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Scale-Dependent Poincaré inequalities, log-Sobolev inequality and the stability of the Heisenberg Uncertainty Principle on the hyperbolic space

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

TL;DR

The paper develops a unifying scale-dependent framework for functional inequalities on the hyperbolic space $H^N$ by proving an abstract identity that couples a weight, a vector field, and a scaling parameter. This yields three Gaussian-type weighted Poincaré inequalities, which in turn drive $L^2$-stability analyses for the Heisenberg Uncertainty Principle on $H^N$ and establish a Gaussian-measure logarithmic Sobolev inequality. The contributions extend Euclidean CK-N and HUP results to negatively curved spaces, provide optimizers in the hyperbolic setting, and broaden the analytic toolkit for geometric analysis on hyperbolic manifolds. Overall, the work links scale-dependent inequalities, uncertainty principles, and entropy inequalities in a cohesive hyperbolic-geometry framework with potential applications to spectral theory and concentration phenomena on curved spaces.

Abstract

We establish a general scale-dependent Poincaré-Hardy type identity involving a vector field on the hyperbolic space. By choosing suitable parameter, potential and vector field in this identity, we can recover, as well as derive new versions of and substantially improve several Poincaré type, Hardy type and Poincaré-Hardy type inequalities in the literature. We also investigate weighted Poincaré inequalities on hyperbolic space, where the weight functions depend on a scaling parameter. This leads to a new family of scale-dependent Poincaré inequalities with Gaussian type measure on the hyperbolic space which is of independent interest. As a result, we derive both scale-dependent and scale-invariant $L^{2}$-stability results for the Heisenberg uncertainty principle in this setting. Finally, we study the logarithmic Sobolev inequality with Gaussian measure on the hyperbolic spaces, that is still missing in the literature.

Scale-Dependent Poincaré inequalities, log-Sobolev inequality and the stability of the Heisenberg Uncertainty Principle on the hyperbolic space

TL;DR

The paper develops a unifying scale-dependent framework for functional inequalities on the hyperbolic space by proving an abstract identity that couples a weight, a vector field, and a scaling parameter. This yields three Gaussian-type weighted Poincaré inequalities, which in turn drive -stability analyses for the Heisenberg Uncertainty Principle on and establish a Gaussian-measure logarithmic Sobolev inequality. The contributions extend Euclidean CK-N and HUP results to negatively curved spaces, provide optimizers in the hyperbolic setting, and broaden the analytic toolkit for geometric analysis on hyperbolic manifolds. Overall, the work links scale-dependent inequalities, uncertainty principles, and entropy inequalities in a cohesive hyperbolic-geometry framework with potential applications to spectral theory and concentration phenomena on curved spaces.

Abstract

We establish a general scale-dependent Poincaré-Hardy type identity involving a vector field on the hyperbolic space. By choosing suitable parameter, potential and vector field in this identity, we can recover, as well as derive new versions of and substantially improve several Poincaré type, Hardy type and Poincaré-Hardy type inequalities in the literature. We also investigate weighted Poincaré inequalities on hyperbolic space, where the weight functions depend on a scaling parameter. This leads to a new family of scale-dependent Poincaré inequalities with Gaussian type measure on the hyperbolic space which is of independent interest. As a result, we derive both scale-dependent and scale-invariant -stability results for the Heisenberg uncertainty principle in this setting. Finally, we study the logarithmic Sobolev inequality with Gaussian measure on the hyperbolic spaces, that is still missing in the literature.

Paper Structure

This paper contains 11 sections, 26 theorems, 204 equations.

Table of Contents

  1. Introduction
  2. An abstract identity on the hyperbolic space $\mathbb{H}^{N}$-Proofs of Theorems \ref{['Thm 1.1']}, \ref{['M1']} and \ref{['M2']}
  3. The first application
  4. The second application: The Caffarelli-Kohn-Nirenberg identities on the hyperbolic space
  5. The third application: Heisenberg Uncertainty Principle Identity
  6. Scale-Dependent Poincaré inequalities with Gaussian type measures on the hyperbolic space-Proofs of Theorems \ref{['1stineq']}, \ref{['2ndineq']} and \ref{['3rdineq']}
  7. The first weighted Poincaré inequality
  8. The second weighted Poincaré inequality
  9. The third weighted Poincaré inequality
  10. $L^{2}$-Stability of the Heisenberg Uncertainty Principle-Proofs of Theorems \ref{['Cor 3.1']}, \ref{['Cor 3.2']} and \ref{['finalstability']}
  11. Log-Sobolev inequality with Gaussian measure-Proof of Theorem \ref{['ThmlogS']}

Key Result

Theorem A

For any smooth vector field $X$ on $\mathbb{H}^{N}$, there exists a unique smooth function on $\mathbb{H}^{N}$, denoted by $\operatorname{div}X$, such that the following identity holds

Theorems & Definitions (46)

  • Theorem A: The Divergence Theorem
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem B: CFNL, Lemma 3.2
  • Theorem C
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • ...and 36 more