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 ≥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.
Christ, M., Liu, H., & Zhang, A. (2016). Sharp Hardy-Littlewood-Sobolev inequalities on quaternionic Heisenberg groups. Nonlinear Analysis, Theory, Methods and Applications, 130, 361–395. https://doi.org/10.1016/j.na.2015.10.018