Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A modified Bakry-Émery $Γ_2$ criterion inequality and the monotonicity of the Tsallis entropy

Xiaohan Cai, Xiaodong Wang

TL;DR

The paper advances sharp functional-analytic inequalities on compact manifolds with positive Ricci curvature by developing a one-parameter family of weighted Bakry-Émery Γ2 inequalities that recovers Ji’s improved constants, and a modified weighted Γ2 inequality incorporating a drift term tied to Tsallis entropy. It then builds an ODE framework for the Tsallis entropy under the heat flow, leading to a monotonicity result and a corresponding family of sharp Sobolev inequalities that extend and unify known constants. The findings connect entropy evolution with geometric-analytic inequalities, offering a broader, entropy-driven pathway to Poincaré, logarithmic Sobolev, and Sobolev-type results. The work also clarifies the ranges of p for which Tsallis-based monotonicity yields sharp constants, and highlights the model-space sharpness on spheres.

Abstract

The Bakry-Émery $Γ_2$ criterion inequality provides a method for establishing the logarithmic Sobolev inequality. We prove a one-parameter family of weighted Bakry-Émery $Γ_2$ criterion inequalities which in the limit case yields the improved constant due to Ji \cite{Ji24}. Furthermore, we establish a modified weighted $Γ_2$ criterion inequality which could be interpreted as a monotonicity of the Tsallis entropy under the heat flow and yields a family of sharp Sobolev inequalities.

A modified Bakry-Émery $Γ_2$ criterion inequality and the monotonicity of the Tsallis entropy

TL;DR

The paper advances sharp functional-analytic inequalities on compact manifolds with positive Ricci curvature by developing a one-parameter family of weighted Bakry-Émery Γ2 inequalities that recovers Ji’s improved constants, and a modified weighted Γ2 inequality incorporating a drift term tied to Tsallis entropy. It then builds an ODE framework for the Tsallis entropy under the heat flow, leading to a monotonicity result and a corresponding family of sharp Sobolev inequalities that extend and unify known constants. The findings connect entropy evolution with geometric-analytic inequalities, offering a broader, entropy-driven pathway to Poincaré, logarithmic Sobolev, and Sobolev-type results. The work also clarifies the ranges of p for which Tsallis-based monotonicity yields sharp constants, and highlights the model-space sharpness on spheres.

Abstract

The Bakry-Émery criterion inequality provides a method for establishing the logarithmic Sobolev inequality. We prove a one-parameter family of weighted Bakry-Émery criterion inequalities which in the limit case yields the improved constant due to Ji \cite{Ji24}. Furthermore, we establish a modified weighted criterion inequality which could be interpreted as a monotonicity of the Tsallis entropy under the heat flow and yields a family of sharp Sobolev inequalities.

Paper Structure

This paper contains 5 sections, 11 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Some integral identities
  3. Weighted Bakry-Émery $\Gamma_2$ criterion inequality
  4. Modified weighted Bakry-Émery $\Gamma_2$ inequality
  5. Monotonicity of the Tsallis entropy

Key Result

Theorem 1

Let $(M^n,g)$ be a compact Riemannian manifold with $Ric\geq n-1$. Denote $\lambda_1$ as the first eigenvalue of $-\Delta$ on $(M^n,g)$. Then for any $0<f\in C^{\infty}(M)$, That is,

Theorems & Definitions (24)

  • Theorem 1: Ji, Ji24
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 1.1
  • Remark 1.1
  • Theorem 5: Fon97DEL14
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 14 more