Entropy analysis and optimal estimation for interpolating specific families of integral inequalities in finite dimensional space

We describe a family of optimal integral inequalities providing an interpolation between the Poincaré and logarithmic Sobolev inequalities in $\mathbb{R}^n$. Through the application of a tensoring method and probabilistic techniques, we extend the validation of these inequalities from the Bernoulli measure to characterizing them for the Gaussian measure. Furthermore, using semigroup techniques, we provide a local version of the inequality, offering finer insights into its behavior in specific regions.