  We give sufficient conditions for a measured length space (X,d,m) to admit
local and global Poincare inequalities. We first introduce a condition DM on
(X,d,m), defined in terms of transport of measures. We show that DM, along with
a doubling condition on m, implies a scale-invariant local Poincare inequality.
We show that if (X,d,m) has nonnegative N-Ricci curvature and has unique
minimizing geodesics between almost all pairs of points then it satisfies DM,
with constant 2^N. The condition DM is preserved by measured Gromov-Hausdorff
limits. We then prove a Sobolev inequality for measured length spaces with
N-Ricci curvature bounded below by K>0. Finally, we prove a sharp global
inequality.
