1$. The modified Dirac operator is introduced for $k\in\mathbb{R}$ by $M_{k}f=Df+k\frac{Q^{\prime}f}{x_{n}}$, where $^{\prime}$ is the main involution and $Qf$ is given by the decomposition $f\left( x\right) =Pf\left( x\right) +Qf\left( x\right) e_{n}$ with $Pf\left( x\right) ,Qf\left( x\right) \in C\ell_{n-1}$. A continuously differentiable function $f:\Omega\rightarrow C\ell_{n}$ is called $k$-hypermonogenic in an open subset $\Omega$ of $\mathbb{R}^{n+1}$, if $M_{k}f\left( x\right) =0$ outside the hyperplane $x_{n}=0$. Note that $0$-hypermonogenic functions are monogenic and $n-1$-hypermonogenic functions are hypermonogenic defined by the author and H. Leutwiler in [10]. The power function $x^{m}$ is hypermonogenic. We prove integral formulas of hypermogenic functions.