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

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.