The Isotonic Cauchy Transform

Abstract.

Starting with an integral representation for the class of continuously differentiable solutions ${f:{\mathbb{R}^{2n}}\, \rightarrow \, \mathbb{C}_{0,n}}$ of the system

$${\partial_ {\underline{x}_{1}}f\, + \, i\tilde{f}\partial_{\underline{x}_{2}}\, = \,0}$$

where ${\mathbb{C}_{0,n}}$ is the complex Clifford algebra constructed over ${\mathbb{R}^{n}, \,{\underline{x}_{1}}, \,{\underline{x}_{2}}}$ are some suitable Clifford vectors and ${{\partial_ {\underline{x}_{1}}, \, \partial_ {\underline{x}_{2}}}}$ their corresponding Dirac operators, we define the isotonic Cauchy transform and establish the Sokhotski-Plemelj formulae. Some consequences of this result are also derived.