Fractional harmonic maps into manifolds in odd dimension n >1

In this paper we consider critical points of the following nonlocal energy $$\begin{array}{ll}{\mathcal{L}}_n(u) = \int_{{I\!\!R}^n}| ({-\Delta})^{n/4} u(x)|^2 dx, \qquad(1)\end{array}$$ where $${u \in \dot{H}^{n/2}({I\!\!R}^n,{\mathcal{N}}), {\mathcal{N}} \subset {I\!\!R}^m}$$ is a compact k dimensional smooth manifold without boundary and n>1 is an odd integer. Such critical points are called n/2-harmonic maps into $${{\mathcal{N}}}$$ . We prove that $${(-\Delta) ^{n/4} u\in L^p_{loc}({I\!\!R}^n)}$$ for every p ≥ 1 and thus $${u \in C^{0,\alpha}_{loc}({I\!\!R}^n)}$$ , for every 0<α