Sharp Regularity for the Integrability of Elliptic Structures

As part of his celebrated Complex Frobenius Theorem, Nirenberg showed that given a smooth elliptic structure (on a smooth manifold), the manifold is locally diffeomorphic to an open subset of Rr×Cn (for some r and n) in such a way that the structure is locally the span of ∂∂t1,…,∂∂tr,∂∂z‾1,…,∂∂z‾n; where Rr×Cn has coordinates (t1,…,tr,z1,…,zn). In this paper, we give optimal regularity for the coordinate charts which achieve this realization. Namely, if the manifold has Zygmund regularity of order s+2 and the structure has Zygmund regularity of order s+1 (for some s>0), then the coordinate charts may be taken to have Zygmund regularity of order s+2. We do this by generalizing Malgrange's proof of the Newlander-Nirenberg Theorem to this setting.