  In this paper, we investigate the sharp Hardy-Littlewood-Sobolev inequalities
on the Heisenberg group. On one hand, we apply the concentration compactness
principle to prove the existence of the maximizers. While the approach here
gives a different proof under the special cases discussed in a recent work of
Frank and Lieb, we generalize the result to all admissible cases. On the other
hand, we provide the upper bounds of sharp constants for these inequalities.
