AbstractWe 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.