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arXiv · 2512.15525

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

Xiaohan Cai·Xiaodong Wang
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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.

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Xiaohan Cai, Xiaodong Wang. 2025-12-17. A modified Bakry-\'Emery $\Gamma_2$ criterion inequality and the monotonicity of the Tsallis entropy. https://arxiv.org/abs/2512.15525

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