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Scale-Dependent Poincar\'{e} inequalities, log-Sobolev inequality and the stability of the Heisenberg Uncertainty Principle on the hyperbolic space

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arxiv 2410.21039 v4 pith:LQRXYMX5 submitted 2024-10-28 math.AP math.DG

Anh Xuan Do , Debdip Ganguly , Nguyen Lam , Guozhen Lu This is my paper · ORCID
classification math.APmath.DG
keywords poincarhyperbolictypeinequalitiesscale-dependentspacehardyderive
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abstract

We establish a general scale-dependent Poincar\'{e}-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 Poincar\'{e} type, Hardy type and Poincar\'{e}-Hardy type inequalities in the literature. We also investigate weighted Poincar\'{e} inequalities on hyperbolic space, where the weight functions depend on a scaling parameter. This leads to a new family of scale-dependent Poincar\'{e} 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.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Stability of the $L^{p}$-Poincar\'e inequality for the Lebesgue measure and Gaussian probability measure with explicit geometric dependence and applications to spectral gaps

    math.AP 2026-02 conditional novelty 7.0 of 10

    An explicit geometric stability constant is derived for the L^p-Poincaré inequality on convex domains, yielding a new but partially non-explicit spectral-gap bound for the p-Laplacian.

  2. Sharp remainder of the $L^{p}$-Poincar\'e inequality for Baouendi-Grushin vector fields

    math.AP 2025-07 reject novelty 3.0 of 10

    An attempted Lp sharp remainder formula for the Baouendi-Grushin Poincaré inequality and a PME application, but the complex-valued formulation is incorrect.

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