The Mizohata complex

This paper deals with the local solvability of systems of first order linear partial differential equations defined by a germ $\omega$ at $0\in \mathbb {R}^{n+1}$ of a $\mathbb {C}$-valued, formally integrable ($\omega \wedge d\omega =0$), 1-form with nondegenerate Levi form. More precisely, the size of the obstruction to the solvability, for $(q-1)$-forms $u$, of the equation

where $\eta$ is a given $q$-form satisfying $d\eta \wedge \omega =0$ is estimated in terms of the De Rham cohomology relative to $\omega$