Sharp stability for the second-order weighted Heisenberg Uncertainty Principle with explicit stability…

Abstract

By using spherical harmonicas decomposition and Gaussian-type Poincaré inequalities, we establish several sharp stability estimates for the following second-order weighted Heisenberg Uncertainty Principle ∫_ℝ^N |Δu|²/|x|^2α dx ∫_ℝ^N |x|^2α+2|∇ u|²dx ≥ ((N+4α+2)²)/4 (∫_ℝ^N |∇ u|² dx)², and ∫_ℝ^N |Δu|²/|x|^2α dx +∫_ℝ^N |x|^2α+2 |∇ u|²dx ≥(N+4α+2) ∫_ℝ^N |∇ u|²dx. We also provide the explicit value and the necessary and sufficient condition for attainability of the sharp stability constants. Moreover, when α=0, our results reduce into those of [\emph{Calc. Var. Partial Differential Equations} \textbf{64} (2025), Paper No. 129], [\emph{J. Funct. Anal.} \textbf{290} (2026), Paper No. 111321] and [arXiv:2510.00453].

Second order Heisenberg Uncertainty Principle for solenoidal fields: sharp constants, optimizers and sharp stability
Anh Xuan Do, Anh Tuan Duong, Nguyen Lam +2
arXiv1 Nov 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
Gromov-Hausdorff Stability and Rigidity of manifolds via Heisenberg-Pauli-Weyl Uncertainty Principle
Mousomi Bhakta, Debdip Ganguly, Debabrata Karmakar
arXiv1 Sep 2026PreprintCC BY-NC-SA 4.0Full textAnalysis & PDEs Geometry & topologyarXiv
Optimal stability hierarchies of the Heisenberg Uncertainty Principle for solenoidal fields and of the second order Caffarelli--Kohn--Nirenberg inequalities
Anh Do, Tuan Duong, Nguyen Lam +2
arXiv25 Sep 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
A Fourier approach to the sharp stability of Heisenberg Uncertainty Principles of higher and fractional orders: a new perspective
Anh Do, Nguyen Lam, Guozhen Lu +1
arXiv25 Sep 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group
Wenjing Chen, Zexi Wang
arXiv14 Sep 2026PreprintCC BY 4.0Full textAnalysis & PDEsarXiv

Cite

BibTeX RIS

BibTeX

@misc{chen2026sharp,
  title = {{Sharp stability for the second-order weighted Heisenberg Uncertainty Principle with explicit stability constants and optimizers}},
  author = {Xiao-Ping Chen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29170},
  eprint = {2609.29170},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29170},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Sharp stability for the second-order weighted Heisenberg Uncertainty Principle with explicit stability constants and optimizers
AU  - Xiao-Ping Chen
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.29170
UR  - https://arxiv.org/abs/2609.29170
AB  - By using spherical harmonicas decomposition and Gaussian-type Poincaré inequalities, we establish several sharp stability estimates for the following second-order weighted Heisenberg Uncertainty Principle ∫_ℝ^N |Δu|^2/|x|^2α dx ∫_ℝ^N |x|^2α+2|∇ u|^2dx ≥ ((N+4α+2)^2)/4 (∫_ℝ^N |∇ u|^2 dx)^2, and ∫_ℝ^N |Δu|^2/|x|^2α dx +∫_ℝ^N |x|^2α+2 |∇ u|^2dx ≥(N+4α+2) ∫_ℝ^N |∇ u|^2dx. We also provide the explicit value and the necessary and sufficient condition for attainability of the sharp stability constants. Moreover, when α=0, our results reduce into those of [\emph{Calc. Var. Partial Differential Equations} \textbf{64} (2025), Paper No. 129], [\emph{J. Funct. Anal.} \textbf{290} (2026), Paper No. 111321] and [arXiv:2510.00453].
ER  -

Abstract

Related

Cite