  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.
