  The $k$-Cauchy-Fueter operator $ D_0^{(k) } $ on one dimensional quaternionic
space $\mathbb{H}$ is the Euclidean version of helicity $\frac k 2$ massless
field operator on the Minkowski space in physics. The $k$-Cauchy-Fueter
equation for $k\geq 2$ is overdetermined and its compatibility condition is
given by the $k$-Cauchy-Fueter complex. In quaternionic analysis, these
complexes play the role of Dolbeault complex in several complex variables. We
prove that a natural boundary value problem associated to this complex is
regular. Then by using the theory of regular boundary value problems, we show
the Hodge-type orthogonal decomposition, and the fact that the non-homogeneous
$k$-Cauchy-Fueter equation $ D_0^{(k) } u=f$ on a smooth domain $\Omega$ in
$\mathbb{H}$ is solvable if and only if $f$ satisfies the compatibility
condition and is orthogonal to the set $\mathscr H^1_{ (k) }(\Omega)$ of
Hodge-type elements. This set is isomorphic to the first cohomology group of
the $k$-Cauchy-Fueter complex over $\Omega$, which is finite dimensional, while
the second cohomology group is always trivial.
