Regularity for quasilinear second‐order subelliptic equations

In this paper, we study the regularity of solutions of the quasilinear equation\documentclass{article}\pagestyle{empty}\begin{document}$$ \sum\limits_{ij = 1}^m {Aij(x,u,Xu)X_i X_j u + B(x,u,Xu) = 0} \end{array} $$\end{document} where X = ( X 1 ,…, X m ) is a system of real smooth vector fields, A ij , B ϵ C ∞(Ω × ℝ R m +1 ). Assume that X satisfies the Hörmander condition and ( A ij ( x, z ,ζ)) is positive definite. We prove that if u ϵ S 2,α (Ω) (see Section 2) is a solution of the above equation, then u ϵ C ∞ (Ω).