A fractional version of Rivière's GL(N)-gauge

Abstract

We prove that for antisymmetric vectorfield $\Omega$ with small $L^2$-norm there exists a gauge $A\in L^{\infty }\cap {\dot{W}}^{1/2,2}({\mathbb{R}}^1,GL(N))$ such that ${\mathrm{div}}_{\frac{1}{2}}(A\Omega -d_{\frac{1}{2}}A)=0$. 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.

Revised version, accepted to AMPA