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$L^2$-Caffarelli--Kohn--Nirenberg inequalities on metric measure spaces

Zhe-Feng Xu, Ye Zhang

TL;DR

The paper develops a unified framework for $L^2$-CKN inequalities on metric measure spaces, revealing that a $k$-uniform constant forces a reverse-volume-type monotonicity of weighted ball measures and, under MCP$(0,N)$, rigidity to volume cones. It shows that the $L^2$-CKN inequalities hold for all $k$ precisely on $N$-volume cones, and it provides a stability theory quantifying deviations from Gaussian extremisers on these cones. A key technical tool is Bernstein's theorem, applied via a Gaussian-test function approach, together with needle decomposition to reduce to 1D analysis on transport rays. The results connect sharp Euclidean constants to geometric structure, offering insight into volume-growth rigidity and stability in non-smooth spaces with curvature-dimension-like bounds, and extend to $L^p$ analogues. This yields a robust link between functional inequalities, geometric measure theory, and the structure of metric measure spaces with MCP-type curvature bounds, with potential applications to analysis on spaces with synthetic Ricci bounds.

Abstract

Motivated by the sharp constants in the $L^2$-Caffarelli--Kohn--Nirenberg (or $L^2$-CKN for short) inequalities on Euclidean spaces, we study, in a unified framework, a sequence of $L^2$-CKN inequalities on metric measure spaces. On a general metric measure space, this sequence implies a reverse volume comparison of Günther type. Moreover, on a subclass of spaces admitting the measure contraction property, we show that this sequence of $L^2$-CKN inequalities are valid if and only if the spaces are volume cones. We also provide a stability result for inequalities of this type on volume cones.

$L^2$-Caffarelli--Kohn--Nirenberg inequalities on metric measure spaces

TL;DR

The paper develops a unified framework for -CKN inequalities on metric measure spaces, revealing that a -uniform constant forces a reverse-volume-type monotonicity of weighted ball measures and, under MCP, rigidity to volume cones. It shows that the -CKN inequalities hold for all precisely on -volume cones, and it provides a stability theory quantifying deviations from Gaussian extremisers on these cones. A key technical tool is Bernstein's theorem, applied via a Gaussian-test function approach, together with needle decomposition to reduce to 1D analysis on transport rays. The results connect sharp Euclidean constants to geometric structure, offering insight into volume-growth rigidity and stability in non-smooth spaces with curvature-dimension-like bounds, and extend to analogues. This yields a robust link between functional inequalities, geometric measure theory, and the structure of metric measure spaces with MCP-type curvature bounds, with potential applications to analysis on spaces with synthetic Ricci bounds.

Abstract

Motivated by the sharp constants in the -Caffarelli--Kohn--Nirenberg (or -CKN for short) inequalities on Euclidean spaces, we study, in a unified framework, a sequence of -CKN inequalities on metric measure spaces. On a general metric measure space, this sequence implies a reverse volume comparison of Günther type. Moreover, on a subclass of spaces admitting the measure contraction property, we show that this sequence of -CKN inequalities are valid if and only if the spaces are volume cones. We also provide a stability result for inequalities of this type on volume cones.
Paper Structure (14 sections, 9 theorems, 104 equations)

This paper contains 14 sections, 9 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main Results
  4. Some Generalisations
  5. Structure of the paper
  6. Preliminaries
  7. Bernstein's Theorem
  8. Measure Contraction Property (MCP)
  9. Proof of main results
  10. Proof of Theorem \ref{['Theorem1']}
  11. Proof of Corollary \ref{['prop1']}
  12. Proof of Theorem \ref{['Theorem2']}
  13. Proof of Proposition \ref{['stability']}
  14. Counterexample

Key Result

Theorem 1.1

Let $(X,\mathrm{d},\mathfrak m)$ be a metric measure space, and fix $x_0 \in X$. Assume that, for any $\lambda > 0$, and that there exists a constant $C>0$, independent of $k$, such that for any $k \in \mathbb N$ and any $u\in \mathrm{Lip}_b(X,\mathrm{d})$ the inequality first holds. Then, for any $\ell \in \mathbb N$, the function is non-decreasing on $(0,{+\infty})$. Here $B_{\rho}(x):=\{y\in

Theorems & Definitions (26)

  • Theorem 1.1: See also Theorem \ref{['thm Lap']}
  • Remark 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Remark 1.7
  • Definition 2.1: Complete monotonicity
  • Theorem 2.2: Bernstein's theorem
  • Definition 2.3: Non-branching geodesic
  • ...and 16 more