A modified Bakry-\'Emery $\Gamma_2$ criterion inequality and the monotonicity of the Tsallis entropy

TL;DR

This paper introduces a modified Bakry-Émery $\Gamma_2$ criterion that leads to improved inequalities and demonstrates the monotonicity of Tsallis entropy under heat flow, resulting in new sharp Sobolev inequalities.

Contribution

It develops a one-parameter family of weighted Bakry-Émery $\Gamma_2$ inequalities and links them to Tsallis entropy monotonicity and sharp Sobolev inequalities.

Findings

Proved a family of weighted $\Gamma_2$ inequalities.

Established monotonicity of Tsallis entropy under heat flow.

Derived a family of sharp Sobolev inequalities.

Abstract

The Bakry-\'Emery $\Gamma_2$ criterion inequality provides a method for establishing the logarithmic Sobolev inequality. We prove a one-parameter family of weighted Bakry-\'Emery $\Gamma_2$ criterion inequalities which in the limit case yields the improved constant due to Ji \cite{Ji24}. Furthermore, we establish a modified weighted $\Gamma_2$ 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.