On the generalized Cauchy transform of power functions (Q1881020)

The authors deal with Clifford analysis in \(\mathbb{R}^m\), which is split up into \(\mathbb{R}^{m_1} \oplus \mathbb{R}^{m_2}\). Also the Dirac operator \(\underline{\partial} = \sum_{j=1}^m \partial / \partial x_j\) is split into \(\underline{\partial} = \underline{\partial}_1 + \underline{\partial}_2\), a vector \(x \in \mathbb{R}\) is split into \(x = \underline{x} + \underline{y}\). A holomorphic function \(f\) with \(\underline{\partial}f=0\) is mapped by the inversion operator \({\mathcal T}f(x) = \frac{\overline{x}}{| x| ^m}f\left(\frac{x}{| x| ^2}\right) \) to a holomorphic function. Here the authors take a homogeneous inner holomorphic polynomial \(P\) from \(\mathbb{R}^{m_1}\), this is also holomorphic in \(\mathbb{R}^m\), then they invert \(P\) in \(\mathbb{R}^m\) and use this new functions as a Cauchy kernel in \(\mathbb{R}^m \setminus \mathbb{R}^{m_2}\). This gives a Cauchy transform which is used to define inner and outer power functions in \(\mathbb{R}^{m_2}\) with many interesting properties.