We prove that for antisymmetric vector field Ω with small L 2 -norm there exists a gauge A ∈ L ∞ ∩ W ˙ 1 / 2 , 2 ( R 1 , G L ( N ) ) such that div 1 2 ( A Ω - d 1 2 A ) = 0 . This extends a celebrated theorem by Rivière to the nonlocal case and provides conservation laws for a class of nonlocal equations with antisymmetric potentials, as well as stability under weak convergence.
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