Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps

Abstract

We consider nonlocal linear Schrödinger-type critical systems of the type \begin{equation}\label{eqabstr} Δ^{1/4} v=Ω\, v~~~\mbox{in $\mathbb{R}.$} \ \end{equation} where $\Omega$ is antisymmetric potential in $L^2(\mathbb{R},so(m))$, $v$ is a ${\mathbb{R}}^m$ valued map and $\Omega v$ denotes the matrix multiplication. We show that every solution $v\in L^2(\mathbb{R},{\mathbb{R}}^m)$ of \rec{eqabstr} is in fact in $L_{loc}^p(\mathbb{R},{\mathbb{R}}^m)$, 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 $C_{loc}^{0,\alpha }$ regularity of weak $1/2$-harmonic maps into $C^2$ compact manifold without boundary.

52 pages

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