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Sharp quantitative integral inequalities for general conformally invariant extensions

Qiaohua Yang, Shihong Zhang

Abstract

In this paper, we develop a refined analysis of hypergeometric functions to establish sharp quantitative integral inequalities for a general family of conformally invariant extension operators and their adjoints. Our results extend the recent work of Frank, Peteranderl, and Read \cite{Frank&Peteranderl&Read} to the full admissible parameter range under the natural index constraints.

Sharp quantitative integral inequalities for general conformally invariant extensions

Abstract

In this paper, we develop a refined analysis of hypergeometric functions to establish sharp quantitative integral inequalities for a general family of conformally invariant extension operators and their adjoints. Our results extend the recent work of Frank, Peteranderl, and Read \cite{Frank&Peteranderl&Read} to the full admissible parameter range under the natural index constraints.
Paper Structure (13 sections, 25 theorems, 321 equations)

This paper contains 13 sections, 25 theorems, 321 equations.

Table of Contents

  1. Introduction
  2. Stability for general integral isoperimetric inequality
  3. Concentration compactness
  4. Spectral gaps
  5. Spectral gaps with weight
  6. Compactness
  7. Comparison of second order variation
  8. Proof of Theorem \ref{['Thm 1']}
  9. The stability of dual operator
  10. Variation and concentration compactness
  11. Spectral gaps
  12. Proof of Theorem \ref{['Thm 2']}
  13. Appendix: conformal invariance

Key Result

Theorem 1.1

For $n\geq 3$, assume that $( \alpha,\beta)$ satisfy Index Condi. Then the following sharp stability inequalities hold:

Theorems & Definitions (34)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.1
  • Remark 1.3
  • Theorem 1.2
  • Remark 1.4
  • Lemma 2.1
  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 24 more