A new proof of the Newlander-Nirenberg theorem (Q1106361)

We give a new nonlinear proof of the local existence of holomorphic coordinates for a formally integrable almost complex structure. These coordinates are obtained as the limit of a sequence of approximately holomorphic coordinate systems, which is constructed and shown to converge by the methods of Kolmogorov, Arnold, and Moser. The linearized problem is solved using the Leray-Koppelman formula on a ball in \({\mathbb{C}}^ n\). We provide new estimates for the operators in this formula, in which a full derivative is gained and the bound (necessarily) blows up at the boundary at a controlled rate. The convergence scheme produces directly a solution of the problem which is sharp as to regularity as measured by Hölder norms.