Regularity of quasi-linear equations with Hörmander vector fields of step two

Abstract

If the smooth vector fields $X_1,\dots ,X_m$ and their commutators span the tangent space at every point in $\Omega \subseteq {\mathbb{R}}^N$ for any fixed $m\le N$, then we establish the full interior regularity theory of quasi-linear equations ${\sum }_{i=1}^m{X}_i^{\ast }{A}_i(X_{1}u,\dots ,X_{m}u)=0$ with $p$-Laplacian type growth condition. In other words, we show that a weak solution of the equation is locally $C^{1,\alpha }$.

46 pages, additions and corrections made