Sharp constants in several inequalities on the Heisenberg group

Details Sharp constants in several inequalities on the Heisenberg group Sharp constants in several inequalities on the Heisenberg group

2010-09-07

arxiv.org/abs/1009.1410v2

Mathematics

Analysis of PDEs

30 pages; addition of Corollary 2.3 and some minor changes; to appear in Annals of Mathematics

Scientific paper

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.

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