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

We consider non-local linear Schrödinger-type critical systems of the type(1)Δ1/4v=Ωvin R, where Ω is antisymmetric potential in L2(R,so(m)), v is an Rm valued map and Ωv denotes the matrix multiplication. We show that every solution v∈L2(R,Rm) of (1) is in fact in Llocp(R,Rm), for every 2⩽p<+∞, in other words, we prove that the system (1) which is a-priori only critical in L2 happens to have a subcritical behavior for antisymmetric potentials. As an application we obtain the Cloc0,α regularity of weak 1/2-harmonic maps into C2 compact sub-manifolds without boundary.