  We study the horizontal Laplacian $\Delta^H$ associated to the Hopf fibration
$S^3\to S^2$ with arbitrary Chern number $k$. We use representation theory to
calculate the spectrum, describe the heat kernel and obtain the complete heat
trace asymptotics of $\Delta^H$. We express the Green functions for associated
Poisson semigroups and obtain bounds for their contraction properties and
Sobolev inequalities for $\Delta^H$. The bounds and inequalities improve as
$|k|$ increases.
