Sub-criticality of non-local Schr\"odinger systems with antisymmetric potentials and applications to half-harmonic maps

Abstract

We consider nonlocal linear Schr\"odinger-type critical systems of the type \begin{equation}\label{eqabstr} \Delta^{1/4} v=\Omega\, v~~~\mbox{in $\R\,.$} \ \end{equation} where $\Omega$ is antisymmetric potential in $L^2(\R,so(m))$, $v$ is a ${\R}^m$ valued map and $\Omega\, v$ denotes the matrix multiplication. We show that every solution $v\in L^2(\R,\R^m)$ of \rec{eqabstr} is in fact in $L^p_{loc}(\R,\R^m)$, for every $2\le p<+\infty$, in other words, we prove that the system (\ref{eqabstr}) which is a-priori only critical in $L^2$ happens to have a subcritical behavior for antisymmetric potentials. As an application we obtain the $C^{0,\alpha}_{loc}$ regularity of weak $1/2$-harmonic maps into $C^2$ compact manifold without boundary.