Sharp Hardy–Littlewood–Sobolev inequalities on quaternionic Heisenberg groups

• We get sharp Hardy–Littlewood–Sobolev inequalities on Quaternionic Heisenberg groups.
• Dual Sobolev and endpoint Log-Sobolev inequalities are given.
• The method is symmetrization-free and center of the group is high dimensional.

In this paper, we get several sharp Hardy–Littlewood–Sobolev-type inequalities on quaternionic Heisenberg groups, using the symmetrization-free method of Frank and Lieb, who considered the analogues on the Heisenberg group. First, we give the sharp Hardy–Littlewood–Sobolev inequality on the quaternionic Heisenberg group and its equivalent on the sphere, for singular exponent of partial range λ≥4λ≥4. The extremal function, as we guess, is “almost” uniquely constant function on the sphere. Then their dual form, a sharp conformally-invariant Sobolev-type inequality involving a (fractional) intertwining operator, and the right endpoint case, a Log-Sobolev-type inequality, are also obtained. Higher dimensional center brings extra difficulty. The conformal symmetry of the inequalities, zero center-mass technique and estimates involving meticulous computation of eigenvalues of singular kernels play a critical role in the argument.