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Bounds for the optimal constant of the Bakry-Émery $Γ_2$ criterion inequality on $ RP^{d-1}$

Sehyun Ji

TL;DR

The paper investigates the optimal Bakry-Émery $\Gamma_2$ constant $\Lambda_d$ for positive symmetric functions on $S^{d-1}$ (equivalently on $RP^{d-1}$), relating it to the logarithmic Sobolev constant $\alpha_d$ and the Poincaré constant $\lambda_d=2d$. It proves a universal lower bound $\Lambda_d \ge d+3-1/(d-1)$ via a refined CD-inequality analysis and Bochner-type identities, improving understanding of Fisher information monotonicity in Landau-type dynamics. For the physically relevant case $d=3$, it presents tight upper bounds: $\Lambda_3 \le 5.73892$ and $\alpha_3 \le 5.8358$, obtained by testing the explicit function class $h(\sigma)=(z^2+t)^2$ on $S^2$. The results illuminate the gap between $\Lambda_d$ and $\alpha_d$, and show the possibility that $\alpha_3$ and $\Lambda_3$ differ, with implications for the monotonicity range of Fisher information in kinetic equations. The work advances the quantitative understanding of functional inequalities on spheres and RP spaces with direct relevance to diffusion and kinetic theory.

Abstract

We prove upper and lower bounds on the optimal constant $Λ_d$ of the Bakry-Émery $Γ_2$ criterion for positive symmetric functions on the unit sphere $S^{d-1}$, which also can be identified as positive functions on the real projective space $RP^{d-1}$. The Bakry-Émery $Γ_2$ criterion inequality was crucially used to prove the monotonicty of the Fisher information for the Landau equation by Guillen and Silvestre recently. Therefore, a better bound on the optimal constant $Λ_d$ expands the range of interaction potentials that exhibits the monotonicity of the Fisher information. In particular, we compute that $Λ_3$ is between $5.5$ and $5.739$.

Bounds for the optimal constant of the Bakry-Émery $Γ_2$ criterion inequality on $ RP^{d-1}$

TL;DR

The paper investigates the optimal Bakry-Émery constant for positive symmetric functions on (equivalently on ), relating it to the logarithmic Sobolev constant and the Poincaré constant . It proves a universal lower bound via a refined CD-inequality analysis and Bochner-type identities, improving understanding of Fisher information monotonicity in Landau-type dynamics. For the physically relevant case , it presents tight upper bounds: and , obtained by testing the explicit function class on . The results illuminate the gap between and , and show the possibility that and differ, with implications for the monotonicity range of Fisher information in kinetic equations. The work advances the quantitative understanding of functional inequalities on spheres and RP spaces with direct relevance to diffusion and kinetic theory.

Abstract

We prove upper and lower bounds on the optimal constant of the Bakry-Émery criterion for positive symmetric functions on the unit sphere , which also can be identified as positive functions on the real projective space . The Bakry-Émery criterion inequality was crucially used to prove the monotonicty of the Fisher information for the Landau equation by Guillen and Silvestre recently. Therefore, a better bound on the optimal constant expands the range of interaction potentials that exhibits the monotonicity of the Fisher information. In particular, we compute that is between and .
Paper Structure (6 sections, 8 theorems, 106 equations)

This paper contains 6 sections, 8 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The logarithmic Sobolev inequality and the Curvature-Dimension condition
  4. Interpretation using the heat flow
  5. The lower bound for $\Lambda_d$
  6. The upper bounds for $\Lambda_3$ and $\alpha_3$

Key Result

Theorem 1.1

Let $f:S^{d-1}\to \mathbb R_{>0}$ be a function defined on $S^{d-1}$ and symmetric, i.e. $f(\sigma)=f(-\sigma)$ for every $\sigma \in S^{d-1}$. The optimal constant $\Lambda_d$ for the inequality eq: 2nd logSobolev has the following lower bound: In particular, $\Lambda_2 \ge 4$ and $\Lambda_3 \ge 5.5$.

Theorems & Definitions (28)

  • Theorem 1.1: The lower bound for $\Lambda_d$
  • Remark 1.2
  • Corollary 1.3: Improvement of Theorem 1.1 of guillen2023global
  • Remark 1.4
  • Theorem 1.5: The upper bound for $\Lambda_3$
  • Theorem 1.6: The upper bound for $\alpha_3$
  • Remark 1.7
  • Definition 2.1: Carré du champ operator
  • Definition 2.2: The Curvature-Dimension condition
  • Theorem 2.3: The lower bound for $\Lambda_M$ under $CD(\rho,n)$
  • ...and 18 more