Cauchy Integral Formulae in $\mathbb{R}^n$

Abstract:

      In this note $p(\underline{D})={\underline{D}}^m+b_1{\underline{D}}^{m-1}+\cdots+b_m$ is a polynomial Dirac operator in $\mathbb{R}^n$, where $\underline{D}=\sum^n_{j=1} e_j\frac{\partial }{\partial x_j}$ is a standard Dirac operator in $\mathbb{R}^n$, $b_j$ are the complex constant coefficients. In this note we discuss all decompositions of $p(\underline{D})$ according to its coefficients $b_j$, and obtain the corresponding explicit Cauchy integral formulae of $f$ which are the solution of $p(\underline{D})f=0$.