Existence of maximizers for Hardy-Littlewood-Sobolev inequalities on the Heisenberg group Xiaosen Han 39B6249J4535R03Hardy-Littlewood-Sobolev inequalities on the Heisenberg groupconcentration compactness principleexistence of maximizersupper bounds of sharp constants In this paper, we investigate the sharp Hardy-Littlewood-So\-bo\-lev inequalities on the Heisenberg group. On the 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 [\textit{Sharp constants in several inequalities on the Heisenberg group}, preprint], we generalize the result to all admissible cases. On the other hand, we provide the upper bounds of sharp constants for these inequalities. Indiana University Mathematics Journal 2013 text pdf 10.1512/iumj.2013.62.4976 10.1512/iumj.2013.62.4976 en Indiana Univ. Math. J. 62 (2013) 737 - 751 state-of-the-art mathematics http://iumj.org/access/