Uhlenbeck’s Decomposition in Sobolev and Morrey–Sobolev Spaces

We present a self-contained proof of Rivière’s theorem on the existence of Uhlenbeck’s decomposition for Ω∈Lp(Bn,so(m)⊗Λ1Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \in L^p(\mathbb {B}^n,so(m)\otimes \Lambda ^1\mathbb {R}^n)$$\end{document} for p∈(1,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in (1,n)$$\end{document}, with Sobolev type estimates in the case p∈[n/2,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \in [n/2,n)$$\end{document} and Morrey–Sobolev type estimates in the case p∈(1,n/2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in (1,n/2)$$\end{document}. We also prove an analogous theorem in the case when Ω∈Lp(Bn,TCO+(m)⊗Λ1Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \in L^p( \mathbb {B}^n, TCO_{+}(m) \otimes \Lambda ^1\mathbb {R}^n)$$\end{document}, which corresponds to Uhlenbeck’s decomposition with conformal gauge group.