Formally Integrable Structures I. Resolution of Solution Sheaves

TL;DR

This paper develops an $L^2$-theory for formally integrable structures, constructing resolutions of solution sheaves and establishing conditions for global exactness and local solvability, especially for Levi flat structures.

Contribution

It introduces a resolution of the solution sheaf for overdetermined systems and proves local solvability of the Treves complex in new cases.

Findings

Constructed a resolution of the solution sheaf for certain overdetermined systems.

Established a sufficient condition for global exactness of the sheaf.

Proved local solvability of the Treves complex for Levi flat structures.

Abstract

This is the first of a series of papers on the $L^2$-theory for formally integrable structures. It is devoted to constructing a resolution of the solution sheaf for a class of overdetermined systems introduced by L. H{\"o}rmander. A sufficient condition for global exactness is obtained, which leads to gluing techniques for local solutions formulated as Cousin type problems. In addition, we also prove the local solvability of the Treves complex for formally integrable structures with vanishing Levi forms, including Levi flat structures as special cases. To the best of the authors' knowledge, nothing more than the elliptic case is known about the local $L^2$-solvability of the Treves complex in the Levi flat case.