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