Logarithmic Sobolev, Poincar\'e and Beckner Inequalities on Hyperbolic Spaces and Riemannian Manifolds

We investigate several functional and geometric inequalities on the hyperbolic space $\mathbb{H}^N$, with a primary emphasis on logarithmic Sobolev inequalities, Poincar\'e inequalities, and Beckner-type inequalities, all studied within the framework of the AB program. The main analytical tool employed throughout this paper is symmetrization. More precisely, our approach relies on an improved version of the P\'olya-Szeg\"o inequality on the hyperbolic space, obtained through a careful comparison of the gradient norms of rearranged functions in the hyperbolic and Euclidean settings. For Beckner-type inequalities, we adopt a semigroup approach based on sharp estimates for the heat semigroup, leading to refined interpolation inequalities between Poincar\'e and logarithmic Sobolev inequalities. Finally, we extend our results beyond hyperbolic space to a class of Riemannian model manifolds $\mathbb{M}^N$ satisfying the centered isoperimetric inequality. This shows that the inequalities and methods developed in this work are robust and rely mainly on geometric and isoperimetric properties, rather than on the specific structure of hyperbolic space itself.