  We derive the sharp constants for the inequalities on the Heisenberg group
H^n whose analogues on Euclidean space R^n are the well known
Hardy-Littlewood-Sobolev inequalities. Only one special case had been known
previously, due to Jerison-Lee more than twenty years ago. From these
inequalities we obtain the sharp constants for their duals, which are the
Sobolev inequalities for the Laplacian and conformally invariant fractional
Laplacians. By considering limiting cases of these inequalities sharp constants
for the analogues of the Onofri and log-Sobolev inequalities on H^n are
obtained. The methodology is completely different from that used to obtain the
R^n inequalities and can be (and has been) used to give a new, rearrangement
free, proof of the HLS inequalities.
