A fractional version of Rivière’s GL(n)-gauge

We prove that for antisymmetric vector field $$\Omega $$ Ω with small $$L^2$$ L 2 -norm there exists a gauge $$A \in L^\infty \cap {\dot{W}}^{1/2,2}({\mathbb {R}}^1,GL(N))$$ A ∈ L ∞ ∩ W ˙ 1 / 2 , 2 ( R 1 , G L ( N ) ) such that $$\begin{aligned} {\text {div}}_{\frac{1}{2}} (A\Omega - d_{\frac{1}{2}} A) = 0. \end{aligned}$$ 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.