Homogenization and convergence of correctors in carnot groups

We consider homogenization of differential operators of the form -Sigma(m)(i,j=1) X-i(a(ij)(delta(1/epsilon)(xi)) X(j)u(epsilon)) = f, where {X-j}(j=1)(m) is a family of linearly independent vector fields in R-N that by commutation generate the Lie algebra of a Carnot group, a(ij)(xi) are periodic functions in the sense of the group, and delta(1/epsilon) are the dilations in the group. We establish Meyers type estimates for the horizontal gradients Xu = (X(1)u,..., X(m)u) of solutions to equations defined with general vector fields satisfying Hormander's condition, and use them to prove convergence of the horizontal gradients of correctors in L2+theta, theta > 0.