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A simple proof of reverse Sobolev inequalities on the sphere and Sobolev trace inequalities on the unit ball

Runmin Gong, Qiaohua Yang, Shihong Zhang

TL;DR

The paper interlinks reverse Hardy-Littlewood-Sobolev inequalities with reverse Sobolev inequalities on the sphere, treating two fractional orders $γ$ separately: for $γ\in(\frac{n}{2},\frac{n}{2}+1)$ the reverse Sobolev inequality is derived from reverse HLS via spherical-harmonic expansion, while for $γ\in(\frac{n}{2}+1,\frac{n}{2}+2)$ a direct Hang-based center-of-mass approach yields a sharp inequality and a quantitative stability bound. It then constructs conformally covariant boundary operators on the Poincaré ball, couples them with scattering theory to obtain higher-order Sobolev trace inequalities on the unit ball, and proves conformal covariance and symmetry properties of the associated Dirichlet forms. The results illuminate a concrete connection between reverse HLS and reverse Sobolev inequalities in the conformal setting, provide explicit extremal cases tied to conformal maps, and extend trace inequalities to higher-order contexts with explicit lower bounds. These contributions yield a coherent framework for stability analysis and trace estimates in conformal geometry and related PDEs.

Abstract

Frank et al. (J. Funct. Anal., 2022) stated that there is no relation between the reversed Hardy-Littlewood-Sobolev (HLS) inequalities and reverse Sobolev inequalities. However, we demonstrate that reverse Sobolev inequalities of order $γ\in(\frac{n}{2},\frac{n}{2}+1)$ on the $n$-sphere can be readily derived from the reversed HLS inequalities. For the case $γ\in(\frac{n}{2}+1,\frac{n}{2}+2)$, we present a simple proof of reverse Sobolev inequalities by using the center of mass condition introduced by Hang. In addition, applying this approach, we establish the quantitative stability of reverse Sobolev inequalities of order $γ\in(\frac{n}{2}+1,\frac{n}{2}+2)$ with explicit lower bounds. Finally, by using conformally covariant boundary operators and reverse Sobolev inequalities, we derive Sobolev trace inequalities on the unit ball.

A simple proof of reverse Sobolev inequalities on the sphere and Sobolev trace inequalities on the unit ball

TL;DR

The paper interlinks reverse Hardy-Littlewood-Sobolev inequalities with reverse Sobolev inequalities on the sphere, treating two fractional orders separately: for the reverse Sobolev inequality is derived from reverse HLS via spherical-harmonic expansion, while for a direct Hang-based center-of-mass approach yields a sharp inequality and a quantitative stability bound. It then constructs conformally covariant boundary operators on the Poincaré ball, couples them with scattering theory to obtain higher-order Sobolev trace inequalities on the unit ball, and proves conformal covariance and symmetry properties of the associated Dirichlet forms. The results illuminate a concrete connection between reverse HLS and reverse Sobolev inequalities in the conformal setting, provide explicit extremal cases tied to conformal maps, and extend trace inequalities to higher-order contexts with explicit lower bounds. These contributions yield a coherent framework for stability analysis and trace estimates in conformal geometry and related PDEs.

Abstract

Frank et al. (J. Funct. Anal., 2022) stated that there is no relation between the reversed Hardy-Littlewood-Sobolev (HLS) inequalities and reverse Sobolev inequalities. However, we demonstrate that reverse Sobolev inequalities of order on the -sphere can be readily derived from the reversed HLS inequalities. For the case , we present a simple proof of reverse Sobolev inequalities by using the center of mass condition introduced by Hang. In addition, applying this approach, we establish the quantitative stability of reverse Sobolev inequalities of order with explicit lower bounds. Finally, by using conformally covariant boundary operators and reverse Sobolev inequalities, we derive Sobolev trace inequalities on the unit ball.

Paper Structure

This paper contains 15 sections, 22 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hypergeometric function
  4. Funk-Hecke formula
  5. The fractional reverse Sobolev inequalities
  6. First case: $\gamma\in \left(\frac{n}{2},\frac{n}{2}+1\right)$
  7. Second case: $\gamma\in \left(\frac{n}{2}+1,\frac{n}{2}+2\right)$
  8. Sobolev trace inequality
  9. Scattering theory in the Poincaré ball
  10. Boundary operators
  11. The boundary operators with small indices
  12. Dirichlet problem
  13. The boundary operators with large indices
  14. Proof of theorem \ref{['Conformal Thm']} and \ref{['Intrinsic Thm']}
  15. Proof of theorem \ref{['Symmetric Thm']} and \ref{['Functional inequ Thm']}

Key Result

Theorem 1.1

For $\gamma\in \left(\frac{n}{2},\frac{n}{2}+1\right)\cup \left(\frac{n}{2}+1,\frac{n}{2}+2\right)$ and $0<f\in H^{\gamma}(\mathbb{S}^n)$, it holds The equality holds if and only if here exists a conformal transformation $\varphi: \mathbb{S}^{n}\rightarrow\mathbb{S}^{n}$ such that

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.1
  • Lemma 3.1
  • Theorem 3.1: Dou and Zhu
  • ...and 15 more