Sharp constants in several inequalities on the Heisenberg group

Abstract

We derive the sharp constants for the inequalities on the Heisenberg group ℍn whose analogues on Euclidean space ℝn 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 ℍn are obtained. The methodology is completely different from that used to obtain the ℝn inequalities and can be (and has been) used to give a new, rearrangement free, proof of the HLS inequalities. © 2011 by the authors.