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 \(\Phi =x_0^{2p}x_1^{2m}x_2^{2n}\).

中文翻译：

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Kravchenko-广义Dirac算子的Cauchy核的存在

本文涉及静态麦克斯韦系统

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

由Kravchenko在四元数分析中以\（\ mathcal {L} F = 0 \）的形式用\（\ mathcal {L} F = DF + F \ alpha \）重新构造该系统。我们考虑系数函数\（\ Phi = \ Phi _0（x_0）\ Phi _1（x_1）\ Phi _2（x_2）\）的特殊情况，并证明了算子\（\ mathcal { L} \）。在\（\ Phi = x_0 ^ {2p} x_1 ^ {2m} x_2 ^ {2n} \）的情况下，我们构造了四个显式的广义柯西内核。