We investigate isoperimetric and functional inequalities for probability measures in
the sub-elliptic setting and more specifically, on groups of Heisenberg type. The
approach we take is based on U-bounds as well as a Laplacian comparison theorem
for H-type groups. We derive different forms of functional inequalities (of [Phi]-entropy
and F-Sobolev type) and show that they can be equivalently stated as isoperimetric
inequalities at the level of sets. Furthermore, we study transportation of measure via
Talagrand-type inequalities. The methods used allow us to obtain gradient bounds for
the heat semigroup. Finally, we examine some properties of more general operators
given in Hormander’s sum of squares form and show that the associated semigroup
converges to a probability measure as t → [infinity].