Sharp Hardy-Littlewood-Sobolev Inequalities on Quaternionic Heisenberg   Groups
  In this paper, we got several sharp Hardy-Littlewood-Sobolev-type inequalities on quaternionic Heisenberg groups (a general form due to Folland and Stein [FS74]), using the symmetrization-free method in a paper of Frank and Lieb [FL12], where they considered the analogues on classical Heisenberg group. First, we give the sharp Hardy-Littlewood-Sobolev inequalities, both on quaternionic Heisenberg group and its equivalent on quaternionic sphere for exponent bigger than 4. The extremizer, as we guess, is almost uniquely constant function on sphere. Then their dual form, sharp conformally-invariant Sobolev inequalities and the right endpoint limit case, Log-Sobolev inequality, are also obtained. For small exponent less 4, constant function is only proved to be a local extremizer. The conformal symmetry of the inequalities and zero center-mass technique play a critical role in the argument. 
Michael Christ, Heping Liu, An Zhang
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