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Horizontal $α$-Harmonic Maps

Given a $C^1$ planes distribution $P_T$ on all ${\mathbb R}^m$ we consider {\em horizontal $α$-harmonic maps}, $α\ge 1/2$, with respect to such a distribution. These are maps $u\in H^α({\mathbb R}^k,{\mathbb R}^m)$ satisfying $P_T\nabla u=\nabla u$ and $P_T(u)(-Δ)^αu=0$ in ${\mathcal D}'({\mathbb R}^k).$ If the distribution of planes is integrable then we recover the classical case of $α$-harmonic maps with values into a manifold. In this paper we shall focus our attention to the case $α=1/2$ in dimension $1$ and $α=2$ in dimension $2$ and we investigate the regularity of the {\em horizontal $α$-harmonic maps}. In both cases we show that such maps satisfy a Schrödinger type system with an antisymmetric potential, that permits us to apply the previous results obtained by the authors. Finally we study the regularity of {\em variational $α$-harmonic} maps which are critical points of $\|(-Δ)^{α/2} u\|^2_{L^2}$ under the constraint to be tangent (horizontal) to a given planes distribution. We produce a convexification of this variational problem which permits to write it's Euler Lagrange equations.