An application of homogenization theory to harmonic analysis on solvable lie groups of polynomial growth

Abstract

Let Q be a connected solvable Lie group of polynomial growth. Let also E1, …, Ep be left invariant vector fields on G that satisfy Hοrmander’s condition and denote by L = -(E1 2 +… + Ep 2) the associated sub-Laplacian and by S(x, t) the ball which is centered at x ∈ 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 |Eiu(x)| ≤ < ct-1u(x), x ∈ Q, t ≥ 1, 1 ≤ i ≤ p, for all u ≥ 0 such that Lu = 0 in S(x, t). This inequality is proved by adapting some ideas from the theory of homogenization. © 1993 by Pacific Journal of Mathematics.