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A sharp Sobolev inequality on the Caffarelli-Kohn-Nirenberg hyperbolic space

Baptiste Devyver, Louis Dupaigne, Pierre-Damien Thizy

TL;DR

This work investigates sharp Sobolev-type inequalities for Caffarelli–Kohn–Nirenberg data in a hyperbolic-cone setting, linking the Euclidean cone Sobolev constant to a hyperbolic analogue via conformal changes. The authors prove a sharp non-improvability result for n≥4 and establish an improved inequality for the critical range n∈[3,4), governed by a threshold λ_* and a mass parameter m_λ that encodes spectral/Green-function data. Central to the analysis are precise Green-function estimates for the cone operator and its hyperbolic perturbation, together with a careful comparison principle that translates Green-function domination into equality of Sobolev constants. The results illuminate how the geometry (via α, n) controls sharp constants, concentration phenomena, and mass-driven thresholds, with both general and radial versions. The techniques combine conformal invariance, variational bubbles, Green-function methods, and Maz'ya–Sobolev-type estimates to characterize when improved inequalities hold and how they fail beyond the threshold.

Abstract

In the Euclidean space $\mathbb{R}^d$, the sharp classical Sobolev inequality is equivalent by conformal invariance to a Sobolev inequality on the hyperbolic space $\mathbb{H}^d$. This inequality is sharp in dimension $d\geq 4$, but it is not in dimension $d=3$ by results of Benguria, Frank and Loss, as well as Mancini and Sandeep. In this article, we investigate a similar phenomenon for the Caffarelli-Kohn-Nirenberg inequality and its hyperbolic analogue. In our setting, the condition for improving the inequality reads $n\in [3,4)$, where $n$ is an ``effective dimension''.

A sharp Sobolev inequality on the Caffarelli-Kohn-Nirenberg hyperbolic space

TL;DR

This work investigates sharp Sobolev-type inequalities for Caffarelli–Kohn–Nirenberg data in a hyperbolic-cone setting, linking the Euclidean cone Sobolev constant to a hyperbolic analogue via conformal changes. The authors prove a sharp non-improvability result for n≥4 and establish an improved inequality for the critical range n∈[3,4), governed by a threshold λ_* and a mass parameter m_λ that encodes spectral/Green-function data. Central to the analysis are precise Green-function estimates for the cone operator and its hyperbolic perturbation, together with a careful comparison principle that translates Green-function domination into equality of Sobolev constants. The results illuminate how the geometry (via α, n) controls sharp constants, concentration phenomena, and mass-driven thresholds, with both general and radial versions. The techniques combine conformal invariance, variational bubbles, Green-function methods, and Maz'ya–Sobolev-type estimates to characterize when improved inequalities hold and how they fail beyond the threshold.

Abstract

In the Euclidean space , the sharp classical Sobolev inequality is equivalent by conformal invariance to a Sobolev inequality on the hyperbolic space . This inequality is sharp in dimension , but it is not in dimension by results of Benguria, Frank and Loss, as well as Mancini and Sandeep. In this article, we investigate a similar phenomenon for the Caffarelli-Kohn-Nirenberg inequality and its hyperbolic analogue. In our setting, the condition for improving the inequality reads , where is an ``effective dimension''.

Paper Structure

This paper contains 27 sections, 39 theorems, 475 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['th1']}
  3. Conformal changes & proof of Theorem \ref{['th3']}
  4. Conformal changes
  5. Computing $\lambda_1(\mathbf{H})$
  6. Improving the inequality by adding a Sobolev term
  7. Theorem \ref{['th2']}: a sufficient condition
  8. Green functions and Sobolev constants
  9. A sufficient condition for the equality of the Sobolev constants of $\mathbf{E}$ and $\mathbf{H}$
  10. Theorem \ref{['th2']}: Green's function estimates
  11. The fundamental solution of $L_\mathbf{E}$
  12. The Green's function of $L_\mathbf{E}$ in ${\mathbf B}^*$
  13. The Green's function of $L_{\mathbf{H},\lambda}$ in the case $\alpha=1$, $n=d$
  14. The Green's function of $L_{\mathbf{H},\lambda}$ in the case $\alpha>0$
  15. Proof of Theorem \ref{['thm:Green_estimate_final']}: regime (i)
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $d\ge 3$ be an integer. Assume standing assumption and Let $A,B\in{\mathbf R}$ be such that for all $F\in C^\infty_c(\mathbf{H})$, there holds Then, $A\le C_{n,\alpha},$ where $C_{n,\alpha}$ is the optimal Sobolev constant in $\mathbf{E}$, given by ckn constant. Moreover, if $A=C_{n,\alpha}$, then

Theorems & Definitions (85)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 2.1
  • Remark 2.2
  • Lemma 4.1
  • Definition 4.2
  • Theorem 4.3
  • ...and 75 more