  We study coercive inequalities on finite dimensional metric spaces with
probability measures which do not have volume doubling property. This class of
inequalities includes Poincar\'e 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.
