2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/230067We study coercive inequalities on finite dimensional metric spaces with probability measures which do not have volume doubling property. This class of inequalities includes Poincaré and Log-Sobolev inequality. Our main result is proof of Log-Sobolev inequality on Heisenberg group equipped with either heat kernel measure or "gaussian" density build from optimal control distance. As intermediate results we prove so called U-bounds.Functional AnalysisProbability22E30; 60E15text