Let Q be a connected solvable Lie group of polynomial growth. Let also E1, ..., E(p) be left invariant vector fields on G that satisfy Hormander's condition and denote by L = -(E1(2) + ... + E(p)2) the associated sub-Laplacian and by S(x, t) the ball which is centered at x is-an-element-of Q and it is of radius t > 0 with respect to the control distance associated to those vector fields. The goal of this article is to prove the following Harnack inequality: there is a constant c > 0 such that \E(i)u(x)\ less-than-or-equal-to ct-1 u(x), x is-an-element-of Q, t greater-than-or-equal-to 1 , 1 less-than-or-equal-to i less-than-or-equal-to p, for all u greater-than-or-equal-to 0 such that Lu = 0 in S(x, t) . This inequality is proved by adapting some ideas from the theory of homogenization.
