The Existence of Cauchy Kernels of Kravchenko-Generalized Dirac Operators

Abstract

This paper deals with the static Maxwell system

$$\begin{aligned} \left\{ \begin{array}{ll} div(\Phi \overrightarrow{E})&{}=0,\\ \ curl\overrightarrow{E}&{}=0,\ (x_0,x_1,x_2)\in \mathbb {R}^3. \end{array} \right. \end{aligned}$$

The system is reformulated in quaternion analysis by Kravchenko in the form $\mathcal {L}F=0$ with $\mathcal {L}F=DF+F\alpha$. We consider special cases of the coefficient function $\Phi =\Phi _0(x_0)\Phi _1(x_1)\Phi _2(x_2)$ and prove the existence of four generalized Cauchy kernels of the operator $\mathcal {L}$. We construct four explicit generalized Cauchy kernels in the case $\Phi =x_0^{2p}x_1^{2m}x_2^{2n}$.