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Improved Sobolev Inequalities on the Quaternionic Sphere

Zongxiong Ren, Zhipeng Yang

Abstract

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

Improved Sobolev Inequalities on the Quaternionic Sphere

Abstract

In this paper we establish improved Sobolev inequalities on the quaternionic sphere under higher-order moment vanishing conditions with respect to the measure . As an application, we give a new proof of the existence of extremals for the sharp Sobolev embedding

Paper Structure

This paper contains 4 sections, 6 theorems, 147 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Improved Sobolev Inequalities on the Quaternionic Sphere
  4. Existence of the Extremals

Key Result

Theorem 1.2

Let $S^{4n+3}$ be the quaternionic unit sphere. Let $Q=4n+6$, let $1<p<Q$, and let $p^*=\frac{Qp}{Q-p}$. Then for every $\varepsilon>0$ there exists a constant $C(\varepsilon)>0$ such that for every $u\in S^{1,p}(S^{4n+3})$ satisfying

Theorems & Definitions (16)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Lemma 3.1
  • proof : Proof of Theorem \ref{['thm:improved-jk']}
  • proof : Proof of Theorem \ref{['thm:improved-l']}
  • Remark 3.2
  • proof : Proof of Corollary \ref{['cor:first-moment']}
  • ...and 6 more