On the regularity of weak subelliptic $F$-harmonic maps

Abstract

Building on work by L. Capogna & D. Danielli & N. Garofalo (cf. [7]-[8] ), G. Citti & N. Garofalo & E. Lanconelli (cf. [10]) and P. Hájlasz & P. Strzelecki (cf. [16]) we study local properties of weak subelliptic $F$-harmonic maps (cf. [4]) of a nondegenerate CR manifold into a sphere $S^{m}$, where $\rho(t)=F^{\prime}(t/2)$, $F\in C^{2},$ $F(t)\geq 0,$ $F^{\prime}(t)>0,$ $t\geq 0$. If $\Omega\subset R^{n}$ is a bounded domain and $X$ is a Hörmander system on $R^{n}$, we show that any weak solution $\phi\in W_{X}^{1,D}(\Omega, S^{m})$ to the nonlinear subelliptic system $-X^{*}\cdot(\rho(|X\phi|^{2})X\phi)=\rho(|X\phi|^{2})\phi|X\phi|^{2}$ is locally Hölder continuous, where $D$ is a homogeneous dimension of $\Omega$ with respect to $X$, provided that $t^{p}/K\leq p(t)\leq Kt^{P}$ for some $0 \lt p \lt (D-2)/2$.