Uhlenbeck's decomposition in Sobolev and Morrey-Sobolev spaces

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 $\Omega \in L^p({\mathbb{B}}^n,TC{O}_{+}(m)\otimes {\Lambda }^1{\mathbb{R}}^n)$, which corresponds to Uhlenbeck's theorem with conformal gauge group.