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We present a self-contained proof of Rivière’s theorem on the existence of Uhlenbeck’s decomposition for $$\Omega \in L^p(\mathbb {B}^n,so(m)\otimes \Lambda ^1\mathbb {R}^n)$$Ω∈Lp(Bn,so(m)⊗Λ1Rn) for $$p\in (1,n)$$p∈(1,n), with Sobolev type estimates in… Click to show full abstract
We present a self-contained proof of Rivière’s theorem on the existence of Uhlenbeck’s decomposition for $$\Omega \in L^p(\mathbb {B}^n,so(m)\otimes \Lambda ^1\mathbb {R}^n)$$Ω∈Lp(Bn,so(m)⊗Λ1Rn) for $$p\in (1,n)$$p∈(1,n), with Sobolev type estimates in the case $$p \in [n/2,n)$$p∈[n/2,n) and Morrey–Sobolev type estimates in the case $$p\in (1,n/2)$$p∈(1,n/2). We also prove an analogous theorem in the case when $$\Omega \in L^p( \mathbb {B}^n, TCO_{+}(m) \otimes \Lambda ^1\mathbb {R}^n)$$Ω∈Lp(Bn,TCO+(m)⊗Λ1Rn), which corresponds to Uhlenbeck’s decomposition with conformal gauge group.
Journal Title: Results in Mathematics
Year Published: 2017
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