Improved Sobolev Inequalities on the Quaternionic Sphere

In this paper we establish improved Sobolev inequalities on the quaternionic sphere under higher-order moment vanishing conditions with respect to the measure $\mathrm{\mid }u{\mathrm{\mid }}^{p^{\ast }}d\xi$. As an application, we give a new proof of the existence of extremals for the sharp Sobolev embedding $$S^{1,2}(S^{4n+3})\hookrightarrow L^{2^{\ast }}(S^{4n+3})\mathrm{.}$$