  We study logarithmic Sobolev inequalities with respect to a heat kernel
measure on finite-dimensional and infinite-dimensional Heisenberg groups. Such
a group is the simplest non-trivial example of a sub-Riemannian manifold. First
we consider logarithmic Sobolev inequalities on non-isotropic Heisenberg
groups. These inequalities are considered with respect to the hypoelliptic heat
kernel measure, and we show that the logarithmic Sobolev constants can be
chosen to be independent of the dimension of the underlying space. In this
setting, a natural Laplacian is not an elliptic but a hypoelliptic operator.
The argument relies on comparing logarithmic Sobolev constants for the
three-dimensional non-isotropic and isotropic Heisenberg groups, and
tensorization of logarithmic Sobolev inequalities in the sub-Riemannian
setting. Furthermore, we apply these results in an infinite-dimensional setting
and prove a logarithmic Sobolev inequality on an infinite-dimensional
Heisenberg group modelled on an abstract Wiener space.
