Horizontal $\alpha$-Harmonic Maps

Understand

Given a $C^1$ planes distribution $P_T$ on all ${\mathbb R}^m$ we consider {\em horizontal $\alpha$-harmonic maps}, $\alpha\ge 1/2$, with respect to such a distribution.

• These are maps $u\in H^{\alpha}({{\mathbb R}}^k,{{\mathbb R}}^m)$ satisfying $P_T\nabla u=\nabla u$ and $P_T(u)(-\Delta)^{\alpha}u=0$ in ${\mathcal D}'({{\mathbb R}}^k).$ If the distribution of planes is integrable then we recover the classical case of $\alpha$-harmonic maps with values into a manifold.
• In this paper we shall focus our attention to the case $\alpha=1/2$ in dimension $1$ and $\alpha=2$ in dimension $2$ and we investigate the regularity of the {\em horizontal $\alpha$-harmonic maps}.
• In both cases we show that such maps satisfy a Schr\"odinger type system with an antisymmetric potential, that permits us to apply the previous results obtained by the authors.