# Uhlenbeck's decomposition in Sobolev and Morrey-Sobolev spaces

## TL;DR

This paper provides a detailed proof of Uhlenbeck's decomposition theorem within Sobolev and Morrey-Sobolev spaces, extending its applicability with new estimates and an analogous theorem for conformal gauge groups.

## Contribution

It offers a self-contained proof of Uhlenbeck's decomposition in Sobolev and Morrey-Sobolev spaces, including new estimates and an extension to conformal gauge groups.

## Key findings

- Established Sobolev type estimates for p in [n/2, n)
- Derived Morrey-Sobolev estimates for p in (1, n/2)
- Proved an analogous theorem for conformal gauge group Ω

## Abstract

We present a self-contained proof of Uhlenbeck's decomposition theorem for $\Omega\in L^p(\mathbb{B}^n,so(m)\otimes\Lambda^1\mathbb{R}^n)$ for $p\in (1,n)$ with Sobolev type estimates in the case $p \in[n/2,n)$ and Morrey-Sobolev type estimates in the case $p\in (1,n/2)$. We also prove an analogous theorem in the case when $\Omega\in L^p( \mathbb{B}^n, TCO_{+}(m) \otimes \Lambda^1\mathbb{R}^n)$, which corresponds to Uhlenbeck's theorem with conformal gauge group.