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Logarithmic Sobolev, Poincaré and Beckner Inequalities on Hyperbolic Spaces and Riemannian Manifolds

Anh Xuan Do, Debdip Ganguly, Nguyen Lam, Guozhen Lu

Abstract

We investigate several functional and geometric inequalities on the hyperbolic space $\mathbb{H}^N$, with a primary emphasis on logarithmic Sobolev inequalities, Poincaré 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 Pólya-Szegö 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 Poincaré 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.

Logarithmic Sobolev, Poincaré and Beckner Inequalities on Hyperbolic Spaces and Riemannian Manifolds

Abstract

We investigate several functional and geometric inequalities on the hyperbolic space , with a primary emphasis on logarithmic Sobolev inequalities, Poincaré 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 Pólya-Szegö 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 Poincaré and logarithmic Sobolev inequalities. Finally, we extend our results beyond hyperbolic space to a class of Riemannian model manifolds 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.
Paper Structure (12 sections, 24 theorems, 276 equations)

This paper contains 12 sections, 24 theorems, 276 equations.

Table of Contents

  1. Introduction
  2. Sobolev and Poincaré inequality on the hyperbolic space.
  3. $AB$ Program.
  4. Logarithmic Sobolev inequalities.
  5. Symmetric decreasing rearrangements: Improvement of Pólya--Szegö principle
  6. Pólya-Szegö principle and its improvement
  7. Poincaré-Sobolev, Poincaré-Gagliardo-Nirenberg and Poincaré-logarithmic Sobolev inequalities on $\mathbb{H}^N$-Proofs of Theorem \ref{['pSob']} and Theorem \ref{['T1']}
  8. Gaussian Poincaré and Gaussian logarithmic Sobolev inequalities on $\mathbb{H}^N$-Proof of Theorem \ref{['T2']}
  9. Extended Beckner inequalities on the hyperbolic space $\mathbb{H}^N$
  10. logarithmic Sobolev inequalities on Cartan-Hadamard model manifolds
  11. Schwarz rearrangements and Pólya-Szegö inequality.
  12. Centered isoperimetric inequality

Key Result

Theorem A

Let $X$ be a smooth vector field on the hyperbolic space $\mathbb{H}^N$. Then there exists a unique smooth function, denoted by $\operatorname{div} X$, such that for every $u \in C_c^\infty(\mathbb{H}^N)$,

Theorems & Definitions (47)

  • Theorem A: Divergence Theorem
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 37 more