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The $k$-Cauchy-Fueter operators and complexes are quaternionic counterpartsof the Cauchy-Riemann operator and the Dolbeault complex in the theory ofseveral complex variables. To develop the function theory of severalquaternionic variables, we need to solve the non-homogeneous $k$-Cauchy-Fueterequation over a domain under the compatibility condition, which naturally leadsto a Neumann problem. The method of solving the $\overline{\partial}$-Neumannproblem in the theory of several complex variables is applied to this Neumannproblem. We introduce notions of $k$-plurisubharmonic functions and$k$-pseudoconvex domains, establish the $L^2$ estimate and solve this Neumannproblem over $k$-pseudoconvex domains in $\mathbb{R}^4$. Namely, we get avanishing theorem for the first cohomology groups of the $k$-Cauchy-Fuetercomplex over such domains.

The Neumann problem for the $k$-Cauchy-Fueter complexes over $k$-pseudoconvex domains in $\mathbb{R}^4$ and the $L^2$ estimate