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) for p∈(1,n), with Sobolev type estimates in the case p∈[n/2,n) and Morrey–Sobolev type estimates in the case p∈(1,n/2). We also prove an analogous theorem in the case when Ω∈Lp(Bn,TCO+(m)⊗Λ1Rn), which corresponds to Uhlenbeck’s decomposition with conformal gauge group.