Global solvability of real analytic involutive systems on compact manifolds

Abstract

The focus of this work is the smooth global solvability of a linear partial differential operator ${\mathbb {L}}$ associated to a real analytic closed non-exact 1-form b—defined on a real analytic closed n-manifold—that may be naturally regarded as the first operator of the complex induced by a locally integrable structure of tube type and co-rank one. We define an appropriate covering projection $\widetilde{M}\rightarrow M$ such that the pullback of b has a primitive $\widetilde{B}$ and prove that the operator is globally solvable if and only if the superlevel and sublevel sets of $\widetilde{B}$ are connected. As a byproduct we obtain a new geometric characterization for the global hypoellipticity of the operator. When M is orientable we prove a connection between the global solvability of ${\mathbb {L}}$ and that of ${\mathbb {L}}^{n-1}$ which is the last non-trivial operator of the complex, in particular, we prove that ${\mathbb {L}}$ is globally solvable if and only if ${\mathbb {L}}^{n-1}$ is globally solvable. In the smooth category, we are able to provide analogous geometric characterizations of the global solvability and the global hypoellipticity when b is a Morse 1-form, i.e., when the structure is of Mizohata type.