# Regularity for The CR Vector Bundle Problem I

## TL;DR

This paper introduces a novel solution to the local integrability problem for CR vector bundles on high-dimensional strictly pseudoconvex hypersurfaces, utilizing a KAM-based approach that improves upon previous methods.

## Contribution

It provides a new, sharper method avoiding Nash-Moser techniques for CR vector bundle integrability on high-dimensional hypersurfaces.

## Key findings

- Achieves sharp H"older continuity in solutions
- Uses KAM rapid convergence instead of Nash-Moser
- Applies to hypersurfaces of dimension seven or greater

## Abstract

We give a new solution to the local integrability problem for CR vector bundles over strictly pseudoconvex real hypersurfaces of dimension seven or greater. It is based on a KAM rapid convergence argument and avoids the previous more difficult Nash-Moser methods. The solution is sharp as to H\"older continuity.