Improved Sobolev Inequalities on the Quaternionic Sphere

TL;DR

This paper proves enhanced Sobolev inequalities on the quaternionic sphere with higher-order moment conditions, providing a new proof for the existence of extremals in the sharp Sobolev embedding.

Contribution

It introduces improved inequalities under higher-order moment vanishing conditions and offers a novel proof for extremal existence in the sharp Sobolev embedding.

Findings

Established improved Sobolev inequalities on quaternionic spheres.

Provided a new proof for the existence of extremals in the sharp Sobolev embedding.

Extended the understanding of Sobolev inequalities with moment conditions.

Abstract

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