An application of homogenization theory to harmonic analysis on solvable Lie groups of polynomial growth
Georgios Alexopoulos · Pacific Journal of Mathematics · 1993
Let Q be a connected solvable Lie group of polynomial growth.Let also E\, ... , E p be left invariant vector fields on G that satisfy Hόrmander's condition and denote by L = -{E\ + + Ej) the associated sub-Laplacian and by S(x, t) the ball which is centered at x E Q and it is of radius / > 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)\ 1, 1 0 such that Lu = 0 in S(x, t).This inequality is proved by adapting some ideas from the theory of homogenization.