Dirichlet problems for the quasilinear second order subelliptic equations

In this paper, we study the Dirichlet problems for the following quasilinear second order sub-elliptic equation, $$\left\{ {\begin{array}{*{20}c} {\sum\limits_{i,j = 1}^m {X_i^* (A_{i,j} (x,u)X_j u) + \sum\limits_{j = 1}^m {B_j (x,u)X_j u + C(x,u) = 0in\Omega ,} } } \\ {u = \varphi on\partial \Omega ,} \\ \end{array} } \right.$$ whereX={X ₁, ...,X _m } is a system of real smooth vector fields which satisfies the Hörmander's condition,A _i,j ,B _j ,C ∈C ^∞( $\bar \Omega$ ×R) and (A _i,j (x,z)) is a positive definite matrix. We have proved the existence and the maximal regularity of solutions in the “non-isotropic” Hölder space associated with the system of vector fieldsX.